1. Four Five Physics Simulators for a Human Body Chris Hecker definition six, inc. checker@d6.com

2. Four^H^H^Hive Physics Simulators for a Human Body Chris Hecker definition six, inc. checker@d6.com

3. Prerequisites comfortable with math concepts, modeling, and equations kinematics vs. dynamics familiar with rigid body dynamics probably have written a physics simulator for a game, or at least read about it in detail

4. Takeaway pros and cons & subtleties of 4 different simulation techniques –all are useful, but different strengths 2 key concepts: –degrees of freedom, configuration space, etc. –stiffness, and why it is important all examples are 2D, but generalize directly to 3D not going to be detailed tutorial

5. A Couple Other Related Talks David Wus talk on mixing kinematics & dynamics Saturday, 2:30pm - 3:30pm Experimental Gameplay Workshop Friday, 4:30pm - 6:30pm

6. Before Physics I Tried... Cyclic Coordinate Descent IK demo –my rock climbing game works okay, but problems: –non-physical movement –no closed loops –no clear path to adding muscle controls

7. Physics Solutions rigid bodies with constraints need to simulate enough to make articulated figure 1st-order dynamics f = mv no inertia/momentum; no force, no movement mouse attached by spring or constraint must be tight control hands/feet attached by springs or constraints must stay locked to the positions

8. I Tried Four Techniques Augmented Coordinates / Explicit Integration Lagrange Multipliers Augmented Coordinates / Implicit Integration Implicit Springs Generalized Coordinates / Explicit Integration Composite Rigid Body Method Generalized Coordinates / Implicit Integration Implicit Recursive Newton Euler demo of all four running at once

9. Obvious Axes of the Techniques Augmented vs. Generalized Coordinates –ways of representing the degrees-of-freedom (DOF) of the systems Explicit vs. Implicit Integration –ways of stepping those DOFs forward in time

10. Augmented Coordinates aka. Lagrange Multipliers, constraint methods calculate constraint forces and apply them simulate each body independently constraint forces keep bodies together 6DOF - 2DOF = 4DOF f

11. Generalized Coordinates aka. reduced coordinates, embedded methods, recursive methods calculate and simulate only the real DOF of the system one rigid body and joints 3DOF + 1DOF = 4DOF

12. Degrees Of Freedom (DOF) DOF is a critical concept in all math find the DOF to understand the system coordinates necessary and sufficient to reach every valid state examples: point in 2D: 2DOF, point in 3D: 3DOF 2D rigid body: 3DOF, 3D rigid body: 6DOF point on a line: 1DOF, point on a plane: 2DOF simple desk lamp: 3DOF (or 5DOF counting head)

13. DOF Continued systems have DOF, equations on those DOF constrain them example, 2D point, on line configuration space is the space of the DOF manifold is the c-space, usually viewed as embedded in the original space (x,y) x = 2y (x,y) = (t,2t)

14. Augmented vs. Generalized Coordinates, Revisited augmented coordinates: dynamics equations + constraint equations general, modular, plugnplay, breakable big (often sparse) linear systems simulating useless DOF generalized coordinates: dynamics equations complicated, custom coded small dense nonlinear systems no closed loops, no nonholonomic constraints

15. Stiffness fast-changing systems are stiff the real world is incredibly stiff rigid body is a simplification to avoid stiffness most game UIs are incredibly stiff the mouse is insanely stiff, IK demo kinematically animating objects can be arbitrarily stiff animating the position with no derivative constraints

16. Handling Stiffness You want to handle as much stiffness as you can! gives designers control can always make things softer, thats easy its very hard to handle explicit integrator will not handle stiff systems without tiny timestep thats almost a definition of numerical stiffness! :)

17. Stiffness Example example: exponential decay demo of increasing spring constant dy/dx = -ydy/dx = -10y

18. Explicit vs. Implicit Integrators explicit jumps forward to next position blindly leap based on current information implicit jumps back from next position find a next position that points to current

19. Four Simulators In More Detail Augmented Coordinates / Explicit Integration Lagrange Multipliers Augmented Coordinates / Implicit Integration Implicit Springs Generalized Coordinates / Explicit Integration Composite Rigid Body Method Generalized Coordinates / Implicit Integration Implicit Recursive Newton Euler spend a few slides on this technique best for game humans?

20. Four Simulators In More Detail Augmented / Explicit Lagrange Multipliers form dynamics equations for bodies form constraint equations solve for constraint forces apply forces to bodies integrate bodies forward in time RK explicit integrator pros: simple, modular, general cons: medium sized matrices, drift, nonstiff references: Baraff, Shabana, Barzel & Barr, my ponytail articles

21. Four Simulators In More Detail Augmented / Implicit Implicit Springs form dynamics equations write constraints as stiff springs use implicit integrator to solve for next state Shampines ode23s adaptive timestep, or semi- implicit Euler pros: simple, modular, general, stiff cons: inexact, big matrices, needs derivatives references: Baraff (cloth), Kass, Lander

22. Four Simulators In More Detail Generalized / Explicit Composite Rigid Body Method form tree structured augmented system traverse tree computing dynamics on generalized coordinates incrementally outward and inward iterations integrate state forward RK pros: small matrices, explicit joints cons: dense, nonstiff, not modular references: Featherstone, Mirtich, Balafoutis

23. Four Simulators In More Detail Generalized / Implicit Implicit Recursive Newton Euler form generalized coordinate dynamics differentiate for implicit integrator fully implicit Euler solve system for new state pros: small matrices, explicit joints, stiff cons: dense, not modular, needs derivatives references: Wu, Featherstone

24. Generalized / Implicit Some Derivation f = f joints + f ext = mv Forward Dynamics Algorithm given joint forces, compute velocities (accelerations) v = (f joints + f ext )/m Inverse Dynamics Algorithm given velocities (accelerations), compute joint forces f joints = mv - f ext you can use an IDA to check for equilibrium given a velocity if f joints = 0, then the current velocity balances the external forces, or f - mv = 0 (which is just a rewrite of f = mv)

25. Generalized / Implicit Some Derivation (cont.) IDA gives F(q,q) when F(q,q) = 0, then system is moving correctly we want F(q 1, q 1 ) = 0, the solution at the new time implicit Euler equation: q 1 = q 0 + h q 1 q 1 = q 0 + h q 1... q 1 = (q 1 - q 0 ) / h plugnchug: F(q 0 + h q 1, q 1 ) = 0 this is a function in q 1, because q 0 is known we can use a nonlinear equation solver to solve F for q 1, then use this to step forward with implicit Euler

26. Problem: Solving F(q 1 ) = 0 can be hard! (but its very well documented) open problem solve vs. minimize?

27. The 5th Simulator Current best: implicit Euler with F(q) = 0 Newton solve lots of wacky subdivision and searching to help find solutions –want to avoid adaptivity, but cant in reality doesnt always work, finds no solution, bails Idea: the ode23s adaptive integrator will find the answer, but slowly the Newton solve sometimes cannot find the answer, no matter how slowly because it lacks info spend time optimizing the ode23s, because at least it has more information to go on

28. Summary simulating an articulated rigid body is hard, and there are a lot of tradeoffs and subtleties there is no single perfect algorithm yet? stiffness is very important to handle for most games generalized coordinates with implicit integration is the best bet so far for run-time maybe augmented explicit (?) for author-time tools Ill put the slides on my page at d6.com