Optimal Path Planning with a Kinematic Airplane Model

Abstract

O PTIMAL path planning is an important problem for robotics and unmanned vehicles. This note describes a method for finding the minimum-time path from an initial position and orientation to a final position and orientation in the two-dimensional plane for an airplane with a bounded turning rate in the presence of a constant wind [1]. This problem was first solved in the no-wind case by Dubins using geometric arguments [2]. Reeds and Shepp extended this result for a vehicle that could reverse its direction [3]. Boissonnat et al. later reproduced the Dubins and Reeds–Shepp results using optimal control methods [4]. These results have also been used for higher-level path planning through multiple points [5,6] and to travel around obstacles [7,8]. The kinematic model assumes a constant velocity airplane traveling in a known constant wind with a magnitude less than the airplane velocity. The control input of the airplane is the turning rate, which is assumed to be bounded. By normalizing the position of the airplane x; y so that the velocity is 1 and defining the orientation with respect to the direction of the wind, the equations of motion can be expressed as