**Zhongliang Deng, Shihao Tang \*, Buyun Jia, Hanhua Wang, Xiwen Deng and Xinyu Zheng**

School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China; dengzhl@bupt.edu.cn (Z.D.); jiabuyun@bupt.edu.cn (B.J.); whh0710@bupt.edu.cn (H.W.); dengxiwen@bupt.edu.cn (X.D.); buptzxy@bupt.edu.cn (X.Z.)

**\*** Correspondence: swift71116@bupt.edu.cn; Tel.: +86-010-6119-8509

Received: 28 September 2020; Accepted: 4 November 2020; Published: 5 November 2020

**Abstract:** Localization estimation and clock synchronization are important research directions in the application of wireless sensor networks. Aiming at the problems of low positioning accuracy and slow convergence speed in localization estimation methods based on message passing, this paper proposes a low-complexity distributed cooperative joint estimation method suitable for dynamic networks called multi-Gaussian variational message passing (M-VMP). The proposed method constrains the message to be a multi-Gaussian function superposition form to reduce the information loss in the variational message passing algorithm (VMP). Only the mean, covariance and weight of each message need to be transmitted in the network, which reduces the computational complexity while ensuring the information completeness. The simulation results show that the proposed method is superior to the VMP algorithm in terms of position accuracy and convergence speed and is close to the sum-product algorithm over a wireless network (SPAWN) based on non-parametric belief propagation, but the computational complexity and communication load are significantly reduced.

**Keywords:** multi-variational message passing (M-VMP); factor graph (FG); second-order Taylor expansion; cooperative localization; joint estimation of position and clock

#### **1. Introduction**

In recent years, wireless sensor network has been widely used in agriculture, warehousing, production safety, emergency rescue and other fields [1]. The information which sensors collected and transmitted is valuable when combined with the sensors' location [2,3]. Therefore, location awareness of wireless sensor networks has become one of the most important directions in the development of wireless sensor networks. Global Navigation Satellite System (GNSS) is able to provide location information for sensor nodes, but it is difficult to apply on sensor networks due to high power consumption and high cost [2–4]. Furthermore, the poor signal penetration capabilities of GNSS lead to inadequate location information in indoor scenes. Cooperative localization can overcome these problems through ranging and exchanging locations between neighbor nodes [5]. Over the past decade, cooperative localization in wireless sensor networks has drawn considerable attention.

In the recent ten years, cooperative positioning in wireless sensor networks has been widely concerned. The multidimensional scaling (MDS) algorithm [6,7] calculates the shortest path distance between nodes according to the connectivity of the network or distance measurement. Then, the relative coordinate diagram of all nodes is constructed by using the multidimension scale algorithm. Finally, the relative coordinate diagram is transformed into an absolute coordinate graph according to the coordinate of the anchor node. The semi-definite programming (SDP) algorithm [8,9] represents geometric constraints among nodes as a set of linear matrix inequalities, then combines the inequality into a semi-definite programming problem and obtains the location of each node by global optimization. The distance-vector (DV) hop algorithm [10,11] firstly measures the minimum number of hops between

the nodes to be located and each anchor node. Then, the average distance of each hop is determined according to the number and coordinate between anchor nodes. After the jump number is converted into distance, the position of the node to be located is obtained according to the trilateral measurement method. The Approximate Perfect Point-In-Triangulation (APIT) algorithm [12,13] firstly selects three anchor nodes connected with the node to be located, and then judges whether they are in the triangle composed of anchor nodes according to the pit test, and then repeats the pit test with a different anchor node combination. Finally, the center of mass position of these triangles is taken as the coordinates of the nodes to be located. The SPAWN algorithm [14] decomposes the posterior probability density function of position variables and represents it as a factor graph. Then, the message is transmitted by particles on the factor graph to calculate the probability of edge distribution of each variable. Finally, position estimation is carried out according to minimum mean square error (MMSE) or map criterion. Although the above cooperative positioning algorithm has high positioning accuracy, it has high computational complexity and high communication cost, which seriously restricts the practical application.

For the above problems, a collaborative location algorithm based on factor graph and VMP is proposed in Reference [15], which has simple message form and small computation. However, Kullback-Leibler (KL) divergence is used to Gauss the non-Gaussian confidence of the nonlinear range measurement model. The first kind of convergence hypergeometric function minimization problem is introduced, which makes the calculation complexity very high. In Reference [16], a hard decision-based cooperative algorithm is proposed to alleviate the effect of the outliers. The geometric relationship between agent position and distance is used to avoid a large deviation coursed by geometric constraints. The authors of Reference [17] propose the distributed particle filtering evolved variational message passing (DPF-E-VMP) algorithm, which improves the convergence speed of positioning estimation by using distributed particle filtering (DPF), but this performance improvement is often accompanied by greater computational consumption. By combining the average consensus method and VMP algorithm, a joint self-localization tracking algorithm called cooperative localization with outlier constraints (CLOC) is proposed in Reference [18], which has better location performance than the separate self-localization algorithm. In Reference [19], KL divergence is minimized by the Newton conjugate gradient method, but the computational complexity is still high. But, the information loss caused by VMP parameterization will have a certain influence on the positioning accuracy. At the same time, the algorithm has been reduced in accuracy and convergence speed due to the clock synchronization between the nodes to be located.

In this paper, a VMP distributed cooperative localization algorithm based on multi-Gaussian is proposed. Based on the non-line of sight (NLOS) environment ranging model, the VMP message passing strategy based on multi-Gaussian is innovated, and the second-order Taylor expansion form of position and time synchronization joint estimation is derived. Its computational complexity is far less than the approximate solution based on KL divergence. The time complexity and communication cost are connected with the traditional VMP algorithm. However, the positioning accuracy and iteration speed have been greatly improved.

The remainder of this paper will be organized as follows: A two-dimensional (2D) wireless network is first established as a system model. Then, a traditional cooperative localization algorithm is introduced and leads to the method proposed in this paper. After that, simulation performance of the proposed method is investigated. Finally, concluding remarks are presented. A list of symbols that are used in the paper is given in Table 1.

Fisher Information Matrix (FIM) of ,

[,, ,]

[,, ,]

**2. System Model** 

(,)

**2. System Model** 

Confluent Hypergeometric Function of the First Type

(∙)

Fisher Information Matrix (FIM) of ,

Fisher Information Matrix (FIM) of ,

(,)

**2. System Model** 

, and the measured value of local clock ̃

, and the measured value of local clock ̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

**2. System Model** 

[,, ,] 

(∙)

**2. System Model** 

**2. System Model** 

(,)

(,)

[,, ,]

[,, ,]

[,, ,]

[,, ,]

(,)

**2. System Model** 


**Table 1.** List of symbols. node , at time of node at time time ̃ time ̃ Slope of local clock of Real time value at time , Slope of local clock of Real time value at time , Real time value at time , , time Measurement value of Measurement value of to be located of node to be located of node ,Set of neighbor anchor nodes of node at time , Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Weight of the -th

Cramer-Rao Lower Bound of ,

Cramer-Rao Lower Bound of ,

Cramer-Rao Lower Bound of ,

Cramer-Rao Lower Bound of ,

Cramer-Rao Lower Bound of ,

Bound of ,

Bound of ,

, Mean of belief (,)

Cramer-Rao Lower Bound ,

CRLB(,)

, Mean of belief (,)

Cramer-Rao Lower Bound of ,

Fisher Information Matrix (FIM) of ,

Weight of the -th Gaussian distribution

(∙)

Confluent Hypergeometric Function of the First Type

Bound of ,

(FIM) of ,

(,)

Fisher Information Matrix (FIM) of

Fisher Information Matrix (FIM) of ,

(FIM) of ,

Fisher Information Matrix(FIM) of ,

#### by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained (,) (,) (,) ,) The anchor nodes are always deployed at the same height, and high vertical dilution of precision (FIM) of , **2. System Model**  , Covariance of belief (,) , , Covariance of belief (,) , , Mean of belief (,) Confluent Hypergeometric Function of the First Type (∙) **2. System Model**

, − ̃

(,)

CRLB(,)

means the system cannot provide reliable vertical positioning results [20], which is usually obtained

location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock **2. System Model**  The anchor are always deployed at the same high dilution of means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this a wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time of sync. The position of at time is <sup>=</sup> [,, ,] , and the measured value of local clock ̃ ,= (). The slope of local clock at time between node and the external standard clock is , (̃ − ̃ )/( − ). The local clock is ,and all communicable node pairs (, ) constitute communicable node set 1:*n*, **2. System Model** The nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of position vector of node at time is , = [,, ,] , and the measured value of local clock , = (. The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1 − −1). The local clock **2. System Model**  The anchor nodes are always deployed at the same and high vertical dilution of means the system cannot provide reliable vertical [20], which is usually obtained other sensors [21]. So, in this paper, a 2D dynamic wireless network is which includes the anchor node with known location and synchronized and the node with inaccurate and local time of sync. The position node at time , = ,] and the value of local clock ̃ ,= (). The slope of local at time ,and all communicable pairs (, ) constitute communicable node set 1:*<sup>n</sup>* by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . (FIM) of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node *i* at time *n* is *xi*,*<sup>n</sup>* = [*xi*,*n*, *yi*,*n*] *T* , and the measured value of local clock ˜*ti*,*<sup>n</sup>* = *ti*(*Tn*). The slope of local clock at time *n* between node *i* and the external standard clock is *<sup>a</sup>i*,*<sup>n</sup>* <sup>=</sup> (˜*ti*,*<sup>n</sup>* <sup>−</sup> ˜*ti*,*n*−1)/(*T<sup>n</sup>* <sup>−</sup> *<sup>T</sup>n*−1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., *ai*,*<sup>n</sup>* = 1∀*i* ∈ *S*. At time *n*, node *i* has neighbor nodes set as *Ni*,*n*, where neighbor anchor node set is *Si*,*n*, node set to be located is *Ci*,*n*, and all communicable node pairs (*i*, *j*) constitute communicable node set Ψ*n*.

, Mean of belief (,)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located of anchor nodes is synchronized the reference clock, i.e., ,<sup>=</sup>. At time node has neighbor nodes set as , where neighbor anchor node set is , , node set to be located of all synchronized with the external reference , = 1∀ ∈ At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located node and the external standard clock is , <sup>=</sup>, − )/( <sup>−</sup>−1). clock of all anchor nodes is synchronized the reference clock, i.e., ,= 1∀ ∈ . At time has nodes ,where neighbor node set is ,, node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time Considering the local clock drift, node *i* measures time of arrival (TOA) from neighbor node *j* at time *n* as follows:

$$\mathfrak{F}\_{\tilde{t}|\mathbf{A}} = \|\mathbf{x}\_{t|\mathbf{A}} - \mathbf{x}\_{f|\mathbf{A}}\| + \zeta\_{\tilde{t}|\mathbf{A}} + \mathcal{L}\mathbf{1}\_{\tilde{t}|\mathbf{A}} + \alpha\_{\tilde{t}|\mathbf{A}^{\mu}} \tag{1}$$

CRLB(,)

,)

Bound of ,

(FIM) of ,

Weight of the -th Gaussian distribution

Fisher Information Matrix

CRLB(,)

CRLB(,)

CRLB(,)

Confluent Hypergeometric Function of the First Type

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . where k *xi*,*<sup>n</sup>* − *xj*,*<sup>n</sup>* k is the Euclidean distance, *dij*,*n*, between nodes *i*, *j*, *T* = *T<sup>n</sup>* − *Tn*−1, ω*ij*,*<sup>n</sup>* is the observed measurement noise, assuming that it obeys the Gaussian distribution, i.e., <sup>ω</sup>*ij*,*<sup>n</sup>* ∼ N 0, σ 2 *d* , and ζ*ij*,*<sup>n</sup>* is NLOS error, which is expressed as follows:

$$\zeta\_{ij,n} = \begin{cases} 0, (i,j) \notin \Theta\_n \\ \lambda e^{-\lambda b\_{ij,n}}, (i,j) \in \Theta\_n \end{cases} \tag{2}$$

,

,

Set of neighbor anchor nodes

of node at time

Relative slope of local clock offset between nodes ,

,

̃,

,

Range measurement between

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of neighbor anchor nodes of node at time

Position vector of node at

Relative slope of local clock offset between nodes ,

Range measurement between

pairs (,) at time

Average velocity of node from time − 1 to time

**Symbol Meaning Symbol Meaning**

̃ ,

,

,

<sup>2</sup> Variance of ,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Range measurement between node , at time

**Table 1.** List of symbols.

,

,

,

pairs (,) at time

time

Average velocity of node from time − 1 to time

time

,

(,)

<sup>2</sup> Variance of , ,

,

̃,

σ

,

σ,

̂,

<sup>2</sup> Variance of ,

,

,

,

,

,

Real time value at time ,

̃,

,

̃,

σ

σ

,

σ,

̂,

,

,

,→,

NLOS error in all node pairs (,) ∈ at time

Message send from , to ,

Confluent Hypergeometric Function of the First Type

Fisher Information Matrix (FIM) of ,

**Symbol Meaning Symbol Meaning**

,

,

,

̃,

σ

,

**Symbol Meaning Symbol Meaning**

,

,

̃,

σ

,

σ,

,

̂,

,

,

,

(∙)

̃,

,

σ

(,)

(,)

,

,

σ

̃,

σ,

(,)

(∙)

(,)

(,)

,→,

,→,

,

σ,

̂,

(∙)

,

(,)

**2. System Model** 

,

(,)

,

̃,

σ

,

σ,

,→,

[,, ,]

(∙)

(,)

,

,

**2. System Model** 

̃,

̃,

σ

σ

σ

(,)

**2. System Model** 

,

,

σ,

σ,

σ,

̂,

̂,

(∙)

(∙)

(∙)

,→,

,→,

[,, ,]

̂,

(∙)

(,)

(,)

(,)

(,)

,→,

**2. System Model** 

,→,

[,, ,]

σ,

,

(∙)

̂,

(,)

(,)

**2. System Model** 

(∙)

**2. System Model** 

(,)

[,, ,]

[,, ,]

[,, ,]

[,, ,]

[,, ,]

**2. System Model** 

**2. System Model** 

**2. System Model** 

(,)

(,)

,→,

̂,

[,, ,]

(,)

(,)

[,, ,]

[,, ,]

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

(∙)

,→,

,

,

,→,

,

σ,

̂,

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

<sup>2</sup> Variance of , ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Estimation result of node at time

Estimation result of node at time

NLOS error in all node pairs (,) ∈ at time

Message send from , to ,

(,)

Confluent Hypergeometric Function of the First Type

Fisher Information Matrix (FIM) of ,

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

<sup>2</sup> Variance of ,

σ

,

σ,

̂,

<sup>2</sup> Variance of ,

(∙)

(,)

**2. System Model** 

[,, ,]

,→,

,

,

,

̃,

,

,

,

̃,

σ

,

,

σ,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

**2. System Model** 

[,, ,]

(,)

̂,

(∙)

(,)

**2. System Model** 

,→,

[,, ,]

(,)

,→,

where *bij*,*<sup>n</sup>* > 0 [22], λ is a constant and Θ*<sup>n</sup>* is the set of all node pairs (*i*, *j*) with NLOS error at time *n*. Where, *aij*,*<sup>n</sup>* is the relative slope of local clock offset between nodes *i*, *j*, defined as follows: Position vector of all nodes at time Average velocity of node at time Clock offset slope of all nodes at time at time Position vector of all nodes at Clock offset slope of all nodes at time at time Position vector of all nodes at time Clock offset slope of all nodes at time pairs (,) at time (,) with NLOS error Position vector of all nodes at time at time Real time value at time , Slope of local clock of time at time Real time value at time , Slope of local clock of time at time Real time value at time , Slope of local clock of at time , Position vector of node at ̃ , Measurement value of local clock of node time at time Real time value at time , Slope of local clock of at time , ̃ , Measurement value of local clock of node Position vector of node at time ̃ , Measurement value of local clock of node

<sup>2</sup> Variance of ,

̃ ,

,

<sup>2</sup> Variance of ,

,

Set of neighbor nodes to be located of node at time

,

̃ ,

,

,

Measurement value of local clock of node at time

Slope of local clock of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor nodes of node at time

Measurement noise of ̃,

**Symbol Meaning Symbol Meaning**

,

̃ ,

,

,

Set of all node pairs (,) with NLOS error

,

Set of neighbor nodes to be located of node at time

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

̃ ,

,

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

Set of neighbor anchor nodes of node at time

**Symbol Meaning Symbol Meaning**

Position vector of node at

of node at time

time

,

Relative slope of local clock offset between nodes ,

Range measurement between

Relative slope of local clock

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error

,

of node at time

**Table 1.** List of symbols.

, NLOS error of ̃,

,

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

̃,

,

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

̃,

Set of neighbor anchor nodes of node at time

σ

Set of neighbor nodes to be located of node

Position vector of node at

σ

, NLOS error of ̃,

**Symbol Meaning Symbol Meaning**

,

,

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

̃ ,

,

at time

,

σ,

,

,

$$a\_{i|j,\mu} \triangleq \begin{pmatrix} \frac{a\_{i\mu}}{a\_{j\mu}}, t\_{i\mu} > t\_{j\mu} \\ \frac{d\_{j\mu}}{a\_{j\mu}}, t\_{i\mu} > t\_{j\mu} \end{pmatrix} \tag{3}$$

,

Set of neighbor nodes

,

Clock offset slope of all nodes at time

node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

̃ ,

,

̃ ,

,

,

,

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node

,

node at time

,

Weight of the -th

Slope of local clock of

Slope of local clock of

Measurement noise of

Measurement noise of

Vector sets of

Vector sets of

Weight of the -th Gaussian distribution

Weight of the -th Gaussian distribution

(,) Belief of variable ,

, Mean of belief (,)

Cramer-Rao Lower Bound of ,

Cramer-Rao Lower Bound of ,

at time

at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs

at time

Clock offset slope of all nodes at time

,

of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Weight of the -th

Cramer-Rao Lower Bound of ,

, Mean of belief (,)

(,) Belief of variable ,

, = (). The slope of local clock at time

local clock of node at time

local clock ofnode at time

Slope of local clock of

Slope of local clock of

Measurement noise of

Measurement noise of

Vector sets of

Vector sets of

Weight of the -th Gaussian distribution

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Cramer-Rao Bound of ,

,−1)/( − −1). The local clock

, NLOS error of ̃,

, NLOS error of ̃,

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor anchor nodes of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node

Set of neighbor nodes to be located of node

node at time

Set of neighbor nodes

Measurement value of local clock of node

Set of all node pairs (,) with NLOS error

Position vector of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Real time value at time ,

Range measurement between node , at time

Set of neighbor nodes to be located of node at time

,

Measurement value of local clock of node

Set of neighbor nodes to be located of node at time

node at time

at time

,

Relative slope of local clock offset between nodes ,

Set of all communicable node pairs (,) at time

> ̃ ,

, NLOS error of ̃,

Set of all communicable node

Range measurement between node , at time

Confluent Hypergeometric Function of the First Type

,

all nodes at time

,

Message send from , to ,

Define *x<sup>n</sup>* , [*x T* 1,*n* , *x T* 2,*n* , . . . , *x T S*+*C*,*n* ] *<sup>T</sup>* as the position vector of all nodes, *a<sup>n</sup>* , [*a*1,*n*, *a*2,*n*, . . . , *aS*+*C*,*n*] T as the clock offset slope of all nodes, ρ˜*<sup>n</sup>* , [. . . , ρ*ij*,*n*, . . .] *<sup>T</sup>* as the range measurement in all node pairs (*i*, *j*) ∈ Ψ*n*, and ζ*<sup>n</sup>* , [. . . , ζ*ij*,*n*, . . .] *<sup>T</sup>* as the NLOS error in all node pairs (*i*, *<sup>j</sup>*) <sup>∈</sup> <sup>Θ</sup>*n*. Moreover, all the vector sets from time 1 to time *n* are defined as follows: ̂, Estimation result of node at time ̃ NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: ,→, (,) Message send from , to , (,) Belief of variable , 1:*<sup>n</sup>* , {*x*1, *x*2, . . . , *xn*}, ̃ Range measurement in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Belief of variable , 1:*<sup>n</sup>* , {*a*1, *a*2, . . . , *an*}, of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , 1:*<sup>n</sup>* , ρ1, ρ2, . . . , ρ*<sup>n</sup>* , of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , 1:*<sup>n</sup>* , {ζ1, ζ2, . . . , ζ*n*}. The goal of the algorithm is to estimate the accurate node location vector *x<sup>n</sup>* and the clock slope *a<sup>n</sup>* by measuring σ, <sup>2</sup> Variance of , , Vector to be estimated ̂, Estimation result of node at time ̃ NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: 1:*<sup>n</sup>* and the information transmitted between nodes. node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time node , at time ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of of node at time , Measurement noise of ̃, , NLOS error of ̃, Set of all node pairs (,) with NLOS error at time Clock offset slope of Relative slope of local clock offset between nodes , of node at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

,

#### (∙) , Mean of belief (,) Confluent Hypergeometric , Confluent Hypergeometric ,→, (,) **3. M-VMP Joint Estimation Algorithm** from time − 1 to time , from time − 1 to time from time − 1 to time Position vector of all nodes at time , from time − 1 to time all nodes at time time Average velocity of node Set of neighbor anchor nodes **Symbol Meaning Symbol Meaning**

Message send from , to ,

offset between nodes ,

of node at time

Range measurement between

,

(,) ∈ at time

,

(FIM) of ,

,

,

Fisher Information Matrix

(FIM) of ,

Position vector of all nodes at time

Range measurement between

offset between nodes ,

Estimation result of node at time

Position vector of all nodes at time

pairs (,) at time

,

,

σ,

σ,

NLOS error in all node pairs (,) ∈ at time

Average velocity of node from time − 1 to time

Message send from , to

(FIM) of ,

,→,

,→,

Message send from , to ,

Confluent Hypergeometric Function of the First Type

, Covariance of belief (,) ,

(∙)

Fisher Information Matrix (FIM) of ,

Fisher Information Matrix (FIM) of ,

(,)

(,)

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

, and the measured value of local clock ̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

between node and the external clock is ,= (̃

[,, ,]

[,, ,]

between node and the standard clock is , = (̃

, Covariance of belief (,) , (,) Fisher Information Matrix (FIM) of , CRLB(,) **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) According to whether the location information is considered as a random variable, the cooperative localization algorithms are divided into two categories: non-Bayesian estimation and Bayesian estimation [5]. In non-Bayesian estimation, the location information is estimated by deterministic methods. The typical algorithms are least square (LS) [23,24] and the maximum likelihood method (ML) [25]. Bayesian estimation is based on the probability model of location information. The typical algorithms include the maximum posterior probability (MAP) estimation [26] and the MMSE method [6]. Location information node *i* estimates by MMSE criteria. The expression is as follows: <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:1:, 1:, 1: Vector sets of , , ̃, from time 1 to time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of of node at time , to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List symbols. **Symbol Meaning Symbol Meaning** ,Set of neighbor anchor nodes , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15**Table** List of symbols. **Symbol Symbol**Set of neighbor anchor nodes ,Set of neighbor to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision ˆ (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* = R (,) Belief of variable , *<sup>i</sup>*,*np*( Message send from , to (,) Belief of variable , *i*,*n*| NLOS error in all node pairs 1:, 1:, 1:, 1: , , ̃, from 1:*n*)*d* Message send from , to (,)Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* (4) , , ̃, from time 1 to time 1:, 1:, 1:, 1: , , ̃, from time 1 to time of node at time Measurement noise of node at time , Set of neighbor nodes at time Measurement value of Measurement of at time Measurement value of at time Measurement value of of node at time at time Measurement value of at time Measurement value of of node at time at time Measurement value of at time Measurement value of

