# Compensation Control Strategy for Orbital Pursuit-Evasion Problem with Imperfect Information
## 2 Mathematical Model of the Orbital Pursuit-Evasion Problem  
The orbital pursuit-evasion problem occurs in the situation that the players are close enough so that they can identify each other with onboard electronic devices [4]. In this situation, the players' nonlinear dynamics can be reduced to the linear C-W equations [3]. Same as in [3], the players' dynamics are described in the local-vertical local-horizontal (LVLH) frame [27] centered on a virtual spacecraft, which follows a circular orbit near the players. Besides, the C-W equations are adopted to describe the players' relative motion.  
As Figure 1 shows, we set a virtual spacecraft O, which is close to the players. The LVLH coordinate system is established on the point O. OX points outwards 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, and OZ is perpendicular to the orbital plane and forms a right-handed frame with OX and OY.  
The C-W equations can be expressed as  
\[\left\{\begin{array}{l}\ddot{x}_{i}=3\omega^{2}x_{i}+2\omega\dot{y}_{i}+u_{xi }\\ \dot{y}_{i}=-2\omega\dot{x}_{i}+u_{yi}\\ \ddot{z}_{i}=-\omega^{2}z_{i}+u_{zi}\end{array}\right.,\quad i=P,E \tag{1}\]  
where \(P\) and \(E\) represent the pursuer and the evader, respectively. \(\omega\) represents the orbital angular velocity of the origin O. \(x_{i}\), \(y_{i}\), and \(z_{i}\) represent the position components of the players in the relative coordinate system. \(u_{xi}\), \(u_{yi}\), and \(u_{zi}\), respectively, represent control variables in the three directions (i.e., \(x\), \(y\), and \(z\) axis).  
The state variables of both players are represented by \(\mathbf{X}_{i}\) as follows:  
\[\mathbf{X}_{i}=\left[x_{i},y_{i},z_{i},\dot{x}_{i},y_{i},z_{i}\right]^{\mathsf{T}},\quad\quad\quad i=P,E. \tag{2}\]  
Thus, the dynamics equations can be written as  
\[\dot{\mathbf{X}}_{i}=\mathbf{A}\mathbf{X}_{i}+\mathbf{B}\mathbf{U}_{i} \tag{3}\]  
Figure 1: The local-vertical local-horizontal (LVLH) coordinate system.  
where  
\[\mathbf{A}=\left[\begin{array}{cccccc}0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 3\omega^{2}&0&0&0&2\omega&0\\ 0&0&0&-2\omega&0&0\\ 0&0&-\omega^{2}&0&0&0\end{array}\right],\quad\mathbf{B}=\left[0_{3\times 3},I_{3 \times 3}\right]^{T} \tag{4}\]  
\(\mathbf{U}_{i}=\left[u_{xi},u_{yi},u_{zi}\right]^{T},\,\left(i=P,E\right)\) is the control variable, which satisfies \(\left\|\mathbf{U}_{P}\right\|_{2}\leq\rho_{P}\), \(\left\|\mathbf{U}_{E}\right\|_{2}\leq\rho_{E},\,\rho_{P}>\rho_{E}>0\). \(\rho_{P}\) and \(\rho_{E}\) are constants.  
Furthermore, we take the relative states of the two spacecraft as state variables. By defining \(\mathbf{X}_{PE}(t)~{}=~{}\mathbf{X}_{P}(t)-\mathbf{X}_{E}(t)\), the state equations are converted to  
\[\dot{\mathbf{X}}_{PE}(t)=\mathbf{AX}_{PE}(t)+\mathbf{BU}_{P}(t)+\mathbf{CU}_{E}(t) \tag{5}\]  
where \(\mathbf{C}=-\mathbf{B}\).  
By taking the terminal distance as the cost, the objective function can be defined as  
\[J=\left\|\mathbf{DX}_{PE}\Big{(}t_{f}\Big{)}\right\| \tag{6}\]  
where \(\mathbf{D}=\left[I_{3\times 3},0_{3\times 3}\right]\).