The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb’s friction: a tutorial. (English) Zbl 1372.70044
Summary: This work deals with the existence and uniqueness of the acceleration and contact forces for Lagrangian systems subject to bilateral and/or unilateral constraints with or without sliding Coulomb’s friction. Sliding friction is known to yield singularities in the system, such as Painlevé’s paradox. Our work aims at providing sufficient conditions on the parameters of the system so that singularities are avoided (i.e., the contact problem is at least solvable). To this end, the frictional problem is treated as a perturbation of the frictionless case. We provide explicit criteria, in the form of calculable upper bounds on the friction coefficients, under which the frictional contact problem is guaranteed to remain well-posed. Complementarity problems, variational inequalities, quadratic programs and inclusions in normal cones are central tools.
Keywords:
Lagrangian systems; unilateral constraint; bilateral constraint; complementarity problem; Coulomb’s friction; Gauss’ principle; KKT conditionsReferences:
| [1] | Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008). doi:10.1007/978-3-540-75392-6 · Zbl 1173.74001 · doi:10.1007/978-3-540-75392-6 |
| [2] | Acary, V., Brogliato, B., Goeleven, D.: Higher order Moreau’s sweeping process: mathematical formulation and numerical simulation. Math. Program. A 113(1), 133-217 (2008). doi:10.1007/s10107-006-0041-0 · Zbl 1148.93003 · doi:10.1007/s10107-006-0041-0 |
| [3] | Addi, K., Brogliato, B., Goeleven, D.: A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems. Applications in electronics. Math. Program. A 126(1), 31-67 (2011). doi:10.1007/s10107-009-0268-7 · Zbl 1229.90224 · doi:10.1007/s10107-009-0268-7 |
| [4] | Anh, L.X.: Dynamics of Mechanical Systems with Coulomb Friction. Foundations of Mechanical Engineering. Springer, Berlin (2003) · Zbl 1094.70001 · doi:10.1007/978-3-540-36516-7 |
| [5] | Bayo, E., Ledesma, R.: Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dyn. 9(1-2), 113-130 (1996). doi:10.1007/BF01833296 · doi:10.1007/BF01833296 |
| [6] | Bernstein, D.: Matrix, Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005) · Zbl 1075.15001 |
| [7] | Blajer, W.: Augmented Lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancy. Multibody Syst. Dyn. 8(2), 141-159 (2002) · Zbl 1022.70004 · doi:10.1023/A:1019581227898 |
| [8] | Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) · Zbl 1058.90049 · doi:10.1017/CBO9780511804441 |
| [9] | Brogliato, B.: Inertial couplings between unilateral and bilateral holonomic constraints in frictionless Lagrangian systems. Multibody Syst. Dyn. 29, 289-325 (2013). doi:10.1007/s11044-012-9317-8 · Zbl 1271.70032 · doi:10.1007/s11044-012-9317-8 |
| [10] | Brogliato, B.: Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst. Dyn. 32, 175-216 (2014). doi:10.1007/s11044-013-9392-5 · Zbl 1351.70014 · doi:10.1007/s11044-013-9392-5 |
| [11] | Brogliato, B., Goeleven, D.: Singular mass matrix and redundant constraints in unilaterally constrained Lagrangian and Hamiltonian systems. Multibody Syst. Dyn. 35, 39-61 (2015). doi:10.1007/s11044-014-9437-4 · Zbl 1335.70035 · doi:10.1007/s11044-014-9437-4 |
