Abstract
Bu makalede, 3-boyutlu Lorentziyen uzayda bir
spacelike yüzeyin bir diğer spacelike yüzey üzerinde yüzeylerin kontak
spacelike yörünge eğrileri boyunca kaymaksızın dönme-yuvarlanma hareketinin
ileri kinematiğini inceledik. Holonomik olmayan bir sistemde ortaya çıkan
dönme-yuvarlanma hareketinin anlık kinematiğini kurmak için bir Darboux çatı
metodu kullanılmıştır. Ardından, kontravaryant vektörler, yuvarlanma hızı ve
geometrik değişmezlere göre hareketli spacelike yüzeyin dönme-yuvarlanma
hareketinin yeni kinematik formülasyonları elde edilmiştir. Keyfi bir noktanın
öteleme hızı formülasyonu ve hareketli spacelike yüzey üzerindeki açısal hız
formülasyonunun denklemi elde edilmiştir.
Geometrik değişmezlerle ifade edilen denklem, her mertebeden
türevlenebilirdir ve keyfi spacelike parametrik yüzeye ve spacelike kontak
yörünge eğrisine kolayca uygulanabilir. Dönme-yuvarlanma kinematiğinin bağıl
eğrilikleri ve burulmasının etkisi açık olarak gösterilmektedir.
Keywords
Lorentziyen 3-Uzay Darboux Çatısı İleri Kinematik Kontak Yuvarlanma Has yuvarlanma Dönme-yuvarlanma
References
- [1] 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.
- [2] 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.
- [3] 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.
- [4] Neimark J. I. and Fufaev N. A., Dynamics of Nonholonomic Systems, Providence, RI: Amer. Math. Soc., 1972.
- [5] Cai C. and Roth B., On the Spatial Motion of Rigid Bodies with Point Contact, In Proc. IEEE Conf. Robot. Autom., 1987; pp. 686–695.
- [6] Cai C. and Roth B., On the Planar Motion of Rigid Bodies with Point Contact, Mech. Mach. Theory, 21 (1986) 453–466.
- [7] Montana D. J., The Kinematics of Multi-fingered Manipulation, IEEE Trans. Robot. Autom., 11-4 (1995) 491–503.
- [8] Li Z. X. and Canny J., Motion of Two Rigid Bodies with Rolling Constraint, IEEE Trans. Robot. Autom., 6-1 (1990) 62–72.
- [9] 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.
- [10] Marigo A. and Bicchi A., Rolling Bodies with Regular Surface: Controllability Theory and Application, IEEE Trans. Autom. Control, 45-9 (2000) 1586–1599.
- [11] 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.
- [12] Chelouah A. and Chitour Y., On the Motion Planning of Rolling Surfaces, Forum Math., 15-5 (2003) 727–758.
- [13] Chitour Y., Marigo A. and Piccoli B., Quantization of the Rolling-Body Problem with Applications to Motion Planning, Syst. Control Lett., 54-10 (2005) 999–1013.
- [14] Tchon K., Repeatability of Inverse Kinematics Algorithms for Mobile Manipulators, IEEE Trans. Autom. Control, 47-8 (2002) 1376–1380.
- [15] Tchon K. and Jakubiak J., An extended Jacobian Inverse Kinematics Algorithm for Doubly Nonholonomic Mobile Manipulators, In Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005; 1548–1553.
- [16] Bottema O. and Roth B., Theoretical Kinematics, North-Holland Publ. Co., Amsterdam, 1979, 556 pp.
- [17] Karger A. and Novak J., Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia, 1978.
- [18] Müller H. R., Kinematik Dersleri, Ankara Üniversitesi Fen Fakültesi Yayınları, 1963.
- [19] 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.
- [20] Birman G. S. and Nomizu K., Trigonometry in Lorentzian Geometry, Ann. Math. Month., 91-9 (1984) 543-549.
- [21] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
- [22] Ratcliffe J. G., Foundations of Hyperbolic Manifolds, Springer, New York, 2006.
- [23] Uğurlu H. H. and Çalışkan A., Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi, Celal Bayar Üniversitesi Yayınları, Manisa, 2012.
- [24] Do Carmo M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
Abstract
In this
paper, we investigate the forward kinematics of spin-rolling motion without
sliding of one spacelike surface on another spacelike 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 spacelike moving surface with respect to
contravariant vectors, rolling velocity, and geometric invariants are obtained.
The translational velocity formulation of an arbitrary point and the equation
of the angular velocity formulation on the spacelike 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.
2010 AMS Subject Classification: 70B10, 53A17, 53A25, 53A35.
Keywords
Darboux Frame Forward Kinematics Lorentzian 3-Space Rolling Contact Pure-rolling Spin-rolling
References
- [1] 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.
- [2] 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.
- [3] 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.
- [4] Neimark J. I. and Fufaev N. A., Dynamics of Nonholonomic Systems, Providence, RI: Amer. Math. Soc., 1972.
- [5] Cai C. and Roth B., On the Spatial Motion of Rigid Bodies with Point Contact, In Proc. IEEE Conf. Robot. Autom., 1987; pp. 686–695.
- [6] Cai C. and Roth B., On the Planar Motion of Rigid Bodies with Point Contact, Mech. Mach. Theory, 21 (1986) 453–466.
- [7] Montana D. J., The Kinematics of Multi-fingered Manipulation, IEEE Trans. Robot. Autom., 11-4 (1995) 491–503.
- [8] Li Z. X. and Canny J., Motion of Two Rigid Bodies with Rolling Constraint, IEEE Trans. Robot. Autom., 6-1 (1990) 62–72.
- [9] 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.
- [10] Marigo A. and Bicchi A., Rolling Bodies with Regular Surface: Controllability Theory and Application, IEEE Trans. Autom. Control, 45-9 (2000) 1586–1599.
- [11] 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.
- [12] Chelouah A. and Chitour Y., On the Motion Planning of Rolling Surfaces, Forum Math., 15-5 (2003) 727–758.
- [13] Chitour Y., Marigo A. and Piccoli B., Quantization of the Rolling-Body Problem with Applications to Motion Planning, Syst. Control Lett., 54-10 (2005) 999–1013.
- [14] Tchon K., Repeatability of Inverse Kinematics Algorithms for Mobile Manipulators, IEEE Trans. Autom. Control, 47-8 (2002) 1376–1380.
- [15] Tchon K. and Jakubiak J., An extended Jacobian Inverse Kinematics Algorithm for Doubly Nonholonomic Mobile Manipulators, In Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005; 1548–1553.
- [16] Bottema O. and Roth B., Theoretical Kinematics, North-Holland Publ. Co., Amsterdam, 1979, 556 pp.
- [17] Karger A. and Novak J., Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia, 1978.
- [18] Müller H. R., Kinematik Dersleri, Ankara Üniversitesi Fen Fakültesi Yayınları, 1963.
- [19] 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.
- [20] Birman G. S. and Nomizu K., Trigonometry in Lorentzian Geometry, Ann. Math. Month., 91-9 (1984) 543-549.
- [21] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
- [22] Ratcliffe J. G., Foundations of Hyperbolic Manifolds, Springer, New York, 2006.
- [23] Uğurlu H. H. and Çalışkan A., Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi, Celal Bayar Üniversitesi Yayınları, Manisa, 2012.
- [24] Do Carmo M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
Details
| Primary Language | English |
|---|---|
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | December 24, 2018 |
| Submission Date | February 3, 2018 |
| Acceptance Date | October 15, 2018 |
| Published in Issue | Year 2018 |