Learning Path: Foundations

Build mathematical and physical foundations for understanding AffineDrift from first principles

Foundations Learning Path

A 12-week guided journey from basics to AffineDrift fundamentals. No prerequisites assumed.

Overview

This path builds from first principles. Start here if you have: - Limited control theory or biomechanics background - Physics intuition but no formal training in robotics - Interest in understanding before diving deep

Total Time: 80–120 hours | Difficulty: Introductory | Prerequisites: High school algebra and trigonometry

Module 1: Calculus & Linear Algebra Essentials

Weeks 1–2 | 8–12 hours

What You’ll Learn

  • Multivariable calculus (gradients, Jacobians, chain rule)
  • Matrix algebra and eigenvalues
  • Differential equations basics

Reading

  • 3Blue1Brown (video series)
  • Grant Sanderson (3Blue1Brown) Linear Algebra: A Geometric Intuition
    • Visualizes concepts: matrices as transformations, eigenvectors, determinants
    • Format: Video (best for intuition)
    • Time: 2–3 hours

Practice

  • Khan Academy exercises on linear algebra (30 min/day, 10 days)
  • Interactive linear algebra visualization tools (Desmos, GeoGebra)

Key Concepts

  • ✓ Vectors and matrices as transformations
  • ✓ Derivatives and partial derivatives
  • ✓ Jacobian matrices (function sensitivities)
  • ✓ Eigenvalues and eigenvectors (mode decomposition)

Module 2: Physics Fundamentals

Weeks 3–4 | 8–10 hours

What You’ll Learn

  • Newton’s laws and force analysis
  • Work, energy, and momentum
  • Rotational dynamics basics

Reading

  • Walter Lewin, MIT OpenCourseWare Physics I (Classical Mechanics)
    • Videos: Lectures 1–10 on forces and dynamics
    • Format: Video lectures
    • Time: 3–4 hours of viewing + 4–6 hours working problems
  • Goldstein, “Classical Mechanics” (Chapter 1–2 overview)
    • Alternative: MIT OpenCourseWare for gentler introduction
    • Key sections: Forces, energy, momentum

Practice

  • Solve 10–15 classical mechanics problems (projectile motion, orbits, pendulums)
  • PhET simulations (interactive physics demos)

Key Concepts

  • ✓ Force, mass, acceleration (F = ma)
  • ✓ Energy conservation
  • ✓ Rotational motion and angular momentum
  • ✓ Moment of inertia and torque

Module 3: Rigid Body Dynamics

Weeks 5–6 | 10–12 hours

What You’ll Learn

  • Rotation matrices and coordinate frames
  • Rigid body equations of motion
  • Multibody chains and constraints

Reading

  • Kevin Lynch & Frank Park, “Modern Robotics” Chapters 3–4
    • Format: Textbook (clear, accessible)
    • Time: 4–5 hours reading + 3–4 hours problems
    • Free online: https://modernrobotics.org/
  • Paul, “Robot Manipulators” Chapter 2 (link frames)
    • Alternatively: Modern Robotics above is preferred
    • Key: Understanding coordinate transformations

Practice

  • Compute forward kinematics for a 2-DOF arm
  • Simulate rigid body in Python (use Drake or PyBullet)
  • Work 5 problems on rotation matrices

Key Concepts

  • ✓ Rotation matrices and homogeneous transforms
  • ✓ Rigid body kinematics (position, velocity, rotation)
  • ✓ Inertia matrices and mass distribution
  • ✓ Euler-Lagrange equations

Module 4: Differential Equations & Dynamical Systems

Weeks 7–8 | 12–15 hours

What You’ll Learn

  • Ordinary differential equations (ODEs) and solutions
  • Nonlinear systems and phase portraits
  • Stability and equilibria

Reading

  • Strogatz, “Nonlinear Dynamics and Chaos” Chapters 1–4
    • Format: Textbook (highly readable)
    • Time: 6–8 hours reading + 4–6 hours problems
    • Best book for intuition
  • MIT OpenCourseWare: Differential Equations (optional video companion)
    • Lectures on linear systems, eigenvalues, phase portraits

