## 3 Target State Estimation and Observer Trajectory Quality Metric
### Observer Trajectory Quality Metric  
A metric is defined in this section which can be used to determine the optimal trajectories of the UAVs for tracking a moving target. In practice, the performance of target-motion estimation depends on the trajectories of the UAVs. That is, the performance of target-motion estimate is a function of the specific observer trajectory, and therefore some trajectories lead to better performance (accuracy, effectiveness, etc.) than others. The goal of the cooperative target tracking mission is to enhance the global estimate performance by determining optimal or near-optimal observer trajectories. To generate good trajectories, it is necessary to define the notion of quality metric for the global estimate of target state. Therefore, the problem of identifying the best trajectory-quality metric is the problem of determining the best description of estimation performance. According to the Cramer-Rao lower bound (CRLB) theorem, the estimation error covariance represents the uncertainty associated with the accuracy of the estimation results, which should be minimized. The CRLB provides a lower bound for the error covariance and any unbiased estimator that achieves this lower bound is considered efficient. Specifically:  
\[\mathbf{P}_{k}=\text{E}[(\hat{x}_{\text{t},k}-\mathbf{x}_{\text{t},k})(\hat{x}_{\text {t},k}-\mathbf{x}_{\text{t},k})^{\text{T}}]\geq\mathbf{C}_{k}=\mathbf{Y}_{k}^{-1} \tag{26}\]  
where \(\mathbf{x}_{\text{t},k}\) is the target state to be estimated, \(\hat{x}_{\text{t},k}\) is the unbiased estimate of \(\mathbf{x}_{\text{t},k}\) given the measurements \(\mathbf{z}_{k}\), and \(\mathbf{P}_{k}\) represents the covariance of the estimation error. The CRLB on \(\mathbf{P}_{k}\) is denoted by \(\mathbf{C}_{k}\) and its inverse \(\mathbf{Y}_{k}=(\mathbf{C}_{k})^{-1}\) is the Fisher information matrix (FIM). CRLB provides a lower bound on the achievable covariance of the estimator. It means minimizing the covariance of the estimation can be achieved by minimizing CRLB. Minimizing the covariance of the estimation will reduce the uncertainty of the estimation results, and then improve the estimator performance. Since the CRLB and the FIM are inversely related, maximizing the FIM will improve the estimator performance and reduce uncertainty. Thus, the optimal sensing trajectory can be generated by maximizing the FIM. According to References [14; 18], the FIM can be predicted at future time (\(k+l\)), based on the current target state estimation results. Prediction of FIM, \(\hat{\mathbf{Y}}_{k+l|k}\), is computed via the following recursive form  
\[\hat{\mathbf{Y}}_{k+l|k}=\left(\mathbf{F}_{k+l-1}\hat{\mathbf{Y}}_{k+l-1|k}^{- 1}\mathbf{F}_{k+l-1}^{\mathrm{T}}+\mathbf{G}_{k+l-1}\mathbf{Q}_{k+l-1}{\mathbf{ G}_{k+l-1}^{\mathrm{T}}}\right)^{-1}+\mathbf{\hat{I}}_{k+l|k} \tag{27}\]  
where the recursive equation starts with \(\hat{\mathbf{Y}}_{k}=\mathbf{Y}_{k\,|k}\). The matrix \(\mathbf{\hat{I}}_{k+l|k}\) is the prediction of the information gain matrix, which is regarded as the expected measurement contribution to the FIM from all the UAVs. On one hand, \(\mathbf{\hat{I}}_{k+l|k}\) is dependent on the planned trajectories of all the UAVs. On the other hand, \(\mathbf{\hat{I}}_{k+l|k}\) is also dependent on the probabilities of the measurement packets being successfully received by the base station. At time \(k\), the information gain from \(A_{i}\) can be defined as  
\[\mathbf{\hat{I}}_{i,k}=\chi_{i,k}\cdot\mathbf{\hat{H}}_{i,k}^{\mathrm{T}} \mathbf{\hat{R}}_{i,k}^{-1}\mathbf{\hat{H}}_{i,k} \tag{28}\]  
where \(\chi_{i,k}\sim\)Bern(\(\beta_{i,k}\)) is a random Bernoulli variable (either 0 or 1). \(\beta_{i,k}\), which is determined by the packet-erasure channel model, indicates the probability of the packet successfully getting through from \(A_{i}\) to the base station. If the measurement packet from \(A_{i}\) is dropped by the base station, the information gain from \(A_{i}\) is \(\mathbf{\hat{I}}_{i,k}=\mathbf{0}\).