Reduced-Complexity Tracking Algorithms on Chip for Real-Time Location Estimation

Comments to the Author

Comments and Suggestions for Authors

Response

The reviewer's comments/suggestions are addressed in this revision (please see responses to individual comments below). In addition, the following points have been added in different sections and paragraphs in order to clarify the concepts in question and to improve the quality of the paper.

Comment

#1

Line89: Does MD means mobile devices? (I see MDs in line103.) Author’d better explain when it first appears in the paper to facilitate reading.

Response

Thanks for reviewer’s careful examinations. We have checked and corrected this revision manuscript as follows.

In Section 1, line 83:

… the model of the mobile devices (MDs) is a nonlinear function of the position and velocity parameters …

In Section 2, line 145:

… observation only depends on the current physical location of MDs …

In Section 2, line 160:

… is used to track the location of an MD …

Comment

#2

Line96: I think “u” in word “using” should be capitalized.

Response

Thanks for reviewer’s careful examinations. We have checked and corrected this revision manuscript as follows.

In Section 2, line 153:

In terms of the recursive Bayesian estimation, …

Comment

#3

Line107: I can’t find w_k in figure1.

Response

Thanks for reviewer’s careful examinations.

w_k should be u_k ,

We have modified the typos as follows.

In Section 2, line 165:

…  is the MD model noise, …

Comment

#4

Figure 2: The figure seems not implying the connections between the variables clearly.

Response

Thanks for reviewer’s careful examinations. We have checked and corrected this revision manuscript. The figure have implied the connections between the variables as follows.

In Section 2, line 176:

Figure 1. The prediction and correction phase of tracking systems.

Comment

#5

Line131: In my perspective, K_k should be the weight to weigh wether to trust estimation or observation while the author write x_k here. So I was wondering if it was just a slip of pen or I didn’t fully understand the algorithm. If the author is confirmed he means x_k here, then I suggest to use another letter since x_k is already used to stand for another meaning in line106.

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 2, line 178 – line 183:

The KG is the weight of the measurement and the weight of the previous estimate. The covariance matrix of the noise of measured value is represented by  (the covariance matrix of v, v: measurement noise). is the state estimate matrix, and K_k is the weight update procedures. As illustrated in (11), when the tracking approach forms a new estimate, it illustrates how much the measurement changes the estimate for the current system state estimat.

Comment

#6

Line189: The KG in (7) is not as obvious as the KG in (6). If formulation(7) and (6) stands for the same principle(I think they are), changing (7) into (6) is recommended to provide a clearer expression.

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 2, line 161 – line 173:

In this article, measurement formula of the observed location and the motion of mobile equation with two-dimensional model describing are taken as follows [10], [30].

= + ,  ~ (0, )                                                (6)

= + ,  ~ (0, ),                                                         (7)

where  is the state vector,  is the state transition matrix,  is the MD model noise also called process noise,  is the MD model noise covariance matrix,  is the actual measurement equation,  is the measurement matrix,  is the measurement noise, and  is the measurement noise covariance matrix. A two-dimensional (2D) model describing the motion of the mobile equations is taken as follows:

= = +                                       (8)

= +                                                     (9)

where  and are the location of x-axis and y-axis, respectively;  and  are speed of x-axis and y-axis, respectively. As illustrated Figure 1, time update can predict current state estimate in advance …

Comment

#7

Line200: Why eight? It’s said that “the the training length of the KG coefficient can 173 be based on iterations of the norm(P)” in line 173, and the figure inserted in figure3 and figure4 shows that norm(P) stabilizes at about 20 iterations. In my opinion, there should also be 20 KGs before it stabilizes, so can the author explain how norm(P) reflect on KG? Or how does the author decide to use 8 KGs? 

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 4.1, line 262 – line 270:

DKF uses the difference between current KG with average of past eight KGs to make judgments. On the basis of the results in Fig. 3, the training lengths are 18, 21, and 23 for the variances of 5, 10, and 15, respectively. According to the simulation results, there is a tradeoff between accuracy and hardware cost, and the window size was eight for the proposed approaches. Namely, after convergence, the sequence values of 8 KGs are averaged in this article. For example, in terms for the variances of 5, the first average value is based on the sampling KG value of iteration number from 18 to 25 for the present state 25, and the second average value is based on the sampling KG value of iteration number from 19 to 26 for the present state 26.

Comment

#8

Line219: The right choice need to be proved bringing better result than the wrong choice. What I’m saying is, comparative experiments are essential if the author want to convince the readers. The author should also illustrate how DKF and PKF perform in the same small environment with same small KG.

