Phun with Physics

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Phun with Physics

• The basic ideas
• Vector calculus
• Mass, acceleration, position,

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Some defs

• Kinematics
• The status of an object
• Position, orientation, acceleration, speed
• Describes the motion of objects without
  considering factors that cause or affect the
  motion
• Dynamics
• The effects of forces on the motion of objects.

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The basics

• Let p(t) be the position in time.
• Well drop the (t) and just say p
• Other values
• v velocity
• a - acceleration

Velocity is the derivative of position Acceleratio
n is the derivative of velocity
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Vector calculus

• Really, p(t) is a triple, right?
• Think of these equations as three equations, one
  for each dimension
• This is vector calculus

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Newtons law and Momentum

• We know this one
• Momentum

Note Force is the derivative of momentum
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What about real objects?

• Think of a real object as a bunch of points, each
  with a momentum
• We can find a center of mass for the object and
  treat the object as a point with the total mass
• We have
• Mass, position vector, acceleration vector,
  velocity vector

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Example Air Resistance

• The resistance of air is proportional to the
  velocity
• F -kv
• We know F ma, so ma -kv

So, how can we solve?
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Eulers method
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What about orientation?

• Well start in 2D
• Let W be the orientation (angle)
• Easy in 2D, not so easy in 3D well be back
• w is the angular velocity around center of
  gravity
• a is the angular acceleration
• Then

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Whats the velocity at a point?
Radians are important, here
Perpendicular to cp Same length! Perpendicularized
radius vector
p
Chasles Theorem Velocity at any point is the
sum of linear and rotational components.
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Angular momentum

• Angular momentum
• At a point?
• For the whole thing?
• Mmv
• Momentum equals mass times velocity
• What does this mean?

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We want to know how much of the momentum is
around the center
Angular Momentum of a point around c
rcp
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Torque Angular force
Torque at a point
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Total angular momentum and moment of inertia
I is the moment of inertia for the object.
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Moment of inertia
What are the characteristics? What does large vs.
small mean? How to we get this value?
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Relation of torque and moment of inertia
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Simple dynamics

• Calculate/define center of mass (CM) and moment
  of inertia (I)
• Set initial position, orientation, linear, and
  angular velocities
• Determine all forces on the body
• Linear acceleration is sum of forces divided by
  mass
• Angular acceleration is sum of torques divided by
  I
• Numerically integrate to update position,
  orientation, and velocities

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Object collisions

• Imagine objects A and B colliding

B
Assume collision is point on plane
A
n
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What happens?

• ???

B
A
n
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Our pal Newton

• Newtons law of restitution for instantaneous
  collisions with no friction
• e is a coefficient of restitution
• 0 is total plastic condition, all energy absorbed
• 1 is total elastic condition, all energy reflected

Whats the consequence of no friction?
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Impulse felt by each object

• Newtons third law equal and opposite forces
• Force on A is jn (n is normal, j is amount of
  force)
• Force on B is jn
• So

B
Well need to know j
A
No rotation for now
n
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Solving for j
Then plug j into
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What about rotation?
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How to determine the time of collision?

• What do I mean?
• Ideas?
• Avoid tunneling objects that move through
  each other in a time step

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Eulers method
We can write
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The problem with Eulers method
Taylor series
Were throwing this away!
Error is O(Dt2)
Half step size cuts error to ¼!
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Improving accuracy

• Supposed we know the second derivative?

But, we would like to achieve this error without
computing the second derivative. We can do this
using the Midpoint Method
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Midpoint Method

• 1. Compute an Euler step
• 2. Evaluate f at the midpoint
• 3. Take a step using the midpoint value.

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Example
Step 1
Step 2
Step 3
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Moving to 3D

• Some things remain the same
• Position, velocity, acceleration
• Just make them 3D instead of 2D
• The killer Orientation
• Its possible to prove that no three-scaler
  parameterization of 3D orientation exists that
  doesnt suck, forsome suitably mathematically
  rigorous definition of suck. Chris Hecker

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Orientation options

• Quaternions
• Well use later
• Matricies
• Well use this here.

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Variables

• x(t)
• Spatial position in time (3D)
• Center of mass at time t
• Assume object has center of mass at (0,0,0)
• v(t)
• Velocity in space
• q(t)
• Orientation in time (quaternion)

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Angular velocity

• w(t)
• Angular velocity at any point in time
• This is a rotation rate times a rotation vector

Strange, huh?
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Note

• Angular velocity is instantaneous
• Well compute it later in equations, but we wont
  keep it around.
• Angular velocity is not necessarily constant for
  a spinning object!
• Interesting! Can you visualize why?

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Angular velocity and vectors

• Rate of change of a vector is
• This is why they like that notation
• Change is orthogonal to the normal and vector
• Magnitude of change is vector length times
  magnitude of w

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Derivative of the rotation quaternion

• Woa

Normalize this sucker and we can take a Newton
step. Be sure to scale w(t) by Dt.
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Velocity of a point (think vertex)

• Not just v(t), must include rotation!!!

Position of point at time t
Just a reminder
Velocity of point at time t
And the magic
Angular part
Linear part
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Force and torque

• Fi(t)
• Force on particle i at time t
• Vector, of course

Torque on point i
Total Force
Total Torque
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Linear Momentum
Momentum
And, the derivative of momentum is force
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Angular Momentum

• Angular momentum is preserved if no torque is
  applied.
• L(t) is the angular momentum

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Inertia Tensor
All that work typing this sucker in and its
pretty much useless. Orientation dependent!
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Inertia Tensor for a base orientation
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Base inertia tensor to current inertia tensor
Relation of angular momentum and inertia tensor
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Bringing it all together

• Current state

Position
Orientation
Linear momentum
Angular momentum
Velocity (may itself be updated)
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3D Collisions

• There are now two possible types of collisions
• Not just vertex to face

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3D collisions

• Vertex to face
• Edge to edge

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We need

• 3D collision detection
• Well do that later
• Determine when the collision occurred
• Binary search for the time
• Bisection technique

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Velocity at a point
Position (center of gravity)
Position of point
Angular velocity
Velocity (linear)
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Relative velocity
Normal to object b Vertex to face normal for
face Edge to edge cross product of edge
directions
Positive vrel means moving apart, ignore
it Negative vrel means interpenetrating, process
it What does zero vrel mean?
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New velocities for a and b
Vector from center of mass to point
Let
Normal
Velocity (linear) update
Mass
Angular velocity update
Inertia tensor inverse
Equivalent equations for b
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And j
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Contrast 2D/3D
2D
3D
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Questions?

• 2D had two different js
• Linear velocity and angular velocity
• 3D gets by with only one.
• Why is this?

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Other issues

• Resting contact
• Bodies in contact, but e lt vrel lt e
• Must deal with motion transfer