Robust node position estimation algorithms for wireless sensor networks based on improved adaptive Kalman filters

Accepted Manuscript

Robust node position estimation algorithms for wireless sensor networks based on improved adaptive Kalman filters Xuming Fang , Lei Nan , Zonghua Jiang , Lijun Chen PII: DOI: Reference:

S0140-3664(16)30555-2 10.1016/j.comcom.2016.11.005 COMCOM 5408

To appear in:

Computer Communications

Received date: Revised date: Accepted date:

12 December 2015 6 November 2016 14 November 2016

Please cite this article as: Xuming Fang , Lei Nan , Zonghua Jiang , Lijun Chen , Robust node position estimation algorithms for wireless sensor networks based on improved adaptive Kalman filters, Computer Communications (2016), doi: 10.1016/j.comcom.2016.11.005

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Xuming Fanga, Lei Nana, Zonghua Jianga, Lijun Chena,b,* a

Department of Computer Science and Technology, Nanjing University, Nanjing 210046, China b

State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210046, China

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Abstract

The Kalman filter (KF) is an optimal state estimator in noisy environments, and it can estimate the current state of a position observation system (such as the location of a node in a network) by measuring the system output with noise [1]. Usually, a process model describing the state change of the system and a measurement model defining the relationship between the internal state and the external output of the system are needed in the KF. For practical positioning scenarios, there exist modeling errors in the process and measurement models, that is, the process noise and the measurement noise. Because the accuracy of the statistical parameter of the process and measurement noise can greatly affect the estimation precision of the KF, obtaining the exact noise parameter is of critical importance for the KF-based positioning algorithm. A wireless sensor network (WSN) is an ad-hoc network made up of a large number of nodes equipped with sensors, microcontrollers and transceivers [2]. WSNs have been applied in various fields, for example, smart homes, health care, industrial automation, and environmental monitoring. For many WSN applications, it is necessary to know the location information of the node in the network. In most WSNs, two nodes communicate with each other by sending and receiving wireless signals. The received signal strength indication (RSSI) between nodes is usually employed to model the inter-node distance. Because the measured RSSI is vulnerable to the interference of the external noise, the KF is often used as the node position estimator of the WSN to estimate the position of the target node (TN) without a known location based on the noisy RSSI measured by the anchor node (AN) with a known location. Although the KF can filter the process noise in the process model and the measurement noise in the measurement model, it requires accurate statistics of the process noise and measurement noise. An inaccurate statistical parameter of the process and measurement noise can cause localization errors in the node position estimation (NPE) algorithm based on the KF (NPE-KF) to increase or even diverge [3]. An unbiased noise statistic estimator (NSE) able to estimate the noise statistic parameter in an online manner is developed [4], which allows the KF to perceive unknown or time-varying noise statistics. The KF has already been used to estimate a WSN node’s position [5,6], but it is only applicable to the linear process and measurement models. Thus, an NPE based on an extended Kalman filter (NPE-EKF) suitable for the nonlinear model is designed [7-9]. Because the NPE-EKF requires linearizing the function in the nonlinear model, it can only have an accuracy of, at most, the first-order Taylor series expansion of the nonlinear function. To overcome the limitation of requiring that the

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Robust node position estimation algorithms for wireless sensor networks based on improved adaptive Kalman filters

1. Introduction

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The Kalman filter (KF) is an optimal state estimator for position observation systems with noise; therefore, it is typically used to estimate the location of a node in a wireless sensor network (WSN) in a noisy environment. The precision of noise statistics largely determines the localization accuracy of the KF, and the statistics of noise are often unknown or time-varying in a real WSN. Therefore, the adaptive Kalman filter (AKF) is utilized for awareness of the statistical parameters of noise. However, the node position estimation (NPE) algorithm based on the state-of-the-art AKF typically lacks robustness and becomes inaccurate in the case of simultaneously perceiving the statistics of process noise and measurement noise. This study proposes a robust NPE algorithm based on an improved adaptive extended Kalman filter (RNPE-IAEKF) and another robust NPE algorithm based on an improved adaptive unscented Kalman filter (RNPE-IAUKF). The RNPE-IAEKF algorithm has low computing complexity, while the RNPE-IAUKF algorithm has high positioning accuracy. Our proposed algorithms solve the problems of poor robustness and low accuracy of the NPE algorithm based on the adaptive extended Kalman filter (NPE-AEKF) and the NPE algorithm based on the adaptive unscented Kalman filter (NPE-AUKF). In addition, the RNPE-IAEKF and the RNPE-IAUKF do not lose robustness upon simultaneous perception of the statistics of process noise and measurement noise, which is strictly proven in theory. The results of practical experiments and numerical simulations demonstrate that regardless of the placement of a static target node, the mobility of a mobile target node, and the number of anchor nodes, the RNPE-IAEKF improves upon the positioning accuracy and convergence speed of the NPE-AEKF by at least 28% and 29%, respectively and that the RNPE-IAUKF increases the localization accuracy and convergence rate of the NPE-AUKF by at least 32% and 37%. Keywords: Robust localization algorithm; Adaptive Kalman filter; Noise statistic estimator; Wireless sensor network; RSSI

* Corresponding author. Tel.: +86-025-83686717. E-mail address: [email protected] (X. Fang), [email protected] (L. Nan), [email protected] (Z. Jiang), [email protected] (L. Chen).

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by a large number of practical experiments and numerical simulations under different placement positions of a static node, different mobilities of a mobile node, and different numbers of anchor nodes. 1.2. Organization

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The organizational structure of this paper is as follows. Section 2 introduces the related work, and Section 3 describes the process and measurement models used in the KF. Section 4 presents the proposed robust NPE algorithm based on the improved AEKF, while Section 5 exhibits the proposed derivative robust NPE algorithm based on the improved AUKF. Section 6 shows the results and analysis of experiments and simulations. Finally, Section 7 presents the paper’s conclusions. 2. Related work

