# Safety Control for Satellite Formation Flying via High-Order Control Barrier Functions
## 4 Simulation Results
The safety controller is synthesized by numerically solving the constrained optimization problem formulated in Equation (10), which systematically enforces state constraints through barrier functions. Figures 14-16 present the relative position trajectories between the reference and accompanying satellites under three operational scenarios, while Figures 17-19 detail the corresponding temporal profiles of safety control signals. Simulation results conclusively demonstrate that the integrated control architecture achieves the following:  
* Constraint Satisfaction: Maintains all relative positions within predefined safety boundaries throughout the formation maneuver (see safety margins in Figures 14-16). * Stability Guarantee: Achieves asymptotic convergence to desired formation configurations without compromising transient performance. * Computational Feasibility: Generates real-time implementable control actions, as evidenced by the smooth signal profiles in Figures 17-19. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]2 The relative position responses under the first stochastic initial condition: safety controller. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]3 The MPC controller responses under the third stochastic initial condition. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]4 The relative position responses under the third stochastic initial condition: safety controller. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]5 The safety controller responses under the first stochastic initial condition. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]6 The relative position responses under the second stochastic initial condition: safety controller. These collective outcomes substantiate the dual capability of the proposed scheme in simultaneously enforcing safety constraints and preserving closed-loop stability, thereby addressing a critical gap in existing satellite formation control paradigms. The proposed high-order control barrier function-based safety control method effectively ensures the safety of satellite formation flying under the tested scenarios. By formulating safety constraints as a QP problem, the method enforces forward invariance of the safe set, preventing collisions. However, the following factors may influence its practical applicability:  
* Sensitivity to Model Uncertainties: The approach assumes an accurate representation of satellite dynamics. Unmodeled perturbations, such as atmospheric drag, solar radiation pressure, or inaccuracies in thruster modeling, may affect performance. Future work will explore incorporating adaptive estimation techniques or robust control modifications to address such uncertainties. * Computational Complexity: The QP-based safety control formulation requires real-time optimization. While the method is computationally feasible for the tested cases, high-dimensional formations or real-time implementations on resource-limited onboard processors may pose challenges. Efficient optimization solvers or approximation methods could enhance practical deployment. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]7 The safety controller responses under the third stochastic initial condition. \[K_{p2} = 0.05,\ K_{d2}=0.2,\]8 The safety controller responses under the second stochastic initial condition. * Actuator Constraints: The method assumes that control inputs can be implemented without saturation. In real-world scenarios, actuator limits may restrict the feasibility of computed control inputs, potentially leading to constraint violations. Introducing actuator-aware constraints within the optimization framework could mitigate this issue. * Robustness to Initial Conditions: The presented results demonstrate feasibility across selected cases, but a broader Monte Carlo analysis is needed to statistically validate robustness across diverse initial conditions. This will be included in future studies to establish a more comprehensive understanding of the method's applicability. Future work will focus on extending the robustness of the approach by incorporating adaptive safety constraints and uncertainty estimation techniques.