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arXiv · 2609.24192

Global Exponential Stabilization of a 3D Nonholonomic Vehicle in Spherical Coordinates

Kwang Hak Kim·Velimir Todorovski·Miroslav Krstic
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Abstract

Many spatial (3D) vehicles, including AUVs and fixed-wing aircraft, are effectively nonholonomic and subject to limited actuation, such that continuous time-invariant stabilization is fundamentally obstructed by Brockett's necessary condition. To overcome this obstruction, we exploit the geometric singularity of spherical coordinates to design a backstepping continuous, time-invariant feedback law that exponentially stabilizes a 3D nonholonomic vehicle actuated solely by forward surge velocity, pitch rate, and yaw rate. The resulting closed-loop region of attraction excludes only the coordinate singularity, codimension-two, measure-zero set of initial conditions in which the vehicle lies on the line through the target orthogonal to the target plane, thereby covering the largest possible domain. We further construct a strict control Lyapunov function to prove global exponential stability of the origin on this domain with a user-specified decay rate, while simultaneously preventing the system from approaching the singular set. Finally, we show that the closed-loop system is exponentially attractive to the origin in Cartesian coordinates. Numerical simulation examples in both spherical and Cartesian coordinates illustrate the effectiveness of the control law.

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Kwang Hak Kim, Velimir Todorovski, Miroslav Krstic. 2026-09-21. Global Exponential Stabilization of a 3D Nonholonomic Vehicle in Spherical Coordinates. https://arxiv.org/abs/2609.24192

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