Tackling Nonconvex Collision Avoidance Constraints for Optimal Trajectory Planning Using Saturation Functions

The problem of optimal trajectory planning with nonconvex collision avoidance constraints is discussed in this paper. The trajectory planning algorithm aims to generate a feasible trajectory with specified performance indexes to reduce collisions of aerospace vehicles in the presence of second-order smooth obstacles. The algorithm design that manages such mission scenarios is challenging because of the stringent avoidance constraints of various nonconvex types, especially considering the requirements of efficient computation. This problem is addressed within the sequential convex programming (SCP) framework by utilizing the characteristics of saturation functions. The introduction of the saturation function is found to significantly improve the convergence of the SCP algorithm by overcoming the drawbacks of potential interruptions from the iteration process and controlling the search directions for successive solutions. Theoretical analyses are performed to guarantee that the original constraints are satisfied and to ensure the iterative solution searches in the directions of the true optimal solution. Under multiple randomly generated test environments, the applications of the proposed approach in highly constrained aircraft path planning problems and spacecraft rendezvous operations are numerically simulated by comparing two representative conventional SCP-based methods to demonstrate the effectiveness, efficiency, and convergence of the proposed methodology.