Hans-Peter Schröcker

Algebraic methods in mechanism analysis and synthesis

Algebraic 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 Studys 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.

Algebraic Geometry and Kinematics

In this overview paper we show how problems in computational kinematics can be translated into the language of algebraic geometry and subsequently solved using techniques developed in this field. The idea to transform kinematic features into the language of algebraic geometry is old and goes back to Study. Recent advances in algebraic geometry and symbolic computation gave the motivation to resume these ideas and make them successful in the solution of kinematic problems. It is not the aim of the paper to provide detailed solutions, but basic accounts to the used tools and examples where these techniques were applied within the last years. We start with Study’s kinematic mapping and show how kinematic entities can be transformed into algebraic varieties. The transformations in the image space that preserve the kinematic features are introduced. The main topic are the definition of constraint varieties and their application to the solution of direct and inverse kinematics of serial and parallel robots. We provide a definition of the degree of freedom of a mechanical system that takes into account the geometry of the device and discuss singularities and global pathological behavior of selected mechanisms. In a short paragraph we show how the developed methods are applied to the synthesis of mechanical devices.

Factorization of rational curves in the study quadric

Abstract For every 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.

The Theory of Bonds: A New Method for the Analysis of Linkages

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.

Construction of Overconstrained Linkages by Factorization of Rational Motions

We prove that for any sufficiently generic rational curve C of degree n in the group of Euclidean displacements, there exists an overconstrained spatial linkage with revolute joints whose linkage graph is the 1-skeleton of the n-dimensional hypercube such that the constrained motion of one of the links is exactly C. The synthesizing algorithm is based on the factorization of polynomials over the dual quaternions. The linkage contains n! open nR chains, so that low degree examples include Bennett’s mechanisms and are related to overconstrained 5R and 6R chains.

Uniqueness results for minimal enclosing ellipsoids

We prove uniqueness of the minimal enclosing ellipsoid with respect to strictly eigenvalue convex size functions. Special examples include the classic case of minimal volume ellipsoids (Lowner ellipsoids), minimal surface area ellipsoids or, more generally, ellipsoids that are minimal with respect to quermass integrals.

Motion Interpolation with Bennett Biarcs

We present an interpolation scheme for first order Hermite motion data (two positions with associated instantaneous screws specifying the tangent vector fields) that is based on a generalization of the classic biarc construction to curves on quadrics. The result is a sequence of Bennett motions. These motions possess several properties that make them particularly useful for motion interpolation, especially for applications requiring collision detection. We suggest methods for choosing the free parameter that determines the interpolating pair of Bennett motions and we demonstrate how to obtain an interpolation algorithm which is invariant with respect to changes in the moving and the fixed coordinate frame.

Kinematics and Algebraic Geometry

This overview paper is a collection of several papers (Husty and Schrocker, Nonlinear Computational Geometry, The IMA Volumes in Mathematics and Its Applications, vol. 151, pp. 85–107, Springer, Berlin, 2010; Husty et al., Mech. Mach. Theory 42:66–81, 2007; Robotica 25:661–675, 2007; Schrocker et al., J. Mech. Des. 129:924–929, 2007; Walter et al., Contemporary Mathematics, vol. 496, pp. 331–346, American Mathematical Society, Providence, 2009; Walter and Husty, Mach. Des. Res. 26:218–226, 2010) that were published by the authors and their collaborators within the last ten years. As basic paradigm we show how problems in computational kinematics can be translated into the language of algebraic geometry and subsequently solved using techniques developed in this field mostly with help of an algebraic manipulation system.

Path Planning in Kinematic Image Space Without the Study Condition

This article proposes a new dual quaternion based approach for motion interpolation . The highlight is that dual quaternions act in the usual way on points, even if the Study condition is not fulfilled. This induces a fibration of kinematic image space into straight lines that describe the same rigid body displacement. This allows to use standard interpolation schemes for (piecewise) rational curves in a linear space rather than on the curved Study quadric.

The theory of bonds II

In this paper, we study closed linkages with six rotational joints that allow a one-dimensional set of motions. We prove that the genus of the configuration curve of such a linkage is at most five, and give a complete classification of the linkages with a configuration curve of genus four or five. The classification contains new families.