Virtual Holonomic Constraints for Euler-Lagrange Systems

This paper investigates virtual holonomic constraints for Euler-Lagrange systems with n degrees-of-freedom and n - 1 controls. In our framework, a virtual holonomic constraint is a relation specifying n - 1 configuration variables in terms of a single angular configuration variable. The enforcement by feedback of such a constraint induces a desired repetitive behavior in the system. We give conditions under which a virtual holonomic constraint is feasible, i.e, it can be made invariant by feedback, and it is stabilizable. We provide sufficient conditions under which the dynamics on the constraint manifold correspond to an Euler- Lagrange system. These ideas are applied to the problem of swinging up an underactuated pendulum while guaranteeing that the second link does not fall over.