Mathematical Methods in Kinematics

Time and Place: Friday 13:45-11:15, T 406, SS 2020. Start: 6.3.2020

Lecture notes.

In kinematics, a linkage is composed of rigid bodies (bars, rods) that are connected by joints (e.g., hinges or spherical joints); examples occur in mechanical engineering and robotics, but also in biology -- the human skeleton may be considered as a quite complex linkage -- and in chemistry, at a miscroscopic scale.

If every link is a bar, and every joint connects two or more bars by a spherical joint (or simply a revolute joint in the plane), then we obtain a bar-joint framework. Each configuration is determined by the position of the centers of the spheres. A mathemtical tool to study the configurations is distance geometry.

Other types of joints are revolute, prismatic, cylindrical, or helical joints. Chains of links connected by revolute or prismatic joints are used in robotics. Mathematically, chains and loops of links can be described by systems of equations over motion groups.

Symmetry -- a finite group acting on the configuration set preserving the geometric parameters -- can often explain paradoxical phenomena such as unexpected mobility.

Stewart platforms also known as multipods consist of two main links, the base and the platform, and bars with one spherical joint on the base and another on the platform. The group of motions of the platform relative to the base can be embedded in a projective space. The geometric parameters defining the bars (length and positions of end points) is contained in another projective space. The two projective spaces are naturally dual to each other, and this leads to an interplay between kinematics and projective geometry.