PYTHON · NUMPY

Track Through a Turn

Estimate a turning aircraft's 2-D position from noisy range fixes with an Extended Kalman Filter — linearizing a nonlinear measurement, the way a real autopilot's EKF2 does.

01Challenge

GPS drops. The aircraft flies a gentle turn while two ground beacons each report a noisy distance to it. Distance is nonlinear in position, so the linear filter's math no longer applies. Build an EKF that linearizes each range measurement and tracks the aircraft far better than dead reckoning.

02Model

Real state estimation is nonlinear. A range to a beacon is √((x−bx)²+(y−by)²) — a curve, not a line — so the plain Kalman filter's linear update breaks. The Extended Kalman Filter's move is simple and powerful: at each step, take the derivative of the measurement with respect to the state (the Jacobian), and use that local linear approximation for the update. Predict the state forward with a constant-velocity model, then correct it by how far each measured range differs from the range you'd expect. This is exactly the machinery inside PX4's EKF2 and every GPS/UWB/visual-inertial estimator — the nonlinear filter that actually flies.

03Guided practice

Worked: simulate a turning truth and noisy ranges to two beacons. Faded: predict with the constant-velocity model, then form the range Jacobian H and update. Independent: confirm the EKF RMSE beats dead reckoning by a wide margin.

04Feedback

PASS WHEN The EKF linearizes each range fix and tracks the turning aircraft far tighter than dead reckoning — nonlinear estimation, the way a real autopilot does it.

If your run fails, check:

FAIL: EKF no better than dead reckoning — the Jacobian H is wrong; each row is the unit vector from beacon to estimate (dx/range, dy/range).
FAIL: filter diverges — the innovation should be measured range minus predicted range (meas - rp), not the reverse.

05Retrieve & space

Why can't the plain Kalman filter use a range measurement directly? (Range is nonlinear in position; the linear update assumes a linear measurement.)
What is the Jacobian doing in an EKF? (Giving a local linear approximation of the nonlinear measurement, recomputed at each step's estimate.)
Where does this exact filter run in the real world? (Inside PX4's EKF2 and every GPS/UWB/visual-inertial estimator on a flying aircraft.)

06Mastery & project

Implements an Extended Kalman Filter that linearizes a nonlinear range measurement to track a maneuvering target (L4: nonlinear state estimation, the working core of a real autopilot).

The EKF is the GPS-denied position estimate the planner and the swarm command against when the field block loses satellite fixes.

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PYTHON · NUMPY

Track Through a Turn

Run an EKF that fuses two nonlinear range measurements to track a turning aircraft.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you