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A Flat Triangular Structure Based on a Multi-Chained Form

Georg Hartl, Conrad Gstöttner, Markus Schöberl

TL;DR

This work advances differential flatness analysis for nonlinear multi-input control-affine systems by introducing a structurally flat triangular form TF_s based on a multi-chained structure. It extends the extended chained form to cases with at least three inputs and provides necessary and sufficient static feedback equivalence conditions for TF_0 and TF_1, yielding constructive methods to compute flat outputs directly from geometric distributions. A key contribution is a modular, invariant procedure that avoids full system transformation while producing valid flat outputs, and it is validated on a three-dimensional gantry crane, where the load position serves as a flat output. The approach broadens the class of systems amenable to flatness-based trajectory planning and control, with explicit procedures for practical realization.

Abstract

Determining whether a nonlinear multi-input system is differentially flat remains challenging. One way to obtain computationally tractable sufficient conditions is to give complete characterizations of flat normal forms. We introduce a structurally flat triangular form for control-affine systems with at least three inputs that is based on a multi-chained form. For two specific instances of this structure, we provide complete geometric characterizations, i.e., necessary and sufficient conditions under which a control-affine system is static-feedback equivalent to the respective triangular form. These characterizations yield sufficient conditions for differential flatness and, in turn, constructive procedures for computing flat outputs.

A Flat Triangular Structure Based on a Multi-Chained Form

TL;DR

This work advances differential flatness analysis for nonlinear multi-input control-affine systems by introducing a structurally flat triangular form TF_s based on a multi-chained structure. It extends the extended chained form to cases with at least three inputs and provides necessary and sufficient static feedback equivalence conditions for TF_0 and TF_1, yielding constructive methods to compute flat outputs directly from geometric distributions. A key contribution is a modular, invariant procedure that avoids full system transformation while producing valid flat outputs, and it is validated on a three-dimensional gantry crane, where the load position serves as a flat output. The approach broadens the class of systems amenable to flatness-based trajectory planning and control, with explicit procedures for practical realization.

Abstract

Determining whether a nonlinear multi-input system is differentially flat remains challenging. One way to obtain computationally tractable sufficient conditions is to give complete characterizations of flat normal forms. We introduce a structurally flat triangular form for control-affine systems with at least three inputs that is based on a multi-chained form. For two specific instances of this structure, we provide complete geometric characterizations, i.e., necessary and sufficient conditions under which a control-affine system is static-feedback equivalent to the respective triangular form. These characterizations yield sufficient conditions for differential flatness and, in turn, constructive procedures for computing flat outputs.

Paper Structure

This paper contains 9 sections, 6 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Known Results
  4. Main Results
  5. Computation of Flat Outputs
  6. A Practical Example
  7. Conclusion
  8. Sketch of the Proof of Theorem \ref{['thm:TFCCS+IC_0']}
  9. Sketch of the Proof of Theorem \ref{['thm:TFCCS+IC_1']}

Key Result

Theorem 1

Consider a control-affine system eq:ai_sys_m with $mk+1$ states, $m+1\ge3$ inputs, the drift $f$ and the input distribution $\mathcal{D} = \operatorname{span}\{g_0,\ldots,g_m\}$. The given system is locally SFE to eq:CCFD if and only if:

Figures (1)

  • Figure 1: Schematic diagram of a gantry crane.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Lemma 3
  • Remark 1: Case $s=1$ vs. $s\ge 2$