According to whether measurement of the distance between nodes is required in the process of localization or not, WSN positioning algorithms are divided into two categories: rangebased and range-free [18-21]. At present, there are two types of factors that cause a WSN localization algorithm to not be robust. One group is induced by the algorithm itself, i.e., internal factors, while the other is not evoked by the algorithm, i.e., external factors. In general, the internal factors that affect the robustness of the algorithm used in a WSN mainly include the network topology, the node density, the environment noise, etc. [22,23]. Due to a design defect of the algorithm, in some special cases, these factors will cause the calculation results of the positioning algorithm to not be correct, and the algorithm cannot be implemented. For range-based WSN localization algorithms, the large measurement noise in the distance between nodes can make the algorithm lose robustness [24,25]. For example, the positioning algorithm using the theory of rigid graphs can produce a flip ambiguity in the presence of measurement noise, which will result in the distortion of the node position calculated by the algorithm. To solve this problem, a robust rigid-graph localization algorithm that resists the measurement noise is designed [26-28]. For range-free WSN positioning algorithms, the particular network topology may lead to the loss of robustness of the algorithm. For instance, there is a large deviation of the node location computed by the algorithm using multidimensional scaling in a non-convex network. Therefore, an improved algorithm, one that is immune to the network topology, is created to increase the positioning accuracy of the node [29,30]. Relative to the internal factors, the external factors that affect the robustness of the algorithm mainly refer to network attacks, missing measurements, link failures and so on [31-33]. In the case of network attacks, a captured node will send forged data to the other nodes. As a result, the node position that the algorithm uses these false data to compute is not trustworthy. To identify captive nodes, a robust localization algorithm

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function in the model be linearized, the NPE based on an unscented Kalman filter (NPE-UKF) exempted from the linearization of the nonlinear function is created [10,11], which can theoretically achieve an accuracy of at least the second-order Taylor series expansion. Because the extended Kalman filter (EKF) and unscented Kalman filter (UKF) algorithms lack the noise-adaptive ability, the adaptive EKF (AEKF) and adaptive UKF (AUKF) are produced by introducing an unbiased NSE into the EKF and the UKF, respectively [12,13]. Accordingly, the NPE based on the AEKF (NPE-AEKF) and the NPE based on the AUKF (NPEAUKF) are obtained [4,14]. However, the unbiased NSE can generally only estimate the process noise statistics or the measurement noise statistics. When two types of noise statistics are simultaneously estimated, the process noise covariance matrix often loses the nonnegative definiteness, and the measurement noise covariance matrix usually misses the positive definiteness, which will lead to the NPE-AEKF and the NPE-AUKF losing robustness [15,16]. A biased NSE that can make sure that the NPE-AEKF and the NPE-AUKF are robust is put forward [17], yet it generally results in them having poor positioning precision. To guarantee the robustness of NPE-AEKF and NPE-AUKF and enable them to have good localization accuracy, we propose a robust NPE algorithm based on an improved AEKF (RNPE-IAEKF) and a derivative robust NPE algorithm based on an improved AUKF (RNPE-IAUKF).

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1.1. Contributions

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Our main contributions are as follows:  A robust NPE algorithm based on an improved AEKF with a novel fault-tolerant NSE is developed, which solves the problem of the existing NPE-AEKF algorithm not being robust and accurate when simultaneously estimating the statistics of the process and measurement noise in a WSN. As far as we know, it is currently the only NPE-AEKF algorithm that does not lose its robustness and that has a positioning accuracy improvement of at least 28% and a convergence rate increase of at least 29% over the state-of-the-art NPEAEKF algorithm.  A robust NPE algorithm based on an improved AUKF with a derivative fault-tolerant NSE is derived from the corresponding RNPE-IAEKF, which addresses the issue that the current NPE-AUKF is not precise and that it can easily lose robustness in the case of simultaneous estimation of the statistical parameters of process noise and measurement noise in a WSN. To the best of our knowledge, it is presently the only NPE-AUKF algorithm that can always remain robust and that improves upon the localization accuracy and the convergence speed of the state-of-the-art NPE-AUKF by at least 32% and 37%, respectively.  The robustness of the RNPE-IAEKF and RNPE-IAUKF algorithms is strictly proved in theory. Moreover, the high positioning accuracy and the fast convergence rate of the RNPE-IAEKF and RNPE-IAUKF algorithms are verified

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against network attack is established [34,35]. Under the actual measurement, the measuring equipment often has some faults that result in the loss of measured data. To reduce the influence of the lost measurement on the operating reliability of the positioning algorithm, a robust algorithm that can tolerate the loss of measurement is developed [36,37]. Likewise, the above-mentioned factors can also cause the Kalman filtering localization algorithm for the WSN to lose robustness [38,39]. For example, an inaccurate parameter of measurement noise easily leads to a large deviation or even divergence of the positioning result of the KF [40,41]. In addition, incomplete measured data can render the KF unable to perform [42,43]. Hence, corresponding robust KFs for WSN localization are proposed. However, compared with the traditional WSN positioning algorithm, there are still a number of special factors that can weaken the robustness of the KF-based localization algorithm. These special factors mainly involve the process noise and the measurement delay [44,45]. The KF needs to utilize the process model to evolve the node's state, but there exists an error, i.e., the process noise, between the actual state and the state evolved by the process model. An inaccurate process noise parameter can greatly increase the positioning error of the KF or even make the KF diverge, which prompts the creation of an adaptive Kalman filter that can perceive the noise parameter [46,47]. Moreover, the KF requires sampling the system measurement at each time step, so the measurement delay will also affect the execution stability of the KF. Because the delay of data can be treated as missing data to some extent, a KF with uncertain measurement is proposed to enhance the robustness [48,49]. In recent years, two classes of AKFs that can perceive the process noise and the measurement noise have been put forward to improve the localization accuracy of the KF algorithm in the case that the process and measurement noise are unknown or time-varying. One embeds the noise covariance estimator into the extended Kalman filter, whose precision can only reach the first-order Taylor series expansion [50,51], and the other introduces a noise covariance estimator into the unscented Kalman filter, whose accuracy can at least achieve the second-order Taylor series expansion [3,52]. However, these two types of AKFs have a common defect in that they can generally only estimate either the process noise or the measurement noise at a given point in time. When the parameters of both the process noise and the measurement noise are estimated simultaneously, the updated noise covariance may lose nonnegative definiteness, which will cause the AKF algorithm to be unable to continue performing [15,16]. Although there has been work to try to ensure the AKF's robustness upon simultaneous sensing of the process and measurement noise, it is at the expense of increasing the positioning error [17].

measurement and the system measurement predicted based on the process and measurement models, as shown in Fig. 1. The form of the KF is defined as follows:

3. Process and measurement models

The position (P) model is a process model utilized to estimate the location of the stationary node by the KF [5]. The state vector in the P model consists of n position components.

 xˆ k  f  xˆ k 1   K  z k  zˆ k  . ˆ  z k  h  f  xˆ k 1  

(1)

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where f    is the state transition function of the system, h    is the measurement function of the system, xˆ k is the state estimate of the system at time step k , z k is the true time-step-k system measurement, zˆ k is the system measurement predicted at time step k , and K is the feedback gain matrix.

