Virtual Holonomic Constraints for Euler-Lagrange Systems

Abstract

This paper investigates virtual holonomic constraints for Euler-Lagrange systems with n degrees-of-freedom and n − 1 controls. The constraints have the form q1 = ϕ1 (qn), …, qn – 1 = ϕn ∓ 1 (qn), where qn is a cyclic configuration variable, so their enforcement corresponds to the stabilization of a desired oscillatory motion. We give conditions under which such a set of constraints is feasible, meaning that it can be made invariant by feedback. We show that it is possible to systematically determine feasible virtual constraints as periodic solutions of a scalar differential equation, the virtual constraint generator. Moreover, under a symmetry assumption we show that the motion on the constraint manifold is a Euler-Lagrange system with one degree-of-freedom, and use this fact to complete characterize its dynamical properties. Finally, we show that if the constraint is feasible then the virtual constraint manifold can always be stabilized using input-output feedback linearization.