The collision avoidance and the controllability for $n$ bodies in dimension one

TL;DR

This paper introduces a control system design for multiple bodies in one-dimensional space and on the circle, ensuring collision avoidance and controllability through geometric control theory.

Contribution

It develops a novel control-affine system framework for collision-free movement of multiple bodies in 1D and on the circle, proving controllability via Lie algebraic methods.

Findings

System is controllable using Lie algebraic spanning conditions.

Control design guarantees collision avoidance among bodies.

Method applies to both real line and circle configurations.

Abstract

We present a method of design of control systems for $n$ bodies in the real line $\Bbb R^1$ and on the unit circle $ S^1$, to be collision-free and controllable. The problem reduces to designing a control-affine system in $\Bbb R^n$ and in $n$-torus $T^n, $ respectively, that avoids certain obstacles. We prove the controllability of the system by showing that the vector fields that define the control-affine system, together with their brackets of first order, span the whole tangent space of the state space, and then by applying the Rashevsky-Chow theorem.