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Geometric Stabilization of Virtual Nonlinear Nonholonomic Constraints

Efstratios Stratoglou, Alexandre Anahory Simoes, Anthony Bloch, Leonardo Colombo

TL;DR

The paper develops a geometric framework for stabilizing mechanical systems around manifolds defined by virtual nonlinear nonholonomic constraints (VNNHC) within an affine-connection setting. It proves the existence and uniqueness of a feedback control law that enforces the constraint and yields exponential convergence to the constraint manifold, extending prior linear results to nonlinear constraint families with tunable gains. The approach leverages velocity-dependent distributions and transversality to render the constrained manifold invariant, shaping the closed-loop dynamics as a constrained connection. The authors validate the theory with two simulations—multi-agent flocking under velocity-alignment constraints and a USV navigating a current—demonstrating rapid constraint satisfaction and favorable energy behavior. Overall, the work advances geometric control methods for nonholonomic-like constraints with potential impact on robotic locomotion and coordinated vehicle systems.

Abstract

In this paper, we address the problem of stabilizing a system around a desired manifold determined by virtual nonlinear nonholonomic constraints. Virtual constraints are relationships imposed on a control system that are rendered invariant through feedback control. Virtual nonholonomic constraints represent a specific class of virtual constraints that depend on the system's velocities in addition to its configurations. We derive a control law under which a mechanical control system achieves exponential convergence to the virtual constraint submanifold, and rendering it control-invariant. The proposed controller's performance is validated through simulation results in two distinct applications: flocking motion in multi-agent systems and the control of an unmanned surface vehicle (USV) navigating a stream.

Geometric Stabilization of Virtual Nonlinear Nonholonomic Constraints

TL;DR

The paper develops a geometric framework for stabilizing mechanical systems around manifolds defined by virtual nonlinear nonholonomic constraints (VNNHC) within an affine-connection setting. It proves the existence and uniqueness of a feedback control law that enforces the constraint and yields exponential convergence to the constraint manifold, extending prior linear results to nonlinear constraint families with tunable gains. The approach leverages velocity-dependent distributions and transversality to render the constrained manifold invariant, shaping the closed-loop dynamics as a constrained connection. The authors validate the theory with two simulations—multi-agent flocking under velocity-alignment constraints and a USV navigating a current—demonstrating rapid constraint satisfaction and favorable energy behavior. Overall, the work advances geometric control methods for nonholonomic-like constraints with potential impact on robotic locomotion and coordinated vehicle systems.

Abstract

In this paper, we address the problem of stabilizing a system around a desired manifold determined by virtual nonlinear nonholonomic constraints. Virtual constraints are relationships imposed on a control system that are rendered invariant through feedback control. Virtual nonholonomic constraints represent a specific class of virtual constraints that depend on the system's velocities in addition to its configurations. We derive a control law under which a mechanical control system achieves exponential convergence to the virtual constraint submanifold, and rendering it control-invariant. The proposed controller's performance is validated through simulation results in two distinct applications: flocking motion in multi-agent systems and the control of an unmanned surface vehicle (USV) navigating a stream.

Paper Structure

This paper contains 10 sections, 4 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Virtual Nonholonomic constraints
  3. Mechanical systems in a coordinate-free setting
  4. Nonlinear nonholonomic mechanics
  5. Virtual nonholonomic constraints
  6. Stabilization of Virtual Nonholonomic Constraints
  7. Applications
  8. Geometric stabilization for flocking motion
  9. Application to the control of a USV navigating a stream
  10. Conclusions and Future Work

Key Result

Theorem 1

A curve $q:I\rightarrow Q$ is a solution of Chetaev's equations for a mechanical type Lagrangian with kinetic energy determined by a Riemannian metric $\mathcal{G}$ on $Q$ and a potential function $V$ if and only if $\Phi(q,\dot{q})=0$ and it satisfies the equation where $S(\dot{q})^{\bot}$ is the orthogonal velocity-dependent distribution to $S(\dot{q})$ with respect to the Riemannian metric $\m

Figures (6)

  • Figure 1: Trajectories of the 4 agents of the flocking closed-loop system. The black points indicate the initial positions.
  • Figure 2: Constraint functions along the same trajectory.
  • Figure 3: Left: Projection of a trajectory of the closed-loop system into the plane $xy$ of the boat on a north-east stream.
  • Figure 4: Left: Constraint $\phi(t)$. Right: Energy of the system.
  • Figure 5: Projection of a trajectory of the closed-loop system into the plane $xy$ of the boat on an anticyclone stream. The black dot indicates the initial position of the boat.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1: VNNHC
  • Definition 1
  • Definition 2
  • Theorem 2: VNNHC
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Remark 1
  • Remark 2