## 2 Mathematical Model of Orbital Pursuit-Evasion-Defense Game  
### Relative Orbital Dynamics  
The orbital pursuit-evasion-defense game occurs in the final phase of the confrontation when the spacecraft are close enough so that they can identify each other with onboard electronic devices [4]. In this type of situation, the motion between the spacecraft can be expressed as relative motion [23]. As is known, Lawden's equations [24] and Clohessy-Wiltshire (C-W) equations [25] are two linearized equations used to describe the relative motion between spacecrafts. Unlike the C-W equations, which can only be applied to circular orbits, the [PERSON]'s equations can describe the relative motion of a spacecraft in elliptical orbits. Same as in [26], the dynamics of the participating spacecraft are described in the local-vertical local-horizontal (LVLH) frame centered at a virtual spacecraft. In addition, [PERSON]'s equations are adopted as the relative dynamic equations of the spacecraft.  
As shown in Figure 1, \(P,D,E\) respectively represents the pursuer, the defender, and the evader. We establish an elliptical fictitious spacecraft \(O\) which is close to the players. The LVLH coordinate system is centered at the point \(O\). \(OX\) is pointing outward along the radius of the Earth, \(OY\) is perpendicular to \(OX\) in the reference orbital plane and points to the front of its flight direction, \(OZ\) is perpendicular to the orbital plane and forms a right-handed frame with \(OX\) and \(OY\).  
The [PERSON]'s equations can be expressed as:  
\[\left\{\begin{array}{l}\ddot{x}_{i}=\alpha^{2}x_{i}+2\omega\dot{y}_{i}+\dot{ \omega}y_{i}+2\frac{\mu x_{i}}{r_{i}^{3}}+T_{i}u_{xi}\\ \ddot{y}_{i}=-2\alpha\dot{x}_{i}-\dot{\alpha}x_{i}+\omega^{2}y_{i}-\frac{\mu y _{i}^{3}}{r_{i}^{3}}+T_{i}u_{yi}\\ \ddot{z}_{i}=-\frac{\mu z_{i}}{r_{i}^{3}}+T_{i}u_{zi}\end{array}\right.,\qquad i =P,D,E \tag{1}\]  
where \(\mu\) is the Earth gravitational constant, \(r_{t}\) is the distance between the origin \(O\) and the Earth core, \(\omega\) and \(\dot{\omega}\) represent the orbital angular velocity and acceleration of the origin \(O\), respectively. \(x_{i}\), \(y_{i}\), and \(z_{i}\) represent the position components of the players in the relative coordinate system. \(T_{i}\) represents the maximum thrust. \(u_{xi}\), \(u_{yi}\) and \(u_{zi}\) respectively represent control variables in three directions (i.e., \(x,y,z\) axis), ranging from 0 to 1.  
The state variables (i.e., position and velocity) of the players are represented by \(\textbf{X}_{i}\) as follows:  
\[\textbf{X}_{i}=\left[x_{i},y_{i},z_{i},\dot{x}_{i},\dot{y}_{i},\dot{z}_{i} \right]^{\text{T}},\qquad\qquad i=P,D,E \tag{2}\]  
Thus, the dynamics equations can be written as:  
\[\dot{\textbf{X}}_{i}=\textbf{A}_{i}\textbf{X}_{i}+T_{i}\textbf{U}_{i} \tag{3}\]  
where  
\[\textbf{A}_{i}(t)=\left[\begin{array}{cccccc}0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ \omega^{2}+\frac{2\mu}{r_{i}^{3}}&\dot{\omega}&0&0&2\omega&0\\ -\dot{\omega}&\omega^{2}-\frac{\mu}{r_{i}^{3}}&0&-2\omega&0&0\\ 0&0&-\frac{\mu}{r_{i}^{3}}&0&0&0\end{array}\right],\quad i=P,D,E \tag{4}\]  
\(\textbf{U}_{i}\)\((i=P,D,E)\) is the control variable, which can be represented by  
\[\textbf{U}_{i}=\left[0,0,0,u_{xi},u_{yi},u_{zi}\right]^{\text{T}},\quad\left\| \textbf{U}_{i}\right\|\leq 1 \tag{5}\]  
Figure 1: The local-vertical local-horizontal (LVLH) coordinate system.