Geometry and Prediction of Drift-Free Trajectories for Redundant Machines Under Pseudoinverse Control

Abstract

In this article, classical differential geometric techniques are used for two purposes: (1) to describe in geometric terms the trajectories to which redundant robots under pseudoinverse control will drift and (2) to formulate from this geometric description a practical and quick method for predicting these limit trajectories, thereby suggesting sets of initial conditions under which no drift will occur. Through this analysis, several useful results were obtained. First, it is demonstrated for a 3R example that the drifting trajectories are geodesics in the constrained joint space, and that as a result of this, all drift-free tra jectories are space curves of zero torsion. Second, it is found that each self-motion manifold in the joint space crosses the limiting (drift-free) trajectories at points where they themselves have zero torsion. Because self-motion manifolds are often more easily computed than exhaustive kinematic simulation of manipulator drift, these results provide a fast, new technique for locating and character izing drift-free trajectories based only on geometry of these self-motion manifolds. This work extends the results of previous work in pseudoinverse control to the point that the drift problem may be circumvented without resort to either exhaustive simulation or symbolic computation of Lie brackets of vector fields.