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Intrinsic Derivation of the Equations of a Snake Robot based on a Cosserat Beam Model

Anis Bousclet, Frederic Boyer, Yacine Chitour, Swann Marx

TL;DR

An intrinsic derivation of the equations ruling the dynamics motion of a snake robot dynamics is presented, based on a Cosserat beam model, which shows that the extended configuration space is a Lie group and provides the constitutive law describing the actuation in this system of PDEs.

Abstract

In this paper, we present an intrinsic derivation of the equations ruling the dynamics motion of a snake robot dynamics. Based on a Cosserat beam model, we first show that the extended configuration space is a Lie group. Endowing it with an appropriate left invariant metric, the corresponding Euler-Poincaré equations can be reduced to a system of hyperbolic PDEs in the Lie algebra $\mathfrak{se}(3)$. We also provide the constitutive law describing the actuation in this system of PDEs.

Intrinsic Derivation of the Equations of a Snake Robot based on a Cosserat Beam Model

TL;DR

An intrinsic derivation of the equations ruling the dynamics motion of a snake robot dynamics is presented, based on a Cosserat beam model, which shows that the extended configuration space is a Lie group and provides the constitutive law describing the actuation in this system of PDEs.

Abstract

In this paper, we present an intrinsic derivation of the equations ruling the dynamics motion of a snake robot dynamics. Based on a Cosserat beam model, we first show that the extended configuration space is a Lie group. Endowing it with an appropriate left invariant metric, the corresponding Euler-Poincaré equations can be reduced to a system of hyperbolic PDEs in the Lie algebra . We also provide the constitutive law describing the actuation in this system of PDEs.
Paper Structure (11 sections, 11 theorems, 79 equations)

This paper contains 11 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The configuration space
  3. Riemannian structure
  4. Euler Poincare equations
  5. Adding a control law
  6. Appendix
  7. Insufficiency of classical medium
  8. The kinetic energy of a Cosserat medium
  9. Basics on Riemannian geometry
  10. Lie Groups and Riemannian geometry
  11. Notations and definitions

Key Result

Proposition 1

The right action $\Phi: M\times G\rightarrow{M}$ defined by is transitive and free.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Definition 5
  • Proposition 2
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 15 more