Five Physics Simulators for Five Physics Simulators for Articulated BodiesArticulated Bodies

Chris Heckerdefinition six, inc.checker@d6.com

PrerequisitesPrerequisites

• 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

TakeawayTakeaway• 2 keykey concepts, vital to understand even if you’re

licensing physics:– degrees of freedomdegrees of freedom, configuration space, etc.– stiffnessstiffness, 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 DomainProblem DomainHighly Redundant IKHighly 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

Solving IK with DynamicsSolving 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 TechniquesI Tried 4 Simulation Techniquesco

ordi

nate

type

integration type

augmentedcoordinates

generalizedcoordinates

explicit integration implicit integration

LagrangeMultipliers

Stiff Springs

Composite RigidBody Method

RecursiveNewton-Euler

• Wacky demo of all 4 running simultaneously...

Obvious Axes of the TechniquesObvious 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)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 ContinuedDOF 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

2DOF - 1DOF = 1DOF

Augmented CoordinatesAugmented 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

f

Generalized CoordinatesGeneralized 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 Augmented vs. Generalized Coordinates, RevisitedCoordinates, 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

StiffnessStiffness• 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 StiffnessHandling 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 ExampleStiffness Example• example: exponential decay

• phase space diagram, position vs. time

• demo of increasing spring constantdy/dx = -y dy/dx = -10y

time

posi

tion

Explicit vs. Implicit IntegratorsExplicit vs. Implicit IntegratorsNon-stiff ProblemNon-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

• 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

Explicit vs. Implicit IntegratorsExplicit vs. Implicit IntegratorsStiff ProblemStiff Problem

Four Simulators In More DetailFour 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 DetailFour Simulators In More Detail Augmented / Explicit Augmented / Explicit

Lagrange MultipliersLagrange 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 DetailFour Simulators In More Detail Augmented / Implicit Augmented / Implicit

Implicit SpringsImplicit 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 DetailFour Simulators In More Detail Generalized / Explicit Generalized / Explicit

Composite Rigid Body MethodComposite 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 DetailFour Simulators In More Detail Generalized / Implicit Generalized / Implicit

Implicit Recursive Newton EulerImplicit 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 / ImplicitGeneralized / ImplicitSome DerivationSome Derivation

Warning: 2 slides of hot and heavy math!Warning: 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 / ImplicitGeneralized / Implicit

Some Derivation (cont.)Some 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 want

F(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(qSolving F(q11’) = 0 can be hard, even impossible!’) = 0 can be hard, even impossible!(but it’s a very well documented impossible problem!)(but it’s a very well documented impossible problem!)

• open problem • solve vs. minimize?

The 5th SimulatorThe 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

SummarySummary

• 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