Singularities of holonomic and non-holonomic robotic systems: A normal form approach

Abstract

This paper is intended to make a contribution to methodology of robotics research with a focus on the analysis of robot singularities. It is well known that the presence of singular configurations impairs the robot’s manipulability/mobility, and invalidates a majority of robot control algorithms. For this reason, the understanding of singularities has been a challenging problem of robotics for decades. The objective of this paper consists in fostering the normal form approach as a tool for analysis of singularities of both holonomic and non-holonomic robots. In order to justify the normal form approach, we have invoked a collection of our existing, earlier and newest results. Intuitively, a normal form of an object is an equivalent object of the same kind as the original, of the simplest possible form. Specifically, for the kinematics of serial manipulators normal forms are defined by means of the LR-equivalence of smooth maps. For mobile robots represented as control systems normal forms are obtained by feedback transformations. A rationale for studying singularities by normal forms stems from the fact that both equivalence relations mentioned above preserve singular configurations. In the paper we shall show that normal forms may facilitate detection and description of singular configurations, as well as explain the robot’s behavior at singularities. This will be illustrated by an application of the normal form approach to example earth and space robots.