## 5 Multi-Drone Guidance Law
### Prediction of the Target's Position after One Sample-Index  
The ship can measure the target's 3D Cartesian coordinates at every sample-index. To track a target with variable velocity, we present how to predict the target one sample-index forward in time. In order to track a maneuvering target, we applied target-tracking filters to [4]. In the inertial reference frame, let \([x_{k}^{t},y_{k}^{t},z_{k}^{t}]\) indicate the vector presenting the 3D coordinates of the target at sample-index \(k\). In addition, \([\dot{x}_{k}^{t},\dot{y}_{k}^{t},\dot{z}_{k}^{t}]^{T}\) denotes the vector presenting the target velocity at sample-index \(k\). Furthermore, \([\dot{x}_{k}^{t},\dot{y}_{k}^{t},\dot{z}_{k}^{t}]^{T}\) defines the vector presenting the target acceleration at sample-index \(k\). Let \(\mathbf{X}_{k}=[\dot{x}_{k}^{t},\dot{x}_{k}^{t},\dot{x}_{k}^{t},\dot{y}_{k}^{t},\dot{y}_{k}^{t},\dot{y}_{k}^{t},\dot{z}_{k}^{t},z_{k}^{t},\dot{z}_{k}^{t}]^{T}\) define the vector presenting the target state. Based on [4], the target's process model is set as  
\[\mathbf{X}_{k+1}=\mathbf{F}*\mathbf{X}_{k}+\mathbf{w}_{k}, \tag{7}\]  
where \(\mathbf{w}_{k}\) is the process noise with following properties: \(\mathbf{w}_{k}\sim\mathcal{N}(\mathbf{0},\mathbf{Q})\). Here, \(\mathcal{N}(\mathbf{0},\mathbf{\alpha})\) denotes a Gaussian distribution with a mean of \(\mathbf{0}\) and a covariance matrix \(\mathbf{\alpha}\). Furthermore, \(\mathbf{F}\) in Equation (7) is  
\[\mathbf{F}=\left( \begin{array}{ccc}\mathbf{M}_{F}&0&0\\ 0&\mathbf{M}_{F}&0\\ 0&0&\mathbf{M}_{F}\end{array} \right), \tag{8}\]  
where we use  
\[\mathbf{M}_{F}=\left( \begin{array}{ccc}1,&T,&\frac{a*T-1+\varepsilon^{-a*T} }{a^{2}}\\ 0,&1,&\frac{1+\varepsilon^{-a*T}}{a^{2}}\\ 0,&0,&e^{-2a*T}\end{array} \right). \tag{9}\]  
In \(\mathbf{w}_{k}\), \(\mathbf{Q}\) is set as  
\[\mathbf{Q}=2*a*\sigma_{m}^{2}*\left( \begin{array}{ccc}\mathbf{M}_{Q}&0&0\\ 0&\mathbf{M}_{Q}&0\\ 0&0&\mathbf{M}_{Q}\end{array} \right), \tag{10}\]  
where  
\[\mathbf{M}_{Q}=\left( \begin{array}{ccc}T^{5}/20,&T^{4}/8,&T^{3}/6\\ T^{4}/8,&T^{3}/6,&T^{2}/2\\ T^{3}/6,&T^{2}/2,&T\end{array} \right). \tag{11}\]  
\(a\) and \(\sigma_{m}\) in Equation (10) are tuning parameters for tracking a maneuvering target. Detailed derivations of Equation (7) appear in [4]. Figure 2: Block diagram of the proposed multi-drone guidance law. At every sample-index \(k\), the ship measures the target's 3D position, say \(\mathbf{m}_{k}\). See the first block of Figure 2. The target measurement model is  
\[\mathbf{m}_{k}=\mathbf{H}*\mathbf{X}_{k}+\mathbf{v}_{k} \tag{12}\]  
where \(\mathbf{H}\) is  
\[\mathbf{H}=\left( \begin{array}{ccccccccc}1&0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0\end{array} \right). \tag{13}\]  
Furthermore, \(\mathbf{v}_{k}\) is the measurement noise, such that \(\mathbf{v}_{k}\sim\mathcal{N}(\mathbf{0},\mathbf{R}_{k})\). We assume that \(\mathbf{R}_{k}\) is known a priori. The Kalman filter (KF) [47] is applied to obtain the estimate vector and its covariance at every sample-index. The KF is composed of the prediction step and the measurement update step. In the KF, the prediction step uses Equation (7), and the measurement update step uses Equation (12). Let \(\hat{\mathbf{X}}_{k|k}\) define the estimation of \(\mathbf{X}_{k}\) derived using all measurements up to sample-index \(k\). Let \(\mathbf{P}_{k|k}\) define the error covariance matrix of \(\hat{\mathbf{X}}_{k|k}\). In the prediction step of the KF, we derive the predicted state vector as  
\[\hat{\mathbf{X}}_{k+1|k}=\mathbf{F}*\hat{\mathbf{X}}_{k|k}, \tag{14}\]  
where Equation (7) is used.