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Simplify Challenging
Software Problems with
Rocket Science
@bendyworks
Brad Grzesiak — @listrophy
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–Ashe Dryden
“Whenever Brad is drinking, you
wanna make sure you're sitting
next to him, because instead of
being obnoxious, he teaches you
orbital mechanics.”
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Heads up:
1 Shaky Movie &
1 Animated GIF
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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we
are
not
superheros
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Scalars & Collections
Primitives & Arrays
• x, y, z = 2, 3, 5
• Math.sqrt(x**2 + y**2 + z**2)
• array.zip(other).map{|(i,j)| i+j}
• [m[0][0]*v[0]+m[0][1]*v[1],
m[1][0]*v[0]+m[1][1]*v[1]]
Vectors & Matrices
• vector = Vector[2, 3, 5]
• vector.magnitude # or vector.r
• vector_1 + vector_2
• matrix * vector
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“matrix”
“times”
“vector?”
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Vector
• Underlying implementation: Array
• Usually Numeric, but really anything
• Native operations are different (c.f. addition)
• Denoted by: ⟨x,y⟩ or x or x or or
⽷x⽹
⽸y⽺
⽷2⽹
⽸0⽺
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Vector Example
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Vector Example
⟨2,0⟩
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Vector Example
⟨2,0⟩
⟨1,3⟩
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Matrices, or
“Linear Transformations”
• Structurally like a 2D Array
• Used to Scale, Rotate, Skew, Perspective, Translate
• Common in CSS3
• Useful for 2D & 3D graphics
• Common in Kinetics & Material Mechanics
• Denoted by M or M or or
⽷a
b⽹
⽸c
d⽺
⽷1
4⽹
⽸9
0⽺
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Matrix Times Vector
1.
⽷a
b⽹⽷x⽹
⽸c
d⽺⽸y⽺
2.
［x
y］
⽷a
b⽹
⽸c
d⽺
3.
⽷ax+by⽹
⽸cx+dy⽺
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Matrix Times Vector
def matrix_times_vector(m, v)
a, b = m[0]
c, d = m[1]
x, y = v
[
a*x + b*y,
c*x + d*y
]
end
require 'matrix'
def matrix_times_vector(m, v)
m * v
end
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Matrix Times Vector
def matrix_times_vector(m, v)
a, b = m[0]
c, d = m[1]
x, y = v
[
a*x + b*y,
c*x + d*y
]
end
require 'matrix'
def matrix_times_vector(m, v)
m * v
end
NOW
WITH DIM
CHECKS!
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Linear Transformation
Examples
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Hooke’s Law
for Anisotropic Materials
σ
=
c
*
ε
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Hooke’s Law
for Anisotropic Materials
⽷σ₁₁⽹
⽷C₁₁
C₁₂
C₁₃
C₁₄
C₁₅
C₁₆⽹⽷
ε₁₁⽹
⾇σ₂₂⾇
⾇C₁₂
C₂₂
C₂₃
C₂₄
C₂₅
C₂₆⾇⾇
ε₂₂⾇
⾇σ₃₃⾇=⾇C₁₃
C₂₃
C₃₃
C₃₄
C₃₅
C₃₆⾇⾇
ε₃₃⾇
⾇σ₂₃⾇
⾇C₁₄
C₂₄
C₃₄
C₄₄
C₄₅
C₄₆⾇⾇2ε₂₃⾇
⾇σ₁₃⾇
⾇C₁₅
C₂₅
C₃₅
C₄₅
C₅₅
C₅₆⾇⾇2ε₁₃⾇