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 4.2, line 286 – line 304:

The basic ideas of derivation processes for the KF algorithm are based on three distinctive features of the innovation approach which are linear, unbiased, and minimum-variance [30]. In terms of the prediction and correction steps, for the state space model with linear and Gaussian, a KF algorithm can be derived and applied to the recursive Bayesian estimation. As a result, the accuracy of KF-based approach depends on the variance of the model noise and the variance of measurement noise. In terms of the same model noise and measurement noise, for KF-based approach, the larger the sampling time is, the smoother the trajectory will be. Namely, the larger sampling time of a KF-based approach is not suitable for dramatic time variances in the corners of hallways. As illustrated in (10), (11), and (20), the weight of the innovation is the KG. For forming a new estimate, the KG is the weight of the measurement and the weight of the previous estimate. For the state space model with same model noise and with different variance of measurement noise, the more closer the actual variance value is, the more accurate the localization approach is. When the measurement uncertainty is high, the KG is low. As illustrated in Figure 3, the convergence of the estimate uncertainty would be slow. In terms of the result in Ref. [32], the KG is close to zero when the measurement uncertainty is high and the estimate uncertainty is low. Namely, if the measurement uncertainty equals the estimate uncertainty, the KG equals to 0.5 [37].

Comment

#9

Figure5: The question is the conclusion I get from figure 5 is PKF’s estimations are closer to the groundtruth than DKF’s estimations under large KGs, however, the paper is trying to prove PKF is fit for small KGs and DKF is fit for large KGs(line238). This is very confusing.

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 4.2, line 307 – line 319:

For applying to the simulation environment of linear systems in large KG, the measurement noise of the PKF-based approach is more closer to the actual variance value of the simulations. Namely, a low measurement uncertainty which is relative to the estimate uncertainty results in a high KG. The new estimate is close to the measurement. On the other hand, a high measurement uncertainty which is relative to the estimate uncertainty results in a low KG. The new estimate would be close to the previous estimate. The KG is the weight of the measurement and the weight of the previous estimate [37]. On the basis of Figure 5, the results illustate the environment with the same large KG. In terms of PFK-based and DKF-based approaches, the results illustrate that the good choice at the right time can get better result. Therefore, the appropriate judgment conditions should be chosen in different environmental factors. As illustrated in Figure 5, the PKF-based approach is suitable for a low measurement noise environments, and the DKF-based approach is suitable for a high measurement noise environments..

Comments to the Author

The Paper presents a very interesting and practical approach to Kalman filtering. Kalman Filters are a very well-known topic in the field and the choice of this subject must present something new. The mentioned practical approach has several flaws, but in the end, final effect of the article is significant:

Response

The reviewer's comments/suggestions are addressed in this revision (please see responses to individual comments below). In addition, the following points have been added in different sections and paragraphs in order to clarify the concepts in question and to improve the quality of the paper.

Comment

#1

the introduction should be enhanced and better stated, it has no references and it's difficult to know what the article will be about

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 1, line 88 – line 134:

Most of today's positioning and tracking systems are developed and implemented with software. However, software implementations often result in speed delay and cannot provide in-time positioning information for LBS applications [17]-[20]. In order to improve the computational speed of positioning systems and the accuracy, there already are lots of improved algorithms in literature [21]-[34]. The calculations of KF approach are involved in the numbers of matrix computations, which including addition, multiplication, and inverse operations. The approach is difficult to implement hardware in traditional device with some practical systems, such as high performances in real time, flexible and convenient implementation [10], [17]-[20], [30]-[32]. In addition, the processors of Digital Signal Processing (DSP) adopt sequential program execution. Therefore, it cannot correspond the require of high-speed and real-time requirement. On the contrary, an FPGA has precision accuracy and high-portability, it can decrease the disadvantages of software for specific integrated circuit approaches. Actually, one possible way to increase performance is through the use of hardware acceleration using field-programmable gate arrays (FPGAs). Implementing functions as hardware have the potential to allow the use of more complex algorithms such as KF, and the approach also allows the implementation of multiple functions-as-a-chip systems on a single FPGA [16]-[20]. Each logical unit of the FPGA is determined at the time of programing, and no instructions are required. As a result, there is a significant advantage in the speed of processing operations with high accuracy and freely programmable characteristic. Namely, it can be easy to develop. Because of these advantages, FPGA is very suitable to be a verification method before tape-out. A chip has the advantage of low cost and fast processing time. Therefore, tape-out is be chosen as the result.

