Efficient multi-sensor path scheduling for cooperative target tracking

: This study deals with path scheduling problem of cooperative target tracking by multiple sensors with bearings only measurements. First, the authors derive a closed-form expression of the determinant (D-optimality criterion) of the Fisher information matrix (FIM) as the cost function, which contains the knowledge of the target and the locations of sensors. Second, a penalty function is introduced to modify the cost function for threats avoidance and physics constraints are applied to limit directions of sensors. Then, an efficient strategy based on steepest descent is proposed to solve the optimisation problem. Finally, the effectiveness of the proposed algorithm is demonstrated both in localisation of a stationary target and tracking a moving target. Simulation results show the trajectories of sensors for cooperative target tracking are almost identical to a grid-based search method; however, the computational complexity is reduced by several orders of magnitude.


Introduction
The availability of real-time high-accuracy tracking is essential for current and future wireless applications. Reliable tracking and navigation with mobile sensors is a critical component for a diverse set of applications including logistics, security tracking, medical services, search and rescue operations, control of home appliances, automotive safety and military systems [1,2]. In recent years, the demand for moving sensors has increased and the research community has been called upon to solve the many issues that arise from using sensor network.
Considerable effort has been dedicated to moving sensors path scheduling with numerous approaches [3]. Generally, path scheduling of cooperative tracking by multiple sensors is implemented based on the information theory [4,5]. The decision processes are determined by minimising or maximising cost function [6,7] associated with measurements. In [8], an approach of maximise the Fisher information matrix (FIM) is proposed to track known target, which is the inverse of Cramer-Rao lower bound (CRLB). In [9], the posterior CRLB (PCRLB) is employed as the cost function obtained from the measurements of multiply sensors with bearings only measurements to find the optimal trajectories. This literature also introduces a search technique to compute the inverse of PCRLB over a grid, and the simulation results show that performances of multi-step scheduling is better than scheduling of single step. More recently, the value of information gain is present in [10] as the objective function while tracking target. In [11], a receding horizon approach is presented to predict the FIM for the potential course of action set. An information theoretic approach in [12] is proposed to control the sensor to search a single stationary target by minimising the entropy of the target distribution. Three reward functions under two information criteria are discussed in [13]. This paper also proposes a grid-based search method to solve the optimisation problem. The literature [14] introduces a cost function related to global position error covariance of cooperatively sensors, which also derives the analytical lower and upper bounds of the target and sensors localisation errors. In [15], a greedy algorithm is developed for emitter localisation using FIM. The steering algorithm is used to determine the next positions of sensors by maximising the determinant of an approximated FIM, which shows that it can get accurate localisation performance.
In this paper, we address the problem of optimal motion scheduling of multiple moving sensors for cooperative tracking with bearings only measurements. Our contributions in this work offers much valuable flexibility not found in existing approaches. First, a closed-form expression of the determinant of the FIM is derived and it is used as the cost function. Second, we introduce a penalty function to modify the cost function for threats avoidance and employ turning angle constraints to limit directions of sensors. Third, an efficient strategy based on steepest descent is proposed to solve the optimisation problem. Finally, we verify the effectiveness of the proposed strategy by comparing it with a grid-based search method.
The remainder of this paper is organised as follows. In Section 2, we describe the target-tracking model in bearings only measurements. In Section 3, we first derive the cost function based on FIM and then introduce the constraints to limit sensors. Section 4 describes the efficient optimisation algorithm based on steepest descent strategy. Section 5 presents the results of the simulation, and finally, Section 6 contains our conclusions.

Target tracking model
As shown in Fig. 1, we consider the problem of a two-dimensional target tracking by n ≥ 1 cooperative moving sensors with bearing only measurements. The moving platform positions at discrete time instants k, k = 1, 2… are denoted by ξ k = p k The dynamical equation of the target is given by where x k = x k , x˙k, y k , y˙k T is the state vector of the target at time step k, which include location and speed of the target in Cartesian coordinates. F is the state transition matrix of the target that explains how current states of the system influences the next step states, which is presented by The process noise v k is contaminated with a Gaussian noise of zero-mean with covariance matrix Q k i.e. v k ∼ N 0, Q k . T s is the time interval between successive measurements. The measurements sequence Z k of cooperative sensors at discrete time k is where the measurement function h x k is described by The measurements noise w k is corrupted with a zero-mean white Gaussian process with covariance matrix R k , i.e. w k ∼ N 0, R k , and R k is expressed as where σ j is the standard deviation of measurements for jth sensor at time k. Furthermore, we assume that the measurement noise processes of the different sensors are independent, which is also not related to processing noise. In this paper, the extended Kalman filter (EKF) algorithm is used to estimate the relative position and velocity of the target. Each sensor uses local measurements to estimate the state of target. Then, these estimates are exchanged and fused to produce global estimates of target. The measurements are combined using the following simple but general fusion relations [16].
where x k i and P k i , i = 1, 2, …, n . are the local state estimates and estimate error covariance matrix. P k fused is the fused estimate error covariance matrix, and x k fused is the fused estimate of the target position. Obviously, the tracking accuracy depends on where the moving sensors cooperative sensing and their trajectories. Therefore, the objective of path scheduling is to find the optimal locations of moving sensors to obtain the best estimates of the target.

