Optimal merging of multiple aircrafts to waypoints via controlled time of arrival windows

This paper presents an algorithm that computes the optimal arrival times and velocities required for maximizing worst case mutual separation in a multiple aircraft optimal merging scenario. The approach leverages the Pontryagin Maximum Principle (PMP) to rapidly generate locally optimal trajectories per aircraft and an outer-loop Particle Swarm Optimization (PSO) scheme to search for globally optimal merging dynamics. The subset of feasible Controlled Time of Arrival (CTA) and velocity windows is estimated based on an asymptotic approximation of the locally optimal trajectory generated by the PMP. The Particle Swarm Optimization (PSO) scheme maximizes the minimum separation between all aircrafts over all instances of the trajectory. The cost functional has been chosen to minimize load factor (enhance comfort) and maximize worst case mutual separation (improve safety). Simulations based on a simplified lateral aircraft model subject to bounded control inputs are provided to illustrate the concept.