Virtual Holonomic Constraints in Motion Planning: Revisiting Feasibility and Limitations

TL;DR

This paper critically examines the limitations of current virtual holonomic constraints (VHCs) in motion planning for underactuated systems, demonstrating that the standard definition excludes many feasible trajectories and proposing a need for redefining VHCs.

Contribution

It reveals the restrictive nature of the conventional VHC definition and provides formal proofs that many feasible trajectories do not satisfy it, suggesting a broader framework is necessary.

Findings

Standard VHC definition excludes feasible trajectories

Formal proof of limitations of current VHC conditions

Call for redefining VHCs to include more trajectories

Abstract

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.