The KF corrects the state estimate of the observed system through the feedback of the residual between the actual system

Fig. 1. The structural diagram of the KF.

The state evolution of the system is described by the process model.

xk  f  xk 1   w k .

(2)

where x k is the state vector of the system at time step k , and w k is the time-step-k process noise vector made up of random variables normally distributed with zero mean and covariance matrix Q k . The predicted measurement of the system is generated by the measurement model.

z k  h  xk   v k .

(3)

where z k is the measurement vector of the system at time step k , and v k is the time-step-k measurement noise vector composed of random variables normally distributed with zero mean and covariance matrix R k . 3.1. Process model

x   x1

xn  . T

(4)

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f  xk 1   I n xk 1.

(5)

where I n is the state transition matrix represented by an n  n identity matrix. The position-velocity (PV) model is a process model employed to estimate the location and the speed of the mobile node by the KF [5]. The state vector in the PV model contains n position components and n corresponding velocity components.

x   x1

xn , v1

vn  . T

(6)

The state transition function in the PV model depicts the change of the location and speed of the moving node.

I f  x k 1    n  0n

tk I n  x . I n  k 1

(7)

3.2. Measurement model

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In most WSNs, the RSSI value can be extracted from the transceiver of the node. The relationship between the RSSI and the distance can be expressed by different models [53]. The widely adopted log-normal shadowing path loss model is used as the measurement model to predict the RSSI value in this work [54]. The log-normal model expresses the RSSI as

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p  d   p0  10 log10  d d0   X  dB .

(8)

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where p is the power in decibels received at distance d from the transmitter, p0 is the mean power received at reference distance d 0 (typically 1 m),  is the path loss exponent, and X  dB is the random variable normally distributed with zero mean and standard deviation  dB . The measurement vector in the log-normal model includes the RSSIs measured by different anchor nodes.

z   pAN1

T

pANL  .

(9)

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where L is the number of anchor nodes. The measurement function in the log-normal model predicts the RSSI between the target node and the anchor node.

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 p0  10 log10 x k , x AN d 0  1   . h  xk       p0  10 log10 x k , x ANL d 0   

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4.1. Existing AEKF The AEKF is a combination of the NSE and the EKF. The EKF iteratively tracks the state of the system through state prediction and state correction. Step 1. State prediction

xˆ k k 1  f  xˆ k 1  . Fk 

f . x xˆ k 1

(11)

Pk k 1  Fk Pk 1FkT  Qk .



(10)

where  is the operator of the Euclidean distance between two nodes.

(12)

(13)

where xˆ k 1 is the time-step-k-1 state estimate of the system, xˆ k k 1 is the system state predicted at time step k , Fk is the Jacobian matrix of f    computed around xˆ k 1 , Pk 1 is the time-step-k-1 state estimate error covariance matrix, and Pk k 1 is the expected state estimate error covariance matrix at time step k . Step 2. State correction

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where tk is the time interval between time step k  1 and time step k , and 0 n is an n  n matrix with all 0 entries.

4. Proposed RNPE-IAEKF

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The state transition function in the P model reflects the evolution of the location of the static node.



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(14)

h . x xˆ k k 1

(15)

zˆ k k 1  h xˆ k k 1 .

Hk 

Sk k 1  Hk Pk k 1HTk  R k .

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K k  Pk k 1HTk Sk1k 1.

(17)

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xˆ k  xˆ k k 1  K k z k  zˆ k k 1 .

(18)

Pk  Pk k 1  K k Hk Pk k 1.

(19)

where zˆ k k 1 is the system measurement predicted at time step k , H k is the Jacobian matrix of h    evaluated around xˆ k k 1 , S k k 1 is the expected measurement error covariance matrix at time step k , K k is the time-step-k Kalman gain matrix, xˆ k is the corrected state estimate, and Pk is the corrected state estimate error covariance matrix. If the process noise covariance matrix Q and the measurement noise covariance matrix R are unknown or time-varying, it will be very difficult for the EKF to accurately estimate the system state. The solution to this problem is to use the NSE to online estimate Q and R in the process of the EKF iterating the system state. Currently, the NSE widely used in the AEKF is an unbiased NSE [14].

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 k  z k  zˆ k k 1.

(20)

dk  1  b  / 1  bk 1  .

(21)

R k 1  1  dk  R k  dk  k  kT  Hk Pk k 1HTk  .

(22)

Qk 1  1  dk  Qk  dk K k  k  kT KTk  Pk  Fk Pk 1FkT  .

(23)

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R k 1 if R k 1 is positive definite   1  d k  R k  d k  k  kT  .  otherwise 

(24)

Q k 1 if Q k 1 is nonnegative definite   1  d k  Q k  d k K k  k  kT K Tk  .  otherwise 

(25)

Fig. 2. The structural diagram of the improved AEKF.

The new fault-tolerant NSE includes a modified biased NSE that ensures the NPE-AEKF’s robustness and meanwhile retains most covariance correction items in the unbiased NSE, whose derivation process is in Appendix A.

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R k 1

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where  k is the residual between the real system measurement and the system measurement predicted at time step k , d k is the time-step-k amnestic factor, and b is a forgetting factor in the range of 0.95 and 0.995, which is set to 0.96 in this paper. However, the matrix subtraction operations in the unbiased NSE usually cause Q to lose the nonnegative definiteness and R to lose the positive definiteness, which will weaken the robustness of the NPE-AEKF. Thus, a fault-tolerant NSE that ensures that the NPE-AEKF is robust is proposed, and it contains an unbiased NSE and a biased NSE [17].

R k 1

R k 1 if R k 1 is positive definite   1  d k  R k  d k diag   k  kT   H k Pk HTk  .  otherwise 

(26)

Q k 1

Q k 1 if Q k 1 is nonnegative definite  1  d k  Q k  .  d k diag(K k  k  kT K Tk )  K k S k k 1K Tk    otherwise 

(27)

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When R computed by the unbiased NSE is not positive definite, it will be recomputed by the biased NSE. Similarly, when Q calculated by the unbiased NSE is not nonnegative definite, it will also be recalculated by the biased NSE.

