Algebraic Methods in Kinematics: Motion Factorisation and Bond Theory

Methods of algebra and algebraic geometry are widely used in kinematics and mechanism science. Since the configuration space of any linkage can be described as the solution set of a system of algebraic equations with parameters, synthesis, analysis, and classification problems for linkages can be reduced to quantifier elimination problems over real closed fields. In principle, elimination problems can be treated by existing algorithms and computer programs. However, the complexity of the involved systems of algebraic equations, especially the number of variables and parameters, is so large that most kinematical problems of interest are clearly out of reach of elimination theory. A more appropriate approach is to take advantage of the well-known fact that the group of direct isometries is isomorphic to the group of unit dual quaternions modulo plus/minus 1. This group isomorphism is fundamental for two algebraic tools we recently introduced in the context of linkages with revolute joints, namely factorisations of motion polynomials and the theory of bonds. In this project, we plan to take up these two techniques. The overall goal is the further development and application of algebraic theories and algorithms to the construction (synthesis), analysis and classification of linkages with special properties. Motion polynomials are left polynomials over the dual quaternions that parametrise rigid body motions. Linear motion polynomials always parametrise motions constrained by a revolute or prismatic joint, and so the factorisation of a motion polynomial into linear factors give rise to the decomposition into a motion constrained by revolute or prismatic joints. Since this factorisation is not unique, one gets several ways to realise the same motion as end effector motion of several open linkages that can be combined, in several ways, to form a closed loop linkage. The combination of rational interpolation techniques, constraint equations in dual quaternions, and motion factorisation could provide a promising toolbox for mechanism synthesis. Bonds have been introduced for the purpose of analysing the algebraic structure of linkages with revolute joints. They are defined as the elements of the boundary of the complexification of the configuration space in a suitable compactification. A closer analysis of these bonds reveals virtual connections between links that are not connected by joints with a geometric meaning that is not apparent but which has implications on the algebraic properties of the linkage. The potential of bond theory for the analysis of closed loops with revolute joints has just began to show; we plan go deeper in this direction, but also broader by extending to other types of linkages.

Mechanical linkages consist of rigid bodies (links) that are connected by revolute, prismatic, helical or spherical joints. A basic problem in kinematics is to compute the possible motions of a given linkage. Based on the answer, one tries to construct linkages that follow a given motion. The investigation of linkages with paradoxical mobility is a special mathematical challenge. In this case, the number of constraint equations is bigger than or equal to the number of degrees of freedom, so that the linkages are not expected to move at all; but still they do. Even though examples are known for a long time, it is often difficult to explain these motions and to classify them. Preliminary to the project, the proposing team developed two algebraic methods for exactly this purpose. These are the theory of motion polynomials and the theory of bonds. In the project, these methods were applied to various classes of linkages, and they turned out to be quite effective. For linkages with up to five revolute, prismatic or helical joints, there is now a complete classification. We also have now a classification of paradoxically moving Stewart platforms with up to five legs. A Stewart platform consists of a base and a platform and several legs, each of which being connected by joints with the base and the platform. Moreover, we developed algorithms for the synthesis of linkages that follow a rationally parametrizable motion. Such a motion might be given by a sequence of poses that have to be interpolated.