Mehmet AYDİNALP, Mustafa KAZAZ, Hasan Hüseyin UĞURLU

The inverse kinematics of rolling contact of timelike curves lying on timelike surfaces

Rolling contact between two surfaces plays an important role in robotics and engineering such as spherical robots, single wheel robots, and multi-ﬁngered robotic hands to drive a moving surface on a fixed surface. The rolling contact pairs have one, two, or three degrees of freedom (DOFs) consisting of angular velocity components. Rolling contact motion can be divided into two categories: spin-rolling motion and pure-rolling motion. Spin-rolling motion has three (DOFs), and pure-rolling motion has two (DOFs). Further, it is well known that the contact kinematics can be divided into two categories: forward kinematics and inverse kinematics. In this paper, we investigate the inverse kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface in the direction of timelike unit tangent vectors of their timelike trajectory curves by determining the desired motion and the coordinates of the contact point on each surface. We get three nonlinear algebraic equations as inputs by using curvature theory in Lorentzian geometry. These equations can be reduced as a univariate polynomial of degree six by applying the Darboux frame method. This polynomial enables us to obtain rapid and accurate numerical root approximations and to analyze the rolling rate as an output. Moreover, we obtain another outputs: the rolling direction and the compensatory spin rate.

Keywords:

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

PDF

___

Agrachev, A. A. and Sachkov, Y. L., An intrinsic approach to the control of rolling bodies, In Proc. 38th IEEE Conf. Decis. Control, Phoenix, AZ, USA (1999), 431--435.
Aydınalp, M., Kazaz, M. and Uğurlu, H. H., The forward kinematics of rolling contact of timelike curves lying on timelike surfaces, (2018), Manuscript submitted for publication.
Birman, G. S. and Nomizu, K., Trigonometry in Lorentzian Geometry, Ann. Math. Month., 91(9), (1984), 543--549.
Borras, J. and Di Gregorio, R., Polynomial solution to the position analysis of two assur kinematic chains with four loops and the same topology, ASME J. Mech. Rob., 1(2), (2009), 021003.
Bottema, O. and Roth, B., Theoretical Kinematics, North-Holland Publ. Co., Amsterdam, 1979, pp 556.
Cai, C. and Roth, B., On the spatial motion of rigid bodies with point contact, In Proc. IEEE Conf. Robot. Autom., (1987), 686--695.
Cai, C. and Roth, B., On the planar motion of rigid bodies with point contact, Mech. Mach. Theory, 21(6), (1986), 453--466.
Chelouah, A. and Chitour, Y., On the motion planning of rolling surfaces, Forum Math., 15(5), (2003), 727--758.
Cui, L. and Dai J. S., A Darboux-frame-based formulation of spin-rolling motion of rigid objects with point contact, IEEE Trans. Rob., 26(2), (2010), 383--388.
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.
Cui, L. and Dai J. S., A polynomial formulation of inverse kinematics of rolling contact, ASME J. Mech. Rob., 7(4), (2015), 041003_041001-041009.
Cui, L., Sun J. and Dai J. S., In-hand forward and inverse kinematics with rolling contact, Robotica, (2017), 1--19, doi:10.1017/S026357471700008X.
Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
Karger, A. and Novak, J., Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia, 1978.
Kerr, J. and Roth, B., Analysis of multifingered hands, Int. J. Rob. Res., 4(4), (1986), 3--17.
Li, Z. X. and Canny, J., Motion of two rigid bodies with rolling constraint, IEEE Trans. Robot. Autom., 6(1), (1990), 62--72.
Li, Z., Hsu, P. and Sastry, S., Grasping and coordinated manipulation by a multifingered robot hand, Int. J. Rob. Res., 8(4), (1989), 33--50.
Marigo, A. and Bicchi, A., Rolling bodies with regular surface: Controllability theory and application, IEEE Trans. Autom. Control, 45(9), (2000), 1586--1599.
McCarthy, J. M., Kinematics, polynomials, and computers--A brief history, ASME J. Mech. Rob., 3(1), (2011), 010201.
Montana, D. J., The kinematics of contact and grasp, Int. J. Rob. Res., 7(3), (1988), 17--32.
Montana, D. J., The kinematics of multi-fingered manipulation, IEEE Trans. Robot. Autom., 11(4), (1995), 491--503.
Müller, H. R., Kinematik Dersleri, Ankara Üniversitesi Fen Fakültesi Yayınları, 1963.
Neimark, J. I. and Fufaev, N. A., Dynamics of Nonholonomic Systems, Providence, RI: Amer. Math. Soc., 1972.
Nelson, E. W., Best, C. L. and McLean, W. G., Schaum's Outline of Theory and Problems of Engineering Mechanics, Statics and Dynamics (5th Ed.), McGraw-Hill, New York, 1997.
O'Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Springer, New York, 2006.
Sarkar, N., Kumar, V. and Yun, X., Velocity and acceleration analysis of contact between three-dimensional rigid bodies, ASME J. Appl. Mech., 63(4), (1996), 974--984.
Uğurlu, H. H. and Çalışkan, A., Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi, Celal Bayar Üniversitesi Yayınları, Manisa 2012.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics-Cover

ISSN: 1303-5991
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 1948
Yayıncı: Ankara Üniversitesi

Arşiv