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4.2. Novel fault-tolerant NSE for an improved AEKF

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Although the fault-tolerant NSE can guarantee the robustness of the NPE-AEKF, the biased NSE abandons most covariance items of the state estimate error and the measurement error to correct Q and R in the unbiased NSE. The missing covariance items can result in the inaccurate estimation of Q and R, and they can lower the accuracy of the NPE-AEKF. To minimize the impact of the fault-tolerant NSE on the precision of the NPE-AEKF, we propose a novel fault-tolerant NSE for the improved AEKF. The structure of the improved AEKF is depicted in Fig. 2.

where diag    is the function returning a matrix that is made up of the diagonal elements of the operated matrix and has the same dimension as the operated matrix. 4.3. Robustness proof of the RNPE-IAEKF algorithm Definition 1. In this paper, the robustness of an algorithm is defined as the absence of any faults for which the computed values of all the parameters violate their constraints or make the algorithm unable to continue running during the calculation. Theorem 1. The node position estimation algorithm based on the improved adaptive extended Kalman filter with the novel fault-tolerant noise statistic estimator is robust. Proof. Please see Appendix B.

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4.4. The pseudocode of the RNPE-IAEKF algorithm Algorithm 1 describes the detailed implementation steps of the IAEKF-based RNPE algorithm. Algorithm 1 RNPE-IAEKF Input: initial state estimate xˆ init , initial state estimate error covariance Pinit , initial process noise covariance Qinit , initial measurement noise covariance R init Output: final estimated state xˆ 1 Parameter initialization by xˆ init , Pinit , Qinit , and R init 2 k←1 3 while k <= maximum time step do 4 State prediction based on Equations (11) to (13) 5 State correction according to Equations (14) to (19) 6 Noise statistic estimation using Equations (26) to (27) 7 k←k+1 8 end while 9 return xˆ

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xˆ k k 1  Wi  m X i , k k 1 . i 0

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X k k 1  f  X k 1  .

(34)

 n    Pk 1  .

where X k k 1 is the point set propagated by the state transfer function at time step k , and the piecewise expressions of the mean weight Wi  m and the covariance weight Wi  c  are defined as follows.

  m  W0  n   c   1   2    W0  n    1   m c , i  1, 2, Wi  Wi  2 n     

.

(35)

, 2n

(28)

where  is a nonnegative parameter used to incorporate part of the prior distribution knowledge. It is set to 2 in this paper because this value is optimal for the normal distribution [58]. Step 4. State correction

(29)

Z k k 1  h X k k 1 .

where xˆ 0 is the expected mean of the initial state x0 , and P0 is the expected covariance matrix of the initial state estimate error. Step 2. Calculate sigma points

xˆ k 1 

 n    Pk 1 i  n

, i  1, 2,

where    2  n     n is a composite scaling parameter, n is the dimensionality of the state vector,  is a positive parameter controlling the spread of the sigma point, and  is a secondary scaling parameter. In this paper,  is set to 0.5, and  is set to 3  n [57]. Step 3. State prediction

The UKF is a linearization-free filter, that is, it does not require calculation of the Jacobian matrices of the nonlinear functions in the process and measurement models [55,56]. Firstly, the UKF captures the mean and covariance of the state distribution by selecting a number of sampling points, i.e., sigma points, uses the state-transition and measurement functions to propagate the sigma points, and finally recounts the mean and covariance of the propagated sigma points. The implementation steps of the UKF algorithm are as follows. Step 1. Initialization

X k 1   xˆ k 1 

 n    Pk 1 i

i 0

5.1. Existing AUKF

T P0  E  x0  xˆ 0  x0  xˆ 0   .  

 

Pk k 1  Wi  c   X i , k k 1  xˆ k k 1   X i , k k 1  xˆ k k 1   Qk .

5. Proposed RNPE-IAUKF

xˆ 0  E  x0 .

  X 0, k 1  xˆ k 1   X i, k 1  xˆ k 1   X  xˆ k 1   i, k 1





(36)

2n

zˆ k k 1  Wi  m Zi , k k 1 .

(37)

i 0

2n

(30)

where the piecewise expression of the sigma point set X k 1 is defined as

Pzz , k k 1  Wi  c   Zi , k k 1  zˆ k k 1   Zi , k k 1  zˆ k k 1   R k . T

(38)

i 0

2n

Pxz , k k 1  Wi  c   X i , k k 1  xˆ k k 1   Zi , k k 1  zˆ k k 1  .

(39)

K k  Pxz , k k 1Pzz1, k k 1 .

(40)

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i 0

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

xˆ k  xˆ k k 1  K k z k  zˆ k k 1 .

(41)

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Pk  Pk k 1  K k PxzT , k k 1.

(42)

  z k  zˆ k k 1.

(43)

dk  1  b  / 1  bk 1  .

(44)

Fig. 3. The structural diagram of an improved AUKF.





 Wi  c  Zi , k k 1  zˆ k k 1 Zi , k k 1  zˆ k k 1 i 0

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 . 

(45)

Q k 1  1  d k  Q k  d k K k  k  kT K Tk  Pk 2n



 Wi  c  X i , k k 1  xˆ k k 1

 X

 xˆ k k 1

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The derivative fault-tolerant NSE contains a novel biased NSE derived from the modified biased NSE in the IAEKF. Likewise, it also retains most covariance correction information. The derivative fault-tolerant NSE for the IAUKF is shown as follows, whose derivation process is in Appendix C.

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R k 1  1  d k  R k  d k  k  kT 2n

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where Z k k 1 is the point set propagated by the measurement function at time step k , Pzz , k k 1 is the expected measurement error covariance matrix, and Pxz , k k 1 is the expected measurement error cross-covariance matrix. The AUKF can be achieved by embedding the NSE into the UKF. It combines the advantage of the UKF with that of the NSE, i.e., it can be applied to a non-differentiable function to be more applicable than the AEKF and adapt to the noise statistic change. The following is a derivative unbiased NSE for the AUKF obtained from the unbiased NSE in the corresponding AEKF [14].

 . 

(46)

R k 1

R k 1 if R k 1 is positive definite  T 1  d k  R k  d k diag   k  k     2n c       Wi Z i , k k 1  zˆ k k 1 Z i , k k 1  zˆ k k 1  i 0  otherwise 

5.2. Derivative fault-tolerant NSE for an improved AUKF

Q k 1

To reduce the error of the AUKF and ensure its robustness, we propose an innovative derivative fault-tolerant NSE for the improved AUKF. The structure of the improved AUKF is as drawn in Fig. 3.

Q k 1 if Q k 1 is nonnegative definite  1  d k  Q k  .  d k diag(K k  k  kT K Tk )  K k Pzz , k k 1K Tk    otherwise 

5.3. Robustness proof of the RNPE-IAUKF algorithm

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Like the unbiased NSE in the AEKF, the derivative unbiased NSE for the AUKF can also cause Q to lose the nonnegative definiteness and R to lose the positive definiteness. At present, only the fault-tolerant NSE in the unimproved AEKF can be utilized to guarantee the robustness of the AUKF. However, it will make the AUKF produce a large estimation error.