Practice

  • Solve ODE problems by hand (linear and simple nonlinear)
  • Draw phase portraits by hand
  • Simulate pendulum, predator-prey, and other classic systems
  • Use Python (scipy.integrate.odeint) to visualize solutions

Key Concepts

  • ✓ Equilibrium points and stability
  • ✓ Linearization around equilibria
  • ✓ Phase space and trajectories
  • ✓ Lyapunov stability concepts

Module 5: Optimal Control Basics

Weeks 9–10 | 10–12 hours

What You’ll Learn

  • Cost functions and optimization
  • Trajectory optimization fundamentals
  • Minimum energy and time-optimal control

Reading

  • Russ Tedrake, “Underactuated Robotics” Chapters 1–2 (free online)
    • Format: Online textbook (very accessible)
    • Time: 4–5 hours reading + 3–4 hours problems
    • Focus on intuition before math
  • Stengel, “Optimal Control and Estimation” Chapter 1 (optional, more rigorous)

Practice

  • Solve simple optimal control problems (straight line distance minimization)
  • Implement gradient descent for trajectory optimization
  • Use Drake or Pydrake to solve a 2D reaching task
  • Interactive notebooks: Google Colab demonstrations

Key Concepts

  • ✓ Cost functions and objective criteria
  • ✓ Gradient descent and optimization
  • ✓ Lagrange multipliers and constraints
  • ✓ Intuition for optimal trajectories

Module 6: Introduction to Control Theory

Weeks 11–12 | 12–15 hours

What You’ll Learn

  • Feedback control and stability
  • PID control and classical methods
  • State space representation
  • Introduction to nonlinear control

Reading

  • Åström & Murray, “Feedback Systems” Chapters 1–6 (free online, Princeton Press)
    • Format: Textbook (excellent pedagogy)
    • Time: 8–10 hours reading + 4–5 hours problems
    • Starts from basics, builds to state space
  • Tedrake, “Underactuated Robotics” Chapter 3 (state machines and control)

Practice

  • Design PID controllers for a DC motor
  • Implement state feedback controller for pendulum
  • Stability analysis using eigenvalues
  • Simulate in Python/Matlab

Key Concepts

  • ✓ Feedback loops and closed-loop stability
  • ✓ PID control and tuning
  • ✓ State space models and Lyapunov stability
  • ✓ Control affine systems

What’s Next?

After completing this path, you’re ready to:

  1. Choose a specialized path:
  2. Read AffineDrift texts directly:
    • Books Roadmap — Choose your textbook path
    • Start with Volume I or The Geometry of Motion articles
  3. Get hands-on:

Common Stumbling Blocks

“Eigenvalues Are Confusing”

Solution: Watch 3Blue1Brown’s eigenvector video. Then compute by hand: find eigenvalues of [[3, 1], [1, 3]].

“I Don’t See Why We Use Jacobians”

Solution: Jacobian = slope of a multivariable function. Compute dF/dx for a 2-input robot and see how it predicts output changes.

“Lagrangian Mechanics Is Scary”

Solution: Skip it for now. Use F=ma first. Come back to Lagrangian after understanding energy.

“Simulation Results Don’t Match My Math”

Solution: Common causes: wrong initial conditions, numerical integration errors, unit mismatches. Debug one variable at a time.

Success Check

By the end of this path, you should be able to:

  • ✓ Write down Newton’s laws for a multi-segment arm
  • ✓ Compute the Jacobian of a forward kinematics function
  • ✓ Draw phase portraits for a nonlinear system
  • ✓ Design a simple PID controller and tune it
  • ✓ Formulate and solve a basic optimal control problem
  • ✓ Explain stability using eigenvalues

If you can do all 6 of these, you’re ready for specialized paths!

Feedback & Customization

  • More visual? Increase 3Blue1Brown and simulation time, decrease textbook time
  • More mathematical? Add Stengel, classical control theory (Ogata)
  • Want applications? Skip to Golf Science after Module 6

Next Steps

  1. Start Module 1 this week: Watch the first 3Blue1Brown algebra video
  2. Set up your workspace: Jupyter, Python, visualization tools
  3. Join the community: Collaborate page
  4. Track your progress: Note what you learn, challenges you face
  5. Questions? Contact or Collaborate

Happy learning! 📚