Game Physics – Part I
Dan Fleck
Coming up: Rigid Body Dynamics
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).
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Example:
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
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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 we’re going to model rigid
bodies – ones that do not deform (not squishy)
We can model articulated rigid bodies – multiple limbs
connected with a joint
Coming up: Bring on calculus
Bring on calculus

Calculus was invented by Newton (and Leibniz) to handle
these problems
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Newton’s Laws
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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
Coming up: F=ma
F=ma
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r=Position, v=Velocity, a=acceleration

Velocity is equal to the change in position over time.
Acceleration is equal to the change in velocity over time.
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Coming up: Intuitive Understanding
Intuitive Understanding
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If every second my position changes by 5m, what is my
velocity?
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Acceleration is the change in velocity over time. If I am
traveling at 5m/s at time t=1, and 6m/s at t=2, my
acceleration is 1m/s^s
Coming up: Integration
Integration
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Integration takes you backwards
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Integrating acceleration over time gives you velocity
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Intuition:
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If you are acceleration at 5m/s^s, then every second you increase
velocity by 5. Integrating ‘sums’ up these changes, so your velocity is

What is “C”?

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At time t=0, what is velocity? C… so C is initial velocity
So, if you are accelerating at 5m/s^s, starting at 7m/s what is your
velocity at time t=3 seconds?
Coming up: Integration
Integration

Similarly, integrating velocity over time gives you position
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Example: If you’re accelerating at a constant 5m/s^s, then:
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So, given you have traveled for 5 seconds starting from point 0,
where are you?
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Plug in the values:
5
r(t)  52  0t  0  62.5
2
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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.

Coming up: Forces
Forces
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But wait… how do we find the acceleration to begin with?
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Linear momentum is denoted as p which is:

To change momentum, we need
a force.Newton says:
p  mv
p  mv

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
So, given a force on a point mass, we can find the acceleration
and then we can find position, velocity… whew, we’re done…
but…..
Coming up: Finding Momentum
Finding Momentum
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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):
Coming up: Center of Mass
Center of Mass
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Using this equation, multiply both
sides by M and take the derivative
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Aha.. .now we have total momentum on the right, but
what is on the left?
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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!
Coming up: Acceleration of CM
Acceleration of CM
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Total linear momentum can be found just using the
velocity of the CM (no summation needed!)
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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
Coming up: Partial Summary
Partial Summary
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We now know, that given an object’s acceleration we can
compute it’s velocity and position by integrating:
a cm 
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FT
M
And to determine acceleration, we can sum forces acting on
the center of mass (CM)
 and divide by total mass
T
F
cm
a 
M
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Current challenge: Integrating symbolically the find v(t) and t(t)
is very hard! Remember differential equations?
Coming up: Differential Equations
Differential Equations
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These equations occur when the dependent variable and it’s
derivative both appear in the equation. Intuitively this occurs
frequently because it means the rate of change of a value
depends on the value.
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Example: air friction.. the faster you are going, the more force
it applies to slow you down:
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f = -v = ma (solve for a)
v
a
m
but a is the derivative of v, so
dv v

dt m
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Solving this analytically is best left to you and your differential

equations professor

Coming up: Numerical Integration of
Ordinary Differential Equations (ODEs)
Numerical Integration of Ordinary
Differential Equations (ODEs)
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Analytically solving these is hard, but solving them
numerically is much simpler. Many methods exist, but
we’ll use Euler’s method.
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Integration is simply summing the area under the curve,
and the derivative is the slope of the curve at any point.
Euler says:
t=3
t=5
Integrating from
t=3 to 5 is
summing the y
values for that
section.
dyn
yn 1  yn  h
dt
Coming up: Euler’s Approximation
Euler’s Approximation
Numerically integrating velocity
and position we get these
equations:
Euler numerical integration is an
approximation (src: Wikipedia)
Coming up: Final Summary of Equations
Final Summary of Equations

Sum up the forces acting on the body at the center of
mass to get current acceleration
T
a

cm
F

M
To get new velocity and position, use your current
acceleration, velocity, position and numerical integration
over some
 small time step (h)
Coming up: Now we can code!
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
Coming up: ForceRegistry
ForceRegistry
√
Coming up: ForceGenerator
ForceGenerator
Coming up: ImpulseForceGenerator
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
Coming up: DragForce generator

DragForce generator
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In order to slow an object down, a drag force can be
applied that works in the opposite direction of velocity.
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typically a simplified drag equation used in games is:
f drag  vˆ(k1 v  k2 v )
2

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k1 and k2 are constants specifying the drag force, and the
direction is in the opposite direction of velocity.
Coming up: DragForce Generator
DragForce Generator
Add force to current forces upon the player
Coming up: Mainloop – Updating Physics
Quantities
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.
Coming up: What’s next?
What’s next?
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Other forces
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Rotational forces
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Spring forces – push and pull
Bungee forces – pull only
Anchored springs/bungees
forces instead of moving the force can also induce rotations on
the object
Collisions
Conversion from 2D to 3D
Coming up: References
References
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These slides are mainly based on Chris Hecker’s articles
in Game Developer’s Magazine (1997).
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The specific PDFs (part 1-4) are available at:
http://chrishecker.com/Rigid_Body_Dynamics
Additional references from:
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http://en.wikipedia.org/wiki/Euler_method
Graham Morgan’s slides (unpublished)
End of presentation