Further Results on Virtual Holonomic Constraints

This paper continues recent work by the authors on virtual holonomic constraints (VHCs) for Euler-Lagrange control systems with n degrees-of-freedom and m control inputs. The focus of the paper is on implicit constraints of the form h(q) = 0. Under suitable regularity conditions, the enforcement of k ≤ m constraints induces constrained dynamics that are described by a reduced-order control system of dimension 2(n – k) with (m – k) control inputs. When m = k = n – 1, conditions are given guaranteeing that the constrained dynamics are Euler-Lagrange. It is shown that the presence of dissipation may have unexpected consequences on the constrained dynamics, turning stable equilibria into unstable ones. Finally, VHCs are applied to the problem of constraining a spherical pendulum to lie on the upper half plane.