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.

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SUMMARYAlgebraic methods in connection with classical multidimensional geometry have proven to be very efficient in the computation of direct and inverse kinematics of mechanisms as well as the explanation of strange, pathological behavior. In this paper, we give an overview of the results achieved within the last few years using the algebraic geometric method, geometric preprocessing, and numerical analysis. We provide the mathematical and geometrical background, like Study's parametrization of the Euclidean motion group, the ideals belonging to mechanism constraints, and methods to solve polynomial equations. The methods are explained with different examples from mechanism analysis and synthesis.

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.

In this paper, we consider the existence of a factorization of a monic, bounded motion polynomial. We prove existence of factorizations, possibly after multiplication with a real polynomial and provide algorithms for computing polynomial factor and factorizations. The first algorithm is conceptually simpler but may require a high degree of the polynomial factor. The second algorithm gives an optimal degree.

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:

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples.

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