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Mechanical feedback linearization of single-input mechanical control systems

Marcin Nowicki, Witold Respondek

TL;DR

This paper develops mechanical feedback linearization (MF-linearization) for single-input mechanical control systems, aiming to preserve the mechanical structure (configurations and velocities) during linearization. It establishes geometric necessary-and-sufficient conditions, expressed via the nested distributions $\mathcal{E}^i$ and compatibility constraints on covariant derivatives and Christoffel symbols, that determine when an MS can be MF-linearized to a controllable linear mechanical system. The main results prove MF-equivalence for $n\ge3$ (with MF1–MF4) and a specialized $n=2$ case (MF1'–MF5'), and are illustrated through three examples, including an inertia wheel pendulum and the TORA3 system, highlighting that MF-linearizable systems form a proper subset of F-linearizable ones. The work emphasizes preserving physical interpretation and sensor requirements, and sets the stage for future comparisons between MF- and classical F-linearization and for broader mechanical control design techniques.

Abstract

We present a new type of feedback linearization that is tailored for mechanical control systems. We call it a mechanical feedback linearization. Its basic feature is preservation of the mechanical structure of the system. For mechanical systems with a scalar control, we formulate necessary and sufficient conditions that are verifiable using differentiations and algebraic operations only. We illustrate our results with several examples.

Mechanical feedback linearization of single-input mechanical control systems

TL;DR

This paper develops mechanical feedback linearization (MF-linearization) for single-input mechanical control systems, aiming to preserve the mechanical structure (configurations and velocities) during linearization. It establishes geometric necessary-and-sufficient conditions, expressed via the nested distributions and compatibility constraints on covariant derivatives and Christoffel symbols, that determine when an MS can be MF-linearized to a controllable linear mechanical system. The main results prove MF-equivalence for (with MF1–MF4) and a specialized case (MF1'–MF5'), and are illustrated through three examples, including an inertia wheel pendulum and the TORA3 system, highlighting that MF-linearizable systems form a proper subset of F-linearizable ones. The work emphasizes preserving physical interpretation and sensor requirements, and sets the stage for future comparisons between MF- and classical F-linearization and for broader mechanical control design techniques.

Abstract

We present a new type of feedback linearization that is tailored for mechanical control systems. We call it a mechanical feedback linearization. Its basic feature is preservation of the mechanical structure of the system. For mechanical systems with a scalar control, we formulate necessary and sufficient conditions that are verifiable using differentiations and algebraic operations only. We illustrate our results with several examples.
Paper Structure (7 sections, 4 theorems, 73 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem statement
  4. Mechanical feedback linearization
  5. Examples
  6. Conclusions
  7. Appendix

Key Result

Theorem 1

Assume $n \geq 3$. A mechanical control system $(\mathcal{MS})$ is, locally around $x_0$, MF-linearizable to a controllable $(\mathcal{LMS})$ if and only if

Figures (1)

  • Figure 1: The TORA3 system

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • proof
  • Theorem 2
  • Remark 3
  • proof
  • Lemma 1
  • Lemma 2
  • ...and 1 more