⽸σ₁₂⽺
⽸C₁₆
C₂₆
C₃₆
C₄₆
C₅₆
C₆₆⽺⽸2ε₁₂⽺
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Hooke’s Law
for Anisotropic Materials
⽷σ₁₁⽹
⽷C₁₁
C₁₂
C₁₃
C₁₄
C₁₅
C₁₆⽹⽷
ε₁₁⽹
⾇σ₂₂⾇
⾇C₁₂
C₂₂
C₂₃
C₂₄
C₂₅
C₂₆⾇⾇
ε₂₂⾇
⾇σ₃₃⾇=⾇C₁₃
C₂₃
C₃₃
C₃₄
C₃₅
C₃₆⾇⾇
ε₃₃⾇
⾇σ₂₃⾇
⾇C₁₄
C₂₄
C₃₄
C₄₄
C₄₅
C₄₆⾇⾇2ε₂₃⾇
⾇σ₁₃⾇
⾇C₁₅
C₂₅
C₃₅
C₄₅
C₅₅
C₅₆⾇⾇2ε₁₃⾇
⽸σ₁₂⽺
⽸C₁₆
C₂₆
C₃₆
C₄₆
C₅₆
C₆₆⽺⽸2ε₁₂⽺
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Our Vector
⟨1,3⟩
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Identity Transform
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Identity Transform
I
=⽷1
0⽹
⽸0
1⽺
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Identity Transform
I
=⽷1
0⽹
⽸0
1⽺
I*v
=⽷1
0⽹⽷1⽹
⽸0
1⽺⽸3⽺
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Identity Transform
I
=⽷1
0⽹
⽸0
1⽺
I*v
=⽷1
0⽹⽷1⽹
⽸0
1⽺⽸3⽺
I*v
=⽷(1*1)
+
(0*3)⽹
⽸(0*1)
+
(1*3)⽺
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Identity Transform
I
=⽷1
0⽹
⽸0
1⽺
I*v
=⽷1
0⽹⽷1⽹
⽸0
1⽺⽸3⽺
I*v
=⽷(1*1)
+
(0*3)⽹
⽸(0*1)
+
(1*3)⽺
I*v
=⽷1⽹
⽸3⽺
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Scale Transform
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Scale Transform
S
=⽷s
0⽹=⽷2
0⽹
⽸0
s⽺
⽸0
2⽺
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Scale Transform
S
=⽷s
0⽹=⽷2
0⽹
⽸0
s⽺
⽸0
2⽺
S*v
=⽷2
0⽹⽷1⽹
⽸0
2⽺⽸3⽺
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Scale Transform
S
=⽷s
0⽹=⽷2
0⽹
⽸0
s⽺
⽸0
2⽺
S*v
=⽷2
0⽹⽷1⽹
⽸0
2⽺⽸3⽺
S*v
=⽷(2*1)
+
(0*3)⽹
⽸(0*1)
+
(2*3)⽺
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Scale Transform
S
=⽷s
0⽹=⽷2
0⽹
⽸0
s⽺
⽸0
2⽺
S*v
=⽷2
0⽹⽷1⽹
⽸0
2⽺⽸3⽺
S*v
=⽷(2*1)
+
(0*3)⽹
⽸(0*1)
+
(2*3)⽺
S*v
=⽷2⽹
⽸6⽺
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Flip Horizontal Transform
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Flip Horizontal Transform
H
=⽷-‐1
0⽹
⽸
0
1⽺
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Flip Horizontal Transform
H
=⽷-‐1
0⽹
⽸
0
1⽺
H*v
=⽷-‐1
0⽹⽷1⽹
⽸
0
1⽺⽸3⽺
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Flip Horizontal Transform
H
=⽷-‐1
0⽹
⽸
0
1⽺
H*v
=⽷-‐1
0⽹⽷1⽹
⽸
0
1⽺⽸3⽺
H*v
=⽷(-‐1*1)
+
(0*3)⽹
⽸(
0*1)
+
(1*3)⽺
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Flip Horizontal Transform
H
=⽷-‐1
0⽹
⽸
0
1⽺
H*v
=⽷-‐1
0⽹⽷1⽹
⽸
0
1⽺⽸3⽺
H*v
=⽷(-‐1*1)
+
(0*3)⽹
⽸(
0*1)
+
(1*3)⽺
H*v
=⽷-‐1⽹
⽸
3⽺
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SkewX Transform
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SkewX Transform
X(θ)
=⽷1
tanθ⽹
⽸0
1⽺
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SkewX Transform
X(θ)
=⽷1
tanθ⽹
⽸0
1⽺
X(30°)*v
=⽷1
.58⽹⽷1⽹
⽸0
1⽺⽸3⽺
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SkewX Transform
X(θ)
=⽷1
tanθ⽹
⽸0
1⽺
X(30°)*v
=⽷1
.58⽹⽷1⽹
⽸0
1⽺⽸3⽺
X(30°)*v
=⽷(1*1)
+
(.58*3)⽹
⽸(0*1)
+
(
1*3)⽺
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SkewX Transform
X(θ)
=⽷1
tanθ⽹
⽸0
1⽺
X(30°)*v
=⽷1
.58⽹⽷1⽹
⽸0
1⽺⽸3⽺
X(30°)*v
=⽷(1*1)
+
(.58*3)⽹
⽸(0*1)
+
(
1*3)⽺