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes Confluent Hypergeometric Function of the First Type , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) ,→, (,) Message send from , to (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) where ˆ (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* is the estimation result and Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* = [*xi*,*n*, *yi*,*n*, *ai*,*n*] *T* is the vector to be estimated. node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, offset between nodes , of node at time Range measurement between Measurement noise of Position vector of node at time ̃ , local clock of node at time , Position vector of node at time ̃ , local clock of node at time Position vector ofnode at time ̃ , local clock of node Position vector of node at time ̃ , local clock of node , Position vector of node at time ̃ , local clock of node , Position vector of node at time ̃ , local clock ofnode , Position vector of node at time ̃ , , Position vector of node at time ̃ ,

,

,

[,, ,]

Range measurement between

Range measurement between

CRLB(,)

**2. System Model** 

,

#### location information and local time out of sync. The position vector of node at time is , = location information and local time out of sync. The position vector of node at time is , = location information and local time out of sync. The position vector of node at time is , = , Covariance of belief (,) , , Covariance of belief (,) , , Covariance of belief (,) , (∙) Confluent Hypergeometric , Covariance of belief (,) , , Mean of belief (,) Confluent Hypergeometric Function of the First Type *3.1. Probability Model* Set of all communicable node node , at time <sup>2</sup> Variance of , Real time value at time , Real time value at time , Real time value at time , Real time value at time , Real time value at time , Real time value at time ,

(FIM) of ,

offset between nodes ,

,

,

offset between nodes ,

Range measurement between

Range measurement between

Fisher Information Matrix

pairs (,) at time

Estimation result of node at time

NLOS error in all node pairs (,) ∈ at time

CRLB(,)

, and the measured value of local clock ̃

pairs (,) at time

pairs (,) at time

Position vector of all nodes at time

Position of all nodes at time

, − ̃

NLOS error in all node pairs (,) ∈ at time

NLOS error in all node pairs (,) ∈ at time

NLOS error in all node pairs (,) ∈ at time

Message send from , to ,

Message send from , to ,

,→,

,→,

Message send from , to ,

Confluent Hypergeometric Function of the First Type

Confluent Hypergeometric Function of the First Type

(∙)

Confluent Hypergeometric Function of the First Type

, Covariance of belief (,) ,

,)

, Covariance of belief (,),

, Covariance of belief (,) ,

Fisher Information Matrix (FIM) of ,

(,)

**2. System Model** 

**2. System Model** 

(,)Fisher Information Matrix(FIM) of ,

Fisher Information Matrix

, and the measured value of local clock ̃

σ,

σ,

,

,

Average velocity of node from time − 1 to time

Average velocity of node from time − 1 to time

[,, ,]

Position vector of all nodes at time

Message send from , to ,

1:, 1:, 1:, 1:

of node at time

of node at time

of node at time

(,)

Relative slope of local clock

NLOS error in all node pairs (,) ∈ at time

between node and the external standard clock is , = (̃ , − ̃ of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , , = (). The slope of local clock at time , − ̃ ,−1)/( − −1). The local clock , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Function of the First Type , Mean of belief (,) Gaussian distribution (,) Fisher Information Matrix CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th , Covariance of belief (,) , Weight of the -th Assume *xi*,*<sup>n</sup>* evolve according to a memory-less Gauss–Markov process, then, there are: pairs (,) at time (,) with NLOS error at time , NLOS error of ̃, Set of all communicable node Set of all node pairs node at time Relative slope of local clock Set of neighbor nodes Real time value at time ,Slope of local clock ofnode at time ,Relative slope of local clock Set of neighbor nodes node at time Relative slope of local clock offset between nodes , node at time Relative slope of local clock Set of neighbor nodes node at time Relative slope of local clock Set of neighbor nodes node at time Relative slope of local clock Set of neighbor nodes Real time value at time , node at time Relative slope of local clock Set of neighbor nodes node at time Relative slope of local clock Set of neighbor nodes

Range measurement between

Range measurement between

, and the measured value of local clock ̃

, Mean of belief (,)

Weight of the -th

at time

at time

Slope of local clock of

Slope of of

Measurement noise of

Measurement noise of

Measurement noise of

̃,

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , Bound of , Bound of , Bound of , , Covariance of belief (,) , Gaussian distribution (FIM) of , Bound of , Gaussian distribution Cramer-Rao Lower Gaussian distribution Cramer-Rao Lower *xi*,*<sup>n</sup>* = *xi*,*n*−<sup>1</sup> + *vi*,*nT* + ω*i*,*n*, (5) Clock offset slope of all nodes at time (,) with NLOS error at time , of node at time , of node at time , Set of neighbor of node at time , of node at time offset between nodes , , of node at time offset between nodes ,, of node at time offset between nodes , , of node at time offset between nodes ,, of node at time

the anchor node with known location and synchronized local time, and the node with inaccurate

Weight of the -th

Vector sets of

time 1 to time

Weight of the -th

Cramer-Rao Lower

Weight of the -th

at time

Set of all node pairs

Measurement noise of

Slope of local clock of

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained (,) (FIM) of , CRLB(,) Bound of , **2. System Model 2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Bound of , (FIM) of , Bound of , where *vi*,*<sup>n</sup>* is the average velocity of node *i* from time *n* − 1 to time *n*, which is measured by the sensor inside nodes. <sup>ω</sup>*i*,*<sup>n</sup>* is Gaussian white noise, and its covariance matrix is *diag*<sup>n</sup> σ 2 *i*,*n* , σ 2 *i*,*n* o . Since the motions of all nodes are independent of each other, there are: Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs ̃,Range measurement between node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs node, at time , ̃, Variance of , ,NLOS error of ̃Set of all communicable node pairs (,) at time Set of all node pairs node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs ̃, node , at time , ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs ̃, node , at time , ̃, σ2 Variance of ,, NLOS error of ̃, Set of all communicable node Set of all node pairs ̃, node , at time , ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs ̃, node , at time , ̃, σ2 Variance of ,, NLOS error of ̃, Set of all communicable node Set of all node pairs

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes *p*(*xn*|*xn*−1) = Y *i p*(*xi*,*n*|*xi*,*n*−1) (6) of node at time ̃ Range measurement in all node pairs <sup>2</sup> Variance of , , Vector to be estimated of node at time (,) with NLOS error at time Clock offset slope of (,) with NLOS error at time Clock offset slope of (,)error at time Clock offset slope of (,) with NLOS error at time Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all at pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all at

$$p(\mathbf{x}\_0|\mathbf{s}\_{n-1}) = \prod\_{\mathbf{i}\mathbf{j}} p(\mathbf{s}\_{i\mathbf{r}}|\mathbf{s}\_{i\mathbf{r}-1}),\tag{7}$$
 
$$\mathbf{x}\_0^T \mathbf{x}\_n = \prod\_{\mathbf{i}\mathbf{j}} p(\mathbf{s}\_{i\mathbf{r}}|\mathbf{s}\_{i\mathbf{r}-1}),\tag{8}$$

between node and the external standard clock is , = (̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , between node and the external standard clock is , = (̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , between node and the external standard clock is , = (̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , , = (). The slope of local clock at time , = (). The slope of local clock at time *p* 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:*n* = *p*(*x*0) Y *n p*(*xn*|*xn*−1), (8) (,) ∈ at time Vector sets of , , , , time <sup>−</sup> <sup>1</sup>to time ,Measurement noise of , , , from time − 1 to time , , from time − 1 to time , ,from time − 1 to time , , from time − 1 to time , ,

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , (,) Belief of variable , *p* 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:*n* = *p*(*a*0) Y *i p*(*an*|*an*−1), (9) <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup>Variance of ,, Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time

Bound of ,

(,) Belief of variable ,

,(,) Belief of variable ,

, Mean of belief (,)

,)

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

CRLB(,)

Vector sets of

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th Message send from , to , (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) where *p*(*x*0) is the prior distribution of all nodes' location information at time 0, which is obtained by GNSS, base station or in other ways [27]. According to the Bayesian rule, the edge probability function in (3) is calculated by the following formula: Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂,Estimation result node at time ̃ Range measurement in all node pairs (,) ∈ at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation resultof node at time ̃ Range measurement in all node pairs (,)∈ at time error in all node pairs ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation resultof node at time ̃ Range measurement in all node pairs (,)∈ at time

$$p(\mathcal{X}\_{1:n}, \mathcal{A}\_{1:n} | \mathcal{P}\_{1:n}) \propto p(\mathcal{P}\_{1:n} | \mathcal{X}\_{1:n}, \mathcal{A}\_{1:n}) p(\mathcal{X}\_{1:n}, \mathcal{A}\_{1:n}) \tag{10}$$

(,) Belief of variable ,

Gaussian distribution

Cramer-Rao Lower Bound of ,

,(,) Belief of variable ,

, = (). The slope of local clock at time

Weight of the -th Gaussian distribution

Weight of the Gaussian distribution

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Bound of ,

Cramer-Rao Lower of ,

CRLB(,)

CRLB(,)

,−1)/( − −1). The local clock

by other sensors [21]. So, in this paper, a 2D wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

The anchor nodes are always deployed at the same height, and high vertical dilution of means the system cannot provide reliable vertical positioning [20], which is usually obtained

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

,−1 − −1)The local clock

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

,−1)/(− −1). The local clock

, = (). The slope of local clock at time

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ . At time ,

, − ̃

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

, = (). The slope of local clock at time

,̃

,̃

,−1)/( − −1). The local clock

)/( − −1). The local clock

, = (). The slope of local clock at time

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

CRLB(,)Cramer-Rao Lower

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

(,) Belief of variable ,

(,) of variable ,

(,) Belief of variable ,

, − ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of means the system cannot provide reliable vertical positioning results [20], which is usually obtained

by other sensors [21]. So, in this paper, a 2D wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is ,=

CRLB(,)

Fisher Information Matrix (FIM) of ,

Fisher Information Matrix (FIM) of ,

, = (). The slope of local clock at time

, = (). The slope of local clock at time

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

of all anchor nodes is synchronized with the external reference clock, i.e., , = . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, = ()The slope of local clock at time

, − ̃

, and the measured value of local clock ̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

between node and the external standard clock is , = (̃

CRLB(,)

Message send from , to ,

Message send from , to

Confluent Hypergeometric

, Covariance of belief (,) ,

CRLB(,)

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

Fisher Information Matrix (FIM) of ,

Fisher Information Matrix (FIM) of ,

, Covariance of belief (,) ,

Message send from , to ,

Message send from , to

Confluent Hypergeometric Function of the First Type

Function of the First Type

CRLB(,)

, − ̃

, − ̃

,− ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

is ,, and all communicable node pairs (, ) constitute communicable node set .

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

The nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate

The anchor nodes are always deployed at the same height, and high vertical of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the node with known location and synchronized local time, and the node with location information and local time out of sync. The position vector of node at time is ,=

(∙)Confluent Hypergeometric

(,)

(,)

is ,, and all communicable node pairs (, ) constitute communicable node set .

(FIM) of ,CRLB(,)

,)

(,)

**2. System Model** 

**2. System Model** 

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, communicable node set .

, and the measured value of local clock ̃

[,, ,]

[,, ,]

between node and the external standard clock is , = (̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

of all is synchronized with the external reference clock, i.e., , = 1∀ ∈ At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

location information and local time out of position vector of node at time is , =[,, ,]

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, and the measured value of local clock ̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

between node and the external standard clock is , = (̃

(,) ∈ at time

(,) ∈ at time

Message send from , to ,

Message send from , to ,

Confluent Hypergeometric Function of the First Type

Confluent Hypergeometric Function of the First Type

<sup>2</sup> Variance of ,

Real time value at time ,

Real time at time ,

Relative slope of local clock offset between nodes ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Range measurement node , at time

Set of all communicable node pairs (,) at time

of all communicable node pairs (,) at time

Position vector of all nodes at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Average velocity of node from time −1 to time

<sup>2</sup> Variance of ,

̃Range measurement between node , at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

of node at time

Position vector of node at time

Position of node at time

,

**Symbol Meaning Symbol Meaning**

**Symbol Meaning Symbol Meaning**

Real time value at time ,

time value at time ,

̃ ,

Relative slope of local clock offset between nodes ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Table 1.** List symbols.

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

**Symbol Meaning Symbol Meaning**

**Symbol Meaning Symbol Meaning**

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

,

,

Set of all communicable node pairs (,) at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Average velocity of node from time − 1to time

Real time value at time ,Slope of local clock of

<sup>2</sup> Variance of , ,

<sup>2</sup> Variance of , ,

,

Estimation result of node at

Estimation result of node at

,

**Symbol Meaning Symbol Meaning**

**Symbol Meaning SymbolMeaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Table 1.** List of symbols.

at time ,

,

,

̃ ,

̃

̃ ,

,

,

,

,

,

̃ ,

̃

̃ ,

, NLOS error of ̃,

,

,

,

,

Set of all node pairs

,

,

,

̃

,

,

,

,

,

,

**Symbol Meaning Symbol Meaning**

,Set of neighbor anchor nodes

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Position vector of node at time

**Symbol Meaning Symbol Meaning**

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Set of neighbor anchor nodes of node at time

Position vector of all nodes at time

<sup>2</sup> Variance of ,

<sup>2</sup> Variance of ,

Set of neighbor anchor nodes of node at time

of node at time

Position vector of node at time

Position vector of node at time

Position vector of node at time

Average velocity of node from time − 1 to time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Real time value at time ,

Relative slope of local clock offset between nodes ,

offset between ,

<sup>2</sup> Variance of , ,

Estimation result of node at

Message send from , to ,

Confluent Hypergeometric Function of the First Type

Position vector of all nodes at time

Position vector of all nodes at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Average velocity of node from time − 1 to time

Average velocity of node from time − 1 to time

Estimation result of node at time

Estimation result of node at time

Estimation result of node at time

NLOS error in all node pairs (,) ∈ at time

(,) ∈ at time

(,) ∈ at time

Message send from , to ,

Message send from , to ,

,

, and the measured value of local clock ̃

, and the measured value of local clock ̃

<sup>2</sup> Variance of ,

σ

,

̃,

,

,

̃,

σ

,

σ,

̂,

(∙)

,→,

(,)

**2. System Model** 

[,, ,]

[,, ,]

(,)

(∙)

(,)

**2. System Model** 

(,)

,→,

between node and the external standard clock is , = (̃

between node and the external standard clock is , = (̃

Fisher Information Matrix (FIM) of ,

<sup>2</sup> Variance of ,

,Set of neighbor anchor nodes

,

,

Set of neighbor anchor nodes of node at time

,

Position vector of node at time

,

Real time value at time ,

̃,

Relative slope of local clock offset between nodes ,

σ

Range measurement between node , at time

σ

Set of all communicable node pairs (,) at time

,

Position vector of all nodes at time

,

σ,

Average velocity of node from time − 1 to time

σ,

Relative slope of local clock

̂,

̂,

time

<sup>2</sup> Variance of ,

<sup>2</sup> Variance of ,

<sup>2</sup> Variance of ,

(∙)

Message send from , to ,

NLOS error in all node pairs (,) ∈ at time

(∙)

(,)

Confluent Hypergeometric Function of the First Type

**2. System Model** 

Fisher Information Matrix (FIM) of ,

**2. System Model** 

NLOS error in all node pairs

[,, ,]

,

,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

**2. System Model** 

[,, ,]

(,)

,→,

(,)

(,)

,→,

(,)

(,)

,→,

,

,

,

,

,

,

,

̃,

̃,

σ

σ

,

,

σ,

σ,

̂,

̂,

,

,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

**2. System Model** 

[,, ,]

,→,

,

̃,

σ

,

,

,

,

,

,

,

̃,

̃,

̃,

σ

σ

σ

<sup>2</sup> Variance of ,

,

σ,

̂,

(∙)

,→,

,→,

(,)

**2. System Model** 

(,)

(,)

(,)

[,, ,]

[,, ,]

[,, ,]

(∙)

(∙)

(,)

(,)

(,)

**2. System Model** 

**2. System Model** 

**2. System Model** 

[,, ,]

[,, ,]

[,, ,]

,→,

,→,

,→,

(,)

(∙)

(∙)

(,)

(,)

**2. System Model** 

**2. System Model** 

,→,

,

,

,

σ,

σ,

σ,

̂,

,

̂,

(,)

The likelihood function *p*( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:*n*| NLOS error in all node pairs (,) ∈ at time , 1:, 1:, 1: , , ̃, from time 1 to time 1:*n*, NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:*n* in (10) is decomposed into the following formula with the independent observations at different times ρ*ij*,*n*: Estimation result of node at ̃ in all node pairs (,) ∈ at time Estimation result of node at time ̃ in all node pairs (,) ∈ at time Vector sets of Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector of Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

to be located of node

,Set of neighbor nodes

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

local clock of node at time

Slope of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

(,) with NLOS error at time

Clock offset slope of all nodes at time

Clock offset slope of all nodes at time

Measurement noise of ,

Measurement noise of ,

Vector to be estimated of node at time

Range measurement

Measurement value of local clock of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Slope of local clock ofnode at time

Set of neighbor nodes of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Clock offset slope of all nodes at time

Measurement noise of ,

Measurement noise of ,

Vector to be estimated of node at time

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of

Vector sets of

Measurement value of local clock of node at time

Slope of local clock of node at time

,Measurement of

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

, NLOS error of ̃,

NLOS error of ̃,

Clock offset slope of all nodes at time

Measurement value of local clock of node at time

, NLOS error of ̃,

, error of ̃,

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Set of all node pairs

local clock of node at time

Slope of local clock of node at time

Slope of local clock of node at time

node at time

Set of neighbor nodes of node at time

Set of neighbor nodes of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Measurement noise of ̃,

Measurement noise of ̃,

Measurement value of local clock of node at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of

, NLOS error of ̃,

,Measurement of

(,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement

, NLOS error of ̃,

, NLOS error of ̃,

, NLOS error of ̃,

Vector sets of

,

,

̃

̃

,

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

,

,

,

̃ ,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

̃ ,

$$p(\mathcal{P}\_{1:n}|\mathcal{X}\_{1:n}, \mathcal{A}\_{1:n}) \propto \prod\_{n} \prod\_{(i,j) \notin \mathcal{B}\_n} p\_{\text{LOS}}(\rho\_{ij,n}) \prod\_{(i,j) \in \mathcal{B}\_n} p\_{\text{NLOS}}(\rho\_{ij,n}),\tag{11}$$

$$p\_{\rm LOS}(\rho\_{ij,n}) \propto \exp\left[-\frac{\left(\rho\_{ij,n} - \|\mathbf{x}\_{i,n} - \mathbf{x}\_{j,n}\| - cTa\_{ij,n}\right)^2}{2\sigma\_d^2}\right] \tag{12}$$

$$p\_{\rm NLCS}(\rho\_{\rm j,n}) \propto \exp\left[-\frac{\left(\rho\_{\rm j,n} - \|\cdot\mathbf{x}\_{\rm j,n} - \mathbf{x}\_{\rm j,n}\| - \mathcal{L}Ta\_{\rm ij,n} - \zeta\_{\rm j,n}\right)^2}{2\sigma\_d^2}\right] \tag{13}$$

CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , According to the above derivation, (10) is expanded to the following formula: (,) ∈ at time (,) ∈ at time (,) ∈ at time

, Mean of belief (,)

,

, Mean of belief (,)

, Mean of belief (,)

$$p(\mathbf{X}\_{1:n}, \mathcal{A} \mathbf{g}\_{1:n} | \mathbf{P}\_{1:n})$$

$$\begin{array}{c} \text{or} \\ \begin{array}{c} \\ \end{array} \begin{array}{c} \text{T} \\ \end{array} \begin{array}{c} \prod \\ (i, j) \oplus \mathbf{S}\_{i} \\ \end{array} p\_{i,0}(\mathbf{x}\_{i,0}) \prod \\ \begin{array}{c} p(\mathbf{x}\_{i,0} | \mathbf{y}(a\_{0})) \\ \end{array} \prod p(\mathbf{x}\_{i,0} | \mathbf{x}\_{i,-1}) p(a\_{i,0} | a\_{i,-1}), \end{array} \tag{14}$$

Vector sets of

Vector sets of

Vector sets of

, = (). The slope of local clock at time

,

,

at time

at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of

̃,

at time

, , ̃, from time 1 to time

, , ̃, from time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

#### , and the measured value of local clock ̃ , = (). The slope of local clock at time , and the measured value of local clock ̃ , and the measured value of local clock , = (). The slope of local clock at time by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes Confluent Hypergeometric Function of the First Type (∙)Confluent Hypergeometric Function of the First Type Confluent Hypergeometric Function of the First Type *3.2. Factor Graph Model* **Symbol Meaning Symbol Meaning Symbol Meaning Symbol Meaning Symbol Meaning Symbol Meaning**

time

offset between nodes ,

node , at time

pairs (,) at time

time

Message send from , to ,

Confluent Hypergeometric Function of the First Type

, Covariance of belief (,) ,

Fisher Information Matrix (FIM) of ,

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , the node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , <sup>=</sup> , and the measured value of local clock ̃ , = ()The slope of local clock at time between node and the external standard clock is , = (̃ ,− ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , of the -th Gaussian distributionFisher Information Matrix (FIM) of ,CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Factor graphs intuitively reflect the spatiotemporal relationship between variables [5]. The factor graph is a dichotomous graph, which contains two kinds of nodes: factor node and variable node. Variable node represents the information to be evaluated, and factor node represents the messages passed between variable nodes. The schematic diagram of cooperative localization is shown in Figure 1. Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at ̃ Measurement value of local clock of node , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Position vector of node at ̃ Measurement value of local clock of node , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Position vector of node at ̃ Measurement value of local clock of node

**Figure 1.** Factor graph model of variational message passing (VMP)-based cooperative localization problem. The circle represents the factor node and the square represents the variable node. *xi*,*<sup>n</sup>* = [*xi*,*<sup>n</sup> yi*,*n*] *T* represents the localization variable of the node to be located, *fi*,*<sup>n</sup>* represents the message that transmits between different times, *fij*,*<sup>n</sup>* represents the distance information that the nodes transmit, ζ*ij*,*<sup>n</sup>* represents the non-sight distance parameter that affects the information between nodes and *bij*,*<sup>n</sup>* represents the non-sight distance error probability function that affects the non-sight distance parameter. from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of

,

,

,

Factor nodes are further assumed as follows: (,) ∈ at time 1:, 1:, 1:, 1: (,) ∈ at time 1:, 1:, 1:, 1: (,) ∈ at time

,

,

Confluent Hypergeometric Function of the First Type

Fisher Information Matrix (FIM) of ,

Confluent Hypergeometric Function of the First Type

Fisher Information Matrix (FIM) of ,

$$f\_{i,0} = p(\mathfrak{z}\_{i,0})\_\prime f\_{i,n} = p(\mathfrak{z}\_{i,n}|\mathfrak{z}\_{i,n-1})\_\prime \tag{15}$$

Weight of the -th Gaussian distribution

, Mean of belief (,)

, Mean of belief (,)

Cramer-Rao Lower Bound of ,

, Mean of belief (,)

CRLB(,)

1:, 1:, 1:, 1:

, = (). The slope of local clock at time

, − ̃

CRLB(,)

,−1)/( − −1). The local clock

, − ̃

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

, , ̃, from time 1 to time

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

CRLB(,)

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

<sup>2</sup> Variance of ,

Position vector of node at time

̃,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of neighbor anchor nodes

<sup>2</sup> Variance of ,

**Table 1.** List of symbols.

σ

Set of neighbor anchor nodes of node at time

Position vector of node at time

Set of neighbor anchor nodes of node at time

,

time

Real time value at time ,

Set of neighbor anchor nodes of node at time

σ

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

,

,

,

,

,

Set of neighbor anchor nodes of node at time

̃,

Position vector of node at

**Table 1.** List of symbols.

,

,

,

**Symbol Meaning Symbol Meaning**

,

,

,

,

,

,

,

̃,

,

̃,

,

σ

Set of neighbor anchor nodes of node at time

Position vector of node at time

,

<sup>2</sup> Variance of ,

Set of neighbor anchor nodes

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

,

Real time value at time ,

Relative slope of local clock offset between nodes ,

,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

,

Position vector of all nodes at time

σ

̃,

Average velocity of node

time

σ,

,

Message send from , to ,

Position vector of node at time

̂,

,→,

(∙)

,→,

Set of all communicable node

,

Real time value at time ,

Position vector of node at time

(∙)

,

**2. System Model** 

pairs (,) at time

NLOS error in all node pairs (,) ∈ at time

̂,

Estimation result of node at time

σ

node , at time

Message send from , to ,

Position vector of all nodes at time

(,)

**2. System Model** 

**2. System Model** 

Position vector of all nodes at time

Average velocity of node

Set of all communicable node pairs (,) at time

NLOS error in all node pairs (,) ∈ at time

Estimation result of node at time

̂,

,→,

,

time

(,) ∈ at time

Function of the First Type

(,)

̃

,

(FIM) of ,

(∙)

,

[,, ,]

σ,

time

[,, ,]

,→,

, and the measured value of local clock ̃

, and the measured value of local clock ̃

Message send from , to ,

Function of the First Type

,→,

, and the measured value of local clock ̃

CRLB(,)

between node and the external standard clock is , = (̃

**2. System Model** 

[,, ,]

Fisher Information Matrix (FIM) of ,

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, Covariance of belief (,) ,

Fisher Information Matrix (FIM) of ,

Confluent Hypergeometric Function of the First Type

(∙)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, and the measured value of local clock ̃

Fisher Information Matrix (FIM) of ,

(,)

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

(,)

,→,

CRLB(,)

between node and the external standard clock is , = (̃

(∙)

, Covariance of belief (,) ,

,→,

[,, ,]

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

(,)

is ,, and all communicable node pairs (, ) constitute communicable node set .

