A Combinatorial Approach to Planar Non-colliding Robot Arm Motion Planning

We propose a combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework. Our approach yields very eÆcient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom, where the known general roadmap techniques would give exponential complexity. It is based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets, for which we provide several equivalent characterization and exhibit rich combinatorial and rigidity theoretic properties. The main application is an eÆcient algorithm for the Carpenter's Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non self-intersecting planar motions. A step in the convexi cation motion consists in moving a pseudotriangulation-based mechanism along its unique trajectory in con guration space until two adjacent edges align. At that point, a local alteration restores the pseudo triangulation. The motion continues for O(n) steps until all the points are in convex position.