# Trajectory Optimisation for Cooperative Target Tracking with Passive Mobile Sensors
###### Abstract  
The paper considers the problem of tracking a moving target using a pair of cooperative bearing-only mobile sensors. Sensor trajectory optimisation plays the central role in this problem, with the objective to minimize the estimation error of the target state. Two approximate closed-form statistical reward functions, referred to as the Expected Renyi information divergence (RID) and the Determinant of the Fisher Information Matrix (FIM), are analysed and discussed in the paper. Being available analytically, the two reward functions are fast to compute and therefore potentially useful for longer horizon sensor trajectory planning. The paper demonstrates, both numerically and from the information geometric viewpoint, that the Determinant of the FIM is a superior reward function. The problem with the Expected RID is that the approximation involved in its derivation significantly reduces the correlation between the target state estimates at two sensors, and consequently results in poorer performance.  
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densities of target, for each possible sensor path and the sensor maneuvers are controlled by the action that maximises the accumulated reward. In general, computing reward functions require the evaluation of future measurements. Of course, it is impossible to give an exact computation of the future reward function since future measurements are inaccessible. One may mitigate this problem by using (conditionally) expected values of the reward at future times, or by using Monte Carlo sampling to numerically evaluate future measurements [15, 16].  
In this paper, we investigate the problem of tracking a moving target via cooperative passive sensors, where the tracking error is a function of sensor trajectories. The focus is on two statistical reward functions for passive sensing, both available in the analytic form. They are referred to as the Expected Renyi information divergence (RID) and the determinant of the Fisher information matrix (FIM). Being available analytically, they are potentially useful for a longer horizon sensor trajectory planning, even on the platforms with limited computational resources. The paper investigates the implications of the approximations involved in derivation of the two reward functions. The analytic expression for the Expected RID is derived assuming a linear-Gaussian case. We show that this approximation significantly reduces the cross-correlation of the tracking error between the two sensor locations, and consequently results in the performance similar to that of the trace of the FIM. On the other hand, the determinant of FIM, where the ground truth is approximated by the prediction of the target state, taking correctly into account the cross-correlation between the two sensors. We also discuss this problem from the information geometric viewpoint by illustrating the geometric properties of the FIM corresponding to the statistical reward function. Finally, we compare the performance of the two reward functions by a numerical example in which a (non-cooperative) target is being chased by two cooperative bearing-only sensors. Preliminary work on this subject was reported in [17].  
The paper is organised as follows: the sensor trajectory scheduling problem of interest is described in Section 2. The analytic expressions for the two reward functions, i.e., the Expected RID and the determinant of FIM are derived and their performance discussed in Section 3. In Section 4, we further compare the performance difference between the two reward functions from the information geometric viewpoint. Simulation results are presented in Section 5. Finally, the concluding remarks are given in Section 7.