1. Introduction
Indoor wireless positioning has attracted much attention in recent years, which is the key important issue arising in robotics, advanced signal processing, social networking, or mobile monitoring of indoor environments, just to mention a few. Most positioning applications provide time-series measurements and demand for continuous estimation of the target’s position [
1]. Particularly, when considering Time-of-Flight (TOF) measurements as ranging measurements, the uncertainty of ranging measurements is coherent with dynamic environments and sensor limitations [
2]. Due to that, the indoor positioning problem is to understand the dynamic of which limited and noisy observations are available. In the main, a considerable amount of research has been put into the purpose of sequential estimations of indoor position, also known as indoor sequential positioning or position tracking [
3,
4,
5].
Although fairly fruitful research has been proposed in sequential positioning algorithms, to achieve accurate indoor position tracking from a multimodal distribution or noisy measurements is still very challenging due to the following difficulties:
Indoor RF ranging uncertainty. Wireless ranging technique is a convenient solution for indoor positioning and tracking, but suffering from severe uncertainty of radio frequency (RF) measurements. Thus, the probability density taking into account more observations may not necessarily lead to better accuracy.
Nonlinear/non-Gaussian dynamic. People or a mobile device can usually take non-uniform and heterogeneous motions, which are difficult to model. Furthermore, indoor RF ranging error is verified to be non-Gaussian in both simulations [
6,
7] and real-world experiments [
8,
9,
10]. Hence, both the observation and state transition are prone to be nonlinear and/or non-Gaussian problems, that the closed-form solution often does not exist.
Priori knowledge. The implementation and performance of the Bayesian framework depend on the priori knowledge of the measurement noise and process models, whereas the prior information is often not accurate.
Computation, real-time and implementation constraints. To be practical, such a sequential tracking solution should be efficient in computation, time delay, and implementation.
Portable devices are typically resources limited, i.e., computing, storage, and communication, etc. Hence, the problem, in using in mobile positioning, is the large storage and computation requirement. Therefore, the objective is to optimize the estimation density for continuous trajectories.
Near real-time position tracking is imposed by a tight delay constraint in the order of milliseconds [
11,
12]. State estimations require to investigate the entire sequence of the observations and state, which are time-consuming; also, the performance of fixed-Lag smoothing is dependent on the size of the smoothing lag (
) [
13].
Instability. It is well known that wireless network resources are scarce and time-varying. Thus, the smoothed tracking has to be robust to the problems of sparse measurements, inconsistent measurements or even losing tracking.
For sequential positioning, the family of recursive Bayesian methods have been subject to extensive research in literature, i.e., the sequential Bayesian filtering and smoothing methods in Hidden Markov Chain (HMC) framework [
14,
15,
16,
17]. Bayesian filtering estimates the state by updating recursively with every new incoming measurement, by applying the time-sequential hidden Markov process as a broad class of Kalman filters [
18] or particle filters (also known as sequential Monte Carlo (MC) methods) [
19]. The framework of filtering is defined as the estimation of the state at the same time as the current measurement. However, in the context of indoor positioning, there is frequent absences of radio measurements. Then, the Kalman filter works only in prediction mode, which degrades the prediction accuracy rapidly with time.
Recently, smoothing has attracted attention mainly in indoor position tracking. In contrast to normal filtering recursions, the smoothing frame is of particular interest as being able to make the state probability from not only the past and present observations (
) but also future observations (
,
) [
20]. The smoothing methods all fall into one of the two categories: joint or marginal. The early surveys have studied the possibilities of smoothing in a Bayesian framework [
21,
22,
23,
24], i.e., the Rauch-Tung-Striebel (RTS) smoothing algorithm [
25] presented by Rauch, Tung and Striebel, the Forward Filtering Backward Smoothing (FFBS) [
26] by Kitagawa and the Two-Filter smoother (TFS) [
27] by Fraser and Potter, and the One Time-step Particle Smoothing by Yuan and Huaming [
28]. Essentially, these smoothing frames tend to achieve improvements of time series analysis by involving the information of more observations. The above smoothing frames can be obtained by recursive formulas, i.e., the Kalman smoother [
29,
30], the extended Kalman smoother [
31], the unscented Kalman smoother [
32], the Gaussian-sum methods [
33]. On the other hand, sequential MC algorithms (known generically as particle smoother) [
34], approximate the time series estimation by a set of weighted particles. The Kalman smoother is an optimal smoother. To reduce the complexity of particle smoothers, some literature [
35] improve the computational complexity which is linear to the number of particles. Furthermore, thanks to massive increases in the computational power of mobile devices, some of these smoothing methods and their variations have been applied in embedded mobile positioning [
36,
37,
38,
39]. Based on these smoothing approaches, different solutions for non-linear sequential densities have been developed, e.g., Nurminen and Ristimaki deployed a fixed-interval forward–backward smoother to improve the position estimation for non-real-time applications [
40]. Most of these methods involve the form of smoothing as an extension to the particle filter, which reweight particles to approximate the smoothing density based on the filter density. For further positioning error mitigation, more schemes can be explored, i.e., multiple observation sub-models [
41], integration positioning [
42], velocity/direction modeling [
43], non-parametric models [
44] and map matching [
45] and etc.
