Kinematic Analysis in 3-Dimensional Generalized Space

Ümit Ziya Savcı

doi:10.17776/csj.1054869

EN

Kinematic Analysis in 3-Dimensional Generalized Space

Abstract

In this paper, we have first obtained the derivatives of spherical and spatial motions by using the special matrix group in generalized space E3(α,β). The rotation matrices and tangent operators were found by using derivatives of one- and multi-parameters motions in E3(α,β). Also, we obtained the angular velocity matrix of the moving body and its linear velocity vector. Finally, we gave some examples including applications of tangent operators and rotation matrices in support of our results.

Keywords

Generalized space, Rigid motion, Kinematics, Tangent operators, Matrix group.

References

[1] Awrejcewicz J., Classical Mechanics; Kinematics and Statics, New York: Springer, (2012).
[2] Agrawal O.P., Hamilton Operators and Dual-number-quaternions in Spatial Kinematics, Mech Mach Theory, 22 (1987) 569-575.
[3] Beggs J.S., Advanced Mechanisms, New York: The Macmillan Company Collier-Macmillan, London (1965).
[4] Herve J.M., The mathematical group structure of the set of displacements, Mech. Mach. Theory, 29 (1994) 73-81.
[5] Hiller M., Woernle C., A Unified Representation of Spatial Displacements, Mech. Mach. Theory, 19(1984) 477-486.
[6] Spong M.W., Hutchison S., Vidyasagar M., Robot Modeling and Control. Hoboken: NJ John Wiley & Sons, (2006).
[7] Altmann S.L., Rotations, Quaternions and Double Groups. Oxford: Oxford University Press, (1986).
[8] Aragon G., Aragon J.L., Rodriguez M.A., Clifford Algebras and Geometric Algebra, Adv Appl Clifford Al., 7(2) (1997) 91-102.
[9] Rosenfeld B., Geometry of Lie Groups. Dordrecht: Kluwer Academic Publishers, (1997).
[10] Uicker J.J., Pennock G.R., Shigley J., Theory of Machines and Mechanisms. New York: Oxford University Press, (2011).

[11] Bottema O., Roth, B., Theoretical Kinematics. New York: North-Holland Press, (1979).
[12] McCarthy J.M., An Introduction to Theoretical Kinematics. Cambridge: MIT Press, (1990).
[13] O’Neill B., Semi-Riemannian Geometry With Applications to Relativity. New York: Academic Press Inc., (1983).
[14] Ryan P.J., Euclidean and non-Euclidean geometry; an analytic approach. Cambridge, New York: Cambridge Univ. Press, (1986).
[15] Ata E., Yıldırım Y.A., Different Polar Representation for Generalized and Generalized Dual Quaternions, Adv Appl Clifford Al., 28 (2010) 193-202.
[16] Ata E., Savcı Ü.Z., Spherical Kinematics in 3-Dimensional Generalized Space Int J Geom Meth Mod Phys., 18(3) (2020) 2150033.
[17] Erdmann K., Skowronski A., Algebras of generalized quaternion type, Advances in Mathematics, 349 (2019) 1036-1116.
[18] Jafari M., Yaylı Y., Generalized Quaternions and Rotation in 3-Space E3αβ, TWMS J. Pure Appl. Math., 6(2) (2015) 224-232.
[19] Lam T.Y., Introduction to Quadratic Forms Over Fields, American Mathematical Society, USA (2005).
[20] Pottman H., Wallner J., Computational line geometry. Berlin Heidelberg, New York: Springer-Verlag, (2000).
[21] Savcı Ü.Z., Generalized Dual Quaternions and Screw Motion in Generalized Space, Konuralp Journal of Mathematics, 10 (1) (2022) 197-202.
[22] Parkin I.A., A third conformation witht the screw systems: finite twist displacements of a directed line and point, Mech. Mach. Theory, 27(2) (1992) 177- 188.
[23] Huang C., Roth B., Analytic expressions for the finite screw systems, Mech. Mach. Theory, 29(2) (1994) 207-222.
[24] Knossow D., Ronfard R., Horaud R., Human motion tracking with a kinematic parameterization of extremal contours, International Journal of Computer Vision, 79 (2008) 247-269.
[25] Durmaz O., Aktaş B., Gündogan H., The derivative and tangent operators of a motion in Lorentzian space, Int. J. Geom. Meth. Mod. Phys., 14(4) (2017) 1750058.
[26] Ward J.P., Quaternions and Cayley numbers algebra and applications. London: Kluwer Academic Publishers, (1997).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Ümit Ziya Savcı ^*
0000-0003-2772-9283
Türkiye

Publication Date

June 29, 2022

Submission Date

January 7, 2022

Acceptance Date

April 22, 2022

Published in Issue

Year 2022 Volume: 43 Number: 2

DOI

https://doi.org/10.17776/csj.1054869

IZ

https://izlik.org/JA57EZ47RA

APA

Savcı, Ü. Z. (2022). Kinematic Analysis in 3-Dimensional Generalized Space. Cumhuriyet Science Journal, 43(2), 299-307. https://doi.org/10.17776/csj.1054869

AMA

1.Savcı ÜZ. Kinematic Analysis in 3-Dimensional Generalized Space. CSJ. 2022;43(2):299-307. doi:10.17776/csj.1054869

Chicago

Savcı, Ümit Ziya. 2022. “Kinematic Analysis in 3-Dimensional Generalized Space”. Cumhuriyet Science Journal 43 (2): 299-307. https://doi.org/10.17776/csj.1054869.

EndNote

Savcı ÜZ (June 1, 2022) Kinematic Analysis in 3-Dimensional Generalized Space. Cumhuriyet Science Journal 43 2 299–307.

IEEE

[1]Ü. Z. Savcı, “Kinematic Analysis in 3-Dimensional Generalized Space”, CSJ, vol. 43, no. 2, pp. 299–307, June 2022, doi: 10.17776/csj.1054869.

ISNAD

Savcı, Ümit Ziya. “Kinematic Analysis in 3-Dimensional Generalized Space”. Cumhuriyet Science Journal 43/2 (June 1, 2022): 299-307. https://doi.org/10.17776/csj.1054869.

JAMA

1.Savcı ÜZ. Kinematic Analysis in 3-Dimensional Generalized Space. CSJ. 2022;43:299–307.

MLA

Savcı, Ümit Ziya. “Kinematic Analysis in 3-Dimensional Generalized Space”. Cumhuriyet Science Journal, vol. 43, no. 2, June 2022, pp. 299-07, doi:10.17776/csj.1054869.

Vancouver

1.Ümit Ziya Savcı. Kinematic Analysis in 3-Dimensional Generalized Space. CSJ. 2022 Jun. 1;43(2):299-307. doi:10.17776/csj.1054869

Cited By

Screw Motion via Matrix Algebra in Three-Dimensional Generalized Space

Symmetry

https://doi.org/10.3390/sym14112235

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