Four-Pose Synthesis of Angle-Symmetric 6R Linkages

Abstract

We use the recently introduced factorization theory of motion polynomials over the dual quaternions for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. Our approach admits either no or a one-parametric family of solutions. We suggest strategies for picking good solutions from this family. IntroductionIn [1], Brunnthaler et al. presented a method for synthesizing closed kinematic chains of four revolute joints whose coupler can visit three prescribed poses (position and orientation). These linkages are also known under the name "Bennett linkage". By that time, it was already known that three poses in general position define a unique Bennett linkage. Other synthesis methods were available [2][3][4][5]. With hindsight, the most important novelty of [1] was the characterization of the coupler motions of Bennett linkages as conics in the Study quadric model of the direct Euclidean displacement group SE(3). Depending on the precise concept of "Bennett linkages", it might be necessary to exclude certain conics, see [6]. This interpretation allows to immediately construct the coupler motion from the three given poses-even before the linkage itself is determined.In this article, we generalize the method of [1] to the four-pose synthesis of special kinematic loops of six revolute joints (6R linkages). Our main tool is the recently developed technique of factoring rational motions [7], or more precisely, motion polynomials. This allows to decompose a rational parametrized equation in Study parameters into the product of linear factors. Each such factorization gives rise to an open kinematic chain. Suitable combinations of open chains produce closed loop linkages whose coupler follows the prescribed rational motion. If we combine this factorization of rational motions with rational interpolation techniques on a quadric, we obtain a framework for mechanism synthesis. It requires the completion of three steps:

Optimal Synthesis of Overconstrained 6R Linkages by Curve Evolution

“…The linear factors are normalized to avoid numeric fluctuation of C(t) without any geometric change. A suitable initial guess for the shape parameter can found by interpolating four poses [5] or, as we did in our example, by assigning random values to the shape parameters. Several attempts might be necessary in order to ensure good convergence.…”

Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions

“…The relevance of polynomial factorization in mechanism science goes beyond above simple examples (see for example [24,5,6,11]). It provides a more or less automatic way to construct linkages from rational motions.…”

The Kinematic Image of 2R Dyads and Exact Synthesis of 5R Linkages

“…The attentive reader will note that we neither provide a synthesis algorithm nor a concrete example. The reason for this (and one of the main points of this paper) is that the synthesis of 5R linkages can be treated as special case of a recently introduced synthesis procedure for 6R linkages (Hegedüs et al, 2014). In this context, 2R spaces play a crucial role.…”

“…Suitable combinations of such factorisations give an overconstrained 6R linkage whose coupler follows the prescribed cubic motion. For details on how to compute cubic interpolants on quadrics, how to factor the resulting motions and how to pick factorisations suitable for linkage synthesis we refer to Hegedüs et al (2014). We just highlight the major differences to the case of 5R linkage synthesis.…”

“…A generalisation of their approach led to a factorisation theory for "motion polynomials" (Hegedüs et al, 2013a). It was used by Hegedüs et al (2014) to synthesise spatial 6R linkages -closed kinematic chains with six revolute joints -to four prescribed poses in general position. More precisely, the four poses determine two families of (not necessary real) rational coupler motions, each giving rise to nine families of overconstrained 6R linkages.…”