Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonholonomic Robot Parking by Feedback -- Part I: Modular Strict CLF Designs

Velimir Todorovski, Kwang Hak Kim, Alessandro Astolfi, Miroslav Krstic

TL;DR

This paper presents a modular framework for globally stabilizing the nonholonomic unicycle parking problem in polar coordinates by decoupling radial distance control (via forward velocity) from steering control. It develops four design families—passivity-based, forwardings (integrator forwarding), and backstepping—each paired with strict, barrier-enforced CLFs that yield explicit convergence rates and enable eigenvalue assignment. The framework is demonstrated across multiple state-space constraints, with barrier CLFs ensuring avoidance of angular wind-up and front-line crossing, while preserving global asymptotic stability in the polar coordinates and attractivity in Cartesian coordinates. The modular composite-CLF approach lays the groundwork for future inverse-optimal and adaptive re-designs (Part II).

Abstract

It has been known in the robotics literature since about 1995 that, in polar coordinates, the nonholonomic unicycle is asymptotically stabilizable by smooth feedback, even globally. We introduce a modular design framework that selects the forward velocity to decouple the radial coordinate, allowing the steering subsystem to be stabilized independently. Within this structure, we develop families of feedback laws using passivity, backstepping, and integrator forwarding. Each law is accompanied by a strict control Lyapunov function, including barrier variants that enforce angular constraints. These strict CLFs provide constructive class KL convergence estimates and enable eigenvalue assignment at the target equilibrium. The framework generalizes and extends prior modular and nonmodular approaches, while preparing the ground for inverse optimal and adaptive redesigns in the sequel paper.

Nonholonomic Robot Parking by Feedback -- Part I: Modular Strict CLF Designs

TL;DR

This paper presents a modular framework for globally stabilizing the nonholonomic unicycle parking problem in polar coordinates by decoupling radial distance control (via forward velocity) from steering control. It develops four design families—passivity-based, forwardings (integrator forwarding), and backstepping—each paired with strict, barrier-enforced CLFs that yield explicit convergence rates and enable eigenvalue assignment. The framework is demonstrated across multiple state-space constraints, with barrier CLFs ensuring avoidance of angular wind-up and front-line crossing, while preserving global asymptotic stability in the polar coordinates and attractivity in Cartesian coordinates. The modular composite-CLF approach lays the groundwork for future inverse-optimal and adaptive re-designs (Part II).

Abstract

It has been known in the robotics literature since about 1995 that, in polar coordinates, the nonholonomic unicycle is asymptotically stabilizable by smooth feedback, even globally. We introduce a modular design framework that selects the forward velocity to decouple the radial coordinate, allowing the steering subsystem to be stabilized independently. Within this structure, we develop families of feedback laws using passivity, backstepping, and integrator forwarding. Each law is accompanied by a strict control Lyapunov function, including barrier variants that enforce angular constraints. These strict CLFs provide constructive class KL convergence estimates and enable eigenvalue assignment at the target equilibrium. The framework generalizes and extends prior modular and nonmodular approaches, while preparing the ground for inverse optimal and adaptive redesigns in the sequel paper.

Paper Structure

This paper contains 55 sections, 18 theorems, 135 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Resuts preview
  3. Literature
  4. Designs in polar coordinates
  5. Time-varying feedbacks
  6. Discontinuous feedback (convergence sans stability)
  7. Hybrid feedback
  8. Contributions and Organization
  9. Modular approach
  10. Strict CLFs
  11. Organization
  12. Unicycle in Cartesian and Polar Representations
  13. Unicycle model
  14. Transformation to "nonholonomic integrator"
  15. Alternative angle definitions
  16. ...and 40 more sections

Key Result

Proposition 1

Consider any continuously differentiable function $(\delta,\gamma) \mapsto V_{\delta\gamma}$ where $(\delta,\gamma)$ belong to any state space $\hat{\mathcal{T}}\in\{\mathcal{T},\mathcal{T}_1,\mathcal{T}_2,\mathcal{T}_3\}$ and such that $\alpha_1(|(\delta, \gamma)|_{\hat{\mathcal{T}}}) \le V_{\delta for some class $\mathcal{K}$ function $\alpha_{\delta\gamma}$. Then, for any function $(r,s) \mapst

Figures (5)

  • Figure 1: Unicycle orientation $(x, y, \theta)$ relative to the goal state $(0, 0, 0)$, and the corresponding polar coordinates $(\rho, \delta, \gamma)$.
  • Figure 2: Cylindrical state spaces $\mathcal{T}_1$ (blue axes) and $\mathcal{T}_2$ (red axes). For $\mathcal{T}_1$, the set $\{\delta \in \mathbb{R}\}\times\{|\gamma| = \pi\}$ is an equilibrium, while for $\mathcal{T}_2$, $\{|\delta| = \pi\}\times\{\gamma \in \mathbb{R}\}$ is the set where the control is undefined.
  • Figure 3: Toroidal state-space $\mathcal{T}_3$ with the undefined control set $\{|\delta|=\pi\}\times \{|\gamma|\leq \pi\}$ (red) and the equilibrium set $\{|\delta|<\pi\}\times \{|\gamma|= \pi\}$ (blue).
  • Figure 4: Cartesian trajectories of GloBa \ref{['eq-bkst-1']} with unity gains.
  • Figure 5: Cartesian trajectories with BAR-FLi \ref{['eq-bkst-3']} (blue) and BAgAl \ref{['eq-control-bounded-in-gamma-delta']} (cyan) with gains $[k_1,k_2,k_3,k_4] = [1,1,0.1,1]$ compared to the GloBa \ref{['eq-bkst-1']} (red).

Theorems & Definitions (21)

  • Definition 1
  • Proposition 1: Composite Lyapunov functions
  • Corollary 1
  • Definition 2: CLF for the unicycle \ref{['eq:unicycle_polar_closed_loop-Gv-1']}
  • Theorem 1
  • Corollary 2
  • Remark 1
  • Theorem 2: BoLSA CLFs
  • Theorem 3: BoPA CLFs
  • Theorem 4: BAgAl CLFs
  • ...and 11 more