1. Human interface Section, P&I Lab, Titech Real-time Rigid Body Simulation for Haptic Interactions Based on Contact Volume of Polygonal Objects Shoichi Hasegawa, Makoto Sato Tokyo institute of technology P&I lab human interface section Shoichi Hasegawa, Makoto Sato Tokyo institute of technology P&I lab human interface section

2. Human interface Section, P&I Lab, Titech qTouch the virtual world qUser feels contact force from haptic interface Haptic interaction VirtualReal

3. Human interface Section, P&I Lab, Titech qTouch the virtual world qUser feels contact force from haptic interface qThe touched object receives force from the user. qThe response : Dynamics Haptic interaction VirtualReal

4. Human interface Section, P&I Lab, Titech Video Darumaw

5. Human interface Section, P&I Lab, Titech Contact force Haptic pointer Contact force to feedback user Contact force for dynamics simulation F=mv N=I  v(t+  t)= v(t) + F/m  t  (t+  t)=  (t)  + I -1 N  t

6. Human interface Section, P&I Lab, Titech Contact model qNormal force qPrevent penetration qFriction force (coulomb’s model) qStatic friction lPrevent sliding motion qDynamic friction lProportional to normal force Normal force Friction force |f f | < |  0 f n | |f f | = |  f n |

7. Human interface Section, P&I Lab, Titech qAnalytical method qDavid Baraff SIGGRAPH `89 … Solving constraints(1) (eq. of motion) (normal) (friction) Advantages q qObject motions are stable. Wide time steps are affordable. q qSolves constraints accurately. Completely rigid. Drawbacks q qMuch computation time for one step. O(n 3 ) q qA virtual coupling is needed to connect a haptic interface. q qCoulomb's friction model comes to NP complete problem.

8. Human interface Section, P&I Lab, Titech Solving constraints(2) qPenalty method penetration d ， penetrating velocity d. SpringDamper slide l ， sliding velocity l. SpringDamper Advantages q qVery fast for one step. O(n) q qDirect connection to haptic interfaces. q qCoulomb’s friction model is easily realized. q qIntegration of other models are easy. (e.g. Featherstone’s method) Drawbacks q qStability and rigidity requires small time steps. (Haptic interfaces also need this.) q qTreatment of large contact area makes instability or takes a lot of computation time.

9. Human interface Section, P&I Lab, Titech Problem on large contact area qWhere should we put spring-damper model? ? On the most penetrating point

10. Human interface Section, P&I Lab, Titech Problem on large contact area qWhere should we put spring-damper model? Top view ? On vertices

11. Human interface Section, P&I Lab, Titech Problem on large contact area qWhere should we put spring-damper model? Many points Will works well. But, it will takes much computation time and memory. ?

12. Human interface Section, P&I Lab, Titech Proposal for the problem qIntegrate forces from distributed model for each triangle. Distributed model !

13. Human interface Section, P&I Lab, Titech Steps qFinding Contact force: 1.Find contact point and normal. 2.Find the shape of the contact volume. 3.Integrate forces over the contact area.

14. Human interface Section, P&I Lab, Titech Contact detection qGilbert, Johnson, and Keerthi (GJK) algorithm. qFind closest points of two convex shapes. lA complex shape can be represented by a set of convex shapes. lAfter the contact, GJK can’t find closest points, So… t=t 0 t=t 1 New closest points

15. Human interface Section, P&I Lab, Titech Contact Analysis qContact part = Intersection of two convexes. qD. E. Muller and F.P.Preparata: “Finding the intersection of two convex” (1978) qFor given two convex and a point in the intersection. qFind the intersection.

16. Human interface Section, P&I Lab, Titech Contact Analysis(2) qFinding the intersection of two convex Half space representation

17. Human interface Section, P&I Lab, Titech Contact Analysis(3) qFinding the intersection of two convex(2) Dual transform Vertex of intersection Dual transform Half space representation Quick hull

18. Human interface Section, P&I Lab, Titech Integration of force qPenalty force qDynamic friction force qMaximum static friction force Integrate forces from distributed model for each triangle.

19. Human interface Section, P&I Lab, Titech Static friction force qSpring-damper model for sliding constraint. Distributed model = = Two (translation and rotation) models

20. Human interface Section, P&I Lab, Titech Evaluation qCompare three simulators qProposed lPenalty method ldistributed model. qPoint based lPenalty method lA model on the most penetrating point. qAnalytic lAnalytical method lOpen Dynamics Engine (Smith R. 2000)

21. Human interface Section, P&I Lab, Titech Computation time 3 5 13 051015 0 10 20 30 40 50 60 Average computation time[ms] Number of blocks Proposed simulator Point based method Analytical method

22. Human interface Section, P&I Lab, Titech Stability on normal force qA cube on a floor. qMeasure angular momentum. angular momentum 2m32m3 g=9.8m/s 0.1rad 012 0 1 time[s] Angular momentum [Nms] Proposed method Point based penalty method Analytical method

23. Human interface Section, P&I Lab, Titech Motion of top

24. Human interface Section, P&I Lab, Titech Stick-slip motion qState transition between static and dynamic friction makes stick-slip motion. Spring 400N/m 2kg 0.0642m/s friction force 2.0kg weight wrapped by paper Tension force Friction force Cardboard floor   =0.265  =0.160 spring 400N/m Velocity 0.0642m/s   =0.265  =0.160

25. Human interface Section, P&I Lab, Titech Result 012 -5 0 5 0 0.1 Force[N] Time[s] Position[m] Real world position of object friction force 012 -5 0 5 0 0.1 Force[N] Time[s] Position[m] Proposed simulator 012 -5 0 5 0 0.1 Force[N] Time[s] Position[m] Point based simulator 012 -5 0 5 0 0.1 Force[N] Time[s] Position[m] Analytical simulator position of object friction force position of object friction force position of object friction force

26. Human interface Section, P&I Lab, Titech Demo

27. Human interface Section, P&I Lab, Titech Conclusion qProposed a real-time rigid body simulator for haptic interaction qPenalty method qFast update rate qPointed out a problem on a large contact area qSolved the problem by integrating penalty over the intersection area qFast and accurate simulation was achieved.

28. Human interface Section, P&I Lab, Titech Thank you for listening qSource codes, demos, movies... http://springhead.info

29. Human interface Section, P&I Lab, Titech Dual Transform qFinding the intersection of two convex(2)  Dual Transformation lDual transformation transform a face into a vertex and a vertex into a face. lDual transformation’s dual transformation is original facet. Dual Transformation OO