Euler Steps and Simple Physics

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Euler Steps and Simple Physics

• We often need the most basic physics
• Position, velocity, force, and acceleration
• Springs
• Limits
• Interpolations
• Simple momentum preservation

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How we describe position

• Vector3 position
• All there really is to it.
• Vector3 position

• Coordinate System?
• Usually described in world coordinates, but could
  be object coordinates

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Scalar Positions

• A position may be scalar if theres only one
  degree of freedom
• As the object moves it can only move in one
  direction.
• Roller coasters, items in a track, etc.

• We have to convert to 3D to display our object
• Position translated to a point in a track system.
  That point is in 3D.

Scalar positions are relatively rare in 3D
graphics systems
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Velocity
Velocity Alternatives

• Velocity Vector
• Velocity is described as a vector
• Vector3 velocity
• Usually means velocity is independent of
  orientation

• Velocity Scalar
• Velocity is a single number
• float velocity
• Usually means velocity is dependent on orientation

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Examples

• Scalar Velocities
• Aircraft
• Automobiles and wheeled vehicles
• Items locked in a track
• Characters
• Velocity Vectors
• Spacecraft
• Ballistic objects
• Items subject to gravity

Velocity vectors are a more general solution in
many applications
TT
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Velocity vs. Position

• Velocity is the derivative of position
• To update the position given the velocity and a
  given amount of time we need to integrate

• Eulers Method
• We update position by adding the velocity times
  the time step duration

This is often called an Euler Step. We are
stepping in time.
Vector3
position velocity (float)gameTime.ElapsedGam
eTime.TotalSeconds
TT
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Scalar Velocities

• Velocity might be a scalar
• float velocity
• You cant add a scalar to a vector we need to
  convert the scalar velocity to a velocity vector

• We had to point the object the right direction
• A transformation matrix must exist.
• We can determine the direction it is going by
  multiplying a vector by this matrix.
• The vector should point onward in the object
  space.

Vector3 velocityDir Vector3.TransformNormal(new
Vector3(0, 0, 1), transform) position
velocityDir velocity (float)gameTime.ElapsedGa
meTime.TotalSeconds
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XNA Transform functions
Vector3.Transform(vector, matrix) Use to
transform a coordinate (point) Vector3.TransformN
ormal(vector, matrix) Use to transform a
vector
Vector3 velocityDir Vector3.TransformNormal(new
Vector3(0, 0, 1), transform) position
velocityDir velocity (float)gameTime.ElapsedGa
meTime.TotalSeconds
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Acceleration

• Acceleration Vector
• Acceleration is described as a vector
• Vector3 acceleration
• Usually means acceleration is independent of
  orientation

• Acceleration Scalar
• Acceleration is a single number
• float acceleration
• Usually means acceleration is dependent on
  orientation

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Common Accelerations

• Thrust
• Gravity
• Drag/Wind resistance
• Braking

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Acceleration

• Acceleration is the derivative of velocity
• To update the velocity based on the current
  acceleration, we need to integrate

• Also uses Euler Steps
• New velocity is old velocity plus acceleration
  times time step

Vector3
velocity acceleration (float)gameTime.Elapsed
GameTime.TotalSeconds
TT
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Scalars vs. Vectors
Once we become a vector, we stay a vector
Which variables need to be persistent? (member
variables)
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Gravity

• Force of gravity is an acceleration vector
• (0, -980, 0) 980cm/sec2
• Clearly, could be different if force is not
  down

• Implementation
• Just add to any other acceleration
• acceleration new Vector3(0, -980, 0)

Be careful, though. Gravity may be canceled if
object is on the ground. Ground is an equal and
opposite force.
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Drag

• Simple Linear Drag
• Called Stokes drag
• Drag is linearly proportional to velocity
• We define a drag coefficient b

• Implementation
• Drag is a force, not an acceleration
• force velocity -drag
• acceleration force / mass

Its common to ignore mass in simple systems.
Just assume the mass is 1 or that the
coefficients are assumed to be divided by the
mass already.
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Force and Mass

• Newtons Second Law
• F ma
• Force equals mass times acceleration
• a F / m
• Again, force may be a scalar or a vector

