Bazı Küresel Mekanizma Hareketlerinin ve Mafsal Tasarımlarının Yeni SLERP İnterpolasyonları ile İncelenmesi

Yıl 2022, Cilt: 25 Sayı: 4, 1513 - 1521, 16.12.2022

Hatice Kusak Samancı , Çetin Kuşçu

https://doi.org/10.2339/politeknik.757983

Cited By: 1

Öz

Küresel mekanizmaların hareketleri robotik çalışmalarda önemli bir yer almaktadır. Çalışmamızda küresel mekanizmalardaki mafsalların tasarımı ve yörünge hareket denklemleri ilk defa tanımladığımız dizisel SLERP, dizisel fast SLERP, De-Moivre ile dizisel SLERP ve modüler SLERP interpolasyon yöntemleri ile verilmiştir. Bununla birlikte, küresel mekanizmalardaki mafsalların konumları için geometrik SLERP interpolasyon eğrisinin Serret-Frenet çatısı ve eğrilikleri hesaplanarak sayısal bir örnek verilmiştir. Ayrıca makalenin sonunda, iki ve üç serbestlik dereceli küresel mekanizmaların hareket denklemlerini üreten interpolasyon denklemleri ile ilgili birkaç sayısal örneklendirme de yapılmıştır.

Anahtar Kelimeler

Kuaterniyon, Dizisel SLERP, Hızlı dizisel SLERP, Modüler SLERP

The Analysis of Some Spherical Mechanism Movements and Joint Designs by The New SLERP Interpolations

Öz

The movements of the spherical mechanisms take an important place in the robotic studies. In this paper, the design and the trajectory motion equations of the joints in the spherical mechanisms were presented by the methods of the sequential SLERP, sequential fast SLERP, sequential SLERP with De-Moivre, and modular SLERP interpolation that we firstly defined. Additionally, the Serret Frenet frame and curvatures of the geometrical SLERP interpolation curve for the locations of the joints in the spherical mechanisms were computed, and then, several numerical examples on the interpolation equations that produce the motion equations of the spherical mechanisms with 2 and 3-degree of freedom were given at the end of the paper.

Anahtar Kelimeler

Quaternion, Sequential SLERP, Sequential Fast SLERP, Modular SLERP, Interpolation

Kaynakça

• [1] Hamilton W.R., “On quaternions, or on a new system of imaginaries in algebra”, Philosophical Magazine, Vol. 25, n 3. p. 489–495, (1844).
• [2] Hamilton W.R, “On quaternions; or on a new system of imaginaries in algebra”, The London, Edinburg, and Dublin Philosophical Magazine and Journal of Science, 33(219), 58-60, (1848).
• [3] Hamilton W.R., “Elements of quaternions”, Chelsea Pub., New York, Vol. I, 3rd Ed., (1969).
• [4] Hanson A. J., "Visualizing quaternions", Elsevier, Morgan Kaufmann, San Francisco. ISBN 0-12-088400-3, (2006).
• [5] Vince J., “Quaternions for computer graphics”, Bournemouth Un., Bournemouth, UK, (2011).
• [6] Hacısalihoğlu, H.H., “Motion geometry and quaternions theory”, Gazi Uni., Ankara, (1983).
• [7] Sheomake K., “Animating rotation with quaternion curves”, San Francısco, 19(3): 245-254, (1985).
• [8] Dam E.B, Koch M. and Lillholm M., “Quaternions, interpolation and animation”, Institute of Computer Science University of Copenhagen, (1998).
• [9] Hast A., Barrera T. and Bengtsson E., “Shading by spherical linear interpolation using de Moivre’s formula”, Proc. 11th Int. Conf. Central Europe on Computer Graphics, Visualization, and Computer Vision, (2003).
• [10] Hast A., Barrera T., Bengtsson E., “Incremental spherical linear interpolation”, In The Annual SIGRAD Conference. Special Theme-Environmental Visualization, 13,7-10. Linköping University Electronic Press, (2004).
• [11] Kremer V.E., “Quaternions and Slerp”, Department for the Computer Science University of Saarbrücken, (2008).
• [12] Jafari M., Molaei H., “Spherical linear interpolation and Bezier curves”, General Scientific Researches, 2(1): 13-17, (2014).
• [13] Kuşak Samancı, H., Çelik S., İncesu M., “The Bishop frame of Bézier curves”, Life Science Journal, 2015, 12(6): 175-180.
• [14] Gündüz H., Kazan A., Karadağ H.B., “Rotational surfaces generated by cubic Hermitian and cubic Bezier curves”, Politeknik Dergisi, 22(4): 1075-1082, (2019).
• [15] Incesu M., “The new characterization of ruled surfaces corresponding dual Bézier curves”, Mathematical Methods in the Applied Sciences, (2021).
• [16] Alizade R., Gezgin E. and Kilit Ö., “A new method in computational kinematics of a spherical wrist motion through quaternions”, Intl. Workshop on Comp. Kinematics, Cassino, p.32, (2005).
• [17] Gezgin E., “Biokinematic analysis of human arm”, Phd Thesis, İzmir Yüksek Teknoloji University, İzmir, (2006).
• [18] Kilit Ö., “The Kinematical design and analysis of the spherical mechanisms”, Phd Thesis, Ege University Institute of Science and Technology, İzmir, (2007).