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Optimal Geodesic Curvature Constrained Dubins' Paths on a Sphere
Optimization and Control · Submitted 30 Mar 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show in this article that if $0<r \le \frac{1}{2}$, the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs.
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The abstract, sentence by sentence
In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius $r <1$, where $r$ depends on the bound, $U_{max}$. We show in this article that if $0<r \le \frac{1}{2}$, the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs. Specifically, if $C$ is the smaller circular arc and $G$ is the great circular arc, then the optimal path can only be $CCC, CGC, CC, CG, GC, C$ or $G$. If $r> \frac{1}{2}$, while paths of the above type may cease to exist depending on the boundary conditions and the value of $r$, optimal paths may be concatenations of more than three circular arcs.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. | background 0.86 | 0.36 | 0.02 | asserted 0.93 |
| s1 | The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius $r <1$, where $r$ depends on the bound, $U_{max}$. | background 0.84 | 0.20 | 0.05 | asserted 1.00 |
| s2 | We show in this article that if $0<r \le \frac{1}{2}$, the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs. | result 0.99 | 0.94 | 0.66 | asserted 1.00 |
| s3 | Specifically, if $C$ is the smaller circular arc and $G$ is the great circular arc, then the optimal path can only be $CCC, CGC, CC, CG, GC, C$ or $G$. | result 1.00 | 0.91 | 0.34 | asserted 0.85 |
| s4 | If $r> \frac{1}{2}$, while paths of the above type may cease to exist depending on the boundary conditions and the value of $r$, optimal paths may be concatenations of more than three circular arcs. | result 0.92 | 0.84 | 0.09 | hedged possibility 0.99 |
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The paper
Original abstract
In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius $r <1$, where $r$ depends on the bound, $U_{max}$. We show in this article that if $0<r \le \frac{1}{2}$, the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs. Specifically, if $C$ is the smaller circular arc and $G$ is the great circular arc, then the optimal path can only be $CCC, CGC, CC, CG, GC, C$ or $G$. If $r> \frac{1}{2}$, while paths of the above type may cease to exist depending on the boundary conditions and the value of $r$, optimal paths may be concatenations of more than three circular arcs.
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