**2. System Model** 

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

[,, ,] 

, and the measured value of local clock ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

between node and the external standard clock is , = (̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

(FIM) of ,

time

(∙)

,

̃

,→,

(,)

,→,

,

,

,

̃,

σ

,

σ,

̂,

time

σ

̃ ,

time

Average velocity of node

̂,

(,)

time

,

,→,

σ

(,)

Set of all communicable node pairs (,) at time

Average velocity of node

time

time

,

Set of all communicable node

,

(∙)

time

,→,

(,)

Relative slope of local clock

(∙)

̂,

(,)

pairs (,) at time

Message send from , to ,

,

Estimation result of node at

NLOS error in all node pairs (,) ∈ at time

Fisher Information Matrix (FIM) of ,

Message send from , to ,

σ

(,)

Estimation result of node at time

Message send from , to ,

**2. System Model** 

Confluent Hypergeometric Function of the First Type

Fisher Information Matrix (FIM) of ,

,

Estimation result of node at time

Message send from , to ,

,

(,)

Confluent Hypergeometric Function of the First Type

Relative slope of local clock offset between nodes ,

(,)

Real time value at time ,

Fisher Information Matrix

(∙)

(,)

(,)

<sup>2</sup> Variance of ,

Position vector of all nodes at time

pairs (,) at time

**2. System Model** 

Position vector of node at time

,

,

,

,

,

,

,

̃,

,

̃,

σ

,

,

,

̃,

σ

,

σ,

̂,

,

σ

,

,

σ,

̂,

(∙)

̂,

(,)

**2. System Model** 

(,)

(∙)

(,)

,→,

̂,

**2. System Model** 

(,)

(,)

[,, ,]

Confluent Hypergeometric Function of the First Type

(FIM) of ,

[,, ,]

,

Function of the First Type

Fisher Information Matrix (FIM) of ,

Estimation result of node at time

[,, ,]

[,, ,]

(,)

**2. System Model** 

(,)

[,, ,]

(,)

(,)

, and the measured value of local clock ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

between node and the external standard clock is , = (̃

[,, ,]

(,)

(,)

(∙)

[,, ,]

(∙)

(,)

(∙)

NLOS error in all node pairs (,) ∈ at time

**2. System Model** 

̂,

Message send from , to ,

(,)

(,)

(,)

**2. System Model** 

(∙)

Confluent Hypergeometric Function of the First Type

**2. System Model** 

(,)

**2. System Model** 

[,, ,]

Fisher Information Matrix (FIM) of ,

(,)

(∙)

**2. System Model** 

[,, ,]

(,)

**2. System Model** 

**2. System Model** 

[,, ,]

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

[,, ,]

(,)

**2. System Model** 

NLOS error in all node pairs (,) ∈ at time

Message send from , to ,

,→,

(,)

,→,

(,)

̂,

Estimation result of node at time

(∙)

σ,

,→,

[,, ,]

**2. System Model** 

(∙)

[,, ,]

Confluent Hypergeometric Function of the First Type

,→,

[,, ,]

Fisher Information Matrix (FIM) of ,

**2. System Model** 

(,)

, Covariance of belief (,) ,

[,, ,]

[,, ,]

,→,

,→,

,

σ,

̂,

σ,

(∙)

,→,

(,)

Real time value at time ,

,

Relative slope of local clock offset between nodes ,

,

Range measurement between node , at time

,

,

̃,

σ

,

σ,

̂,

Set of all communicable node pairs (,) at time

,

Position vector of all nodes at time

̃,

Set of neighbor anchor nodes of node at time

Average velocity of node

σ

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

time

time

,

Message send from , to ,

σ,

node , at time

Confluent Hypergeometric Function of the First Type

̂,

Set of all communicable node pairs (,) at time

,→,

(,)

(∙)

,→,

(,)

**2. System Model** 

(,)

Estimation result of node at time

(,)

**2. System Model** 

Message send from , to ,

NLOS error in all node pairs (,) ∈ at time

Fisher Information Matrix (FIM) of ,

(∙)

(,)

Position vector of all nodes at time

<sup>2</sup> Variance of ,

,

,

,

,

,

,

,

,

̃,

σ

,

Set of neighbor anchor nodes of node at time

,

̃,

,

σ

,

σ,

,

Position vector of node at time

,

Real time value at time ,

σ,

Relative slope of local clock offset between nodes ,

,

̂,

σ

̃,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

,→,

̂,

(∙)

Average velocity of node from time − 1 to time

σ,

,

,→,

<sup>2</sup> Variance of , ,

,→,

,

̂,

Estimation result of node at time

σ,

,→,

,

̃,

,

,

σ

,

,

̃,

σ

,

̃,

σ

,

,

,

̃,

Set of neighbor anchor nodes of node at time

,

σ

,

Position vector of node at time

̃,

,

σ

<sup>2</sup> Variance of ,

Real time value at time ,

̃,

,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

,

,

σ,

̂,

Set of all communicable node pairs (,) at time

,

,

σ,

,

σ,

̂,

σ

(∙)

σ,

,

(,)

̂,

(,)

(∙)

,→,

̂,

Average velocity of node from time − 1 to time

,

Position vector of all nodes at time

̃,

σ

σ,

**2. System Model** 

,→,

(,)

(∙)

,→,

σ

,

̃,

,

,

,

,

,

,

̃,

σ

,

σ,

̂,

,

,

,

̃,

σ

,

,

σ,

,

̂,

,

,

,

,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

**2. System Model** 

,→,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

(,)

**2. System Model** 

[,, ,]

[,, ,]

,→,

̃,

σ

,

,→,

(∙)

(,)

,

**2. System Model** 

<sup>2</sup> Variance of ,

σ,

̂,

,

(,)

σ,

̂,

(∙)

(,)

**2. System Model** 

[,, ,]

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

[,, ,]

[,, ,]

(∙)

(,)

(,)

**2. System Model** 

,→,

[,, ,]

[,, ,]

[,, ,]

,→,

(,)

,

(∙)

(,)

̃,

,

<sup>2</sup> Variance of ,

**2. System Model** 

σ

(,)

,→,

*fij*,*<sup>n</sup>* = *<sup>p</sup>LOS* ρ*ij*,*<sup>n</sup>* ,(*i*, *j*) < Θ*<sup>n</sup> <sup>p</sup>NLOS* ρ*ij*,*<sup>n</sup>* ,(*i*, *j*) ∈ Θ*<sup>n</sup>* , (16) (,) with NLOS error at time Clock offset slope of (,) with NLOS error at time Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error Position vector of all nodes at Clock offset slope of , NLOS error of ̃, Set of all node pairs (,) with NLOS error , NLOS error of ̃, Set of all node pairs (,) with NLOS error <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error , Measurement noise of ̃, , NLOS error of ̃, Range measurement between , Measurement noise of ̃, , NLOS error of ̃, Range measurement between node , at time , Measurement noise of ̃, σ2 Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of σ2 Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of <sup>2</sup> Variance of , , NLOS error of ̃, , Set of neighbor nodes of node at time Measurement noise of , Set of neighbor nodes of node at time Measurement noise of ̃ , local clock of node at time Slope of local clock of ̃ , local clock of node at time Slope of local clock of to be located of node at time Measurement value of at time ̃ Measurement value of local clock of node Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning**

,

all nodes at time

̃,

Vector to be estimated

$$\dot{\zeta}\_{\bar{i}|\bar{n}} = \dot{p} \dot{\zeta}\_{\bar{i}|\bar{n}},\tag{17}$$
 
$$\text{According to the above definition, also convolutionality of localization is shared calculated in the}$$

,

,

at time

Clock offset slope of

all nodes at time

Measurement noise of

**Table 1.** List of symbols.

Vector to be estimated

,

Vector sets of

, NLOS error of ̃,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Slope of local clock of node at time

Measurement noise of ̃,

Set of neighbor nodes of node at time

Set of neighbor nodes to be located of node at time

,

,

̃ ,

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

,

̃ ,

,

Slope of local clock of node at time

Set of neighbor nodes of node at time

,

Set of neighbor nodes to be located of node at time

,

̃ ,

Measurement value of local clock of node at time

,

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement value of local clock of node at time

Measurement noise of ̃,

Slope of local clock of node at time

Set of all node pairs

Set of neighbor nodes of node at time

at time

̃,

,

at time

at time

Slope of local clock of

at time

Range measurement in all node pairs

Measurement noise of ,

all nodes at time

Vector sets of

time 1 to time

at time

local clock of node at time

node at time

of node at time

̃,

Set of all node pairs

at time

Clock offset slope of all nodes at time

,

of node at time

in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Weight of the -th

Cramer-Rao Lower Bound of ,

in all node pairs

̃,

Vector sets of

time 1 to time

at time

Bound of ,

,

Bound of ,

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement value of local clock of node at time

Measurement noise of ̃,

Slope of local clock of node at time

Set of all node pairs

̃,

Set of neighbor nodes of node at time

,

Range measurement

Measurement noise of ,

all nodes at time

at time

̃,

,

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement value of local clock of node at time

Measurement noise of ̃,

Slope of local clock of node at time

Set of all node pairs

Set of neighbor nodes of node at time

, NLOS error of ̃,

Range measurement

Measurement noise of

Set of neighbor nodes

all nodes at time

node at

Set of neighbor nodes to be located of node

Set of neighbor nodes

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement value of local clock of node at time

,

Slope of local clock of node at time

̃ ,

,

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

Measurement noise of ̃,

Slope of local clock of node at time

Set of all node pairs

Set of neighbor nodes of node at time

, NLOS error of ̃,

, NLOS error of ̃,

Range measurement

Measurement noise of

, = (). The slope of local clock at time

CRLB(,)

, = (). The slope of local clock at time

Cramer-Rao Lower Bound of ,

Weight of the -th Gaussian distribution

,−1)/( − −1). The local clock

, − ̃

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

, − ̃

, = (). The slope of local clock at time

, Mean of belief (,)

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

,−1)/( − −1). The local clock

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate

,−1)/( − −1). The local clock

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

, − ̃

, Mean of belief (,)

, = (). The slope of local clock at time

, − ̃

all nodes at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

̃ ,

Set of neighbor nodes to be located of node at time

,

Slope of local clock of node at time

Set of neighbor nodes of node at time

̃ ,

Measurement value of local clock of node at time

Slope of local clock of node at time

,

Measurement noise of ̃,

Set of neighbor nodes of node at time

,

Slope of local clock of node at time

Set of all node pairs

Set of neighbor nodes of node at time

,

, NLOS error of ̃,

Measurement noise of ̃,

of node at time

all nodes at time

,

,

Vector to be estimated

Slope of local clock of

at time

Vector sets of

Set of neighbor nodes

Vector to be estimated

at time

Vector sets of

Range measurement

According to the above definition, edge probability of localization is clearly calculated in the factor graph. Next, a localization algorithm based on the VMP will be introduced. from time − 1 to time , , <sup>2</sup> Variance of , , Vector to be estimated from time − 1 to time , , <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time , , σ, <sup>2</sup> Variance of , , Vector to be estimated , from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Position vector of all nodes at time all nodes at time Average velocity of node , Measurement noise of time all nodes at time Average velocity of node , Measurement noise of time all nodes at time , Average velocity of node , Measurement noise of time all nodes at time , Average velocity of node , Measurement noise of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time at time Position vector of all nodes at Clock offset slope of pairs (,) at time at time Position vector of all nodes at Clock offset slope of pairs (,) at time at time Position vector of all nodes at Clock offset slope of pairs (,) at time Position vector of all nodes at Clock offset slope of <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs (,) with NLOS error <sup>2</sup> Variance of , Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between Measurement noise of offset between nodes , , of node at time Range measurement between , Measurement noise of Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes at time Real time value at time , Slope of local clock of Position vector of node at ̃ , Measurement value of local clock of node at time Position vector of node at ̃ , Measurement value of local clock of node Position vector of node at time ̃ , Measurement value of local clock of node at time , Position vector of node at ̃ , Measurement value of local clock of node , Position vector of node at time ̃ , Measurement value of local clock of node at time , Position vector of node at ̃ , Measurement value of local clock of node , Position vector of node at time ̃ , Measurement value of local clock of node of node at time , to be located of node at time Measurement value of of node at time , to be located of node at time Measurement value of **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

time

time

time

Set of neighbor nodes to be located of node at time

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Position vector of node at time

Set of neighbor anchor nodes of node at time

**Symbol Meaning Symbol Meaning**

Real time value at time ,

Relative slope of local clock offset between nodes ,

,

Position vector of node at time

Range measurement between node , at time

̃ ,

Real time value at time ,

,

Set of all communicable node

,

Relative slope of local clock offset between nodes ,

time

,

,

,

,

Set of neighbor nodes to be located of node at time

Set of neighbor anchor nodes of node at time

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Position vector of node at time

Set of neighbor anchor nodes of node at time

,

Set of neighbor anchor nodes of node at time

Position vector of node at time

Set of neighbor anchor nodes of node at time

,

**Symbol Meaning Symbol Meaning**

,

Real time value at time ,

̃ ,

Set of neighbor nodes to be located of node at time

Relative slope of local clock offset between nodes ,

Position vector of node at time

Range measurement between node , at time

,

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of

**Symbol Meaning Symbol Meaning**

Real time value at time ,

,

Set of neighbor nodes to be located of node

Slope of local clock of node at time

Set of all communicable node

Relative slope of local clock offset between nodes ,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes to be located of node at time

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

**Table 1.** List of symbols.

Set of neighbor nodes to be located of node at time

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Set of neighbor nodes to be located of node at time

Measurement noise of ̃,

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

̃ ,

,

,

̃ ,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

**Table 1.** List of symbols.

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement value of

̃ ,

Real time value at time ,

Relative slope of local clock offset between nodes ,

Position vector of node at time

Range measurement between node , at time

Real time value at time ,

Set of all communicable node

Relative slope of local clock offset between nodes ,

,

̃ ,

̃ ,

,

,

,

,

,

Set of neighbor nodes

Measurement value of local clock of node at time

Position vector of node at time

Set of neighbor anchor nodes of node at time

,

**Symbol Meaning Symbol Meaning**

Slope of local clock of node at time

̃ ,

Real time value at time ,

,

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes of node at time

̃ ,

Relative slope of local clock offset between nodes ,

Position vector of node at time

> Measurement noise of ̃,

Range measurement between node , at time

,

,

Set of all node pairs

**Table 1.** List of symbols.

,

<sup>2</sup> Variance of ,

Real time value at time ,

Set of all communicable node

Relative slope of local clock offset between nodes ,

Measurement value of local clock of node at time

,

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Slope of local clock of node at time

̃ ,

,

**Symbol Meaning Symbol Meaning**

̃ ,

Set of neighbor nodes of node at time

,

Measurement noise of ̃,

̃ ,

Set of all node pairs

<sup>2</sup> Variance of ,

,

,

at time

,

,

, NLOS error of ̃,

<sup>2</sup> Variance of ,

#### *3.3. VMP-Based Localization Algorithm* of node at time Range measurement Estimation result of node at Range measurement Estimation result of node at Estimation result of node at Estimation result of node at from time − 1 to time from time − 1 to time <sup>2</sup> Variance of , , σ, <sup>2</sup> Variance of , , σ, <sup>2</sup> Variance of , , time time Average velocity of node time time Average velocity of node pairs (,) at time Position vector of all nodes at node , at time , node , at time <sup>2</sup> Variance of , offset between nodes , , offset between nodes , Range measurement between Relative slope of local clock Slope local clock of Position vector of node at time ̃ , Position vector of node at time ̃ , , Set of neighbor anchor nodes **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Table 1.** List of symbols.

(∙)

(,) ∈ at time

(FIM) of ,

̃

,

σ,

(FIM) of ,

is ,, and all communicable node pairs (, ) constitute communicable node set .

**2. System Model** 

[,, ,]

(,)

(,) ∈ at time

(,)

is ,, and all communicable node pairs (, ) constitute communicable node set .

(,)

(∙)

is ,, and all communicable node pairs (, ) constitute communicable node set .

**2. System Model** 

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

[,, ,]

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

[,, ,]

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

Confluent Hypergeometric Function of the First Type

,

, − ̃

Fisher Information Matrix (FIM) of ,

[,, ,]

, Covariance of belief (,) ,

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

CRLB(,)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, = (). The slope of local clock at time

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, = (). The slope of local clock at time

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, − ̃

,−1)/( − −1). The local clock

is ,, and all communicable node pairs (, ) constitute communicable node set .

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

(,)

(,)

̂,

,

(∙)

Estimation result of node at

,

at time

(∙)

,

,

̃ ,

the anchor node with known location and synchronized local time, and the node with inaccurate

1:, 1:, 1:, 1:

1:, 1:, 1:, 1:

, − ̃

CRLB(,)

(,)

(∙)

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

, = (). The slope of local clock at time

, Covariance of belief (,) ,

, − ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

**2. System Model** 

CRLB(,)

, and the measured value of local clock ̃

Fisher Information Matrix

CRLB(,)

Function of the First Type

between node and the external standard clock is , = (̃

Gaussian distribution

Cramer-Rao Lower Bound of ,

,−1)/( − −1). The local clock

[,, ,]

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Symbol Meaning SymbolMeaning**

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

,

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

**Table 1.** List of symbols.

Position vector of node at time

,

**Symbol Meaning Symbol Meaning**

,

,

Real time value at time ,

Real time value at time ,

̃,

Relative slope of local clock offset between nodes ,

σ

,

,

̃,

node , at time

σ

̃ ,

of node at time

from time − 1 to time

,

,

Position vector of node at time

Set of neighbor anchor nodes of node at time

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Symbol Meaning Symbol Meaning**

Real time value at time ,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of node at time

Relative slope of local clock offset between nodes ,

Position vector of node at time

,

,

**Table 1.** List of symbols.

̃,

σ

,

**Table 1.** List of symbols.

̃,

Range measurement between node , at time

**Symbol Meaning Symbol Meaning**

Relative slope of local clock offset between nodes ,

Real time value at time ,

Set of all communicable node

offset between nodes ,

node , at time

time

Average velocity of node

,

Set of all communicable node

Position vector of all nodes at

Set of neighbor anchor of node at time

,

from time − 1 to time

time

̃ ,

**Table 1.** List of symbols.

,

Set of neighbor anchor nodes of node at time

,

**Symbol Meaning Symbol Meaning**

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Position vector of node at time

̃,

**Table 1.** List of symbols.

σ

̃,

,

**Table 1.** List of symbols.

offset between nodes ,

,

σ

Set of all communicable node

,

node , at time

,

**Symbol Meaning Symbol Meaning**

Real time value at time ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

̃ ,

Position vector of node at time

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Position vector of node at time

Set of neighbor anchor nodes of node at time

Relative slope of local clock offset between nodes ,

Real time value at time ,

Set of neighbor anchor nodes

node , at time

Position vector of node at time

offset between nodes ,

Range measurement between node , at time

<sup>2</sup> Variance of ,

Set of all communicable node

Relative slope of local clock

Set of all communicable node pairs (,) at time

Real time value at time ,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Real time value at time ,

,

Relative slope of local clock offset between nodes ,

**Table 1.** List of symbols.

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

node , at time

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Position vector of all nodes at

,

Set of all communicable node pairs (,) at time

VMP uses the exponential model to deliver messages, which greatly reduces communication consumption. According to the VMP message rules based on the factor graph proposed in Reference [28] and the assumptions in Section 3.2, the message delivered from factor node to parameter node at time n is as follows: Estimation result of node at ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, Estimation result of node at time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:,1: Vector sets of , , ̃, from time 1 to time ̂, time ̃ (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from <sup>2</sup> Variance of , , of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Range measurement from time − 1 to time , , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at Range measurement , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Range measurement , from time − 1 to time , , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at Range measurement ,Average velocity of node from time − 1 to time , ,σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at Range measurement ,Average velocity of node from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at Range measurement , from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation resultof node at Range measurement Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Vector to be estimated time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs (,) with NLOS error node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Real time value at time , node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between , Measurement noise of Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between , Measurement noise of Real time value at time , node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between , Measurement noise of Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of Real time value at time , node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of Real time value at time , node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of at time Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time at time Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time of node at time at time , Position vector of node at time ̃ , Measurement value of local clock of node at time , to be located of node at time ̃ , Measurement value of local clock of node Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at Measurement value of **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes node at time , Set of neighbor nodes to be located of node **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning**

