# PROJECTED GAUSS–SEIDEL SUBSPACE MINIMIZATION METHOD FOR INTERACTIVE RIGID BODY DYNAMICS - Improving Animation Quality using a Projected Gauss–Seidel Subspace Minimization Method

### Morten Silcowitz, Sarah Niebe, Kenny Erleben

#### Abstract

In interactive physical simulation, contact forces are applied to prevent rigid bodies from penetrating and to control slipping between bodies. Accurate contact force determination is a computationally hard problem. Thus, in practice one trades accuracy for performance. This results in visual artifacts such as viscous or damped contact response. In this paper, we present a new approach to contact force determination. We formulate the contact force problem as a nonlinear complementarity problem, and discretize the problem to derive the Projected Gauss–Seidel method. We combine the Projected Gauss–Seidel method with a subspace minimization method. Our new method shows improved qualities and superior convergence properties for specific configurations.

#### References

- Anitescu, M. and Potra, F. A. (1997). Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems.
- Arechavaleta, G., E.Lopez-Damian, and Morales, J. (2009). On the use of iterative lcp solvers for dry frictional contacts in grasping. In International Conference on Advanced Robotics 2009, ICAR 2009.
- Billups, S. C. (1995). Algorithms for complementarity problems and generalized equations. PhD thesis, University of Wisconsin at Madison.
- Cottle, R., Pang, J.-S., and Stone, R. E. (1992). The Linear Complementarity Problem. Academic Press.
- Erleben, K. (2007). Velocity-based shock propagation for multibody dynamics animation. ACM Trans. Graph., 26(2).
- Erleben, K. and Ortiz, R. (2008). A Non-smooth Newton Method for Multibody Dynamics. In American Institute of Physics Conference Series.
- Featherstone, R. (1998). Robot Dynamics Algorithms. Kluwer Academic Publishers, second printing edition.
- Guendelman, E., Bridson, R., and Fedkiw, R. (2003). Nonconvex rigid bodies with stacking. ACM Trans. Graph.
- Hahn, J. K. (1988). Realistic animation of rigid bodies. In SIGGRAPH 7888: Proceedings of the 15th annual conference on Computer graphics and interactive techniques.
- Kaufman, D. M., Sueda, S., James, D. L., and Pai, D. K. (2008). Staggered projections for frictional contact in multibody systems. ACM Trans. Graph., 27(5).
- Milenkovic, V. J. and Schmidl, H. (2004). A fast impulsive contact suite for rigid body simulation. IEEE Transactions on Visualization and Computer Graphics, 10(2).
- Mirtich, B. V. (1996). Impulse-based dynamic simulation of rigid body systems. PhD thesis, University of California, Berkeley.
- Morales, J. L., Nocedal, J., and Smelyanskiy, M. (2008). An algorithm for the fast solution of symmetric linear complementarity problems. Numer. Math., 111(2).
- O'Sullivan, C., Dingliana, J., Giang, T., and Kaiser, M. K. (2003). Evaluating the visual fidelity of physically based animations. ACM Trans. Graph., 22(3).
- Redon, S., Kheddar, A., and Coquillart, S. (2003). Gauss least constraints principle and rigid body simulations. In In proceedings of IEEE International Conference on Robotics and Automation.
- Silcowitz, M., Niebe, S., and Erleben, K. (2009). Nonsmooth Newton Method for Fischer Function Reformulation of Contact Force Problems for Interactive Rigid Body Simulation. In VRIPHYS 09: Sixth Workshop in Virtual Reality Interactions and Physical Simulations, pages 105-114. Eurographics Association.
- Stewart, D. E. (2000). Rigid-body dynamics with friction and impact. SIAM Review.
- Stewart, D. E. and Trinkle, J. C. (1996). An implicit timestepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. International Journal of Numerical Methods in Engineering.
- Trinkle, J. C., Tzitzoutis, J., and Pang, J.-S. (2001). Dynamic multi-rigid-body systems with concurrent distributed contacts: Theory and examples. Philosophical Trans. on Mathematical, Physical, and Engineering Sciences.

#### Paper Citation

#### in Harvard Style

Silcowitz M., Niebe S. and Erleben K. (2010). **PROJECTED GAUSS–SEIDEL SUBSPACE MINIMIZATION METHOD FOR INTERACTIVE RIGID BODY DYNAMICS - Improving Animation Quality using a Projected Gauss–Seidel Subspace Minimization Method** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)* ISBN 978-989-674-026-9, pages 38-45. DOI: 10.5220/0002830700380045

#### in Bibtex Style

@conference{grapp10,

author={Morten Silcowitz and Sarah Niebe and Kenny Erleben},

title={PROJECTED GAUSS–SEIDEL SUBSPACE MINIMIZATION METHOD FOR INTERACTIVE RIGID BODY DYNAMICS - Improving Animation Quality using a Projected Gauss–Seidel Subspace Minimization Method},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)},

year={2010},

pages={38-45},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002830700380045},

isbn={978-989-674-026-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)

TI - PROJECTED GAUSS–SEIDEL SUBSPACE MINIMIZATION METHOD FOR INTERACTIVE RIGID BODY DYNAMICS - Improving Animation Quality using a Projected Gauss–Seidel Subspace Minimization Method

SN - 978-989-674-026-9

AU - Silcowitz M.

AU - Niebe S.

AU - Erleben K.

PY - 2010

SP - 38

EP - 45

DO - 10.5220/0002830700380045