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.  

(47)

(48)

Theorem 2. The node position estimation algorithm based on the improved adaptive unscented Kalman filter with the derivative fault-tolerant noise statistic estimator is robust. Proof. Please see Appendix D. 5.4. The pseudocode of the RNPE-IAUKF algorithm Algorithm 2 describes the execution steps of the IAUKFbased RNPE algorithm in detail.

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assumed initial values, that is, all the diagonal elements of Qinit are 10−4, Rinitʹs diagonal elements are all 10−2, the diagonal elements of Pinit are the same as those of Rinit, and their nondiagonal elements are all zero. xinit is provided by the multilateration method, the mean received signal strength p0 measured at the reference distance is −62, and the signal attenuation exponent α is 2, which is obtained through model fitting based on the mean value of the RSSI measured at different distances. For each experiment, every anchor node measures 100 RSSIs of the target node, that is, the number of time steps for location estimation is 100.

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Algorithm 2 RNPE-IAUKF Input: initial state estimate xˆ init , initial state estimate error covariance Pinit , initial process noise covariance Qinit , initial measurement noise covariance R init Output: final estimated state xˆ 1 Parameter initialization by xˆ init , Pinit , Qinit , and R init 2 k←1 3 while k <= maximum time step do 4 Sigma point calculation by Equations (30) 5 State prediction based on Equations (32) to (34) 6 State correction according to Equations (36) to (42) 7 Noise statistic estimation using Equations (47) to (48) 8 k←k+1 9 end while 10 return xˆ 6. Practical experiment and numerical simulation

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6.1. Experiment results and analysis

Fig. 5. The node deployment in the static scenario experiment.

In this paper, the root mean square error (RMSE) between the estimated and real positions of the target node is used to evaluate the accuracy of the positioning algorithm.

RMSE=

 x1  xˆ1    x2  xˆ2  2

2

.

(49)

where  x1 , x2  and  xˆ1 , xˆ2  are the true coordinates and the estimated ones, respectively. Fig. 6(a) shows the RMSEs of the node positions estimated by the NPE-EKF, NPE-AEKF, and RNPE-IAEKF when the target node is placed at (0.65,1.50). Correspondingly, the RMSEs of the NPE-UKF, NPE-AUKF, and RNPE-IAUKF are exhibited in Fig. 6(b).

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In this subsection, we verify the validity of the proposed algorithms in a real WSN testbed made up of target nodes that are mounted on a mobile robot and send a radio signal every 0.1 s and anchor nodes that are used to measure the RSSI of the signal emitted by the target node, as shown in Fig. 4. The hardware platforms that the target node and the anchor node use are off-the-shelf MicaZ Motes, whose programs are developed based on the ZigBee network protocol and the TinyOS operating system.

Fig. 4. The WSN testbed for the experiment.

In the experiment, 8 anchor nodes are evenly deployed around a region of 5 m × 5 m, and 20 target nodes are randomly placed in 100 different grids with an edge length of 0.5 m, in turn, to confirm the effectiveness of the proposed algorithms, as shown in Fig. 5. Because the statistical parameters of the process noise and the measurement noise related to the KF are unknown, all the algorithms are evaluated based on the same

Fig. 6. The RMSEs produced by the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF upon estimating the position of the target node located at (0.65,1.50) in a static scenario experiment.

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respectively. To compare the positioning accuracy of the algorithm under different anchor nodes, we change the number of anchor nodes from 3 to 8 in the experiment, as shown in Fig. 8.

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As can be observed from Fig. 6(a), the positioning error of the NPE-EKF basically remains unchanged in the entire process of position estimation. That is because the accuracy of the NPE-EKF without the noise-adaptive capability completely depends on the initially assigned values of the noise statistic parameters. The NPE-AEKF has an RMSE curve with slow convergence and large errors. The main reason is that the ability to correct the noise parameter for its NSE is too weak. As shown in Fig. 6(a), when the initial values of the noise statistics seriously deviate from the true ones, the accuracy of the NPEAEKF is even worse than that of the NPE-EKF. However, the RNPE-IAEKF always maintains an RMSE curve with fast convergence and small errors whether the initial value is accurate or not. From Fig. 6(b), we can see that the nearly invariable noise statistics in the experiment result in the NPE-UKF's RMSE remaining constant throughout the entire process of location estimation. In Fig. 6(b), the final RMSE of the NPEAUKF is larger than that of the NPE-AEKF, mainly attributed to how the NPE-AUKF is sensitive to the noise statistics, making it easier to affect via an inaccurate noise parameter than the NPE-AEKF. Compared with the NPE-UKF and the NPEAUKF, the RNPE-IAUKF has higher localization accuracy.

Fig. 8. Different numbers of anchor nodes in the experiment.

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Fig. 7. The average RMSEs and the convergence times of the NPE-EKF, NPEAEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF in the static scenario experiment.

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In Fig. 7(a), the average RMSEs obtained from the NPEEKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF after performing 500 static scenario experiments are described, which are 1.03 m, 1.19 m, 0.65 m, 1.04 m, 1.29 m, and 0.58 m, respectively. From the experimental results, upon simultaneous estimation of process noise and measurement noise, the positioning accuracies of the NPE-AEKF and NPE-AUKF become even worse compared with the NPEEKF and the NPE-UKF. In contrast, the RNPE-IAEKF increases upon the localization accuracy of the NPE-EKF and NPE-AEKF by 36% and 45%, while the RNPE-IAUKF reduces the position estimate errors of the NPE-UKF and NPEAUKF by 44% and 55%. Fig. 7(b) reveals the mean convergence times of the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF upon the estimated node location converging to the vicinity of the true value. Experimental results show that the convergence rate of the RNPE-IAEKF is 55% and 46% faster than those of the NPEEKF and NPE-AEKF, respectively, and the RNPE-IAUKF is 59% and 53% quicker than the NPE-UKF and the NPE-AUKF,

Fig. 9. The average RMSEs produced by the NPE-EKF, NPE-AEKF, RNPEIAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF under different numbers of anchor nodes in the static scenario experiment.

