The forward kinematics of dual rolling contact of one dual unit sphere on another with dual Darboux frame based equation

Yıl 2021, Cilt: 50 Sayı: 6, 1583 - 1594, 14.12.2021

Mehmet Aydınalp

https://doi.org/10.15672/hujms.556270

Cited By: 1

Öz

In this paper, we investigate the forward kinematics of dual rolling contact motion without sliding of one dual unit sphere $\tilde{S}_{m}^{2}$ on the fixed sphere $\tilde{S}_{f}^{2}$ along their dual spherical curves, which correspond to ruled surfaces generated by the straight lines in the real line space $\mathbb{E}^{3}$. We adopt a dual Darboux frame method to develop instantaneous kinematics of dual rolling motion. We obtain some new kinematic equations of rolling motion of the moving sphere $\tilde{S}_{m}^{2}$ with regards to dual unit vectors, dual rolling velocity, and dual geometric invariants. Namely, the dual translational velocity equation of an arbitrary dual point and the equation of the dual angular velocity on the moving sphere $\tilde{S}_{m}^{2}$ are derived. The equation represented by geometric invariants can be handily generalized to suit arbitrary dual spherical curve on $\tilde{S}_{m}^{2}$ and can be differentiated to any order.

Anahtar Kelimeler

Dual Darboux frame, dual rolling contact, dual spherical curve, dual unit sphere, ruled surface

Kaynakça

• [1] P. Azariadis and N. Aspragathos, Computer graphics representation and transforma- tion of geometric entities using dual unit vectors and line transformations, Comput. Graph. 25 (2), 195–209, 2001.
• [2] W. Blaschke, Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie, Dover, New York, 1945.
• [3] O. Bottema and B. Roth, Theoretical Kinematics, North-Holland Publ. Co., Amster- dam, 1979.
• [4] H.H. Cheng, Programming with dual numbers and its applications in mechanisms design, Eng. Comput. 10 (4), 212–229, 1994.
• [5] W.K. Clifford, Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. 4 (64), 381–395, 1873.
• [6] L. Cui and J.S. Dai, A Darboux-frame-based formulation of spin-rolling motion of rigid objects with point contact, IEEE Trans. Rob. 26 (2), 383–388, 2010.
• [7] F.M Dimentberg, The Screw Calculus and its Applications in Mechanics, English translation: AD680993, Clearinghouse for Federal and Scientific Technical Informa- tion, (Izdat. Nauka, Moscow, USSR), 1965.
• [8] A. Karger and J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, Prague, 1978.
• [9] A.P. Kotelnikov, Screw Calculus and Some Applications to Geometry and Mechanics, Annals of Imperial University of Kazan, 1895.
• [10] J.M. McCarthy, Geometric Design of Linkages, Springer, 2000.
• [11] J.M. McCarthy and B. Ravani, Differential kinematic of spherical and spatial motions using kinematic mapping, J. Appl. Mech. 53 (1), 15–22, 1986.
• [12] H.R. Müller, Kinematik Dersleri, Ankara Üniversitesi Fen Fakültesi Yaynlar, 1963.
• [13] E.W. Nelson, C.L. Best and W.G. McLean, Schaums Outline of Theory and Problems of Engineering Mechanics, Statics and Dynamics, (5th Ed.), McGraw-Hill, New York, 1997.
• [14] E. Study, Geometrie der Dynamen, Leibzig, 1903.
• [15] H.H. Uğurlu, The principle of transference between real and dual Lorentzian spaces and dual Lorentzian angles, 16th International Geometry Symposium, Celal Bayar University, Manisa, Turkey, 2018.
• [16] G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mech. Mach. Theory, 11 (2), 141–156, 1976.
• [17] A.T. Yang and F. Freudenstein, Application of dual number quaternions algebra to the analysis of spatial mechanisms, J. Appl. Mech. 31 (2), 300–308, 1964.
• [18] X.F. Zha, A new approach to generation of ruled surfaces and its applications in engineering, Int. J. Adv. Manuf. Technol. 13 (3) 155–163, 1997.