<sup>µ</sup>*fij*,*n*<sup>→</sup> time 1 to time (,) Belief of variable , *i*,*n* ( (,) Belief of variable , *<sup>i</sup>*,*n*) <sup>=</sup> exp(Z µ*bij*,*n*→*fij*,*<sup>n</sup> bij*,*<sup>n</sup>* µ time 1 to time Message send from , to , (,) Belief of variable , *<sup>j</sup>*,*n*→*fij*,*<sup>n</sup>* (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *j*,*n* ln *fij*,*<sup>n</sup>* ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *i*,*n*, ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *j*,*n d* ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *j*,*n* ) , (18) NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time time ̃ in all node pairs (,) ∈ at time Vector sets of time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of time ̃ in all node pairs (,) ∈ at time Vector sets of time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of ̂, time ̃ (,) ∈ at time NLOS error in all node pairs <sup>2</sup> Variance of , , of node at time Estimation result of node at ̃ Range measurement in all node pairs of node at time Estimation result of node at time ̃ Range measurement in all node pairs time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , time Average velocity of node from time − 1 to time , Measurement noise of , pairs (,) at time at time Position vector of all nodes at Clock offset slope of at time Position vector of all nodes at Clock offset slope of all nodes at time Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time ̃, , NLOS error of ̃, Set of all node pairs node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs node , at time σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs node , at time σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Real time value at time , Slope of local clock of node at time , Relative slope of local clock , Set of neighbor nodes at time Slope of local clock of node at time ̃ , local clock of node at time Real time value at time , Slope of local clock of ̃ , Measurement value of local clock of node at time Position vector of node at ̃ , Measurement value of local clock of node at time , Position vector of node at time ̃ , Measurement value of at time Measurement value of local clock of node Set of neighbor anchor nodes of node at time , to be located of node at time Set of neighbor anchor nodes of node at time , to be located of node at time , Set of neighbor anchor nodes of node at time , to be located of node at time

Set of neighbor nodes

Estimation result of node at

, NLOS error of ̃,

at time

local clock of node

of node at time

at time

̃,

at time

Slope of local clock of

Average velocity of node

time

local clock of node

of node at time

, Mean of belief (,) , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Function of the First Type , Mean of belief (,) Function of the First Type , Mean of belief (,) Function of the First Type , Mean of belief (,) Function of the First Type , Mean of belief (,) <sup>µ</sup>*fi*,*n*<sup>→</sup> (,) Belief of variable , *i*,*n* ( (,) Belief of variable , *<sup>i</sup>*,*n*) = *p*( Message send from , to (,) Belief of variable , *i*,*n*| Message send from , to (,) Belief of variable , *<sup>i</sup>*,*n*−1), (19) 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:, 1:, 1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:,1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:,1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from (,) ∈ at time Vector sets of (,) ∈ at time Vector sets of <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time all nodes at time Measurement noise of , Measurement noise of , Position vector of all nodes at time Clock offset slope of all nodes at time (,) with NLOS error at time (,) with NLOS error at time (,) with NLOS error at time Set of all communicable node pairs (,) at time (,) with NLOS error at time pairs (,) at time (,) with NLOS error at time Set of all communicable node pairs (,) at time (,) with NLOS error at time Set of all communicable node pairs (,) at time (,) with NLOS error Set of all node pairs (,) with NLOS error Set of all node pairs (,) with NLOS error offset between nodes , of node at time Range measurement between , Measurement noise of Set of neighbor nodes of node at time node at time Set of neighbor nodes Slope of local clock ofnode at time Slope of local clock of node at time Real time value at time , Slope of local clock of Slope of local clock of Measurement value of local clock of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. Position vector of node at ̃ , Measurement value of local clock of node Position vector of node at time ̃ , Measurement value of local clock of node

, Covariance of belief (,) , Weight of the -th Gaussian distribution Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix CRLB(,) Cramer-Rao Lower Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th µ*fij*,*n*→*bij*,*<sup>n</sup> bij*,*<sup>n</sup>* <sup>=</sup> exp(Z µ Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*→*fij*,*<sup>n</sup>* Message send from , to , (,) Belief of variable , Confluent Hypergeometric *i*,*n* µ Message send from , to , (,) Belief of variable , *<sup>j</sup>*,*n*→*fij*,*<sup>n</sup>* (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *j*,*n* ln *fij*,*<sup>n</sup>* ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *i*,*n*, ,→, (,) Message send from , to , (,)Belief of variable , Confluent Hypergeometric *j*,*n d* ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *j*,*n* ) , (20) NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Estimation result of node at ̃ Range measurement in all node pairs Estimation result of node at time ̃ Range measurement in all node pairs from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time from time − 1 to time <sup>2</sup> Variance of , , Vector to be estimated of node at time , Average velocity of node from time − 1 to time , Measurement noise of , Vector to be estimated Clock offset slope of all nodes at time Position vector of all nodes at Clock offset slope of all nodes at Position vector of all nodes at time Clock offset slope of all nodes at time Position vector of all nodes at time Clock offset slope of all nodes at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Position vector of all nodes at time Clock offset slope of all nodes at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node pairs (,) at time at time Position vector of all nodes at Clock offset slope of pairs (,) at time at time Position vector of all nodes at Clock offset slope of ̃, node , at time ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs Measurement noise of ̃, , NLOS error of ̃, offset between nodes , , of node at time Range measurement between , Measurement noise of ̃, , Set of neighbor nodes of node at time Measurement noise of Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between Measurement noise of , Relative slope of local clock offset between nodes , , Set of neighbor nodes Range measurement between Measurement noise of node at time Set of neighbor nodes of node at time at time Real time value at time , Slope of local clock of node at time **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes Set of neighbor nodes time at time Real time value at time , Slope of local clock of node at time at time Real time value at time , Slope of local clock of node at time

Vector sets of

CRLB(,) Bound of , (FIM) of , Bound of , (,) (FIM) of , CRLB(,) Bound of , **2. System Model**  (,) (FIM) of , Bound of , **2. System Model**  (FIM) of , Bound of , **2. System Model**  (FIM) of , Bound of , **2. System Model**  (FIM) of , **2. System Model**  , Covariance of belief (,) , Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution where the cooperative messages µ Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>j</sup>*,*n*→*fij*,*<sup>n</sup>* Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *j*,*n* are divided into two categories: one is related to the neighbor anchor node *j* ∈ *Si*,*n*, and the other is related to the neighbor node *k* ∈ *Ci*,*<sup>n</sup>* to be located. The calculations are performed below: time (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of σ, <sup>2</sup> Variance of , , of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , Average velocity of node from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Vector to be estimated time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Vector to be estimated Set of all communicable node pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of Set of all node pairs (,) with NLOS error at time node , at time <sup>2</sup> Variance of , ,NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error Range measurement between node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs ̃, node , at time , σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node , Measurement noise of ̃, , NLOS error of ̃, Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between , Measurement noise of of node at time , to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution , Covariance of belief (,) , Weight of the -th Gaussian distribution <sup>µ</sup>*fij*,*n*<sup>→</sup> (,) Belief of variable , *i*,*n* ( (,) Belief of variable , *<sup>i</sup>*,*n*) ∝ N ρ*ij*,*n*|k *xi*,*<sup>n</sup>* − *xj*,*<sup>n</sup>* k, σ 2 *d* (21) 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time NLOS error in all node pairs Vector sets of Range measurement in all node pairs ̃ Range measurement in all node pairs ̃ Range measurement in all node pairs Estimation result of node at time ̃ Range measurement in all node pairs Estimation result of node at time ̃ Range measurement in all node pairs Estimation result of node at time ̃ Range measurement in all node pairs ̂, Estimation result of node at time ̃ Range measurement in all node pairs <sup>2</sup> Variance of , , of node at time Range measurement <sup>2</sup> Variance of , , of node at time Range measurement time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Clock offset slope of all nodes at time at time Clock offset slope of all nodes at time (,) with NLOS error at time Clock offset slope of (,) with NLOS error at time Clock offset slope of pairs (,) at time (,) with NLOS error Position vector of all nodes at Set of all node pairs (,) with NLOS error at time ̃, , NLOS error of ̃, Set of all node pairs Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes node , at time ̃, <sup>2</sup> Variance of ,, NLOS error of ̃, Set of all node pairs node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs

$$\mu\_{\tilde{f}\_{\tilde{f}|\mu},\mathfrak{T}\_{\text{tr}}}(\mathfrak{T}\_{0|\mu}) = \exp[\mathbb{E}\_{\theta|\tau\_{0}}[\mathcal{N}[\rho\_{\mathbf{J}|\mu} \parallel \mathbf{r}\_{0|\mu} - \mathbf{r}\_{\text{tr},\mu} \parallel \sigma\_{\mathbf{J}}^{2}]],\tag{22}$$
 
$$\mu\_{\tilde{f}|\mu}, \mathfrak{T}\_{\text{tr}}, \mathfrak{T}\_{\text{tr}}, \{\mathcal{N}\_{0}, \dots, \mathcal{N}\_{\text{tr}}, \{\mathcal{N}\_{0}, \sigma\_{\mathbf{J}}^{2} - \mathbf{r}\_{\text{tr},\mu} \parallel \sigma\_{\mathbf{J}}^{2}\}\},\tag{23}$$

, , ̃, from

Set of all node pairs

location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time , and the measured value of local clock ̃ , = (). The slope of local clock at time location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained , Covariance of belief (,) , Weight of the -th Gaussian distribution , Covariance of belief (,) , Gaussian distribution Confluent Hypergeometric , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) where E*f*(·) [*g*(·)] means the mean of *g*(·) with respect to *f*(·). The belief of ,→, (,) , (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* is obtained as follows: 1:, 1:, 1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs Vector sets of NLOS error in all node pairs Vector sets of Range measurement <sup>2</sup> Variance of , , of node at time from time − 1 to time <sup>2</sup> Variance of , , Vector to be estimated Average velocity of node from time − 1 to time , , Average velocity of node from time − 1 to time , , , from time − 1 to time , , Measurement noise of Position vector of all nodes at Clock offset slope of node , at time ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Position vector of all nodes at Clock offset slope of Position vector of all nodes at Clock offset slope of

Estimation result of node at

between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃, − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Bound of , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix CRLB(,) Cramer-Rao Lower *b*( (,) Belief of variable , , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *<sup>i</sup>*,*n*) = <sup>1</sup> *Z* <sup>µ</sup>*fi*,*n*<sup>→</sup> Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th *i*,*n* ( Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *<sup>i</sup>*,*n*) Y *j*∈*Si*,*<sup>n</sup>* <sup>µ</sup>*fij*,*n*<sup>→</sup> Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th *i*,*n* ( Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *<sup>i</sup>*,*n*) Y *k*∈*Ci*,*<sup>n</sup>* µ*f ik*,*n*→ ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th *i*,*n* ( ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *<sup>i</sup>*,*n*) ∝ 1 *Z*N time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *i*,*n*| (,) ∈ at time time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*n*−<sup>1</sup> + *vi*,*nT*, σ 2 *i*,*n* Y *j*∈*Si*,*<sup>n</sup>* N ρ*ij*,*n*|k *xi*,*<sup>n</sup>* − *xj*,*<sup>n</sup>* k, σ 2 *d* × Y *k*∈*Ci*,*<sup>n</sup>* expn E*b*(*x<sup>k</sup>* ) h N ρ*ik*,*n*|k *xi*,*<sup>n</sup>* − *xk*,*<sup>n</sup>* k, σ 2 *d* io (23) ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Range measurement in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time σ, <sup>2</sup> Variance of , , Vector to be estimated ̂, Estimation result of node at time ̃ Range measurement , Vector to be estimated of node at time Range measurement in all node pairs all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Set of all communicable node pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time time all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time

, , ̃, from

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate (FIM) of , Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , where *Z* is the normalization constant. Obviously, it can be seen that *b*( ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *<sup>i</sup>*,*n*) is not a Gaussian function about time 1 to time (,) Belief of variable , , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *<sup>i</sup>*,*n*, and it is difficult to transmit directly between nodes. To reduce communication overhead, Reference [19] approximates *b*( (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*n*) by minimizing KL divergence: where G represents a Gaussian function, and KLD(*q*( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*)|b( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*)) is the KL divergence between Gaussian function *q*( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*) and belief *b*( (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Belief of variable , *<sup>i</sup>*,*n*), the calculation formula is as follows: Estimation result of node at time ̃Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, Estimation resultof node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, = (). The slope of local clock at time

, and the measured value of local clock ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

, = (). The slope of local clock at time

between node and the external standard clock is , = (̃

, = (). The slope of local clock at time

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, Mean of belief (,)

Cramer-Rao Lower Bound of ,

is ,, and all communicable node pairs (, ) constitute communicable node set .

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

is ,, and all communicable node pairs (, ) constitute communicable node set .

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

,−1)/( − −1). The local clock

, − ̃

, − ̃

Cramer-Rao Lower Bound of ,

Bound of ,

Weight of the -th Gaussian distribution

Fisher Information Matrix (FIM) of ,

,−1)/( − −1). The local clock

Cramer-Rao Lower Bound of ,

, Mean of belief (,)

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

(FIM) of ,CRLB(,)

,−1)/( − −1). The local clock

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, − ̃

, = (). The slope of local clock at time

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

, = (). The slope of local clock at time

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

,−1)/( − −1). The local clock

, − ̃

Fisher Information Matrix (FIM) of ,

Function of the First Type

CRLB(,)

, Covariance of belief (,) ,

is ,, and all communicable node pairs (, ) constitute communicable node set .

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, − ̃

, − ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

, − ̃

between node and the external standard clock is , = (̃

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

location information and local time out of sync. The position vector of node at time is , =

, = (). The slope of local clock at time

, and the measured value of local clock ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Gaussian distribution Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Cramer-Rao Lower , Mean of belief (,) Weight of the -th Gaussian distribution , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Gaussian distribution , Mean of belief (,) Weight of the -th *b*( (,) Belief of variable , *<sup>i</sup>*,*n*) = **argmin** *q*( time 1 to time (,) Message send from , to (,) Belief of variable , *<sup>i</sup>*,*n*)∈G KLD *q* (*m*) ( Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*)|*b* (*m*) ( Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*) (24)

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

CRLB(,)

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

Cramer-Rao Lower Bound of ,

Weight of the -th Gaussian distribution

, = (). The slope of local clock at time

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes as ,, where neighbor anchor node set is , , node set to be located

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

, = (). The slope of local clock at time

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

is ,, and all communicable node pairs (, ) constitute communicable node set .

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

between node and the external standard clock is , = (̃

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, and the measured value of local clock ̃

,

,

(FIM) of ,

̃

CRLB(,)

,

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

<sup>2</sup> Variance of ,

,

Average velocity of node from time − 1 to time

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

CRLB(,)

Position vector of node at time

Set of neighbor anchor nodes of node at time

node , at time

pairs (,) at time

,

time

,

Average velocity of node from time − 1 to time

CRLB(,)

,

,

̃,

̃ ,

σ

,

node , at time

̃,

σ

,

pairs (,) at time

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

[,, ,]

is ,, and all communicable node pairs (, ) constitute communicable node set .

between node and the external standard clock is , = (̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, and the measured value of local clock ̃

, − ̃

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

(,)

[,, ,]

, and the measured value of local clock ̃

CRLB(,)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

**2. System Model** 

between node and the external standard clock is , = (̃

location information and local time out of sync. The position vector of node at time is , =

[,, ,]

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

(∙)

is ,, and all communicable node pairs (, ) constitute communicable node set .

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

is ,, and all communicable node pairs (, ) constitute communicable node set .

, and the measured value of local clock ̃

[,, ,]

between node and the external standard clock is , = (̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

[,, ,]

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

Fisher Information Matrix (FIM) of ,

[,, ,]

, and the measured value of local clock ̃

, and the measured value of local clock ̃

Fisher Information Matrix (FIM) of ,

between node and the external standard clock is , = (̃

location information and local time out of sync. The position vector of node at time is , =

(,)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

**2. System Model** 

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

between node and the external standard clock is , = (̃

**2. System Model** 

, − ̃

, and the measured value of local clock ̃

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, − ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

[,, ,]

, and the measured value of local clock ̃

, = (). The slope of local clock at time

, and the measured value of local clock ̃

, − ̃

between node and the external standard clock is , = (̃

between node and the external standard clock is , = (̃

CRLB(,)

̃ ,

,

,

,

̃,

σ

,

,

,

,

,

Estimation result of node at

, = (). The slope of local clock at time

, = (). The slope of local clock at time

Relative slope of local clock

of node at time

Real time value at time ,

Clock offset slope of

Set of all communicable node

Estimation result of node at time

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

,

,

̃ ,

,

,

Set of neighbor nodes to be located of node at time

,

,

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

̃,

σ

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error

Measurement value of local clock of node at time

,

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

̃

Measurement value of local clock of node at time

1:, 1:, 1:, 1:

Set of neighbor nodes of node at time

,

,

Slope of local clock of node at time

Set of all node pairs

̃ ,

,

,

,

̃ ,

,

̃

,

,

1:, 1:, 1:, 1:

, NLOS error of ̃,

,

̃ ,

,

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

̃ ,

,

,

to be located of node at time

CRLB(,)

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of

Set of all node pairs (,) with NLOS error at time

̃ ,

, NLOS error of ̃,

Slope of local clock of node at time

of node at time

at time

Slope of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

,

in all node pairs

time 1 to time

Gaussian distribution

, Mean of belief (,)

Cramer-Rao Lower Bound of ,

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

,

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

,

Clock offset slope of all nodes at time

Measurement noise of ,

Vector sets of

in all node pairs

time 1 to time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

,−1)/( − −1). The local clock

, NLOS error of ̃,

**Symbol Meaning Symbol Meaning**

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

time

,

to be located of node at time

of node at time

to be located of node at time

Measurement value of local clock of node at time

**Table 1.** List of symbols.

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes

at time

Slope of local clock of node at time

̃ ,

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

at time

̃,

,

,

at time

̃,

Set of all node pairs

at time

all nodes at time

,

Vector sets of

to be located of node at time

to be located of node at time

**Table 1.** List of symbols.

of node at time

of node at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , ̃, from time 1 to time

<sup>2</sup> Variance of ,

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Set of neighbor nodes to be located of node at time

, NLOS error of ̃,

Clock offset slope of all nodes at time

Measurement value of local clock of node at time

Measurement noise of ,

Slope of local clock of node at time

Vector to be estimated of node at time

Set of neighbor nodes of node at time

Range measurement in all node pairs (,) ∈ at time

Measurement noise of ̃,

> Vector sets of , , ̃, from time 1 to time

Set of all node pairs (,) with NLOS error

, NLOS error of ̃,

, NLOS error of ̃,

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning**

Position vector of all nodes at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning**

̃ ,

Average velocity of node from time − 1 to time

<sup>2</sup> Variance of , ,

,

̃ ,

,

,

,

,

,

,

̃ ,

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Set of neighbor anchor nodes of node at time

<sup>2</sup> Variance of ,

Position vector of all nodes at time

Position vector of node at time

Average velocity of node from time − 1 to time

<sup>2</sup> Variance of , ,

Real time value at time ,

Estimation result of node at time

Real time value at time ,

Range measurement between node , at time

Position vector of node at time

Relative slope of local clock offset between nodes ,

Set of neighbor anchor nodes of node at time

Message send from , to ,

Position vector of all nodes at time

pairs (,) at time

Range measurement between node , at time

offset between nodes ,

Confluent Hypergeometric Function of the First Type

Average velocity node from time − 1 to time

Set of all communicable node pairs (,) at time

Fisher Information Matrix (FIM) of ,

Set of neighbor anchor nodes of node at time

Position vector of node at

,

Average velocity of node from time − 1 to time

Position vector of all nodes at time

Estimation result of node at

<sup>2</sup> Variance of , ,

Fisher Information Matrix (FIM) of ,

Fisher Information Matrix (FIM) of ,

<sup>2</sup> Variance of ,

, Covariance of belief (,) ,

Set of all communicable node pairs (,) at time

σ

̃,

,

,

Position vector of all nodes at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Average velocity of node from time − 1 to time

,

̃,

,

Function of the First Type

<sup>2</sup> Variance of ,

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

time

,

,

,

̃,

σ

,

time

Function of the First Type

<sup>2</sup> Variance of ,

Fisher Information Matrix (FIM) of ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

, and the measured value of local clock ̃

,

σ,

̂,

time

, and the measured value of local clock ̃

time

,

σ,

, Covariance of belief (,) ,

Position vector of node at time

Real time value at time ,

Range measurement between node , at time

Real time value at time ,

Set of all communicable node pairs (,) at time

Range measurement between node , at time

Real time value at time ,

̃,

σ

Relative slope of local clock offset between nodes ,

,

Position vector of all nodes at time

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Average velocity of node from time − 1 to time

Set of all communicable node pairs (,) at time

time

time

Average velocity of node

Message send from , to ,

time

̂,

Confluent Hypergeometric

Estimation result of node at

NLOS error in all node pairs (,) ∈ at time

Fisher Information Matrix (FIM) of ,

Message send from , to ,

Confluent Hypergeometric Function of the First Type

Fisher Information Matrix (FIM) of ,

(,)

Confluent Hypergeometric Function of the First Type

(∙)

,

(,)

,→,

NLOS error in all node pairs (,) ∈ at time

, and the measured value of local clock ̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

between node and the external standard clock is , = (̃

[,, ,]

[,, ,]

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

<sup>2</sup> Variance of , ,

Set of neighbor anchor nodes

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

,

,

,

̃,

σ

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Position vector of node at time

,

Real time value at time ,

σ,

Real time value at time ,

Relative slope of local clock offset between nodes ,

,

Range measurement between node , at time

,

̂,

Range measurement between node , at time

Relative slope of local clock offset between nodes ,

Set of all communicable node pairs (,) at time

Set of all communicable node pairs (,) at time

,

(∙)

Average velocity of node from time − 1 to time

Position vector of all nodes at time

(,)

<sup>2</sup> Variance of , ,

<sup>2</sup> Variance of , ,

**2. System Model** 

Estimation result of node at time

,

,

,

,

̃,

Set of neighbor anchor nodes of node at time

σ

Position vector of node at time

<sup>2</sup> Variance of ,

Real time value at time ,

,

<sup>2</sup> Variance of ,

Relative slope of local clock offset between nodes ,

σ,

Range measurement between node , at time

̂,

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

NLOS error in all node pairs (,) ∈ at time

,

Estimation result of node at time

NLOS error in all node pairs (,) ∈ at time

̂,

Message send from , to ,

(,)

,

Confluent Hypergeometric Function of the First Type

(∙)

Confluent Hypergeometric Function of the First Type

,

Message send from , to ,

, Covariance of belief (,) ,

Fisher Information Matrix (FIM) of ,

, Covariance of belief (,) ,

,

σ

<sup>2</sup> Variance of ,

,

,

,

̃,

σ

σ,

̂,

̃,

<sup>2</sup> Variance of ,

Fisher Information Matrix (FIM) of ,

(,)

**2. System Model** 

(,)

<sup>2</sup> Variance of ,

,

,

̃,

σ

,

,

,

,

,

,

,

̃,

σ

,

σ,

̂,

̃,

σ

,

σ,

̂,

(∙)

(∙)

,→,

(,)

(,)

**2. System Model** 

**2. System Model** 

[,, ,]

[,, ,]

(,)

(,)

,

,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

,→,

,→,

,

,

σ,

,

,

̂,

̃,

,

σ

<sup>2</sup> Variance of ,

,

̃,

σ

(∙)

,

σ,

,

σ,

,

̂,

(,)

**2. System Model** 

̂,

(∙)

(∙)

(,)

(,)

**2. System Model** 

,

**2. System Model** 

[,, ,]

,→,

̃,

σ

,

σ,

, and the measured value of local clock ̃

(∙)

,→,

(,)

(,)

,

σ,

̂,

(∙)

(,)

**2. System Model** 

(,)

**2. System Model** 

[,, ,]

(,)

[,, ,]

,→,

[,, ,]

**2. System Model** 

(,)

**2. System Model** 

[,, ,]

[,, ,]

[,, ,]

[,, ,]

**2. System Model** 

[,, ,]

̂,

, and the measured value of local clock ̃

<sup>2</sup> Variance of ,

(,)

(∙)

,→,

,

σ,

̂,

(,)

**2. System Model** 

,→,

(,)

**2. System Model** 

(∙)

Fisher Information Matrix (FIM) of ,

[,, ,]

(,)

Average velocity of node from time − 1 to time

(∙)

,→,

Estimation result of node at time

(,)

**2. System Model** 

NLOS error in all node pairs (,) ∈ at time

Message send from , to ,

(,)

Confluent Hypergeometric

(,)

time

NLOS error in all node pairs (,) ∈ at time

,

Confluent Hypergeometric Function of the First Type

(∙)

Fisher Information Matrix (FIM) of ,

(,)

**2. System Model** 

,→,

σ

(,)

[,, ,] 

,

̃,

,

[,, ,]

[,, ,] 

σ

[,, ,]

,

,→,

,

,

,

,

,

̃,

σ

,

σ,

̂,

,→,

,

,

̃,

,

σ

,

̃,

σ

,

̃,

,

σ,

̂,

,

σ,

σ,

,

̂,

(∙)

̂,

(,)

(,)

(∙)

**2. System Model** 

(∙)

(,)

,→,

(∙)

,→,

,→,

(,)

(,)

**2. System Model** 

,→,

(,)

σ

̃,

Position vector of all nodes at time

Average velocity of node from time − 1 to time

,

,

σ,

,→,

,→,

<sup>2</sup> Variance of ,

$$\begin{split} \text{KLD}(\boldsymbol{\phi}(\mathbf{z}\_{(x)})|\boldsymbol{h}(\mathbf{z}\_{(x)}) \\ = \frac{\|\mathbf{x}\_{(x-\mathbf{x}\_{0})}\|^{2} + 2\alpha\_{x}^{2}}{2\alpha\_{0}^{2}} \cdot \ln \alpha\_{x,u}^{2} \\ + \sum\_{j\in\mathcal{G}\_{0}} \ln \frac{-2(\boldsymbol{\phi}\_{(j,u-\square\varpi\_{0})})\sqrt{2\frac{\omega\_{j,u}^{2}}{\omega\_{j}^{2}}} (-\frac{1}{2}\boldsymbol{\lambda}\_{-}\frac{(\mathbf{x}\_{j}-\mathbf{x}\_{0})^{2}}{\omega\_{j}^{4}}) \cdot |\mathbf{x}\_{(x)} - \boldsymbol{x}\_{j,0}| + 2\alpha\_{x}^{2}}{2\alpha\_{x}^{2}} \\ + \sum\_{j\in\mathcal{G}\_{0}} \ln \frac{-2(\boldsymbol{\phi}\_{(j,u-\square\varpi\_{0})})\sqrt{\frac{\omega\_{j,u}^{2}}{\omega\_{j}^{4}}} (-\frac{1}{2}\boldsymbol{\lambda}\_{-}\frac{(\mathbf{x}\_{j,u}-\mathbf{x}\_{0})^{2}}{\omega\_{j}^{2}}) + |\mathbf{x}\_{(x)} - \boldsymbol{x}\_{j,0}| + 2\alpha\_{j}^{2}}{2\alpha\_{j}^{2}} \\ \end{split} \tag{25}$$

local clock of node at time

**Table 1.** List of symbols.

,

#### *3.4. M-VMP-Based Joint Estimation Algorithm* (,) ∈ at time time (,) ∈ at time ̃ in all node pairs of node at time Range measurement of node at time Range measurement time ̃ , of node at time of node at time of node at time of node at time of node at time **Table 1.** List of symbols. **Table 1.** List of symbols. **Table 1.** List of symbols. **Table 1.** List of symbols.

