Game Physics

1
Game Physics Part I

• Dan Fleck

2
Rigid Body Dynamics

• Kinematics is the study of movement over time
• Dynamics is the study of force and masses
  interacting to cause movement over time (aka
  kinematic changes).
• Example
• How far a ball travels in 10 seconds at 50mph is
  kinematics
• How far the same ball travels when hit by a bat
  and under the force of gravity is dynamics
• Additionally for simplification were going to
  model rigid bodies ones that do not deform (not
  squishy)
• We can model articulated rigid bodies multiple
  limbs connected with a joint

3
Bring on calculus

• Calculus was invented by Newton (and Leibniz) to
  handle these problems
• Newtons Laws
• 1. An object at rest stays at rest and an object
  in uniform motion stays in the same motions
  unless acted upon by outside forces (conservation
  of inertia)
• 2. Force Mass Acceleration
• 3. For every action there is an equal and
  opposite reaction

4
Fma

• rPosition, vVelocity, aacceleration
• Velocity is equal to the change in position over
  time.
• Acceleration is equal to the change in velocity
  over time.

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Intuitive Understanding

• If every second my position changes by 5m, what
  is my velocity?
• Acceleration is the change in velocity over time.
  If I am traveling at 5m/s at time t1, and 6m/s
  at t2, my acceleration is 1m/ss

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Integration

• Integration takes you backwards
• Integrating acceleration over time gives you
  velocity
• Intuition
• If you are acceleration at 5m/ss, then every
  second you increase velocity by 5. Integrating
  sums up these changes, so your velocity is
• What is C?
• At time t0, what is velocity? C so C is initial
  velocity
• So, if you are accelerating at 5m/ss, starting
  at 7m/s what is your velocity at time t3 seconds?

7
Integration

• Similarly, integrating velocity over time gives
  you position
• Example If youre accelerating at a constant
  5m/ss, then
• So, given you have traveled for 5 seconds
  starting from point 0, where are you?
• Plug in the values
• So, given initial position, initial velocity, and
  acceleration you can find the new position,
  velocity.
• We will do this every frame, using values from
  the previous frames.

8
Forces

• But wait how do we find the acceleration to
  begin with?
• Linear momentum is denoted as p which is
• To change momentum, we need a force.Newton says
• So, given a force on a point mass, we can find
  the acceleration and then we can find position,
  velocity whew, were done but..

9
Finding Momentum

• On a rigid body, we have mass spread over an area
• We compute momentum by treating each point on the
  object discretely and summing them up
• Lets try to simplify this by introducing the
  center of mass (CM). Define CM as (where M is the
  total mass of the body)

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Center of Mass

• Using this equation, multiply bothsides by M and
  take the derivative
• Aha.. .now we have total momentum on the right,
  but what is on the left?
• Because M is a constant it comes out of the
  derivative and then we have change in position
  over time of the center of mass or velocity of
  CM!

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Acceleration of CM

• Total linear momentum can be found just using the
  velocity of the CM (no summation needed!)
• So, finally the acceleration of the entire body
  can be calculated by assuming the forces are all
  acting on the CM and computing the acceleration
  of CM

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Partial Summary

• We now know, that given an objects acceleration
  we can compute its velocity and position by
  integrating
• And to determine acceleration, we can sum forces
  acting on the center of mass (CM) and divide by
  total mass
• Current challenge Integrating symbolically the
  find v(t) and t(t) is very hard! Remember
  differential equations?

13
Differential Equations

• These equations occur when the dependent variable
  and its derivative both appear in the equation.
  Intuitively this occurs frequently because it
  means the rate of change of a value depends on
  the value.
• Example air friction.. the faster you are going,
  the more force it applies to slow you down
• f -v ma (solve for a) but a is the
  derivative of v, so
• Solving this analytically is best left to you and
  your differential equations professor

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Numerical Integration of Ordinary Differential
Equations (ODEs)

• Analytically solving these is hard, but solving
  them numerically is much simpler. Many methods
  exist, but well use Eulers method.
• Integration is simply summing the area under the
  curve, and the derivative is the slope of the
  curve at any point. Euler says

Integrating from t3 to 5 is summing the y values
for that section.
t3
t5
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Eulers Approximation
Numerically integrating velocity and position we
get these equations
Euler numerical integration is an approximation
(src Wikipedia)
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Final Summary of Equations

• Sum up the forces acting on the body at the
  center of mass to get current acceleration
• To get new velocity and position, use your
  current acceleration, velocity, position and
  numerical integration over some small time step
  (h)

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Now we can code!

• ForceRegistry stores which forces are being
  applied to which objects
• ForceGenerator virtual (abstract) class that all
  Forces implement
• Mainloop
• for each entry in Registry
• add force to accumulator in object
• for each object
• compute acceleration using resulting
  total force
• compute new velocity using acceleration
• compute new position using velocity
• reset force accumulator to zero

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ForceRegistry

• v

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ForceGenerator
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ImpulseForceGenerator
Warning This code is actually changing the
acceleration, it should just update the forces
and the acceleration should be computed at the
end of all forces
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DragForce generator

• In order to slow an object down, a drag force can
  be applied that works in the opposite direction
  of velocity.
• typically a simplified drag equation used in
  games is
• k1 and k2 are constants specifying the drag
  force, and the direction is in the opposite
  direction of velocity.

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DragForce Generator
Add force to current forces upon the player
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Mainloop Updating Physics Quantities
Inside mainloop
After the forces have been updated, you must then
apply the forces to create acceleration and
update velocity and position.
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Whats next?

• Other forces
• Spring forces push and pull
• Bungee forces pull only
• Anchored springs/bungees
• Rotational forces
• forces instead of moving the force can also
  induce rotations on the object
• Collisions
• Conversion from 2D to 3D

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References

• These slides are mainly based on Chris Heckers
  articles in Game Developers Magazine (1997).
• The specific PDFs (part 1-4) are available
  athttp//chrishecker.com/Rigid_Body_Dynamics
• Additional references from
• http//en.wikipedia.org/wiki/Euler_method
• Graham Morgans slides (unpublished)