The Forward Kinematics of Rolling Contact of Timelike Surfaces With Spacelike Trajectory Curves


  • Mehmet Aydinalp Manisa Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, 45140, Manisa
  • Mustafa Kazaz Manisa Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, 45140, Manisa
  • Hüseyin Uğurlu Gazi University, Faculty of Education, Department of Secondary Education Science and Mathematics Teaching, Mathematics Teaching Program, 06560, Ankara



Lorentzian 3-Space, Darboux frame, Forward kinematics, Pure-rolling, Rolling contact, Spin-rolling


In this paper, we investigate the forward kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface along the spacelike contact trajectory curves of the surfaces in Lorentzian 3-space. A Darboux frame method is adopted to develop instantaneous kinematics of spin-rolling motion, which occurs in a nonholonomic system. Then, new kinematic formulations of spin-rolling motion of timelike moving surface with regards to contravariant vectors, rolling velocity, and geometric invariants are obtained. Namely, the translational velocity formulation of an arbitrary point and the equation of the angular velocity formulation on the timelike moving surface are derived. The equation, which is represented with geometric invariants, can be easily applied to arbitrary spacelike parametric surface and spacelike contact trajectory curve and can be differentiated to any order. The influence of the relative curvatures and torsion on spin-rolling kinematics is clearly presented.


Download data is not yet available.


A. A. Agrachev and Y. L. Sachkov, An intrinsic approach to the control of rolling bodies, in Proc. 38th IEEE Conf. Decis. Control, 1999, 431–435, Phoenix, AZ, USA, DOI: 10.1109/CDC.1999.832815.

G. S. Birman and K. Nomizu, Trigonometry in Lorentzian geometry, Ann. Math. Month. 91(9) (1984), 543 – 549, DOI: 10.1080/00029890.1984.11971490.

O. Bottema and B. Roth, Theoretical Kinematics, North-Holland Publ. Co., Amsterdam (1979), 556 p., DOI: 10.1137/1022104

C. Cai and B. Roth, On the spatial motion of rigid bodies with point contact, in Proc. IEEE Conf. Robot. Autom., 1987, pp. 686 – 695, DOI: 10.1109/ROBOT.1987.1087971.

C. Cai and B. Roth, On the planar motion of rigid bodies with point contact, Mech. Mach. Theory 21 (1986), 453 – 466, DOI: 10.1016/0094-114X(86)90128-X.

A. Chelouah and Y. Chitour, On the motion planning of rolling surfaces, Forum Math. 15(5) (2003), 727 – 758, DOI: 10.1515/form.2003.039.

Y. Chitour, A. Marigo and B. Piccoli, Quantization of the rolling-body problem with applications to motion planning, Syst. Control Lett. 54(10) (2005), 999 – 1013, DOI: 10.1016/j.sysconle.2005.02.012.

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) (2010), 383 – 388, DOI: 10.1109/TRO.2010.2040201.

L. Cui, 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).

L. Cui and J. S. Dai, A polynomial formulation of inverse kinematics of rolling contact, ASME J. Mech. Rob. 7(4) (2015), 041003_041001-041009, DOI: 10.1115/1.4029498.

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, New Jersey (1976).

A. Karger and J. Novak, Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia (1978).

Z. X. Li and J. Canny, Motion of two rigid bodies with rolling constraint, IEEE Trans. Robot. Autom. 6(1) (1990), 62 – 72, DOI: 10.1109/70.88118.

A. Marigo and A. Bicchi, Rolling bodies with regular surface: Controllability theory and application, IEEE Trans. Autom. Control 45(9) (2000), 1586 – 1599, DOI: 10.1109/9.880610.

D. J. Montana, The kinematics of multi-fingered manipulation, IEEE Trans. Robot. Autom. 11(4) (1995), 491 – 503, DOI: 10.1109/70.406933.

H. R. Müller, Kinematik Dersleri, Ankara íœniversitesi Fen Fakültesi Yayınları (1963).

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Amer. Math. Soc., Providence, RI (1972).

E.W. Nelson, C. L. Best andW. G. McLean, Schaum's Outline of Theory and Problems of Engineering Mechanics, Statics and Dynamics, 5th ed., McGraw-Hill, New York (1997).

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London (1983).

J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, New York (2006), DOI: 10.1007/978-1-4757-4013-4.

N. Sarkar, V. Kumar and X. Yun, Velocity and acceleration analysis of contact between threedimensional rigid bodies, ASME J. Appl. Mech. 63(4) (1996), 974 – 984, DOI: 10.1115/1.2787255.

K. Tchon, Repeatability of inverse kinematics algorithms for mobile manipulators, IEEE Trans. Autom. Control 47(8) (2002), 1376 – 1380, DOI: 10.1109/TAC.2002.801192.

K. Tchon and J. Jakubiak, An extended Jacobian inverse kinematics algorithm for doubly nonholonomic mobile manipulators, in Proc. IEEE Int. Conf. Robot. Autom., 2005, 1548 – 1553, Barcelona, Spain, DOI: 10.1109/ROBOT.2005.1570334.

H. H. Ugurlu and A. í‡alıskan, Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi, Celal Bayar íœniversitesi Yayınları, Manisa (2012).




How to Cite

Aydinalp, M., Kazaz, M., & Uğurlu, H. (2019). The Forward Kinematics of Rolling Contact of Timelike Surfaces With Spacelike Trajectory Curves. Journal of Informatics and Mathematical Sciences, 11(2), 133–146.



Research Articles