This article implements the localization algorithms using high-speed hardware devices. To implement the localization algorithms, the filtering formulas are derived and simplified from the filtering models and matrix operations, and then the hardware-based approaches using FPGAs are implemented. The FPGA-based implementation of the localization algorithm can improve the traditional matrix operation approach and can be applied to practical systems. The hardware implementation has advances of high portability, low cost, high accuracy and wide extension in scope. In addition, under a stationary environment, the coefficients of the α-β filter is extract from the KF algorithm, which is a degenerate form of the KF algorithm [15], [31]-[32],. The proposed training and tracking scheme replace the decision mode of the KF algorithm with an alpha-beta (α-β) algorithm to avoid repeatedly calculating the Kalman gain (KG). In this article, an adaptive low-complexity location-estimation approach combined α-β filtering algorithm with the KF algorithm is proposed and implemented. The proposed tracking approach is based on the α-β filtering algorithm extracted the coefficients of the KG from the KF algorithm according to different judgements. In addition, this article proposed the hardware implementation for proposed filtering algorithm. The concept is performed by chip, which has the features of real-time processing and pipeline structure. Under a stationary environment, as compared with software implementation for location tracking, the proposed chip training, decision, and tracking approach can not only implement location accuracy which is similar to KF tracking approach, but it possess computational complexity and provides better performance than that of non-tracking approaches. The approach with low complexity and high performance is proposed. Compared to the software implementation and the previous researches, because of its low complexity and faster processing speed, the proposed approach on chip has more benefits on LBS services or IoT applications.

Comment

#2

Subsection 2 and 3 - information are well known, nothing new here, in my opinion, it could be shortened

Response

The authors thank Reviewer 2 for his / her comments.

We have shortened this revision manuscript to improve the quality of the paper.

Comment

#3

Please add information about the more complex Kalman filter types: EKF, UKF and why are they unsuitable for the application

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

In Section 1, line 37 – line 87:

Given the rapid advance in wireless communication and mobile network technologies, the development of micro-electromechanical systems, the extensive use of satellite navigation technologies, the increased demand for intelligent nursing care systems, and the maturing technologies in cloud computing and Internet of Things (IoT), the database systems can collect the big data with mobile devices, and then they analyze, compute, and extract the useful information from the disorganized data in the database [1[-[6]. After-ward, they can supply the useful information via location-based services (LBSs) for different purposes. Recently, the people have paid more attention to the rapid and friendly approaches. Namely, the expectations in terms of market opportunities are the continuously and rapidly increasing attention on the mobile computing, IoT, smart agriculture, and smart city issues. In order to improve the positioning and tracking accuracy, fusion approaches between the absolute approach and relative approach can be considered an important technique for positioning and tracking systems [7]-[13]. Namely, to improve localization accuracy, most filters utilize Bayesian estimators such as the Kalman filter (KF), extended KF (EKF), or unscented KF (UKF) applied to linear and nonlinear systems. KF approaches are based on the state space equation of the linear system, and provides a recursive solution to the linear optimal filtering problem. The solution is on the basis of recursive, and it is suitable for stationary and non-stationary environments. In the light of KF-based approaches, each updated state is computed based on the previous estimate and new input data. Namely, KF-based approaches only need to store the previous estimated data, it is not needing to store the entire past observation data [7]-[10], [14]. KF-based approaches are suitable for dealing with the estimation of the state vector in the linear model of the dynamic system. If the nonlinear model is generated by the linearization program, the filter is on the basis of the EKF algorithm generally [7], [10], [14]-[16]. EKF-based algorithms are on the basis of KF algorithms with excellent accuracy. EKF-based algorithms are used as techniques to perform recursive nonlinear estimation, and these approaches are described in terms of difference equations in the case of discrete-time systems. There-fore, EKF-based approaches are one of the state estimation algorithms suitable for many embedded systems. For nonlinear systems, an EKF-based algorithm only provides an optimal approximation for performing recursive nonlinear estimation. Nevertheless, the UKF-based algorithm can perform better than the EKF-based algorithm. The differences between EKF and UKF approaches are represented for propagating through system dynamics based on Gaussian random variables (GRV) [7], [14], [17]. For EKF-based approaches, the state distribution is approximated by GRV and then propagated analytically through the first-order linearization of the nonlinear system. Namely, EKF-based approaches may introduce large errors in the true posterior mean and covariance of the transformed GRV, and it leads to reduce the performance of the systems. The UKF algorithm is based on a deterministic sampling approach. The state distribution is approximated by a GRV. For propagating through the true nonlinear system, the sample points completely capture the true mean and covariance of the GRV [14]. Namely, for the non-linear system, UKF-based approaches capture the posterior mean and covariance to sec-ond order series expansion. UKF-based approaches do not require an explicit Jacobian calculation, and the computational complexity of UKF algorithms are the same as that of EKF algorithms [14]. The fusion approach based on the traditional KF tracking algorithm requires inverse operations which are high computational complexity, and direct implementation of the KF algorithm may be too complex to be applied to the practical systems. Furthermore, the model of the mobile devices (MDs) is a nonlinear function of the position and velocity parameters [10]. Nonlinear models in state equation assumption may lead to non-optimal solutions for linearized schemes. In terms of these services and applications, the techniques of positioning and tracking approaches have become particularly important and increased dramatically under these developments.

