# Presentation: 056139

## Simulating Multi-Rigid-Body Dynamics with Contact and Friction

- 	Mihai Anitescu
- SIAM OPTIMIZATION MEETING
- May 10, 1999

## Rigid Multi-Body Dynamics Simulation Applications

- Industrial simulation and design for  aircraft, ships, heavy vehicles (product design -- production, 5 years).
- Flexible Manufacturing Systems design and redesign  (commissioning+tuning, 3 years).
- Appropriate simulation could reduce lead times by up to 50%.
- Simulating virtual environments (+real time = interactive).

## Virtual Environment Application Example

- Setting up the real installation for training is very expensive.
- The virtual environment can be easily modified for different scenarios, but simulating it in real-time is essential.
- Simplified model: rigid bodies with joints and contacts.

- Insert Picture

**Notes:**
INSERT HARD SLIDE

## Initial model requirements

- Acceleration based Newton laws.
- Rigid bodies.
- No inter penetration, nonnegative contact impulses and passive contact
- Joint constraints
- Coulomb friction or reasonable approximations.
- Impact resolution (Poisson hypothesis).

## Classical Constrained Dynamics

## Inconsistency example

- State space representation
- Constraint               , initially

## Frictional Inconsistency

- What went wrong? Newton’s laws rather than Coulomb’s model.
- Configuration inconsistent, not only singular.
- P. Painleve: Governing equations of motion do not have a solution in general( Comptes Rendus Acad. Sci. Paris, 1895).
- Approaches for the continuous case.
- Solutions in a distributional sense.
*Differential inclusions* instead of differential equations (Moreau, 1986; Monteiro-Marques, 1993).

## Avoiding Frictional Inconsistencyin time-stepping schemes

- Time-stepping: If a continuous model exists, how should the associated numerical integration scheme look like?
- Friction is estimated, and the frictionless problem is solved (Lotstedt,1982). Problems: starting configuration and the impact case.
- The inconsistent configuration is treated as a collision(Baraff,1993). Problems: arbitrary collisions and the impact case.
- Newton's law with impulses and velocities (Stewart & Trinkle 1995; Anitescu & Potra, 1996).

## Revised model requirements

*Velocity* based Newton laws, by combining * time stepping  *with *impulse resolution*.
- Rigid bodies.
- No inter penetration, nonnegative contact impulses and passive contact
- Joint constraints
- Coulomb friction or reasonable approximations.
- Impact resolution (Poisson hypothesis).

## Framework and notations used

- The rigid multi-body system is described by its *generalized coordinates*, q, *and generalized velocities*, v.
- The contacts (joints) are described by inequalities (equalities).
- The ``tangent plane'' for a contact constraint is defined by the cone generators
- M: *mass matrix*, symmetric positive definite; h: the *time step *k: *external force*.

**Insert Model Slides**

## Contact Model

- Contact configuration described by the (generalized) distance function                 ,    which allows for some interpenetration.
- Feasible set                 .
- Contact impulses are compressive
- The normal velocity for non inter penetration is velocity-based (as opposed to Stewart & Trinkle, where it is position-based):

- Complementarity guarantees energetic passivity of contact:

## Contact Description

-     is the normal impulse and     is the tangential impulse
-      is the total impulse in Newton Euler world coordinates .
- Classical Coulomb model requires the 3D contact impulse  to lie in a circular cone.
- If relative motion exists, the frictional impulse has to be opposite to the relative velocity (Maximum Dissipation)

## Polygonal approximation of Coulomb cone

## The Integration Step

## Matrix Form of The Integration Step

## Linear Complementarity Problems  (LCPs)

- Linear and Quadratic Programming are LCPs, via primal-dual formulations.
- If 					, a solution exists, (by a Brouwer fixed point argument), and can be computed by Lemke’s algorithm (pivotal, similar to simplex).
- M has to be copositive

## Interior Point Algorithms (IP)

- It takes		     corrected Newton steps to compute a solution if  M is positive semi definite. The *central path* is well defined.
- In the convex case IP algorithms have *polynomial* complexity,  but other extensions are unclear.

