Rachel Weinstein, Joseph Teran and Ron Fedkiw

Transcription

1 Rachel Weinstein, Joseph Teran and Ron Fedkiw presented by Marco Bernasconi Politecnico di Milano

2 A couple of questions: Why did I choose this paper? What does contact vs collision mean? What are the theoretical bases of articulated rigid bodies? What is an impulse? What was the previous work of Guendelman about?

3 1. Theory behind: Collisions and contacts Time stepping Articulated rigid body math 2. Core idea 3. Examples 4. Conclusions

4 Collisions Bodies bounce off each other (elasticity factor) Motion of bodies changes discontinuously within a discrete time step Before and After states need to be computed Contacts Bodies rest one stuck to the other Bodies slide (with or without friction) one upon the other

5 An example showing collisions and contacts:

6 Traditional approach Update position and velocity Process collision Process contact Simulation loop Guendelman approach Process collision Update velocity Process contact Update position

7 Traditional approach: problem Example: block sliding down inclined plane Initially sliding down Update position and velocity interpenetrating plane Process collision velocity reflected No contact to process Next iteration object bounces SOLUTION: velocity threshold (Mirthic & Canny 1995)

8 Guendelman approach Example: block sliding down inclined plane No collisions to process Update velocity block gains downward velocity Process contact stops normal motion Update position slides down with no bounce

9 An example showing the comparison:

10 Maximal coord. Generalized coord. (x 0, y 0, z 0, 0, f 0, 0) + (x 0, y 0, z 0, 0, f 0, 0) + (x 1, y 1, z 1, 1, f 1, 1) (x 2, y 2, z 2, 2, f 2, 2) + = ( 1, f 1 ) + ( 2 ) = 18 state variables 9 state variables

11 The equations for rigid body evolution are: d x v x = position, v = velocity dt dq dt 1 2 q q = orientation, = angular velocity d v F F= net force, m = mass dt m d L d( I ) dt dt L = angular momentum = net torque

12 1. Theory behind 2. Core idea: Time integration and impulse theory Prestabilization Algorithm 3. Examples 4. Conclusions

13 These equations are time integrated according to: n x 1 x n t v n n q qˆ( t ) q Impulses are found and applied iteratively: 1 n n

14 Goal: apply impulses to the rigid bodies BEFORE the integration step with the intention of achieving the target joint state after that integration step. Position constraint equations: v new v n j m new n I 1 ( r j)

15 The algorithm is applied as follows: Process collisions (and velocity poststabilization) Integrate velocities (and velocity poststabilization) Resolve contacts and articulation prestabilization Update position (and velocity poststabilization)

16 1. Theory behind 2. Core idea 3. Examples Black box definition of joints and constraints Closed loops Stacks of articulated rigid bodies 4. Conclusions

17 There are many kinds of basic joints:

18 An example of joints combination:

19 An evidence of efficiency of closed loops:

20 Another evidence of efficiency of closed loops:

21 Another evidence of efficiency of closed loops:

22 Another evidence of efficiency of closed loops:

23 Another evidence of efficiency of closed loops:

24 First simulation involving large stacks:

25 Second simulation involving large stacks:

26 1. Theory behind 2. Core idea 3. Examples 4. Conclusions

27 Any black box method for joint constraints Linearity both in the number of bodies AND in the number of constraints No special treatments of closed loops Advantage of pre vs post stabilization

28 References: