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

2025-09-29

AI Summary

This paper addresses the problem of achieving global asymptotic stabilization for a unicycle system modeled in polar coordinates. The proposed method introduces a modular framework for constructing strict control Lyapunov functions (CLFs), integrating polar-coordinate modeling, bidirectionally smooth feedback design, and a variant of barrier functions—ensuring almost-global stability while excluding a zero-measure subset of the rotational manifold. The resulting CLF enables inverse-optimal reconstruction and guarantees infinite gain margin, with an explicitly characterized convergence rate bound. Key contributions include: (i) the first modular CLF construction paradigm that balances theoretical rigor with engineering implementability; and (ii) the first simultaneous achievement—in the unicycle’s polar-coordinate model—of efficient parking maneuvers, explicit convergence-rate guarantees, and strong robustness properties (notably infinite gain margin). This work establishes a novel, structured approach to CLF design for nonlinear mechanical systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Designing smooth feedback laws for global unicycle stabilization in polar coordinates
• Constructing modular strict control Lyapunov functions with explicit convergence rates
• Developing bidirectional controllers enabling efficient parking maneuvers with stability guarantees

Innovation

• Modular framework designs smooth feedback laws
• Bidirectional laws enable efficient parking maneuvers
• Strict CLFs guarantee global asymptotic stability