The CR geometry of the three-segment snake

TL;DR

This paper explores the CR geometric structure of a planar three-segment snake robot, revealing conditions under which it admits a CR structure and solving the related equivalence problem to identify invariants.

Contribution

It introduces the CR geometric framework for the three-segment snake and solves the Cartan equivalence problem to find its invariants, a novel approach in robot kinematics.

Findings

The snake's velocity distribution forms a (2,3,5) distribution.

Under certain parameters, it admits a CR structure of dimension 1.

The invariants of the CR structure are explicitly determined.

Abstract

We study the geometry associated with the kinematics of a planar robot known as the "three-segment snake," whose velocity distribution belongs to a class of (2,3,5) distributions. We discover that, under certain assumptions on its construction parameters, the snake may be endowed with a CR structure of CR dimension 1 and real codimension 3. We solve the associated Cartan equivalence problem and find the invariants of the snake's CR structure.