Reach–Avoid Control Synthesis for a Quadrotor UAV With Formal Safety Guarantees

Accomplishing reach-avoid tasks for quadrotor unmanned aerial vehicles (UAVs) while ensuring formal safety guarantees remains a challenging problem despite significant research advancements. In this paper, we address this challenge by adopting an efficient planning-tracking paradigm that integrates geometric control for tracking and Bézier curves for polynomial trajectory generation. Our framework explicitly addresses tracking errors during trajectory synthesis by incorporating closed-form, time-varying error bounds that are independent of the specific trajectory, as long as it meets easily imposed constraints. This approach enables efficient trajectory synthesis while ensuring formal safety guarantees. To derive tight time-varying tracking error bounds, we revisit the stability analysis of the closed-loop quadrotor system under geometric tracking control. Our analysis demonstrates that the tracking error dynamics exhibit local exponential stability for any positive control gains. Additionally, we establish sufficient conditions on the desired trajectory, utilizing the derived bounds to ensure the well-posedness of the closed-loop system. We propose a novel and efficient algorithm for constructing a safe tube or corridor using sampling-based planning and safe hyper-rectangular set computations, explicitly incorporating the time-varying bounds. The trajectory is then synthesized within this safe tube while considering the error bounds, using a heuristic and computationally efficient approach based on linear programming. To validate the effectiveness of the proposed framework, we present numerical simulations of a reach-avoid control task in a cluttered environment and compare the performance of our method against two state-of-the-art optimization-based tracking approaches.