1. Forschungsschwerpunkte Fakultät für Bauingenieurwesen Symbolic Computation in Mechanism and Machine Science Manfred L. Husty Inst. f. Basic Sciences in Engineering Unit Geometry and CAD

2. Overview Introduction Kinematic Mapping, Study Model of SE(3) Constraint Varieties for Mechanism Analysis and Synthesis Application to Mechanism Analysis –Direct Kinematics of Stewart-Gough Platforms –Assembly modes of a nine-bar structure – Dixon’s mechanisms –Architecture Singularity and Self Motions of Platform Manipulators (Griffis- Duffy Platforms) –6-AS-Manipulator –Discussion of the Inverse Kinematics of General 6-R Manipulators –3UPU Analysis Application to Mechanism Synthesis –Synthesis of Bennett Mechanisms Conclusions H-P. Schröcker, K. Brunnthaler, M. Pfurner, Dominic Walter, Jim Robinson

3. Examples

4. Basics Design parameter: parameters that determine the geometry and topology of the mechanism Joint coordinates: Lengths or angles of the actuators Cartesian coordinates: Parameters that determine the relative position and orientation of end- effector frame to base frame Workspace: set of all poses that can be reached by the manipulator in presence of limits of active and passive joints Direct Kinematics: Given: Design of the manipulator and joint parameter Find: possible positions and orientations of the manipulator Inverse Kinematics Given: Design of the manipulator and the pose of the end-effector Find: joint parameters

5. Study-Model of SE(3)

6. Kinematic Mapping

8. kinematic image space (P 7 ) moving coordinate system fixed coordinate system Study quadric S 6 2

9. Properties of S 6 2

10. Constraint Varieties