Fast Implementation of Lemke

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Fast Implementation of Lemkes Algorithm for
Rigid Body Contact Simulation
John E. Lloyd
Computer Science Department University of British
Columbia Vancouver, Canada
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Applications mechanical simulation, animation,
haptics
Haptics requires speed (1 Khz) and accuracy
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Extended contact can result in many contact points
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Contributions
Most exact solution method is Lemkes
algorithm with expected complexity

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

number of contacts, number of
bodies
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Problem formulation
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Constraints
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Results in a Linear Complementarity Problem (LCP)
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Solving Contact LCPs

• Iterative techniques includes impulse methods
  Mirtich Canny 95, Guendelman 03
  gt Accuracy, convergence?
  
• Pivoting methods Lemkes algorithm Anitescu
  Potra 97, Stewart Trinkle 96 gt
  Speed, robustness?

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Pivoting exchange subsets of z and w
Generally, one variable exchange per pivot
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Once per pivot compute

• Involves solving
• Complexity , and typically
  pivots
• Hence total expected complexity

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Peg in hole test case
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How to improve performance?

• 1 Solve more efficiently
• 2 Reduce the number of pivots

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1 Solving
Ignoring Lemke covering vector in this discussion
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This reduces the system to
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This yields the final system

• Reduced matrix has size
• Hence per-pivot computation is
• So total expected complexity

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2 Reducing the number of pivots
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So start with a frictionless LCP
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Ideally, final system rank is
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Results peg in hole
Standard Structural Reduced
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Results sample contacts
Standard Structural Reduced
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Results block stack
Standard Structural Reduced
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Results
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Conclusions Improved pivoting method for contact
simulation

• 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

http//www.cs.ubc.ca/lloyd/fastContact.html
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Future Work

• Exploit temporal coherence (give solver an
  advanced starting point)
• More efficient solution for reduced equation
• Robust pivot selection (minimum ratio test)

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LCP matrix can be quite large
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Larger number of needed for accurate
friction computation
f
v
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Closeup sample contacts