Skip to main content

The kinematic image of RR, PR, and RP dyads

  • Tudor-Dan Rad (a1), Daniel F. Scharler (a1) and Hans-Peter Schröcker (a1)

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.

Corresponding author
*Corresponding author. E-mail:
Hide All
1. Bottema, O. and Roth, B., Theoretical Kinematics (Dover Publications, New York, 1990).
2. Selig, J., Geometric Fundamentals of Robotics, 2nd ed. (Monographs in Computer Science, Springer, New York, 2005).
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.
4. Klawitter, D., “Clifford algebras,” In: Geometric Modelling and Chain Geometries with Application in Kinematics (Springer Spektrum, Wiesbaden, 2015).
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).
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. 377384.
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), 953968 (2006).
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. 145157.
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.
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).
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).
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), 6681 (2007).
13. Selig, J. M., “Some rigid-body constraint varieties generated by linkages,” In: Latest Advances in Robot Kinematics, (Springer, Netherlands, 2012) pp. 293300.
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, 407424 (2013).
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), 167180 (2015).
16. Selig, J. M. and Husty, M., “Half-turns and line symmetric motions,” Mech. Mach. Theory 46 (2), 156167 (2011).
17. Selig, J. M., “On the geometry of point-plane constraints on rigid-body displacements,” Acta Applicandae Mathematicae 116 (2), 133 (2011).
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), 142152 (2013).
19. Stigger, T., Algebraic Constraint Equations of Simple Kinematic Chains Master Thesis (University of Innsbruck, 2015).
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).
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).
22. Casas-Alvero, E., Analytic Projective Geometry (European Mathematical Society, Zürich, Switzerland, 2014).
23. Coxeter, H. S. M., Non-Euclidean Geometry, 6th ed. (Cambride University Press, Washington, 1998).
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).
25. Ravani, B. and , R. B., “Mappings of spatial kinematics,” J. Mech. Trans. Autom. 106 (3), 341347 (1984).
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).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0263-5747
  • EISSN: 1469-8668
  • URL: /core/journals/robotica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 1 *
Loading metrics...

Abstract views

Total abstract views: 8 *
Loading metrics...

* Views captured on Cambridge Core between 27th June 2018 - 20th July 2018. This data will be updated every 24 hours.