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Understand how a robot works.

Build working knowledge of robotics and machine learning through examples you can change, calculations you can check, and data you can inspect.

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148 lessons found

Foundations first

  1. Foundations / 10 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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.