Generalization of Optimal Geodesic Curvature Constrained Dubins’ Path on Sphere With Free Terminal Orientation

In this letter, 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, where r represents the radius of the tightest left or right turn the vehicle can take on the sphere. Noting that $r \in $ (0, 1) and can affect the trajectory taken by the vehicle, it is desired to determine the candidate optimal paths for r ranging from nearly zero to close to one to attain a desired final location. 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 CG and CC paths, which includes C, G paths, for $r \leq {}\frac {1}{2}$ . Here, $C~\in $ { $L, R$ } 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 letter, 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 CG and CC paths for $r \leq {}\frac {\sqrt {3}}{2}$ . Noting that at most two LG paths and two RG paths can exist for a given final location, this letter 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 $r \leq {}\frac {\sqrt {3}}{2}$ . Additional conditions for the optimality of CC paths are also derived in this letter.