## 2 Mathematical Model of Orbital Pursuit-Evasion-Defense Game
### Dimension-Reduction  
According to Equation (3), each player has 6 state variables, so the total number of state variables in the game is 18. In the numerically solving process, co-state variables associated with state variables are introduced, and the problem converts to a 36-dimensional TPBVP. However, solving this high-dimensional TPBVP is quite difficult and possesses high computational demands [27]. In order to improve computational efficiency, the dimension of the system needs to be reduced. This process is performed in two steps. First, the relative state variables between the spacecraft are used to replace the system states. Then, the ZEM variables are applied to further reduce the number of variables and equations in the system.  
In the first step, the game can be divided into two parts: One is the game between the pursuer \(P\) and the evader \(E\); the other is the game between the defender \(D\) and the pursuer \(P\). The relative state variables in two parts, \(\mathbf{X}_{\text{PE}}\) and \(\mathbf{X}_{\text{DP}}\), can be respectively represented by  
\[\left\{\begin{array}{l}\mathbf{X}_{\text{PE}}=\mathbf{X}_{\text{P}}-\mathbf{X}_{\text{E }}\\ \mathbf{X}_{\text{DP}}=\mathbf{X}_{\text{D}}-\mathbf{X}_{\text{P}}\end{array}\right. \tag{6}\]  
Substituting Equation (6) into Equation (3), the state equations are converted to:  
\[\left\{\begin{array}{l}\dot{\mathbf{X}}_{\text{PE}}=\mathbf{A}\mathbf{X}_{\text{PE}}+T_ {\text{P}}\mathbf{U}_{\text{P}}-T_{\text{E}}\mathbf{U}_{\text{E}}\\ \dot{\mathbf{X}}_{\text{DP}}=\mathbf{A}\mathbf{X}_{\text{DP}}+T_{\text{D}}\mathbf{U}_{\text{ D}}-T_{\text{P}}\mathbf{U}_{\text{P}}\end{array}\right. \tag{7}\]  
where \(\mathbf{A}=\mathbf{A}_{\text{P}}=\mathbf{A}_{\text{E}}=\mathbf{A}_{\text{D}}\).  
In the second step, according to the linear system theory, the zero-input state transfer matrix \(\mathbf{\Phi}(t_{\text{f}},t)\) of the state equation is defined as:  
\[\left\{\begin{array}{l}\mathbf{\Phi}(t_{\text{f}},t)=-\Phi(t_{\text{f}},t)\text {A}\\ \mathbf{\Phi}(t_{\text{f}},t_{\text{f}})=\mathbf{I}_{6}\end{array}\right. \tag{8}\]  
where \(t_{\text{f}}\) is the terminal time and \(\mathbf{I}_{6}\) is the \(6\times 6\) unit matrix.  
Although two factors, the relative position and the relative velocity are involved in the game, only the first factor needs to be considered at the end of the game. The ZEM is the miss distance if both players do not apply any control from the current moment to the end of the game. Thus, the ZEM variables are introduced to reduce the dimension of the system and defined as:  
\[\left\{\begin{array}{l}\mathbf{Z}_{\text{PE}}(t)=\mathbf{D}\mathbf{\Phi}(t_{\text{f}},t )\mathbf{X}_{\text{PE}}\\ \mathbf{Z}_{\text{DP}}(t)=\mathbf{D}\mathbf{\Phi}(t_{\text{f}},t)\mathbf{X}_{\text{DP}}\end{array}\right. \tag{9}\]  
where \(\mathbf{D}=[I_{3\times 3},0_{3\times 3}]\).  
Substituting Equation (9) into Equation (7), the state equations are reduced to:  
\[\left\{\begin{array}{l}\dot{\mathbf{Z}}_{\text{PE}}=\mathbf{D}\mathbf{\Phi}T_{\text{P}} \mathbf{U}_{\text{P}}-\mathbf{D}\mathbf{\Phi}T_{\text{E}}\mathbf{U}_{\text{E}}\\ \dot{\mathbf{Z}}_{\text{DP}}=\mathbf{D}\mathbf{\Phi}T_{\text{D}}\mathbf{U}_{\text{D}}-\mathbf{D} \mathbf{\Phi}T_{\text{P}}\mathbf{U}_{\text{P}}\end{array}\right. \tag{10}\]