×

zbMATH — the first resource for mathematics

From A to B: new methods to interpolate two poses. (English) Zbl 1426.70003
Summary: We present two methods to interpolate between two given rigid body displacements. Both are based on linear interpolation in the ambient space of well-known curved point models for the group of rigid body displacements. The resulting motions are either vertical Darboux motions or cubic circular motions. Both are rational of low degree and lie in the cylinder group defined by the two input poses. We unveil the essential parameters in the construction of these motions and discuss some of their properties.

MSC:
70B10 Kinematics of a rigid body
51N10 Affine analytic geometry
53A17 Differential geometric aspects in kinematics
65D17 Computer-aided design (modeling of curves and surfaces)
PDF BibTeX XML Cite
Full Text: arXiv Link
References:
[1] O. Bottema, B. Roth: Theoretical Kinematics. Dover Publications, 1990. · Zbl 0747.70001
[2] A. Grünwald: Die kubische Kreisbewegung eines starren Körpers. Z. Math. Physik 55, 264–296 (1907).
[3] M. Hamann: Line-symmetric motions with respect to reguli. Mech. Mach. Theory 46(7), 960–974 (2011). · Zbl 1337.70003
[4] M. Husty, H.-P. Schröcker: Kinematics and Algebraic Geometry. In J. M. McCarthy (ed.): 21st Century Kinematics. The 2012 NSF Workshop. Springer, London 2012, pp. 85–123.
[5] D. Klawitter: Clifford Algebras. Geometric Modelling and Chain Geometries with Application in Kinematics. Springer Spektrum, Wiesbaden 2015. · Zbl 1310.15037
[6] J. Krames: Zur aufrechten Ellipsenbewegung des Raumes (Über symmetrische Schrotungen III). Monatsh. Math. Physik 46(1), 38–50 (1937). · JFM 63.0723.02
[7] J. Krames: Zur Geometrie des Bennett’schen Mechanismus (Über symmetrische Schrotungen IV). Sitzungsber., Abt. II, österr. Akad. Wiss., Math.-Naturw. Kl. 146, 159–173 (1937). · JFM 63.0723.04
[8] M. Pfurner, H.-P. Schröcker, M. Husty: Path Planning in Kinematic Image Space Without the Study Condition. In J. Lenarčič, J.-P. Merlet (eds.): Proceedings of Advances in Robot Kinematics 2016, Springer International Publishing AG 2018.
[9] A. Purwar, J. Ge: 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, pp. 1761–1772.
[10] T.-D. Rad, D.F. Scharler, H.-P. Schröcker: The Kinematic Image of RR, PR, and RP Dyads. Submitted for publication, 2016.
[11] B. Ravani, B. Roth: Mappings of Spatial Kinematics. J. Mech., Trans., and Automation 106(3), 341–347 (1984).
[12] J. Selig: Geometric Fundamentals of Robotics. Monographs in Computer Science, 2nd ed., Springer 2005. · Zbl 1062.93002
[13] W. Wunderlich: Kubische Zwangläufe. Sitzungsber., Abt. II, österr. Akad. Wiss., Math.-Naturw. Kl. 193, 45–68 (1984).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.