Dubins path

Abstract: The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are specified? One need consider only paths with at most 2 cusps or reversals

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.

Abstract: In this paper, we explore the problem of generating the optimal time path from an initial position and orientation to a flnal position and orientation in the two-dimensional plane for an aircraft with a bounded turning radius in the presence of a constant wind. Following the work of Boissonnat, we show using the Minimum Principle that the optimal path consists of periods of maximum turn rate or straight lines. We demonstrate, however, that unlike the no wind case, the optimal path can consist of three arcs where the length of the second arc is less than …. A method for generating the optimal path is also presented which iteratively solves the no wind case to intercept a moving virtual target. I. Introduction Optimal path planning is an important problem for robotics and unmanned vehicles. In this paper, we explore a method for flnding the shortest path from an initial position and orientation to a flnal position and and orientation in the two-dimensional plane for an aircraft with a bounded turning radius in the presence of a constant wind. This work was motivated by our group’s work with our ∞eet of small autonomous aircraft. Each modifled Sig Rascal aircraft ∞ies under the combined control of an ofi-the-shelf Piccolo avionics package for low level control and an onboard PC104 computer for higher level tasks. These aircraft ∞y at a nominal speed of 20 m/s, and wind speeds of over 5 m/s have been encountered during ∞ight testing. The problem described above was flrst solved in the case of no wind by Dubins using geometric arguments 8