A generalization for the kinematics of sliding-rolling motion in the semi-euclidean space $\mathbb{R}_\varepsilon^3$
Abstract
UDC 531
We propose a generalization for the sliding-rolling motion, which leads to meaningful physical and mathematical results as the general metric includes the time dimension. We also investigate the kinematics of the relative motion of two rigid objects, which maintain sliding-rolling contact, by using the general adjoint approach in the semi-Euclidean space $\mathbb{R}_{\varepsilon}^{3},$ where $\varepsilon \in \{0,1\}.$ This generalization gives the geometric kinematic equations of the sliding-rolling motion in Minkowski and Euclidean spaces. In these spaces, we get a set of overconstrained equations. Solving this system, we determine translational and angular velocities of the moving surface.
Finally, we illustrate the results by two examples.
References
C. Cai, B. Roth, On the spatial motion of rigid bodies with point contact, in: IEEE Conference on Robotics and Automation (1987), p. 686–695.
D.J. Montana, The kinematics of contact and grasp, Int. J. Robot. Res., 7, № 3, 17–32 (1988). DOI: https://doi.org/10.1177/027836498800700302
Z. Li, P. Hsu, S. Sastry, Grasping and coordinated manipulation by a multi-fingered robot hand, Int. J. Robot. Res., 8, № 4, 33–50 (1989). DOI: https://doi.org/10.1177/027836498900800402
M. R. Cutkosky, Robotic grasping and fine manipulation, Kluwer Academic, Dordrecht (1985). DOI: https://doi.org/10.1007/978-1-4684-6891-5
R. S. Fearing, Simplified grasping and manipulation with dextrous robot hand, IEEE J. Robot. Automat., 2, № 4, 188–195 (1986). DOI: https://doi.org/10.1109/JRA.1986.1087063
J. R. Kerr, An analysis of multi fingered hands, PhD Dissertation, Stanford University (1984).
J. Kerr, B. Roth, An analysis of multi fingered hands, Int. J. Robot. Res., 4, № 4, 3–17 (1986).
Y. Nakamura, K. Nagai, T. Yoshikawa, Dynamics, stability in coordination of multiple robotic mechanisms, Int. J. Robot. Res., 8, № 2, 44–61 (1989). DOI: https://doi.org/10.1177/027836498900800204
E. Paljug, X. Yun, V. Kumar, Control of rolling contacts in multi-arm manipulation, IEEE Trans. Robot. Automat., 10, № 4, 441–452 (1994). DOI: https://doi.org/10.1109/70.313095
G. P. Starr, Experiments in assembly using a dextrous hand, IEEE Trans. Robot. Automat., 6, № 3, 342–347 (1990). DOI: https://doi.org/10.1109/70.56666
J. C. Trinkle, J. M. Abel, R. P. Paul, An investigation of frictionless enveloping grasping in the plane, Int. J. Robot. Res., 7, № 3, 33–50 (1988). DOI: https://doi.org/10.1177/027836498800700303
J. C. Trinkle, A quasi-static analysis of dextrous manipulation with sliding and rolling contacts, in: Proceedings 1989 International Conference on Robotics and Automation, 2 (1989), p.~788–793. DOI: https://doi.org/10.1109/ROBOT.1989.100080
M. Zribi, J. Chen, M. S. Mahmoud, Coordination and control of multi-fingered robot hands with rolling and sliding contacts, J. Intelligent and Robot. Systems, 24, 125–149 (1999).
D. L. Brock, Enhancing the dexterity of a robot hand using controlled slip, in: Proceedings 1988 IEEE International Conference on Robotics and Automation, 1 (1988), p.~249–251. DOI: https://doi.org/10.1109/ROBOT.1988.12056
R. C. Brost, Automatic grasp planning in the presence of uncertainty, Int. J. Robot. Res., 7, № 1, 3–50 (1988). DOI: https://doi.org/10.1177/027836498800700101
C. Cai, B. Roth, On the planar motion of a rigid body with point contact, Mech. Mach. Theory, 21, № 6, 453–466 (1986). DOI: https://doi.org/10.1016/0094-114X(86)90128-X
N. Sarkar, V. Kumar, X. Yun, Velocity and acceleration equations for three-dimensional contact, J. Appl. Mech., 63, 974–984 (1996). DOI: https://doi.org/10.1115/1.2787255
A. Marigo, A. Bicchi, Rolling bodies with regular surface: controllability theory and application, IEEE Trans. Automat. Control, 45, 1586–1599 (2000). DOI: https://doi.org/10.1109/9.880610
M. Xiao, Y. Ding, Contact kinematics between three-dimensional rigid bodies with general surface parameterization, J. Mech. Robot.; DOI: https://doi.org/10.1115/1.4048916. DOI: https://doi.org/10.1115/1.4048916
L. Cui, D. Wang, J. S. Dai, Kinematic geometry of circular surfaces with a fixed radius based on Euclidean in variants, J. Mech. Des., 131, 101001–101008 (2009). DOI: https://doi.org/10.1115/1.3212679
L. Cui, J. S. Dai, From sliding-rolling loci to instantaneus kinematics: an adjoint approach, Mech. and Mach. Theory, 85, 161–171(2015). DOI: https://doi.org/10.1016/j.mechmachtheory.2014.11.015
L. Cui, S. J. Dai, A polynomial formulation of inverse kinematics of rolling contact, J. Mech. Robot., 7, № 4 (2015); https://doi.org/10.1115/1.4029498. DOI: https://doi.org/10.1115/1.4029498
E. Cartan, Riemannian geometry in an orthogonal frame, World Scientific Press, Singapore (2002). DOI: https://doi.org/10.1142/4808
