Given a generic rational curve C in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly C. Our construction is based on the factorization of polynomials over dual quaternions. Low degree examples include the Bennett mechanisms and contain new types of overconstrained 6R-chains as sub-mechanisms.

In this paper we introduce a new technique, based on dual quaternions, for the analysis of closed linkages with revolute joints: the theory of bonds. The bond structure comprises a lot of information on closed revolute chains with a one-parametric mobility. We demonstrate the usefulness of bond theory by giving a new and transparent proof for the well-known classification of overconstrained 5R linkages.

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:

Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω1 the pth primitive unity root, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _1 : = e^{\tfrac{{2\pi i}} {p}}$$ \end{document}.Define ωi ≔ ω1i for 0 ≦ i ≦ p − 1 and B ≔ {1, ω1 , …, ωp −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν1 , …, νk ∈ B such that for any vector w ∈ B , there is an i , 1 ≦ i ≦ k , such that νi · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A1 , …, Am of {1, …, 4 n } with | Ai | = 2 n for each 1 ≦ i ≦ n , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ i ≦ m , with Ai ∩ B having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.