# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 2 Problem Formulation  
### System Model  
In spacecraft PE games, the relative dynamics equations of the spacecraft are described in the Local Vertical Local Horizontal (LVLH) coordinate system, as illustrated in Figure 1. The LVLH frame is centered on a virtual reference spacecraft following a nominal reference orbit. The \(x\)-axis points from the Earth's center toward the origin, the \(y\)-axis lies within the orbital plane, perpendicular to the \(x\)-axis and aligned with the direction of the spacecraft's motion, while the \(z\)-axis is determined by the right-hand rule, coinciding with the direction of the orbital angular momentum. \begin{table}
\begin{tabular}{l l l l l l} \hline
**Category** & **Liu [24]** & \multicolumn{2}{l}{**Proposed Method**} & \multicolumn{2}{l}{**Advancement**} \\ \hline \multirow{4}{*}{J\({}_{2}\) Handling} & \(\bullet\) & Linearized & \(\bullet\) & Nonlinear coupling & \(\bullet\) & 62\% lower position \\  & \(\bullet\) & approximation & & preservation & & error \\  & \(\bullet\) & Offline & \(\bullet\) & Real–time adaptive & \(\bullet\) & Exact sin\({}^{2}(kt)\) terms \\  & & pre–compensation & & rejection & & \\ \hline \multirow{3}{*}{Control Strategy} & \(\bullet\) & Fixed–gain controller & \(\bullet\) & Dynamic gain & \multirow{3}{*}{\(\bullet\)} & \multirow{3}{*}{\(\bullet\)} \\  & \(\bullet\) & Single–step prediction & & & & \\ \cline{1-1}  & & & & adjustment & & 22\% fuel saving \\ \cline{1-1}  & \(\bullet\) & Single–step prediction & & & Multi–horizon & \(\bullet\) \\ \cline{1-1}  & & & & optimization & & \\ \hline \multirow{3}{*}{Stability} & \multirow{3}{*}{Local convergence} & \multirow{3}{*}{Global UUB (Theorem 1)} & \(\bullet\) & Guaranteed \(O(Nn^{3})\) \\  & & & & & & PE-conditioned \\ \cline{1-1}  & & & & & \(\bullet\) & 6.42\% faster than SSE \\ \cline{1-1}  & & & & & \(\bullet\) & Parallelizable \\ \hline \end{tabular}
\end{table} 
Table 1: Comparative analysis with state–of–the–art approaches. The dynamics of a single spacecraft in the Local Vertical Local Horizontal (LVLH) coordinate system can be described as follows [24]:  
\[ \begin{cases}\ddot{x}-2n_{c}\dot{y}-(5c^{2}-2)n^{2}x=-3n^{2}J_{2}\left(\frac{R_{ c}}{r_{ref}}\right)^{2}\times\left\{\frac{1}{2}-A-B\right\}+a_{x}\\ \ddot{y}+2n_{c}\dot{x}=-3n^{2}J_{2}\left(\frac{R_{c}}{r_{ref}}\right)^{2}\sin^ {2}(i_{ref})\sin(kt)\cos(kt)+a_{y}\\ \ddot{z}+q^{2}z=2 lq\cos(qt+\phi)+a_{z}\end{cases}  \tag{1}\]  
These equations represent the relative motion in the LVLH frame, including the primary effects of the Earth's oblateness (\(J_{2}\) perturbation). The terms involving \(n^{2}\) describe linearized gravitational and centrifugal forces, while the terms with \(2n_{c}\) represent Coriolis forces. The \(J_{2}\) perturbation introduces additional forces that cause long-term drift, which are captured by the trigonometric terms and constants defined in Appendix A. We denote the state and control input for the pursuer as \(\mathbf{f}_{p}\) and \(\mathbf{a}_{p}\), and for the evader as \(\mathbf{f}_{e}\) and \(\mathbf{a}_{e}\), respectively. The control inputs \(\mathbf{a}_{p},\mathbf{a}_{e}\in\mathbb{R}^{3}\) represent the thrust acceleration vectors. The relative state of the PE game is defined as \(\mathbf{\tilde{f}}=\mathbf{f}_{e}-\mathbf{f}_{p}\). After feedback linearization and discretization, the relative dynamics of the PE game can be expressed as:  
\[\mathbf{\tilde{f}}_{t+1}=\mathbf{X}_{d}\mathbf{\tilde{f}}_{t}+\mathbf{Y}_{d}\mathbf{a}_{p,t+1}+Y_{e }\mathbf{a}_{e,t} \tag{2}\]  
The state vector \(\mathbf{\tilde{f}}_{t}\in\mathbb{R}^{6}\) and control inputs \(\mathbf{a}_{p,t},\mathbf{a}_{e,t}\in\mathbb{R}^{3}\) imply that the control gain matrices \(\mathbf{L}_{p,t},\mathbf{L}_{e,t}\) map \(\mathbb{R}^{6}\rightarrow\mathbb{R}^{3}\); hence, their dimensions are \(3\times 6\). The adaptive gain matrices \(\mathbf{\Gamma}_{p},\mathbf{\Gamma}_{e}\in\mathbb{R}^{6\times 6}\) are positive definite to ensure compatibility with the state-space update laws (Equation (28)).