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QP-Collide: A New Approach to Collision Treatment

Explore a new technique for collision treatment that ensures accurate corrections and enhancements for challenging physical simulations. Watch our teaser video for an in-depth look at the process. Our method allows for implicit or explicit mechanics solving, formulating collisions as linear constraints, and employing quadratic programming for corrections. Discover the strategy behind handling multiple collisions effectively and avoiding looping issues. This innovative approach offers solutions to complex collision problems and provides insights into improving numerical precision.

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QP-Collide: A New Approach to Collision Treatment

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  1. QP-Collide: A New Approach to Collision Treatment Laks Raghupathi François Faure Co-encadre par Marie-Paule CANI EVASION/GRAVIR INRIA Rhône-Alpes, Grenoble GTAS, Toulouse, Juin 15-16, 2006

  2. Teaser Video GTAS, Toulouse, Juin 15-16, 2006

  3. Classical Physical Simulation • Advance time-step (solve ODE for Δv) • Detect collisions at • Make corrections to and/or GTAS, Toulouse, Juin 15-16, 2006

  4. Problem at Hand • Detect and handle multiple collisions • Discrete methods miss thin, fast objects • Displacement correction results in large deformation for stiff objects • Avoidloopingproblems in the case of multiple collisions • Responding one-by-one will provoke further collisions GTAS, Toulouse, Juin 15-16, 2006

  5. Our Strategy • Solve mechanics implicit (stable) or explicit form • Formulate collisions as linear inequality constraints • Detect both at discrete and continuous time • Solve mechanic equations + constraint inequalities • collision correction via quadratic programming (QP) GTAS, Toulouse, Juin 15-16, 2006

  6. Previous Techniques • [Baraff94] – contact force computation of rigid bodies using LCP/QP • Requires inertial matrix inversion M-1 • Not always possible to compute K-1 for deformable objects • Others use iterative correction [Volino00] and [Bridson02] GTAS, Toulouse, Juin 15-16, 2006

  7. Overall Collision Approach • Perform broad-phase rejection tests • E.g. Box-Box test between Cable and Object • If i. is true, enter narrow-phase Recursively test Cable Primitives and Object-Children (e.g. ray-box) Primitive tests between Cable and Object primitives (e.g. ray-triangle) Apply Response for each valid collision GTAS, Toulouse, Juin 15-16, 2006

  8. Broad Phase Detection 1-subdivision 3-subdivisions Polygonal model GTAS, Toulouse, Juin 15-16, 2006

  9. Why Continuous? No interpenetration at t and t + t => Undetected by static methods GTAS, Toulouse, Juin 15-16, 2006

  10. Case for Continuous • Static collision detection missed if: RelativeVelocityt > ObjectSize • Robust detection for thin, fast moving objects • Permits large time step simulations • Continuous methods detect first contact during collision detection GTAS, Toulouse, Juin 15-16, 2006

  11. Narrow Phase – Vertex-Triangle • Point P(t) colliding with triangle (A(t), B(t), C(t)) and normalN(t) • Cubic equation in tc • Find valid tc ε [t0, t0+dt] and u,wε [0, 1], u+w 1 GTAS, Toulouse, Juin 15-16, 2006

  12. Quadratic Form (1) • Mechanical equation • Implicit (Baraff98) • Quadratic form with linear constrains • Minimize: • Subject to: GTAS, Toulouse, Juin 15-16, 2006

  13. Quadratic Form (2) • Lagrange form • Such that • - set of “active” constraints • - set of “non-active” constraints GTAS, Toulouse, Juin 15-16, 2006

  14. Active Set Method (1) • Find ¢vin • Detect Collisions (populate J and c matrices) • Classify constraints • Find new ¢vand for all constraint Ji2 J[1 …m] do if Ji¢vi> cithen Add ithconstraint to A else Add ithconstraint to A’ GTAS, Toulouse, Juin 15-16, 2006

  15. Active Set Method (2) • Verify Lagrange multiplier sign • Verify if solution satisfies constraints for all q(k), q 2 A do if q(k)> 0then Move q from A to A’(active to non-active) endloop = false for all ¢vp, p 2 A do if Jpv(k) p> cpthen Move p from A’to A (non-active to active) endloop = false GTAS, Toulouse, Juin 15-16, 2006

  16. On Convergence • Using iterative method such as conjugate gradient • Should converge within 3n + m steps • ‘n’ number vertices, ‘m’ number of constraints • Watch out for numerical errors • Toggling Constraints GTAS, Toulouse, Juin 15-16, 2006

