Simplify Challenging Software Problems with Rocket Science @bendyworks Brad Grzesiak — @listrophy 

–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.” 

Heads up: 1 Animated GIF 

Agenda • Reduce Cognitive Load • Visualize Simulations with Graphics • Control and Respond to Real World Processes 

Agenda • Reduce Cognitive Load • Visualize Simulations with Graphics • Control and Respond to Real World Processes 

we are not superheros 

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
 

“matrix” “times” “vector?” 

Linear Transformations • Scale, Rotate, Skew, Translate (with a trick) • Common in CSS3 • Common in Kinetics & Kinematics 

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   

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! 

Rotation Matrix R(θ)  =  ⽷cosθ  -­‐sinθ⽹                ⽸sinθ    cosθ⽺   R(30deg)*⟨1,0⟩  =  ⟨0.87,0.5⟩   R(90deg)*⟨1,0⟩  =  ⟨0,1⟩ 

Rotation Matrix R(-­‐30deg)*                        =   

Agenda • Reduce Cognitive Load • Visualize Simulations with Graphics • Control and Respond to Real World Processes 

Why Graphic Simulations? • Complex Systems • High Sensitivity • Pattern Recognition Required • Seeking “Good Enough” 

In Aerospace Engineering • Launch Vibration Testing • “Kick” tests • Thermal Equilibrium • Designing to Safety Factor 

In Aerospace Engineering • Launch Vibration Testing • “Kick” tests • Thermal Equilibrium • Designing to Safety Factor • N-Body Orbital Trajectories 

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) 

The n-Body Problem Gravity Force = x Newton’s 2nd Law Force = mass ⨉ acceleration Newton’s 3rd Law Force₁ = -Force₂ Velocity-Acceleration Position-Velocity 

Wait a minute. Derivatives? 

Formal Derivative 

Our Approximation 

Our Approximation 

Our Approximation 

The n-Body Problem Gravity Force = x Newton’s 2nd Law Force = mass ⨉ acceleration Newton’s 3rd Law Force₁ = -Force₂ Velocity-Acceleration Position-Velocity 

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 

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 

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 

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 

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 

(demo) 

Agenda • Reduce Cognitive Load • Visualize Simulations with Graphics • Control and Respond to Real World Processes 

The World is Complicated • “Assume no air resistance” • “Assume a frictionless surface” • “Assume perfect DC voltage” • “Assume no impact from general or special relativity” 

You simply cannot predefine a specific solution to a problem that is perturbed by the real world 

–Helmuth von Moltke the Elder “No battle plan ever survives contact with the enemy.” 

The Answer: React 

Feedback Controller 

Feedback Controller 

Feedback Controller 

PID Controller 

Controller P I D roportional ntegral erivative 

Feedback Controller diff == error 

Proportional P  =  error  =  desired  -­‐  actual 

Derivative D  =  (error  -­‐  prev_error)/dt 

Integral I  =  I  +  error*dt 

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 

(demo) 

Agenda • Reduce Cognitive Load • Visualize Simulations with Graphics • Control and Respond to Real World Processes 

Photo Credits • “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
 

Thank You! Brad Grzesiak [email protected] @listrophy