X(30°)*v
=⽷2.73⽹
⽸3
⽺
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SkewX Transform
X(θ)
=⽷1
tanθ⽹
⽸0
1⽺
X(30°)*v
=⽷1
.58⽹⽷1⽹
⽸0
1⽺⽸3⽺
X(30°)*v
=⽷(1*1)
+
(.58*3)⽹
⽸(0*1)
+
(
1*3)⽺
X(30°)*v
=⽷2.73⽹
⽸3
⽺
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SkewX Transform
X(θ)
=⽷1
tanθ⽹
⽸0
1⽺
X(30°)*v
=⽷1
.58⽹⽷1⽹
⽸0
1⽺⽸3⽺
X(30°)*v
=⽷(1*1)
+
(.58*3)⽹
⽸(0*1)
+
(
1*3)⽺
X(30°)*v
=⽷2.73⽹
⽸3
⽺
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Rotation Transform
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Rotation Transform
R(θ)
=⽷cosθ
-‐sinθ⽹
⽸sinθ
cosθ⽺
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Rotation Transform
R(θ)
=⽷cosθ
-‐sinθ⽹
⽸sinθ
cosθ⽺
R(90°)*v
=⽷0
-‐1⽹⽷1⽹
⽸1
0⽺⽸3⽺
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Rotation Transform
R(θ)
=⽷cosθ
-‐sinθ⽹
⽸sinθ
cosθ⽺
R(90°)*v
=⽷0
-‐1⽹⽷1⽹
⽸1
0⽺⽸3⽺
R(90°)*v
=⽷(0*1)
+
(-‐1*3)⽹
⽸(1*1)
+
(
0*3)⽺
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Rotation Transform
R(θ)
=⽷cosθ
-‐sinθ⽹
⽸sinθ
cosθ⽺
R(90°)*v
=⽷0
-‐1⽹⽷1⽹
⽸1
0⽺⽸3⽺
R(90°)*v
=⽷(0*1)
+
(-‐1*3)⽹
⽸(1*1)
+
(
0*3)⽺
R(90°)*v
=⽷-‐3⽹
⽸
1⽺
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Rotation Transform
R(30°)*
=
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Rotation Transform
R(30°)*
=
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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Why Graphic Simulations?
• Complex Systems
• High Sensitivity
• Pattern Recognition Required
• Seeking “Good Enough”
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Why Graphic Simulations?
• Complex Systems
• High Sensitivity
• Pattern Recognition Required
• Seeking “Good Enough”
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Why Graphic Simulations?
• Complex Systems
• High Sensitivity
• Pattern Recognition Required
• Seeking “Good Enough”
Tests?
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Why Graphic Simulations?
• Complex Systems
• High Sensitivity
• Pattern Recognition Required
• Seeking “Good Enough”
Tests?
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In Aerospace Engineering
• Launch Vibration Testing
• “Kick” tests
• Thermal Equilibrium
• Designing to Safety Factor
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In Aerospace Engineering
• Launch Vibration Testing
• “Kick” tests
• Thermal Equilibrium
• Designing to Safety Factor
• N-Body Orbital Trajectories
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The n-Body Problem
• Newton’s Laws + Gravity
• Solved for N=2
• For N>2, “no general analytical solution,” except for
special cases (e.g., Lagrangian Points)
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The n-Body Problem
Gravity Force = x
Newton’s 2nd Law Force = mass ⨉ acceleration
Newton’s 3rd Law Force₁ = -Force₂
Velocity-Acceleration
Position-Velocity
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Wait a minute.
Derivatives?
