Point-based Surface Modeling - Computer Graphics at Stanford
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Quasi-Rigid Objects in Contact Mark Pauly Dinesh Pai Leo Guibas Stanford University Rutgers University Stanford University Contacts in Simulation • Bio-medical applications: • surgery simulation • artifical joints, dental implants • Mechanical design: • wear and tear of industrial parts • Physics-based animation: • movies • games Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Existing Models • Rigid body dynamics • small number of state variables • efficient collision detection • contact sensitivity problem (a stool with hundreds of legs) • Fully deformable (e.g. FEM, mass-spring) • accurate modeling of complex materials (elasticity, plasticity) • too costly for models that hardly deform Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Quasi-Rigid Objects • Physical model • point force applied to object only leads to small, local deformation • analytical system response model to define displacements due to point force • linear elasticity: Global system response by superposition • forces and displacements evaluated on surface only Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Quasi-Rigid Objects • Surface model • point cloud representation for modeling consistent, highly dynamic contact surface Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Physical Model • Boussinesq approximation • infinite elastic half-space Poisson’s ratio force at x 1 p( x ) u( x, y) 2G x y displacement at y due to force at x shear modulus Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Physical Model • Boussinesq approximation • system response function 1 1 f (r ) 2G r r xy Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Physical Model • Linear elasticity • superposition 1 p( x ) u( y) dx 2G S x y total displacement at y Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Volume Preservation • Condition: f (r )rdr 0 0 Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Volume Preservation Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Discretization • Approximate system response at discrete nodes (point samples) shape function N p(x ) p j j ( x ) j 1 displacement at node i force at node j j (x) 1 ui pj dx 2G j x qi S Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Discretization system response matrix u Rp vector of displacements [u1,...,uN]T vector of tractions [p1,...,pN]T 1 j (x ) Rij dx 2G S qi x matrix coefficient Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Contact • Collision detection • static bounding volume hierarchies (small deformations) • Contact resolution • compute forces and displacements that resolve contact • Contact surface • find contact surface that is consistent for both models Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Contact Resolution • Collision detection determines points that potentially experience displacements (active nodes) active nodes corresponding nodes • find corresponding point for each active node Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Contact Resolution • Separation of active nodes • initial separation A B s~i qi qi • final separation si ui ui qi qi A Mark Pauly B A Quasi-Rigid Objects in Contact B SCA 04 Contact Resolution • Condition for contact resolution: • non-negative separation: si ≥ 0 • non-negative force: pi ≥ 0 Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Contact Resolution • Linear Complementarity Problem (LCP) s Rp q s0 p0 sT p 0 • solved using Lemke’s method Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Contact Surface • Consistent conforming contact surface • Adaptive moving least squares (MLS) approximation requires no re-meshing or zippering Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Simulation • Treat objects as rigid while in free motion • Integrate contact forces to compute total wrench Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Example • Model acquisition • laser-range scan • Hierarchy construction • recursive clustering • efficient multi-level computation Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Example • Simulation Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Example • Validation Measurement Simulation X2 FootSensor (xSensor Corp.) 37 x 13 sensors, 1.94 sensors/cm2 Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Bio-medical Applications • Simulate friction effects to predict attrition design of artificial joints Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Computer Animation • Quasi-rigid body simulation Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Computer Animation • Explicit representation of contact surface allows accurate simulation of friction effects Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Computer Animation • Explicit representation of contact surface allows accurate simulation of friction effects Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Conclusion • Quasi-rigid objects bridge the gap between rigid bodies and fully deformable models • Simple and efficient model for contact resolution • Limitations: • small deformations • linear elasticity • sharp corners Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Future Work • Coupling with low-resolution FEM model • Acquired system response functions • Incorporate friction into LCP • Application: Contact simulation in knee joint Mark Pauly Quasi-Rigid Objects in Contact SCA 04 Acknowledgements • NSF grants CARGO-0138456, ITR-0205671, IIS-0308157, EIA-0215887, ARO grant DAAD19-03-1-0331 • Anonymous reviewers Mark Pauly Quasi-Rigid Objects in Contact SCA 04
