Inertial state estimation

The glass-lio course

Nine core lessons, a mathematical toolkit, and practical exercises.

Each lesson follows one idea through the maths and into the glass-lio source, including the bugs it caused. Every lesson has quizzes, a from-memory recall box and a short reference card to revise from.

Allow 30–45 minutes per lesson on first study (45–60 for the capstone), plus the exercises; the 15–20 minute estimates are for revision. Start with the mathematical toolkit if matrices, derivatives, or covariance are unfamiliar. Use the practical worksheet at the checkpoints linked from the lessons.

By the end, aim to derive the IMU updates, explain covariance and bias correction, differentiate a point-to-plane residual, identify unobservable motions, and audit the complete solve.

Take them in order: each one builds on the last. Terms are defined in the glossary.

  1. Start here · 45–60 minutesThe mathematical toolkitFrames, rotations, Jacobians, covariance, and the normal equations, with worked examples.
  2. Lesson 1 · 30–45 minutes first studyWhat an IMU actually measuresTwo equations, one sign, and the bug it caused in glass-lio.

    Reference card

  3. Lesson 2 · 30–45 minutes first studyIntegrating one IMU stepWhere gravity comes back in, and the update-order trap.

    Reference card

  4. Lesson 3 · 30–45 minutes first studyWhy preintegrateFrom eq. (31) iterated, to three deltas reusable when the starting pose changes.

    Reference card

  5. Lesson 4 · 30–45 minutes first studyThe bias correctionWhy the deltas go stale when the bias moves, and the first-order fix.

    Reference card

  6. Lesson 5 · 30–45 minutes first studyThe noise preintegration carriesOne covariance, propagated a sample at a time, and why it decides who wins.

    Reference card

  7. Lesson 6 · 30–45 minutes first studyThe point-to-plane factorOne residual per laser return, its Jacobian, and how thousands of them build one linear system.

    Reference card

  8. Lesson 7 · 30–45 minutes first studyPlanes and degeneracyOne skill, used twice: reading the smallest eigenvalue of a 3×3 matrix.

    Reference card

  9. Lesson 8 · 30–45 minutes first studyFilter, window, or factorWhat each estimator keeps, what it re-linearises, and what marginalization buys.

    Reference card

  10. Lesson 9 · 45–60 minutes first studyThe tight solve, end to endOne scan through glass-lio, every term justified, and each one traced to the lesson it came from.

    Reference card

  11. Apply what you learned · 60–90 minutesPractical worksheetIntegrate IMU samples, check a Jacobian, find an invisible motion, and audit covariance. Hints and worked solutions included.