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A contact force solution for non-colliding contact dynamics simulation. (English) Zbl 1207.70006
Summary: Rigid-body impact modeling remains an intensive area of research spurred on by new applications in robotics, biomechanics, and more generally multibody systems. By contrast, the modeling of non-colliding contact dynamics has attracted significantly less attention. The existing approaches to solve non-colliding contact problems include compliant approaches in which the contact force between objects is defined explicitly as a function of local deformation, and complementarity formulations in which unilateral constraints are employed to compute contact interactions (impulses or forces) to enforce the impenetrability of the contacting objects. In this article, the authors develop an alternative approach to solve the non-colliding contact problem for objects of arbitrary geometry in contact at multiple points. Similarly to the complementarity formulation, the solution is based on rigid-body dynamics and enforces contact kinematics constraints at the acceleration level. Differently, it leads to an explicit closed-form solution for the normal forces at the contact points. Integral to the proposed formulation is the treatment of tangential contact forces, in particular the static friction. These friction forces must be calculated as a function of microslip velocity or displacement at the contact point. Numerical results are presented for four test cases: (1) a thin rod sliding down a stationary wedge; (2) a cube pushed off a wedge by an applied force; (3) a cube rotating off the wedge under application of an external moment; and (4) the cube and the wedge both moving under application of a moment. To ascertain validity and correctness, the solutions to frictionless and frictional scenarios obtained with the new formulation are compared to those generated by using a commercial simulation tool MSC ADAMS.

##### MSC:
 70E55 Dynamics of multibody systems 70-08 Computational methods for problems pertaining to mechanics of particles and systems 70F40 Problems involving a system of particles with friction
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 [1] Hertz, H.: {Miscellaneous Papers. MacMillan, London (1896) [2] Goldsmith, W.: {Impact: The Theory and Physical Behavior of Colliding Solids. Edward Arnold, London, UK (1960) · Zbl 0122.42501 [3] Flügge, W.: {Viscoelasticity. Blaisdell Publishing, Waltham, MA (1967) [4] Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. {ASME Ser. E, J. Appl. Mech. 42, 440–445 (1975) [5] Mirza, K., Hanes, M.D., Orin, D.E.: Dynamic simulation of enveloping power grasps. In: {Proceedings of 1993 IEEE International Conference on Robotics and Automation, pp. 430–435. Atlanta, GA (1993) [6] Kraus, P.R., Kumar, V.: Compliant contact models for rigid body collisions. In: {Proceedings of 1997 IEEE International Conference on Robotics and Automation, pp. 1382–1387. Albuquerque, New Mexico (1997) [7] Vukobratovic, M.K., Potkonjak, V.: Dynamics of contact tasks in robotics. Part I: General model of robot interacting with environment. {Mech. Mach. Theory 34, 923–942 (1999) · Zbl 1049.70623 [8] Herbert, R.G., McWhannell, D.C.: Shape and frequency composition of pulses from an impact pair. {ASME J. Eng. Ind. 99, 513–518 (1977) [9] Lee, T.W., Wang, A.C.: On the dynamics of intermittent-motion mechanisms. Part 1: Dynamic model and response. {ASME J. Mech., Transm. Autom. Des. 105, 534–540 (1983) [10] Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multi-body systems. {ASME J. Mech. Des. 112, 369–376 (1990) [11] Zhang, Y., Sharf, I.: Compliant force modeling for impact analysis. In: {Proceedings of 2004 ASME International Design Technical Conference. Salt Lake City, UT, DETC 2004–57220 (2004) [12] Gonthier, Y., McPhee, J., Lange, C., Piedboeuf, J.: A regularized contact model with asymmetric damping and dwell-time dependent friction. {Multibody Syst. Dyn. 11, 209–233 (2004) · Zbl 1143.74344 [13] McDevitt, T.W.: Treatment of frictional contact problems in MSC.ADAMS. In: {Proceedings of 2002 North American MSC.ADAMS Users Conference. Scottsdale, AZ (2002) [14] MSC Software Corporation: {ADAMS 2003 Online Manual: Using ADAMS/Solver [15] Brogliato, B., Ten Dam, A.A., Paoli, L., Genot, F., Abadie, M.: Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. {Appl. Mech. Rev. 55, 107–150 (2002) [16] Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. {SIAM J. Appl. Mech. 42, 281–296 (1982) · Zbl 0489.70016 [17] Baraff, D.: Analytical methods for dynamic simulation of non-penetrating rigid bodies. {Comput. Graph. 