# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 4 Simulation Experiment
The MSE strategy has the fastest growing cost function, rising 53.71% faster than the SSE strategy in the 2000 s at the beginning of the game, and eventually stabilizing, suggesting that this strategy is advantageous in terms of cost-effectiveness, and it can also be seen that this strategy permits the evader to take a leadership position in the game, and by making positive adjustments in the early stages, it can reduce the distance to the evader more quickly, thereby reducing long-term costs. Figure 6b shows the fuel consumption of the pursuer for the three strategies. During the initial phase of the MSE strategy, fuel consumption increases rapidly, consistent with the strategy taking more aggressive action in the initial phase to respond quickly to the motion of the evader. Over time, the increase in fuel consumption slows down and stabilizes at a relatively low level. This demonstrates the advantage of the algorithm in the presence of fuel consumption constraints in spacecraft operations. Figure 7 illustrates the relative distance between the two spacecraft over time. The relative distance decreases rapidly and tends to zero under the MSE strategy, and its speed of reducing the distance between the pursuing two spacecraft in 1200 s is 25.46% faster than the SSE strategy, and about 32.14% faster than the NE strategy, which shows its advantage in the efficiency of the pursuit. The rapid decrease in relative velocity helps the pursuer to match the evader's velocity more quickly, leading to an effective pursuit. Figure 8 shows the relative velocity changes between the two spacecraft. Consistent with the pursuit efficiency shown in Figure 7, the MSE strategy achieves the fastest reduction in relative velocity, decreasing 10.47% faster than the SSE strategy in the initial 3000 s. This rapid decrease is crucial for a successful capture, as it allows the pursuer to quickly match the evader's velocity vector and stabilize the relative dynamics. Figure 6: Cost function and fuel consumption of pursuit spacecraft over time. (**a**) Pursuit spacecraft cost function over time. (**b**) Pursuit spacecraft fuel consumption curve over time. Figure 7: The relative distance between the two spacecraft varies with time. Figure 9 compares the computational time distributions of SSE, NE, MSE, and MPC, based on 100 Monte Carlo trials for each scenario. The boxplot shows that the median time (24.79 ms) of MSE is 7.36% and 39.11% lower than that of SSE (26.75 ms) and NE (40.7 ms), respectively, and the variance is smaller (MSE standard deviation 0.452 vs. 0.804 for NE), indicating its computational efficiency and stability advantages. In order to further verify the anti-interference performance of the studied algorithm, the following data are set in the input signal of the algorithm to simulate the interference signals in the spacecraft PE game. The simulation parameters of the interference signals are shown in Table 3. The position measurement noise is the interference inserted by the pursuit spacecraft when acquiring the position of the evader; the signal loss interference is the 5% probability that the pursuit spacecraft loses the signal from the evader at each iteration of the algorithm, and the signal delay interference denotes the time delay that occurs in receiving the motion state of the evader by the pursuit spacecraft. Figure 8: Relative velocity variation. Figure 9: Time consumption in generating three strategies. \begin{table}
\begin{tabular}{c c} \hline \hline
**Interference Factors** & **Description** \\ \hline Position measurement noise & \(X\sim\mathcal{N}(50,100)\) \\ The probability of signal loss & 5\% \\ Signal delay time & \(Y\sim\text{Uniform}(20,40)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Interference factor settings. Figure 10a compares the success rates of the three algorithms under different interference conditions. In the 'No Interference' case, the MSE algorithm achieves a lower success rate than the SSE algorithm.