Generalization of Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914949919408128 |
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| author | Kumar, Deepak Prakash Darbha, Swaroop Manyam, Satyanarayana Gupta Casbeer, David |
| author_facet | Kumar, Deepak Prakash Darbha, Swaroop Manyam, Satyanarayana Gupta Casbeer, David |
| contents | 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 $r \leq \frac{1}{2}.$ 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 $r \leq \frac{\sqrt{3}}{2}.$ 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 $r \leq \frac{\sqrt{3}}{2}.$ Additional conditions for the optimality of $CC$ paths are also derived in this study. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_09954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalization of Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation Kumar, Deepak Prakash Darbha, Swaroop Manyam, Satyanarayana Gupta Casbeer, David Optimization and Control 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 $r \leq \frac{1}{2}.$ 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 $r \leq \frac{\sqrt{3}}{2}.$ 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 $r \leq \frac{\sqrt{3}}{2}.$ Additional conditions for the optimality of $CC$ paths are also derived in this study. |
| title | Generalization of Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2409.09954 |