Robotics and control
Motion is derivatives: velocity, acceleration, Jacobians from joints to hands, dynamics as ODEs, and feedback control that keeps it all stable.
8 topics
A robot is calculus with motors. Two chains summarize most of it:
position ──d/dt──► velocity ──d/dt──► acceleration (kinematics)
joint angles q ──► Jacobian J(q) ──► end-effector velocity ṗ = J(q) q̇
On top come dynamics (forces and torques as differential equations), state estimation (fusing noisy sensors with a model, the Kalman filter) and control (feedback laws such as PID, designed with Laplace transforms and stability analysis).
Topics
Kinematics: position, velocity, acceleration
Describing motion without its causes: velocity is the derivative of position, acceleration the derivative of velocity. Robots, drones, cameras and game characters all plan and track trajectories with these derivatives.
Robot Jacobian (velocity kinematics)
Forward kinematics gives the hand position from the joint angles; its Jacobian maps joint velocities to hand velocities, , and joint torques to hand forces, . Where the arm is singular.
Inverse kinematics
Find joint angles that place the hand at a target: solve . Usually iteratively with the Jacobian — Newton/Gauss–Newton steps , or damped least squares near singularities. Game engines use it to plant feet and aim hands.
Robot dynamics
How torques produce motion: the Euler–Lagrange equations of the Lagrangian give , a system of second-order ODEs simulated and inverted by controllers.
PID control
The most used controller in the world: correct proportionally to the error (P), to its accumulated integral (I, removes steady offset) and to its derivative (D, anticipates and damps). Calculus in one line, running in thermostats, drones, cruise control and hard-disk heads.
Control theory
Designing feedback so a dynamical system does what we want, robustly. Classical control works with transfer functions (Laplace); modern control with state-space ODEs , optimal control (LQR, MPC) and stability theory (eigenvalues, Lyapunov).
Kalman filter
Optimal state estimation for linear systems with Gaussian noise: predict with the model, correct with the measurement, weighting each by its uncertainty. The extended version linearizes with Jacobians. Used in every GPS receiver, phone, drone and spacecraft (Apollo).
Trajectory optimization and MPC
Choose the whole motion that minimizes a cost (time, energy, jerk) subject to dynamics, limits and obstacles — a constrained optimization over functions, discretized into a large nonlinear program. Model predictive control re-solves it every few milliseconds over a short horizon.