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Five Physics Simulators for Articulated Bodies. Chris Hecker definition six, inc. 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 l.jpg

Five Physics Simulators for Articulated Bodies

Chris Hecker

definition six, inc.

Prerequisites l.jpg


  • 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

Takeaway l.jpg


  • 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

Problem domain highly redundant ik l.jpg

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

Solving ik with dynamics l.jpg

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

  • I tried 4 simulation techniques l.jpg

    I Tried 4 Simulation Techniques

    integration type

    explicit integration

    implicit integration





    Stiff Springs

    coordinate type





    Composite Rigid

    Body Method

    • Wacky demo of all 4 running simultaneously...

    Obvious axes of the techniques l.jpg

    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

    Degrees of freedom dof l.jpg

    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)

  • Dof continued l.jpg

    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)


    x = 2y

    (x,y) = (t,2t)

    2DOF - 1DOF = 1DOF

    Augmented coordinates l.jpg

    Augmented Coordinates

    • aka. Lagrange Multipliers, constraint methods

    • simulate each body independently

    • calculate constraint forces and apply them

      • constraint forces keep bodies together


    3DOF + 3DOF - 2DOF = 4DOF

    Generalized coordinates l.jpg

    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

    Augmented vs generalized coordinates revisited l.jpg

    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

  • Stiffness l.jpg


    • 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

  • Handling stiffness l.jpg

    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!

  • Stiffness example l.jpg

    Stiffness Example

    • example: exponential decay

      • phase space diagram, position vs. time

  • demo of increasing spring constant

  • position


    dy/dx = -y

    dy/dx = -10y

    Explicit vs implicit integrators non stiff problem l.jpg

    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

  • Slide17 l.jpg

    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

  • Four simulators in more detail l.jpg

    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?

  • Four simulators in more detail augmented explicit lagrange multipliers l.jpg

    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

  • Four simulators in more detail augmented implicit implicit springs l.jpg

    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

  • Four simulators in more detail generalized explicit composite rigid body method l.jpg

    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

  • Four simulators in more detail generalized implicit implicit recursive newton euler l.jpg

    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

  • Generalized implicit some derivation warning 2 slides of hot and heavy math l.jpg

    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”)

  • Generalized implicit some derivation cont l.jpg

    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

  • Solving f q 1 0 can be hard even impossible but it s a very well documented impossible problem l.jpg

    Solving F(q1’) = 0 can be hard, even impossible!(but it’s a very well documented impossible problem!)

    • open problem

    • solve vs. minimize?

    The 5th simulator l.jpg

    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

  • Summary l.jpg


    • 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

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