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Velocity = Derivative of Position
0 km 60 km
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Velocity = Derivative of Position
0 km 60 km
t = 1 hour
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Velocity = Derivative of Position
0 km 1 km
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Velocity = Derivative of Position
0 km 1 km
t = 1 minute
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Velocity = Derivative of Position
0 km 16.67 meters
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Velocity = Derivative of Position
0 km 16.67 meters
t = 1 second
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Velocity = Derivative of Position
0 km 1.667 centimeters
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Velocity = Derivative of Position
0 km 1.667 centimeters
t = 1 millisecond
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Velocity = Derivative of Position
now 1 hour later
60 km
0 km
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Velocity = Derivative of Position
now 1 hour later
60 km/hr
0 km/hr
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Velocity = Derivative of Position
now 1 hour later
60 km/hr
0 km/hr
1 minute later
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Formal Derivative
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Derivative Applied
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Derivative Applied
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Derivative Applied
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Derivative Applied
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Derivative Applied
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Dimensional Analysis,
or “Fixing Units”
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Dimensional Analysis,
or “Fixing Units”
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Our Approximation
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Our Approximation
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Our Approximation
velocity = [position(now) - position(recent)] / (now - recent)
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Our Approximation
velocity = [position(now) - position(recent)] / (now - recent)
acceleration = [velocity(now) - velocity(recent)] / (now - recent)
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Our Approximation
velocity = [position(now) - position(recent)] / (now - recent)
acceleration = [velocity(now) - velocity(recent)] / (now - recent)
position(now) = position(recent) + velocity * (now - recent)
velocity(now) = velocity(recent) + acceleration * (now - recent)
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Our Approximation
position(now) = position(recent) + velocity * (now - recent)
velocity(now) = velocity(recent) + acceleration * (now - recent)
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The n-Body Problem
Gravity Force = x
Newton’s 2nd Law Force = mass ⨉ acceleration
Newton’s 3rd Law Force₁ = -Force₂
Velocity-Acceleration
Position-Velocity
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The n-Body Problem
Gravity Force = x
Newton’s 2nd Law Force = mass ⨉ acceleration
Newton’s 3rd Law Force₁ = -Force₂
Velocity-Acceleration vnow = vprev + anow dt
Position-Velocity xnow = xprev + vnow dt
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The n-Body Problem
Gravity Force = x
Newton’s 2nd Law acceleration = Force ÷ mass
Newton’s 3rd Law Force₁ = -Force₂
Velocity-Acceleration vnow = vprev + anow dt
Position-Velocity xnow = xprev + vnow dt
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Gravity Fnow = x
Newton’s 2nd Law anow = Fnow ÷ m
Newton’s 3rd Law (loop over each mass-pair)
Velocity-Acceleration vnow = vprev + anow dt
Position-Velocity xnow = xprev + vnow dt
The n-Body Problem
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The n-Body Problem
bodies.map do |body|
[body, gravity(body, bodies-[body])]
end.each do |(body, force)|
accel = force / body.mass
body.velocity += accel * dt
body.position += body.velocity * dt
end
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The n-Body Problem
def gravity(body, others)
others.reduce(Vector[0,0]) do |memo, other|
offset = other.position - body.position
memo + offset * G * body.mass * other.mass / offset.r**3
end
end
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(demo)
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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The World is Complicated
• “Assume no air resistance”
• “Assume a frictionless surface”
• “Assume perfect DC voltage”
• “Assume no impact from general or special
relativity”
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You simply cannot predefine a
specific solution to a problem that
is perturbed by the real world
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–Helmuth von Moltke the Elder
“No battle plan ever survives
contact with the enemy.”
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The Answer: React
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Feedback Controller
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Feedback Controller
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Feedback Controller
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PID Controller
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Controller
P
I
D
roportional
ntegral
erivative
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Feedback Controller
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Feedback Controller
diff == error
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Proportional
P
=
error
=
desired
-‐
actual
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Derivative
D
=
(error_now
-‐
error_recent)/dt
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Integral
I
+=
error*dt
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Our PID Controller
loop do
sleep 0.2
curr_time = Time.now
dt = curr_time - prev_time
p = error = desired - actual
i = i + error * dt
d = (error - prev_error)/dt
setThrust(0.1*p + 0.003*i + 0.08*d)
prev_error, prev_time = error, curr_time
end
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(demo)
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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Agenda
• Reduce Cognitive Load
• Visualize Simulations with Graphics
• Control and Respond to Real World Processes
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Photo Credits
• “LPE II” (Bradley Grzesiak)
• “Zero G Robot” (Bradley Grzesiak)
• “Outredgeous Lettuce” (Public Domain):
https://www.nasa.gov/mission_pages/station/
research/news/meals_ready_to_eat/
• “ABS/EUTELSAT-1 Launch” (Public Domain):
http://www.spacex.com/media-gallery/detail/
127081/4896
• “Beer Actor” (CC-BY):
https://www.flickr.com/photos/cogdog/
21127813750/
• “Up, up and away!” (CC-BY-SA):
https://www.flickr.com/photos/kindercapes/
5694816038
• “I’m confused” (CC-BY-SA):
https://www.flickr.com/photos/doctabu/342220423
• “Cookie Monster Waiting” (no idea):
https://www.google.com/search?q=cookie+monster
+waiting
• “Elmer Pump Heat Equation” (CC-BY-SA):
https://commons.wikimedia.org/wiki/File:Elmer-
pump-heatequation.png
• “Owl Stuck” (CC-BY):
https://www.flickr.com/photos/eben-frostey/
7147015007
• Car (Public Domain):
https://thenounproject.com/search/?q=car&i=13094
• Curing Sausage (Used with permission):
https://www.flickr.com/photos/aaronp/7532277486/