Title: Virtual Holonomic Constraints for Euler-Lagrange Systems (1 hour)
Abstract: The notion of virtual holonomic constraint is a useful paradigm for the control of oscillations in mechanical systems. It is one of the underlying mechanisms in the technique developed by Grizzle and collaborators to stabilise walking motion in bipedal robots. This talk explores 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 stabilisation of a desired oscillatory motion. We will present conditions under which such a set of constraints is feasible, meaning that it can be made invariant by feedback. We will show that it is possible to systematically determine feasible virtual constraints as periodic solutions of a scalar differential equation. Further, under a symmetry assumption we will show that the motion on the constraint manifold is a Euler-Lagrange system with one degree-of-freedom, and use this fact to completely characterize its dynamical properties. Finally, we will show that if the constraint is feasible then the virtual constraint manifold is exponentially stabilisable.