# A Closed-Form Method for Simultaneous Target Localization and UAV Trajectory Optimization
## 3 Simultaneous Target Localization and Unmanned Aerial Vehicle (UAV) Trajectory Optimization
### Simultaneous Target Localization and UAV Trajectory Optimization
#### 3.2.2 UAV Trajectory Optimization  
Although the EIF can estimate the target position consistently, the sensor-target geometry also significantly influences the performance of target localization. Based on the decoupling structure, the UAV trajectory should be optimized at each waypoint. Assuming the previous waypoints have existed at time \(k\), and the next waypoint at time \(k+1\) is to be optimized. According to relevant theory in the literature, making the uncertainty of the target position estimate minimized is equivalent to maximizing the posterior FIM after observation at time \(k+1\) in some measure, which can be expressed mathematically as:  
\[\max\mathcal{T}[\mathbf{Y}_{k+1}] \tag{23}\]  
where \(\mathcal{T}\) denotes some measure of the FIM. On the other hand, the UAV maneuverability constraints should also be taken into account:  
\[\left\{ \begin{array}{c}\gamma_{\mathrm{h}}\leq\ \gamma_{\mathrm{h,max}}\\ \gamma_{\mathrm{v}}\leq\ \gamma_{\mathrm{v,max}}\\ v_{\mathrm{min}}\leq\|\mathbf{v}_{k}\|\leq v_{\mathrm{max}}\\ z_{u,k+1}-z_{0}\geq h_{\mathrm{min}}\end{array} \right.,\forall k=1,2,\cdots \tag{24}\]  
where \(\mathbf{v}_{k}\) is the velocity vector of the UAV; \(v_{\mathrm{min}}\),\(v_{\mathrm{max}}\) denote the minimum and maximum velocity values, respectively; \(\gamma_{\mathrm{h}}\),\(\gamma_{\mathrm{v}}\) denote the horizontal turn rate and vertical turn rate with \(\gamma_{\mathrm{h,max}}\),\(\gamma_{\mathrm{v,max}}\) as their maximum values, respectively; \(z_{u,k+1}\) denotes the third component of \(\mathbf{y}_{k+1}\), the UAV position vector in \(\mathbb{C}^{g}\); \(z_{0}\) represents the height of ground plane (i.e., the third component of target position vector). The first constraint is about the maximum turn rate of UAV; the second constraint is about the maximum velocity of UAV; the third equality is to restrict the relative height between the target and the UAV. Combining (23) and (24), for UAV trajectory optimization, we have the following optimization problem:  
\[ \begin{array}{c}\max\ \mathcal{T}[\mathbf{Y}_{k+1}]\\ \mathrm{s.t.}\ \mathbf{Y}_{k+1}=\mathbf{Y}_{k+1|k}+\mathbf{I}_{k+1}\\ \gamma_{\mathrm{h}}\leq\ \gamma_{\mathrm{h,max}}\\ \gamma_{\mathrm{v}}\leq\ \gamma_{\mathrm{v,max}}\\ v_{\mathrm{min}}\leq\|\mathbf{v}_{k}\|\leq v_{\mathrm{max}}\\ z_{u,k}-z_{0}\geq h_{\mathrm{min}}\forall k=&1,2,\cdots\end{array}  \tag{25}\]  
where the predicted information matrix \(\mathbf{Y}_{k+1|k}\) and information gain \(\mathbf{I}_{k+1}\) at time step \(k+1\) can be computed by (14) and (18) using the current information matrix and information state vector \(\mathbf{Y}_{k}\). Note that in order to optimize the UAV trajectory, \(\mathbf{I}_{k+1}\) should be expressed in terms of the target position, and as the target position is unknown, we replaced it with the predicted state, \(\hat{\mathbf{k}}_{k+1|k}\). Before proceeding to solve the optimization problem of (25), the valid FIM of the target state for the trajectory optimization should be illustrated. As the Fisher information is a way of measuring the amount of information that some measurement carries about an unknown variable, upon which the probability of the measurement depends, the FIM is a way of measuring the information of an unknown vector. Each element of the FIM corresponds to the amount of information about the corresponding variable element in the vector to be estimated. For example, the element in entry (1,1) of the \(\mathbf{Y}_{k+1}\) represents the information of the component \(x_{g,k+1}\) of the posterior target state estimate vector from the prior estimate and the current measurement, and the element in entry (1,2) of \(\mathbf{Y}_{k+1}\) represents the information relevant with both component \(x_{g,k+1}\) and component \(y_{g,k+1}\) of the target state vector. As the measurement model (12) only involves the position variables of the target state vector, the essential variables to be estimated is the position variables, according to which the velocity variables can be inferred.