In Section 1, line 111 – line 134:

This article implements the localization algorithms using high-speed hardware devices. To implement the localization algorithms, the filtering formulas are derived and simplified from the filtering models and matrix operations, and then the hardware-based approaches using FPGAs are implemented. The FPGA-based implementation of the localization algorithm can improve the traditional matrix operation approach and can be applied to practical systems. The hardware implementation has advances of high portability, low cost, high accuracy and wide extension in scope. In addition, under a stationary environment, the coefficients of the α-β filter is extract from the KF algorithm, which is a degenerate form of the KF algorithm [15], [31]-[32],. The proposed training and tracking scheme replace the decision mode of the KF algorithm with an alpha-beta (α-β) algorithm to avoid repeatedly calculating the Kalman gain (KG). In this article, an adaptive low-complexity location-estimation approach combined α-β filtering algorithm with the KF algorithm is proposed and implemented. The proposed tracking approach is based on the α-β filtering algorithm extracted the coefficients of the KG from the KF algorithm according to different judgements. In addition, this article proposed the hardware implementation for proposed filtering algorithm. The concept is performed by chip, which has the features of real-time processing and pipeline structure. Under a stationary environment, as compared with software implementation for location tracking, the proposed chip training, decision, and tracking approach can not only implement location accuracy which is similar to KF tracking approach, but it possess computational complexity and provides better performance than that of non-tracking approaches. The approach with low complexity and high performance is proposed. Compared to the software implementation and the previous researches, because of its low complexity and faster processing speed, the proposed approach on chip has more benefits on LBS services or IoT applications.

Comment

#4

Figures 10 and 12 - low resolution and please check descriptions of the axes in the drawings

Response

We have checked and corrected Figure 10 and Figure 12 of this revision manuscript as follows.

Figure 10

Figure 12

Comment

#5

From the above figures: errors can be calculated: the basic descriptive statistics, other than figure 11

Response

We have checked and corrected this revision manuscript as follows, and the below description has been modified and added.

Figure 11

In Section 6.2, line 464 – line 472:

Table 7 illustrated the results of the proposed localization approaches. The location ac-curacy corresponds to the different fixed values of α and β coefficients, where the different fixed coefficients of α and β are tuning constants between number 0 and number 1. On the basis of the results, the location accuracy of proposed the α-β training-tracking method based on the coefficients of the KG is almost the same as the location accuracy of the KF tracking approach. Therefore, to use the coefficients of the KG obtained from the proposed approaches has a considered effect on the performance of the α-β training-tracking algorithm.

Table 7 The Cumulative distribution function of the error distances, KF software-based, α-β soft-ware-based, KF VLSI-based, and proposed approach with α-β chip-based tracking approaches.

Comment

#6

Conclusions are well stated

Response

The authors thank Reviewer 2 for his / her comments.

Comments to the Author

Putting forward a low complexity filtering algorithm for simpler filter design, the work is practical and can be published after some editing. For example, will it be OK if a senior scientist was coaching his/her postdocs/doctors how to publish a scientific paper while he/she can do what they want? The title is usually within 10 words as 'Reduced complexity tracking algorithms on chip for real-time location estimation'.

Response

The reviewer's comments/suggestions are addressed in this revision.

This manuscript is on the basis of our conference paper: “Design FPGA-Based Implementation of Reduced-Complexity Filtering Algorithm for Real-Time Location Tracking”, The 17th IEEE International Conference on Dependable, Autonomic and Secure Computing (IEEE DASC 2019) [18]. In the extended version of this manuscript, there are four broad changes: a new title, a new abstract, additional theory and idea, and additional experimentations We hereby certify that this paper consists of original work, which is not under consideration for publication elsewhere.

In addition, the title of this article has been changed to improve the quality of the paper.