Pedestrian Dead Reckoning (Phone IMU)

Each sensor does exactly half the job, and neither half is useful alone. The accelerometer sees the impact of every footfall, so counting those peaks and multiplying by a step length gives distance — but it says nothing about which way you were pointed. The gyroscope measures rotation rate, so integrating it gives heading — but nothing about how far you travelled. Distance with no direction and direction with no distance are both worthless; position needs the two fused together, one heading per step.

The obvious alternative — double-integrate acceleration straight into position, the textbook approach — was tried and abandoned. A constant bias integrated once grows the error linearly, which is survivable over a two-minute walk. Integrated twice it grows quadratically, and on this data that path produced over 50 % error: unusable. That is the reason real pedestrian systems count steps instead of integrating acceleration, and it is the design decision the rest of the pipeline is built around.

The phone was not rigidly mounted, so its orientation drifts during the walk and no single axis reliably carries the footfall signal. Taking the magnitude √(x² + y² + z²) sidesteps that entirely — magnitude is rotation-invariant, so a tilted phone reads the same as a level one. Subtracting the mean removes the gravity DC term and leaves the walking oscillation centred on zero.

An FFT of the walking segment put the cadence at about 1.4 Hz, so the band-pass was set to 0.7–2.1 Hz around it: low enough to reject residual gravity and postural sway, high enough to reject sensor noise and arm jitter, leaving only the stepping rhythm. It runs as a 4th-order Butterworth applied with filtfilt — forward and backward — so the filter contributes zero phase lag and the detected peaks stay where the footfalls actually happened. Peak detection with a minimum spacing of fs / 2.1 samples then places one detection per footfall.

Step length is derived, not assumed. Dividing the known 76.2 m route by the detected step count gives 0.508 m/step for Run A. Assuming a textbook stride would have quietly hidden the step-detection error described below — deriving it from ground truth is what made that error visible.

Heading comes from the gyroscope's z-axis — rotation about vertical, which is yaw. A MEMS gyro reads a small non-zero rate even when it is completely still, and that bias is what eventually destroys the heading estimate: integrated over two minutes, a constant offset becomes a steadily growing angular error. The ~20 s stationary block at the start of each recording exists to measure it. Averaging gyro-Z over that window gives the bias, which is subtracted from the whole trace before anything is integrated.

The corrected rate is then integrated once by the trapezoidal rule to turn rate into cumulative heading. As a sanity check, five laps of a rectangle should total 1800° of turning; Run A's integrated heading came to 1820°, about 1 % off, which confirms the integration and bias correction are behaving.

Dead reckoning is then the fusion step: interpolate the heading at each detected footfall time, and advance one step length in that direction, accumulating (x, y). Because the walk ends on the exact start mark, true displacement is zero, so the magnitude of the final position vector is the drift — no reference system, no alignment, just the length of the vector that shouldn't exist.

Run A reconstructs as five laps that visibly rotate instead of stacking on top of each other, ending 5.14 m from the true start — 6.7 % of the 76.2 m walked. That rotation is the drift made visible: each lap is laid down against a heading estimate that has decayed a little further than the last one.

One run proves nothing, so the whole pipeline was repeated on a second independent recording. Run B drifted 6.00 m, 7.9 % — the same ballpark, but a visibly different path shape, because each run carries its own gyro bias and its own phone tilt. That difference is itself the finding: drift is not a fixed offset you calibrate away once.

The stronger check is held-out. Step length is fitted per run, so evaluating it on the same run it was fitted to proves nothing. Taking Run B's step count and multiplying it by the step length derived from Run A gives 78.7 m against a true 76.2 m — about 3 % off. The parameter transfers between runs, so it isn't overfit to a single recording.

The derived step length landed around 0.5 m, which is short for a normal stride, and that discrepancy is the tell. Since step length is total distance divided by step count, a step length that comes out too short means the step count came out too high: the peak detector was reading spurious peaks as footfalls and over-counting. So the reconstruction carries a real distance error from step detection, not purely a heading error as the first pass concluded.

Heading is still drifting too — gyro bias plus phone tilt rotate each successive lap, which is what stops the loop from closing. Both error sources are in play, and separating them cleanly is exactly the kind of thing the failure analysis was for.

The fixes are algorithmic, not experimental. First, re-estimate the gyro bias during the walk instead of once at the start: bias drifts with time and temperature, and every footfall is a brief moment where the phone is near-stationary and the bias can be re-measured and updated. Second, add the magnetometer. The gyro is smooth and accurate short-term but drifts long-term; a compass is noisier moment to moment but gives an absolute heading that never drifts. Weighting each sensor where it is strongest — a complementary or Kalman filter — cancels the long-term drift. The takeaway is that you cannot fix MEMS drift by walking more carefully. You fix it in the algorithm.