# Geometry and combinatorics for revolute-jointed robot arms

# Geometry and combinatorics for revolute-jointed robot arms

We present a complete theoretical characterization and a method for calculating the reachable workspace boundary for all serial manipulators with revolute joints having any pair of consecutive joint axes coplanar. The number of joints is arbitrary. The workspace boundary is a surface of revolution, obtained by rotating a planar bounded real semi-algebraic curve, called the planar workspace boundary, about the first joint axis. We show that the planar boundary is composed of a finite set of circular arcs. Their connectivity is controlled by an underlying combinatorial structure which is fully identified.The workspace boundary can be decomposed into a max-boundary and a min-boundary, which are defined as those parts that can be traced in the reference plane as the maximum reach, respectively non-zero minimum reach of the end-point distance function between an arbitrary base-point on the first joint axis and the end-effector. Their exact determination is based on recent results of the authors on extremal configurations for robot arms with revolute joints.

This is joint work with I. Streinu.