23, 223–232 (1989) [18] Baraff, D.: Curved surfaces and coherence for non-penetrating rigid body simulation. {Comput. Graph. 24, 19–28 (1990) [19] Pfeiffer, F., Glocker, C.: {Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996) · Zbl 0922.70001 [20] Glocker, C.: Formulation of spatial contact situations in rigid multibody systems. {Comput. Methods Appl. Mech. Eng. 177, 199–214 (1999) · Zbl 0952.70007 [21] Glocker, C.: {Set-Valued Force Laws – Dynamics of Non-smooth Systems. Springer, Berlin (2001) · Zbl 0979.70001 [22] Baraff, D.: Issues in computing contact forces for non-penetrating rigid bodies. {Algorithmica 10, 292–352 (1993) · Zbl 0777.70006 [23] Trinkle, J.C., Pang, J.S., Sudarsky, S., Lo, G.: On dynamic multi-rigid-body contact problems with coulomb friction. {Zeithschrift fur Angewandte Mathematik und Mechanik 77, 267–279 (1997) · Zbl 0908.70008 [24] Pfeiffer, F.: Unilateral problems of dynamics. {Arch. Appl. Mech. 69, 503–527 (1999) · Zbl 0953.70008 [25] Trinkle, J.C.: Formulation of multibody dynamics as complementarity problems. In: {Proceedings of 2003 ASME Design Engineering Technical Conferences, pp. 361–370. Chicago, IL (2003) [26] Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. {Int. J. Numer. Methods Eng. 39, 2673–2691 (1996) · Zbl 0882.70003 [27] Anitescu, M., Potra, E.A., Stewart, D.E.: Time-stepping for three-dimensional rigid-body dynamics. {Comput. Methods Appl. Mech. Eng. 117, 183–197 (1999) · Zbl 0967.70003 [28] Stewart, D.E.: Rigid-body dynamics with friction and impact. {SIAM Rev. 42, 3–39 (2000) · Zbl 0962.70010 [29] Pfeiffer, F.: Applications of unilateral multibody dynamics. {Philos. Trans. R. Soc. Lond., Ser. A (Math. Phys. Eng. Sci.) 359, 2609–2628 (2001) · Zbl 1014.70006 [30] Gilbert, E.G., Johnson, D.W., Keerthi, S.S.: A fast procedure for computing the distance between objects in three-dimensional space. J. Robot. Autom. 4, 193–203 (1988) [31] Ma, O., Nahon, M.: General method for computing the distance between two moving objects using optimization techniques. {ASME Des. Eng. Div. (Publ.) DE, Adv. Des. Autom. 44, 109–117 (1992) [32] Redon, S., Kheddar, A., Coquillart, S.: Fast continuous collision detection between rigid bodies. In: {Computer Graphics Forum, 23rd Annual Conference of the Eurographics Association, pp. 279–287. Saarbrucken, Germany (2002) [33] Hughes, P.C.: {Spacecraft Attitude Dynamics. Wiley, New York (1986) [34] Bauchau, O.A., Rodriquez, J., Battasso, C.J.: Modelling of unilateral contact conditions with application to aerospace systems involving backlash, freeplay and friction. {Mech. Res. Commun. 28, 571–599 (2001) · Zbl 1008.74527 [35] Leamy, M.J., Wasfy, T.M.: Transient and steady-state dynamic finite element modeling of belt-drives. {ASME J. Dyn. Syst., Meas. Control 24, 575–581 (2002) · Zbl 1110.74531 [36] Wasfy, T.M.: Asperity spring friction model with application to belt-drives. In: {Proceedings ASME Design Engineering Technical Conference, pp. 371–378. Chicago, IL, USA (2003) [37] Haessig, D.A. Jr., Friedland, B.: On the modeling and simulation of friction. {ASME J. Dyn. Syst., Meas. Control 113, 354–362 (1991) [38] Armstrong-Helouvry, B., Dupont, P., Canudas De Wit, C.: A survey of models, analysis tools and compensation methods for the control of machines with friction. {Automatica 30, 1083–1138 (1994) · Zbl 0800.93424 [39] Ma, O., Buhariwala, K., Roger, N., MacLean, J., Carr, R.: MDSF – a generic development and simulation facility for flexible, complex robotic systems. {Robotica 15, 49–62 (1997) [40] Johnson, K.L.: {Contact Mechanics. Cambridge University Press, Cambridge, UK (1987) · Zbl 0599.73108 [41] Tenaglia, C.A., Orin, D.E., LaFarge, R.A., Lewis, C.: Toward development of a generalized contact algorithm for polyhedral objects. In: {Proceedings of 1999 IEEE International Conference on Robotics and Automation, pp. 2887–2892. Detroit, MI (1999) [42] Lin, A.: Friction identification from robotic insertion tasks. Master’s thesis, University of Victoria, Victoria, BC, Canada (2001)
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