Generation of natural motions for redundant multi-joint systems: A differential-geometric approach based upon the principle of least actions

This article challenges Bernstein's problem of redundant degrees of freedom (DOF) that remains unsolved from both the standpoints of physiology and robotics. A rather simpler but difficult control problem of movements of human-like multi-joint reaching with excess DOF is analyzed from Newtonian mechanics and differential geometry. It is shown that, regardless of ill-posedness of inverse kinematics for such a redundant system, a simpler control signal composed of a well-tuned (synergistic) combination of task-space position feedback (corresponding to spring-like forces) and joint velocity feedback (viscous-like forces) leads to a skilled motion of reaching in a natural way without solving inverse kinematics or dynamics. Fundamental characteristics of human skilled multi-joint movements such as (1) generation of a quasi-straight line trajectory of the endpoint and (2) a little "variability" in task space but notable "variability" in joint space are analyzed from the concepts of "stability on an EP (equilibrium-point) manifold" and "transferability to an EP submanifold." It is claimed that the control signal exerts torques on joints of the whole arm just like a single virtual-spring drawing the endpoint of the arm to the target while giving a specified viscosity to each joint. This leads to an interpretation that skilled reaching movements emerge through formation of a set of neuro-motor signals exerting relevant group of muscles to generate a total potential energy equivalent to that of the spring. Discussions are presented on how such control signals in case of human reaching can be generated in a feedforward manner with capability of anticipatory adjustments of stiffness. (C) 2005 Wiley Periodicals, Inc.