| [12] | Brogliato, B., Thibault, L.: Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems. J. Convex Anal. 17(3-4), 961-990 (2010) · Zbl 1217.34026 |
| [13] | Brüls, O., Arnold, M.: Convergence of the generalized α \(\alpha\) scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185-202 (2007) · Zbl 1121.70003 · doi:10.1007/s11044-007-9084-0 |
| [14] | Chen, X., Xiang, S.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18(4), 1250-1265 (2007) · Zbl 1172.90018 · doi:10.1137/060653019 |
| [15] | Cottle, R.: On a problem in linear inequalities. J. Lond. Math. Soc. 1(1), 378-384 (1968) · Zbl 0181.04001 · doi:10.1112/jlms/s1-43.1.378 |
| [16] | Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Computer Science and Scientific Computing. Academic Press, San Diego (1992) · Zbl 0757.90078 |
| [17] | ten Dam, A., Dwarshuis, E., Willems, J.: The contact problem for linear continuous-time dynamical systems: a geometric approach. IEEE Trans. Autom. Control 42(4), 458-472 (1997) · Zbl 0886.93031 · doi:10.1109/9.566656 |
| [18] | Dopico, D., González, F., Cuadrado, J., Kövecses, J.: Determination of holonomic and nonholonomic constraint reactions in an index-3 augmented Lagrangian formulation with velocity and acceleration projections. J. Comput. Nonlinear Dyn. 9(4), 041006 (2014) · doi:10.1115/1.4027671 |
| [19] | Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Operations Research, vol. 1. Springer, New York (2003) · Zbl 1062.90002 |
| [20] | Fraczek, J., Wojtyra, M.: On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction in joints. Mech. Mach. Theory 46(3), 312-334 (2011). doi:10.1016/j.mechmachtheory.2010.11.003 · Zbl 1336.70020 · doi:10.1016/j.mechmachtheory.2010.11.003 |
| [21] | Génot, F., Brogliato, B.: New results on Painlevé paradoxes. Eur. J. Mech. A, Solids 18(4), 653-677 (1999). doi:10.1016/S0997-7538(99)00144-8 · Zbl 0962.70019 · doi:10.1016/S0997-7538(99)00144-8 |
| [22] | Glocker, C.: The principles of d’Alembert, Jourdain, and Gauss in nonsmooth dynamics, part I: scleronomic multibody systems. Z. Angew. Math. Mech. 78, 21-37 (1998) · Zbl 0901.70014 · doi:10.1002/(SICI)1521-4001(199801)78:1<21::AID-ZAMM21>3.0.CO;2-W |
| [23] | Glocker, C.: Set-Valued Force Laws: Dynamics of Non-Smooth Systems. Springer, Berlin (2001) · Zbl 0979.70001 · doi:10.1007/978-3-540-44479-4 |
| [24] | Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Series in Computational Mathematics, vol. 14. Springer, Berlin (1996), second revised edn. · Zbl 0859.65067 · doi:10.1007/978-3-642-05221-7 |
| [25] | Hjiaj, M., de Saxcé, G., Mroz, Z.: A variational inequality-based formulation of the frictional contact law with a non-associated sliding rule. Eur. J. Mech. A, Solids 21, 49-59 (2002) · Zbl 1005.74043 · doi:10.1016/S0997-7538(01)01183-4 |
| [26] | Hurtado, J., Sinclair, A.: Lagrangian mechanics of overparameterized systems. Nonlinear Dyn. 66, 201-212 (2011) · Zbl 1331.70048 · doi:10.1007/s11071-010-9921-1 |
| [27] | Ivanov, A.: Singularities in the dynamics of systems with non-ideal constraints. J. Appl. Math. Mech. 67(2), 185-192 (2003) · Zbl 1067.70503 · doi:10.1016/S0021-8928(03)90004-9 |
| [28] | Izmailov, A., Kurennoy, A., Solodov, M.: Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption. Comput. Optim. Appl. 1(30), 111-140 (2014). doi:10.1007/s10589-014-9658-8 · Zbl 1338.90408 · doi:10.1007/s10589-014-9658-8 |
