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A Proper Definition of Higher Order Rigidity

Tomohiro Tachi

Abstract

[Connelly and Servatius, 1994] shows the difficulty of properly defining n-th order rigidity and flexiblity of a bar-and-joint framework for higher order (n >= 3) through the introduction of a cusp mechanism. The author proposes a "proper" definition of the order of rigidity by the order of elongation of the bars with respect to the arclength along the path in the configuration space. We show that the classic definition using formal n-th derivative of the length constraint is a sufficient condition for the n-th flexiblity in the proposed definition and also a necessary condition only for n = 1, 2.

A Proper Definition of Higher Order Rigidity

Abstract

[Connelly and Servatius, 1994] shows the difficulty of properly defining n-th order rigidity and flexiblity of a bar-and-joint framework for higher order (n >= 3) through the introduction of a cusp mechanism. The author proposes a "proper" definition of the order of rigidity by the order of elongation of the bars with respect to the arclength along the path in the configuration space. We show that the classic definition using formal n-th derivative of the length constraint is a sufficient condition for the n-th flexiblity in the proposed definition and also a necessary condition only for n = 1, 2.

Paper Structure

This paper contains 5 sections, 3 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition
  3. Flex and Flexibility
  4. Cusp Mechanism
  5. Discussion

Key Result

Theorem 3

By definition, $n$-th order rigidity implies $n+1$-th order rigidity. $n$-th order rigidity implies finite rigidity.

Figures (2)

  • Figure 1: Cusp mechanism constructed by connecting a pair of Watt's mechanisms.
  • Figure 2: The bifurcation of the motion of horizontal bar.

Theorems & Definitions (8)

  • Definition 1: Finite Flexibility of a bar-and-joint framework
  • Definition 2: Proposed Definition of $n$-th order rigidity
  • Theorem 3: Proper
  • Definition 4: Classic Definition
  • Theorem 5: Classic $n$-th Order Flexiblity is Sufficient
  • proof
  • Theorem 6: First and Second Order
  • proof