Anholonomy in the Control of Kinematically Redundant Mechanisms with Flexible Components

Abstract We study robot mechanisms which feature kinematic redundancy together with elastic components. For instance, we discuss manipulators in which there are more controlled (joint space) degrees of freedom than task space degrees of freedom and in which at least one of these controlled joints is connected to an adjacent link through an elastic coupling. It will be shown that there is a wide class of examples for which it is impossible to find joint space motions which track a given end effector trajectory and do not at the same time excite elastic dynamics. No assumptions on the smoothness of the prescribed end effector trajectory or the degree of redundancy alter this conclusion. For certain congenial mechanism geometries, however, it is shown that it is possible to plan motions which store no elastic energy. These motions involve differential-algebraic constraints relating accelerations of the end effector along the work space path to joint space velocity and configuration variables. Planning motions which interpolate between given points in the work space for kinematically redundant flexible robots is similar to recently studied problems in the control theory of nonholonomic systems. Interesting open questions involve the existence and multiplicity of interpolating work space paths, the geometry of the reachable work space, and how closely arbitrarily prespecified paths in the work space may be tracked.