| [29] | Jain, A.: Operational space for closed-chain robotic systems. ASME J. Comput. Nonlinear Dyn. 9(2), 021015 (2014). doi:10.1115/1.4025893 · doi:10.1115/1.4025893 |
| [30] | de Jalon, J.G., Callejo, A., Hidalgo, A.: Efficient solution of Maggi’s equations. ASME J. Comput. Nonlinear Dyn. 7, 021003 (2012) · doi:10.1115/1.4005238 |
| [31] | de Jalon, J.G., Gutierrez-Lopez, M.: Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraints forces. Multibody Syst. Dyn. 30(3), 311-341 (2013). doi:10.1007/s11044-013-9358-7 · Zbl 1274.70010 · doi:10.1007/s11044-013-9358-7 |
| [32] | Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001-). URL http://www.scipy.org/. [Online; accessed 2016-05-04] · Zbl 1005.74043 |
| [33] | Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, San Diego (1985) · Zbl 0558.15001 |
| [34] | Laulusa, A., Bauchau, O.: Review of classical approaches for constraint enforcement in multibody systems. ASME J. Comput. Nonlinear Dyn. 3(1), 011004 (2008) · doi:10.1115/1.2803257 |
| [35] | Leine, R., Brogliato, B., Nijmeijer, H.: Periodic motion and bifurcations induced by the Painlevé paradox. Eur. J. Mech. A, Solids 21(5), 869-896 (2002). doi:10.1016/S0997-7538(02)01231-7. http://www.sciencedirect.com/science/article/pii/S0997753802012317 · Zbl 1023.70009 · doi:10.1016/S0997-7538(02)01231-7 |
| [36] | Leine, R., van de Wouw, N.: Stability and Convergence of Mechanical Systems with Unilateral Constraints. Lecture Notes in Applied and Computational Mechanics, vol. 36. Springer, Berlin (2008) · Zbl 1143.70001 |
| [37] | Lötstedt, P.: Coulomb friction in two-dimensional rigid body systems. Z. Angew. Math. Mech. 61, 605-615 (1981) · Zbl 0495.73095 · doi:10.1002/zamm.19810611202 |
| [38] | Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42(2), 281-296 (1982) · Zbl 0489.70016 · doi:10.1137/0142022 |
| [39] | Lunk, C., Simeon, B.: Solving constrained mechanical systems by the family of Newmark and α \(\alpha \)-methods. J. Appl. Math. Mech., Z. Angew. Math. Mech. 86(10), 772-784 (2006) · Zbl 1116.70008 · doi:10.1002/zamm.200610285 |
| [40] | Anitescu, M., Potra, F.A.: A time-stepping method for stiff multibody dynamics with contact and friction. Int. J. Numer. Methods Eng. 55(7), 753-784 (2002). doi:10.1002/nme.512 · Zbl 1027.70001 · doi:10.1002/nme.512 |
| [41] | Matrosov, V., Finogenko, I.: Right-hand solutions of the differential equations of dynamics for mechanical systems with sliding friction. J. Appl. Math. Mech. 59(6), 837-844 (1995) · Zbl 0925.70142 · doi:10.1016/0021-8928(95)00116-6 |
| [42] | Matrosov, V.M., Finogenko, I.: The theory of differential equations which arise in the dynamics of a system of rigid bodies with Coulomb friction. Monogr. Acad. Nonlinear Sci., Adv. Nonlinear Sci. 2, 16-106 (2008) |
| [43] | Moreau, J.: Les liaisons unilatérales et le principe de Gauss. C. R. Acad. Sci. 256(4), 871-874 (1963) |
| [44] | Moreau, J.: Quadratic programming in mechanics: dynamics of one-sided constraints. SIAM J. Control 4(1), 153-158 (1966) · doi:10.1137/0304014 |
| [45] | Murua, A.: Partitioned half-explicit Runge-Kutta methods for differential-algebraic systems of index 2. Computing 59(1), 43-61 (1997) · Zbl 0881.65071 · doi:10.1007/BF02684403 |
