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The rational motion of minimal dual quaternion degree with prescribed trajectory. (English) Zbl 1417.53012

Summary: We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the trajectory’s degree minus its circularity. Hence, it is lower than the degree of a trivial curvilinear translation for circular curves.

MSC:

53A17 Differential geometric aspects in kinematics
70B10 Kinematics of a rigid body
70B15 Kinematics of mechanisms and robots
65D17 Computer-aided design (modeling of curves and surfaces)
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[1] Abbott, T. G., Generalizations of Kempe’s universality theorem, (2008), Massachusetts Institute of Technology, Master’s thesis
[2] Bottema, O.; Roth, B., Theoretical kinematics, (1990), Dover Publications · Zbl 0747.70001
[3] Demaine, E. D.; O’Rourke, J., Geometric folding algorithms: linkages, origami, polyhedra, (2007), Cambridge University Press · Zbl 1135.52009
[4] Hamann, M., Line-symmetric motions with respect to reguli, Mech. Mach. Theory, 46, 7, 960-974, (2011) · Zbl 1337.70003
[5] Hegedüs, G.; Schicho, J.; Schröcker, H.-P., Factorization of rational curves in the study quadric and revolute linkages, Mech. Mach. Theory, 69, 1, 142-152, (2013)
[6] Jüttler, B., Über zwangläufige rationale bewegungsvorgänge, Sitz.ber. - Österr. Akad. Wiss. Math.-Nat.wiss. Kl., II Math. Astron. Phys. Meteorol. Tech., 202, 117-132, (1993) · Zbl 0806.53011
[7] Li, Z.; Schicho, J.; Schröcker, H.-P., 7R Darboux linkages by factorization of motion polynomials, (Chang, Shuo-Hung, Proceedings of the 14th IFToMM World Congress 2015, (2015)), ISBN 979-986-04-6098-8. In press
[8] Li, Z., Schicho, J., Schröcker, H.-P., 2015b. Factorization of motion polynomials. Submitted for publication. arXiv:1502.07600 [cs.SC].
[9] Li, Z.; Schicho, J.; Schröcker, H.-P., Spatial straight-line linkages by factorization of motion polynomials, ASME J. Mech. Robot, (2015), In press
[10] Ore, O., Theory of non-commutative polynomials, Ann. Math. (2), 34, 3, 480-508, (1933) · Zbl 0007.15101
[11] Peternell, M.; Gruber, D.; Sendra, J., Conchoid surfaces of spheres, Comput. Aided Geom. Des., 30, 1, 35-44, (2013) · Zbl 1255.65055
[12] Selig, J., Geometric fundamentals of robotics, Comput. Sci. Monogr, (2005), Springer · Zbl 1062.93002
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