Five Physics Simulators for Articulated Bodies

1
Five Physics Simulators for Articulated Bodies

• Chris Hecker
• definition six, inc.
• checker_at_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
  youre 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

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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

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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)

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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
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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
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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
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Augmented vs. Generalized Coordinates, Revisited

• augmented coordinates dynamics equations
  constraint equations
• general, modular, plugnplay, breakable
• big (often sparse) systems
• simulating useless DOF, drift
• generalized coordinatesdynamics 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

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Handling Stiffness

• You want to handle as much stiffness as you can!
• gives designers control
• you can always make things softer, thats easy
• its very hard to handle stiffness robustly
• explicit integrator will not handle stiff systems
  without tiny timestep
• thats sometimes used as a definition of
  numerical stiffness!

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Stiffness Example

• example exponential decay
• phase space diagram, position vs. time
• demo of increasing spring constant

position
time
dy/dx -y
dy/dx -10y
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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

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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

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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?

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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

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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. Shampines ode23s adaptive timestep, or
  semi-implicit Euler
• pros simple, modular, general, stiff
• cons inexact, big matrices, needs derivatives
• references Baraff (cloth), Kass, Lander

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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

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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

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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)

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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
• plugnchug 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

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Solving F(q1) 0 can be hard, even
impossible!(but its a very well documented
impossible problem!)

• open problem
• solve vs. minimize?

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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
• 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

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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