Abstract
This article proposes a new dual quaternion based approach for motion interpolation . The highlight is that dual quaternions act in the usual way on points, even if the Study condition is not fulfilled. This induces a fibration of kinematic image space into straight lines that describe the same rigid body displacement. This allows to use standard interpolation schemes for (piecewise) rational curves in a linear space rather than on the curved Study quadric.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bottema, O., Roth, B.: Theoretical Kinematics. North-Holland Series in Applied Mathematics and Mechanics, vol. 24. North-Holland Publishing Company, Amsterdam (1979)
Gfrerrer, A.: On the construction of rational curves on hyperquadrics. Grazer Math. Berichte 340, 1–69 (2000)
Gfrerrer, A.: Study’s kinematic mapping. In: Lenarcic, J., Stanisic, M. (eds.) Recent Advances in Robot Kinematics, pp. 7–16. Kluwer Academic Publishers, Dordrecht (2000)
Hofer, M., Pottmann, H., Ravani, B.: From curve design algorithms to the design of rigid body motions. Visual Comput. 20(5), 279–297 (2004)
Husty, M., Pfurner, M., Schröcker, H.P.: Algebraic methods in mechanism analysis and synthesis. Robotica 25(6), 661–675 (2007)
Husty, M., Schröcker, H.P.: Algebraic geometry and kinematics. In: Emiris, I.Z., Sottile, F., Theobald, T. (eds.) Nonlinear Computational Geometry. The IMA Volumes in Mathematics and its Applications, vol. 151, pp. 85–107. Springer, Berlin (2010). Chap. Algebraic Geometry and Kinematics
Jaklic, G., Jüttler, B., Krajnc, M., Vitrih, V., Žager, E.: Hermite interpolation by rational \(g^k\) motions of low degree. J. Comput. Appl. Math. 240, 20–30 (2013)
Jüttler, B.: Über zwangläufige rationale Bewegungsvorgänge. Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 202(1–10), 117–232 (1993)
Kecskeméthy, A., Hiller, M.: Einflüsse der Parametrisierung auf die Interpolation von Drehbewegungen. Z. Angew. Math. Mech. 73(4–5), T129–T131 (1993)
Krajnc, M., Pockaj, K., Vitrih, V.: Construction of low degree rational Lagrange motions. J. Comput. Appl. Math. 256, 92–103 (2014)
Rad, T.D., Scharler, D.F., Schröcker, H.P.: The kinematic image of RR, PR, and RP dyads. Submitted for publication (2016)
Schröcker, H.P., Jüttler, B.: Motion interpolation with Bennett biarcs. In: Kecskeméthy, A., Müller, A. (eds.) Proceedings of Computational Kinematics (CK 2009), pp. 101–108. Springer, Berlin (2009)
Study, E.: Geometrie der Dynamen. B. G. Teubner, Leipzig (1903)
Acknowledgements
This work was supported by the Austrian Science Fund (FWF): I 1750-N26, Kinematic analysis of lower-mobility parallel manipulators using efficient algebraic tools.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Pfurner, M., Schröcker, HP., Husty, M. (2018). Path Planning in Kinematic Image Space Without the Study Condition. In: Lenarčič, J., Merlet, JP. (eds) Advances in Robot Kinematics 2016. Springer Proceedings in Advanced Robotics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-56802-7_30
Download citation
DOI: https://doi.org/10.1007/978-3-319-56802-7_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56801-0
Online ISBN: 978-3-319-56802-7
eBook Packages: EngineeringEngineering (R0)