## Theorem

- Consider the mixed LCP above. If M is a symmetric positive definite matrix, N a copositive matrix and b a nonnegative vector, then the mixed LCP has a solution. Lemke’s algorithm (with anti cycling strategy) will find a solution of the LCP obtained by eliminating x,y.

## Properties of the formulation

- The model is *solvable *for any configuration.
- Discrete *Maximum Dissipation Principle* (MDP): The frictional impulse               maximizes the dissipation over all feasible contact impulses, given
- If the mass matrix is constant, the kinetic energy at the new step cannot exceed the the kinetic energy of the system with no constraint enforced (contacts are passive).

## Energy/ stability properties

*Assumption*: M constant.
- If the external force is of the form
- c depends only on M and d. h has to be sufficiently small but depending only on c, T.

## Constraint stability

- There are  constants C,H  depending on the problem data, but not on the time-step h,  such that
- for any h < H (q(t) is obtained by linear interpolation).

## Difficult Friction Configuration

- For any friction coefficient we obtain multiple solutions.
- Does this result in undesirable properties of the LCP matrix?

## Column sufficiency?

## Row sufficiency?

## Properties of the LCP matrix.

- The matrix Q is neither row, nor column sufficient.
- As a result, it cannot be in the        class, which makes it difficult to use polynomial-time interior-point algorithms.
- Lemke’s algorithm appears to be the only one which is guaranteed to provide a solution to this model.
- However, further analysis may reveal good matrix properties for pointed friction cones.
- Since model needs to solve a succession of these LCPs anyway, other approaches may provide (computationally) better matrices.

## Differential Inclusions

-   v has  bounded variation
- K(t) is closed and convex set valued

## Theorem

- For a *pointed *friction cone, there exists
- The quantities                are obtained by linear interpolation. (D. Stewart)

## Theorem(II)

- For one contact, maximal dissipation holds

## Accomplishments

*The time-stepping model to has a solution regardless of the configuration and dimension of the problem. Impulsive solutions are accommodated.*
- Static and dynamic friction are treated in an unitary manner.
- The constraints are guaranteed to be satisfied within any given tolerance.
- The numerical scheme is dissipative and, as a result the velocities are uniformly bounded.
- A differential inclusion model with energy dissipation satisfied in the limit.

## 2-dimensional examples

- Long bar with multiple contacts:
-  low_[ ](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml6.wrl)__[friction](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml6.wrl)_ (mu=0.05, e=0.6)
-  high_[ ](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml8.wrl)__[friction](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml8.wrl)_.(mu=0.2, e=0.6)
- 2 blocks on table
- low _[friction](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml2.wrl)_ (mu=0.02, e=0.2)
- high _[friction](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml1.wrl)__[ ](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml1.wrl)_(mu=0.1, e=0.2)
- Multiple collisions with high restitution, _[high](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml4.wrl)__[ ](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml5.wrl)_friction_[ ](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml4.wrl)_(mu=0.3, e=0.9).
- Three blocks on table, low _[friction](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml5.wrl)__[ ](http://www.mcs.anl.gov/otc/Guide/CaseStudies/contacts/myfrexpl/SIMULATIONS/myvrml5.wrl)_(mu=0.05, e=0.6).
- Several blocks, (mu=0.2, e=1.0)

- VRML: to start animation click on the base object after browser is loaded.

## Open Problems

- Can a consistent differential inclusion approach be formulated for general configurations?
- Under  which conditions does the  maximum dissipation principle apply in the multiple contact case?
- Can other properties of the time-stepping scheme be used in the continuous time context (such as the bound on the square of the variation of the _[velocities](slide19.xml)_)?
- Is the solution set of the friction LCP convex? (it is not true for arbitrary copositive matrices),
- Can the LCP be recast as a convex optimization problem?

## Future Research

- Treat the Coulomb cone directly rather than using a polygonal approximation (partly solved  APS98 ).
-  Design stable numerical algorithms to handle stiffness (implicit or linearly implicit, work in progress).
- Nonlinear formulations (NCP).
- Investigate Interior-Point Algorithms for solving the friction LCP or NCP.
- Are higher-order methods relevant?
- Formulate an efficient piecewise DAE strategy.