## 5 Simulation
### Example 2
Due to the inverse relation between the covariance and F  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Observation Point** & \( \begin{array}{c}\det(\mathbf{FIM_{k}^{1}})-\det(\mathbf{FIM_{k}^{4}})>0(i=1,2,3 )\end{array} \) for trajectory \(i\) at two observations. \\ \hline \multirow{2}{*}{ \begin{tabular}{c} mean of \(L(\hat{\mathbf{T}})(k=10)\) \\ mean of \(\hat{L}(\hat{\mathbf{T}})(k=15)\) \\ \end{tabular}  } & \( \begin{array}{c}\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k}})\end{array} \) 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. Due to the inverse relation between the covariance and F \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Trajectory 1** & **Trajectory 2** & **Trajectory 3** & **Trajectory 4** \\ \hline \multirow{2}{*}{
\begin{tabular}{c} mean of \(L(\hat{\mathbf{T}})(k=15)\) \\ mean of \(\hat{L}(\hat{\mathbf{T}})(k=15)\) \\ \end{tabular} } & 91.16 & 90.58 & 89.62 & 90.49 \\  & 93.42 & 92.86 & 88.85 & 90.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean values of \(L(\hat{\mathbf{T}})\) for four trajectories at two observation points. Trajectory 4 also utilizes more past information than Trajectory 3. However, compared with Trajectories 1 and 2, not all the information is used. Comparison of the mean of \(\det(\mathbf{FIM_{k}})\) for four trajectories can be seen intuitively in Figure 9. When the sampling step increases, \(\det(\mathbf{FIM_{k}})\) of the proposed method increases more. In addition, more \(\det(\mathbf{FIM_{k}})\) can be obtained when the angles are updated in Trajectory 1 at the cost of increasing the calculation amount. Less computation time is consumed by generating Trajectory 2. When the measurement noise is small, the difference of the determinant obtained by Trajectories 1 and 2 is subtle as seen in the inner plot of Figure 9.  
The trajectories simulation results are shown in Figure 10. In the process of trajectory planning, the predicted sensor position keeps getting closer to the target. When the sensor motion area is separated from the target safety area, \(\mathbf{S}_{k+1}\) sits on the movement boundary sphere centering at \(\mathbf{S}_{k}\). As shown in Figure 10, Trajectories 1 and 2 approach the target much faster because the trajectory planning method proposed in this paper takes into account sensor position  
Figure 10: Sensor trajectories in typical runs. The trajectories in other runs are similar to the ones in this run. Trajectories 1, 2, 3 and 4 are represented by circle, cross, upper triangle and square, respectively. The black sphere is the boundary of the target safety area. The target position represented by the five pointed star. Figure 9: Comparison of the mean of \(\det(\mathbf{FIM_{k}})\) for four trajectories over 100 Monte Carlo runs. information at all past moments and have higher FIM determinant. Since Trajectories 3 and 4 correspond to lower and mild FIM determinants, their curves reach the target in a slow and mild speed, respectively, amongst four trajectories; although it seems that Trajectory 3 distances itself from the target. For each scenario, the whole computation complexity includes the following parts: initialization, Kalman-related localization, objective function construction and optimization. The average runtime for each trajectory is given in Table 3 to describe the complexity.