## 2 Mathematical Model of Orbital Pursuit-Evasion-Defense Game
### Design of Objective Function Based on Fuzzy Comprehensive Evaluation  
In the orbital pursuit-evasion-defense game, the pursuer must survive from the defender's interception before it can successfully access the evader. Therefore, the pursuer-evader game and the defender-pursuer game must be considered and weighed in the objective function of the pursuer. As noted by [PERSON] [19], the fuzzy comprehensive evaluation is an effective way to quantify various factors that are difficult to evaluate. Thus, it is used to obtain the weights corresponding to the two games. Detailed design is shown as follows. By taking the terminal miss distance as the cost, the objective function of the three players can be defined as:  
\[\left\{ \begin{array}{l}J_{\text{E}}=-\|\mathbf{Z}_{\text{PE}}(t_{\text{f}})\|\\ I_{\text{D}}=\|\mathbf{Z}_{\text{DP}}(t_{\text{f}})\|\\ J_{\text{P}}=k_{1}\|\mathbf{Z}_{\text{PE}}(t_{\text{f}})\|-k_{2}\|\mathbf{Z}_{ \text{DP}}(t_{\text{f}})\|\end{array} \right. \tag{11}\]  
where the parameter \(k_{i}\), \(i=1,2\) is the weight factor, which satisfies \(k_{i}\geq 0\). \(k_{1}>k_{2}\) indicates that the pursuer prefers to reduce the terminal miss distance of the pursuer-evader game, while \(k_{1}<k_{2}\) indicates that the pursuer prefers to increase the terminal miss distance of the defender-pursuer game. The value of \(k_{i}\) is divided into 11 scales, which are shown in Table 1.  
According to the analysis above, two factors need to be evaluated, one is the urgency of intercepting the evader at the very moment, denoted by \(u_{1}\); the other is the urgency of evading the defender at the very moment, denoted by \(u_{2}\). The effect of the factor \(u_{1}\) increases as the ZEM distance of the pursuer-evader game decreases. The effect of the factor \(u_{2}\) increases as the ZEM distance of the defender-pursuer game decreases. According to this rule, \(\mathbf{Z}_{\text{PE}}(t)\) and \(\mathbf{Z}_{\text{DP}}(t)\) are used to construct the weights of the two factors, which are given by:  
\[\left\{ \begin{array}{l}a_{1}=1-\left(\frac{\|\mathbf{Z}_{\text{PE}}(t)\|}{ \|\mathbf{Z}_{\text{PE}}(t)\|+\|\mathbf{Z}_{\text{DP}}(t)\|}\right)^{3}\\ a_{2}=1-a_{1}\end{array} \right. \tag{12}\]  
where \(a_{1}\) and \(a_{2}\) represent the weights of the factor \(u_{1}\) and the factor \(u_{2}\) respectively. Then the weight vector is expressed as: \(\mathbf{A}=[a_{1},a_{2}]\). In order to establish the relationship between weighting factors and evaluation scales, the membership degree of each factor is calculated by the non-linear membership function which is written as follows:  
\[\left\{ \begin{array}{l}u_{1}(x)=\left(k(x-1)\right)^{3}\\ u_{2}(x)=1-\left(k(x-1)\right)^{3}\end{array} \right. \tag{13}\]  
where \(k=0.1\), \(x=1,\cdots,11\) are the corresponding evaluation scales. Let \(r_{ij}=u_{i}(j)\), where \(i=1,2,j=1,\cdots 11\), the fuzzy evaluation matrix can be obtained as:  
\[\mathbf{R}=\left[r_{ij}\right]_{2\times 11} \tag{14}\]  
The fuzzy comprehensive evaluation result vector is generated by the fuzzy synthetic operation of the weight vector and the fuzzy evaluation matrix. The fuzzy synthetic formula is defined as follows:  
\[\mathbf{B}=\mathbf{A}\circ\mathbf{R}=\left(b_{1},b_{2},\cdots,b_{11}\right) \tag{15}\]  
where \"\(\circ\)\" is a fuzzy synthetic operator. In this paper, the weighted average fuzzy arithmetic operator is adopted, which can make full use of the information of \(\mathbf{R}\). It is specifically expressed as:  
\[b_{j}=\text{min}\Bigg{\{}1,\sum_{i=1}^{2}a_{i}r_{ij}\Bigg{\}},\;\;j=1,2, \cdots,11 \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(v_{i}\)1 & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline \(k_{1}\) & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ \hline \(k_{2}\) & 1 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 & 0.4 & 0.3 & 0.2 & 0.1 & 0 \\ \hline \end{tabular}  
* \(v_{i},i=1,\cdots,11\) represents the corresponding scales, respectively.  
\end{table} 
Table 1: The evaluation scales.