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  1. 1. Quasi-Rigid Objects in Contact Mark Pauly Dinesh Pai Leo Guibas Stanford University Rutgers University Stanford University
  2. 2. Contacts in Simulation <ul><li>Bio-medical applications: </li></ul><ul><ul><li>surgery simulation </li></ul></ul><ul><ul><li>artifical joints, dental implants </li></ul></ul><ul><li>Mechanical design: </li></ul><ul><ul><li>wear and tear of industrial parts </li></ul></ul><ul><li>Physics-based animation: </li></ul><ul><ul><li>movies </li></ul></ul><ul><ul><li>games </li></ul></ul>
  3. 3. Existing Models <ul><li>Rigid body dynamics </li></ul><ul><ul><li>small number of state variables </li></ul></ul><ul><ul><li>efficient collision detection </li></ul></ul><ul><ul><li>contact sensitivity problem (a stool with hundreds of legs) </li></ul></ul><ul><li>Fully deformable (e.g. FEM, mass-spring) </li></ul><ul><ul><li>accurate modeling of complex materials (elasticity, plasticity) </li></ul></ul><ul><ul><li>too costly for models that hardly deform </li></ul></ul>
  4. 4. Quasi-Rigid Objects <ul><li>Physical model </li></ul><ul><ul><li>point force applied to object only leads to small, local deformation </li></ul></ul><ul><ul><li>analytical system response model to define displacements due to point force </li></ul></ul><ul><ul><li>linear elasticity: Global system response by superposition </li></ul></ul><ul><ul><li>forces and displacements evaluated on surface only </li></ul></ul>
  5. 5. Quasi-Rigid Objects <ul><li>Surface model </li></ul><ul><ul><li>point cloud representation for modeling consistent, highly dynamic contact surface </li></ul></ul>
  6. 6. Physical Model <ul><li>Boussinesq approximation </li></ul><ul><ul><li>infinite elastic half-space </li></ul></ul>Poisson’s ratio shear modulus force at x displacement at y due to force at x
  7. 7. Physical Model <ul><li>Boussinesq approximation </li></ul><ul><ul><li>system response function </li></ul></ul>
  8. 8. Physical Model <ul><li>Linear elasticity </li></ul><ul><ul><li>superposition </li></ul></ul>total displacement at y
  9. 9. Volume Preservation <ul><li>Condition: </li></ul>
  10. 10. Volume Preservation
  11. 11. <ul><li>Approximate system response at discrete nodes (point samples) </li></ul>Discretization force at node j shape function displacement at node i
  12. 12. Discretization system response matrix vector of displacements [u 1 ,...,u N ] T vector of tractions [p 1 ,...,p N ] T matrix coefficient
  13. 13. Contact <ul><li>Collision detection </li></ul><ul><ul><li>static bounding volume hierarchies (small deformations) </li></ul></ul><ul><li>Contact resolution </li></ul><ul><ul><li>compute forces and displacements that resolve contact </li></ul></ul><ul><li>Contact surface </li></ul><ul><ul><li>find contact surface that is consistent for both models </li></ul></ul>
  14. 14. Contact Resolution <ul><li>Collision detection determines points that potentially experience displacements (active nodes) </li></ul><ul><li>find corresponding point for each active node </li></ul>active nodes corresponding nodes
  15. 15. Contact Resolution <ul><li>Separation of active nodes </li></ul><ul><ul><li>initial separation </li></ul></ul><ul><ul><li>final separation </li></ul></ul>
  16. 16. Contact Resolution <ul><li>Condition for contact resolution: </li></ul><ul><ul><li>non-negative separation: s i ≥ 0 </li></ul></ul><ul><ul><li>non-negative force: p i ≥ 0 </li></ul></ul>
  17. 17. <ul><li>Linear Complementarity Problem (LCP) </li></ul><ul><li>solved using Lemke’s method </li></ul>Contact Resolution
  18. 18. Contact Surface <ul><li>Consistent conforming contact surface </li></ul><ul><li>Adaptive moving least squares (MLS) approximation requires no re-meshing or zippering </li></ul>
  19. 19. Simulation <ul><li>Treat objects as rigid while in free motion </li></ul><ul><li>Integrate contact forces to compute total wrench </li></ul>
  20. 20. Example <ul><li>Model acquisition </li></ul><ul><ul><li>laser-range scan </li></ul></ul><ul><li>Hierarchy construction </li></ul><ul><ul><li>recursive clustering </li></ul></ul><ul><ul><li>efficient multi-level computation </li></ul></ul>
  21. 21. Example <ul><li>Simulation </li></ul>
  22. 22. Example <ul><li>Validation </li></ul>Measurement Simulation X2 FootSensor (xSensor Corp.) 37 x 13 sensors, 1.94 sensors/cm2
  23. 23. Bio-medical Applications <ul><li>Simulate friction effects to predict attrition </li></ul><ul><li> design of artificial joints </li></ul>
  24. 24. Computer Animation <ul><li>Quasi-rigid body simulation </li></ul>
  25. 25. Computer Animation <ul><li>Explicit representation of contact surface allows accurate simulation of friction effects </li></ul>
  26. 26. Computer Animation <ul><li>Explicit representation of contact surface allows accurate simulation of friction effects </li></ul>
  27. 27. Conclusion <ul><li>Quasi-rigid objects bridge the gap between rigid bodies and fully deformable models </li></ul><ul><li>Simple and efficient model for contact resolution </li></ul><ul><li>Limitations: </li></ul><ul><ul><li>small deformations </li></ul></ul><ul><ul><li>linear elasticity </li></ul></ul><ul><ul><li>sharp corners </li></ul></ul>
  28. 28. Future Work <ul><li>Coupling with low-resolution FEM model </li></ul><ul><li>Acquired system response functions </li></ul><ul><li>Incorporate friction into LCP </li></ul><ul><li>Application: Contact simulation in knee joint </li></ul>
  29. 29. Acknowledgements <ul><li>NSF grants CARGO-0138456, ITR-0205671, IIS-0308157, EIA-0215887, ARO grant DAAD19-03-1-0331 </li></ul><ul><li>Anonymous reviewers </li></ul>