Abstract
The increasing demand for real-time high-fidelity multibody dynamics simulations in several modern fields such as robotics and computer game industries has motivated many researches to propose novel approaches to model multibody systems with several contacts. The possibility of different contact conditions in a system with several contacts yields a combinatorial problem of potentially large size. Rigid contact model which is the most common model used for real-time simulations yields a non-smooth dynamic formulation. The solution of such a system can be governed using different methods. In this paper a comparison between the complementarity approaches and the augmented Lagrangian based formulations to deal with non-smooth contact models is presented via numerical examples, and the advantages and shortcomings of each method are discussed.
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Abbreviations
- \(\mathbf{q}\) :
-
generalized coordinate vector
- \(\mathbf{q}_{M}\) :
-
generalized coordinate vector at the midpoint of the time step
- \(\mathbf{q}_{A}\) :
-
generalized coordinate vector at the beginning of the time step
- \(\mathbf{q}_{E}\) :
-
generalized coordinate vector at the end of the time step
- \(\mathbf{M}\) :
-
system mass matrix
- \(t\) :
-
time
- \(\lambda_{N}\) :
-
normal contact force
- \(\lambda_{T}\) :
-
tangential contact force
- \(\mathbf{W}_{N}\) :
-
normal contact force direction vector
- \(\mathbf{W}_{T}\) :
-
tangential contact force direction vector
- \(P_{N}\) :
-
non-smooth potential
- \(\mathbf{h}\) :
-
vector containing all smooth forces and Coriolis acceleration
- \(g_{N}\) :
-
gap function at the contact
- \(\gamma_{T}\) :
-
tangential relative velocity of contacting surfaces
- \(\mu \) :
-
friction coefficient
- \(\mathbf{r}_{u}\) :
-
vector containing regularization factors for unilateral constraints
- \(\mathbf{r}_{b}\) :
-
vector containing regularization factor for bilateral constraints
- \(\mathbf{r}_{T}\) :
-
vector containing regularization factor for tangential constraints
- \(\varLambda \) :
-
contact impulses vector
- \(\mathbf{u}_{E}\) :
-
velocity vector at the end of the time-step
- \(\mathbf{u}_{A}\) :
-
velocity vector at the beginning of the time-step
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Acknowledgements
The research reported in this paper was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), CMLabs Simulations, Inc., and Mitacs, Inc. The support is gratefully acknowledged.
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Jalali Mashayekhi, M., Kövecses, J. A comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problem. Multibody Syst Dyn 40, 327–345 (2017). https://doi.org/10.1007/s11044-016-9510-2
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DOI: https://doi.org/10.1007/s11044-016-9510-2