Feasible Dubins Paths in Presence of Unknown, Unsteady Velocity Disturbances

A Dubins path is the planar path of minimum length and bounded curvature that connects two specified endpoints with specified approach angles [1]. Because of this optimality property, and because there is a simple, geometric procedure for constructing them, Dubins paths are often used for guidance of constant-speed, planar vehicles [2] [3] [4]. For such a vehicle, often referred to as a “Dubins car,” a time-optimal path consists of maximum rate turns and straight line segments. However, a Dubins path that is constructed using the true maximum turn rate cannot be tracked in the presence of disturbances, because feedback commands may exceed the turn rate limit. If there is sufficient control authority, and the disturbances are perfectly known (whether steady [5], [6], [7] or unsteady [8]), then minimum time trajectories to the goal state can be planned that account for these disturbances explicitly. In general, though, disturbances are unknown and unavailable for planning and control purposes. One approach to dealing with uncertain disturbances is to dispense with planning altogether and to use feedback control to drive the system toward a desired end goal. In [9], for example, an optimal control law is presented that drives the vehicle to a target set (with a free final course angle) in the presence of a stochastically varying wind. In some applications, however, such as directional sensing or vehicle recovery operations, attaining a desired position with a prescribed course angle may be important. This note describes how the Dubins path planning method can be modified to construct sub-optimal paths that remain feasible in the presence of a bounded, unsteady disturbance. The path is planned using an artificially reduced “maximum” turn rate that is a function of the (known) upper bound on the disturbance magnitude. Though the resulting path is longer, it could be tracked by the vehicle if the disturbance were known. For unknown disturbances, the reserve control authority enables a path following algorithm to force convergence to the desired path.