The Forward Kinematics of Rolling Contact of Timelike Curves Lying on Timelike Surfaces

Yıl 2020, Cilt: 16 Sayı: 2, 215 - 224, 24.06.2020

Mehmet Aydınalp Mustafa Kazaz Hasan Hüseyin Uğurlu

Öz

The aim of the
present study is to investigate the forward kinematics of spin-rolling contact
motion of one timelike surface on another
timelike surface along their timelike trajectory curves in Lorentzian 3-space.
This study does not take sliding motion into consideration but applies a new
Darboux frame method to establish the kinematics of spin-rolling motion.

Anahtar Kelimeler

Darboux frame, forward kinematics, Lorentzian 3-space, pure-rolling, rolling contact, spin-rolling

Kaynakça

• [1]. Agrachev, AA, Sachkov, YL. An intrinsic approach to the control of rolling bodies, in Proc. 38th IEEE Conf. Decis. Control, Phoenix, AZ, USA; 1999, pp 431–435.
• [2]. Birman, GS, Nomizu, K. 1984. Trigonometry in Lorentzian Geometry, Ann. Math. Month., 91(9): 543-549.
• [3]. Bottema, O, Roth, B. Theoretical Kinematics; North-Holland Publ. Co.: Amsterdam, 1979; pp 556.
• [4]. Cai, C, Roth, B. On the spatial motion of rigid bodies with point contact, in Proc. IEEE Conf. Robot. Autom.; 1987, pp 686–695.
• [5]. Cai, C, Roth, B. 1986. On the planar motion of rigid bodies with point contact, Mech. Mach. Theory, 21: 453–466.
• [6]. Chelouah, A, Chitour, Y. 2003. On the motion planning of rolling surfaces, Forum Math., 15(5): 727–758.
• [7]. Chitour, Y, Marigo, A, Piccoli, B. 2005. Quantization of the rolling-body problem with applications to motion planning, Syst. Control Lett., 54(10): 999–1013.
• [8]. Cui, L, Dai, JS. 2010. A Darboux-Frame-Based Formulation of Spin-Rolling Motion of Rigid Objects With Point Contact, IEEE Trans. Rob., 26(2): 383–388.
• [9]. Cui, L. Differential Geometry Based Kinematics of Sliding-Rolling Contact and Its Use for Multifingered Hands, Ph.D. thesis, King’s College London, University of London, London, UK, 2010.
• [10]. Cui, L, Dai, JS. 2015. A Polynomial Formulation of Inverse Kinematics of Rolling Contact, ASME J. Mech. Rob., 7(4): 041003_041001-041009.
• [11]. Do Carmo, MP. Differential Geometry of Curves and Surfaces; Prentice-Hall: Englewood Cliffs, New Jersey, 1976.
• [12]. Karger, A, Novak, J. Space Kinematics and Lie Groups; STNL Publishers of Technical Lit.: Prague, Czechoslovakia, 1978.
• [13]. Li, ZX, Canny, J. 1990. Motion of two rigid bodies with rolling constraint, IEEE Trans. Robot. Autom., 6(1): 62–72.
• [14]. Marigo, A, Bicchi, A. 2000. Rolling bodies with regular surface: Controllability theory and application, IEEE Trans. Autom. Control, 45(9): 1586–1599.
• [15]. Montana, DJ. 1995. The kinematics of multi-fingered manipulation, IEEE Trans. Robot. Autom., 11(4): 491–503.
• [16]. Müller, HR. Kinematik Dersleri; Ankara Üniversitesi Fen Fakültesi Yayınları, 1963.
• [17]. Neimark, JI, Fufaev, NA. Dynamics of Nonholonomic Systems; Providence, RI: Amer. Math. Soc., 1972.
• [18]. Nelson, EW, Best CL, McLean, WG. Schaum’s Outline of Theory and Problems of Engineering Mechanics, Statics and Dynamics (5th Ed.); McGraw-Hill: New York, 1997.
• [19]. O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity; Academic Press: London, 1983.
• [20]. Ratcliffe, JG. Foundations of Hyperbolic Manifolds; Springer: New York, 2006.
• [21]. Sarkar, N, Kumar, V, Yun, X. 1996. Velocity and Acceleration Analysis of Contact Between Three-Dimensional Rigid Bodies, ASME J. Appl. Mech., 63(4): 974–984.
• [22]. Tchon, K. 2002. Repeatability of inverse kinematics algorithms for mobile manipulators, IEEE Trans. Autom. Control, 47(8): 1376– 1380.
• [23]. Tchon, K, Jakubiak, J. An extended Jacobian inverse kinematics algorithm for doubly nonholonomic mobile manipulators, in Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain; 2005, pp 1548–1553.
• [24]. Uğurlu HH, Çalışkan, A. Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi; Celal Bayar Üniversitesi Yayınları: Manisa, 2012.