×

Space kinematics and projective differential geometry over the ring of dual numbers. (English) Zbl 1489.53017

Summary: We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers. Some seemingly weird phenomena in this space have lucid kinematic interpretations. An example is the existence of non-straight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also suggest geometrically meaningful ways to select osculating conics of a curve in this projective space and illustrate their corresponding motions. Furthermore, we investigate factorizability of these special motions and use the obtained results for the construction of overconstrained linkages.

MSC:

53A20 Projective differential geometry
70B10 Kinematics of a rigid body
16S99 Associative rings and algebras arising under various constructions
PDFBibTeX XMLCite
Full Text: Link

References:

[1] J. Angeles:Rational Kinematics. Springer, 1989. · Zbl 0706.70003
[2] K. Brunnthaler,H.-P. Schröcker, andM. Husty:A New Method for the Synthesis of Bennett Mechanisms. InProceedings of CK 2005, International Workshop on Computational Kinematics. Cassino, 2005.
[3] M. Hamann:Line-symmetric motions with respect to reguli. Mech. Mach. Theory46(7), 960-974, 2011. · Zbl 1337.70003
[4] G. Hegedüs,J. Schicho, andH.-P. Schröcker:Factorization of Rational Curves in the Study Quadric and Revolute Linkages. Mech. Mach. Theory69(1), 142-152, 2013.
[5] M. HustyandH.-P. Schröcker:Kinematics and Algebraic Geometry. InJ. M. McCarthy, ed.,21st Century Kinematics. The 2012 NSF Workshop, 85-123. Springer, London, 2012.
[6] Z. Li, ,T. Rad,J. Schicho, andH.-P. Schröcker:Factorization of Rational Motions: A Survey with Examples and Applications. InS.-H. Chang et al., ed., Proc. IFToMM 14, 833-840. 2015.
[7] M. Pfurner,H.-P. Schröcker, andM. Husty:Path Planning in Kinematic Image Space Without the Study Condition. InJ. LenarčičandJ.-P. Merlet, eds.,Proceedings of Advances in Robot Kinematics. 2016.https://hal.archives-ouvertes.fr/h al-01339423.
[8] A. PurwarandJ. Ge:Kinematic Convexity of Rigid Body Displacements. InProceedings of the ASME 2010 International Design Engineering Technical Conference & Computers and Information in Engineering Conference IDETC/CIE. Montreal, 2010.
[9] D. Scharler:Characterization of Lines in Extended Kinematic Image Space. Master thesis, University of Innsbruck, 2017.
[10] H.-P. Schröcker:From A to B: New Methods to Interpolate Two Poses. J. Geom. Graphics22(1), 87-98, 2018. · Zbl 1426.70003
[11] C. Segre:Le geometrie proiettive nei campi di numeri duali. InCorrado Segre, Opere, vol. II, 396-431. Unione Matematica Italiana, Roma, edizione cremonese ed., 1912.
[12] J. Siegele,M. Pfurner, andH.-P
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.