Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-03T19:20:08.550Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected]
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: [email protected], [email protected]
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: [email protected], [email protected]
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bottema, O. and Roth, B., Theoretical Kinematics (Dover Publications, New York, 1990).Google Scholar
2. Selig, J., Geometric Fundamentals of Robotics, 2nd ed. (Monographs in Computer Science, Springer, New York, 2005).Google Scholar
3. Husty, M. and Schröcker, H.-P., “Kinematics and algebraic geometry,” In Proceedings of the 21st Century Kinematics, The 2012 NSF Workshop (McCarthy, J. M., ed.) (Springer, London, 2012), pp. 85–123.Google Scholar
4. Klawitter, D., “Clifford algebras,” In: Geometric Modelling and Chain Geometries with Application in Kinematics (Springer Spektrum, Wiesbaden, 2015).Google Scholar
5. Brunnthaler, K., Schröcker, H.-P. and Husty, M. “A New Method for the Synthesis of Bennett Mechanisms,” Proceedings of CK 2005, International Workshop on Computational Kinematics, Cassino (2005).Google Scholar
6. Brunnthaler, K., Schröcker, H.-P. and Husty, M., “Synthesis of Spherical Four-Bar Mechanisms Using Spherical Kinematic Mapping,” In: Advances in Robot Kinematics (Lenarčič, J. and Roth, B., eds.) (Springer, Dordrecht, The Netherlands, 2006) pp. 377–384.Google Scholar
7. Perez-Gracia, A. and McCarthy, J. M., “Kinematic synthesis of spatial serial chains using Clifford algebra exponentials,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 220 (7), 953–968 (2006).Google Scholar
8. Ge, Q. J., Zhao, P. and Purwar, A., “Decomposition of Planar Burmester Problems using Kinematic Mapping,” In: Advances in Mechanisms, Robotics and Design Education and Research (Kumar, V., Schmiedeler, J., Sreenivasan, S. V. and Su, H. -J., eds.) (Springer International Publishing, Heidelberg, 2013) pp. 145–157.Google Scholar
9. Hegedüs, G., Schicho, J. and Schröcker, H. -P., “Four-pose synthesis of angle-symmetric 6R linkages,” ASME J. Mech. Robot. 7 (4) (2015) 7 pages.Google Scholar
10. Zhu, L., Zhao, P., Zi, B., Purwar, A. and Ge, Q. J., “Simultaneous Type and Dimensional Synthesis Approach with Expandable Solution Space for Planar Linkages,” Proceedings of the 14th IFToMM World Congress (2015). http://www.iftomm2015.tw/IFToMM2015CD/PDF/OS2-029.pdf.Google Scholar
11. Schröcker, H.-P., Husty, M. and McCarthy, J. M., “Kinematic Mapping Based Evaluation of Assembly Modes for Planar Four-bar Synthesis,” Proceedings of ASME 2005 29th Mechanism and Robotics Conference, Long Beach (2005).Google Scholar
12. Husty, M., Pfurner, M. and Schröcker, H. -P., “A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator,” Mech. Mach. Theory 42 (1), 66–81 (2007).Google Scholar
13. Selig, J. M., “Some rigid-body constraint varieties generated by linkages,” In: Latest Advances in Robot Kinematics, (Springer, Netherlands, 2012) pp. 293–300.Google Scholar
14. Hegedüs, G., Schicho, J. and Schröcker, H. -P., “The theory of bonds: A new method for the analysis of linkages,” Mech. Mach. Theory 70, 407–424 (2013).Google Scholar
15. Hegedüs, G., Li, Z., Schicho, J. and Schröcker, H.-P., “The theory of bonds II: Closed 6R linkages with maximal genus,” J. Symb. Comput. 68 (2), 167–180 (2015).Google Scholar
16. Selig, J. M. and Husty, M., “Half-turns and line symmetric motions,” Mech. Mach. Theory 46 (2), 156–167 (2011).Google Scholar
17. Selig, J. M., “On the geometry of point-plane constraints on rigid-body displacements,” Acta Applicandae Mathematicae 116 (2), 133 (2011).Google Scholar
18. Hegedüs, G., Schicho, J. and Schröcker, H. -P., “Factorization of rational curves in the Study quadric and revolute linkages,” Mech. Mach. Theory 69 (1), 142–152 (2013).Google Scholar
19. Stigger, T., Algebraic Constraint Equations of Simple Kinematic Chains Master Thesis (University of Innsbruck, 2015).Google Scholar
20. Pfurner, M., Schröcker, H.-P. and Husty, M., “Path Planning in Kinematic Image Space Without the Study Condition,” Proceedings of Advances in Robot Kinematics (J., Lenarčič and Merlet, J.-P., eds.) (2016). https://hal.archives-ouvertes.fr/hal-01339423.Google Scholar
21. Rad, T.-D. and Schröcker, H.-P., “The Kinematic Image of 2R Dyads and Exact Synthesis of 5R Linkages,” Proceedings of the IMA Conference on Mathematics of Robotics, ISBN 978-0-90-509133-4 (2015).Google Scholar
22. Casas-Alvero, E., Analytic Projective Geometry (European Mathematical Society, Zürich, Switzerland, 2014).Google Scholar
23. Coxeter, H. S. M., Non-Euclidean Geometry, 6th ed. (Cambride University Press, Washington, 1998).Google Scholar
24. Li, Z., Schicho, J. and Schröcker, H. -P., “7R Darboux Linkages by Factorization of Motion Polynomials,” Proceedings of the 14th IFToMM World Congress (Chang, S.-H., ed.) (2015).Google Scholar
25. Ravani, B. and , R. B., “Mappings of spatial kinematics,” J. Mech. Trans. Autom. 106 (3), 341–347 (1984).Google Scholar
26. Purwar, A. and Ge, J., “Kinematic Convexity of Rigid Body Displacements,” Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, Montreal (2010).Google Scholar