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41 lessons found

Foundations first

  1. Foundations / 11 min read

    Pseudoinverse: choose the smallest least-squares solution

    Understand the Moore–Penrose pseudoinverse through exact and inconsistent systems. Separate residual error from solution norm, inspect projectors, and see how an SVD cutoff changes the problem.

  2. Foundations / 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. Foundations / 12 min read

    Manifolds and tangent spaces: move along a constraint

    Use the unit circle to understand local coordinates and tangent vectors. Compare straight steps, exact rotation, and normalization, then examine why averaging headings and rotations needs care.

  11. Foundations / 13 min read

    Geodesics: shortest arcs and longer routes on a circle

    Compare a shortest circle arc, a longer constant-speed geodesic, and a straight chord. Work through angle wrapping, antipodal ties, coincident endpoints, and the metric that defines distance.

  12. Foundations / 12 min read

    Lie groups and Lie algebras: connect robot poses to local motions

    Use planar robot poses to understand SE(2), its tangent space se(2), and the Lie bracket. Compare motion order, shrink a commutator loop, and reproduce the calculations in Python.