Cost function based on fisher information matrix
In this section, we derive the cost function based on FIM, which contains the contributions of all moving sensors measurements and their positions. We also modify the cost function for threats avoidance and apply physics constraints to limit directions of sensors.

Signal model
As the measurement model is independent of target velocity, we only consider target position in the derivation of FIM. We use the target state vector x k = x k , y k to denote the target position component only when the closed-form expression is derived of FIM as follows [13]. Under certain conditions, any estimator has a lower bound of variance, the variance of the estimate can only be greater than or equal to this lower bound, this lower bound is called the Cramer-Rao lower bound. If x k is the unbiased estimate of x k based on the measurement at time k, then The CRLB for the error covariance is defined as the inverse of the FIM G k [17], which is The inequality in (9) means that CRLB k x − G k −1 is a semi-definite matrix. Generally, FIM is calculated by (10) in which α is the observation, x is the position of target which has to be estimated, and p α x is the probability density function of the observations.
If p α x is subject to a Gaussian distribution, it is easy to get FIM [15] by (11) where R k is the covariance matrix of the measurement noises and J is the Jacobian matrix. The measuring error e k is calculated by (12).
where Ẑ k is the vector of bearings only measurements and Z k x is the vector of exact angle values. On the basic of (12), the Jacobian matrix can be written as If the derivation of (13) is calculated by x k i and y k i , thus each component of it can be expressed as where d k i is the distance between target and ith sensor, i.e.
Using (11), (13), and (14), (11) can be rewritten as The most important step of path scheduling is the choice of cost function that requires low real-time computation. To formulate an appropriate cost function, we use the determinant of FIM (product of the eigenvalues of FIM) as the cost function. In optimal experiments theory, it is referred to as D-optimal criterion [14]. A D-optimality design for parameters will minimise the volume of the uncertainty ellipsoid generated by CRLB. The determinant of the FIM is a widespread criterion used for path scheduling [18], and it is more comprehensive than the trace of the FIM [2] (Aoptimality criterion) since it takes into account all the terms of FIM. The cost function at k state based on FIM is Note that the determinant of FIM, abbreviated as Ψ, is a function of the target state, the noise distribution, and the position of the sensors. As we assumed, the standard deviation of measurement noise σ is a constant, the calculation of (16) only requires the knowledge of target location x k , which in practice is approximated by the track estimate x k . In this paper, we use EKF to obtain the local estimate of the target by each tracking sensor, then the distributed fusion relationships (6) and (7) is employed to obtain the fused estimate x k .

Threats and sensor physics constraints
In electronic warfare environment, sensors need to avoid some threats because of a priori knowledge. Due to the inherent physical constraints of sensors, sensors must not perform abrupt changes of direction. Therefore, we need to solve the optimisation problem subject to threats constraints and sensor physics constraints. The objective of path optimisation is determining the trajectories of the mobile sensors to maximise tracking performance. In our case, the cost function need to be maximised is the determinant of FIM. We finally establish the path optimisation problem that needs to be solved in each time step is where ξ k = p k 1 p k 2 , … p k n T is the vector containing the positions of sensors at time k. θ k i is the turning angle of ith sensor at time k.
φ max is the maximum turning angle of each sensor. Υ j is the lth threat location and m is the number of threats. In practice, solving the optimisation problem (17) with constraints is a challenge. Our strategy transforms this constrained optimisation problem into an unconstrained optimisation problem. In order to keep the tracking sensors away from threats, we introduce a penalty function to modify the cost function. The modified cost function including threats constraints is where Ψ k Z x is the modified cost function and ρ j is the jth threat intensity. The larger ρ j the larger threat of Υ j will be. When sensors are far from the threats, then Ψ k Z ≃ Ψ k , i.e. the threats do not make an effect to move sensors trajectories. However, Ψ k Z will become larger when the distance is near, which has a huge influence on moving sensors trajectories.
To address the inherent physical constraints of sensors, we impose the turning angle constraint by (19).
When θ k i > φ max , the ith sensor can only rotate with the maximum steering angle φ max , i.e. θ k i = φ max .We perform the physics constraints last because they have an advantage over other constraints.

