Learning library
Make the foundations concrete.
Build working knowledge of robotics and machine learning through examples you can change, calculations you can check, and data you can inspect.
Start with the map of machine learning fields to connect tasks, data, learning signals, and model choices.
99 lessons to explore
Foundations firstFoundations / 9 min read
Coordinate frames: read the same point from a robot and the world
Convert a fixed landmark between robot and world coordinates. Learn frame conventions, translation and rotation, inverse transforms, and point versus displacement.
Foundations / 12 min read
Rotation matrices: turn vectors, track frames, and check the order
Build rotation matrices that preserve length and handedness. Compare fixed-axis rotations in 3D, distinguish rotating a vector from changing its coordinates, and undo a rotation with its transpose.
Foundations / 12 min read
Homogeneous transformations: map sensor coordinates into the world
Combine rotation and translation in one matrix. Follow a sensor-to-robot-to-world frame chain, distinguish points from displacements, and calculate the inverse.
Foundations / 10 min read
Euler angles and gimbal lock: when different angles mean the same orientation
Explore roll, pitch, and yaw with full rotation matrices. Compare equivalent orientations at ±90° pitch and separate Euler angle rates from angular velocity.
Foundations / 12 min read
Axis-angle rotation: build Rodrigues’ formula from three vector terms
Rotate a vector around any nonzero axis. Normalize the direction, follow Rodrigues’ parallel and perpendicular terms, and understand the equivalent descriptions at zero and 180 degrees.
Foundations / 12 min read
Unit quaternions: compose rotations and understand the sign
Rotate vectors with Hamilton quaternions, check composition order, and see why q and minus q describe the same orientation. Includes an interactive experiment and Python.
Foundations / 10 min read
Derivatives and the chain rule: predict a small change
Understand derivatives as local rates, compare secant and tangent slopes, and multiply the correct factors through nested functions. Test a smooth curve and a corner with an interactive experiment.
Foundations / 13 min read
Partial derivatives and gradients: predict a multivariable change
Hold one input fixed to find a partial derivative, combine the partials into a gradient, and compare a directional derivative with the actual change from a finite step.
Foundations / 13 min read
Jacobian matrices: from joint rates to robot tip velocity
Read a Jacobian by its rows and columns, calculate a two-link arm's tip velocity, and compare a local prediction with a finite move. Includes singularities and the multivariate chain rule.
Foundations / 11 min read
Hessians: measure curvature in every direction
Differentiate a gradient to build the Hessian, calculate directional curvature, and classify stationary points. Explore coupled quadratics, saddles, and the limits of zero eigenvalues.
Foundations / 13 min read
Taylor expansion and linearization: predict locally and check the error
Build constant, linear, and quadratic approximations around a chosen center. Compare their errors, calculate a Taylor remainder bound, and connect the same idea to gradients, Hessians, and Jacobians.
Foundations / 12 min read
Ordinary differential equations: turn a rate law into a time course
Solve a cooling initial-value problem, compare its exact solution with Euler steps, and separate model behavior from numerical accuracy and stability.