Nevertheless, the key issues of applying a smoothing frame in RF position tracking are to derive tractable solutions to ameliorate the smoothing efficiency. To reduce the establishment cost and improve the accuracy, this paper focuses on implementing a smoothing frame with the observations after the present, namely, Marginalized Particle Smoother (MPS). Besides, the nonlinear and non-Gaussian 2D positioning recursion is not analytically solvable. Towards this end, we propose a lightweight MPS on a Sequential Monte Carlo (SMC) method. The proposed MPS hypothesizes that the smoothing density allows a better probability propagation, which continuously estimates the target’s position based on the posterior derived from the general smoothing density. In the scenarios of a relatively high modeling uncertainty of the experimental measurements and target’s motion, the results demonstrate that our proposed MPS is able to considerably improve the performance. In particular, the contribution of this study is summarized in the following.
Implement a lightweight marginalized particle smoother (MPS) on the SMC frame, which provides a trackable solution to the nonlinear and non-Gaussian indoor range-based positioning.
Propose the marginal smoothed smoothing that dynamically derives the posterior from a backward smoothing density in an efficient way. The virtue of MPS is that it can be applied to a very wide class of SMC methods.
Implement two popular nonlinear smoothing solutions: Forward Filtering Backward Smoothing (FFBS) and Two-filter Smoothing (TFS).
Combine two linear smoother (Moving Average (MA) smoother and Rauch-Tung-Striebel (RTS) smoother) with a nonlinear filtering output (Generic Particle Filter (GPF)).
The aforementioned smoothing algorithms are evaluated over indoor CSS–TOF (Chirp-Spread-Spectrum Time-of-Flight) test-bed. Experimental results validate the effectiveness and efficiency of the MPS framework on real-world indoor position tracking.
The remainder of this paper is structured as follows.
Section 2 presents the problem statement and system models.
Section 3 introduces the Bayesian smoothing formulas of FFBS, TFS, and MPS in detail. The moving average and Kalman smoother are introduced in
Section 4. The performance evaluation of real-world indoor tracking experiments are provided in
Section 5.
Section 6 concludes this work.
3. Filtering and Smoothing
The sequential position estimation or position tracking is defined as the time-series estimation of the posterior given all available observations. The 2D position can be estimated by either filtering or smoothing frames.
Filtering : to estimate the distribution of the state conditionally to the observations up to t.
Smoothing : to estimate the distribution of the state conditionally to the observations up to T (with ).
To smooth the estimated trajectory, improve accuracy, and deal with the sparsity problem, the sequential smoothing can be promising. However, the complexity of the smoothing method is proportional to the smoothing lag (): the larger the is, the higher the computation and latency are. Consequently, it is highly relevant to implement the Bayesian smoothing methods involving only a few future observations.
3.1. Bayesian Filtering
Filtering is to calculate sequentially the filter distributions
by the receipt of observation
. To recur the Bayesian frame, it essentially applies a hidden Markov model (HMM) of order one [
52] as follows.
State transition model
where
is the 2D coordinates;
denotes the process noise. The state propagation from
to
t is
, assuming that the state at time
t is stochastically dependent on the state at
.
Measurement model
with
being the measurement noise. It assumes the observation at
t is conditionally independent given the state, leading to
. The vector of the current observations (
) from
reachable anchors (
) is
with
for the ranging measurement from the
lth anchor at
t.
The analytical solution of nonlinear and non-Gaussian models is intractable in general. Sequential MC methods, like the aforementioned particle filters, use an approximate numerical posterior represented by
samples (
) with weights (
w)
3.2. Forward Filtering Backward Smoothing (FFBS)
The smoothing density can be deduced from a forward–backward recursive expression, namely, Forward Filtering Backward Smoothing (FFBS) [
53]. The smoothing density of FFBS frame is
The filtering density (
) can be computed by any forward filter, such as the Generic Particle Filter (GPF) [
54] approximated as
where
is the filtering weight of the
ith particle at
T.
The smoothing density (8) known as the forward filtering backward sampling [
55,
56], which can be numerically represented as
with the
ith smoothing weight
Then, the position estimation at
t by the smoothing density is
The FFBS (
12) consists of the filtering distribution (
) and the backward re-weighting probability from the future (
). The pseudo-code of FFBS is in Algorithm 1.
