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Five Physics Simulators for Articulated Bodies

Five Physics Simulators for Articulated Bodies. Chris Hecker definition six, inc. checker@d6.com. Prerequisites. comfortable with math concepts, modeling, and equations kinematics vs. dynamics familiar with rigid body dynamics

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Five Physics Simulators for Articulated Bodies

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  1. Five Physics Simulators for Articulated Bodies Chris Hecker definition six, inc. checker@d6.com

  2. 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, even a hacky one, • or at least read about it in detail, • or are using a licensed simulator at a low level

  3. Takeaway • 2 key concepts, vital to understand even if you’re licensing physics: • degrees of freedom, configuration space, etc. • stiffness, and why it is important for games • pros and cons & subtleties of 4 different simulation techniques • all are useful, but different strengths • all examples are 2D, but generalize directly to 3D • not going to be detailed physics tutorial

  4. Problem DomainHighly Redundant IK • Simulate a human figure under mouse control • for a game about rock climbing...demo • Before going to physics, I tried... • Cyclic Coordinate Descent (CCD) IK • works okay, simple to code, but problems: • non-physical movement • no closed loops • no clear path to adding muscle controls • gave a GDC talk about CCD problems, slides on d6.com

  5. Solving IK with Dynamics • 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

  6. I Tried 4 Simulation Techniques integration type explicit integration implicit integration augmented coordinates Lagrange Multipliers Stiff Springs coordinate type generalized coordinates Recursive Newton-Euler Composite Rigid Body Method • Wacky demo of all 4 running simultaneously...

  7. 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, and how to deal with system stiffness

  8. 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)

  9. DOF Continued • systems have DOF, equations on those DOF constrain them and subtract DOF • example, 2D point, on line • “configuration space” is the space ofthe DOF • “manifold” is the c-space, usuallyviewed as embedded in theoriginal space (x,y) 2DOF x = 2y (x,y) = (t,2t) 2DOF - 1DOF = 1DOF

  10. Augmented Coordinates • aka. Lagrange Multipliers, constraint methods • simulate each body independently • calculate constraint forces and apply them • constraint forces keep bodies together f 3DOF + 3DOF - 2DOF = 4DOF

  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 q 3DOF + 1DOF = 4DOF

  12. Augmented vs. Generalized Coordinates, Revisited • augmented coordinates: dynamics equations + constraint equations • general, modular, plug’n’play, breakable • big (often sparse) systems • simulating useless DOF, drift • generalized coordinates:dynamics equations • small systems, no redundant DOFs, no drift • complicated, custom coded • dense systems • no closed loops, no nonholonomic constraints

  13. 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 • FPS control is stiff, 3rd person move change, etc. • kinematically animating objects can be arbitrarily stiff • animating the position with no derivative constraints • have keyframes drag around a ragdoll closely

  14. Handling Stiffness • You want to handle as much stiffness as you can! • gives designers control • you can always make things softer, that’s easy • it’s very hard to handle stiffness robustly • explicit integrator will not handle stiff systems without tiny timestep • that’s sometimes used as a definition of numerical stiffness!

  15. Stiffness Example • example: exponential decay • phase space diagram, position vs. time • demo of increasing spring constant position time dy/dx = -y dy/dx = -10y

  16. Explicit vs. Implicit IntegratorsNon-stiff Problem • explicit jumps forward to next position • blindly leap forward based on current information • implicit jumps back from next position • find a next position that points back to current

  17. Explicit vs. Implicit IntegratorsStiff Problem • explicit jumps forward to next position • blindly leap forward based on current information • implicit jumps back from next position • find a next position that points back to current

  18. 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?

  19. Four Simulators In More Detail Augmented / ExplicitLagrange Multipliers • form dynamics equations for bodies • form constraint equations • solve for constraint forces given external forces • the constraint forces are called “Lagrange Multipliers” • apply forces to bodies • integrate bodies forward in time • forward euler, RK explicit integrator, etc. • pros: simple, modular, general • cons: medium sized matrices, drift, nonstiff • references: Baraff, Shabana, Barzel & Barr, my ponytail articles

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

  21. Four Simulators In More Detail Generalized / ExplicitComposite 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

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

  23. Generalized / ImplicitSome DerivationWarning: 2 slides of hot and heavy math! • f = fint + fext = mv • Forward Dynamics Algorithm • given forces, compute velocities (accelerations) • v = m-1(fint + fext) • Inverse Dynamics Algorithm • given velocities (accelerations), compute forces • fint = mv - fext • Insight: you can use an IDA to check for equilibrium given a velocity • if fint = 0, then the current velocity balances the external forces, or f - mv = 0 (which is just a rewrite of “f = mv”)

  24. Generalized / ImplicitSome Derivation (cont.) • IDA computes F(q,q’) (ie. forces given state) • when F(q,q’) = 0, then system is moving correctly • we want to do implicit integration, so we wantF(q1, q1’) = 0, the solution at the new time • implicit Euler equation: q1 = q0 + h q1’ • q1 = q0 + h q1’ ... q1’ = (q1 - q0) / h • plug’n’chug: F(q0 + h q1’, q1’) = 0 • this is a function in q1’, because q0 is known • we can use a nonlinear equation solver to solve F for q1’, then use this to step forward with implicit Euler

  25. Solving F(q1’) = 0 can be hard, even impossible!(but it’s a very well documented impossible problem!) • open problem • solve vs. minimize?

  26. 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 can’t in reality • doesn’t always work, finds no solution, bails • Idea: • an adaptive implicit 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 adaptive integrator, because at least it has more information to go on

  27. 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 • I’ll put the slides on my page at d6.com

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