In Fig. 9, the mean RMSEs obtained through each algorithm performing 500 static scenario experiments under different numbers of anchor nodes are depicted. We can observe from

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accurate, and so the localization error of the RNPE-IAEKF is smaller than those of the NPE-EKF and NPE-AEKF, even in the case of node maneuvering. In Fig. 11(b), the RMSEs of the node locations estimated by the NPE-UKF, NPE-AUKF, and RNPE-IAUKF algorithms are exhibited. Similar to the experimental results of the NPE-EKF, NPE-AEKF, and RNPE-IAEKF, target maneuvering can also lead to an error increase of the NPE-UKF, NPE-AUKF, and RNPE-IAUKF. However, the RNPE-IAUKF possesses the most accurate NSE, which allows its position estimate error to remain the smallest at all times.

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the figure that, when the anchor nodes are increased from 3 to 8, the errors of all the algorithms are decreased. Because the NPE-AEKF and the NPE-AUKF have an inaccurate NSE, their localization errors are always larger than those of the RNPEIAEKF and RNPE-IAUKF algorithms. In contrast, the RNPEIAEKF and the RNPE-IAUKF maintain a much smaller error relative to the other corresponding algorithms.

Fig. 12. The average RMSEs and the convergence times of the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF in the mobile scenario experiment.

Fig. 10. The trajectory of the moving target node in a mobile scenario experiment.

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To verify the effectiveness of the proposed algorithms for the maneuvering target, we control the robot carrying the target node such that it makes 500 curve movements at randomly selected starting points. In each curve motion, the target node moves at a speed of 1 m/s for 100 time steps and randomly performs 3 turning maneuvers.

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Fig. 11. The RMSEs generated by the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF in a mobile scenario experiment.

Fig. 11(a) describes the RMSEs of the node positions estimated by the NPE-EKF, NPE-AEKF, and RNPE-IAEKF algorithms in a target-tracking experiment. As found in the figure, when the target node maneuvers at three turns, the positioning errors of all the algorithms are increased. The main reason is that the target maneuvering often causes a process model mismatch, which will make the position estimated by the algorithm using this model greatly deviate from the true one. In addition, the noise parameter deviation can also enlarge the location estimation error of the node. Compared to the NPE-EKF and the NPE-AEKF, the noise statistics of the RNPE-IAEKF are more

Fig. 13. The average RMSEs produced by the NPE-EKF, NPE-AEKF, RNPEIAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF under different numbers of anchor nodes in the mobile scenario experiment.

In Fig. 12(a), the average RMSEs of the NPE-EKF, NPEAEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPEIAUKF algorithms running 500 curve movement experiments are presented, which are 1.55 m, 1.22 m, 0.54 m, 1.55 m, 1.15 m, and 0.53 m, respectively. From the experimental data, when estimating the location of the maneuvering target node, the RNPE-IAEKF greatly improves upon the positioning accuracy of the NPE-EKF and NPE-AEKF by 65% and 55%, and the RNPE-IAUKF reduces the localization errors of the NPE-UKF and NPE-AUKF by 65% and 53%. Fig. 12(b) describes the average numbers of time steps required for the convergence of the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-

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6.2. Simulation results and analysis

Fig. 15. The RMSEs produced by the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF upon estimating the location of the target node located at (10.05,-8.55) in a static scenario simulation.

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In this subsection, we employ extensive simulations using different parameter settings to further confirm the superiorities of the proposed algorithms. In the static scenario simulation, eight anchor nodes are uniformly distributed around an area of 40 m by 40 m, and the target node sending a radio signal every second is randomly deployed in 400 grids with an edge length of 2 m, as illustrated in Fig. 14. In the actual WSN localization scenario, the statistical parameters of the process noise and the measurement noise for the KF are generally unknown, so the assumed initial values of parameters are usually inaccurate. In view of this, the initial measurement noise covariance is set to 102  102  while the true covariance of the simulated measurement noise is 104 , ,104  where  represents diagonal matrix notation. The initial covariance of the process noise and state estimate error are 104  104  and 1, ,1 respectively. The values of the parameters for the measurement model come from a real WSN positioning experiment [59], i.e., the signal attenuation exponent  is 2.62 and the mean received signal strength p0 at the reference distance is −45.55.

In Fig. 15(a), the RMSEs of the NPE-EKF, NPE-AEKF and RNPE-IAEKF algorithms after estimating the location of the target node located at (10.05,-8.55) are revealed. As seen from the figure, the estimated error of the NPE-EKF has a slow decline compared with the NPE-AEKF and the RNPE-IAEKF. This is because the NPE-EKF has a constant and inaccurate noise parameter. Relative to the NPE-EKF, the NPE-AEKF and the RNPE-IAEKF have RMSE curves with fast convergence. The major reason is that they possess the ability to correct the imprecise noise parameter. However, the ability to correct the noise parameter of the NPE-AEKF is too weak, so the error of the NPE-AEKF is larger than that of the RNPE-IAEKF. When the values of the noise statistic parameters greatly deviate from the true ones, the positioning accuracy of the NPEAEKF is even worse than that of the NPE-EKF. In summary, with its ability to accurately perceive the noise statistics, the RNPE-IAEKF maintains the smallest RMSE throughout the localization. Correspondingly, the RMSEs of the NPE-UKF, NPE-AUKF, and RNPE-IAUKF algorithms are exhibited in Fig. 15(b). From the figure, we can observe that the RMSE of NPE-AUKF is much bigger than those of the NPE-UKF and RNPE-IAUKF, mainly attributed to how the NPE-AUKF with the poor NSE is easier to affect via an inaccurate noise parameter. Because the RNPE-IAUKF has a good NSE and no linearization error, its localization accuracy is highest among all the evaluated algorithms.

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AUKF, and RNPE-IAUKF. The results of the experiments indicate that the RNPE-IAEKF improves upon the convergence rates of the NPE-EKF and NPE-AEKF by 55% and 43% and that the RNPE-IAUKF reduces the convergence times of the NPE-UKF and NPE-AUKF by 56% and 50%. The average RMSEs of the NPE-EKF, NPE-AEKF, RNPEIAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF algorithms performing 500 mobile scenario experiments with different numbers of anchor nodes are shown in Fig. 13. As is displayed in the figure, with the increase of anchor nodes, the positioning errors of all the algorithms present a downward trend from beginning to end. However, when the anchor nodes reach a certain scale, this downward trend starts to slow, which implies that the effect of increasing the anchor nodes to improve the localization accuracy of the algorithm is limited.

Fig. 16. The average RMSEs and the convergence times of the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF in the static scenario simulation. Fig. 14. The node deployment in the static scenario simulation.

In Fig. 16(a), the average RMSEs of the NPE-EKF, NPEAEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-

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Fig. 18. The average RMSEs produced by the NPE-EKF, NPE-AEKF, RNPEIAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF under different numbers of anchor nodes in the static scenario simulation.

