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A global approach for the redefinition of higher-order flexibility and rigidity

Georg Nawratil

Abstract

The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively, as they attest the cusp configuration of the mechanism a third-order rigidity, which conflicts with its continuous flexion. Some attempts were done to resolve the dilemma but they could not settle the problem. As cusp mechanisms demonstrate the basic shortcoming of any local mobility analysis using higher-order constraints, we present a global approach inspired by Sabitov's finite algorithm for testing the bendability of a polyhedron, which allows us (a) to compute iteratively configurations with a higher-order flexion and (b) to come up with a proper redefinition of higher-order flexibility and rigidity. The presented approach is demonstrated on several examples (double-Watt mechanisms and Tarnai's Leonardo structure). Moreover, we determine all configurations of a given 3-RPR manipulator with a third-order flexion and present a corresponding joint-bar framework of flexion order 23.

A global approach for the redefinition of higher-order flexibility and rigidity

Abstract

The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively, as they attest the cusp configuration of the mechanism a third-order rigidity, which conflicts with its continuous flexion. Some attempts were done to resolve the dilemma but they could not settle the problem. As cusp mechanisms demonstrate the basic shortcoming of any local mobility analysis using higher-order constraints, we present a global approach inspired by Sabitov's finite algorithm for testing the bendability of a polyhedron, which allows us (a) to compute iteratively configurations with a higher-order flexion and (b) to come up with a proper redefinition of higher-order flexibility and rigidity. The presented approach is demonstrated on several examples (double-Watt mechanisms and Tarnai's Leonardo structure). Moreover, we determine all configurations of a given 3-RPR manipulator with a third-order flexion and present a corresponding joint-bar framework of flexion order 23.
Paper Structure (10 sections, 2 theorems, 46 equations, 12 figures, 1 table)

This paper contains 10 sections, 2 theorems, 46 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Algebraic approach to rigidity theory
  3. Review on higher-order flexibility and rigidity
  4. Structures studied with respect to higher-order flexibility
  5. Redefinition of a higher-order flexibility and rigidity
  6. Computational Aspects
  7. Computing 3-RPR configurations with a higher-order flexion
  8. Bar-plate framework
  9. Bar-joint framework
  10. Final remarks, open problems and future work

Key Result

lemma 1

Every regular point of $V_1$ has to have a single non-trivial instantaneous flexion.

Figures (12)

  • Figure 1: Double-Watt mechanism of Connelly and Servatius in its cusp configuration; i.e. the mechanism has an instantaneous standstill. The dimensions of each Watt mechanism are as follows: the arms have length $1$ and the coupler is of length $\sqrt{2}$. The midpoints ${\mathbf x}_1$ and ${\mathbf x}_2$ of both couplers are connected by a bar of length $3$.
  • Figure 2: Reduction of the double-Watt mechanism of Connelly and Servatius to a two-point guidance problem.
  • Figure 3: Double-Watt mechanism of Stachel in a branching configuration; i.e. it corresponds to a double point in the configuration space.
  • Figure 4: Reduction of Stachel's double-Watt mechanism to a two-point guidance problem.
  • Figure 5: Stachel's double-Watt mechanism extended by a Kempe-mechanism (Figure by courtesy of Hellmuth Stachel stachel_aim).
  • ...and 7 more figures

Theorems & Definitions (21)

  • definition 1
  • remark 1
  • definition 2
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • lemma 1
  • proof
  • ...and 11 more