## 4 Sensor Trajectory Design
### Optimal Solution Region  
Through the analysis of (62), we have:  
**Theorem 4**. : \[\det(\mathbf{FIM}_{\mathbf{k+1}})-\det(\mathbf{FIM}_{\mathbf{k}})=\sum_{ \begin{subarray}{c}c_{i}\in\{p_{7},\ p_{8},\ p_{9}\}\\ c_{i}\
otin\{p_{13},\ p_{14}\}\end{subarray} }\sum_{j\in\{1,\ \ldots,\ 3\}}\left(\prod_{i=1}^{3}\text{sgn}(j)L_{i,j_{i}}^{(c_{i})} \right),\] (75)  
_where the set \(\{p_{7},\ p_{8},\ p_{9}\}\) is the combination selected from \(\{1,\ \ldots\ 2(k+1)\}\) with \(\exists p_{j}(7\leq j\leq 9)\)\(\in\{2k+1,2(k+1)\}\). The set \(\{p_{13},\ p_{14}\}\) is the combination selected from \(\{p_{7},\ p_{8},\ p_{9}\}\). \(j\) represents the permutation of \(\{j_{1},\ j_{2},\ j_{3}\}\). The sign function \(\text{sgn}(j)=-1\) when the number of exchanges that \(\{j_{1},\ j_{2},\ j_{3}\}\) takes to be the standard order \(\{1,\ 2,\ 3\}\) is odd. Otherwise, \(\text{sgn}(j)=1\). \(L_{i,j_{i}}^{(c_{i})}\) is the ith row and \(j_{i}\)th column element in \(\mathbf{V}_{\epsilon_{i}}\)._  
Proof of Theorem 4.: From Theorem 3 and the Leibniz formula,  
\[\det(\mathbf{FIM}_{\mathbf{k+1}})-\det(\mathbf{FIM}_{\mathbf{k}})=\sum_{c_{i} \in\{p_{7},\ p_{8},\ p_{9}\}}\sum_{j_{i}\in\{1,\ \ldots,\ 3\}}\left(\prod_{i=1}^{3}\text{sgn}(j)L_{i,j_{i}}^{(c_{i})} \right). \tag{76}\]  
In addition,  
\[\sum_{\{p_{13},\ p_{14}\}}\det(\mathbf{V}_{\mathbf{p_{13}}}+\mathbf{V}_{ \mathbf{p_{14}}})=\sum_{c_{i}\in\{p_{13},\ p_{14}\}}\sum_{j_{i}\in\{1,\ \ldots,\ 3\}}\left(\prod_{i=1}^{3}\text{sgn}(j)L_{i,j_{i}}^{(c_{i})} \right)=0. \tag{77}\]  
Deducting (77) from (76) leads to (75). \(\hat{\Upsilon}^{k}\) is outside the sensor motion area and \(\triangle_{TAB}\) is not contained by the target safety area, define the part of the sensor motion boundary inside \(\triangle_{TAB}\) with the section inside the target safety area replaced by the section of the target safety boundary inside the sensor movement area, such as surface 2 in Figure 2(b). \(E\) and \(F\) (\(E\) and \(F\) are symmetrically distributed) are the points where the two spheres intersect; (ii) when the \(\hat{\Upsilon}^{k}\) is outside the sensor motion area and if \(\triangle_{TAB}\) is contained by the target safety area, define the part of the target safety boundary inside the sensor motion area as surface 3 in Figure 2(c); (iii) when the \(\hat{\Upsilon}^{k}\) is inside the sensor motion area, define the part of the target safety boundary inside the sensor motion area as surface 4 in Figure 2(d); 3. when the sensor motion area contains the target safety area, the feasible region is the sensor motion area where the interior section of the target safety area is excluded. The target safety boundary is defined as surface 5 in Figure 2(e). Figure 3: Surface 1 (red part in (**a**)) when the two areas are separated, surface 2 (red part in (**b**)), surface 3 (red part in (**c**)) and surface 4 (red part in (**d**)) when the two areas intersect, the target safety boundary (red part in (**e**)) when the target safety area is in the sensor movement area. **Proposition 1**. : _In case 1, the maximizer of the \(f\left(\mathbf{S}_{k+1}\right)\) exists on the spherical crown (surface 1). In case 2(i), the maximizer of the \(f\left(\mathbf{S}_{k+1}\right)\) exists on surface 2 where the two areas intersect in the sensor motion area, and the side of the circular truncated cone (lower bottom circle with AB as the diameter and upper bottom circle with EF as the diameter). In cases 2(ii) and 2(iii), the maximizer of the \(f\left(\mathbf{S}_{k+1}\right)\) exists on surfaces 3, 4 where the two areas intersect. In case 3, the maximizer of \(f\left(\mathbf{S}_{k+1}\right)\) is on the boundary of the target safety area (surface 5)._  
Proof of Proposition 1.: Regardless of the relative location of the two areas, a straight blue dotted line is drawn between \(\hat{\mathbf{T}}^{k}\) and any point in the feasible region; all the points on this line have the same \(\hat{\boldsymbol{\phi}}_{k+1}(\hat{\mathbf{T}}^{k})\) and \(\hat{\boldsymbol{\phi}}_{k+1}(\hat{\mathbf{T}}^{k})\) as shown.