To further verify the performance of tracking the maneuvering target of the proposed algorithms, we carry out 500 simulations of mobile scenarios with different parameter settings. In each simulation, the target node begins to move from a randomly selected starting point and then performs 3 random turning maneuvers. The entire process of each movement lasts 100 time steps, and the speed of the node is 1 meter per second. Fig. 19 depicts the movement trajectory of the target node in a mobile scenario simulation.

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IAUKF when implementing 500 static scenario simulations are depicted, which are 0.81 m, 0.91 m, 0.52 m, 0.70 m, 1.15 m, and 0.45 m, respectively. Similar to the result of the static scenario experiment, when the process noise and the measurement noise are estimated at the same time, the localization errors of the NPE-AEKF and NPE-AUKF even exceed those of the NPE-EKF and NPE-UKF. From the averaged result, the RNPE-IAEKF improves upon the positioning precision of the NPE-EKF and NPE-AEKF by 36% and 43%, and the RNPEIAUKF reduces the localization errors of the NPE-UKF and NPE-AUKF by 35% and 60%. Fig. 16(b) reveals the average convergence time required by the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF. The results show that the convergence time of the RNPEIAEKF is 66% and 37% less than that of the NPE-EKF and NPE-AEKF, respectively, and that of the RNPE-IAUKF is 66% and 42% smaller than that of the NPE-UKF and the NPEAUKF, respectively.

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Fig. 17. Different numbers of anchor nodes in the simulation.

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To confirm that the proposed algorithms still hold the best performance for the scenario with different parameter configurations under different numbers of anchor nodes, we add 4 different quantities of anchor nodes, as shown in Fig. 17. In Fig. 18, the average RMSEs generated by all the algorithms executing 500 static scenario simulations under different numbers of anchor nodes are revealed. As we can see from the figure, when the number of anchor nodes is equal to 20 or 24, the positioning errors of all the algorithms become relatively larger. This is because the anchor nodes deployed in the center of the area are very close to the target node, which makes the state estimation error components of these anchor nodes in the KF tend to zero and causes a computer with limited calculation accuracy to generate the rounding error, eventually leading to larger and larger accumulation error of the KF. However, even in such a case, the localization accuracy of the RNPE-IAEKF and RNPE-IAUKF is still better than that of the other corresponding algorithms.

Fig. 19. The trajectory of the moving target node in a mobile scenario simulation.

Fig. 20(a) describes the RMSEs formed by the NPE-EKF, NPE-AEKF, and RNPE-IAEKF algorithms in a target-tracking simulation. From the figure, upon the target node performing the turning maneuver, the localization errors of the NPE-EKF, NPE-AEKF, and RNPE-IAEKF become relatively larger. Nevertheless, the error of the RNPE-IAEKF is much smaller than those of the other two algorithms in the case of node maneu-

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Fig. 20. The RMSEs generated by the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF in a mobile scenario simulation.

figure, with the increase in the number of anchor nodes, the localization errors of all the algorithms first decline and then rise. The major reason is that the anchor nodes located at the inside of the region easily make the position estimation error components related to these anchor nodes in the KF tend to zero, which causes the computer to produce the rounding error, ultimately leading to the larger location estimate deviation of the KF. In the presence of the side effect caused by inappropriate node placement, the RNPE-IAEKF and the RNPE-IAUKF still maintain relatively slow error growth compared with the other corresponding algorithms.

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vering, which shows that the RNPE-IAEKF has better performance in tracking the maneuvering target. Fig. 20(b) reveals the RMSEs produced by the NPE-UKF, NPE-AUKF, and RNPE-IAUKF algorithms in the same target tracking simulation. Similar to the simulation results of the NPE-EKF, NPE-AEKF, and RNPE-IAEKF, target maneuvering still results in larger errors of the NPE-UKF, NPE-AUKF, and RNPE-IAUKF. However, the RNPE-IAUKF with the most accurate NSE maintains the best localization accuracy at all times.

Fig. 21. The average RMSEs and the convergence times of the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF in the mobile scenario simulation.

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In Fig. 21(a), the average RMSEs of the NPE-EKF, NPEAEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPEIAUKF algorithms executing 500 curve movement simulations are presented, which are 1.53 m, 0.62 m, 0.44 m, 1.52 m, 0.51 m, and 0.34 m, respectively. From the simulation results of estimating the position of the maneuvering target, the RNPEIAEKF raises the localization accuracy of the NPE-EKF and NPE-AEKF by 71% and 28%, and the RNPE-IAUKF decreases the positioning errors of the NPE-UKF and NPE-AUKF by 77% and 32%. Fig. 21(b) displays the mean convergence times required by the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPEUKF, NPE-AUKF, and RNPE-IAUKF. Simulation results indicate that the convergence time of the RNPE-IAEKF is 56% and 29% shorter than that of the NPE-EKF and NPE-AEKF, respectively, and that of the RNPE-IAUKF is 64% and 37% less than that of the NPE-UKF and the NPE-AUKF, respectively. Fig. 22 presents the mean RMSEs obtained after all the evaluated algorithms carry out 500 mobile scenario simulations under different numbers of anchor nodes. As observed from the

Fig. 22. The average RMSEs produced by the NPE-EKF, NPE-AEKF, RNPEIAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF under different numbers of anchor nodes in the mobile scenario simulation.

6.3. Computational complexity analysis All the algorithms are evaluated via MATLAB implementations executed on a computer equipped with a 1.86-GHz CPU. The average execution times per time step of the NPE-EKF, NPE-AEKF, RNPE-IAEKF, NPE-UKF, NPE-AUKF, and RNPE-IAUKF algorithms are calculated according to all the results obtained from the experiments and the simulations, and the respective values are 0.58 ms, 0.96 ms, 1.05 ms, 2.18 ms, 3.58 ms, and 3.62 ms. Although the average running time per time step of the RNPE-IAUKF grows a little relative to the RNPE-IAEKF, it is still much less than the signal transmission cycle of 100 ms in the experiment. Thus, it can be observed that both the RNPE-IAEKF and the RNPE-IAUKF can meet the real-time demand of practical WSN localization. 7. Conclusions In this paper, we proposed a robust NPE algorithm based on an improved AEKF with a novel fault-tolerant NSE for WSNs and its derivative, i.e., a robust NPE algorithm based on an improved AUKF with an innovative derivative fault-tolerant NSE for WSNs. Our proposed algorithms greatly enhance the ability to adapt to the process and measurement noise of the

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By Equation (17),

Hk Pk k 1  Sk k 1KTk .

