Configurations and Path Planning of Convex Planar Polygonal Loops

Abstract

Polygonal loops are interesting both as classical geometric objects and in modeling practical engineering systems, e.g., grasping systems with fingers having planar revolute joints. Convex loop configurations and path planning between them are important since many naturally occurring manipulation poses for human and robotic hands are convex or close to convex, and current collision-free path planning methods for polygonal loops use convex configurations in intermediate steps. We prove that, in a set of triangle-based parameters, the space CConvex of convex configurations of a planar polygonal loop with fixed edge lengths and orientation, and one link pinned to the plane, is star-shaped with respect to an easily computed triangular configuration; with a further condition on edge lengths, CConvex is actually a convex polyhedron. Thus reconfiguration between identically oriented convex configurations of a planar polygonal loop can be achieved by one or two straight-line motions within CConvex. We conjecture that, in our parameter space, the straight-line motion joining any two such configurations passes through only non-self-intersecting configurations, although it may leave CConvex. These results are substantially simpler and more efficient than prior work, and demonstrate the importance of suitable system parametrization.