1. q qWhere should we put spring damper models? q qPut on the most penetrating point? q qIntegrate penetration over the contact area q qFind collision normal and common point Gilbert, Johnson, and Keerthi (GJK) algorithm Real-time Rigid Body Simulation Based on Volumetric Penalty Method Real-time Rigid Body Simulation Based on Volumetric Penalty Method Shoichi Hasegawa, Nobuaki Fujii, Yasuharu Koike, Makoto Sato Precision and Intelligence Laboratory, Tokyo Institute of Technology http://sklab-www.pi.titech.ac.jp hase@hi.pi.titech.ac.jp qGoal q qReal-time rigid body simulator 200Hz or faster update rate for Haptic. q qFor natural virtual object manipulations. q qHaptic interfaces have been developed. It requires fast update. q qObjects should move under the law of motion. q qFor simple virtual world for educational / training / entertainment applications. qChoice of algorithm for contact solver q qD. Baraff: “Analytical methods for dynamic simulation of non-penetrating rigid bodies” (1989) q qB. Mirtich: “Impulse-based Simulation of Rigid Bodies” (1995) q qPenalty methods, which convert constraints into penalty force by spring and damper model. H. Keller: ”Virtual Mechanics” (1993) … qInvention q qPrevious penalty method regards contacts occurs at a point. We regards contact area and integrate forces: 1 Contact detection Unstable Stable q qFind penetrating part = Intersection of two convexes 2 Contact Analysis Dual transform Vertex of intersection Dual transform Half space representation qProposed simulator q qThe simulation step is consists from following procedure Normal forces q qWhere should we put Coulomb ‘s friction model? q qPut on the application point of the normal force? q qIntegrate forces from Coulomb model over the contact area Friction forces Contact area （ current ） Friction force Can’t calculate moment of friction force Contact area(previous) Friction force Contact area(previous) Contact area （ current ） q qThe intersection is convex polyhedron We integrate penalty by each face. 3 Integrate normal forces and torques h0h0 h 0 +h b h 0 +h a a b o h 0 +h b h 0 +h a p0p0 h0h0 n (collision normal) ＝ ー Upper boundLower bound Intersecting part (F s : force from spring models, M s : torque from spring models) D. E. Muller and F.P.Preparata 1978 Convex hull

2. q qDynamic friction   Integrate dynamic friction force ( f dy ) over the contact area q qStatic friction q qObject does not slip ＝ Constraint → Convert into penalty force. q qState Transition 4 Friction forces and torques qProposed simulator (continue) vpvp f dy Spring-damper model for rotation Spring-damper model for translation Object at previous step Object at current step Spring-damper model Distributed models can be replaced by two models Static frictionDynamic friction Static friction force > maximum friction force (=    Dynamic friction force) Static friction force < dynamic friction force qEvaluation q qCompare stability on two simulators q qProposed: Integrate forces over the intersection. q qSimple: Spring model is put on the most penetrating point. q qSimulate same virtual world: Measure the angular momentum 2m×2m×2m 1kg gravity:9.8m/s 0.1rad Angular Momentum [Nm] 0100200 -0.1 -0.05 0 0.05 Number of step [10ms] Simple method -0.05 0100200 -0.1 0 0.05 Proposed method Number of step [10ms] Angular Momentum [Nm] q qResult qHaptic interaction q qChoose a virtual object as a proxy object of haptic interface Display force and torque Set position and orientation qConclusion q qWe proposed a rigid body motion simulator which can: q qRun at haptic rate (200Hz or faster) with simple virtual world. q qTreat dynamic and friction forces. Haptic interface(SPIDAR)