Guidance, Navigation & Control
Before a vehicle can perceive or plan, it must know where it is and how it is oriented. GNC is the discipline of estimating that state from imperfect sensors and acting on it. This section builds the estimation backbone from the ground up — the inertial core, the filter that fuses everything, and the two great absolute references: the stars and the satellites.
Symbols used on this page are defined in the Nomenclature → notation conventions.
Eight pages, grouped below in reading order. Each one carries derivations and at least one simulation you can run in the browser.
1 · Orientation & the inertial core
Start here. These two pages define the vocabulary — frames, rotations, and the sensor that gives you a self-contained but drifting estimate of motion.
Attitude & rotations
Orientation done right: Euler, axis-angle, quaternions, Rodrigues (CRP/MRP), the group SO(3) with its exp/log maps, and how to carry a covariance for something that isn't a vector space.
IMU & inertial navigation
Accelerometers plus gyroscopes give a self-contained, jam-proof estimate — that inevitably drifts. The physics of dead reckoning and why position error grows like t².
2 · Alignment on the ground
Before an INS can navigate it has to know which way it is pointed, and on the pad the only references available are gravity and the Earth's own rotation. These two pages work that problem end to end — the estimator, and the test that decides whether to trust it.
Fine alignment & gyrocompassing
Finding north from a 15°/hr signal: the two accuracy floors, the error-state filter and what each state is for, why tilt and accelerometer bias are inseparable while stationary, and what consider states are for.
Ground alignment & the drift test
Propagate the aligned state with no aiding and watch it walk. Deriving the pass/fail threshold from sensor specs, why an inherited threshold from a rotational test is loose, and how to tell pad sway from a failing IMU.
3 · Estimation theory
How to fuse a model with measurements, which filter to pay for, and the flight-program practices that keep a real filter honest.
Filtering: KF · EKF · UKF · PF
The recursive Bayesian ladder: linear Kalman, Extended KF, Unscented KF (sigma points) and particle filters — when to pay compute for fidelity, with a live sim of each.
Extended Kalman filter
The workhorse that fuses a motion model with noisy nonlinear measurements, tracking both the state and its uncertainty. Predict, linearize, update — and the ways it diverges.
NASA nav-filter best practices
Flight-proven rules from NASA/TP-2018-219822: Joseph-form covariance, FOGM bias models, measurement underweighting, the consider filter, UDU factorization, and the MEKF.
4 · Absolute references
An inertial core alone always loses. These are the measurements that bound its error — and what you do when the best of them is denied to you.
Star tracker
Absolute 3-axis attitude from a photo of the sky — centroid the stars, match the pattern to a catalog, solve for orientation. The most accurate attitude sensor flown.
GNSS
Absolute position from satellite time-of-flight. Pseudoranges, trilateration, the clock-bias unknown, and why geometry (GDOP) makes or breaks accuracy.
Alt-PNT (GPS-denied)
Navigating without GPS: a cascading architecture of INS, terrestrial RF, signals of opportunity, MagNav, celestial and atomic timing that bounds drift when GNSS is jammed or spoofed.
5 · Why fuse them?
Each sensor has a complementary failure mode. The IMU is fast and always available but drifts unboundedly. GNSS is absolute but slow, jammable, and unavailable indoors. A star tracker is exquisitely accurate for attitude but needs a clear sky and is low-rate. The EKF is what welds them into a single estimate that is better than any one alone — high-rate from the IMU, bounded by the absolute references.
high-rate, drifts
predict + update
absolute, low-rate
6 · Where this leads
Perception stack →
The same estimation machinery, now carrying learned detections into tracks under a 100 ms deadline.
Rocket navigation →
These pieces assembled into one flight filter: ground align → GNSS-INS → landing.
The hardware →
What the IMU grades in this section actually cost in mass, power and dollars.