Null-forms of conic systems in $\mathbb{R}^3$ are determined by their symmetries

TL;DR

This paper characterizes null-forms of 3D conic control systems using their symmetry Lie algebras, providing a direct method to identify these systems among all control-affine systems.

Contribution

It introduces a novel symmetry-based characterization that uniquely determines null-forms of conic systems in three dimensions.

Findings

Lie algebra of symmetries uniquely identifies null-forms

Provides a direct characterization method

Enhances understanding of conic control systems

Abstract

We address the problem of characterisation of null-forms of conic $3$-dimensional systems, that is, control-affine systems whose field of admissible velocities forms a conic (without parameters) in the tangent space. Those systems have been previously identified as the simplest control systems under a conic nonholonomic constraint or as systems of zero curvature. In this work, we propose a direct characterisation of null-forms of conic systems among all control-affine systems by studying the Lie algebra of infinitesimal symmetries. Namely, we show that the Lie algebra of infinitesimal symmetries characterises uniquely null-forms of conic systems.