  17. Collisions as Linear Constraints • Vertex-Triangle condition for non-penetration: • Left-hand side of J matrix: • Similar for edge-edge and other constraints GTAS, Toulouse, Juin 15-16, 2006

  18. Demos GTAS, Toulouse, Juin 15-16, 2006

  19. Known Problems • Degradation of matrix conditioning • Susceptible to numerical errors • Disadvantage vis-à-vis penalty approaches • Linearly dependent constraints • High condition number • Simple fix in lieu of QR decomposition • Looping • Constraints toggle between “active” and “non-active” • Suppress them – not theoretically justified GTAS, Toulouse, Juin 15-16, 2006

  20. Linearly dependent constraints • Add minor “perturbation” to avoid singularity: L = diagonal matrix (l1, …lm), li ~ 10-3 GTAS, Toulouse, Juin 15-16, 2006

  21. Summary • New approach to handling multiple collisions and other linear constraints • Works well for moderately difficult cases • Further theoretical work needed for better numerical precision GTAS, Toulouse, Juin 15-16, 2006

  22. Questions/Comments? Et bien sur, allez les Bleus! GTAS, Toulouse, Juin 15-16, 2006

  23. Demo: Mechanical Simulation (1) Cable stiffness = 1.38E+6 N/m Gravity = -10 m/s2 Mechanics: Implicit Euler resolved by conjugate gradient Run-time: 40 Hz in Pentium 4 3.0 GHz, 1GB RAM, GeForce 3 Pulley 13722 triangles Cable with 100 particles Particle Mass 0.7 kg initial velocity = -2.0 m/s Endpoint Masses 10000 kg GTAS, Toulouse, Juin 15-16, 2006

  24. Primary Accomplishments • Fast octree-based method for collision detection (rigid & deformable) • Continuous collision detection • Numerically stable ODE solver • Demo: Simulation of interaction cable and rigid mechanical parts • Relevance to medical applications GTAS, Toulouse, Juin 15-16, 2006

  25. Relevant apps: thin tissue cutting Scalpel cutting a thin tissue GTAS, Toulouse, Juin 15-16, 2006

  26. Thin tissue cutting (2) GTAS, Toulouse, Juin 15-16, 2006

  27. Bounding box tests • Ray-Box Test • 6 Ray-Plane Tests Why ray? • Ray is trajectory of positions between x(t) and x(t+dt) GTAS, Toulouse, Juin 15-16, 2006

  28. Recursive BB test • If Ray collides with Box, check with all the 8-children of the boxes recursively till the end Colliding AABBs GTAS, Toulouse, Juin 15-16, 2006

  29. Primitive Tests • Test primitives within the smallest box • Continuous test: • Cubic equation for two objects moving • Quadratic equation for one fixed mobile Ray-Triangle Tests GTAS, Toulouse, Juin 15-16, 2006

  30. Demo: Mechanical Simulation (2) Cable stiffness = 1E+4 N/m Gravity = -10 m/s2 Mechanics: Implicit Euler resolved by conjugate gradient Run-time: 35-40 Hz in Pentium 4 3.0 GHz, 1GB RAM, GeForce 3 Pulley 13722 triangles Cable with 100 particles Particle Mass 0.7 kg GTAS, Toulouse, Juin 15-16, 2006

  31. Demo: Mechanical Simulation (3) Cable stiffness = 1.38E+6 N/m Gravity = -10 m/s2 Mechanics: Implicit Euler resolved by conjugate gradient Run-time: 35-40 Hz in Pentium 4 3.0 GHz, 1GB RAM, GeForce 3 Pulley 13722 triangles Funnel 11712 triangles Cable with 250 particles Particle Mass 0.3 kg initial velocity = -1.5 m/s Endpoint Masses 10000 kg GTAS, Toulouse, Juin 15-16, 2006

  32. Work in Progress • So far: Proof of concept • Approach works for Rigid Object + Cable • Now, extend it to deformable objects • Extensions: • Deformable Octree for organ (trivial) • Cubic continuous test (just need a bit more CPU power) • Unified treatment of mechanics, collisions and other constraints (present challenge) GTAS, Toulouse, Juin 15-16, 2006

  33. Coming Soon! • Deformable organs being disconnected by cutting-off the interstitial tissues GTAS, Toulouse, Juin 15-16, 2006

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