Download Rigid Body Dynamics - Essential Math for Games Programmers

Rigid Body Dynamics
Jim Van Verth ([email protected])
Rigid Body Dynamics
• Simplest form of physical simulation
• Gets you a good way towards making a
more realistic looking game
• Not that hard, either
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Rigid Body
• Objects we simulate will not deform
• Brick vs. clay
• Fixed model: only change position and
orientation
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Dynamics
• Want to move objects through the game
world in the most realistic manner
possible
• Applying velocity not enough – need
ramp up, ramp down – acceleration
• Same with orientation
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Calculus Review
• Have function y(t)
• Function y'(t) describes how y changes
as t changes (also written dy/dt)
• y'(t) gives slope at time t
y
y(t)
y'(t)
t
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Calculus Review
• Our function is position:
• Derivative is velocity:
• Derivative of velocity is acceleration
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Basic Newtonian Physics
• All objects affected by forces
 Gravity
 Ground (pushing up)
 Other objects pushing against it
• Force determines acceleration (F = ma)
• Acceleration changes velocity (
)
• Velocity changes position (
)
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Basic Newtonian Physics
• Assume acceleration constant, then
• Similarly
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Basic Newtonian Physics
• Key equations
• Note: force is derivative of momentum P
 Remember for later – easier for angular
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Basic Newtonian Physics
• General approach
 Compute all forces on object, add up
 Compute acceleration
• (divide total force by mass)
 Compute new position based on old
position, velocity, acceleration
 Compute new velocity based on old
velocity, acceleration
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Newtonian Physics
• Works fine if acceleration is constant
• Not good if acceleration dependant on
position or velocity – changes over time
step
• E.g. spring force: Fspring = –kx
• E.g. drag force: Fdrag = –mv
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Analytic Solution
• Can try and find an analytic solution
 I.e. a formula for x and v
 In case of simple drag:
 But not always a solution
 Or may want to try different simulation
formulas
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Numeric Solution
• Problem: Physical simulation with force
dependant on position or velocity
• Start at x(0) = x0, v(0) = v0
• Only know:
• Basic solution: Euler’s method
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Euler’s Method
• Idea: we have the derivative (x or v)
• From calculus, know that
• Or, for sufficiently small h:
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Euler’s Method
• Can re-arrange as:
• Gives us next function value in terms of
current value and current derivative
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Final Formulas
• Using Euler’s method with time step h
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What About Orientation?
• Force (F) applies to center of mass* of
object – creates translation
• Torque () applies to offset from center
of mass – creates rotation
• Add up torques just like forces
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Force vs. Torque (cont’d)
• To compute torque, take cross product
of vector r (from CoM to point where
force is applied), and force vector F
• Applies torque ccw around vector
r
F
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Other Angular Equivalents
•
•
•
•
•
Force F vs. torque 
Momentum P vs. angular momentum L
Velocity v vs. angular velocity 
Position x vs. orientation 
Mass m vs. moments of inertia J
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Why L?
• Difficult to compute angular velocity
from angular acceleration
• Compute ang. momentum by integrating
torque
• Compute ang. velocity from momentum
• Since
then
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Moments of Inertia
• Moments of inertia are 3 x 3 matrix, not
single scalar factor (unlike m)
• Many factors because rotation depends
on shape and density
• Describe how object rotates around
various axes
• Not easy to compute
• Change as object changes orientation
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Computing J
• Can use moments of inertia for closest
box or cylinder
• Can use sphere (one factor: 2mr2/5)
• Or, can just consider rotations around
one axis and fake(!) the rest
• With the bottom two you end up with
just one value… can simplify equations
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Computing J
• Alternatively, can compute based on
geometry
• Assume constant density, constant
mass at each vertex
• Solid integral across shape
• See Eberly for more details
 Also at www.geometrictools.com
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Using J in World Space
• Remember,
• J computed in local space, must
transform to world space
• If using rotation matrix , use formula
• If using quaternion, convert to matrix
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Computing New Orientation
• Have matrix  and vector 
• How to integrate?
• Convert  to give change in 
 Change to linear velocity at tips of basis
vectors
 One for each basis gives 3x3 matrix
 Can use Euler's method then
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Computing New Orientation
• Example:
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Computing New Orientation
•   r gives linear velocity at r
• Could do this for each basis vector
• Better way:
 Use symmetric skew matrix to compute
cross products
 Multiply by orientation matrix
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Computing New Orientation
• If have matrix , then
where
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Computing New Orientation
• If have quaternion q, then
where
• See Baraff or Eberly for derivation
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Computing New Orientation
• We can represent wq as matrix
multiplication
where
• Assumes q = (w, x, y, z)
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Angular Formulas
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Reducing Error
• Keep time step as small as possible
• Clamp accelerations, velocities to
maximum values – avoid large forces
• If velocity, acceleration very small, set to
zero (avoids little shifts in position)
• Damping acceleration based on velocity
(i.e. friction) can help
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Improving Performance
• If not moving, don’t simulate
• Only do as much as you have to
• If you can fake it, do so
 objects on ground, don’t bother with gravity
 only rotate around z, don’t bother with J
 simple drag instead of full friction model
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References
• Burden, Richard L. and J. Douglas Faires, Numerical
Analysis, PWS Publishing Company, Boston, MA, 1993.
• Hecker, Chris, “Behind the Screen,” Game Developer,
Miller Freeman, San Francisco, Dec. 1996-Jun. 1997.
• Witken, Andrew, David Baraff, Michael Kass,
SIGGRAPH Course Notes, Physically Based Modelling,
SIGGRAPH 2002.
• Eberly, David, Game Physics, Morgan Kaufmann, 2003.
Essential Math for Games