Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.24192

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

Kwang Hak Kim·Velimir Todorovski·Miroslav Krstic
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.OCcs.SYeess.SY

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Subpath-Based Column Generation for Electric Vehicle Routing Problems

Motivated by widespread electrification targets, this paper studies an Electric Vehicle Routing Problem with Time Windows and Nonlinear Charging (EVRPTWNL) that jointly optimizes routing-scheduling decisions and charging decisions given vehicle capacities, time windows and battery capacities. We develop a column generation scheme with a subpath-based label-setting algorithm that decomposes the pricing problem into two phases: (i) generating subpaths between charging stations, and (ii) combining subpaths into paths while optimizing charging decisions in between. We formalize a domination framework to establish the convergence and exactness of the algorithm, and prove that the methodology can solve a range of EVRP variants (e.g., with vehicle capacities, time windows, and nonlinear charging) and relaxation-tightening strategies (e.g., ng-relaxations and subset-row cuts). Computational results show improvements over path-based benchmarks in both computational time and solution quality, especially when time windows become wider, when vehicles can perform multiple tasks on a single charge and when vehicles still need to recharge several times across the planning horizon. Ultimately, the methodology can scale to otherwise intractable instances with up to 100 customers, thereby enhancing fleet management capabilities across electrified logistics areas.

math.OC↗
arXiv2026

Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they do not address \emph{complexity}: for broad operator classes described only through regularity (e.g.\ uniform continuity or $C^r$-regularity), information-theoretic lower bounds imply that minimax-optimal NO approximation rates scale \emph{exponentially} in the reciprocal accuracy $1/\varepsilon$. This has shifted the focus of NO theory toward identifying additional problem-specific structure, beyond regularity, under which suitably tailored NO architectures can leverage to unlock polynomial scaling in $1/\varepsilon$. We exhibit the first polynomial-scaling regime for NO approximations of solution operators in stochastic analysis; by identifying structured families of \emph{non-Markovian} BSDEs with randomized terminal condition parameterized by the Sobolev-regular terminal condition and by Sobolev-regular additive nonlinear perturbations of the generator. We prove that their solution operator can be approximated (uniformly over the family) by a tailored NO whose number of trainable parameters grows \emph{polynomially} in $1/\varepsilon$. We unlock this polynomial scaling regime by \emph{informing the NO's inductive bias} by factoring out the singular part of the associated semilinear elliptic PDE Green's function and by incorporating the Doléans--Dade exponential of the BSDE's common non-Markovian factor into the NO's decoding layers. As a byproduct, we extend polynomial-scaling guarantees from families of linear elliptic PDEs on regular domains to the semilinear setting.

math.OC↗
arXiv2026

The Competive Spectral Radius of Families of Nonexpansive Mappings

We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Mañe's lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the second author (2012), concerning the asymptotic behavior of nonexpansive mappings. We discuss two special cases of this game: order preserving and positively homogeneous self-maps of a cone equipped with Funk's and Thompson's metrics, and translations of a finite dimensional normed space.

math.OC↗