Module 4: Calculus, Motion & Optimization

Welcome to Module 4: Calculus, Motion & Optimization. In this module, we explore the mathematics of motion and optimization—from speedometer rates of change and elevator jerk to area integration and gradient descent.

Concepts in this Module

• Concept 01: Rates of Change & Derivatives
  • The Everyday Problem: How does an optical encoder turn wheel tick counts into instantaneous robot velocity?
  • Code & Math: Secant slope vs. tangent slope, numerical differentiation dx / dt, velocity from position.
• Concept 02: Acceleration, Jerk & S-Curves
  • The Everyday Problem: Why does slamming an elevator motor to full power snap roller chains and spill boiling coffee?
  • Code & Math: Higher derivatives (da/dt), Newton’s law F = m·a, S-curve motion profiling with bounded jerk.
• Concept 03: Accumulation, Area & Numerical Integration
  • The Everyday Problem: How does a robot track where it is on the field by adding up speed measurements over time?
  • Code & Math: Area under a velocity curve, Forward Euler vs. Trapezoidal integration accuracy.
• Concept 04: Multivariable Gradients & Hill Climbing
  • The Everyday Problem: How does an autonomous robot steer around obstacles using artificial potential fields?
  • Code & Math: Partial derivatives (∂f/∂x, ∂f/∂y), gradient vector ∇f (steepest uphill), and Gradient Descent -∇f.