Analysis of kinematics of mechanisms containing revolute joints

TL;DR

This paper presents a method to simplify kinematic analysis of mechanisms with revolute joints by leveraging the non-prime nature of their constraint ideals, reducing computational complexity using algebraic geometry techniques.

Contribution

It introduces a novel approach to simplify polynomial computations in mechanism kinematics by selecting prime components of the constraint ideal, applicable to various revolute joint mechanisms.

Findings

Reduced computational complexity in kinematic analysis

Effective application to Bennett's and Bricard's mechanisms

Generalizable method for mechanisms with revolute joints

Abstract

Kinematics of rigid bodies can be analyzed in many different ways. The advantage of using Euler parameters is that the resulting equations are polynomials and hence computational algebra, in particular Gr\"obner bases, can be used to study them. The disadvantage of the Gr\"obner basis methods is that the computational complexity grows quite fast in the worst case in the number of variables and the degree of polynomials. In the present article we show how to simplify computations when the mechanism contains revolute joints. The idea is based on the fact that the ideal representing the constraints of the revolute joint is not prime. Choosing the appropriate prime component reduces significantly the computational cost. We illustrate the method by applying it to the well known Bennett's and Bricard's mechanisms, but it can be applied to any mechanism which has revolute joints.