Open-Source CRS Solver Finds Fastest Paths for Robots Rolling on Spheres
A new characterization gives closed-form time-optimal paths on a sphere for vehicles that can drive forward, reverse, and turn in place.
PRO- First complete characterization of time-optimal paths for a Reeds-Shepp vehicle on a sphere, extending Dubins-only prior work.
- Optimal paths use at most six segments from three primitives: tight turns, great circles, turn-in-place motions.
- Full classification yields 23 sufficient path types with closed-form segment angles derived analytically.
- Applies to underactuated two-wheel satellites, spherical rolling robots, and mobile robots on gently curved terrain.
- Uses Pontryagin's Maximum Principle plus phase-portrait analysis on the SO(3) cotangent bundle.
- Open-source Python code and visualization available at the project repository.
Planning the fastest route for a car that can only turn so sharply is a classic problem. On a flat surface, it was solved decades ago: the Dubins model handled forward-only vehicles, and the Reeds-Shepp model extended it to vehicles that can also reverse. Put that same vehicle on a curved surface like a sphere, and the geometry changes in a way that breaks the planar solutions. A new paper closes that gap for the sphere.
The authors work out the complete structure of time-optimal paths for a convexified Reeds-Shepp (CRS) vehicle rolling on a unit sphere, with bounded speed and bounded turning rate. Using Pontryagin's Maximum Principle and a phase-portrait analysis, they show that the optimal path connecting a given initial configuration to a desired terminal one consists of at most six segments drawn from three motion primitives: tight turns, great circular arcs, and turn-in-place motions. The full classification produces 23 path types with closed-form segment angles, plus open-source Python code to compute them.
Why a Sphere Breaks the Planar Recipe
On a flat plane, translation and rotation are independent. On a sphere they are not: the intrinsic curvature couples translational and rotational motion, so even locally straight travel induces a change in orientation, fundamentally altering the structure of optimal paths. Driving in a "straight line" on a sphere means following a great circle, which itself rotates the vehicle's heading. That coupling is why the planar Reeds-Shepp solutions do not transfer.
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