Analysis of kinematics of mechanisms containing revolute joints

TL;DR

The paper addresses the computational burden of analyzing kinematics for mechanisms with revolute joints using polynomial representations via Euler parameters. By decomposing revolute-joint constraints into two prime components and selecting the physically relevant branch, the authors reduce polynomial degrees and eliminate variables, enabling efficient Gröbner-basis analysis for complex mechanisms. Applied to Bricard's and Bennett's mechanisms, the method yields complete-intersection descriptions, explicit parametrizations, and regular configuration spaces, clarifying mobility and enabling reliable dynamics simulations. The approach provides a general, scalable framework for algebraic-geometry-based kinematic analysis of revolute-joint mechanisms with potential for broader applicability in mechanism design and simulation.

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 Gröbner bases, can be used to study them. The disadvantage of the Gröbner 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.

Figures (2)

• Figure 1: The two curves of Bricard's variety both represent the same physical situation.
• Figure 2: The projection of Bennett's variety to $(a_0,a_2,c_0)$ space with $m=(7/4,5/6,1/3)$.