# Dynamic Virtual Holonomic Constraints for Stabilization of Closed Orbits in Underactuated Mechanical Systems

## TL;DR

This paper presents a novel control method for enforcing virtual holonomic constraints and stabilizing closed orbits in underactuated mechanical systems, demonstrated on a PVTOL aircraft to follow a circular path without rollover.

## Contribution

It introduces a new approach using a double-integrator parameterization of VHCs to achieve both constraint enforcement and orbit stabilization in underactuated systems.

## Key findings

- Successfully stabilizes closed orbits on the constraint manifold.
- Ensures the aircraft follows a circular path with a specified speed.
- Prevents rollover for suitable initial conditions.

## Abstract

This article investigates the problem of enforcing a virtual holonomic constraint (VHC) on a mechanical system with degree of underactuation one while simultaneously stabilizing a closed orbit on the constraint manifold. This problem, which to date is open, arises when designing controllers to induce complex repetitive motions in robots. In this paper, we propose a solution which relies on the parameterization of the VHC by the output of a double-integrator. While the original control inputs are used to enforce the VHC, the control input of the double-integrator is designed to asymptotically stabilize the closed orbit and make the state of the double-integrator converge to zero. The proposed design is applied to the problem of making a PVTOL aircraft follow a circle on the vertical plane with a desired speed profile, while guaranteeing that the aircraft does not roll over for suitable initial conditions.