Algorithm 1 Forward Filtering Backward Smoothing (FFBS) |
Output and input: Setting: , , , Initialization:
- •
- •
- •
- •
- 1:
Importance sampling - 2:
Update the filtering weights (9) and normalization - 3:
Assign the smoothing weights (12), and normalization - 4:
Estimate position (13) - 5:
If , then resampling - 6:
Set and iterate to item 1
|
3.3. Two Filter Smoothing (TFS)
The two-filter smoothing (TFS) [
57] is a well-established alternative to FFBS, which obtains the smoothing density (
) from two independent filters (the forward and the backward filters).
Given observations up to
T, the smoothing density of TFS is
The first filter is the forward filter, which calculates the posterior distribution ; the second filter calculates a series of backward functions , that in the time-series is . Together, these two filters construct the smoothing density of TFS.
An important requirement of TFS is that
should be a probability density, in other words, the integral of this function is finite. Thus, the smoothing density of (14) is rewritten as
where
and
are the normalization factors of the smoothing and backward density, respectively. The smoothing density is represented as (9) with the weights
The pseudo-code of the TFS is described in Algorithm 2.
3.4. Marginalized Particle Smoother (MPS)
The FFBS and TFS formulate the smoothing density () from the current () and future () density. They are theoretically sound, as taking into account future measurements. The shortcoming is that the smoothing density only influence the point estimation in (13) rather than improving the density propagation.
Since FFBS and TFS have not incorporated the smoothing density into the state recursion, we propose to propagate the posterior from the smoothing density, formulated as
namely, Marginalized Particle Smoother (MPS). Indeed, the only difference to FFBS and TFS is that instead of propagating the posterior from the prediction density, the MPS is derived from the smoothing density.
Form a Markov process of order one, it means that
The factor
can be derived by (18), alternatively, by the approximation
leading to
It is based on two facts of indoor RF positioning: (1) the difference of the state at neighborhood time-series is very small, in other words, the target has a low velocity; (2) the uncertainty of the ranging measurements is much larger than that of the position estimation.
Algorithm 2 Two Filter Smoothing (TFS) |
Output and input: Setting: , , , Initialization:
- •
- •
- •
- •
- 1:
Importance sampling - 2:
Assign filtering weights (9) and normalization - 3:
Assign smoothing weights as (16) - 4:
Estimate position (13) - 5:
If , - 6:
Set and iterate to item 1
|
Similar to the TFS, the smoothing density of MPS is
Hence, Equation (
17) is reformulated as
Note that the posterior
in (17) is a filtering density derived from the smoothing density, which can be numerically represented as
with the smoothing density deduced from (20) as
The
in (
21) can either be performed from the component of (
15), or simply be approximated
as the target’s motion is slow.
Differing from that the FFBS and TFS estimate the state based on the smoothing density
, the MPS estimation is by the filtering density (
) as
The pseudo-code of the MPS is described in Algorithm 3.
Algorithm 3 Marginalized Particle Smoother (MPS) |
Output and input: Setting: , , , Initialization:
- •
- •
- •
- •
- 1:
Estimate position ( 24) - 2:
Assign smoothing weights ( 23) - 3:
Importance sampling - 4:
Update filtering weights ( 22) and normalization - 5:
If , then resampling - 6:
Set and iterate to item 1
|
The smoothed posterior involves future observations in the density propagation, which are powerful information to mitigate the estimation instability and sparsity problem.
6. Conclusions
Drawing from a smoothing density and finding an analytical solution is in most cases a difficult task, especially in case of non-linear and non-Gaussian errors. Due to the severe uncertainty of the indoor wireless environment and sensor system, range-based position tracking often encounters the LOS and NLOS measurements. To combat the uncertainty, this paper applies the Bayesian smoothing frame to the nonlinear non-Gaussian SMC models, including the FFBS, TFS, MPS, GPF + MA, and GPF + RTS. Furthermore, to be aware of the real-time constraint, we focus on the smoothing frame using a lightweight recursion.
The experiments defenestrate that the GPF+RTS and GPF+MA are more a sense to improve the representation of the estimated trajectory, whereas, the MPS filters out the estimation bias. The MPS algorithm fits the scenarios of typical motions (the mobile target with a general acceleration) and high measurement uncertainty. Since MPS requires no other assumptions, offline training, or high complexity, it is practical for its performance and efficiency. Our theoretical derivations and experimental verifications provide a better understanding of the time-series smoothing on 2D position tracking. Future work will investigate how to adaptively set the smoothing lag (the number of future observations) that achieved tradeoff of the smoothing performance and complexity. It is also useful to use the smoothing density to improve the sample quality of the SMC frame. The future work will investigate how to adaptively choose the value of the smoothing lag (the number of future observations) that leverages the tradeoff of the smoothing performance and complexity.