# Trajectory Optimisation for Cooperative Target Tracking with Passive Mobile Sensors
## 5 Simulation Example  
In this section, we illustrate the effectiveness of sensor scheduling using an example of tracking a maneuvering target by two bearing-only sensors. Target motion follows a hidden Markov process which contains the states of a left turn (\(-5^{\circ}\)), a right turn (\(5^{\circ}\)) and straightline (\(0^{\circ}\)), and it is moving at a constant speed of 10 m/s. The initial target state is \(\mathbf{x}_{0}=[600,560,10*\cos(30^{\circ}),10*\sin(30^{\circ})]^{\prime}\). In the first 50 of a total of 100 scans, the target is in the left turn state and it shifts to a right turn state for the rest of scans. Initially, the two bearing-only sensors are moving at a constant speed of 10m/s from the locations \(S_{1}=(1900,560)\) m and \(S_{2}=(1950,560)\) m heading in the directions \(0^{\circ}\) and \(60^{\circ}\), respectively. They acquire bearing measurements of the target at a sampling rate of T = 5 s. During each scan, they will be steered to one of five directions with respect to their current headings according to the trajectory optimisation decision. The 5 directions are \(-90^{\circ}\), \(-45^{\circ}\), \(0^{\circ}\), \(45^{\circ}\) and \(90^{\circ}\). Therefore, for a \(N\)-Step ahead decision process, the number of action hypotheses yielded for the two cooperative sensors are \((5^{2})^{N}\). For example, the number of action hypotheses for \(N=1\) is \(5^{2}=25\) and for \(N=3\) is \((5^{2})^{3}=15,625\). We illustrate this example in Figure 5.  
In practice, the maximum number of hypothesis histories allowed to steer future measurement at each epoch is constrained by a fixed number \"MaxH\" to provide a feasible computational overhead for real-time operation. In our simulation, we set MaxH = 1000 and such a choice does not affect the comparison of the optimisation performance under different reward functions. We observed from simulation that under this constraint the track error difference between \(N>3\) and \(N=3\) is statistically close to zero and thus is negligible. Therefore, in the performance comparison versus Monte Carlo runs, we only consider \(N\leq 3\). Figure 4: Illustration of the circles of statistical manifold spanned by the FIM with constant valued off-diagonal elements for given sensor locations (**a**) sensor 1 is at (1500, 400) and sensor 2 (973, 387); (**b**) sensor 1 is at (1500, 400) and sensor 2 moved to (1219, 544) in a circle around the target on the left. Colorbar value represents the changes of the amount of information gained by the sensor system if the target moves between the center and a manifold circle boundary. An EKF tracker [28] is implemented to estimate the posterior density of target state from the target bearing measurements taken by the two sensors. We assume that the target bearing measurements are corrupted with a Gaussian noise of zero-mean with standard deviation of two degrees. Figures 6 and 7 show the typical trajectories of the two bearing-only sensors in the target tracking experiment under the Determinant of FIM (DetFIM), the Expected RID (ExpReward), and the Trace of FIM (TrFIM), respectively. By maximising DetFIM, the two sensors move in a way such that the angle formed by sensor 1, target, sensor 2 is approximately \(\pi/2\) while approaching the target. Thus, the measurements taken by the two sensors along these computed trajectories minimise the error covariance of the underlying tracker (Figure 6). On the other hand, the sensor trajectories scheduled by maximising ExpReward (or TrFIM) show no cooperative movement between the sensors while they are approaching the target. This is because none of correlations between the two sensors with respect to the target are used in the evaluation of either ExpReward or TrFIM criterion as discussed in Section 3.