The Algebra of Motions in 3-Space

Algebra is playing an increasingly important role for understanding and explaining phenomena related to motions of rigid bodies in 3-space, in particular aspects that depend only on the geometry of the objects and not on physical forces. This is a proposal to extend the known algebraic toolbox and use it to treat problems/questions arising in kinematics and robotics. Concrete leverage points are two theories that have been introduced in kinematics recently, namely the theory of factorizations of motion polynomials and the theory of bonds. Motion polynomials are known to be useful for decomposing polynomially parametrized motions into rotational components around fixed axes. We propose to extend the factorization theory of such polynomials in various ways. The perspective is to construct linkages that follow a predescribed motion. For planar bounded rational motions, such a construction was recently developed in a predecessor project. Bond theory is a technique that allows to understand unexpected mobility of linkages consisting of links that are cyclically connected by rotational joints. We plan to use this tool for analyzing linkages with a more complicated structure, such as flexible realizations of graphs in the plane or polyhedra in 3-space.

Motions in 3-space can be composed in several ways. The composition which was investigated in this project, and which we call kinematic composition, reflects the fact that a motion describes the relative position of one rigid body with respect to another rigid body at a certain time. If we compose the motion describing the position of the body A in relation to body B with the motion describing the position of body B in relation to body C, then we obtain the motion that describes the position of body A in relation to body C. On the other hand, we can model motions in 3-space as functions from the set of real numbers -- time -- to the group of Euclidean displacements. If this function is a polynomial or rational function, then it can be expressed as a polynomial in one variable over the skew algebra of dual quaternions. A basic observation (for our project) is that the non-commutative multiplication of these polynomials reflects exactly the kinematic composition of motions in 3-space. The correspondence between kinematics and algebra made it possible to give surprisingly simple and explicit solutions to some challenging tasks in kinematics, such as the synthesis of kinematic chains that follow a prescribed motion. A quite general mathematical description of a linkage is a moving graph. One thinks of the edges of the graph as its rigid body links, which are joined at the vertices. In 2-space, these are revolute joints, and in 3-space, these are spherical joints. For moving graphs in the plane and in the sphere, we developed a theory of edge colorings of the graph which determine mobility and provide an algebraic way to parametrize the motions. We also adapted this approach to flexible polyhedra by scaling the directed edges of the polyhedron so that they become unit vectors and the flexibility question is reduced to the mobility of a moving graph on the sphere.