Generalization of Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation
Abstract
In this paper, motion planning for a Dubins vehicle on a unit sphere to attain a desired final location is considered. The radius of the Dubins path on the sphere is lower bounded by r. In a previous study, this problem was addressed, wherein it was shown that the optimal path is of type CG, CC, or a degenerate path of the same for r ≤ 12. Here, C = L, R denotes an arc of a tight left or right turn of minimum turning radius r, and G denotes an arc of a great circle. In this study, the candidate paths for the same problem are generalized to model vehicles with a larger turning radius. In particular, it is shown that the candidate optimal paths are of type CG, CC, or a degenerate path of the same for r ≤ 32. Noting that at most two LG paths and two RG paths can exist for a given final location, this article further reduces the candidate optimal paths by showing that only one LG and one RG path can be optimal, yielding a total of seven candidate paths for r ≤ 32. Additional conditions for the optimality of CC paths are also derived in this study.
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