,

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time In Section 3.3, the clock drift of node or model loss with single Gaussian distribution are not considered, which affects localization accuracy [19]. In this paper, a joint estimation algorithm of time synchronization and localization based on multi-Gaussian distribution VMP is proposed, that the belief *b*( 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Belief of variable , , Mean of belief (,) *<sup>i</sup>*,*n*) of the variable NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*<sup>n</sup>* to be estimated is approximately a multi-Gaussian function ˆ*b*( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*n*) <sup>∝</sup> <sup>P</sup> η*M*,*ij*N( (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Belief of variable , *<sup>k</sup>*.*i*,*n*|*E* Estimation result of node at ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *i*,*n* , *V* Estimation result of node at time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *i*,*n* ), and the belief of <sup>ζ</sup>*ij*,*<sup>n</sup>* is <sup>ˆ</sup>*b*(ζ*ij*,*n*) ∝ N(ζ*ij*,*n*|ρ˜*ij*,*<sup>n</sup>* <sup>−</sup> *cTaij*,*<sup>n</sup>* − k *<sup>x</sup>i*,*<sup>n</sup>* <sup>−</sup> *<sup>x</sup>j*,*<sup>n</sup>* <sup>k</sup>, <sup>σ</sup> 2 *d* ). The factor graph between nodes *i* and *j* at the time *n* is shown in Figure 2. Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between node , at time , Measurement noise of ̃, Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes , Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes , Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at ̃ , Measurement value of local clock of node at time **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Position vector of node at time ̃ , Measurement value of local clock of node **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Position vector of node at time ̃ , Measurement value of local clock of node at time **Symbol Meaning Symbol Meaning** , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Position vector of node at time ̃ , Measurement value of local clock of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning**Set of neighbor anchor nodes Set of neighbor nodes *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

of node at time

at time

to be located of node at time

location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located the anchor node with known location synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . **Figure 2.** FG of a node pair (*i*, *j*) at time n. *xi*,*n*, *yi*,*n*, *ai*,*<sup>n</sup>* represent the localization and time synchronization parameter of the node to be estimated, *fi*,*<sup>n</sup>* represents the message that transmits between different times, *fij*,*<sup>n</sup>* represents the distance information that the nodes transmit, *bij*,*<sup>n</sup>* represents the non-sight distance parameter that affects the information between nodes and ζ*ij*,*<sup>n</sup>* represents the non-sight distance error probability function that affects the non-sight distance parameter. Estimation result of node at time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of , Measurement noise of ̃, , NLOS error of ̃, Set of all node pairs (,) with NLOS error Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs (,) with NLOS error , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs ̃, Range measurement between node , at time , σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, , Relative slope of local clock offset between nodes , , ̃, Range measurement between node , at time , Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between , Measurement noise of

Range measurement

offset between nodes ,

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . In order to approximate *b*( Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*) to a Gaussian function, Formula (23) is rewritten as follows: NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:,1: , , ̃, from NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from Estimation result of node at Range measurement Estimation result of node at Range measurement in all node pairs Estimation result of node at Range measurement of node at time Range measurement Estimation result of node at Range measurement of node at time Estimation result of node at Range measurement from time − 1 to time , Measurement noise of , time all nodes at time Average velocity of node Measurement noise of at time pairs (,) at time at time Set of all communicable node pairs (,) at time (,) with NLOS error pairs (,) at time σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs σ <sup>2</sup> Variance of , , NLOS error of ̃, <sup>2</sup> Variance of , , NLOS error of ̃,

time 1 to time

in all node pairs

time 1 to time

, Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *b*( time 1 to time (,) Belief of variable , , Mean of belief (,) *<sup>i</sup>*,*n*) ∝ *fi*,*n*( time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *i*,*n*| time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*−1) + X *j*∈*Si*,*<sup>n</sup> fj*( time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*) + X*k*∈*Ci*,*<sup>n</sup> fk* ( time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*) , (26) ̃ (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̃ (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̃ (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time ̃ (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̂, time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at Range measurement all nodes at time Measurement noise of , Vector to be estimated all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Vector to be estimated Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of Vector to be estimated pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time pairs (,) at time Position vector of all nodes at time Set of all node pairs (,) with NLOS error at time Clock offset slope of

,

Position vector of all nodes at

̃,

Vector sets of

Clock offset slope of

of node at time

Vector sets of

in all node pairs

Set of all communicable node

Real time value at time ,

time 1 to time

Fisher Information Matrix (FIM) of , Function of the First Type Function of the First Type Function of the First Type (∙) where: Message send from , to Message send from , to (,) time ̂, <sup>2</sup> Variance of , , <sup>2</sup> Variance of , , Position vector of all nodes at time Average velocity of node

from time − 1 to time

(,)

(∙)

,→,

**2. System Model** 

,→,

Estimation result of node at time

(,) ∈ at time

Confluent Hypergeometric Function of the First Type

, Covariance of belief (,) ,

,→,

Set of all communicable node pairs (,) at time

node , at time

Confluent Hypergeometric Function of the First Type

,

σ,

,

,

σ,

Position vector of all nodes at time

(,)

,

,

σ,

̂,

,→,

,

̂,

time

(∙)

(FIM) of ,

(FIM) of ,

(,)

(,) ∈ at time

time

where: 
$$f\_{\left(\mu\right)}\left(\mathfrak{Z}\_{i\mu}\mid\mathfrak{Z}\_{i,\mu-1}\right)\triangleq\sum\_{M}\frac{-\eta\_{Mj}}{2}\left(\mathfrak{Z}\_{i\mu}-\mathbb{E}\_{\mathfrak{Z}\_{i,\mu-1}M}\right)^{T}V\_{i,\mu-1,M}^{-1}\left(\mathfrak{Z}\_{i,\mu}-\mathbb{E}\_{\mathfrak{Z}\_{i,\mu-1}M}\right)\tag{27}$$
 
$$(\ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \downarrow^{2}$$

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained CRLB(,) Bound of , CRLB(,) Bound of , CRLB(,) Bound of , (FIM) of , CRLB(,) Bound of , (,) (FIM) of , CRLB(,) Bound of , **2. System Model**  , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *fj*( time 1 to time (,) Belief of variable , , Mean of belief (,) *<sup>i</sup>*,*n*) , X *M* −η*M*,*ij* ρ˜*ij*,*<sup>n</sup>* − k *xi*,*<sup>n</sup>* − *x<sup>j</sup>* k − *cTEaij*,*n*−1,*<sup>M</sup>* − *bij*,*<sup>n</sup>* 2 2σ 2 (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , *ij*,*n* , (28) (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from ̂,Estimation result of node at time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of time (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1:Vector sets of , , ̃, from of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs

, = (). The slope of local clock at time

Function of the First Type

Fisher Information Matrix (FIM) of ,

, − ̃

,−1)/( − −1). The local clock

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, = (). The slope of local clock at time

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, = (). The slope of local clock at time

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

,−1)/( − −1). The local clock

, − ̃

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, − ̃

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

CRLB(,)

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

, = (). The slope of local clock at time

, Mean belief (,)

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

CRLB(,)

, = (). The slope of local clock at time

, − ̃

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

Cramer-Rao Lower Bound of ,

Weight of the -th Gaussian distribution

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes CRLB(,) Cramer-Rao Lower Bound of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *fk* ( time 1 to time (,) Belief of variable , *<sup>i</sup>*,*n*) , X *M* <sup>−</sup>η*M*,*ik* <sup>Z</sup> *b* time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric *k*,*n* ρ˜*ij*,*<sup>n</sup>* − k *xi*,*<sup>n</sup>* − *xk*,*<sup>n</sup>* k − *cTEaik*,*n*−1,*<sup>M</sup>* − *bik*,*<sup>n</sup>* 2 2σ 2 (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , *ik*,*n d* ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>k</sup>*,*n*, (29) NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from

, = (). The slope of local clock at time

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

Fisher Information Matrix

location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, − ̃

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

CRLB(,)

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

Gaussian distribution

Cramer-Rao Lower Bound of ,

location information and local time out of sync. The position vector of node at time is , =

CRLB(,)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, − ̃

, and the measured value of local clock ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

between node and the external standard clock is , = (̃

, − ̃

, − ̃

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Because the existence of two non-linear terms, k *xi*,*<sup>n</sup>* − *x<sup>j</sup>* k in *fj*( ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*) and k *xi*,*<sup>n</sup>* − *Exk*,*n*−1,*<sup>M</sup>* k in *f<sup>k</sup>* ( ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*n*), results in non-Gaussian of *b*( time 1 to time Message send from , to , (,) Belief of variable , is ,, and all communicable node pairs (, ) constitute communicable node set . *i*,*n*), the two non-linear terms are expanded by a second-order Taylor

, = (). The slope of local clock at time

is ,, and all communicable node pairs (, ) constitute communicable node set .

The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, = (). The slope of local clock at time

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, − ̃

Gaussian distribution

Cramer-Rao Lower Bound of ,

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

Gaussian distribution

Cramer-Rao Lower Bound of ,

, − ̃

Function of the First Type

CRLB(,)

is ,, and all communicable node pairs (, ) constitute communicable node set .

Fisher Information Matrix (FIM) of ,

, = (). The slope of local clock at time

between node and the external standard clock is , = (̃

, and the measured value of local clock ̃

, = (). The slope of local clock at time

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

, = (). The slope of local clock at time

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

,−1)/( − −1). The local clock

[,, ,]

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

location information and local time out of sync. The position vector of node at time is , =

, = (). The slope of local clock at time

Fisher Information Matrix (FIM) of ,

, Covariance of belief (,) ,

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

location information and local time out of sync. The position vector of node at time is , =

(,)

**2. System Model** 

CRLB(,)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

, − ̃

(∙)

,−1)/( − −1). The local clock

**2. System Model** 

(,)

, and the measured value of local clock ̃

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

,−1)/( − −1). The local clock

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

,−1)/( − −1). The local clock

, − ̃

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

, Mean of belief (,)

**Table 1.** List of symbols.

Relative slope of local clock

̃

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Estimation result of node at

,

,

Range measurement between

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

Position vector of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

,

Real time value at time ,

,

,

,

**Symbol Meaning Symbol Meaning**

̃,

Real time value at time ,

Relative slope of local clock offset between nodes ,

,

Range measurement between node , at time

σ

,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

,

,

,

Set of neighbor anchor nodes of node at time

**Symbol Meaning Symbol Meaning**

Position vector of node at time

,

Set of all communicable node pairs (,) at time

,

**Table 1.** List of symbols.

Position vector of node at time

Position vector of all nodes at time

̃ ,

Average velocity of node from time − 1 to time

Relative slope of local clock offset between nodes ,

Set of neighbor anchor nodes of node at time

Real time value at time ,

̃,

σ

,

Range measurement between node , at time

,

Position vector of node at time

<sup>2</sup> Variance of , ,

<sup>2</sup> Variance of ,

,

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

̃ ,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Relative slope of local clock offset between nodes ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Range measurement between node , at time

**Symbol Meaning Symbol Meaning**

,

**Table 1.** List of symbols.

Real time value at time ,

**Symbol Meaning Symbol Meaning**

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

̃,

σ

,

,

<sup>2</sup> Variance of ,

,

Average velocity of node from time − 1 to time

<sup>2</sup> Variance of , ,

<sup>2</sup> Variance of ,

,

Relative slope of local clock offset between nodes ,

Estimation result of node at time

,

,

,

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

,

,

Set of neighbor anchor nodes of node at time

Position vector of node at time

,

Set of neighbor anchor nodes of node at time

̃,

Position vector of node at time

σ

Relative slope of local clock offset between nodes ,

Real time value at time ,

,

Range measurement between node , at time

,

Relative slope of local clock offset between nodes ,

,

Relative slope of local clock offset between nodes ,

,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

,

Position vector of all nodes at time

σ

̃,

Average velocity of node from time − 1 to time

,

<sup>2</sup> Variance of , ,

,

Range measurement between node , at time

,

,

,

σ,

pairs (,) at time

time

̃,

,

σ

,

,

,

̂,

time

,

̃,

,

σ

,

̃,

σ

,

Set of neighbor anchor nodes of node at time

,

,

Position vector of node at time

,

,

,

,

Real time value at time ,

σ,

̃,

Relative slope of local clock offset between nodes ,

,

̂,

σ

,

<sup>2</sup> Variance of ,

,

Set of neighbor anchor nodes of node at time

,

,

,

,

,

,

,

,

,

̃,

,

̃,

σ

,

σ,

̂,

σ

,

,

̃,

σ

,

σ,

̂,

(∙)

(,)

,→,

(∙)

**2. System Model** 

(,)

,→,

[,, ,]

(,)

,

̂,

,

,

,

̃,

σ

̃,

(∙)

̃,

(,)

σ

(∙)

,

(,)

σ,

**2. System Model** 

̂,

(,)

,→,

σ

,

σ,

̂,

σ,

**2. System Model** 

,

(,)

**2. System Model** 

̂,

[,, ,]

(,)

(∙)

(,)

[,, ,]

(∙)

,→,

(∙)

(,)

(∙)

(,)

**2. System Model** 

(,)

**2. System Model** 

**2. System Model** 

[,, ,]

[,, ,]

[,, ,]

[,, ,]

,→,

,→,

[,, ,]

,→,

(,)

,

,→,

̃,

,→,

,

σ,

̂,

σ

σ,

,

,

,

̃,

,

,

σ

,

,

̃,

σ

,

σ,

̂,

(∙)

σ,

̂,

(∙)

(,)

**2. System Model** 

,→,

[,, ,]

(,)

**2. System Model** 

[,, ,]

[,, ,]

[,, ,]

(,)

,→,

,→,

,

Real time value at time ,

Position vector of node at time

,

,

̃,

Relative slope of local clock offset between nodes ,

σ

Range measurement between node , at time

,

,

,

,

,

̃,

σ

,

σ,

<sup>2</sup> Variance of ,

̃,

σ

σ

,

,

̃,

,

<sup>2</sup> Variance of ,

<sup>2</sup> Variance of ,

,

,

,

̃,

σ

,

,

,

,

,

σ,

̂,

,

Set of all communicable node pairs (,) at time

,

Position vector of all nodes at time

σ,

Average velocity of node from time − 1 to time

̂,

<sup>2</sup> Variance of , ,

<sup>2</sup> Variance of ,

Range measurement between node , at time

̃,

Estimation result of node at time

(∙)

Message send from , to ,

Confluent Hypergeometric Function of the First Type

,→,

Position vector of all nodes at time

Average velocity of node from time − 1 to time

,

NLOS error in all node pairs (,) ∈ at time

(,)

,

, Covariance of belief (,) ,

Estimation result of node at time

,

̂,

,→,

NLOS error in all node pairs (,) ∈ at time

̃,

,

Message send from , to ,

̃,

σ

Confluent Hypergeometric Function of the First Type

(∙)

(,)

,

σ,

**2. System Model** 

Fisher Information Matrix (FIM) of ,

,→,

,

<sup>2</sup> Variance of ,

σ,

(,)

σ,

̂,

, Covariance of belief (,) ,

(∙)

σ

,

̂,

, and the measured value of local clock ̃

**2. System Model** 

,

̂,

Fisher Information Matrix (FIM) of ,

,→,

,

(,)

**2. System Model** 

̂,

σ,

,→,

,→,

[,, ,]

,→,

σ,

̂,

,

(∙)

,→,

σ,

[,, ,]

̂,

(∙)

(∙)

**2. System Model** 

(,)

**2. System Model** 

(∙)

(,)

(,)

**2. System Model** 

(,)

,→,

[,, ,]

(∙)

(,)

[,, ,]

**2. System Model** 

**2. System Model** 

, and the measured value of local clock ̃

, and the measured value of local clock ̃

[,, ,]

, and the measured value of local clock ̃

[,, ,]

, and the measured value of local clock ̃

(,)

**2. System Model** 

[,, ,] 

**2. System Model** 

(,)

(∙)

(,)

,→,

[,, ,]

(,)

,

σ,

(,)

σ

̃,

,

[,, ,]

(∙)

(,)

(∙)

, and the measured value of local clock ̃

,

[,, ,]

̂,

, and the measured value of local clock ̃

,→,

[,, ,]

(∙)

**2. System Model** 

[,, ,]

(,)

[,, ,]

[,, ,] 

**2. System Model** 

[,, ,]

[,, ,]

**2. System Model** 

(,)

[,, ,]

[,, ,]

[,, ,]

[,, ,]

**2. System Model** 

(,)

[,, ,]

[,, ,]

[,, ,]

(∙)

(,)

(,)

,→,

(∙)

**2. System Model** 

(,)

**2. System Model** 

[,, ,] 

**2. System Model** 

[,, ,]

[,, ,]

between node and the external standard clock is , = (̃

,

σ,

σ,

̂,

̂,

(,)

[,, ,]

**2. System Model** 

**2. System Model** 

(,)

,→,

(,)

,→,

(,)

(,)

,

,

,

̃,

(∙)

σ

σ,

,

(,)

̃,

σ

̂,

<sup>2</sup> Variance of , ,

,

,

̃,

**2. System Model** 

,

σ

,→,

,

σ,

<sup>2</sup> Variance of ,

(,)

,

σ,

(,) ∈ at time

,

̂,

̃,

,

,

σ

Confluent Hypergeometric Function of the First Type

, Covariance of belief (,) ,

,

(∙)

NLOS error in all node pairs (,) ∈ at time

Estimation result of node at time

(,)

NLOS error in all node pairs (,) ∈ at time

Estimation result of node at time

Position vector of node at time

<sup>2</sup> Variance of , ,

̃,

,

,

σ

̃,

,

,→,

,

,

σ

,

̃,

̂,

node , at time

,

σ

σ,

offset between nodes ,

(,)

**2. System Model** 

Confluent Hypergeometric Function of the First Type

,

σ,

Fisher Information Matrix (FIM) of ,

Set of all communicable node pairs (,) at time

σ,

σ,

,

̂,

̂,

,

(∙)

Average velocity of node from time − 1 to time

(,)

,→,

σ,

Fisher Information Matrix (FIM) of ,

(,)

,→,

,

σ,

[,, ,]

̂,

(,)

Estimation result of node at time

NLOS error in all node pairs (,) ∈ at time

σ,

(∙)

**2. System Model** 

(∙)

(,)

(,)

(,)

,

(∙)

Function of the First Type

(FIM) of ,

**2. System Model** 

**2. System Model** 

,→,

(,)

Fisher Information Matrix (FIM) of ,

Fisher Information Matrix (FIM) of ,

(∙)

[,, ,]

(,)

Fisher Information Matrix (FIM) of ,

[,, ,]

(FIM) of ,

(∙)

,

**2. System Model** 

(,)

(,)

(,)

̂,

,→,

(,)

,→,

(∙)

(∙)

(,)

(,)

**2. System Model** 

(∙)

,→,

**2. System Model** 

(,)

,→,

Fisher Information Matrix (FIM) of ,

(∙)

̃,

,

σ

(,)

,

,

(,)

(,)

σ

̃,

σ,

**2. System Model** 

̂,

̂,

,→,

,

Estimation result of node at time

,

,

,

Set of all communicable node pairs (,) at time

σ

,

,

,

,

,→,

,

̂,

,

(∙)

,

(,)

̃,

,

̃,

σ

**2. System Model** 

σ

,→,

,

,

,

σ,

,

̂,

σ,

̂,

,

**Symbol Meaning Symbol Meaning**

,

̃,

σ

Set of neighbor anchor nodes of node at time

<sup>2</sup> Variance of ,

<sup>2</sup> Variance of ,

**Symbol Meaning Symbol Meaning**

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

<sup>2</sup> Variance of ,

Real time value at time ,

,

Set of neighbor anchor nodes of node at time

̃,

Position vector of node at time

σ

,

Relative slope of local clock offset between nodes ,

Real time value at time ,

,

,

node , at time

σ,

,

̂,

,

̃,

time

Average velocity of node from time − 1 to time

Set of neighbor anchor nodes of node at time

,

Position vector of node at time

σ

time

,

time

Position vector of node at time

Set of neighbor anchor nodes of node at time

,

σ,

Position vector of node at time

Real time value at time ,

̂,

Real time value at time ,

,

Relative slope of local clock offset between nodes ,

<sup>2</sup> Variance of ,

Range measurement between node , at time

Relative slope of local clock offset between nodes ,

Set of all communicable node

node , at time

(,)

σ

,

<sup>2</sup> Variance of ,

<sup>2</sup> Variance of ,

,

,

̃,

Average velocity of node from time − 1 to time

Fisher Information Matrix (FIM) of ,

Position vector of node at time

,

,

̃,

<sup>2</sup> Variance of ,

Message send from , to ,

,

Set of neighbor anchor nodes of node at time

time

Confluent Hypergeometric Function of the First Type

Real time value at time ,

,

,→,

,

Position vector ofnode at time

Relative slope of local clock

Fisher Information Matrix (FIM) of ,

Range measurement between node , at time

Estimation result of node at time

offset between nodes ,

,

pairs (,) at time

Range measurement between node , at time

σ

time

,→,

NLOS error in all node pairs (,) ∈ at time

(,)

Position vector of all nodes at time

̃

Average velocity of node from time − 1 to time

Set of all communicable node pairs (,) at time

(∙)

,→,

,

,

σ

(,)

(,) ∈ at time

̃,

Position vector of node at time

pairs (,) at time

̂,

pairs (,) at time

,

offset between nodes ,

CRLB(,)

,

<sup>2</sup> Variance of ,

Relative slope of local clock

Estimation result of node at time

<sup>2</sup> Variance of ,

(∙)

,→,

,

σ

, − ̃

time

,

time

,

(,)

,

̂,

(∙)

time

(,)

,

(,)

̂,

time

σ,

,→,

(,)

(,)

(∙)

̂,

,→,

[,, ,]

σ,

,→,

time

̂,

[,, ,]

**2. System Model** 

[,, ,]

(,)

(FIM) of ,

**2. System Model** 

**2. System Model** 

(,)

(FIM) of ,

[,, ,]

is ,, and all communicable node pairs (, ) constitute communicable node set .

