On the Generation of Virtual Holonomic Constraints for Mechanical Systems With Underactuation Degree One

This paper introduces the notion of a virtual constraint generator (VCG) for underactuated mechanical systems with underactuation degree one. The VCG is a control system whose state space is the configuration manifold of the mechanical system, and whose orbits are all possible regular virtual holonomic constraints of the mechanical system. Leveraging this new tool, we propose to pose the design of virtual constraints as a suitable control specification for the VCG. We take a first step in this direction by presenting two such control designs aimed at producing virtual holonomic constraints whose associated constrained dynamics correspond to a through-motion with bounded speed and oscillatory behaviour, respectively. We apply our results to the control of an acrobot mounted on a cart, with the goal of designing constraints that make the robot "kneel" under an obstacle while traversing the real line, or oscillate back and forth under it.