## 4 Sensor Trajectory Design
### Analytical Derivation for the Global Maximizer
\begin{array}{ccccc}S_{15}(y_{\mathbf{s}_{k+1}},z_{ \mathbf{s}_{k+1}})&0&1&0&...&0\\ S_{14}(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})&S_{15}(y_{\mathbf{s}_{k+1}},z _{\mathbf{s}_{k+1}})&-2x_{\mathbf{s}_{k}}&1&...&0\\ S_{13}(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})&S_{14}(y_{\mathbf{s}_{k+1}},z _{\mathbf{s}_{k+1}})&\eta(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})&-2x_{ \mathbf{s}_{k}}&...&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ S_{1}(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})&S_{2}(y_{\mathbf{s}_{k+1}},z _{\mathbf{s}_{k+1}})&0&0&...&1\\ S_{0}(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})&S_{1}(y_{\mathbf{s}_{k+1}},z _{\mathbf{s}_{k+1}})&0&0&...&-2x_{\mathbf{s}_{k}}\\ 0&S_{0}(y_{\mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}})&0&0&...&\eta(y_{\mathbf{s} _{k+1}},z_{\mathbf{s}_{k+1}})\end{array} \right). \tag{88}\]  
The Sylvester resultant \(\det(\mathbf{Syl_{1}}((\ref{eq:sylvester}),(\ref{eq:sylvester});y_{ \mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}}))\) is a bivariate polynomial with an upper bound of the 58 th order for \(y_{\mathbf{s}_{k+1}}\) and the 46 th order for \(z_{\mathbf{s}_{k+1}}\). Similarly, the joint solution of (85) and (87) can be developed as the \((15+2)\times(15+2)\) Sylvester matrix of (85) and (87) is similar to (88). \(\det(\mathbf{Syl_{2}}((\ref{eq:sylvester}),(\ref{eq:sylvester});y_{ \mathbf{s}_{k+1}},z_{\mathbf{s}_{k+1}}))\) is also a bivariate polynomial with an upper bound of the 58 th order for \(y_{\mathbf{s}_{k+1}}\) and the 46 th order for \(z_{\mathbf{s}_{k+1}}\). The real solution can be obtained by solving the equations given by the determinants of the two Sylvester matrices. Comparing only the real solutions lying on surface 1 in terms of \(f(\mathbf{S}_{k+1})\) gives the global maximizer. For surface 2, the real solutions of the surface equation and (86) are compared with the two intersection points to generate the global maximizer. For surfaces 3 and 4, the real solutions of the surface equation and (86) on the surface are compared with the two intersection points to provide \(\mathbf{S}_{k+1}\). When the movement area contains the safety area, the real solutions of the safety constraint and the (86) that has the biggest \(f(\mathbf{S}_{k+1})\) are the global maximizer. **Remark 1**. : _The position of a sensor should not be collinear with the target position in the \(z\)-axis direction so that the target is observable. It is necessary to set the cylindrical as a non-flying area in Figure 4. It means to insert a cylinder centering at the target position with the radius of \(\rho\) in the target safety area, and the \(z\)-axis direction extends to infinity. The constraint is formulated as:_  
\[(x_{k+1}-\mathbf{\hat{T}}^{k}(1))^{2}+(y_{k+1}-\mathbf{\hat{T}}^{k}(2))^{2} \geqslant\rho^{2}. \tag{89}\]  
_If the maximum solution of \(f\) (\(\mathbf{S}_{k+1}\)) falls into the section where the surface intersects the cylinder, we choose the other one which leads to the second largest \(f\) (\(\mathbf{S}_{k+1}\)), and so forth. If all the maximum solutions of \(f\) (\(\mathbf{S}_{k+1}\)) fall within the intersection section, a point is randomly selected from the feasible region as the solution of \(f\) (\(\mathbf{S}_{k+1}\))._  
Figure 4: Sensor motion area, target safety area and cylindrical non-flying area.