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Modular Design of Strict Control Lyapunov Functions for Global Stabilization of the Unicycle in Polar Coordinates

Velimir Todorovski, Kwang Hak Kim, Miroslav Krstic

TL;DR

This work develops a modular framework for globally stabilizing a unicycle in polar coordinates, circumventing Cartesian Brockett obstructions by combining forward-velocity decoupling with angular control via backstepping and passivity. It constructs multiple composite control Lyapunov functions and barrier Lyapunov functions to guarantee global asymptotic stability or almost-global stabilization, with distinct controllers (GloBa, BAR-FLi, BoLSA, BAgAl) offering various region-of-attraction properties and barrier-enforced safety. The designs yield explicit convergence guarantees and CLFs, enabling systematic extension and barrier-based safety, with a companion paper extending to inverse-optimal redesigns. The results provide practical, modular designs for smooth stabilization and safe parking maneuvers of unicycle-like vehicles in polar coordinates, highlighting the trade-offs imposed by topology and barrier constraints.

Abstract

Since the mid-1990s, it has been known that, unlike in Cartesian form where Brockett's condition rules out static feedback stabilization, the unicycle is globally asymptotically stabilizable by smooth feedback in polar coordinates. In this note, we introduce a modular framework for designing smooth feedback laws that achieve global asymptotic stabilization in polar coordinates. These laws are bidirectional, enabling efficient parking maneuvers, and are paired with families of strict control Lyapunov functions (CLFs) constructed in a modular fashion. The resulting CLFs guarantee global asymptotic stability with explicit convergence rates and include barrier variants that yield "almost global" stabilization, excluding only zero-measure subsets of the rotation manifolds. The strictness of the CLFs is further leveraged in our companion paper, where we develop inverse-optimal redesigns with meaningful cost functions and infinite gain margins.

Modular Design of Strict Control Lyapunov Functions for Global Stabilization of the Unicycle in Polar Coordinates

TL;DR

This work develops a modular framework for globally stabilizing a unicycle in polar coordinates, circumventing Cartesian Brockett obstructions by combining forward-velocity decoupling with angular control via backstepping and passivity. It constructs multiple composite control Lyapunov functions and barrier Lyapunov functions to guarantee global asymptotic stability or almost-global stabilization, with distinct controllers (GloBa, BAR-FLi, BoLSA, BAgAl) offering various region-of-attraction properties and barrier-enforced safety. The designs yield explicit convergence guarantees and CLFs, enabling systematic extension and barrier-based safety, with a companion paper extending to inverse-optimal redesigns. The results provide practical, modular designs for smooth stabilization and safe parking maneuvers of unicycle-like vehicles in polar coordinates, highlighting the trade-offs imposed by topology and barrier constraints.

Abstract

Since the mid-1990s, it has been known that, unlike in Cartesian form where Brockett's condition rules out static feedback stabilization, the unicycle is globally asymptotically stabilizable by smooth feedback in polar coordinates. In this note, we introduce a modular framework for designing smooth feedback laws that achieve global asymptotic stabilization in polar coordinates. These laws are bidirectional, enabling efficient parking maneuvers, and are paired with families of strict control Lyapunov functions (CLFs) constructed in a modular fashion. The resulting CLFs guarantee global asymptotic stability with explicit convergence rates and include barrier variants that yield "almost global" stabilization, excluding only zero-measure subsets of the rotation manifolds. The strictness of the CLFs is further leveraged in our companion paper, where we develop inverse-optimal redesigns with meaningful cost functions and infinite gain margins.

Paper Structure

This paper contains 16 sections, 5 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Modular Feedback Design for the Unicycle
  3. Forward velocity feedback
  4. State spaces and stability definitions
  5. Composite Lyapunov functions
  6. Families of CLFs for the Unicycle
  7. Backstepping-based design
  8. GloBa ($\mathcal{T}$) and BAR-FLi ($\mathcal{T}_1$) controllers
  9. Passivity-inspired design
  10. BoLSA controller ($\mathcal{T}_2$)
  11. BAgAl controller ($\mathcal{T}_3$)
  12. Barrier CLFs and "nearly global" feedbacks
  13. Barrier CLFs
  14. Feedback actions near and at the barriers
  15. Topological behavior of BAgAl controller \ref{['eq-control-bounded-in-gamma-delta']}
  16. ...and 1 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 (4)

  • Figure 1: Unicycle orientation $(x,y,\theta)$ with respect to the goal state $(0,0,0)$, along with the polar coordinate transformation $(x,y,\theta)\mapsto (\rho,\delta,\gamma)$.
  • Figure 2: Visualization of the torus $\{|\delta|\leq\pi\}\times \{|\gamma|\leq \pi\}$ with the undefined control set $\{|\delta|=\pi\}\times \{|\gamma|\leq \pi\}$ (red) and the equilibrium set $\{|\delta|<\pi\}\times \{|\gamma|= \pi\}$ (blue).
  • Figure 3: Closed-loop trajectories of GloBa \ref{['eq:backstepping_controllers']} with \ref{['eq:Delta_globa']}.
  • Figure 4: Closed-loop trajectories with BAR-FLi (blue) controller \ref{['eq:backstepping_controllers']} with \ref{['eq:Delta_Barfli']} and BAgAl (cyan) controller \ref{['eq-control-bounded-in-gamma-delta']} compared to the GloBa controller \ref{['eq:backstepping_controllers']} with \ref{['eq:Delta_globa']} (red).

Theorems & Definitions (9)

  • Definition 1
  • Proposition 1: Composite Lyapunov functions
  • Definition 2: CLF for the unicycle \ref{['eq:unicycle_polar_closed_loop-Gv-1']}
  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3