Geometric tools for analysis of parallel robots

Parallel manipulators are characterized by the fact that more than one kinematic chains are acting in parallel to move the end effector of a robot. They have become popular within the last twenty years as flight simulators, milling machines, telescopes robots in surgery and motion simulators for entertainment industry. Although some of the most important theoretical problems (like direct and inverse kinematics) of most of the proposed designs have been solved, there are still a lot of open problems in kinematics, dynamics and design of parallel robots. This research project aims in attacking some of the open problems with help of geometric methods. The main goal of the project is to define and evaluate quality indices that allow to distinguish between different designs of a class of parallel manipulators. The main type of investigated manipulator will be the Stewart-Gough-Platform, a six legged platform, where the platform is moved by changing the lengths of the six legs. In a first step different quality functions will be developed to evaluate workspace and singularity performance of a parallel manipulator. For the workspace evaluation and the trajectory planning and verification a method will be developed that heavily relies on the Study parametrization of Euclidean motions and the geometric interpretation of this parametrization. It is believed that this method will help to give better answers to the following questions: - given the orientation or a range of orientations, what is the positional workspace? - Given a position or a range (in 3D) of positions, what is the (singularity free) orienting capability? - Given a trajectory, whether it is completely inside the workspace and singularity free