[,, ,]

[,, ,]

is ,, and all communicable node pairs (, ) constitute communicable node set .

[,, ,]  (,)

[,, ,]

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

[,, ,]

,→,

,

time

,

σ,

[,, ,]

(∙)

σ,

**2. System Model** 

Estimation result of node at time

(,)

Estimation result of node at time

Real time value at time ,

Real time value at time ,

,

̃,

̃,

σ

,

Function of the First Type

̂,

pairs (,) at time

pairs (,) at time

time

time

Average velocity of node from time − 1 to time

Confluent Hypergeometric Function of the First Type

Message send from , to ,

Fisher Information Matrix (FIM) of ,

,

Fisher Information Matrix (FIM) of ,

(,)

,→,

[,, ,]

σ,

(∙)

̂,

time

time

̂,

(,) ∈ at time

(,)

,→,

,

,

(∙)

,→,

(,)

(∙)

(,)

,

[,, ,]

σ

Set of all communicable node pairs (,) at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Position vector of all nodes at time

(,)

Message send from , to ,

̂,

pairs (,) at time

Position vector of all nodes at time

σ

̃,

Average velocity of node from time − 1 to time

Position vector of all nodes at time

**2. System Model** 

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Confluent Hypergeometric Function of the First Type

(,)

,

σ,

Average velocity of node from time − 1 to time

Average velocity of node from time − 1 to time

(,)

Estimation result of node at time

σ,

̂,

(,)

̂,

NLOS error in all node pairs (,) ∈ at time

Estimation result of node at time

,

(∙)

,→,

(∙)

,

Estimation result of node at time

(,)

Estimation result of node at time

(,)

(,) ∈ at time

̂,

,→,

**2. System Model** 

,→,

,

(,)

,

(,) ∈ at time

(,)

Fisher Information Matrix (FIM) of ,

(∙)

,

Estimation result of node at time

Real time value at time ,

,

Relative slope of local clock

Range measurement between node , at time

,

σ,

,

(,)

Relative slope of local clock offset between nodes ,

,

Real time value at time ,

<sup>2</sup> Variance of , ,

,

Position vector of node at time

Real time value at time ,

expansion to obtain the mean and covariance of *b*( ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*). For discussion purposes, define *gj*( ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*) , k *xi*,*<sup>n</sup>* − *x<sup>j</sup>* k − *cTEaij*,*n*−1,*<sup>M</sup>* and *g<sup>k</sup>* (,) ∈ at time time 1 to time Message send from , to , (,) Belief of variable , *i*,*n*, (,) ∈ at time time 1 to time Message send from , to , (,) Belief of variable , *k*,*n* , k *xi*,*<sup>n</sup>* − *xk*,*<sup>n</sup>* k − *cTEaik*,*n*−1,*<sup>M</sup>* . NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at ̃ Range measurement in all node pairs σ, <sup>2</sup> Variance of , , of node at time ̂, Estimation result of node at ̃ Range measurement in all node pairs σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at ̃ Range measurement in all node pairs from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time from time − 1 to time , , <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time , , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , from time − 1 to time , , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node , Measurement noise of Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node , Measurement noise of time all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , Position vector of all nodes at time Clock offset slope of , Average velocity of node , Measurement noise of at time Position vector of all nodes at time Clock offset slope of all nodes at time at time Position vector of all nodes at time Clock offset slope of all nodes at time at time Position vector of all nodes at time Clock offset slope of all nodes at time Position vector of all nodes at time Clock offset slope of all nodes at time (,) with NLOS error at time Clock offset slope of Set of all communicable node (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of Set of all communicable node pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error offset between nodes , , of node at time Range measurement between node , at time , Measurement noise of ̃, offset between nodes , , of node at time Range measurement between node , at time , Measurement noise of ̃, ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between , Measurement noise of , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of Real time value at time , node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Real time value at time , node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time at time Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Real time value at time , Slope of local clock of node at time , Relative slope of local clock , Set of neighbor nodes Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Position vector of node at time ̃ , local clock of node at time Real time value at time , Slope of local clock of Position vector of node at time ̃ , local clock of node at time Slope of local clock of Position vector of node at time ̃ , Measurement value of local clock of node at time at time Position vector of node at time ̃ , Measurement value of local clock of node at time , Position vector of node at time ̃ , Measurement value of local clock of node at time , Position vector of node at time ̃ , Measurement value of local clock of node at time , Position vector of node at time ̃ , local clock of node at time Real time value at time , Slope of local clock of , Position vector of node at time ̃ , local clock of node Slope of local clock of , Position vector of node at time ̃ , Measurement value of local clock of node at time at time , Position vector of node at time ̃ , Measurement value of local clock of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Set of neighbor anchor nodes of node at time , , Set of neighbor anchor nodes of node at time , **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor nodes **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Real time value at time ,

Set of neighbor anchor nodes of node at time

> Set of all node pairs (,) with NLOS error at time

Position vector of node at time

Relative slope of local clock offset between nodes ,

Clock offset slope of all nodes at time

**Symbol Meaning Symbol Meaning**

<sup>2</sup> Variance of ,

Relative slope of local clock offset between nodes ,

Measurement noise of ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

,

Vector to be estimated of node at time

Range measurement in all node pairs

̃ ,

Set of all communicable node pairs (,) at time

Set of neighbor anchor nodes of node at time

Average velocity of node

offset between nodes ,,

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Position vector of node at time

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

,

Real time value at time ,

,

̃,

,

Set of neighbor anchor nodes of node at time

Relative slope of local clock offset between nodes ,

Position vector of node at time

Set of neighbor nodes to be located of node at time

̃ ,

Range measurement between node , at time

Set of neighbor anchor nodes of node at time

σ

̃,

Position vector of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Real time value at time ,

Set of all communicable node pairs (,) at time

Position vector of node at time

,

<sup>2</sup> Variance of ,

Measurement value of local clock of node at time

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

**Symbol Meaning Symbol Meaning**

Slope of local clock of node at time

Real value at time ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of all nodes at time

Set of neighbor nodes of node at time

Set of neighbor anchor nodes of node at time

,

, NLOS error of ̃,

,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Relative slope of local clock offset between nodes ,

Set of neighbor nodes to be located of node at time

**Table 1.** List of symbols.

Real value at time ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of node at time

,

**Table 1.** List of symbols.

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

<sup>2</sup> Variance of ,

Position vector of all nodes at

Range measurement between node , at time

Measurement value of local clock of node at time

**Symbol Meaning Symbol Meaning**

Set of all communicable node pairs (,) at time

,

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes of node at time

Set of neighbor anchor nodes of node at time

Slope of local clock of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of all nodes at time

<sup>2</sup> Variance of ,

Average velocity of node

Measurement noise of ̃,

̃ ,

Set of neighbor anchor nodes of node at time

Real time value at time ,

Average velocity of node from time − 1 to time

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

,Relative slope of local clock

,

Measurement noise of ̃,

<sup>2</sup> Variance of , ,

**Symbol Meaning Symbol Meaning**

Position vector of all nodes at time

Position vector of node at time

̃ ,

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

σ,

Slope of local clock of node at time

**Table 1.** List of symbols.

,

Range measurement between node , at time

Real time value at time ,

̂,

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

<sup>2</sup> Variance of ,

̃ ,

Position vector of all nodes at time

,

**Table 1.** List of symbols.

Set of all node pairs

Estimation result of node at time

<sup>2</sup> Variance of ,

Average velocity of node from time − 1 to time

,

Relative slope of local clock

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Set of all node pairs (,) with NLOS error at time

,

, NLOS error of ̃,

Set of all communicable node pairs (,) at time

1:, 1:, 1:, 1:

Set of neighbor nodes

at time

Position vector of node at

Set of neighbor anchor nodes

,

Set of neighbor nodes to be located of node at time

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Slope of local clock of node at time

Set of neighbor nodes of node at time

**Symbol Meaning Symbol Meaning**

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

,

Set of all node pairs (,) with NLOS error at time

Set of neighbor nodes to be located of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Measurement value of local clock of node at time

̃ ,

Clock offset slope of all nodes at time

,

Slope of local clock of node at time

Real time value at time ,

Measurement value of local clock of node at time

**Table 1.** List of symbols.

Measurement noise of ,

Set of neighbor nodes of node at time

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor nodes to be located of node at time

, NLOS error of ̃,

,

Slope of local clock of node at time

Relative slope of local clock offset between nodes ,

Real time value at time ,

Vector to be estimated of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Measurement noise of ̃,

**Table 1.** List of symbols.

Range measurement in all node pairs (,) ∈ at time

̃ ,

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Vector sets of , , ̃, from

̃ ,

Set of all node pairs (,) with NLOS error at time

Set of neighbor nodes to be located of node at time

Measurement value of

Slope of local clock of node at time

Set of neighbor nodes

Measurement value of local clock of node at time

Slope of local clock of node at time

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes of node at time

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Measurement noise of ̃,

**Symbol Meaning Symbol Meaning**

Set of all node pairs (,) with NLOS error at time

Set of neighbor anchor nodes of node at time

> Clock offset slope of all nodes at time

Position vector of node at time

,

Position vector of node at time

,

**Symbol Meaning Symbol Meaning**

at time Real time value at time ,

Measurement noise of ,

Vector to be estimated of node at time

, NLOS error of ̃,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

̃ ,

Set of neighbor nodes of node at time

Measurement noise of ̃,

**Symbol Meaning Symbol Meaning**

Set of all node pairs

,

,

Set of all communicable node pairs (,) at time

, 1:, 1:, 1:

Set of neighbor nodes

,

, NLOS error of ̃,

Relative slope of local clock

Position vector of node at

Range measurement in all node pairs (,) ∈ at time

Range measurement between node , at time

> Vector sets of , , ̃, from

Set of all communicable node pairs (,) at time

̃ ,

**Table 1.** List of symbols.

Position vector of node at time

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of neighbor nodes to be located of node at time

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Relative slope of local clock offset between nodes ,

**Symbol Meaning Symbol Meaning**

Real time value at time ,

**Table 1.** List of symbols.

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

<sup>2</sup> Variance of ,

**Table 1.** List of symbols.

Set of neighbor anchor nodes of node at time

̃ ,

, NLOS error of ̃,

Range measurement between node , at time

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor nodes to be located of node at time

, NLOS error of ̃,

Set of neighbor anchor nodes of node at time

̃ ,

Position vector of node at time

,

Set of neighbor anchor nodes of node at time

**Symbol Meaning Symbol Meaning**

Set of all communicable node pairs (,) at time

**Table 1.** List of symbols.

Position vector of node at time

,

Real time value at time ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of all nodes at time

Real time value at time ,

**Symbol Meaning Symbol Meaning**

Slope of local clock of node at time

Set of neighbor anchor nodes of node at time

Measurement value of local clock of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Average velocity of node from time − 1 to time

offset between nodes ,

Set of neighbor nodes of node at time

Slope of local clock of node at time

Measurement noise of ̃,

**Symbol Meaning Symbol Meaning**

Position vector of node at time

Set of neighbor nodes of node at time

Range measurement between node , at time

̃

, NLOS error of ̃,

<sup>2</sup> Variance of , ,

,

**Table 1.** List of symbols.

Measurement noise of ,

Vector to be estimated of node at time

Clock offset slope of all nodes at time

1:, 1:, 1:, 1:

̃ ,

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

Real time value at time ,

Estimation result of node at time

Set of all communicable node pairs (,) at time

Set of all node pairs

Relative slope of local clock

, NLOS error of ̃,

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

̃ ,

to be located of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

Set of all node pairs (,) with NLOS error at time

Measurement value of local clock of node at time

Clock offset slope of all nodes at time

Slope of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement noise of ,

Set of neighbor nodes of node at time

Vector to be estimated of node at time

Measurement noise of ̃,

Measurement value of local clock of node at time

Range measurement in all node pairs (,) ∈ at time

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node

Measurement value of

at time

at time

Set of all node pairs (,) with NLOS error

Slope of local clock of node at time

Set of neighbor nodes

,

(,) with NLOS error

̃,

at time

̃,

Set of neighbor nodes

̃,

time 1 to time

at time

(,) with NLOS error at time

of node at time

Clock offset slope of

Measurement noise of ̃,

,

,

at time

,

of node at time

at time

,

1 to time

,

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

,Set of neighbor nodes

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

,

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

Set of all node pairs (,) with NLOS error at time

Measurement value of local clock of node at time

, NLOS error of ̃,

Clock offset slope of all nodes at time

Measurement noise of ,

Slope of local clock of node at time

Vector to be estimated

at time

Set of neighbor nodes

Set of neighbor nodes

, NLOS error of ̃,

Slope of local clock of node at time

̃ ,

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

,

Set of all node pairs (,) with NLOS error at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

,

Clock offset slope of all nodes at time

Measurement value of local clock of node at time

, NLOS error of ̃,

Slope of local clock of node at time

Measurement noise of

**Table 1.** List of symbols.

,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Measurement value of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Slope of local clock of node at time

Set of all node pairs (,) with NLOS error at time

, NLOS error of ̃,

Measurement noise of ̃,

Set of neighbor nodes of node at time

Clock offset slope of all nodes at time

Set of all node pairs (,) with NLOS error

Set of neighbor nodes to be located of node at time

Measurement noise of

Slope of local clock of node at time

,

Set of neighbor nodes of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

, NLOS error of ̃,

Clock offset slope of

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

, NLOS error of ̃,

Measurement noise of

Measurement value of

Measurement noise of ̃,

Set of neighbor nodes to be located of node at time

Set of all node pairs (,) with NLOS error at time

Measurement value of local clock of node

, NLOS error of ̃,

Set of neighbor nodes of node at time

Measurement noise of ̃,

Slope of local clock of node at time

,

Clock offset slope of all nodes at time

Slope of local clock of node at time

Measurement noise of ,

Set of neighbor nodes of node at time

Set of all node pairs (,) with NLOS error at time

Set of neighbor nodes to be located of node at time

̃

, NLOS error of ̃,

Vector to be estimated of node at time

Measurement noise of ̃,

Clock offset slope of all nodes at time

Measurement value of local clock of node at time

Measurement noise of ,

Set of neighbor nodes

Slope of local clock of node at time

, , ̃, from

Range measurement in all node pairs (,) ∈ at time

, NLOS error of ̃,

Set of all node pairs (,) with NLOS error

1:, 1:, 1:, 1:

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Gaussian distribution

Cramer-Rao Lower Bound of ,

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

, NLOS error of ̃,

Set of neighbor nodes to be located of node at time

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

all nodes at time

,

in all node pairs

Vector sets of , , ̃, from time 1 to time

, NLOS error of ̃,

Vector sets of

time 1 to time

Bound of ,

Bound of ,

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node at time

,

̃ ,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

,

Set of neighbor nodes of node at time

Measurement noise of ̃,

̃ ,

Set of neighbor nodes to be located of node at time

Set of all node pairs (,) with NLOS error at time

, NLOS error of ̃,

Measurement value of local clock of node at time

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Clock offset slope of all nodes at time

Slope of local clock of node at time

Measurement noise of ,

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes of node at time

Set of neighbor nodes to be located of node at time

, NLOS error of ̃,

,

,

, , ̃, from

at time

,−1)/( − −1). The local clock

Gaussian distribution

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

means the system cannot provide reliable vertical positioning results [20], which is usually obtained

, = (). The slope of local clock at time

CRLB(,)

CRLB(,)

Weight of the -th

Weight of the -th

,

,

,

̃ ,

, NLOS error of ̃,

,

,

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

̃ ,

̃

**Table 1.** List of symbols.

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

̃ ,

Set of all node pairs (,) with NLOS error at time

,

Clock offset slope of all nodes at time

̃ ,

Measurement noise of ,

Vector to be estimated of node at time

,

, NLOS error of ̃,

,

,

,

,

Range measurement in all node pairs (,) ∈ at time

Set of neighbor nodes to be located of node at time

,

, NLOS error of ̃,

Vector sets of , , ̃, from time 1 to time

Measurement value of local clock of node at time

**Symbol Meaning Symbol Meaning**

, NLOS error of ̃,

**Symbol Meaning Symbol Meaning**

,

̃ ,

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

Measurement value of local clock of node at time

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor nodes to be located of node at time

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

,

̃ ,

,

Slope of local clock of node at time

,

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

,

,

,

̃ ,

Set of neighbor nodes to be located of node at time

Set of all node pairs (,) with NLOS error at time

Measurement value of local clock of node at time

̃

Slope of local clock of

,

Clock offset slope of all nodes at time

Measurement noise of

Set of neighbor anchor nodes of node at time

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of node at time

Real time value at time ,

̃ ,

Relative slope of local clock offset between nodes ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor nodes to be located of node at time

̃ ,

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node at time

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes to be located of node at time

Range measurement between node , at time

Measurement value of local clock of node at time

**Table 1.** List of symbols.

,

Set of all communicable node pairs (,) at time

,

Slope of local clock of node at time

̃ ,

Position vector of all nodes at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

Slope of local clock of node at time

,

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes of node at time

̃ ,

Measurement noise of ̃,

,

Set of neighbor nodes to be located of node at time

̃ ,

, NLOS error of ̃,

Set of all node pairs (,) with NLOS error at time

Measurement value of local clock of node at time

,

Clock offset slope of all nodes at time

Measurement noise of

,

Slope of local clock of

Set of neighbor nodes of node at time

Average velocity of node from time − 1 to time

**Table 1.** List of symbols.

Measurement noise of ̃,

,

<sup>2</sup> Variance of , ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor nodes to be located of node at time

Set of all node pairs (,) with NLOS error at time

,

Estimation result of node at time

,

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes to be located of node at time

,

Clock offset slope of all nodes at time

Measurement value of local clock of node at time

**Table 1.** List of symbols.

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

NLOS error in all node pairs (,) ∈ at time

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node

,

Measurement noise of ,

Slope of local clock of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor nodes to be located of node

Set of neighbor nodes to be located of node at time

,

Set of neighbor nodes

Measurement value of

, , ̃, from

̃,

at time

Set of neighbor nodes

,

(,) with NLOS error

Set of neighbor nodes

1:, 1:, 1:, 1:

Set of neighbor nodes

Range measurement

**Symbol Meaning Symbol Meaning**

,

̃ ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

Set of neighbor anchor nodes of node at time

,

,

Position vector of node at time

̃ ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

,

Set of neighbor anchor nodes of node at time

,

<sup>2</sup> Variance of ,

Set of all communicable node pairs (,) at time

,

̃

, NLOS error of ̃,

,

Average velocity of node from time − 1 to time

,

,

Position vector of all nodes at time

̃ ,

,

Position vector of node at time

1:, 1:, 1:, 1:

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Relative slope of local clock offset between nodes ,

Position vector of node at

Set of neighbor nodes

Set of neighbor anchor nodes of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

**Symbol Meaning Symbol Meaning**

̃ ,

,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor anchor nodes of node at time

Position vector of node at time

**Symbol Meaning Symbol Meaning**

,

Set of neighbor anchor nodes of node at time

,

,

Set of neighbor anchor nodes of node at time

,

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Position vector of node at time

**Symbol Meaning Symbol Meaning**

,

Position vector of node at time

,

̃ ,

Measurement value of local clock of node at time

,

Set of neighbor anchor nodes of node at time

**Symbol Meaning Symbol Meaning**

**Table 1.** List of symbols.

Set of neighbor nodes to be located of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

Position vector of node at time

Set of neighbor nodes to be located of node at time

Slope of local clock of node at time

Position vector of node at time

̃ ,

,

,

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

Real time value at time ,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

<sup>2</sup> Variance of ,

Real value at time ,

,

Relative slope of local clock offset between nodes ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

̃ ,

**Table 1.** List of symbols.

Measurement value of local clock of node at time

Range measurement between node , at time

Slope of local clock of node at time

Set of neighbor anchor nodes of node at time

Set of all node pairs (,) with NLOS error at time

**Table 1.** List of symbols.

, NLOS error of ̃,

,

Range measurement between node , at time

**Symbol Meaning Symbol Meaning**

Relative slope of local clock offset between nodes ,

,

,

,

**Symbol Meaning Symbol Meaning**

̃,

Clock offset slope of all nodes at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Set of neighbor anchor nodes of node at time

<sup>2</sup> Variance of ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Set of all communicable node pairs (,) at time

,

**Symbol Meaning Symbol Meaning**

,

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

<sup>2</sup> Variance of ,

σ

from time − 1 to time

**Table 1.** List of symbols.

,

,

Position vector of all nodes at time

̃,

Measurement noise of ,

**Table 1.** List of symbols.

Position vector of node at time

̃

Real time value at time ,

σ

Set of all communicable node pairs (,) at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Measurement noise of ̃,

Set of neighbor nodes of node at time

Position vector of all nodes at time

Position vector of node at time

<sup>2</sup> Variance of ,

,

Vector to be estimated of node at time

**Table 1.** List of symbols.

Range measurement in all node pairs

Relative slope of local clock

of node at time

1:, 1:, 1:, 1:

Average velocity of node

Set of all node pairs

Real time value at time ,

,

**Symbol Meaning Symbol Meaning**

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Relative slope of local clock offset between nodes ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

Range measurement between node , at time

,

̃,

,

σ

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

,

̃ ,

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

, NLOS error of ̃,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Real time value at time ,

,

,

,

<sup>2</sup> Variance of ,

̃,

̃ ,

σ

**Symbol Meaning Symbol Meaning**

,

**Symbol Meaning Symbol Meaning**

,

,

,

**Table 1.** List of symbols.

Average velocity of node from time − 1 to time

**Table 1.** List of symbols.

<sup>2</sup> Variance of , ,

,

**Symbol Meaning Symbol Meaning**

,

σ,

,

,

̃,

Set of neighbor anchor nodes of node at time

,

σ

̂,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

**Table 1.** List of symbols.

̃ ,

,

**Symbol Meaning Symbol Meaning**

<sup>2</sup> Variance of ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Estimation result of node at time

