# 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
#### 3.2.1 Target Localization Using EIF  
Assume the target has a constant velocity motion model as follows:  
\[\mathbf{x}_{k}=\boldsymbol{\Phi}_{k|k-1}\mathbf{x}_{k-1}+\boldsymbol{\Gamma}_{k |k-1}\boldsymbol{v}_{k-1} \tag{9}\]  
where \(\mathbf{x}_{k}=\left[x_{g,k},y_{g,k},v_{x,k},v_{y,k}\right]^{\top}\) denotes the target state at time step \(k\), of which the first two components represent the horizontal position of the target and the last two components are the velocities in the horizontal plane; and  
\[\boldsymbol{\Phi}_{k|k-1}=\left( \begin{array}{cc}\boldsymbol{I}_{2\times 2}& \Delta t\boldsymbol{I}_{2\times 2}\\ \boldsymbol{0}_{2\times 2}&\boldsymbol{I}_{2\times 2}\end{array} \right) \tag{10}\]  
denotes the state transition matrix of the target from time \(k-1\) to time \(k\), where \(\boldsymbol{I}_{2\times 2}\),\(\boldsymbol{0}_{2\times 2}\) denote the identity matrix and zero matrix of order 2, respectively; \(\Delta t\) denotes the time interval of measurements; and  
\[\boldsymbol{\Gamma}_{k|k-1}=\left( \begin{array}{c}\Delta t\boldsymbol{I}_{2 \times 2}\\ \boldsymbol{I}_{2\times 2}\end{array} \right) \tag{11}\]  
denotes the noises input matrix; \(\boldsymbol{v}_{k-1}=(n_{x},n_{y})^{\top}\) denotes the process noise assumed as zero mean Gaussian with covariance matrix \(\boldsymbol{Q}_{k-1}\). Based on (8), in the previous section, we have the measurement model of the form:  
\[\mathbf{z}_{k}=\boldsymbol{h}(\mathbf{x}_{k})+\boldsymbol{w}_{k} \tag{12}\]  
where \(\mathbf{z}_{k}\),\(\boldsymbol{w}_{k}\) denote the measurement vector and measurement noise vector at time \(k\), respectively; and \(\boldsymbol{w}_{k}\) is assumed as zero mean Gaussian with covariance matrix \(\boldsymbol{R}_{k}\) and independent from \(\boldsymbol{v}_{k-1}\). Given the initial state \(\hat{\mathbf{x}}_{0}\) and covariance matrix \(\mathbf{P}_{0}\), the filtering methods such as EKF and ascent Kalman filtering (UKF) can be used to iteratively and consistently estimate target  
Figure 4: Simultaneous target localization and trajectory optimization. (**a**) Coupled problem. (**b**) Decoupled problem. state, performing the real-time localization to the moving target. However, to optimize the UAV trajectory so that the target localization uncertainty can be minimized, the predicted FIM is needed in the trajectory optimization phase. To circumvent the inverse computation of predicted covariance matrix in each iteration, the information filtering corresponding to EKF is used, which can express the predicted FIM of the target state explicitly in each iteration. Furthermore, compared with the existing methods, using the predicted FIM, the process noise of the target model was also taken into consideration, which can optimize the next waypoint of UAV more effectively. EIF is an information form of the extended Kalman filtering, of which the target state estimation and covariance matrix are replaced with the information state vector and information matrix, respectively. The relation between them can be expressed as follows:  
\[\left\{ \begin{array}{l}\mathbf{Y}_{k}=\mathbf{P}_{k}^{-1}\\ \mathbf{\hat{y}}_{k}=\mathbf{P}_{k}^{-1}\mathbf{\hat{x}}_{k}\end{array} \right. \tag{13}\]  
where \(\mathbf{Y}_{k}\mathbf{\hat{y}}_{k}\) denote the information matrix and information state vector at time \(k\), respectively; \(\mathbf{P}_{k}\mathbf{\hat{x}}_{k}\) denote the covariance matrix and target state estimation at time step \(k\), respectively. Using (13), the initial information matrix \(\mathbf{Y}_{0}\) and the information state vector \(\mathbf{\hat{y}}_{0}\) can be obtained. Given the information matrix \(\mathbf{Y}_{k-1}\) and information state vector \(\mathbf{\hat{y}}_{k-1}\) at time instant \(k-1\), the EIF advances with time as follows:  
1.