Minimum-Time Path Planning for Unmanned Aerial Vehicles in Steady Uniform Winds

Abstract

This paper is concerned with time-optimal path planning for a constant-speed unmanned aerial vehicle flying at constant altitude in steady uniform winds. The unmanned aerial vehicle is modeled as a particle moving at a constant air-relative speed and with symmetric bounds on turn rate. It is known from the necessary conditions for optimality that extremal paths comprise only straight segments and maximum-rate turns. An essential observation is that maximum-rate turns correspond to trochoidal path segments, as observed from an Earth-fixed inertial frame. The path-planning problem therefore reduces to identifying the switching points at which straight and trochoidal path segments join to form a feasible path and choosing the true minimum-time solution from the resulting set of candidate extremals. The paper's primary contribution is a simple analytical solution for a subset of candidate extremal paths: those for which an initial maximum-rate turn is followed by a straight segment and then a second maximum-rate turn in the same direction as the first. The solution is easy to compute and is suitable for real-time implementation onboard an unmanned aerial vehicle with limited computational power. The remaining candidate extremal paths may be found using a simple numerical root-finding routine. The paper also shows that, for some candidate extremal paths, no corresponding Dubins path exists in the (moving) air-relative frame.