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The kinematic image of RR, PR, and RP dyads

Published online by Cambridge University Press:  27 June 2018

Tudor-Dan Rad ,
Daniel F. Scharler  and
Hans-Peter Schröcker Open the ORCID record for Hans-Peter Schröcker [Opens in a new window]
Tudor-Dan Rad
Affiliation:
Unit Geometry and CAD, Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria E-mails: tudor-dan.rad@uibk.ac.at, daniel.scharler@uibk.ac.at
Daniel F. Scharler
Affiliation:
Unit Geometry and CAD, Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria E-mails: tudor-dan.rad@uibk.ac.at, daniel.scharler@uibk.ac.at
Hans-Peter Schröcker*
Affiliation:
Unit Geometry and CAD, Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria E-mails: tudor-dan.rad@uibk.ac.at, daniel.scharler@uibk.ac.at
*
*Corresponding author. E-mail: hans-peter.schroecker@uibk.ac.at
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Summary

We provide necessary and sufficient conditions for all projective transformations of the projectivized dual quaternion model of rigid body displacements that are induced by coordinate changes in moving and/or fixed frame. These transformations fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. Moreover, we fully characterize the constraint varieties of dyads with revolute and prismatic joints in the dual quaternion model. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a “null quadrilateral.” If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterize such dyads. We provide a complete characterization by introducing a new invariant, the “Study fibre projectivity,” and we present examples that demonstrate its potential to explain hitherto not sufficiently well-understood phenomena.

Keywords

Kinematic mapDual quaternionStudy quadricNull coneRevolute jointPrismatic jointStudy fibre projectivityVertical Darboux motion
Type
Articles
Information
Robotica , Volume 36 , Issue 10 , October 2018 , pp. 1477 - 1492
Copyright
Copyright © Cambridge University Press 2018 

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