An Analytic Solution to the Optimal Spherical Dubins Path Problem with Geodesic Curvature Constraints
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914232855953408 |
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| author | Li, Linhong Feng, Qi Liang, Yangang Li, Kebo |
| author_facet | Li, Linhong Feng, Qi Liang, Yangang Li, Kebo |
| contents | Computing shortest paths for curvature-constrained Dubins vehicles on the unit sphere is fundamental to many engineering applications, including long-range flight planning, persistent surveillance patterns, and global routing problems where great circles are natural routes. Numerical optimization methods on $\SO(3)$ suffer from sensitivity to initialization, may converge to local minima, and often miss feasible solution branches. This paper proposes a unified analytic computational approach for spherical Dubins CGC and CCC paths that overcomes these limitations. By exploiting the axis-fixing property of rotations and developing a closed-form back-substitution method using geometric projection, the three-dimensional boundary value problem is reduced to solving a quadratic polynomial equation. The proposed analytic solver achieves machine precision accuracy with errors on the order of $10^{-16}$, is approximately $717$ times faster than numerical methods under the same computational environment, and systematically enumerates all feasible solution branches without requiring exhaustive multi-start initialization. The method provides closed-form solutions for optimal path computation in the regime where turning radius $\Rturn \in (0, 1/2]$, corresponding to $U_{\max} \geq \sqrt{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01174 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Analytic Solution to the Optimal Spherical Dubins Path Problem with Geodesic Curvature Constraints Li, Linhong Feng, Qi Liang, Yangang Li, Kebo Geophysics Computing shortest paths for curvature-constrained Dubins vehicles on the unit sphere is fundamental to many engineering applications, including long-range flight planning, persistent surveillance patterns, and global routing problems where great circles are natural routes. Numerical optimization methods on $\SO(3)$ suffer from sensitivity to initialization, may converge to local minima, and often miss feasible solution branches. This paper proposes a unified analytic computational approach for spherical Dubins CGC and CCC paths that overcomes these limitations. By exploiting the axis-fixing property of rotations and developing a closed-form back-substitution method using geometric projection, the three-dimensional boundary value problem is reduced to solving a quadratic polynomial equation. The proposed analytic solver achieves machine precision accuracy with errors on the order of $10^{-16}$, is approximately $717$ times faster than numerical methods under the same computational environment, and systematically enumerates all feasible solution branches without requiring exhaustive multi-start initialization. The method provides closed-form solutions for optimal path computation in the regime where turning radius $\Rturn \in (0, 1/2]$, corresponding to $U_{\max} \geq \sqrt{3}$. |
| title | An Analytic Solution to the Optimal Spherical Dubins Path Problem with Geodesic Curvature Constraints |
| topic | Geophysics |
| url | https://arxiv.org/abs/2601.01174 |