## 4 Sensor Trajectory Design
### Sensor Trajectory Planning Strategies
According to Theorem 3, it is still a reasonable choice because the design of \(\mathbf{S}_{k+1}\) considers information given by \(\mathbf{S}_{k+1}\) and all the past sensor locations jointly. Defining  
\[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]0  
as the objective function of solving \(\mathbf{S_{k+1}}\). Formally defining  
**Trajectory 1**. : _the location \(\mathbf{S}_{k+1}\) on the trajectory is generated pointwisely through the maximization of_  
\[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]1 \[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]2 (70) \[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]3  
_with respect to \(\mathbf{S}_{k+1}\) given \(\mathbf{S}_{1},\ldots,\mathbf{S}_{k}(k\geq 3)\)._  
When optimizing \(f(\mathbf{S}_{k+1})\), unknown \(\mathbf{T}\), \(\phi_{i}\), \(\phi_{i}\), \(\sigma^{2}\) and \(\xi^{2}\ (i=1,\ldots,k)\) are replaced by the estimated \(\mathbf{\hat{T}^{k}}\), \(\hat{\phi}_{i}(\mathbf{\hat{T}^{k}})\) and \(\hat{\theta}_{i}(\mathbf{\hat{T}^{k}})\), \(\sigma^{2}(\mathbf{\hat{T}^{k}})\) and \(\xi^{2}(\mathbf{\hat{T}^{k}})\). This method updates target location and re-estimates all the historical data in each step. To circumvent this weakness, we present  
**Trajectory 2**. : _the location \(\mathbf{S}_{k+1}\) on the trajectory is generated pointwisely through the maximization of_  
\[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]4 \[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]5 (73) \[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\]6  
_with respect to \(\mathbf{S}_{k+1}\) given \(\mathbf{S}_{1},\ldots,\mathbf{S}_{k}(k\geqslant 3)\)._  
When optimizing \(\hat{f}(\mathbf{S}_{k+1})\), \(\sigma^{2}\) and \(\xi^{2}\) are replaced by \((\sum_{i=1}^{k}\hat{\sigma}_{i}^{2}(\hat{\mathbf{T}}^{i-1})+\hat{\sigma}_{k+1} ^{2}(\hat{\mathbf{T}}^{k}))/(k+1)\) and \((\sum_{i=1}^{k}\xi_{i}^{2}(\hat{\mathbf{T}}^{i-1})+\xi_{k+1}^{2}(\hat{\mathbf{ T}}^{k}))/(k+1)\), respectively. Distinct with trajectory \(1\), \(\hat{\phi}_{i}\) and \(\hat{\theta}_{i}\) (\(i=1,\ldots,k\)) are not updated by \(\hat{\mathbf{T}}^{k}\). In Trajectory 1 and 2, the computation time to generate the sensor position at the \((k+1)\)th moment includes the time to obtain the objective function and the time to optimize the objective function. Trajectory 1 needs \((k+1)*(49\) additions/multiplications, three square roots, nine trigonometric function calculations) and 18 additions/multiplications for the calculation of \(\det(\mathbf{FIM}_{\mathbf{k+1}})\) to obtain the objective function, while trajectory 2 requires only 49 additions/multiplications, three square roots, nine trigonometric functions and 18 additions/multiplications for the calculation of \(\det(\mathbf{FIM}_{\mathbf{k+1}})\). The generation of trajectory 2 reduces \(k*(49\) additions/multiplications, three square root, nine trigonometric function calculations) calculations compared to trajectory 1. In addition, the objective function structure is similar, the optimization time of the objective function is almost equal depending on the optimization method.