A unified geometric approach to modeling and control of constrained mechanical systems

Dynamic control of constrained mechanical systems, such as robotic manipulators under end-effector constraints, parallel manipulators, and multifingered robotic hands under closure constraints have been classic problems in robotics research. In this paper, we provide a unified geometric framework for modeling, analysis, and control of constrained mechanical systems. Starting with the constraint, we define two canonical subspaces, namely the subspace of constraint forces and the tangent space of the constraint manifold for holonomic constraint. Using the kinetic energy metric, we define the remaining subspaces and show explicitly the relations among these subspaces. We project the Euler-Lagrange equation of a constrained mechanical system into two orthogonal components and give geometric and physical interpretations of the projected equations. Based on the projected equations, a unified and asymptotically stable hybrid position/force-control algorithm is proposed, along with experimental results for several practical examples. In the case of nonholonomic constraints, we show that the equations can be projected to the distribution/codistribution associated with the constraints and the control law reduces to hybrid velocity/force control.