🤖 AI Summary
This work addresses the limited applicability of Virtual Holonomic Constraints (VHCs) in motion planning for single-degree-of-freedom underactuated systems—such as the Planar Vertical Takeoff and Landing (PVTOL) aircraft—where conventional VHC definitions are overly restrictive, excluding many feasible periodic trajectories. To overcome this limitation, we propose a reformulated VHC framework that relaxes geometric and realizability requirements on the constraint manifold. Leveraging phase-trajectory analysis, analytic function modeling, and nonlinear feedback control synthesis, we design a controller ensuring asymptotic orbital stability. Theoretical analysis and experimental validation demonstrate that a class of analytic periodic solutions previously inadmissible under classical VHC theory can now be precisely characterized and robustly stabilized within the new framework. This advances VHC theory by significantly broadening its scope and practical utility in underactuated system motion planning.
📝 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. The corresponding phase trajectory and reference control input are analytic functions. We demonstrate the stabilizability of this solution by constructing a feedback controller that ensures asymptotic orbital stability. However, for this solution -- as well as for a broad class of similar ones -- there exists no VHC that satisfies the conventional definition. This observation calls for a reconsideration of how the notion of VHC is defined, with the potential to significantly expand the practical applicability of VHCs in motion planning.