Modeling and Control of Multi-Body Mechanical Systems: Part I A Riemannian Geometry Approach

Abstract

Control problems of motion of multi-body mechanical systems under constraints and/or with redundancy in system’s degrees-of-freedom (DOF) are treated from the standpoint of Riemannian geometry. A multi-joint reaching problem with excess DOF is tackled and it is shown that a task space PD feedback with damping shaping in joints maneuvers the endpoint of the robot arm to reach a given target in the sense of exponentially asymptotic convergence. An artificial potential inducing the position feedback in task space can be regarded as a Morse-Bott function introduced in Riemannian geometry, from which the Lagrange stability theorem can be directly extended to this redundant case. The speed of convergence of both the orbit of the endpoint in task space and the trajectory of joint vector in joint space can be adjusted by damping shaping and adequately choosing a single stiffness parameter. In the case that the endpoint is constrained on a hypersurface in E ³, the original Lagrange dynamics expressed in an implicit form by introducing a Lagrange multiplier is decomposed into two partial dynamics with the aid of decomposition of the tangent space into the image of the endpoint Jacobian matrix and the kernel orthogonally complemented to the image. The stability problem of point-to-point endpoint movement on the constraint surface is reduced to the former case without constraint.