## 4 Motion Planning Algorithm Based on Co-Optimization of Communication and Sensing
### Receding Horizon Optimizing Model for Multi-UAVs Cooperative Target Tracking  
The receding horizon optimization model for multiple UAV cooperative target tracking at time \(k\) is:  
\[\mathbf{u}^{*}[k+1:k+T_{\rm P}]=\arg\max_{u}\left\{\sum_{l=1}^{T_{\rm P}}(\lambda_{ k+l}J_{k+l})\right\}=\arg\max_{u}\left\{\sum_{l=1}^{T_{\rm P}}\lambda_{k+l} \cdot\tr(\hat{\mathbf{Y}}_{k+l|k})\right\} \tag{33}\]  
subject to  
\[\mathbf{x}_{\rm t}(k+1)=\mathbf{f}_{\rm t}(\mathbf{x}_{\rm t}(k),\mathbf{w}(k)) \tag{34}\]  
\[\mathbf{x}_{i}(k+1)=\mathbf{f}_{\rm u}(\mathbf{x}_{i}(k),\mathbf{u}_{i}(k)) \tag{35}\]  
\[\mathbf{z}_{i}(k)=\mathbf{h}_{i}(k,\mathbf{x}_{i}(k),\mathbf{x}_{\rm t}(k))+\mathbf{v}_{i}(k) \tag{36}\]  
\[\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}\geq d_{\rm safe },i\
eq j \tag{37}\]  
where \(\mathbf{u}[k+1:k+T_{\rm P}]=\{\mathbf{u}_{1}[k+1:k+T_{\rm P}]\}\), \(\mathbf{u}_{2}[k+1:k+T_{\rm P}],\,\ldots,\,\mathbf{u}_{N}[k+1:k+T_{\rm P}]\}\) denotes the vector of control inputs to all the UAVs over the time horizon \([k+1:k+T_{\rm P}]\). According to the UAV dynamic model, which is described by Equation (1), the control input \(\mathbf{u}\) is the desired horizontal flight path angle (\(\chi^{c}\)) of the aircraft. In this receding horizon optimization model, the purpose is to maximize the weighted sum of the trace of \(\hat{\mathbf{Y}}_{k+l|k}\) over the time horizon \([k+1:k+T_{\rm P}]\), and \(l=1,2,\,\ldots,\,T_{\rm P}\). The weight factors are given by \(\lambda(k+l)\). Equations (34)-(36) denote the target dynamical model, the UAV dynamical model, and the sensor model, respectively. Equation (37) ensures collision avoidance between the UAVs, where \(d_{\rm safe}\) is the safety distance for collision avoidance. For maximizing the weighting sum of the trace of \(\hat{\mathbf{Y}}_{k+l|k}\) over the time horizon \([k+1:k+T_{\rm P}]\), an approximate solution is to maximize the trace of \(\hat{\mathbf{Y}}_{k+l|k}\) at time \((k+l)\). Thus, Equation (33) becomes  
\[\mathbf{u}^{*}(k+l)=\arg\max_{u(k+l)}\{J_{k+l}\}=\arg\max_{u(k+l)}\left\{  
contained in the measurements of all the UAVs. \(\hat{\mathbf{Y}}_{k+l|k}^{-}\) is not related to \(\mathbf{u}^{*}(k+\hat{l})\), so \(\hat{\mathbf{Y}}_{k+l|k}^{-}\) can be ignored, and the objective function \(J_{k+l}\) in Equation (38) can be reduced to:  
\[ \begin{array}{ll}\widetilde{J}_{k+l}&=\text{tr}\big{(}\sum\limits_{i=1}^{N} \beta_{i,k+l}\cdot\hat{\mathbf{H}}_{i,k+l}^{\text{T}}\mathbf{R}_{i,k+l}^{-1}\hat{\mathbf{H} }_{i,k+l}\big{)}=\sum\limits_{i=1}^{N}\left[\beta_{i,k+l}\cdot\text{tr}(\hat{ \mathbf{H}}_{i,k+l}^{\text{T}}\mathbf{R}_{i,k+l}^{-1}\hat{\mathbf{H}}_{i,k+l})\right]\\ &\leq\underbrace{\left(\sum\limits_{i=1}^{N}\beta_{i,k+l}\right)}_{k+l}\cdot \underbrace{\left(\sum\limits_{i=1}^{N}\text{tr}(\hat{\mathbf{H}}_{i,k+l}^{\text{ T}}\mathbf{R}_{i,k+l}^{-1}\hat{\mathbf{H}}_{i,k+l})\right)}_{k+l}\end{array}  \tag{40}\]  
In Equation (40), the objective function \(\widetilde{J}_{k+l}\) contains communication optimization item \(J_{k+l}^{\text{communication}}\) and sensing optimization item \(J_{k+l}^{\text{sensing}}\). Therefore, the co-optimization of communication and sensing is reflected in the tradeoffs between communication reliability and sensing utility, aims to maximize the overall information gain delivered to the base station. In the framework of distributed receding horizon optimization, as shown in Figure 3, at time \(k\), each UAV \(A_{i}\) receives the planned control inputs \(\overline{\mathbf{U}}_{-i}(k)\) and the trajectories \(\overline{\mathbf{X}}_{-i}(k)\) from the other UAVs. Then \(A_{i}\) optimizes its control inputs \(\overline{\mathbf{U}}_{i}(k)\) and plans its own trajectory \(\overline{\mathbf{X}}_{i}(k)\) based on the estimates of the target state and the future state predictions of the other UAVs. \(A_{i}\) also can share the optimal control inputs \(\overline{\mathbf{U}}_{i}(k)\) and the planned trajectory \(\overline{\mathbf{X}}_{i}(k)\) with the other UAVs.