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99 lessons to explore

Foundations first

  1. Foundations / 12 min read

    Numerical integration: compare drift, phase, and step cost

    Advance an oscillator with forward Euler, velocity-first symplectic Euler, and classical RK4. Compare each method with the analytic solution and separate energy drift, phase error, step cost, and stability.

  2. Foundations / 12 min read

    ODE stability: equilibria, attraction, and basins

    Classify equilibria of a nonlinear rate law, use a phase line to find basins of attraction, and compare exact trajectories without confusing model stability with numerical stability.

  3. 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.

  4. 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.

  5. 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.

  6. Foundations / 13 min read

    Exponential and logarithm maps: turn a body twist into a pose

    Exponentiate a constant planar body twist, calculate its coupled translation, and recover a chosen logarithm. Explore straight-motion limits, half-turn branch choices, and information lost in a full turn.

  7. Foundations / 13 min read

    Twists and screw axes: connect point velocities to rigid motion

    Build a six-component twist from a screw axis, calculate point velocities, and compare exact helical motion with a tangent prediction. Separate pitch, accumulated displacement, current rate, and pure translation.

  8. Foundations / 12 min read

    Adjoint transformations: express a twist in another frame

    Transform angular-first twists between body and space frames. Derive the origin-shift term, distinguish linear twist coordinates from point velocity, and check a planar example with an interactive adjoint matrix.

  9. Foundations / 14 min read

    Product of exponentials: build a robot arm’s forward kinematics

    Build a two-joint arm's tool pose from fixed home screw axes and matrix exponentials. Check the result against geometry, inspect multiplication order, and connect space and body formulas.

  10. Foundations / 14 min read

    Wrenches: combine force, moment, and power across frames

    Calculate a force's moment about a chosen origin, include a free couple, and transform a moment-first wrench between frames. Use a worked planar load to check the inverse-transpose rule and power invariance.

  11. Foundations / 14 min read

    Space and body Jacobians: map joint rates to rigid motion

    Build space and body Jacobians from joint screw axes, recover the physical tool velocity, and compare their ranks with a position-only task. Explore a planar two-link arm and verify its derivatives in Python.

  12. Foundations / 14 min read

    Robot statics: turn tool loads into holding torques

    Use virtual work and a Jacobian transpose to calculate a robot arm's joint loads. Distinguish external and holding torque, check space and body frames, and interpret zero-torque loads.