Research Article
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Kinematic Analysis in 3-Dimensional Generalized Space

Year 2022, Volume: 43 Issue: 2, 299 - 307, 29.06.2022

Ümit Ziya Savcı

https://doi.org/10.17776/csj.1054869
https://izlik.org/JA57EZ47RA

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).

Year 2022, Volume: 43 Issue: 2, 299 - 307, 29.06.2022

Ümit Ziya Savcı

https://doi.org/10.17776/csj.1054869
https://izlik.org/JA57EZ47RA

Abstract

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).
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Ümit Ziya Savcı 0000-0003-2772-9283

Submission Date January 7, 2022
Acceptance Date April 22, 2022
Publication Date June 29, 2022
DOI https://doi.org/10.17776/csj.1054869
IZ https://izlik.org/JA57EZ47RA
Published in Issue Year 2022 Volume: 43 Issue: 2

Cite

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

Cited By

Screw Motion via Matrix Algebra in Three-Dimensional Generalized Space
Symmetry
https://doi.org/10.3390/sym14112235

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