## 5 Simulation
### Example 2  
The third simulation is about sensor trajectory planning. For fairness and completeness, we use the pseudolinear Kalman filter method in [30] for target localization as the platform to compare the performance of trajectories. One fitness function,  
\[L(\mathbf{\Upsilon})=100*(1-\frac{||\mathbf{T}-\mathbf{\Upsilon}||_{2}}{|| \mathbf{\Upsilon}||_{2}}), \tag{92}\]  
Figure 8: Bias norm with increasing sensor number at the difference of 40 for each method. Figure 7: MSE with increasing sensor number at the difference of 40 for each method. in [31] is used to demonstrate the performance of each estimator over \(n\) runs. It could even be much less than zero when the estimators deviate far from the real values. To downplay the effect of the extremum points, \(L(\hat{\mathbf{T}})\) is modified into:  
\[\hat{L}(\hat{\mathbf{T}})=\max(0,100*(1-\frac{||\mathbf{T}-\hat{\mathbf{T}}||_{ 2}}{||\mathbf{T}||_{2}})). \tag{93}\]  
**Trajectory 3**. : _The location \(\mathbf{S}_{k+1}\) on the trajectory is generated through the maximization of \(det(FIM_{k+1})\)-\(det(FIM_{k-2})\) with respect to \(\mathbf{S}_{k+1}\) recursively. The design of \(\mathbf{S}_{k+1}\) is given by the \((k+1)\)th measurement jointly with its two temporal nearest predecessors._  
**Trajectory 4**. : _The location \(\mathbf{S}_{k+1}\) on the trajectory is generated through the minimization of the state estimate covariance._  
The four trajectories planning methods are simulated and compared. The positions of the target and sensors are located at \(\mathbf{T}= \begin{bmatrix}20&35&55\end{bmatrix} ^{\prime}\) m, \(\mathbf{S}_{1}= \begin{bmatrix}10&25&20\end{bmatrix} ^{\prime}\) m, \(\mathbf{S}_{2}= \begin{bmatrix}30&20&40\end{bmatrix} ^{\prime}\) m, \(\mathbf{S}_{3}= \begin{bmatrix}70&50&15\end{bmatrix} ^{\prime}\) m, \(\mathbf{S}_{4}= \begin{bmatrix}12&80&70\end{bmatrix} ^{\prime}\) m, \(\mathbf{S}_{5}= \begin{bmatrix}59&22&18\end{bmatrix} ^{\prime}\) m, \(\mathbf{S}_{6}= \begin{bmatrix}120&150&90\end{bmatrix} ^{\prime}\) m, respectively. For each test, the four trajectories share the same six initial sensor locations. The noise of the angle measurements are Gaussian independently distributed with zero mean and the standard deviation is \(\sqrt{0.1}\) radian. The anti-collision distance and the maximum distance of sensor movement are set to \(R=5\) m, \(r=10\) m, respectively. The radius of the cylindrical non-flying area is set to \(\rho=0.5R\). One hundred independent Monte Carlo experiments are run. Performance of the four trajectories is measured and displayed at two observation points when \(k=10\) and \(k=15\). Table 1 shows that Trajectory 1 and 2 contribute to greater \(\det(\mathbf{FIM_{k+1}})\) in over 90% of tests where trajectory 4 is used as a benchmark. This is consistent with the numerical results given in Table 2 where Trajectory 1 has the highest \(\hat{L}(\hat{\mathbf{T}})\) at \(k=10\) and \(k=15\). Compared with Trajectory 3 and 4, the target estimations are more accurate under Trajectory 1 and 2 as the corresponding \(\hat{L}(\hat{\mathbf{T}})\) values are closer to 100. The accuracy of target localization is related to \(\det(\mathbf{FIM_{k}})\). The larger the \(\det(\mathbf{FIM_{k}})\), the higher the positioning accuracy. From Theorem 3, \(\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k}})\) for trajectory 1 and \(\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k-2}})\) for Trajectory 3 both are parts of \(\det(\mathbf{FIM_{k+1}})\). Since Trajectory 3 only utilizes the sensor location of the past two moments, \(\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k-2}})\) weighs less in \(\det(\mathbf{FIM_{k+1}})\) than \(\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k}})\). The maximizer of \(\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k-2}})\) is less likely to be close to the global optimizer of \(\det(\mathbf{FIM_{k+1}})\). Under such circumstances, Trajectory 3 contributes a smaller information increment than Trajectories 1 and 2. Trajectory 4 is achieved by minimizing the trace of the state estimation covariance matrix.