Efficient Polynomial Sum-Of-Squares Programming for Planar Robotic Arms

Collision-avoiding motion planning for articulated robotic arms is one of the major challenges in robotics. The difficulty of the problem arises from its high dimensionality and the intricate geometry of the feasible space. Our goal is to seek large convex domains in configuration space, which contain no obstacles. In these domains, simple linear trajectories are guaranteed to be collision free, and can be leveraged for further optimization. To find such domains, practitioners have harnessed a methodology known as Sum-Of-Squares (SOS) Programming. SOS programs, however, are notorious for their poor scaling properties, which makes it challenging to employ them for complex problems. In this paper, we explore a simple formulation for a two-dimensional arm, which results in smaller SOS programs than previous suggested ones. We show that this formulation can express a variety of scenarios in a unified manner.

I. Introduction

Finding collision free trajectories for robotic arms is a fundamental problem in robotics. A wide variety of strategies are available, ranging from gradient-based trajectory optimization methods to random sampling techniques such as the Rapidly Exploring Random Tree approach. One notable approach aims to identify large convex domains in Configuration space (C-space) that are devoid of obstacles. A linear path between two points in such domains is guaranteed to be collision-free. By combining several such domains, it is possible to find paths between any two points. Furthermore, these domains can be leveraged in order to find initial trajectories for further optimization or for a more efficient sampling.