<sup>2</sup> Variance of ,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Symbol Meaning Symbol Meaning**

̃,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

σ

**Table 1.** List of symbols.

<sup>2</sup> Variance of ,

,

,

**Table 1.** List of symbols.

Position vector of node at time

,

Real time value at time ,

Relative slope of local clock offset between nodes ,

̃,

,

Range measurement between node , at time

σ

Set of all communicable node pairs (,) at time

,

Set of neighbor anchor nodes of node at time

Set of all communicable node pairs (,) at time

̃,

,

Position vector of all nodes at

Position vector of node at

σ

Average velocity of node

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

time

Position vector of all nodes at time

̃

,

(,) ∈ at time

node , at time

,

,

,

(,)

node , at time

Relative slope of local clock

Position vector of node at time

of node at time

,→,

time

time

node , at time

from time − 1 to time

,

,

̃,

,

Real time value at time ,

σ

,

̃ ,

,

,

̃ ,

̃,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Average velocity of node

Position vector of node at

σ

,

,

,

Set of neighbor anchor nodes of node at time

Set of neighbor anchor nodes of node at time

,

,

σ,

σ

,

Set of neighbor anchor nodes

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Relative slope of local clock

**Table 1.** List of symbols.

of node at time

̃,

,

Position vector of node at time

,

(,)

̃

̂,

̃ ,

̃ ,

(∙)

time

(,)

,→,

time

,

,

,

σ

,

,

̃

(,)

̃

time

,

̃ ,

̂,

,

̃,

,

̃ ,

,

σ

time

̃,

,

,

Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution Function of the First Type , Covariance of belief (,) , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th a. Second-order Taylor series expansion for nonlinear term *gj*( ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*) :*gj*( ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *<sup>i</sup>*,*n*) is expanded at time 1 to time (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* = *E* time 1 to time Message send from , to (,) Belief of variable , *i*,*n*−1 by a second-order Taylor to get: NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time time (,) ∈ at time NLOS error in all node pairs Vector sets of time (,) ∈ at time NLOS error in all node pairs Vector sets of time (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from time (,) ∈ at time NLOS error in all node pairs Vector sets of Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated of node at time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Range measurement from time − 1 to time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated , Average velocity of node from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated , Average velocity of node from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated all nodes at time , Measurement noise of , all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , time all nodes at time Average velocity of node from time − 1 to time ,Measurement noise of , at time Position vector of all nodes at Clock offset slope of all nodes at time at time Position vector of all nodes at time Clock offset slope of all nodes at time <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error node , at time ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , offset between nodes , , of node at time Range measurement between , Measurement noise of offset between nodes , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Real time value at time , node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Real time value at time , Slope of local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Real time value at time , node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock Set of neighbor nodes Real time value at time , Slope of local clock of node at time **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Position vector of node at ̃ , Measurement value of local clock of node **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Position vector of node at time ̃ , Measurement value of local clock of node , Position vector of node at time ̃ , , Position vector of node at time ̃ , of node at time , to be located of node at time Position vector of node at Measurement value of of node at time , to be located of node at time Position vector of node at Measurement value of of node at time , to be located of node at time Position vector of node at Measurement value of Set of neighbor anchor nodes of node at time , to be located of node at time Measurement value of , of node at time , to be located of node at time Position vector of node at Measurement value of , Set of neighbor anchor nodes of node at time , to be located of node at time Measurement value of , Set of neighbor anchor nodes of node at time , to be located of node at time Measurement value of , Set of neighbor anchor nodes of node at time , to be located of node at time Measurement value of Set of neighbor anchor nodes of node at time ,Set of neighbor nodes to be located of node at time Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time , Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Table 1.** List of symbols. **Symbol Meaning Symbol Meaning Symbol Meaning Symbol Meaning** Set of neighbor nodes

Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision (,) Fisher Information Matrix (FIM) of , CRLB(,) **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Mean of belief (,) Weight of the -th Gaussian distribution Cramer-Rao Lower Bound of , , Mean of belief (,) Weight of the -th Gaussian distribution Cramer-Rao Lower *gj*( (,) Belief of variable , , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *<sup>i</sup>*,*n*) = k *xi*,*<sup>n</sup>* − *x<sup>j</sup>* k − *cTEaij*,*n*−1,*<sup>M</sup>* ≈ *g<sup>j</sup> E* time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *i*,*n*−1 + ∇ *<sup>T</sup> g<sup>j</sup> E* (,) ∈ at time time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *i*,*n*−1 (,) ∈ at time time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*<sup>n</sup>* − *E* (,) ∈ at time time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *i*,*n*−1 +<sup>1</sup> 2 NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* − *E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , *i*,*n*−1 *T* ∇ 2 *gj E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , *i*,*n*−1 NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* − *E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , *i*,*n*−1 , (30) Range measurement in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̃ Range measurement in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Estimation result of node at time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, Estimation result of node at time ̃ Range measurement (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs 1:, 1:1:, 1: , , ̃, from Vector to be estimated of node at time Range measurement in all node pairs (,) ∈ at time Vector sets of Vector to be estimated of node at time Range measurement in all node pairs (,) ∈ at time <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time Measurement noise of , Vector to be estimated of node at time Range measurement , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Range measurement at time Clock offset slope of all nodes at time , Measurement noise of , pairs (,) at time at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , pairs (,) at time at time Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of Set of all communicable node Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time , NLOS error of ̃, Set of all communicable node Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of , Measurement noise of ̃, , NLOS error of ̃, Set of all node pairs (,) with NLOS error Range measurement between , Measurement noise of ̃, , NLOS error of ̃, Set of all communicable node Set of all node pairs (,) with NLOS error , of node at time , Measurement noise of ̃, , NLOS error of ̃, Set of all node pairs offset between nodes , , of node at time Range measurement between , Measurement noise of ̃, , NLOS error of ̃, Set of all node pairs offset between nodes , of node at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs offset between nodes , of node at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time ̃, Range measurement between node , at time , Measurement noise of σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error offset between nodes , , of node at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs , offset between nodes , , of node at time ̃, Range measurement between node , at time , Measurement noise of σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs to be located of node at time Measurement value of local clock of node at time Slope of local clock of to be located of node at time Measurement value of local clock of node at time Slope of local clock of Slope of local clock of node at time Set of neighbor nodes of node at time Measurement noise of Set of neighbor anchor nodes , to be located of node at time Position vector of node at ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of Set of neighbor anchor nodes of node at time , to be located of node at time Position vector of node at ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of Real time value at time , Slope of local clock ofnode at time Relative slope of local clock , Set of neighbor nodes of node at time Range measurement between Measurement noise of Real time value at time , , Relative slope of local clock offset between nodes , , ̃, Range measurement between , Real time value at time , , Relative slope of local clock offset between nodes , , ̃, Range measurement between , local clock of node at time Slope of local clock of node at time Set of neighbor nodes of node at time ̃ , local clock of node at time Real time value at time , Slope of local clock of node at time , Set of neighbor nodes of node at time ̃ , local clock of node at time Real time value at time , Slope of local clock ofnode at time , Set of neighbor nodes of node at time ̃ , local clock of node at time Real time value at time , Slope of local clockof node at time Relative slope of local clock , Set of neighbor nodes time ̃ , local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time time ̃ , local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock , Set of neighbor nodes time ̃ , local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock , Set of neighbor nodes , time ̃ , local clock of node Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes Measurement value of local clock of node at time Slope of local clock of node at time ,Measurement value of local clock of node at time Slope of local clock of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Position vector of node at ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time Set of neighbor anchor nodes of node at time , to be located of node at time Position vector of node at ̃ , Measurement value of local clock of node at time Set of neighbor anchor nodes of node at time , to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time of node at time , to be located of node at time Position vector of node at time ̃, Measurement value of local clock of node at time

Range measurement

(,) with NLOS error

, NLOS error of ̃,

**Table 1.** List of symbols.

at time

means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model 2. System Model**  CRLB(,) Bound of , Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution , Covariance of belief (,) , Weight of the -th Gaussian distribution , Covariance of belief (,) , Weight of the -th Gaussian distribution , Covariance of belief (,) , Weight of the -th Gaussian distribution Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th Confluent Hypergeometric Function of the First Type , Mean of belief (,) (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) where ∇*g<sup>j</sup> E* Message send from , to (,) Belief of variable , *i*,*n*−1 and ∇ 2 *gj E* Message send from , to , (,) Belief of variable , *i*,*n*−1 are the first and second steps of *gj*( ,→, (,) Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*) at *E* ,→, (,) Message send from , to , (,) Belief of variable , *i*,*n*−1 . (,) ∈ at time time 1 to time Message send from , to (,) ∈ at time time 1 to time Message send from , to (,) ∈ at time Message send from , to (,) ∈ at time Message send from , to 1:, 1:, 1:, 1: , , ̃, from time 1 to time 1:, 1:, 1:,1: Vector sets of , ,̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̃ in all node pairs (,) ∈ at time Vector sets of ̃ in all node pairs (,) ∈ at time Vector sets of <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time σ, of node at time Estimation result of node at Range measurement σ, <sup>2</sup> Variance of , ,Vector to be estimated of node at time Estimation result of node at Range measurement σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time σ, <sup>2</sup> Variance of , , Vector to be estimated Average velocity of node from time − 1 to time , Measurement noise of , Average velocity of node from time − 1 to time , Measurement noise of , time all nodes at time Average velocity of node , Measurement noise of time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , time all nodes at time , Average velocity of node , Measurement noise of at time Position vector of all nodes at time Clock offset slope of all nodes at time pairs (,) at time at time Position vector of all nodes at Clock offset slope of Set of all communicable node pairs (,) at time (,) with NLOS error at time Set of all communicable node pairs (,) at time (,) with NLOS error at time pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of Position vector of all nodes at time Clock offset slope of Position vector of all nodes at pairs (,) at time (,) with NLOS error Set of all communicable node pairs (,) at time node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time , ̃, , NLOS error of ̃, node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, node , at time σ <sup>2</sup> Variance of , , NLOS error of ̃, node , at time σ <sup>2</sup> Variance of , offset between nodes , Range measurement between , Measurement noise of ̃, offset between nodes , Range measurement between node , at time , Measurement noise of ̃, offset between nodes , Range measurement between node , at time , Measurement noise of ̃, offset between nodes , of node at time Range measurement between node , at time , Measurement noise of ̃, Range measurement between node , at time , Measurement noise of ̃, , offset between nodes , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, , offset between nodes , of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, Range measurement between node , at time , Relative slope of local clock , Set of neighbor nodes of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Real time value at time , Slope of local clock of node at time Real time value at time , Slope of local clock of node at time Real time value at time , Slope of local clock of node at time Relative slope of local clock

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Gaussian distribution Fisher Information Matrix CRLB() Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix CRLB(,) Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower Confluent Hypergeometric Function of the First Type , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) b. Second-order Taylor series expansion for nonlinear term *g<sup>k</sup>* ,→, (,) , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *i*,*n*, ,→, (,) , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *k*,*n* : *g<sup>k</sup>* ,→, (,) , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *i*,*n*, ,→, (,) , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *k*,*n* is expanded at (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* = *E* Message send from , to (,) Belief of variable , *i*,*n*−1 and Message send from , to (,) Belief of variable , *<sup>k</sup>*,*<sup>n</sup>* = *E* Message send from , to , (,) Belief of variable , *k*,*n*−1 by a second-order Taylor to get: NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Estimation result of node at time ̃ Range measurement in all node pairs Estimation result of node at time ̃ Range measurement in all node pairs ̂, time ̃ in all node pairs (,) ∈ at time Vector sets of ̂, time ̃ in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs <sup>2</sup> Variance of , , Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time from time − 1 to time , <sup>2</sup> Variance of , , Vector to be estimated σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time from time − 1 to time , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Average velocity of node from time − 1 to time ,Measurement noise of , time all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of Position vector of all nodes at time Clock offset slope of all nodes at time time all nodes at time , Average velocity of node Measurement noise of time all nodes at time Average velocity of node Measurement noise of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , time all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of Position vector of all nodes at time Clock offset slope of all nodes at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error Set of all communicable node pairs (,) at time Set of all communicable node pairs (,) at time node , at time <sup>2</sup> Variance of , ,NLOS error of ̃, Set of all communicable node Set of all node pairs <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node Set of all node pairs <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs σ <sup>2</sup> Variance of , , NLOS error of ̃, Set all communicable node Set of all node pairs σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all node pairs Range measurement between , Measurement noise of ̃, , NLOS error of ̃, Range measurement between node , at time ,Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Range measurement between node , at time ,Measurement noise of ̃, <sup>2</sup> Variance of , Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, ̃, Range measurement between node , at time , Measurement noise of ̃, σ <sup>2</sup> Variance of , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between Measurement noise of , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between Measurement noise of , offset between nodes , , Set of neighbor nodes of node at time Measurement noise of

between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . (FIM) of , Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , (FIM) of , CRLB(,) Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , (FIM) of , CRLB(,) Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , (FIM) of , Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , (,) (FIM) of , CRLB(,) Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time , Covariance of belief (,) , Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix(FIM) of , CRLB(,) Cramer-Rao Lower **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = Function of the First Type , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , Mean of belief (,) Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes Confluent Hypergeometric , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes *gk* (,) Belief of variable , , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained *i*,*n*, (,) Belief of variable , , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained *k*,*n* = k *xi*,*<sup>n</sup>* − *xk*,*<sup>n</sup>* k − *cTEaik*,*n*−1,*<sup>M</sup>* ≈ *g<sup>k</sup> E* (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:,1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *i*,*n*−1 , *E* (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *k*,*n*−1 + ∂*g<sup>k</sup> E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  *i*,*n*−1 ,*n* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *k*,*n*−1 ∂ (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *i*,*n* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *<sup>i</sup>*,*<sup>n</sup>* − *E* (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *i*,*n*−1 + ∂ *<sup>T</sup> g<sup>k</sup> E* Estimation result of node at time ̃ in all node pairs (,)∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *i*,*n*−1 , *E* Estimation result of node at time ̃ in all node pairs (,)∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *k*,*n*−1 ∂ of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th *k*,*n* Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *<sup>k</sup>*,*<sup>n</sup>* − *E* Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,)∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *k*,*n*−1 + 1 2 <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*<sup>n</sup>* − *E* <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *i*,*n*−1 from time − 1 to time , , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric *<sup>k</sup>*,*<sup>n</sup>* − *E* Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , *k*,*n*−1 *T* ∇ 2 *gk E* from time − 1 to time , , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *i*,*n*−1 , *E* , from time − 1 to time , , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) <sup>∈</sup> at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *k*,*n*−1 σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*<sup>n</sup>* − *E* σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) *i*,*n*−1 , from time − 1 to time , , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , () Belief of variable , Confluent Hypergeometric *<sup>k</sup>*,*<sup>n</sup>* − *E* , Average velocity of node from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , *k*,*n*−1 , (31) Set of all node pairs (,) with NLOS error at time Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Set of all communicable node Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time at time Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ,*ESensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,)∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Position vector of all nodes at time , Average velocity of node from time − 1 to time , σ, <sup>2</sup> Variance of , , ̂, Estimation result of node at time ̃ NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Position vector of all nodes at time , Average velocity of node from time − 1 to time , σ, <sup>2</sup> Variance of , , ̂, Estimation result of node at time ̃ NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: (,) with NLOS error at time Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node from − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of Set of all communicable node pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs Vector sets of Set of all communicable node pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) <sup>∈</sup> at time Vector sets of Set of all communicable node pairs (,) at time (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂,Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of Set of all communicable node pairs (,) at time (,) with NLOS error Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time Vector sets of Set of all node pairs (,) with NLOS error at time Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time Set of all node pairs (,) with NLOS error at time Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at ̃ Range measurement in all node pairs (,) ∈ at time , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time Estimation result of node at time ̃ Range measurement in all node pairs node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time <sup>−</sup> <sup>1</sup> to time , Measurement noise of ,<sup>2</sup> Variance of , , Vector to be estimated of node at time ̃, node , at time , ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃, Range measurement between node , at time , ̃, σ <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time , Average velocity of node from time − 1 to time , Measurement noise of , σ, <sup>2</sup> Variance of , , Vector to be estimated of node at time Range measurement

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , where ∂*g<sup>k</sup> E* (,) Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower *i*,*n*−1 ,*E* (,) Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower *k*,*n*−1 ∂ time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th *i*,*n* and ∂*g<sup>k</sup> E* (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower *i*,*n*−1 (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower *k*,*n*−1 ∂ Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) Weight of the -th *k*,*n* are the first-order partial derivatives of *g<sup>k</sup>* ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , *i*,*n*, ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) ,Weight of the -th *k*,*n* with respect to NLOS error in all node pairs 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*<sup>n</sup>* and NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>k</sup>*,*<sup>n</sup>* at NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>i</sup>*,*<sup>n</sup>* = *E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , Confluent Hypergeometric , Mean of belief (,) *i*,*n*−1 , and (,) ∈ at time 1:, 1:1:, 1: , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) *<sup>k</sup>*,*<sup>n</sup>* = *E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *k*,*n*−1 , ∇ 2 *gk E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *i*,*n*−1 , *E* NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric , Mean of belief (,) *k*,*n*−1 is the Hessian matrix of *g<sup>k</sup>* NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , *i*,*n*, NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , *k*,*n* at NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* = *E* (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , *i*,*n*−1 and NLOS error in all node pairs (,) ∈ at time 1:, 1: 1:, 1: Vector sets of , , ̃, from time 1 to time Message send from , to , (,) Belief of variable , *<sup>k</sup>*,*<sup>n</sup>* = *E* (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , *k*,*n*−1 . Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time NLOS error in all node pairs (,)∈ at time 1:, 1: 1:, 1: Vector sets of , , ̃, from time 1 to time ̂, Estimation result of node at time ̃ in all node pairs (,) ∈ at time NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time

Gaussian distribution

by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes

means the system cannot provide reliable vertical positioning results [20], which is usually obtained

Gaussian distribution

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ location information and local time out of sync. The position vector of node at time is , = by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision **2. System Model 2. System Model 2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision **2. System Model 2. System Model 2. System Model**  CRLB(,) Bound of , (FIM) of , CRLB(,) Bound of , , Covariance of belief (,) , Gaussian distribution Fisher Information Matrix Cramer-Rao Lower (FIM) of , CRLB(,) Bound of , (FIM) of , CRLB(,) Bound of , , Covariance of belief (,) , Gaussian distribution Fisher Information Matrix Cramer-Rao Lower (,) Fisher Information Matrix (,) Fisher Information Matrix , Covariance of belief (,) , Weight of the -th , Covariance of belief (,) , Weight of the -th , Covariance of belief (,) , Weight of the -th Function of the First Type , Covariance of belief (,) , Weight of the -th , Covariance of belief (,) , Weight of the -th Function of the First Type , Covariance of belief (,) , Weight of the -th Function of the First Type , Covariance of belief (,) , Function of the First Type , Covariance of belief (,) , Confluent Hypergeometric , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) Confluent Hypergeometric Function of the First Type , Mean of belief (,) (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) After calculation, we can get the mean *E* Message send from , to , (,) Belief of variable , *i*,*n* and covariance *V* ,→, (,) Message send from , to , (,) Belief of variable , *i*,*n* of *b*( ,→, (,) Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*):

The anchor nodes are always deployed at the same height, and high vertical dilution of precision

means the system cannot provide reliable vertical positioning results [20], which is usually obtained

is ,, and all communicable node pairs (, ) constitute communicable node set .

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time ,

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is ,= (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock CRLB(,) Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock (FIM) of , CRLB(,) Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes (FIM) of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes (FIM) of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Fisher Information Matrix CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , Confluent Hypergeometric Function of the First Type , Mean of belief (,), Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution (,) Fisher Information Matrix (FIM) of ,CRLB(,) Cramer-Rao Lower Bound of , *Vzi*,*<sup>n</sup>* = h *V* −1 *<sup>z</sup>i*,*n*−<sup>1</sup><sup>+</sup> X *j*∈*Si*,*<sup>n</sup>* X *M*<sup>1</sup> η*M*<sup>1</sup> ,*ij* σ 2 *d I* − ρ˜*ij*,*<sup>n</sup>* − *cTEaij*,*n*−<sup>1</sup> − ζ*ij*,*<sup>n</sup>* ∇ 2 *gj Ezi*,*n*−<sup>1</sup> + X *k*∈*Ci*,*<sup>n</sup>* X *M*2 η*M*2,*ik* σ 2 *d* (*I* − ρ˜*ik*,*<sup>n</sup>* − *cTEaik*,*n*−<sup>1</sup> − ζ*ik*,*<sup>n</sup>* ∇ 2 *gk Ezi*,*n*−<sup>1</sup> , *Ezk*,*n*−<sup>1</sup> i−<sup>1</sup> , (32)

Gaussian distribution

means the system cannot provide reliable vertical positioning results [20], which is usually obtained

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time ,

is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is ,= (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ ,̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = [,, ,] , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock *E*ˆ *<sup>z</sup>i*,*<sup>n</sup>* = *Vzi*,*<sup>n</sup> V* −1 *zi*,*n*−<sup>1</sup> *Ezi*,*n*−<sup>1</sup> +<sup>X</sup> *j*∈*Si*,*<sup>n</sup>* X *M*<sup>1</sup> η*M*<sup>1</sup> ,*ij* σ2 *d xj* − ρ˜*ij*,*<sup>n</sup>* − *cTEaij*,*n*−<sup>1</sup> − ζ*ij*,*<sup>n</sup>* ∇*g<sup>j</sup> Ezi*,*n*−<sup>1</sup> − ∇<sup>2</sup> *<sup>g</sup>j Ezi*,*n*−<sup>1</sup> *Ezi*,*n*−<sup>1</sup> + X *k*∈*Ci*,*<sup>n</sup>* X *M*2 η*M*<sup>2</sup> ,*ik* σ2 *d* (*x<sup>k</sup>* − ρ˜*ik*,*<sup>n</sup>* − *cTEaik*,*n*−<sup>1</sup> − ζ*ik*,*<sup>n</sup>* ∇*g<sup>j</sup> Ezi*,*n*−<sup>1</sup> −∇<sup>2</sup> *<sup>g</sup><sup>k</sup> Ezi*,*n*−<sup>1</sup> , *Ezk*,*n*−<sup>1</sup> *Ezi*,*n*−<sup>1</sup> , (33)

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes to be located of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

Slope of local clock of node at time

**Symbol Meaning Symbol Meaning**

Set of neighbor nodes of node at time

Measurement noise of ̃,

,

̃ ,

,

Set of all node pairs (,) with NLOS error at time

Set of neighbor anchor nodes

̃ ,

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Position vector of all nodes at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

Average velocity of node from time − 1 to time

Real time value at time ,

Relative slope of local clock offset between nodes ,

node , at time

time

offset between nodes ,

<sup>2</sup> Variance of , ,

Real time value at time ,

Position vector of node at time

<sup>2</sup> Variance of ,

Set of neighbor anchor nodes of node at time

,

̃ ,

,

Position vector of node at time

Real time value at time ,

,

Position vector of node at

Real time value at time ,

Range measurement between

Relative slope of local clock

Average velocity of node

time

Initialization:

σ

Set of neighbor anchor nodes of node at time

,

,

**Table 1.** List of symbols.

**Symbol Meaning Symbol Meaning**

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

̃

, NLOS error of ̃,

1:, 1:, 1:, 1:

Set of neighbor anchor nodes

**Symbol Meaning Symbol Meaning**

Set of neighbor anchor nodes of node at time

Position vector of node at time

,

,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

**Symbol Meaning Symbol Meaning**

̃,

σ

,

,

,

σ,

,

,

,

̃,

σ

,

σ,

,

̃,

σ

,

̂,

,

̃,

σ

,

(∙)

,

,

(,)

<sup>2</sup> Variance of ,

is ,, and all communicable node pairs (, ) constitute communicable node set .

