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Fast Implementation of Lemke’s Algorithm for Rigid Body Contact Simulation Computer Science Department University of British Columbia Vancouver, Canada John E. Lloyd
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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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* = 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
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d 21 n 1 d 11 f x Problem formulation n 2
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Constraints n i d i4 d i3 d i1 d i5 d i6 d i2 Non-penetration: Friction opposite velocity: Friction cone:
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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] => Accuracy, convergence? Pivoting methods: Lemke’s algorithm [Anitescu & Potra ’97, Stewart & Trinkle ’96] => Speed, robustness?
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Pivoting: exchange subsets of z and w If, then we have a solution Generally, one variable exchange per pivot
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Involves solving Complexity, and typically pivots Hence total expected complexity Once per pivot: compute
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Peg in hole test case
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1 Solve more efficiently 2 Reduce the number of pivots How to improve performance?
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1 Solving can be partitioned into:* Eliminate * Ignoring Lemke covering vector in this discussion
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This reduces the system to Eliminate
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Reduced matrix has size Hence per-pivot computation is So total expected complexity This yields the final system
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2 Reducing the number of pivots Observation: Only need to compute and for active contacts; i.e, those for which
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So start with a frictionless LCP and expand it as become active:
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Ideally, final system rank is Hence pivots So total expected complexity
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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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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
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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 … for 4 friction directions, size
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Larger number of needed for accurate friction computation v f
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Closeup: sample contacts
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