Animation CS 551 / 651

1. AnimationCS 551 / 651 Rigid-body SimulationMirtich Paper Mirtich, 2000

2. Timewarp Rigid Body Simulation • Presented at SIGGRAPH 2000 • Brian Mirtich • MERL, A Mitsubishi Electric Research Lab (Cambridge, MA)

3. What stages are in a simulation? • Integration • Collision checking (impacts) • Force computation • Constraints (non penetration) • Control • What is this?

4. How much time do they take? • What’s so slow? • Why do multibodies exacerbate this?

5. Big Idea • Simulations are slow • What part is so slow??? • Simulations are discontinuous • What’s the source of discontinuities? • Collisions, contacts, rolling sliding, etc. • But not discontinuities are unpredictable and only affect parts of simulation

6. A simple game of Pong • What goes on in this sim? • Integration • Collision checking • Collision resolution • Would this scale well if balls didn’t collide with each other?

7. General rigid body simulator • Nontrivial geometries • All pairs can potentially collide • Second-order physics equations of motion • Could be deployed on multiprocessor (simulation farm)

8. Discontinuities • When do they occur? • When the state or equations of motion change • Collisions • New contacts • Changing from rolling to sliding dynamics

9. Discontinuities • From §2.1 Root finding method localizes the exact moment of a discontinuity • What is a root? • How do we find the roots?

10. Newton-Raphson • Find the zero of an equation (where it crosses the x-axis) • If pn is close to pn-1then return pn • else set pn-1 = pn and repeat

11. Newton-Raphson • Why set • y-intercept equation of line: y = mx + b • let y-intercept, b = f (pn-1) • let slope, m = f’(pn-1) • we want to find x-value where y-value = 0 • let y = 0 • 0 = f’(pn-1) * x + f(pn-1) • solve for x: -f(pn-1) / f’(pn-1)

12. Newton-Raphson • Be aware that Newton-Raphson could set pn to a value not defined by curve • f’(pn-1) could be 0 (cannot divide by 0)

13. Discontinuities • What trouble do they cause? • Integrator must be stopped and restarted with new state, forces, or equations • Main control loop constantly checking for discontinuities so appropriate handling takes place • Two common methods • Retroactive detection and conservative advancement

14. A general simulation problem • §3 references DES, Discrete Event Simulations • Much more than graphics simulations encounter these challenges • Many simulations are represented as DES and thus this technology is relevant

15. Retroactive Detection • Catch discontinuities after the happen • Detect when two objects go through one another • Use root finding to detect exact time of impact • Roll back the clock to time of impact • All objects or just some? • Continue simulation only up to impact

16. Retroactive Detection • What’s bad about this algorithm? • You’re repeating work • Simulate, back up, simulate again • Additional collisions may be encountered during subsequent smaller simulation timesteps (and others won’t take place after all) • Synchronization requires repeated simulation of all objects if they are involved in collision or not • Doesn’t scale well in number of discontinuities

17. Conservative Advancement • Don’t simulate past discontinuities • Heuristics predict future discontinuities and impending events are stored in a priority queue • Upon approaching time of event in priority queue, take small steps

18. Conservative Advancement • What’s bad about this method? • Conservative, but slow as number of discontinuities increases • All simulated objects move forward at small timestep • Difficult to compute tight bounds on discontinuities

19. Timewarp Algorithm • Derived from a 1985 paper by David Jefferson • Be optimistic • Each object simulates as quickly as possible • Be responsive • If an object is informed of an event in the past, it must reset its state to that previous time

20. What is STATE? • What is the state in the rigid body sims?

21. Timewarp • Multiple processes • Each models part of the system • Each has its own “time” called Local Virtual Time (LVT) • Process clocks are not synchronized

22. Timewarp • Message Passing • When a process experiences an event • message is sent to all processes with timestamp = LVT of sending process • When a process receives a message • If (message time > recipient’s LVT) • Message is placed on time-based queue • Process will advance clock to next event on queue • What if (message time < recipient’s LVT)?

23. Rollback • Occurs when a recipient has simulated “past” the time at which another process has generated an event • Go back in time (how?) • Put all processed events back on queue • “unsend” events sent to other processes • For each message sent, an antimessage is stored on a queue that can be easily flushed

24. Rollback • Overhead • Keeping track of previous states • Keeping track of processed events • Keeping track of submitted events • Global Virtual Time (GVT) • The minimum of all LVTs and safe times (why?) • All states before GVT can be purged • Monotonically increasing

25. Timewarp for rigid body sim • Every simulated body is a process • The state = position & velocity • Global event queue stores all bodies’ events • Event = timestamp and list of affected bodies • Main loop • Pull front item from event queue • Simulate affected bodies to that time • This can be put on a uniprocessor

26. Types of events • Collision Check • Between pairs of bodies • Group Check • Collision check for contacting bodies • Splitting groups of bodies when no longer contacting • Redraw • At every new rendered frame, copy positions to buffer • Callback • User-defined functions to be executed

27. Collisions • Safe-time • Each body keeps track of the last time it was known not to be in collision with other bodies • This helps limit how far a body must roll back upon detection of a collision • Safe-time is included in collision/group check events • Upon collision checking, safe-time is updated

28. Collisions (A collides with B) • Vertical connections join post-collision states • Gray states are new states after A-B collision check event

29. Collision (between A and B)

30. Multibodies • Jointed (articulated) bodies like a human • All bodies must be treated in unison • Easily integrated into their system

31. Contact Groups • Bodies in persistent contact • Simulated as a unit • Membership of group is dynamic • Complicated state keeping so, upon rollback, a body can be detached from its group and simulated

32. Collision checks • Active Pairs • The set of all bodies that must be checked for potential collisions • We want to keep this set small

33. Active Pairs • Looking for collisions • Each body has an axis-aligned bounding box • For each timestep between GVT and LVT • Record location of bounding box • Compute bounding box (swept volume) of all bounding boxes over time • Using six heaps for max/min {x, y, z} • GVT increment causes volume to shrink

34. Looking for collisions • If swept volumes do not intersect • Collision free between [GVT, tmin (A, B)] • Depending on LVT of A and B • A and B not an active pair

35. Looking for collisions • If newest timestep for B causes overlapping swept volume with A • Collision check at time tmin (A, B) • B, before integration (last known free time) • Rollback will resolve collisions

36. Results • Applications • Atoms: 200 spheres and 100 water molecules in divided box • Cars: 4 multibody vehicles and 400 pendula • Robots: 20 eight-link manipulators throwing blocks • Avalanche: 300 rocks tumbling down mountainside

37. Results • Note average integration step vs. time between discontinuities • Look at cars vs. robots

38. Results • Atoms for 2 seconds • Average integration step • Conservative advancement slows down

39. Results • Atoms for 2 seconds • Total “work done” per simulated object • Retroactive detection repeats a lot of work

40. Results • Total time • 100 atoms • RD-1/30 = 0.142, TW = 0.147, CA = 1.15 • 200 atoms • RD -1/30 = 2.61, TW = .388, CA = 4.74

41. Discussion • §4.2 – If rollback time is between safe time and current time, nothing is done… • Jefferson proved this was what? • Figure 2 • No global state