Fast Implementation of Lemke’s Algorithm for

Rigid Body Contact Simulation

Computer Science Department

University of British Columbia

Vancouver, Canada

John E. Lloyd

Applications: mechanical simulation, animation,

haptics

Haptics requires speed (1 Khz) and accuracy

Extended contact can result in many contact

points

* = number of contacts, = number of bodies

• Use problem structure to speed up solution• Reduce complexity to nearly *• complexity for fixed number of bodies

Contributions Most exact solution method is

Lemke’s algorithm with expected complexity

d21

n1

d11

fx

Problem formulation

n2

Constraints

ni

di4

di3

di1

di5 di6

di2

Non-penetration:

Friction opposite velocity:

Friction cone:

Results in a Linear Complementarity Problem

(LCP):

Solving Contact LCPs

• Iterative techniques; includes impulse methods [Mirtich & Canny ’95, Guendelman ’03] => Accuracy, convergence?

• Pivoting methods: Lemke’s algorithm [Anitescu & Potra ’97, Stewart & Trinkle ’96] => Speed, robustness?

Pivoting: exchange subsets of z and w

If , then we have a solution

Generally, one variable exchange per pivot

• Involves solving

• Complexity , and typically pivots

• Hence total expected complexity

Once per pivot: compute

Peg in hole test case

1 Solve more efficiently

2 Reduce the number of pivots

How to improve performance?

1 Solving

can be partitioned into:*

Eliminate

*Ignoring Lemke covering vector in this discussion

This reduces the system to

Eliminate

• Reduced matrix has size

• Hence per-pivot computation is

• So total expected complexity

This yields the final system

2 Reducing the number of pivots

Observation: Only need to compute and for active contacts; i.e,those for which

So start with a frictionless LCP

and expand it as become active:

Ideally, final system rank is

• Hence pivots

• So total expected complexity

Results: peg in hole

Standard

Structural

Reduced

Results: sample contacts

Standard

Structural

Reduced

Results: block stack

Standard

Structural

Reduced

Results

• Fast: exploit problem structure• Better complexity: nearly • for fixed number of bodies• Efficient: no need to compute • More robust: smaller system to solve

each pivot

Conclusions: Improved pivoting method for contact

simulation

http://www.cs.ubc.ca/~lloyd/fastContact.html

Future Work

• Exploit temporal coherence (give solver an advanced starting point)

• More efficient solution for reduced equation

• Robust pivot selection (minimum ratio test)

LCP matrix can be quite large …

for 4 friction directions, size

Larger number of needed for accurate friction

computation

v

f

Closeup: sample contacts