## 4 Sensor Trajectory Design
### Analytical Derivation for the Global Maximizer  
According to (78), the partial derivative of \(f(\mathbf{S}_{k+1})\) over \(\mathbf{x}_{\mathbf{s}_{k+1}},y_{\mathbf{s}_{k+1}}\) and \(z_{\mathbf{s}_{k+1}}\) can be written into:  
\[\frac{\partial f(\mathbf{S}_{k+1})}{\partial\mathbf{x}_{\mathbf{s}_{k+1}}}= \frac{\sum\limits_{h=0}^{13}\sum\limits_{q=0}^{14}\sum\limits_{l=0}^{14}\sum \limits_{q^{\prime}=1}^{8}\alpha_{hgl}^{1}(y_{\mathbf{s}_{k+1}}^{q},z_{ \mathbf{s}_{k+1}}^{l})x_{\mathbf{s}_{k+1}}^{h}}{\sum\limits_{h^{\prime}=0}^{ 20}\sum\limits_{q^{\prime}=0}^{20}\sum\limits_{l^{\prime}=0}^{12}\sum\limits _{b^{\prime}=0}^{+}\beta_{h^{\prime}+q^{\prime}+l^{\prime}}^{h^{\prime}}x_{ \mathbf{s}_{k+1}}^{h^{\prime}}y_{\mathbf{s}_{k+1}}^{q^{\prime}}z_{\mathbf{s} _{k+1}}^{l^{\prime}}}, \tag{81}\]  
\[\frac{\partial f(\mathbf{S}_{k+1})}{\partial y_{\mathbf{s}_{k+1}}}=\frac{ \sum\limits_{h=0}^{14}\sum\limits_{q=0}^{14}\sum\limits_{l=0}^{14}\sum \limits_{q^{\prime}=0}^{7}\alpha_{hgl}^{3}(y_{\mathbf{s}_{k+1}}^{q},z_{ \mathbf{s}_{k+1}}^{l})x_{\mathbf{s}_{k+1}}^{h}}{\sum\limits_{h^{\prime}=0}^{ 20}\sum\limits_{q^{\prime}=0}^{20}\sum\limits_{l^{\prime}=0}^{12}\beta_{h^{ \prime}+q^{\prime}+l^{\prime}}^{h^{\prime}}x_{\mathbf{s}_{k+1}}^{h^{\prime}} y_{\mathbf{s}_{k+1}}^{q^{\prime}}z_{\mathbf{s}_{k+1}}^{l^{\prime}}}, \tag{82}\]  
\[\frac{\partial f(\mathbf{S}_{k+1})}{\partial z_{\mathbf{s}_{k+1}}}=\frac{ \sum\limits_{h=0}^{14}\sum\limits_{q=0}^{14}\sum\limits_{l=0}^{7}\alpha_{hgl} ^{3}(y_{\mathbf{s}_{k+1}}^{q},z_{\mathbf{s}_{k+1}}^{l})x_{\mathbf{s}_{k+1}}^{ h}}{\sum\limits_{h^{\prime}=0}^{20}\sum\limits_{q^{\prime}=0}^{20}\sum\limits_{l^{ \prime}=0}^{12}\beta_{h^{\prime}+q^{\prime}+l^{\prime}}^{h^{\prime}}x_{ \mathbf{s}_{k+1}}^{h^{\prime}}y_{\mathbf{s}_{k+1}}^{q^{\prime}}z_{\mathbf{s} _{k+1}}^{l^{\prime}}}, \tag{83}\]  
where the coefficient \(\alpha_{hgl}^{1}\), \(\alpha_{hgl}^{2}\), \(\alpha_{hgl}^{3}\) and \(\beta_{h^{\prime}+q^{\prime}+l^{\prime}}^{h^{\prime}}\) can be zero. From Proposition 1, the feasible region always lies on the area boundary. Take the case when the two areas are separated as an example. \(\left[\partial f(\mathbf{S}_{k+1})/\partial x_{\mathbf{s}_{k+1}}\quad \partial f(\mathbf{S}_{k+1})/\partial y_{\mathbf{s}_{k+1}}\quad\partial f( \mathbf{S}_{k+1})/\partial z_{\mathbf{s}_{k+1}}\right]^{\prime}\) should be colinear with the normal vector \(\left[x_{\mathbf{s}_{k+1}}-x_{\mathbf{s}_{k}}\quad y_{\mathbf{s}_{k+1}}-y_{ \mathbf{s}_{k}}\quad z_{\mathbf{s}_{k+1}}-z_{\mathbf{s}_{k}}\right]^{\prime}\) at the optimal, i.e.,  
\[\frac{\partial f(\mathbf{S}_{k+1})}{\partial x_{\mathbf{s}_{k+1}}}(y_{ \mathbf{s}_{k+1}}-y_{\mathbf{s}_{k}})-\frac{\partial f(\mathbf{S}_{k+1})}{ \partial y_{\mathbf{s}_{k+1}}}(x_{\mathbf{s}_{k+1}}-x_{\mathbf{s}_{k}})=0, \tag{84}\]  
\[\frac{\partial f(\mathbf{S}_{k+1})}{\partial x_{\mathbf{s}_{k+1}}}(z_{\mathbf{s }_{k+1}}-z_{\mathbf{s}_{k}})-\frac{\partial f(\mathbf{S}_{k+1})}{\partial z_{ \mathbf{s}_{k+1}}}(x_{\mathbf{s}_{k+1}}-x_{\mathbf{s}_{k}})=0. \tag{85}\]  
Otherwise, any infinitesimal move along the tangent vector that constructs an obtuse angle with \(-\left[\partial f(\mathbf{S}_{k+1})/\partial x_{\mathbf{s}_{k+1}}\quad \partial f(\mathbf{S}_{k+1})/\partial y_{\mathbf{s}_{k+1}}\quad\partial f( \mathbf{S}_{k+1})/\partial z_{\mathbf{s}_{k+1}}\right]^{\prime}\) increases the value of \(f(\mathbf{S}_{k+1})\), (84) is equivalent to:  
\[\sum\limits_{h=0}^{15}g_{h}(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})x_{\mathbf{ s}_{k+1}}^{h}=0, \tag{86}\]  
where \(g_{h}(y_{\mathbf{s}_{k+1}}  
The joint solution of (84) and (87) can be developed as follows. The \((15+2)\times(15+2)\) Sylvester matrix of (84) and (87) with respect to \(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}}\) is:  
\[\mathbf{Syl_{1}}((\ref{eq:sylvester}),(\ref{eq:sylvester});y_{ \mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})=\\ \left(