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The application of Euler-Rodrigues formula over hyper-dual matrices. (English) Zbl 1526.22014


MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
15A66 Clifford algebras, spinors
70E17 Motion of a rigid body with a fixed point

Software:

DNAD
Full Text: DOI

References:

[1] [1] Aslan, S., Bekar, M., Yaylı, Y.: Hyper-dual split quaternions and rigid body motion, Journal of Geometry and Physics. 158, 103876 (2020). · Zbl 1444.11022
[2] [2] Bottema, O., Roth, B.: Theoretical Kinematics, North-Holland Publishing Company, New York, (1979). · Zbl 0405.70001
[3] [3] Cantún-Avila, K. B., González-Sánchez, D., Díaz-Infante, S., Peñuñuri, F.: Optimizing functionals using differential evolution, Engineering Applications of Artificial Intelligence. 97, 104086 (2021).
[4] [4] Chasles, M.: Note sur les propriétés générales du système de deux corps semblables entr’eux, Bulletin des Sciences, Mathématiques, Astronomiques, Physiques et Chemiques. 4, 321-326 (1830).
[5] [5] Clifford, W.K.: Preliminary sketch of biquaternions, Proc. London Mathematical Society. 4, 381-395 (1873). · JFM 05.0280.01
[6] [6] Cohen, A., Shoham, M.: Application of hyper-dual numbers to multi-body kinematics, Journal of Mechanisms and Robotics. 8, 011015 (2016).
[7] [7] Cohen, A., Shoham, M.: Application of hyper-dual numbers to rigid bodies equations of motion, J. Mech. Mach. Theory. 111, 76-84 (2017).
[8] [8] Cohen, A., Shoham, M.: Hyper dual quaternions representation of rigid bodies kinematics, J. Mech. Mach. Theory. 150, 103861 (2020).
[9] [9] Dai, J.S.: Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections, Mech. Mach. Theory. 92, 144-152 (2015).
[10] [10] Euler, L.: Formulae generales pro translatione quacunque corporum rigidorum, Novi Comm. Acad. Sci. Imp. Petrop. 20, 189-207 (1776).
[11] [11] Fike, J., Alonso, J.: The development of hyper-dual numbers for exact second derivative calculations,49th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition. (2011).
[12] [12] Fischer, I.: Dual-number methods in kinematics, statics and dynamics. CRC press, (1998).
[13] [13] Griewank, A.: On automatic differentiation, Mathematical Programming: recent developments and applications, 6(6), 83-107 (1989). · Zbl 0696.65015
[14] [14] Gromov, N.A.: Possible quantum kinematics. J. Math. Phys. 47(1), 013502 (2006). · Zbl 1111.81098
[15] [15] Hall, B.: Lie groups, Lie algebras, and representations: an elementary introduction (Vol. 222). Springer, (2015). · Zbl 1316.22001
[16] [16] Imoto, Y., Yamanaka, N., Uramoto, T., Tanaka, M., Fujikawa, M., Mitsume, N.: Fundamental theorem of matrix representations of hyper-dual numbers for computing higher-order derivatives, JSIAM Letters. 12, 29-32 (2020). · Zbl 07246802
[17] [17] Kahveci, D., Gök, ˙I., Yaylı, Y.: Some variations of dual Euler-Rodrigues formula with an application to point-line geometry, Journal of Mathematical Analysis and Applications. 459(2), 1029-1039 (2018). · Zbl 1380.51011
[18] [18] Kisil, V.V.: Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R), Imperial College Press, London, (2012). [19] McCarthy, J.M.: Introduction to Theoretical Kinematics, MIT Press, (1990).
[19] [20] Müller, A.: Coordinate mappings for rigid body motions. Journal of Computational and Nonlinear Dynamics. 12(2), (2017).
[20] [21] Palais, B., Palais, R.: Euler’s fixed point theorem: The axis of a rotation, Journal of Fixed Point Theory and Applications. 2(2), 215-220 (2007). · Zbl 1140.55001
[21] [22] Palais, B., Palais, R.: Chasles’ fixed point theorem for Euclidean motions. Journal of Fixed Point Theory and Applications, 12(1-2), 27-34 (2012). · Zbl 1266.54093
[22] [23] Pottmann, H., Wallner, J.: Computational Line Geometry, Springer-Verlag, (2001). · Zbl 1006.51015
[23] [24] Ramis, Ç., Yaylı, Y.: Dual Split Quaternions and Chasles’ Theorem in 3-Dimensional Minkowski Space E31, Advances in Applied Clifford Algebras, 23(4), 951-964 (2013). · Zbl 1290.53011
[24] [25] Rall, L.B.: Automatic differentiation - techniques and applications, Springer Lecture Notes in Computer Science, Vol.120, (1981). · Zbl 0473.68025
[25] [26] Ravani, B., Wang, J.W.: Computer aided geometric design of line constructs, ASME Journal of Mechanical Design. 113 (3), 363-371 (1991).
[26] [27] Study, E.: Geometry der Dynamen, Leipzig, (1901). · JFM 31.0691.02
[27] [28] Yu, W., Blair, M.: DNAD, a simple tool for automatic differentiation of Fortran codes using dual numbers, Comput. Phys. Comm. 184, 1446-1452 (2013). · Zbl 1315.65023
[28] [29] Zhang, Y., Ting, K.L.: On point-line geometry and displacement, Mech. Mach. Theory 39, 1033-1050 (2004). · Zbl 1143.70356
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