H. Cartan, Differential forms, Dover Publisher, New York (1996).
L. Cui, J. S. Dai, Reciprocity-based singular value decomposition for inverse kinematic analysis of the metamorphic multi fingered hand, J. Mech. Robot., 4, 034502–034506 (2012). DOI: https://doi.org/10.1115/1.4006187
A. Weiss, R. G. Langlois, M. J. D. Hayes, Unified treatment of the kinematic interface between a sphere and omnidirectional wheel actuators, J. Mech. Robot., 3, 041001–041009 (2011). DOI: https://doi.org/10.1115/1.4004888
M. Tosun, M. A. Gungor, I. Okur, On the one-parameter Lorentzian spherical motions and Euler–Savary formula, J. Appl. Mech., 74, № 5, 972 (2007). DOI: https://doi.org/10.1115/1.2722775
M. Aydı nalp, M. Kazaz, H.H. Uǧurlu, The invers kinematics of rolling contact of timelike curves lying on timelike surfaces in Lorentzian 3-space, Commun. Fac. Sci. Univ. Ank. S'er. A1 Math. Stat., 68, 1879–1894 (2019). DOI: https://doi.org/10.31801/cfsuasmas.461781
J. S. Dai, D. Wang, L. Cui, Orientation and workspace analysis of the multifingered metamorphic hand-metahand, IEEE Trans. Robot., 25, № 4, 942–947 (2009). DOI: https://doi.org/10.1109/TRO.2009.2017138
L. Cui, U. Cupcic, J. S. Dai, An optimization approach to teleoperation of the thumb of a humanoid robot hand: kinematic mapping and calibration, J. Mech. Des., 136, 091005–091005 (2014). DOI: https://doi.org/10.1115/1.4027759
J. Hu, M. Wang, T. Zan, The kinematics of ball-screw mechanisms via the slide-roll ratio, Mech. and Mach. Theory, 79, 158–172 (2014). DOI: https://doi.org/10.1016/j.mechmachtheory.2014.04.017
G. Fekete, P. De Baets, M. Wahab, B. Csizmadia, G. Katona, L. V. Vanegas-Useche, J. A. Solanilla, Sliding-rolling ratio during deep squat with regard to different knee prostheses, Acta Polytech. Hungar., 9, № 5, 5–24 (2012).
W. Pu, D. Zhu, J. Wang, Q. J. Wang, Rolling-sliding contact fatigue of surfaces with sinusoidal roughness, Int. J. Fatigue, 90, 57–68 (2016). DOI: https://doi.org/10.1016/j.ijfatigue.2016.04.007
J. Xie, J. Xu, W. Shang, K. Zhang, Mode selection between sliding and rolling for droplet on inclined surface: effect of surface wettability, Int. J. Heat and Mass Transfer, 122, 45–58 (2018). DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.098
Y. Liu, M. Kong, N. Wan, P. Ben-Tzvi, A geometric approach to obtain the closed-form forward kinematics of H4 parallel robot, J. Mech. Robot., 10, № 5 (2018). DOI: https://doi.org/10.1115/1.4040703
A. T. Yang, G. R. Pennock, L. Hsia, Instantaneous invariants and curvature analysis of a planar Four–Link mechanism, J. Mech. Des., 116, № 4, 1173–1176 (1994); https://doi.org/10.1115/1.2919504. DOI: https://doi.org/10.1115/1.2919504
C. L. Chan, K. L. Ting, Curvature theory on contact and transfer characteristics of enveloping curves, in: Proceedings of the ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 5B, Quebec City, Quebec, Canada, August 26–29 (2018); https://doi.org/10.1115/DETC2018-86378. DOI: https://doi.org/10.1115/DETC2018-86378
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