Virtual Holonomic Constraints in Motion Planning: Revisiting Feasibility and Limitations

This paper addresses the feasibility of virtual holonomic constraints (VHCs) in the context of motion planning for underactuated mechanical systems with a single degree of underactuation. While existing literature has established a widely accepted definition of VHC, we argue that this definition is overly restrictive and excludes a broad class of admissible trajectories from consideration. To illustrate this point, we analyze a periodic motion of the Planar Vertical Take-Off and Landing (PVTOL) aircraft that satisfies all standard motion planning requirements, including orbital stabilizability. However, for this solution -- as well as for a broad class of similar ones -- there exists no VHC that satisfies the conventional definition. We further provide a formal proof demonstrating that the conditions imposed by this definition necessarily fail for a broad class of trajectories of mechanical systems. These findings call for a reconsideration of the current definition of VHCs, with the potential to significantly broaden their applicability in motion planning.