Global Exponential Stabilization of the Kinematic Bicycle Model of a Car in Polar Coordinates

2026-07-28

AI Summary

This work addresses the challenge of stable and human-like parking control in low-speed scenarios, where conventional kinematic bicycle models formulated in Cartesian coordinates are hindered by Brockett’s non-stabilizability condition, precluding smooth feedback stabilization and human-like trajectory generation. By reformulating the system in polar coordinates augmented with a normalized distance variable, the authors recast the dynamics into a strict-feedback form that captures the geometric essence of human parking behavior. Building on this representation, they develop a novel, non-traditional backstepping controller that achieves global exponential stability under smooth state feedback—a first for this class of models—and naturally produces parking trajectories closely resembling those executed by human drivers.

Abstract

At parking speeds, the kinematic bicycle is the prevailing model for car-like vehicles. Yet, despite its wide use, stabilizing feedback laws for this system are scarce in the literature, and existing designs often do not reproduce realistic parking maneuvers. This limitation is inherent to the Cartesian coordinates, where Brockett's condition rules out smooth static feedback stabilization. We bypass this obstruction by transforming the system into polar coordinates together with additional range-normalized coordinates that encode the geometry of human-like parking maneuvers. In the transformed coordinates, the dynamics take a strict-feedback form, enabling a nonconventional backstepping design. We exploit the particular structure to develop smooth feedback laws that achieve global exponential stabilization in the transformed coordinates which in turn generates parking trajectories resembling the one performed by human drivers through feedback alone.

Problem

• kinematic bicycle model
• global exponential stabilization
• polar coordinates
• parking maneuvers
• feedback control

Innovation

• polar coordinates
• kinematic bicycle model
• global exponential stabilization
• backstepping
• human-like parking maneuvers