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Rational motions with generic trajectories of low degree. (English) Zbl 1448.70006
Summary: The trajectories of a rational motion given by a polynomial of degree $$n$$ in the dual quaternion model of rigid body displacements are generically of degree $$2n$$. In this article we study those exceptional motions whose trajectory degree is lower. An algebraic criterion for this drop of degree is existence of certain right factors, a geometric criterion involves one of two families of rulings on an invariant quadric. Our characterizations allow the systematic construction of rational motions with exceptional degree reduction and explain why the trajectory degrees of a rational motion and its inverse motion can be different.

##### MSC:
 70B10 Kinematics of a rigid body
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##### References:
 [1] Bottema, O.; Roth, B., Theoretical Kinematics (1990), Dover Publications · Zbl 0747.70001 [2] Cheng, C. C.A.; Sakkalis, T., On new types of rational rotation-minimizing frame space curves, J. Symb. Comput., 74, 400-407 (2016) · Zbl 1332.53016 [3] Hamann, M., Line-symmetric motions with respect to reguli, Mech. Mach. Theory, 46, 960-974 (2011) · Zbl 1337.70003 [4] Hegedüs, G.; Schicho, J.; Schröcker, H. P., Factorization of rational curves in the Study quadric and revolute linkages, Mech. Mach. Theory, 69, 142-152 (2013) [5] Jüttler, B., Über zwangläufige rationale Bewegungsvorgänge, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II, 202, 117-232 (1993) [6] Li, Z.; Schicho, J.; Schröcker, H. P., 7R Darboux linkages by factorization of motion polynomials, (Chang, S. H., Proceedings of the 14th IFToMM World Congress (2015)) [7] Li, Z.; Schicho, J.; Schröcker, H. P., The rational motion of minimal dual quaternion degree with prescribed trajectory, Comput. Aided Geom. Des., 41, 1-9 (2016) · Zbl 1417.53012 [8] Pfurner, M.; Schröcker, H. P.; Husty, M., Path planning in kinematic image space without the Study condition, (Lenarčič, J.; Merlet, J. P., Proceedings of Advances in Robot Kinematics (2016)) [9] Pottmann, H.; Wallner, J., Computational Line Geometry. Mathematics and Visualization (2010), Springer [10] Rad, T. D.; Scharler, D. F.; Schröcker, H. P., The kinematic image of RR, PR, and RP dyads, Robotica, 36, 1477-1492 (2018) [11] Röschel, O., Rational motion design – a survey, Comput. Aided Des., 30, 169-178 (1998) · Zbl 0906.68175 [12] Selig, J. M., Geometric Fundamentals of Robotics. Monographs in Computer Science (2005), Springer · Zbl 1062.93002 [13] Wunderlich, W., Kubische Zwangläufe, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II, 193, 45-68 (1984) · Zbl 0555.53006
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