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Symbolic Computation in Mechanism and Machine

Science

Manfred L. Husty

Inst. f. Basic Sciences in Engineering Unit Geometry and CAD

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

Examples

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 KinematicsGiven: Design of the manipulator and the pose of the end-effector

Find: joint parameters

Study-Model of SE(3)

Kinematic Mapping

Kinematic Mapping

Kinematic Mapping

kinematic image space (P7)

moving coordinate system

fixed coordinate system

Study quadric S62

Properties of S62

Constraint Varieties