(A.6)

Substitute Equation (A.6) into Equation (A.5),

Qk 1  1  d k  Qk  d k diag(xˆ k  xˆ k k 1 ) 2  K k S k k 1K Tk  .

(A.7)

By Equation (18) and Equation (20),

(xˆ k  xˆ k k 1 )2  K k  k  kT KTk .

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state-of-the-art NPE algorithms based on the KF such that the robustness and accuracy of those algorithms are improved. In addition, the robustness of the two proposed algorithms is strictly proved theoretically. The results of both practical experiments and numerical simulations show that, no matter how the static target node is placed, and whether the mobile target node maneuvers or not, the positioning accuracy of the proposed algorithms is at least 28% and 32% higher than that of the two existing corresponding algorithms, respectively, and the convergence rate of the proposed algorithms is at least 29% and 37% faster.

(A.8)

Equation (A.8) is substituted into Equation (A.7),

Acknowledgments

Qk 1  1  d k  Qk

Appendix A. Derivation process of NSE for the IAEKF

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(A.1)

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Qk 1  1  d k  Qk



M

The modified biased NSE is acquired on the basis of a slightly modified unbiased NSE [59].Equation Section (Next)

R k 1  1  dk  R k  dk diag z k  zˆ k k 1 





 d k diag(xˆ k  xˆ k k 1 ) 2  Pk  Pk k 1  Q k  .  

(A.2)

CE

PT

The Q estimated by the modified unbiased NSE still carries the risk of losing the nonnegative definitiveness. To ensure that the estimated Q is nonnegative definitive and loses the correction information as little as possible, we alter the formula for estimating the Q. By Equation (19),

AC

Pk  Pk k 1  K k Hk Pk k 1.

 d k diag(K k  k  kT K Tk )  K k S k k 1K Tk  .

(A.3)

(A.9)

Substitute Equation (20) into Equation (A.1),

R k 1  1  dk  R k  dk diag  k  kT   Hk Pk HTk  .

AN US

This research is financially supported by the National Natural Science Foundation of China (Nos. 60873026 and 61272418), the National Science and Technology Support Program of China (No. 2012BAK26B02), the Future Network Prospective Research Program of Jiangsu Province (No. BY2013095-5-02), and the Lianyungang City Science and Technology Project (Industry Development) (No. CG1420).

(A.10)

Appendix B. Proof of the robustness of the RNPE-IAEKF Proof. The initial R k is positive definite, but it is possible for  k  kT  Hk Pk k 1HTk to be not nonnegative definite, so R k 1 calculated based on Equation (22) can violate the constraint of positive definiteness, even leading to S k 1k 1 becoming unsolvable, i.e., the algorithm terminates. However, diag  k  kT   Hk Pk HTk is always nonnegative definite. Thus, R k 1 recalculated based on Equation (A.10) is positive definite at all times. Similarly, the initial Q k is nonnegative definite, yet K k  k  kT KTk  Pk  Fk Pk 1FkT can lose the nonnegative definiteness, so Q k 1 computed via Equation (23) can violate the constraint of nonnegative definiteness. Nevertheless, diag(K k  k  kT KTk )  K k Sk k 1KTk is nonnegative definite. Therefore, Q k 1 recomputed via Equation (A.9) is nonnegative definite all the time. To summarize, the novel fault-tolerant noise statistic estimator can not cause R and Q to violate the constraint or terminate the algorithm, that is, the node position estimation algorithm based on the improved adaptive extend Kalman filter is robust. Equation Section (Next)

Equation (A.3) is substituted into Equation (A.2),

Qk 1  1  d k  Qk

 d k diag(xˆ k  xˆ k k 1 ) 2  K k H k Pk k 1  Q k  .

(A.4)

Appendix C. Derivation process of NSE for the IAUKF

Because the matrix subtraction can lead the process noise covariance matrix to lose the nonnegative definiteness, we exclude the negative Q in the correction term.

By comparing (17) with (40), the process noise covariance correction item for the biased NSE in the derivative faulttolerant NSE can be found from the corresponding part of the modified biased NSE in the IAEKF.Equation Section (Next)

Qk 1  1  d k  Qk

Sk k 1  Pzz , k k 1.

 d k diag(xˆ k  xˆ k k 1 )  K k H k Pk k 1  . 2

(A.5) Substitute (C.1) into (A.9),

(C.1)

ACCEPTED MANUSCRIPT Author name / 00 (2016) 000–000

However, thus far, there is no expression that corresponds to the measurement noise covariance correction item of the modified biased NSE in the IAEKF and can be substituted into the biased NSE in the derivative fault-tolerant NSE. Thus, converting the correction term of the measurement noise covariance in the modified biased NSE into the corresponding part of the biased NSE in the derivative fault-tolerant NSE is the key. Comparing (16) with (38), Equation (C.3) can be deduced.



2n



H k Pk k 1HTk  Wi  c  Zi , k k 1  zˆ k k 1 Zi , k k 1  zˆ k k 1 i 0



T

.

(C.3)

According to (30), (36), (37) and (C.3), we attain the measurement noise covariance correction term for the biased NSE in the derivative fault-tolerant NSE.

xˆ k k 1 



 n    Pk  .

(C.4)



Z k k 1  h X k k 1 .

(C.5)

2n

zˆ k k 1  Wi  m Zi, k k 1 .

(C.6)

i 0



2n



H k Pk HTk  Wi  c  Zi, k k 1  zˆ k k 1 Zi, k k 1  zˆ k k 1

T

.

(C.7)

M

i 0



2n





ED

where X k k 1 is the recalculated sigma point set based on the corrected state estimate error covariance at time step k . Equation (C.7) is substituted into (A.10),

R k 1  1  d k  R k  d k diag   k  kT 

CE

i 0



T

 . 

(C.8)

PT

 Wi  c  Z i, k k 1  zˆ k k 1 Z i, k k 1  zˆ k k 1

Appendix D. Proof of the robustness of the RNPE-IAUKF Proof. From Theorem 1, it is impossible for the novel faulttolerant noise statistic estimator to lead R and Q to break the constraint or to interrupt the algorithm. Because the derivative fault-tolerant noise statistic estimator for the improved adaptive unscented Kalman filter is derived from the novel fault-tolerant noise statistic estimator in the improved adaptive extend Kalman filter, it also cannot do that, i.e., it is impossible to cause the algorithm to lose the robustness. Thus, the node position estimation algorithm based on the improved adaptive unscented Kalman filter with the derivative fault-tolerant noise statistic estimator is robust.

AC

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X k k 1   xˆ k k 1 

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