,→,

[,, ,]

time

NLOS error in all node pairs (,) ∈ at time

Function of the First Type

(∙)

(,)

**2. System Model** 

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

is ,, and all communicable node pairs (, ) constitute communicable node set .

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

(,)

<sup>2</sup> Variance of , ,

[,, ,] 

time

Average velocity of node from time − 1 to time

,

σ,

**2. System Model** 

pairs (,) at time

[,, ,]

(,)

<sup>2</sup> Variance of , ,

[,, ,]

,→,

̂,

(,)

,→,

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

<sup>2</sup> Variance of , ,

Estimation result of node at time

(,) ∈ at time

,

̃,

<sup>2</sup> Variance of ,

,

,

,

,

,

̃,

σ

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

,

,

,

̃,

σ

,

σ,

σ

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Position vector of all nodes at time

Average velocity of node from time − 1 to time

Estimation result of node at time

,

,

NLOS error in all node pairs (,) ∈ at time

σ,

<sup>2</sup> Variance of ,

̂,

(∙)

time

̂,

NLOS error in all node pairs (,) ∈ at time

(,)

**2. System Model** 

,→,

,

(,)

Function of the First Type

(∙)

Fisher Information Matrix (FIM) of ,

(,)

**2. System Model** 

, and the measured value of local clock ̃

between node and the external standard clock is , = (̃

[,, ,]

[,, ,]

(,)

,

,

,

̃,

σ

,

σ,

̂,

[,, ,]

σ

,

σ,

̂,

(∙)

(,)

**2. System Model** 

[,, ,]

[,, ,]

(,)

,→,

,

σ,

̂,

<sup>2</sup> Variance of ,

(∙)

(,)

**2. System Model** 

σ

,

σ,

̂,

(∙)

(,)

**2. System Model** 

,→,

(,)

,→,

where η*M*,*ij* = *Z*/ <sup>ρ</sup>*ij*·*tr V* Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , *j*,*n*−1 , and *Z* is the normalization constant. Compared with the approximation method based on KL divergence minimization, the proposed method based on multi-Gaussian approximation with second-order Taylor series expansion greatly reduces the complexity of approximate calculation. When *b*( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: Vector sets of , , ̃, from time 1 to time (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , Weight of the -th Gaussian distribution *<sup>i</sup>*,*n*) is approximated to a Gaussian function ˆ*b*( NLOS error in all node pairs (,) ∈ at time 1:, 1:, 1:, 1: ,→, (,) Message send from , to , (,) Belief of variable , (∙) Confluent Hypergeometric Function of the First Type , Mean of belief (,) , Covariance of belief (,) , *<sup>i</sup>*,*n*), each node only needs to send its own position vector and covariance matrix to its neighbor nodes, and the communication overhead is much lower than that of the particle message-based method. In addition, since each node has three parameters to be estimated, the network is required to include at least three anchor nodes. The flow of the M-VMP localization algorithm (Algorithm 1) is as follows: Clock offset slope of all nodes at time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated of node at time ̃ Range measurement in all node pairs (,) ∈ at time node at time Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , <sup>2</sup> Variance of , , Vector to be estimated node , at time , ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node from time − 1 to time , Measurement noise of , Range measurement between node , at time , Measurement noise of ̃, <sup>2</sup> Variance of , , NLOS error of ̃, Set of all communicable node pairs (,) at time Set of all node pairs (,) with NLOS error at time Position vector of all nodes at time Clock offset slope of all nodes at time Average velocity of node Measurement noise of , of node at time , to be located of node at time , Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time ̃, Range measurement between node , at time , Measurement noise of ̃, of node at time at time , Position vector of node at time ̃ , Measurement value of local clock of node at time Real time value at time , Slope of local clock of node at time , Relative slope of local clock offset between nodes , , Set of neighbor nodes of node at time Range measurement between Measurement noise of *Sensors* **2020**, *20*, x FOR PEER REVIEW 9 of 15 communication overhead is much lower than that of the particle message-based method. In addition, since each node has three parameters to be estimated, the network is required to include at least three anchor nodes. The flow of the M-VMP localization algorithm (Algorithm 1) is as follows: **Algorithm 1**: M-VMP joint estimation algorithm **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at time ̃ , Measurement value of local clock of node at time *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols. **Symbol Meaning Symbol Meaning** Set of neighbor anchor nodes of node at time , Set of neighbor nodes to be located of node at time Position vector of node at ̃ Measurement value of local clock of node *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 *Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

Measurement noise of ,

,

,

<sup>2</sup> Variance of , , NLOS error of ̃,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Set of neighbor nodes to be located of node at time

Set of neighbor anchor nodes of node at time

Position vector of node at time

Real time value at time ,

Relative slope of local clock offset between nodes ,

Range measurement between node , at time

Set of all communicable node pairs (,) at time

Set of neighbor nodes to be located of node at time

Position vector of all nodes at time

Measurement value of local clock of node at time

Average velocity of node from time − 1 to time

<sup>2</sup> Variance of , ,

Set of neighbor nodes of node at time

Measurement noise of ̃,

Slope of local clock of node at time

Estimation result of node at time

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

**Table 1.** List of symbols.

,

̃ ,

,

,

,

̃

CRLB(,)

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Vector sets of , , ̃, from time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower Bound of ,

, NLOS error of ̃,

**Symbol Meaning Symbol Meaning**

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Set of neighbor nodes to be located of node at time

Measurement noise of ,

Measurement value of local clock of node at time

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Set of neighbor nodes

Set of neighbor nodes to be located of node

Cramer-Rao Lower

Slope of local clock of node at time

,

Set of all node pairs

̃,

, , ̃, from time 1 to time

Weight of the -th

Cramer-Rao Lower Bound of ,

Range measurement in all node pairs

, Mean of belief (,)

time 1 to time

Weight of the -th Gaussian distribution

Cramer-Rao Lower

Fisher Information Matrix

Vector to be estimated

at time

of node at time

, NLOS error of ̃,

Set of neighbor nodes to be located of node at time

Measurement value of local clock of node at time

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of

<sup>2</sup> Variance of ,

,

,

,

̃ ,

,

,

,

,

̃ ,

̃

,

Slope of local clock of node at time

Set of neighbor nodes of node at time

Measurement noise of ̃,

Set of all node pairs (,) with NLOS error at time

Clock offset slope of all nodes at time

Measurement noise of ,

Vector to be estimated of node at time

Range measurement in all node pairs (,) ∈ at time

Set of neighbor nodes to be located of node at time

Vector sets of , , ̃, from time 1 to time

Measurement value of local clock of node at time

Slope of local clock of

,

, NLOS error of ̃,

,

̃,

σ

,

̃ , ,

̃ ,

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15

,

**Table 1.** List of symbols.

*Sensors* **2020**, *20*, x FOR PEER REVIEW 3 of 15 **Table 1.** List of symbols.

,

,

σ,

̂,

,

Fisher Information Matrix (FIM) of , CRLB(,) (,) **Algorithm 1**: M-VMP joint estimation algorithm Vector sets of from time − 1 to time <sup>2</sup> Variance of , , from time − 1 to time Set of all communicable node node , at time Initialize node location distribution Real time value at time , **Table 1.** List of symbols. **Table 1.** List of symbols. **Table 1.** List of symbols.


,

#### by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model**  (FIM) of , , Covariance of belief (,) , **4. Simulation Analysis and Test Results** (,) ∈ at time **4. Simulation Analysis and Test Results**  Estimation result of node at time <sup>2</sup> Variance of , , Set of all communicable node Set of all node pairs Set of all communicable node Set of all communicable node

Fisher Information Matrix

location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision means the system cannot provide reliable vertical positioning results [20], which is usually obtained **2. System Model**  The anchor nodes are always deployed at the same height, and high vertical dilution of precision (,) (FIM) of , CRLB(,) Bound of , **2. System Model**  In this section, Cramer-Rao Lower Bound (CRLB) of ,→, (,) Message send from , to , (,) Belief of variable , Confluent Hypergeometric *<sup>i</sup>*,*<sup>n</sup>* is derived first, then the performance of the proposed joint algorithms is analyzed through simulation. In this section, Cramer-Rao Lower Bound (CRLB) of , is derived first, then the performance of the proposed joint algorithms is analyzed through simulation. (,) ∈ at time NLOS error in all node pairs 1:, 1:, 1:, 1: Vector sets of , , ̃, from Estimation result of node at time ̃ Range measurement in all node pairs (,) ∈ at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of pairs (,) at time (,) with NLOS error at time Position vector of all nodes at Clock offset slope of

̃

, NLOS error of ̃,

CRLB(,)

Set of all node pairs

,−1)/( − −1). The local clock

all nodes at time

Range measurement

#### between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , , = (). The slope of local clock at time between node and the external standard clock is , = (̃ the anchor node with known location and synchronized local time, and the node with inaccurate by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes means the system cannot provide reliable vertical positioning results [20], which is usually obtained The anchor nodes are always deployed at the same height, and high vertical dilution of precision Function of the First Type *4.1. CRLB Lower Bound of 4.1. CRLB Lower Bound of* , *i*,*n* (,) ∈ at time NLOS error in all node pairs time time

̃

1:, 1:, 1:, 1:

, − ̃

is ,, and all communicable node pairs (, ) constitute communicable node set .

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

(FIM) of ,

, and the measured value of local clock ̃

<sup>2</sup> Variance of ,

(∙)

(,)

, and the measured value of local clock ̃

Message send from , to ,

with

[,, ,] 

time

NLOS error in all node pairs (,) ∈ at time

Function of the First Type

Fisher Information Matrix (FIM) of ,

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes , Covariance of belief (,) , Gaussian distribution For node , the Fisher Information Matrix (,) can be derived as: For node *i*, the Fisher Information Matrix *F*( Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*) can be derived as: (,) ∈ at time 1:, 1:, 1:, 1: , , ̃, from time 1 to time , Measurement noise of , Average velocity of node from time − 1 to time , Measurement noise of , Average velocity of node from time − 1 to time , Measurement noise of ,

Range measurement

all nodes at time

, − ̃

between node and the external standard clock is , = (̃

̃

1:, 1:, 1:, 1:

Fisher Information Matrix

, NLOS error of ̃,

is ,, and all communicable node pairs (, ) constitute communicable node set . between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time Fisher Information Matrix (FIM) of , CRLB(,) Cramer-Rao Lower Bound of , (,) = (… , σ 2 , … ) T , (34) Confluent Hypergeometric Function of the First Type , Mean of belief (,) *F*( (,) Belief of variable , *<sup>i</sup>*,*n*) <sup>=</sup> **<sup>A</sup>***idiag* . . . , σ 2 *d* , . . . **A** T *i* , (34) Vector to be estimated of node at time Vector to be estimated of node at time <sup>2</sup> Variance of , , Vector to be estimated of node at time

, − ̃

, − ̃

Bound of ,

Set of all node pairs

Vector to be estimated of node at time

Vector sets of

all nodes at time

Range measurement

1:, 1:, 1:, 1:

, NLOS error of ̃,

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located between node and the external standard clock is , = (̃ **2. System Model**  Confluent Hypergeometric Function of the First Type with Estimation result of node at Estimation result of node at Estimation result of node at

is ,, and all communicable node pairs (, ) constitute communicable node set . of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located The anchor nodes are always deployed at the same height, and high vertical dilution of precision , = [… , ,, … ], ∈ , (35) , − , , Covariance of belief (,) , Gaussian distribution Cramer-Rao Lower , Mean of belief (,) , Covariance of belief (,) , Weight of the -th **A***i*,*<sup>n</sup>* = h . . . , *<sup>u</sup>ij*,*n*, . . .<sup>i</sup> , *j* ∈ *Ni*,*<sup>n</sup>* (35) in all node pairs (,) ∈ at time in all node pairs (,) ∈ at time ̃ in all node pairs (,) ∈ at time

,−1)/( − −1). The local clock

Weight of the -th

$$\mathbf{u}\_{ij,n} = \frac{1}{\rho\_{ij,n}} \begin{bmatrix} \mathbf{x}\_{l,n} - \mathbf{x}\_{j,n} \\ y\_{l,n} - y\_{j,n} \\ cTa\_{lj,n} \end{bmatrix}. \tag{36}$$

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ *4.2. Simulation Scenario and Result Analysis*  The anchor nodes are always deployed at the same height, and high vertical dilution of precision and CRLB of Message send from , to (,) Belief of variable , *<sup>i</sup>*,*<sup>n</sup>* is CRLB( Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*) = *F* −1 ( Message send from , to , (,) Belief of variable , *<sup>i</sup>*,*n*).

Fisher Information Matrix (FIM) of ,

#### [,, ,] between node and the external standard clock is , = (̃ To analyze the performance of the proposed method, we have built a simulation scenario means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes The anchor nodes are always deployed at the same height, and high vertical dilution of precision Confluent Hypergeometric , Mean of belief (,) Confluent Hypergeometric , Mean of belief (,) Confluent Hypergeometric , Mean of belief (,) *4.2. Simulation Scenario and Result Analysis*

node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

, = (). The slope of local clock at time

the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , =

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located

,− ̃

,−1)/( − −1). The local clock

, = (). The slope of local clock at time

, − ̃

of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . according to the real scene of the zone 1, underground parking lot, Beijing University of Posts and Telecommunications. The real scene is shown in Figure 3a and the top view is shown in Figure 3b. According to the actual size of zone 1, the simulation scene is a rectangular area of 20 × 24 m. By the anchor node with known location and synchronized local time, and the node with inaccurate location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time means the system cannot provide reliable vertical positioning results [20], which is usually obtained by other sensors [21]. So, in this paper, a 2D dynamic wireless network is considered, which includes the anchor node with known location and synchronized local time, and the node with inaccurate , Covariance of belief (,) , Weight of the -th Gaussian distribution Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower , Covariance of belief (,) , Weight of the -th Gaussian distribution Fisher Information Matrix Cramer-Rao Lower To analyze the performance of the proposed method, we have built a simulation scenarioaccording to the real scene of the zone 1, underground parking lot, Beijing University of Posts and Telecommunications. The real scene is shown in Figure 3a and the top view is shown in Figure 3b.

default, the number of anchor nodes is 4, the number of nodes to be located is 20 and the initial position distribution of nodes in the site conforms to the uniform distribution. In order to limit the communication area of nodes to less than 1/2 of the whole simulation area, the maximum communication distance is set to 10 m. The performance of the proposed method is obtained in the between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , node has neighbor nodes set as ,, where neighbor anchor node set is , , node set to be located is ,, and all communicable node pairs (, ) constitute communicable node set . location information and local time out of sync. The position vector of node at time is , = , and the measured value of local clock ̃ , = (). The slope of local clock at time between node and the external standard clock is , = (̃ , − ̃ ,−1)/( − −1). The local clock of all anchor nodes is synchronized with the external reference clock, i.e., , = 1∀ ∈ . At time , CRLB(,) Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision (FIM) of , CRLB(,) Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision (FIM) of , CRLB(,) Bound of , The anchor nodes are always deployed at the same height, and high vertical dilution of precision According to the actual size of zone 1, the simulation scene is a rectangular area of 20 × 24 m. By default, the number of anchor nodes is 4, the number of nodes to be located is 20 and theinitial position distribution of nodes in the site conforms to the uniform distribution. In order to limit the communication area of nodes to less than 1/2 of the whole simulation area, the maximum

, = (). The slope of local clock at time

,−1)/( − −1). The local clock

,−1)/( − −1). The local clock

communication distance is set to 10 m. The performance of the proposed method is obtained in the line of sight (LOS) environment, except the last simulation, which shows localization accuracy with different NLOS probability. All simulation results are the average of 1000 independent runs.

**Figure 3.** (**a**) The real scene of the underground parking lot. (**b**) Simulation scenario. Zone 1 is the simulation area.

The initial position measurement error of the anchor node conforms to the Gaussian distribution with the standard deviation of 0.1 m. The initial position measurement error distribution of the node to be located is composed of Gaussian distributions with the standard deviation of 10 m. According to the performance of the crystal oscillator used in the hardware (TG5032CFN), the clock drift of the node to be located conforms to uniform distribution from [−1 ppm, 1 ppm], that is, the maximum distance measurement error caused by the clock drift between adjacent time (1 s) is 300 m. In the process of simulation, anchor nodes remain stationary, and the velocity of the node to be located is consistent with the uniform distribution of the maximum value of 3 m/s, and the direction is random every time step. The position movement measurement of the node conforms to the Gaussian distribution with the standard deviation of 1 m. The default process is 15 s, 10 iterations per second. Root mean square error (RMSE) and cumulative distribution functions (CDF) are used to measure the performance of algorithms.

In Figure 4a, the relationship between localization RMSE and initial location error is shown. The CLOC method [16] treats the uncertainties of nodes' positions and clock offsets as measurement noise and thus suffers performance degradation. The SPAWN [26] method is implemented by using 4000 particles to represent the messages on FG. The localization accuracy of the VMP method [17] and the proposed M-VMP method are better than the CLOC method, but slightly worse than the SPAWN method.

**Figure 4.** (**a**) Root mean square error (RMSE) of position error with different initial location error. (**b**) Cumulative distribution function (CDF) of position error with different initial location error.

The position error of VMP and the proposed M-VMP method are compared in Figure 4b. The cumulative distribution function of the VMP method and the proposed method under different initial position errors is given in Figure 4b. It is found that the probability of position error less than 1.6 m of the two methods under different initial position errors is basically the same, while the proposed method has a better effect of restraining position error. The result is consistent with the conclusion in Figure 4a.

Figure 5 shows the relationship among the clock drift slope, RMSE of localization accuracy and the number of iterations, respectively. It is seen that three methods converge in the finite iterations, which proves that the algorithms are feasible. Compared with the VMP method [17], the proposed method improves the convergence speed of localization and time synchronization by adding time synchronization parameters into the localization process.

**Figure 5.** (**a**) RMSE of the clock drift slope versus iterations and (**b**) RMSE of position error versus iterations.

In Figure 6, the CDF of the proposed method under different values of the number of Gaussian distribution M are compared. It is seen that the localization accuracy (3σ) of the proposed method is better than that of the VMP method when M is greater than 2. With the increase of M, the localization accuracy is also improved.

**Figure 6.** CDF of position error with different values of the number of Gaussian distribution M in line of sight (LOS) environment.

Figure 7 reflects the influence of communication distance on the localization accuracy of the proposed method under the same node density. Obviously, localization accuracy increases with the increase of communication distance, which increases the connectivity of nodes and the redundancy of information. At the same time, it can be seen that the improvement of localization accuracy is not obvious when the communication distance is 20 and 30 m. The difference of node connectivity between those two conditions is little, so the localization accuracy cannot be greatly improved. Because the communication distance is proportional to the transmission power of the node, it is necessary to determine the appropriate communication distance in practical application by considering the localization accuracy, node power consumption and other indicators.

**Figure 7.** CDF of position error with different communication distance.

In the simulation of Figure 8, in order to amplify the influence of node density on positioning accuracy, we reset some of the simulation parameters, and set the communication distance to 15 m and the variance of the distance observation and the initial position error of the node to be located to 20 m<sup>2</sup> . It can be seen from Figure 8 that as the density of nodes increases, the overall positioning accuracy is decreasing. However, when the density of nodes to be located is too large, the positioning error will increase. At the same time, when there are more nodes to be located, the speed at which the positioning error decreases with more anchor nodes also decreases. This is due to the increase in the density of nodes to be located, and the overall weight of the information contained in neighboring nodes to be located in the proposed method becomes larger. When the number of anchor nodes is insufficient, the increase of neighboring nodes to be located helps to improve the positioning accuracy, but the effect is limited.

**Figure 8.** Relationship between position error and nodes' density.

In Figure 9, the relationship between the RMSE of localization accuracy and NLOS occurrence probability of the VMP method and the proposed method is compared. Furthermore, it compares with the method that only deals with LOS. It is seen that both the VMP and the proposed method effectively suppress NLOS error. Because the two-way NLOS parameters between nodes are added in the proposed method, the NLOS error is better suppressed.

**Figure 9.** (**a**) RMSE of position accuracy with different NLOS probability. (**b**) CDF of position error with different NLOS probability.

#### *4.3. Computational Complexity and Communication Overhead Analysis*

Because the proposed method in this paper and the comparison methods are distributed algorithms, the computation and communication are done by the nodes independently, so only one node's time complexity and communication overhead need to be considered in the analysis.

The time complexity is evaluated by the number of operations in local computing, and the communication cost is evaluated by the number of information parameters broadcast by nodes. The time complexity of the CLOC method is related to its neighbor nodes number Ninb, and its communication cost is 3·O(1). The SPAWN method is based on particles. When the number of particles in each message is Np, its time complexity is O N<sup>p</sup> + N 2 p ·Ninb , and its communication cost is O N<sup>p</sup> + 2·O(1). Compared with the VMP method, the number of Gaussian messages in the proposed method is M times, and every message from neighbor nodes needs to be processed once. The time complexity of dimension reduction depends on Ninb, so its time complexity is O(M·Ninb) + O(Ninb). In the proposed method, nodes transmit a localization vector including xi,n, yi,n, ai,n and a covariance matrix, so the communication cost is 3O(M) + O(1). The computational complexity, run-time and communication Overhead of the three algorithms is shown in Table 2.

**Table 2.** Comparisons of different methods for each node at each iteration.


#### *4.4. Future Research Directions*

In the next stage of research, we will mainly focus on the following aspects: First, analyze the performance of the positioning and simultaneous joint estimation problem and its influencing factors in principle. Secondly, expand the positioning scene from 2 dimensions to 3 dimensions, and reduce the problem of positioning accuracy degradation in dense scenes with nodes to be located through methods such as signal quality screening. Finally, the proposed algorithm will be implemented based on the hardware platform, and the measured results will be compared with the simulation results to further improve the performance of the proposed method.

### **5. Conclusions**

This paper presented a M-VMP-based TOA localization and time synchronization joint estimation algorithm for mixed LOS and NLOS environments. Firstly, according to the VMP method, a message propagation model based on a factor graph was constructed for localization and time synchronization. Owing to the existence of nonlinear terms, it is difficult to represent the message in a closed form, so Taylor expansion was used for the linearization of nonlinear terms. Moreover, to reduce the communication cost, all messages were expressed in the form of multi-Gaussian distribution, and only the mean value, covariance and weight of each Gaussian distribution need to be transferred in the message transmission. The simulation results showed that the accuracy and convergence speed of the proposed method were close to that of SPAWN method, and the time complexity and communication cost were greatly reduced: the run-time was only 1.5% that of SPAWN.

**Author Contributions:** Conceptualization, S.T.; Data curation, S.T.; Formal analysis, S.T., B.J. and H.W.; Funding acquisition, Z.D. and S.T.; Investigation, S.T., B.J., H.W. and X.D.; Project administration, Z.D., S.T., B.J., H.W. and X.Z.; Resources, Z.D. and S.T.; Software, S.T., B.J., H.W. and X.D.; Supervision, B.J., H.W. and X.Z.; Validation, S.T. and X.D.; Visualization, S.T.; Writing—original draft, S.T.; Writing—review and editing, S.T., B.J., H.W., X.D. and X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Key Research & Development Program: 2016YFB0502003. **Conflicts of Interest:** The authors declare no conflict of interest.
