## 4 Assumptions and Definitions  
This section discusses assumptions and definitions in our paper. \(max(a,b)\) returns a bigger value between two variables (\(a\) and \(b\)). In addition, \(min(a,b)\) returns a smaller value between two variables (\(a\) and \(b\)). In our paper, bold characters are used to denote vectors and matrices. \(\angle(\mathbf{v}_{1},\mathbf{v}_{2})\) is the angle formed by two vectors (\(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\)). Mathematically, \(\angle(\mathbf{v}_{1},\mathbf{v}_{2})=\arccos(\frac{\mathbf{v}_{1},\mathbf{v} _{2}}{\|\mathbf{v}_{1}\|\cdot\|\mathbf{v}_{2}\|})\). Here, \(0\leq\angle(\mathbf{v}_{1},\mathbf{v}_{2})\leq\pi\). \(l(\mathbf{A},\mathbf{B})\) is the line segment connecting two locations \(\mathbf{A}\) and \(\mathbf{B}\). Furthermore, \(\|l(\mathbf{A},\mathbf{B})\|\) indicates the length of \(l(\mathbf{A},\mathbf{B})\). This article uses the discrete-time system, where \(T\) denotes the sample duration. In this article, all drones make a planar grid formation to protect against the incoming target. The grid formation can be considered as a \"net\" structure for capturing the incoming target. Let \(M\) indicate the total number of drones. \(M\) is selected such that  
\[M=G^{2}, \tag{4}\]  
where \(G\geq 1\) is a positive integer. In the case where \(G=1\), we use only one drone. In this case, the grid formation cannot be used, and the waypoint of the drone is set as the virtual agent. In the inertial reference frame, let \(\mathbf{r}_{0,k}\) define the 3D Cartesian coordinates of the virtual agent. In the inertial reference frame, let \(\mathbf{r}_{i,k}\) (\(i\in\{1,2,\ldots,M\}\)) denote the 3D Cartesian coordinates of the \(i\)-th drone at sample-index \(k\). Note that the subscript \(k\) indicates the sample-index \(k\). In the inertial reference frame, let \(\mathbf{r}_{k}^{t}\) denote the target's 3D Cartesian coordinates at sample-index \(k\). In the inertial reference frame, let \(\mathbf{r}_{k}^{s}\) denote the ship's 3D Cartesian coordinates at sample-index \(k\). Let \(v_{k}^{t}\) denote the target's speed at sample-index \(k\). Let \(v_{ijk}\) denote the speed of the \(i\)-th drone at sample-index \(k\). Let \(v_{max}\) indicate the maximum speed of a drone or the virtual agent. Note that \(v_{k}^{t}\), \(v_{i,k}\), and \(v_{max}\) are scalar values. We say that the target is _captured_ when the relative distance between the target and any drone is less than a constant, say \(\Delta\). The motion model of the \(i\)-th drone (\(i\in\{1,2,\ldots,M\}\)) is  
\[\mathbf{r}_{i,k+1}=\mathbf{r}_{i,k}+T*v_{i,k}*\mathbf{u}_{i,k}. \tag{5}\]  
Here, \(\mathbf{u}_{i,k}\) indicates the \(i\)-th drone's heading vector at sample-index \(k\). Note that \(\mathbf{u}_{ijk}\) is a unit vector presenting the \(i\)-th drone's heading direction. The motion model in Equation (5) is commonly used in multi-drone systems [40; 41; 42; 43; 44; 45; 46]. In Equation (5), \(\mathbf{r}_{i,k+1}\) generates the high-level reference position signal at every sample-index \(k\). For letting the \(i\)-th drone move towards \(\mathbf{r}_{i,k+1}\) at every sample-index \(k\), one utilizes low-level controls in [1; 2; 3]. The motion model of the virtual agent is  
\[\mathbf{r}_{0,k+1}=\mathbf{r}_{0,k}+T*v_{0,k}*\mathbf{u}_{0,k}. \tag{6}\]  
Recall that the virtual agent is at the center of the grid formation. We say that the virtual agent is _in the lineState_ at sample-index \(k\) if the line segment \(l(\mathbf{r}_{k}^{t},\mathbf{r}_{k}^{s})\) meets the virtual agent position \(\mathbf{r}_{0,k}\). At every sample-index \(k\), \(v_{0,k}\) and \(\mathbf{u}_{0,k}\) are set so that the virtual agent is in the lineState. Let \(L_{k}\) denote the infinite line crossing both \(\mathbf{r}_{k}^{s}\) and \(\mathbf{r}_{k}^{t}\) at sample-index \(k\). Let \(\bar{L}_{k+1}\) denote the infinite line crossing both \(\mathbf{r}_{k+1}^{s}\) and \(\mathbf{r}_{k+1}^{t}\). \(\mathbf{c}_{k}\) is the point on \(\bar{L}_{k+1}\), which is the closest to \(\mathbf{r}_{0,k}\).