• Mass is sometimes ignored
• Assume any thrust is measured in cm/sec2
• Step 2 works this way

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Order of Operations
Accumulate Forces
Vector3 force engineDirection engineThrust
velocity
-drag Vector3 acceleration force / mass
new Vector3(0, -980,
0) velocity acceleration
deltaTime position velocity deltaTime
Accumulate Accelerations
Velocity Euler Step
Position Euler Step
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Limits

• Clearly, if sitting on the ground, gravity is
  canceled
• Have to think of how to handle that situation
• Both chairs are subject to gravity, but only one
  will be accelerated

• Something we are touching cancels all
  acceleration and velocity in that direction
• Easy when that direction is (0, -1, 0)

Vector3 acceleration force / mass
new Vector3(0, -980, 0) if (positionlt 0)
acceleration.Y 0
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A More Complete Example
Vector3 force engineDirection
engineThrust velocity -drag
Vector3 acceleration force / mass new
Vector3(0, -980, 0) if (position.Y
lt 0 acceleration.Y lt 0)
acceleration.Y 0 velocity
acceleration deltaTime if
(position.Y lt 0 velocity.Y lt 0)
velocity.Y 0 position
velocity deltaTime if (position.Y
lt 0) position.Y 0
Corrected slide!
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Common interpolations get to point A

• You need to get something to point A in T seconds

• Two things to keep track of
• Where we are going
• How much time is left

Remember, we dont have a loop to compute the
locations. When update is called, we only know
where we are and where we are going and any
another other state we have decided to save.
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Example solution
if (timeRemaining gt 0) float delta
gameTime.ElapsedGameTime.TotalSeconds if
(delta gt timeRemaining) delta
timeRemaining Vector3 direction target -
position float distance direction.Length()
if (distance 0)
timeRemaining 0 else
position direction delta / timeRemaining
timeRemaining - delta
How much to step?
What Direction andhow far?
Take the step.
Consume the time
Works for scalars and vectors. Scalars may be
angles.
TT
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Opening/Closing Actions
float delta gameTime.ElapsedGameTim
e.TotalSeconds if (deployed
wingAngle lt 0.20f)
wingAngle (float)(0.20 delta /
wingDeployTime) if(wingAngle gt
0.20f) wingAngle 0.20f
else if(!deployed
wingAngle gt 0)
wingAngle - (float)(0.20 delta /
wingDeployTime) if(wingAngle lt
0) wingAngle 0

Code from Step 2
The difference is we expect a constant movement
rate rather than a constant time
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General Linear Interpolation

• Going from A to B in T seconds
• Amount of time that has elapsed t
• Where should we be?

Youll use this a lot! A, B, and p can be vectors
or scalars
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Simple Momentum (really velocity) Preservation

• Rock becomes bunch of rocks
• How do we ensure this looks right?

• Cloud of rock should be moving at same velocity
  as original rock
• But, each has its own velocity
• How would you determine the velocity of the cloud?

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Velocity Preservation

• Average the velocity vectors for all objects
• Sum and divide by the count

• Add to every object
• desiredVelocity - currentVelocity

How would real momentum preservation be different
than this?
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Springs

• Springs are a handy thing
• Springs allow us to attach things softly rather
  than rigidly. Well use this to make a chase
  camera

Desired Position

• Spring Parameters
• Stiffness How much force per unit of length
• Damping How much resistance there is to the
  spring collapsing

Current Position
Damping keeps the spring from oscillating
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Spring Equations
Desired Position

• Force is the sum of a positive pull and negative
  damping
• Damping is dependent on velocity

// Calculate spring force
Vector3 stretch desiredPosition - position
Vector3 force stiffness stretch -
damping velocity // Apply
acceleration Vector3 acceleration
force / mass velocity
acceleration elapsed // Apply
velocity position velocity
elapsed
Current Position

• Example

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Simple Collision Detection

• Bounding Sphere
• Minimum sphere that contains an object

• Advantages
• Invariant under rotation
• XNA support

Well examine more complex methods later.
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Physics well do later

• This is the most basic stuff, all linear
  positions and velocities
• Later well do
• Rotation (10x as hard as translation)
• Impact/bounce
• More general mass/spring systems