## 4 Sensor Trajectory Design
### Sensor Trajectory Planning Strategies  
The position of sensor is deployed to improve the subsequent target positioning accuracy. According to [19], define:  
\[\mathbf{M_{1}}=\frac{1}{\sigma^{2}}\sum_{i=1}^{k+1}\frac{1}{||\mathbf{t}- \mathbf{s}_{i}||^{2}} \begin{bmatrix}\sin^{2}\phi_{i}&-\frac{1}{2}\sin 2\phi_{i}&0 \\ -\frac{1}{2}\sin 2\phi_{i}&\cos^{2}\phi_{i}&0\\ 0&0&0\end{bmatrix} , \tag{57}\]  
\[\mathbf{M_{2}}=\frac{1}{\xi^{2}}\sum_{i=1}^{k+1}\frac{1}{||\mathbf{T}- \mathbf{S}_{i}||^{2}} \begin{bmatrix}\sin^{2}\theta_{i}\cos^{2}\phi_{i}&\frac{ 1}{2}\sin^{2}\theta_{i}\sin 2\phi_{i}&-\frac{1}{2}\sin 2\theta_{i}\cos\phi_{i} \\ \frac{1}{2}\sin^{2}\theta_{i}\sin 2\phi_{i}&\sin^{2}\theta_{i}\sin^{2}\phi_{i}&- \frac{1}{2}\sin 2\theta_{i}\sin\phi_{i}\\ -\frac{1}{2}\sin 2\theta_{i}\cos\phi_{i}&-\frac{1}{2}\sin 2\theta_{i}\sin\phi_{i}& \cos^{2}\theta_{i}\end{bmatrix} , \tag{58}\]  
where the matrix \(\mathbf{M_{1}}\) represents the azimuth FIM over \(k+1\) measurements, as it corresponds to the 2D projection of the localization problem onto \(xy\) plane, and the matrix \(\mathbf{M_{2}}\) represents the elevation FIM over \(k+1\) measurements. Therefore, the FIM over \(k+1\) measurements in 3D space is:  
\[\mathbf{FIM}_{\mathbf{k+1}}=\mathbf{M_{1}}+\mathbf{M_{2}}. \tag{59}\]  
**Lemma 3**. : \[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}}),\] (60)  
_where the set \(\{p_{1},\ p_{2},\ p_{3}\}\) is the combination selected from \(\{1,\ \ldots\,2(k+1)\}\)._  
\(\mathbf{V_{2i-1}}\) and \(\mathbf{V_{2i}}\)\((i=1,2,3,...,k+1)\) are the azimuth FIM and the elevation FIM, respectively. Proof of Lemma 3.: By the definition of matrix determinant, we have:  
\[\det(\mathbf{FIM}_{\mathbf{k+1}})=\sum_{\{p_{1},\,p_{2},\,p_{3}\}}^{(2^{(k+1) })}\det(\mathbf{V_{p_{1}}}+\mathbf{V_{p_{2}}}+\mathbf{V_{p_{3}}})\]  
\[+\Psi_{1}\sum_{\{p_{\boldsymbol{\mu}},\,p_{5}\}}^{(2^{(k+1)})}\det(\mathbf{V_{p_ {4}}}+\mathbf{V_{p_{5}}})+\Psi_{2}\sum_{\{p_{6}\}}^{(2^{(k+1)}_{1})}\det( \mathbf{V_{p_{6}}}), \tag{61}\]  
where the set \(\{p_{4},\ p_{5}\}\) and \(\{p_{6}\}\) are the combinations selected from \(\{1,\ \ldots\,2(k+1)\}\). \(\Psi_{1}\) and \(\Psi_{2}\) are constant factors decided by the combination numbers. Under \(\text{rank}(\mathbf{V_{p_{4}}}+\mathbf{V_{p_{5}}})\leq 2\) and \(\text{rank}(\mathbf{V_{p_{6}}})=1\), \(\det(\mathbf{V_{p_{4}}}+\mathbf{V_{p_{5}}})=\det(\mathbf{V_{p_{6}}})=0\). Then (61) turns into (60). **Theorem 3**. : \[\det(\mathbf{FIM_{k+1}}) =\det(\mathbf{FIM_{k}})+\sum_{\{p_{7},\ p_{8}\}}^{\binom{2k}{2}}\det (\mathbf{V_{2k+1}}+\mathbf{V_{p7}}+\mathbf{V_{ps}})\] (62) \[\quad+\sum_{\{p_{9},\ p_{10}\}}^{\binom{2k+1}{2}}\det(\mathbf{V_{2( k+1)}}+\mathbf{V_{ps}}+\mathbf{V_{ps10}}),\] (63)  
_where the set \(\{p_{7},\ p_{8}\}\) and \(\{p_{9},\ p_{10}\}\) are the combination selected from \(\{1,\ldots,2k\}\) and \(\{1,\ldots,\ 2\ (k+1)\}\)._  
Proof of Theorem 3.: From Lemma 3,  
\[\det(\mathbf{FIM_{k}})=\sum_{\{p_{11},\ p_{12},\ p_{13}\}}^{\binom{2k}{3}}\det (\mathbf{V_{p_{11}}}+\mathbf{V_{p_{12}}}+\mathbf{V_{p_{13}}}), \tag{64}\]  
where the set \(\{p_{11},\ p_{12},\ p_{13}\}\) is the combination selected from \(\{1,\ \ldots\ 2k\}\). \[\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k}}) =\sum_{\{p_{7},\ p_{8}\}}^{\binom{2k}{2}}\det(\mathbf{V_{2k+1}}+ \mathbf{V_{p7}}+\mathbf{V_{ps}}) \tag{65}\] \[\quad+\sum_{\{p_{9},\ p_{10}\}}^{\binom{2k+1}{2}}\det(\mathbf{V_{ 2(k+1)}}+\mathbf{V_{ps}}+\mathbf{V_{p10}}) \tag{66}\]  
naturally follows. Let the location \(\mathbf{S}_{k+1}\) on the trajectory is generated pointwisely through the maximization of  
\[\max_{\mathbf{S_{k}}\leftarrow\mathbf{S_{n}}}\det(\mathbf{FIM_{n}})\geq\sum_{k =4}^{n}\max_{\mathbf{S_{k+1}}}(\det(\mathbf{FIM_{k+1}})-\det(\mathbf{FIM_{k}}) )+\det(\mathbf{FIM_{k}}), \tag{67}\]  
which shows the trajectory is suboptimal.