ÖK Ömer KahyaoğluMathematical Engineering · ITU
Released · v1.1.0MTH 407 term project · May 2026, revised Oct 2026

MTH Tracker: time-based target localisation and tracking

A two-dimensional TOA/TDOA tracker, least-squares initialiser and extended Kalman filter, validated against the posterior Cramér–Rao bound, with a real-time interactive simulator.

Extended Kalman filterEstimation theoryCramér–Rao boundTOA / TDOAMonte CarloTkinterPython
Urban clutter: walls, trees, two base stations, 40 pedestrians and light rain. The map-aware filter stays consistent.

Overview

A sensor network measures when a signal from a moving target arrives (TOA), or the differences between arrival times (TDOA). MTH Tracker turns those times into a position and velocity track and asks how close it comes to the best any unbiased estimator could do.

The answer is very close: on matched data the Monte Carlo RMSE is between 0.967 and 1.004 of the posterior Cramér–Rao bound for constant-velocity, constant-acceleration, Singer and coordinated-turn models, and the filter's self-reported uncertainty is consistent to within 3 %.

The report derives the mathematics, proves three propositions and presents ten experiments: geometry and GDOP, sensor count, noise sweeps, TOA vs TDOA, manoeuvres, occlusion and an environment ladder. Every figure and number in it is regenerated by one command.

What I built

  • Two plug-in axes: any motion model (CV, CA, Singer with exact Van Loan discretisation, CT) combines with any measurement model, and the EKF never branches on their type.
  • TOA and TDOA as one model z = D r(p)/c + D w; TDOA noise is simulated per sensor and then differenced, so its correlation appears in the data rather than being imposed.
  • Closed-form initialisation (Chan–Ho for three sensors), Gauss–Newton refinement and explicit resolution of ambiguous geometries; Joseph-form updates with one Cholesky factorisation and no explicit inverses.
  • NIS-gated adaptive process noise for manoeuvres and map-aware measurement noise from an environment model; 76 tests that run in under 10 s.

The interactive simulator (python -m mth_tracker gui)

Fig. 1A six-sensor ring tracking an abrupt zigzag with a CV model and NIS-gated adaptive process noise.
Fig. 2The default scene: figure-eight target, four sensors, live NEES, NIS and GDOP.
Fig. 3Monte Carlo RMSE against the posterior Cramér–Rao bound.
Fig. 4Reference trajectories and their estimated tracks.
Fig. 5Sensor geometry and geometric dilution of precision.
Fig. 6TOA vs TDOA inside the array and as the target leaves it.
Fig. 7Adaptive process noise on a manoeuvring target.
Fig. 8The environment ladder: from free space to full clutter.

Run it

$ pip install -e .
$ python -m mth_tracker gui
$ python -m mth_tracker run --trajectory coordinated_turn --motion CT --runs 50
$ python -m mth_tracker reproduce   # every figure and number, < 3 min