Extended Kinematic Mappings and Application to Motion Design

Within the project Extended Kinematic Mappings and Application to Motion Design (EKIMAP) we explore a new approach for the simple and interactive design of motions, that is, for example, useful in computer graphics or for planing the movement of robots. The basic idea is to extend and adapt well- known techniques for interactive drawing and manipulation of freeform curves or surfaces on a computer screen. The user interactively picks points, the software automatically draws a smooth approximating or interpolating curve or surface. In motion design, similar algorithms do exist but using them can be quite cumbersome. Naive versions of these algorithms distort objects during the motion. More refined versions are hard to compute. Moreover, a precise concept of approximation in the context of moving objects is not obvious. While the curves and surfaces usually lie in our three-dimensional ambient space, the motions must be embedded into higher dimensional spaces. In order to determine the position of a rigid body in space, we need three coordinates to fix a point but three more coordinates to also fix its spatial orientation. This demonstrates that we need at least six dimensions. A basic idea of our approach is to use even higher dimensions (seven or even twelve). This does not complicate mathematical descriptions but actually simplifies them. In this way we gain a lot of additional geometric and algebraic structure that motion design benefits from. We envisage that an engineer using our methods may specifiy a number of key poses of a robot. The motion connecting these poses is then automatically computed, taking into account a number of given constraints, for example collisions with ambient objects, equilibrium and maximal velocity of the robot etc. This is very useful for innovative manufactoring methods where robots have to perform diverse and complicated tasks. Being able to design excellent motions in an intuitive and efficient manner is crucial in this context.

The research project Extended Kinematic Mappings and Application to Motion Design explores a special mathematical description of Euclidean motions, the so called kinematic or extended kinematic mappings. These describe motions of real objects as curves or manifolds in a high dimensional space, the kinematic image space. In case of kinematic mapping all these geometric objects lie on a curved manifold, the so-called Study quadric. The mapping and inverse mapping is unique but hardly usable for different tasks (e.g. motion design) because of the curved nature of this special quadric. The extended kinematic mapping takes the whole ambient space into account instead. Loosing on the one hand uniqueness of the mapping yields linearity on the other hand. This gives simpler mathematical usability of the description. In a lower degree example one could think of a sphere and the whole 3-space around. It would mathematically be much simpler to specify a random curve in 3-space compared to a curve entirely on the sphere. To apply motion design in the extended kinematic image space we have to understand the space and the inverse mapping much better. Furthermore, the connection of the motions and generating mechanisms must be revealed more extensive. Therefore, we restricted in this project to rational motions and their properties. These kinds of motions generate point trajectories, which are rational curves in 3- space. The mathematical description in kinematic image space can be achieved by special polynomials. They can be, as in case of polynomials known from middle school, factorized in linear factors. In case of extended kinematic mapping additional factors could be found, which yield a new kind of joints, the so called Darboux joints. This is a combination of a rotation around and a harmonical oscillation along the same axis. Unfortunately, there exists no suitable mechanical realization of such a joint. But during the project a single loop closed mechanism could be found, which is able to fulfil such a motion and consists of revolute and prismatic joints only. In the inverse mapping from kinematic image space to spatial motions the degree of the point trajectories usually doubles, but there exist exceptions. Many of them were already know, but not fully understood. This project could give algebraic and geometric conditions on motions to explain this exceptional behaviour. A further consideration of the kinematic image space as a mathematical construct over the dual numbers (most likely comparable to complex numbers) yields more statements about motions and their approximations. Although some weird phenomena occur this perspective is the correct geometric framework to approach systematic considerations of special local approximations for motions.