Transcript Game Physics

Game Physics
Chris Miles
The Goal

To learn how to create game objects with
realistic physics models

To learn how to simulate aspects of reality
in order to make them behave more
realistically
Overview
Newton’s laws
 Data types
 Particle implementation
 Rigid body implementation
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Newton’s Laws of Motion
1.
Inertia
2.
F = ma
3.
Equal and opposite
How They Apply
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Velocity remains constant unless acted on
by a force.

Objects accelerate via f = ma
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Interactions between objects should affect
both similarly
How They Don’t Apply
1.
Velocities that reduce over time give
better numerical stability
2.
Objects are often moved instantaneously
3.
Many interactions are one-sided, there is
no reason to simulate the other side.
An Object – Rigid Body
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An object that moves but does not
change shape
“Mass properties” define how these
objects are affected by forces
Mass
2. Moment of inertia
3. Center of gravity
1.
Simulation

We want to write a simulation describing the
motion of the object

look at the universe after 1 second, 2 seconds,
so on and so on.

integrate the equations of motion between those
steps
Data Types
Scalar
 Vector
 Matrix
 Quaternion
 Euler Angles
 Rotation Matrix
 Tensor
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Scalars
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Scalars are a single number
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They represent simple characteristics,
such as mass
Vectors
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[2,5]
[1,5,17]
[-.14,5,-7]
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Vectors usually represent a location, both
in world and local space
Quaternion
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Quaternion’s contain an orientation
Explaining how they work is long and fruitless
Know that they hold an orientation in some
undecipherable way
Can quickly rotate vectors, cumulate easily, and
interpolate smoothly
Used by virtually every modern game engine
(Laura Croft gets points for pioneering)
Particles

First we will look at particles, which are a
reduced form of Rigid Bodies

Particles differ in that they do not have
orientation, they just have positions
Euler angles

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What quaternions were developed to replace
Vector containing rotations around x,y,z
Suffers from gimbal lock - makes it impossible to
go from some orientations to others in a single
step
Interpolates poorly - very ugly animations /
camera movement
Understandable
Rotation matrixes
Another alternative to quaternions
 Too large, requires 9 variables vs. 4
 Numerically unstable – unnormalizes over
time
 Slower

Tensor
Don’t worry about them, math term
 Generalization of scalars/vectors/matrix’s
 Scalars are rank 0 tensor, vectors 1,
matrix’s 2
 Concerned with rank 2 tensors, for
moment of inertia
 Can also accurately simulate drag
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Particle Class Definition
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Class Particle
–
–
–
–
–
–
vector position
vector velocity
vector resultantForce
scalar mass
Function Integrate( dtime )
Called every time step to update position
Function Integrate( dtime )
//dtime = how much time has passed in this time step
position += velocity * dtime
velocity += force / mass
force = 0
Improvements


Violate law #1, have velocities slowly decrease
– greatly increases numerical stability
Original:
–
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New:
–
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velocity += force / mass * dtime
velocity += (force/mass – velocity*cf)* dtime
cf should be a coefficient of on the order of .999
or so
Implicit Integration
We are doing explicit (forward) euler
integration so far in our modeling.
 If stability is an issue you can use implicit
integration


http://www.mech.gla.ac.uk/~peterg/software/MT
T/examples/Simulation_rep/node89.html is a
good overview
Runge-Kutta
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Precise integration technique
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Sample several times for each time step.
t = 0, t = .25, t = .5, t = .75 to find value at t = 1
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Provides very accurate results
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http://mathworld.wolfram.com/Runge-KuttaMethod.html
Rigid Body
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Position and Orientation
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Linear and Angular movements are
independent and analagous
Rigid Body Class Definition

Class RigidBody
–
vector
vector
vector
scalar
position
velocity
resultantForce
mass
–
quaternion
vector
vector
tensor
orientation
rotationalVelocity
resultantTorque
momentOfInertia
–
Function Integrate( dtime )
–
–
–
–
–
–
Moments of Inertia

The moment of inertia is the rotational equivalent to
mass, it describes how hard an object is to rotate
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Depends on the axis of rotation so it is a tensor,
mapping from a direction to a magnitude
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3x3 matrix
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Just using a simple scalar in this example7
Calculating Moments of Inertia
Producing the values that go inside the
moment of inertia is non trivial.
 Assign point masses to the object,
representing where things are distributed
 Then Σmr2
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Ideal Integrate function
position += velocity * dtime
velocity += force / mass * dtime
force = 0
orientation += rotationalVelocity * dtime
rotationalVelocity += torque / momentOfInertia *
dtime
torque = 0
Complications
Quaternion operators are odd so to add
rotationalVelocity
 from
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–
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orientation += rotationalVelocity * dtime
to
–
o += (rv * o) / 2.0f * dtime;
Point Forces
To calculate the torque caused by point
forces
 Use the vector cross product
 force X position = torque
 Where position is the vector from the
center of gravity to the point
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