Borrowing elementary ideas from solid mechanics and differential geometry, this presentation shows that the volume swept by a regular solid undergoing a wide class of volume-preserving deformations induces a rather natural metric structure with well-defined and computable geodesics on its configuration space. This general result applies to concrete classes of articulated objects such as robot manipulators, and we demonstrate as a proof of concept the computation of geodesic paths for a free flying rod and planar robotic arms as well as their use in path planning with many obstacles.

Motion planning is a fundamental robotics problem [2, 3] with numerous applications in mobile robot navigation [4], industrial robotics [5], humanoid robotics [6] and other domains. Sampling-based methods such as Rapidly Exploring Random Trees (RRT) [7] and Probabilistic Roadmaps (PRM) [8] have been shown successful for finding a collision-free path in complex environments with many obstacles. Such methods are able to solve the so-called piano mover problem [9] and typically assume static environments and prior knowledge about the shape and location of obstacles. In many practical applications, however, it is often difficult or even impossible to obtain detailed a-priori knowledge about the real state of environments. It is therefore desirable to design methods relying on partial observations obtained from sensor measurements and enabling motion planning in unknown and possibly dynamic environments.