Kempe’s universality theorem for rational space curves. (English) Zbl 1430.70006

Li, Zijia; Schicho, Josef; Schröcker, Hans-Peter

Found. Comput. Math. 18, No. 2, 509-536 (2018).

Summary: We prove that every bounded rational space curve of degree \(d\) and circularity \(c\) can be drawn by a linkage with \(\frac{9}{2} d-6c+1\) revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage.

MSC:

70B05	Kinematics of a particle
14H50	Plane and space curves
65D17	Computer-aided design (modeling of curves and surfaces)
68U07	Computer science aspects of computer-aided design

Keywords:

dual quaternion; motion polynomial; factorization; Bennett flip; linkage

References:

[1]	T. G. Abbott, Generalizations of Kempe’s Universality Theorem, Master’s thesis, Massachusetts Institute of Technology, Cambridge 2008. · Zbl 1372.65120
[2]	K. Abdul-Sater, M. M. Winkler, F. Irlinger and T. C. Lueth, Three-position synthesis of origami-evolved, spherically constrained spatial revolute-revolute chains, ASME J. Mechanisms Robotics 8 (2016), no. 1, doi:10.1115/1.4030370. · Zbl 1332.53016
[3]	I. I. Artobolevskii, Mechanisms for the generation of plane curves, Pergamon Press, Oxford, 1964.
[4]	J. E. Baker, On the motion geometry of the Bennett linkage, In Proceedings of the 8th International Conference on Engineering Computer Graphics and Descriptive Geometry, Austin, 1998, 433-437. · Zbl 1404.70007
[5]	Bennett, GT, A new mechanism, Engineering, 76, 777-778, (1903)
[6]	G. T. Bennett, The skew isogramm-mechanism, Proc. London Math. Soc. s2-13 (1914), 151-173.
[7]	W. Blaschke and H. R. Müller, Ebene Kinematik, Oldenbourg, München, 1956. · Zbl 0071.14204
[8]	Cheng, CC-A; Sakkalis, T, On new types of rational rotation-minimizing frame space curves, J. Symb. Comput., 74, 400-407, (2016) · Zbl 1332.53016
[9]	Corless, RM; Watt, SM; Zhi, L, QR factoring to compute the gcd of univariate approximate polynomials, IEEE Transactions on Signal Processing, 52, 3394-3402, (2004) · Zbl 1372.65120
[10]	E. D. Demaine and J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, Cambridge, 2007. · Zbl 1135.52009
[11]	P. Dietmaier, Einfach übergeschlossene Mechanismen mit Drehgelenken, Habilitation thesis, Graz University of Technology, Graz, 1995. · Zbl 1056.14077
[12]	Gallet, M; Koutschan, C; Li, Z; Regensburger, G; Schicho, J; Villamizar, N, Planar linkages following a prescribed motion, Math. Comput., 86, 473-506, (2016) · Zbl 1404.70007
[13]	Gao, X-S; Zhu, C-C; Chou, S-C; Ge, J-X, Automated generation of kempe linkages for algebraic curves and surfaces, Mech. Machine Theory, 36, 1019-1033, (2001) · Zbl 1140.70332
[14]	Hegedüs, G; Schicho, J; Schröcker, H-P, Factorization of rational curves in the study quadric and revolute linkages, Mech. Machine Theory, 69, 142-152, (2013)
[15]	G. Hegedüs, J. Schicho and H.-P. Schröcker, Four-pose synthesis of angle-symmetric 6R linkages, J. Mechanisms Robotics 7 (2015), no. 4. doi:10.1115/1.4029186.
[16]	Huang, L; So, W, Quadratic formulas for quaternions, Appl. Math. Lett., 15, 533-540, (2002) · Zbl 1011.15010
[17]	Husty, M; Schröcker, H-P; Emiris, Z (ed.); Sottile, F (ed.); Theobald, T (ed.), Algebraic geometry and kinematics, 85-107, (2009), New York · Zbl 1185.70005
[18]	Jüttler, B, Über zwangläufige rationale bewegungsvorgänge, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B., II, 117-232, (1993) · Zbl 0806.53011
[19]	E. Kaltofen, Z. Yang and L. Zhi, Approximate Greatest Common Divisors of Several Polynomials with Linearly Constrained Coefficients and Singular Polynomials, In Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (J.-G. Dumas, eds.), ACM, New York, 2006, pp. 169-176. · Zbl 1356.12011
[20]	Kapovich, M; Millson, JJ, Universality theorems for configuration spaces of planar linkages, Topology, 41, 1051-1107, (2002) · Zbl 1056.14077
[21]	A. B. Kempe, On a general method of describing plane curves of the nth degree by linkwork, Proc. London Math. Soc. s1-7 (1876), 213-216. · Zbl 0806.53011
[22]	A. Kobel, Automated generation of Kempe linkages for algebraic curves in a dynamic geometry system, Bachelor’s thesis, Saarland University, 2008. · Zbl 1404.70007
[23]	Z. Li, Sharp linkages, In Advances in Robot Kinematics (J. Lenarčič and O. Khatib, eds.), Springer, Cham, 2014, pp. 131-138.
[24]	Li, Z; Schicho, J, Classification of angle-symmetric 6R linkages, Mech. Machine Theory, 70, 372-379, (2013)
[25]	Z. Li and J. Schicho, Three types of parallel 6R linkages, In Computational Kinematics: Proceedings of the 6th International Workshop on Computational Kinematics (CK2013) (F. Thomas and A. Perez Gracia, eds.), Springer, Dordrecht, 2014, pp. 111-119.
[26]	Z. Li, J. Schicho and H.-P. Schröcker, 7R Darboux linkages by factorization of motion polynomials, In Proceedings of the 14th IFToMM World Congress (S.-H. Chang, ed.), 2015, doi:10.6567/IFToMM.14TH.WC.OS2.014
[27]	Z. Li, J. Schicho and H.-P. Schröcker, Factorization of motion polynomials, arXiv:1502.07600, 2015.
[28]	Li, Z; Schicho, J; Schröcker, H-P, The rational motion of minimal dual quaternion degree with prescribed trajectory, Comput. Aided Geom. Design, 41, 1-9, (2016)
[29]	A. J. Perez, Analysis and Design of Bennett Linkages, Ph.D. thesis, University of California, Irvine, 2004. · Zbl 1372.65120
[30]	Saxena, A, Kempe’s linkages and the universality theorem, Resonance, 16, 220-237, (2011)