# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 4 Simulation Experiment  
The reference orbital radius in the spacecraft model is \(a_{0}=8\times 10^{6}\:\mathrm{km}\), the orbital angular velocity \(\omega=7.2722\times 10^{-5}\:\mathrm{rad/s}\), the Earth's gravitational constant \(\mu=3.986\times 10^{14}\:\mathrm{m}^{2}/\mathrm{s}^{2}\), and the sampling period \(T=0.1\mathrm{s}\). The algorithm is set with the parameter \(V_{p}=V_{N,p}=\mathbf{I}\), \(W_{1,p}=\mathbf{I}\), \(W_{1,e}=100\mathbf{I}\), \(W_{2,p}=\mathbf{I}\), \(W_{2,e}=10\mathbf{I}\) where \(\mathbf{I}\) denotes the unit matrix with appropriate dimensions. The initial state of the model is  
\[\mathbf{\tilde{f}}_{0}=1000\Big{[}-76\quad-107\quad 48\quad 8\quad-6.1\quad-2.5 \Big{]}^{\mathrm{T}}\]  
Number of iterations N = 100. According to the algorithm, the control gain can be obtained as follows:  
\[\hat{\mathbf{L}}_{e,100}=\left[ \begin{array}{cccccc}7.66&-0.09&-0.13&0.10&0.31& -0.02\\ 15.6&-0.16&-0.11&0.13&0.67&-0.10\\ 0.00&0.00&-0.28&0.00&0.00&0.07\end{array} \right]\]  
The Persistent Excitation condition is naturally satisfied in our simulation due to the evader's dynamic maneuvers and the pursuer's continuous adaptation, ensuring sufficient signal richness for parameter convergence. The initial relative states of the PE spacecraft are set as shown in Table 2.  
All simulation results presented are based on the initial conditions specified in Table 2 unless otherwise noted. For the performance comparison plots (Figures 4-9), a single representative simulation run is shown to illustrate the typical dynamic behavior of the strategies. Figure 4 demonstrates the simulation results of the model with reference to the MSE strategy algorithm. The figure describes the change rule of the relative motion state of the pursuer, and the results show that with the increase in the iteratio \begin{table}
\begin{tabular}{c c c} \hline \hline
**Relative State** & **Pursuit** & **Evader** \\ \hline \(x\) (km) & 10 & 0 \\ \(y\) (km) & 0 & 0 \\ \(z\) (km) & 0 & 0 \\ \(\dot{x}\) (m/s) & 1.7 & 0 \\ \(\dot{y}\) (m/s) & 4.3 & 0 \\ \(\dot{z}\) (m/s) & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial relative state of the pursuit and evader spacecraft. position finally tends to 0. This indicates that, under the designed game control strategy, by dynamically adjusting the control strategy, the pursuer, as a follower, is able to adapt to the actions of the evader, and gradually reduces the distance between the two. The pursuer finally achieves the pursuit of the evader. In order to further verify the effectiveness of the proposed control algorithm, the model constructed in this paper is compared with the Stackelberg equilibrium-based (MSE) model reference adaptive control (MSE) algorithm and the single-step predictive Stackelberg equilibrium (SSE) algorithm [24] as well as the non-zero-sum Nash equilibrium (NE) algorithm [18], and the simulation results are now presented as follows. Figure 5 illustrates the acceleration variations of the spacecraft in the three directions for the three strategies. The MSE algorithm takes into account the dynamic interaction between the pursuer and the evader, allowing the pursuer to respond quickly and take more aggressive actions in the early stages of the game, resulting in a large acceleration in the initial phase, followed by a rapid decrease and stabilization, which suggests that the strategy quickly adjusts the spacecraft state to match the motion of the evader in the initial phase. Figure 4: The law of the relative motion state of the Pursuit–Evasion spacecraft. **(a)** The law of variation of relative position with time steps. **(b)** Trajectory diagram of pursuit and evader. Figure 5: Acceleration changes of spacecraft under three strategies. **(a)** The law of variation of \(a_{x}\). **(b)** The law of variation of \(a_{y}\). **(c)** The law of variation of \(a_{z}\). **(d)** The law of variation in the magnitude of acceleration. Figure 6a illustrates the changes in the cost function of spacecraft pursuit over time for the three strategies.