## 5 Multi-Drone Guidance Law
### Guidance Law of the Virtual Agent
Here, recall that \(\bar{L}_{k+1}\) denotes the infinite line crossing both \(\mathbf{r}_{k+1}^{s}\) and \(\mathbf{r}_{k+1}^{t}\). In the inertial reference frame, let \(\mathbf{I}_{L}\) denote the 3D coordinates of the intersection between \(L_{k}\) and \(\bar{L}_{k+1}\). For instance, if the ship is static, then \(L_{k}\) and \(\bar{L}_{k+1}\) meet at the ship position \(\mathbf{r}_{k}^{s}=\mathbf{r}_{k+1}^{s}\). Suppose that \(\mathbf{r}_{0,k}\) lies on \(l(\mathbf{r}_{k}^{t},\mathbf{r}_{k}^{s})\). Let \(\mathbf{c}_{k}^{t}\) denote the point on \(\bar{L}_{k+1}\), which is the closest to \(\mathbf{r}_{k}^{t}\). Let \(l_{k}=\|\mathbf{c}_{k}-\mathbf{r}_{0,k}\|\) for convenience. Figure 3 depicts the case where \(L_{k}\) and \(\bar{L}_{k+1}\) meet at \(\mathbf{I}_{L}\). Using the geometry in this figure, we have  
\[\|\mathbf{c}_{k}^{t}-\mathbf{r}_{k}^{t}\|=\frac{l_{k}*\|\mathbf{I}_{L}- \mathbf{r}_{k}^{t}\|}{\|\mathbf{I}_{L}-\mathbf{r}_{0,k}\|}. \tag{28}\]  
Since the target speed is \(v_{k}^{t}\), we have  
\[v_{k}^{t}*T\geq\|\mathbf{c}_{k}^{t}-\mathbf{r}_{k}^{t}\|. \tag{29}\]  
Using Equations (28) and (29), we have  
\[\mathbf{g}_{k} = \mathbf{c}_{k}+\frac{\mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}}{\| \mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}\|}*\delta_{k}. \tag{22}\]0  
Suppose that the drone's maximum speed \(v_{max}\) satisfies  
\[\mathbf{g}_{k} = \mathbf{c}_{k}+\frac{\mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}}{\| \mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}\|}*\delta_{k}. \tag{22}\]1  
\[\mathbf{g}_{k} = \mathbf{c}_{k}+\frac{\mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}}{\| \mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}\|}*\delta_{k}. \tag{22}\]2 The case where \(L_{k}\) and \(\bar{L}_{k+1}\) meet at \(\mathbf{I}_{L}\). The next theorem addresses the condition for remaining in the lineState at every sample-index. **Theorem 2**: _Suppose that \(L_{k}\) and \(\bar{L}_{k+1}\) meet at a point, say \(\mathbf{I}_{L}\). Suppose that \(\mathbf{r}_{0,k}\) lies on \(l(\mathbf{r}_{k}^{t},\mathbf{r}_{k}^{s})\). The target speed is \(v_{k}^{t}\). If the drone's maximum speed satisfies Equation (31), then \(\mathbf{r}_{0,k+1}\) lies on \(l(\mathbf{r}_{k+1}^{t},\mathbf{r}_{k+1}^{s})\)._  
Suppose the drone's maximum speed satisfies Equation (31). Then, Equations (30) and (31) lead to  
\[\mathbf{g}_{k} = \mathbf{c}_{k}+\frac{\mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}}{\| \mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}\|}*\delta_{k}. \tag{22}\]3  
This implies that Equation (21) is met. In the case where Equation (21) is met, Theorem 1 makes \(\mathbf{r}_{0,k+1}\) exist on \(l(\mathbf{r}_{k+1}^{t},\mathbf{r}_{k+1}^{s})\). The proof is complete. In practice, the ship moves much slower than the target. Thus, the ship position is close to \(\mathbf{I}_{L}\), as plotted in \[\mathbf{g}_{k} = \mathbf{c}_{k}+\frac{\mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}}{\| \mathbf{r}_{k+1}^{t}-\mathbf{c}_{k}\|}*\delta_{k}. \tag{22}\]4 Consider the case where the virtual agent is sufficiently close to the ship. In this case, \(\|\mathbf{I}_{L}-\mathbf{r}_{0,k}\|\) is small to satisfy Equation (31). Then, using Corollary 2, \(\mathbf{r}_{0,k+1}\) lies on \(l(\mathbf{r}_{k+1}^{t},\mathbf{r}_{k+1}^{s})\). Thus, the virtual agent remains in the lineState at sample-index \(k+1\). In the case where the virtual agent stays in the lineState at all times, the target cannot reach the ship without being captured by the virtual agent. In order to stay in the lineState at each sample-index, it is desirable that the virtual agent does not become too far from the ship. In the case where \(\|\mathbf{r}_{k}^{s}-\mathbf{r}_{0,k}\|\geq\beta\) is met, the guidance point is set using Equation (24) instead of Equation (22). In this way, drones stay close to the ship while staying in the lineState at all times. Note that Equation (31) can be satisfied even when \(v_{max}<v_{k}^{t}\). This implies that even slow drones can capture a fast target when the drones stay close to the ship while staying in the lineState at all times.