Efficient strategy based on steepest descent
In this section, an efficient path scheduling strategy based on steepest descent is proposed and the computational complexity is also analysed. Fig. 2 is the flow chart of efficient path planning algorithm based on steepest descent for cooperative target tracking. Gradient controllers are at the core of our solution, which contains the modified cost function, the calculation of gradient, and turning angle constraints.

Path scheduling strategy based on steepest descent
The next position of moving sensors at time k + 1 is (20) where u k is the control input of moving sensors which include step size and their directions. As the number of cooperative sensors increase, obtaining the control input quickly would be very sophisticated. We propose a strategy based on steepest descent akin to [19] to solve the difficult problem, which implies that the update rule is where η k is a time varying step size, and the direction of each moving sensor is where α k i is the ith sensor direction at time k and it take one of a set of finite values. To implement conveniently, we assume that the speed v k = v 0 1 , v 0 2 , ⋯, v 0 n T of sensors is a constant, where v 0 i is the speed of ith sensor at time k. Thus η k can be expressed as Using (21) and (23), (20) can be rewritten as In our strategy, each sensor first takes measurements to estimate target state, then the fused centre obtain the fused estimates. After that, each sensor maximises the cost function to obtain the gradient direction, which causes rapid increase of cost function. Finally, each sensor updates its path in the gradient direction to catch new measurements.

Computational complexity analysis
In order to prove the effectiveness of our proposed algorithm, we compare it with the grid-based search method [20], which is similar to traversal search. The grid-based search method is that sensors steer themselves to the best course of ℵ directions with respect to their current directions at each frame. Assuming that our approach calculating the trajectories of three sensors at each frame requires I a addition operation and I m multiplication operation, thus the total calculation is I a + I m . However, the total calculation of grid-based search method is ℵ 3 I a + I m , which calculates >10 3 values during each optimisation.

Simulation results
In this section, simulations are designed to verify the performance of the steepest descent strategy both in localisation of a stationary target and tracking a moving target. In order to better reflect the effectiveness, the method in [20] is provided for comparison and we define it as the grid-based search method. As shown in Fig. 3, we use three bearings, only sensors starting from  Fig. 3 shows the trajectories of multiple moving sensors using the proposed strategy and the grid-based search method. From Fig. 3, we see that the three sensors move apart from each other from the initial locations and then close to the target gradually. We also find our method is relatively smoother than the grid-based search method, which indicate our technique is more feasible to actual flight. Moreover, we note that the sensors trajectories of our approach is almost identical to the grid-based search method; however, our strategy requires very little computation.
As shown in Fig. 4, the moving target moves from 1700 m, 1700 m to the northeast at the speed of 30 m/s in x-axis and 10 m/s in y-axis. There are two threats at Υ 1 = 1600 m, 1450 m and Υ 2 = 1800 m, 1300 m with threat intensities ρ 1 = ρ 2 = 50. The maximise turning angle φ max = 30°. Filter initial value is initialised to x 0 = 1725 m, 30 m/s, 1680 m, 10 m/s T and P 0 = 5I. The initial position of the sensors is the same as Fig. 3. Fig. 4a shows typical trajectories of three cooperative sensors track a moving target. We see that cooperative sensors can avoid threats with smooth trajectories, by comparing Figs. 4a with Fig. 4b. That is because when the sensors are close to the threat, the cost function is penalised and the sensors are forced to move away from threats. Fig. 5 depicts the performance comparison of the proposed steepest descent strategy, the grid-based search method, and random search method. Fig. 5a shows the cost of our strategy is close to the grid-based search method, and they are far superior to random search method. We also note that the determinant of the FIM curve increases monotonously, which means cooperative sensors gradually take more available information. Fig. 5b depicts the proposed algorithm takes almost 0.45 ms to solve the optimisation problem for each frame, while exhaustive search spends almost 55 ms, which verifies the effectiveness of our method. Fig. 6 shows the root-mean-squared position error comparison under the three methods. We see that the proposed strategy exhibit an acceptable performance loss with respect to the grid-based search method but a significant performance gain over random search method.

Conclusion
In this paper, we have studied the path scheduling problem for cooperative target tracking by multiply sensors with bearings only measurements. First, we derived a closed-form expression of the determinant of the FIM as the cost function, which contains the knowledge of the target and the locations of sensors. Next, we introduce a penalty function for threats avoidance and employ   turning angle constraints to limit the directions of sensors. Then, an efficient strategy based on steepest descent is proposed to solve the optimisation problem. Finally, we have evaluated the performances of the proposed algorithm both in localisation of a stationary target and tracking a moving target. The results highlight that the proposed strategy exhibit an acceptable performance loss with respect to the grid-based search method but a significant performance gain over random search method with less computational complexity.