| [46] | Negrut, D., Jay, L., Khude, N.: A discussion of low-order numerical integration formulas for rigid and flexible multibody dynamics. ASME J. Comput. Nonlinear Dyn. 4, 021008 (2009) · doi:10.1115/1.3079784 |
| [47] | Painlevé, P.: Leçons sur le Frottement. Hermann, Paris (1895) · JFM 26.0781.01 |
| [48] | Pang, J., Trinkle, J.: Complementarity formulation and existence of solutions of dynamic rigid-body contact problems with Coulomb friction. Math. Program. 73(2), 199-226 (1996) · Zbl 0854.70008 · doi:10.1007/BF02592103 |
| [49] | Pang, J., Trinkle, J., Lo, G.: A complementarity approach to a quasistatic multi-rigid-body contact problem. J. Comput. Optim. Appl. 5(2), 139-154 (1996) · Zbl 0859.90123 · doi:10.1007/BF00249053 |
| [50] | Pfeiffer, F.: On non-smooth multibody dynamics. Proc. Inst. Mech. Eng., Part K, J. Multi-Body Dyn. 226(2), 147-177 (2012) |
| [51] | Rockafellar, R.T.: Convex Analysis, vol. 28. Princeton University Press, Princeton (1970). doi:10.1142/9789812777096 · Zbl 0193.18401 · doi:10.1142/9789812777096 |
| [52] | Ruzzeh, B., Kövecses, J.: A penalty formulation for dynamics analysis of redundant mechanical systems. J. Comput. Nonlinear Dyn. 6(2), 021008 (2011) · doi:10.1115/1.4002510 |
| [53] | de Saxcé, G., Feng, Z.: New inequality and functional for contact friction: the implicit standard material approach. Mech. Struct. Mach. 19, 301-325 (1991) · doi:10.1080/08905459108905146 |
| [54] | van der Schaft, A., Schumacher, J.: Complementarity modeling of hybrid systems. IEEE Trans. Autom. Control 43(4), 483-490 (1998) · Zbl 0899.93002 · doi:10.1109/9.664151 |
| [55] | Shabana, A.: Euler parameters kinetic singularity. Proc. Inst. Mech. Eng., Part K, J. Multi-Body Dyn. (2014). doi:10.1177/1464419314539301 · doi:10.1177/1464419314539301 |
| [56] | Simeon, B.: Computational Flexible Multibody Dynamics. A Differential-Algebraic Approach. Springer, Berlin (2013). Differential-Algebraic Equations Forum · Zbl 1279.70002 · doi:10.1007/978-3-642-35158-7 |
| [57] | Terze, Z., Müller, A., Zlatar, D.: Lie-group integration method for constrained multibody systems in state space. Multibody Syst. Dyn. 34(3), 275-305 (2015). doi:10.1007/s11044-014-9439-2 · Zbl 1339.70037 · doi:10.1007/s11044-014-9439-2 |
| [58] | Udwadia, F., Kalaba, R.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (1996) · Zbl 0875.70100 · doi:10.1017/CBO9780511665479 |
| [59] | Udwadia, F., Phohomsiri, P.: Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc. R. Soc. Lond. A 462, 2097-2117 (2006) · Zbl 1149.70311 · doi:10.1098/rspa.2006.1662 |
| [60] | Udwadia, F., Schutte, A.: Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech. 213(1-2), 111-129 (2010) · Zbl 1320.70016 · doi:10.1007/s00707-009-0272-2 |
| [61] | Wojtyra, M.: Joint reactions in rigid body mechanisms with dependent constraints. Mech. Mach. Theory 44(12), 2265-2278 (2009). doi:10.1016/j.mechmachtheory.2009.07.008 · Zbl 1247.70026 · doi:10.1016/j.mechmachtheory.2009.07.008 |
| [62] | Wojtyra, M., Fraczek, J.: Comparison of selected methods of handling redundant constraints in multibody systems simulations. ASME J. Comput. Nonlinear Dyn. 8, 0210007 (2013) |
| [63] | Wojtyra, M., Fraczek, J.: Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints. Multibody Syst. Dyn. 30(2), 153-171 (2013). doi:10.1007/s11044-013-9352-0 · doi:10.1007/s11044-013-9352-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
