Log-based tuning

Why this matters

UAT 303 used an FFT to find where vibration sits in frequency. This module uses the same data to set filters so that the control loop does not respond to motor vibration. Without filtering, the rate loop’s D term amplifies noise until the motors run hot; with too much filtering, the loop gains delay and oscillates (Module 1). The drone knowledge base’s unit on analysing filters from flight logs teaches reading spectra and filter effects offline, and its PX4 Flight Review and PlotJuggler units show log tools.

Notch and harmonic notch

A notch filter reduces a narrow band around frequency $f_0$ and passes the rest. The prototype from the W3C Audio EQ Cookbook is $H(s)=(s^2+{\omega }_0^2)/(s^2+({\omega }_0/Q)s+{\omega }_0^2)$ with bandwidth $BW=f_0/Q$. ArduPilot uses a harmonic notch, placing filters at the fundamental and its harmonics, set with INS_HNTCH_ENABLE, INS_HNTCH_FREQ (default 80 Hz), INS_HNTCH_BW (40 Hz, typically half the base frequency), INS_HNTCH_ATT (40 dB depth) and INS_HNTCH_MODE, which selects the frequency source, for example 1 = throttle, 3 = ESC telemetry (which the documentation calls generally the best) and 4 = in-flight FFT. Motor frequency changes with throttle, so a fixed notch can miss when flying differently from when it was set.

import cmath
import math

F0, BW = 80.0, 40.0
Q = F0 / BW

def notch(f):
    s = 1j * f / F0
    return (s * s + 1) / (s * s + s / Q + 1)

for f in (8, 60, 80, 83, 100, 160):
    h = notch(f)
    db = 20 * math.log10(max(abs(h), 1e-5))
    print(f"{f:>3} Hz: {db:7.1f} dB, phase {math.degrees(cmath.phase(h)):6.1f} deg")

  8 Hz:    -0.0 dB, phase   -2.9 deg
 60 Hz:    -2.4 dB, phase  -40.6 deg
 80 Hz:  -100.0 dB, phase    0.0 deg
 83 Hz:   -16.7 dB, phase   81.6 deg
100 Hz:    -3.5 dB, phase   48.0 deg
160 Hz:    -0.5 dB, phase   18.4 deg

Low-pass filters and phase

Copter’s INS_GYRO_FILTER defaults to 20 Hz and is a second-order low-pass filter in the source code. Texas Instruments’ note states that a second-order low-pass has 90° of phase lag at the cutoff and approaches 180° well above it. Even below the cutoff, the filter adds phase lag to the loop.

import math

F_CROSS = 8.0
for fc in (40, 20, 10):
    r = F_CROSS / fc
    phase = -math.degrees(math.atan2(math.sqrt(2) * r, 1 - r * r))
    print(f"gyro filter {fc:>2} Hz: phase at {F_CROSS:.0f} Hz = {phase:6.1f} deg")

gyro filter 40 Hz: phase at 8 Hz =  -16.4 deg
gyro filter 20 Hz: phase at 8 Hz =  -34.0 deg
gyro filter 10 Hz: phase at 8 Hz =  -72.3 deg

The log-based tuning process

ArduPilot’s tuning instructions follow an order: fix mechanical vibration first, set filters from the spectrum, then tune the rate and angle loops. After every change, fly a new log and compare with the previous one. Tools such as ArduPilot WebTools (FilterReview), PX4 Flight Review and PlotJuggler show spectra before and after filtering and compare desired with actual values.

Module lab

Common mistakes

Summary

• A notch cuts a narrow band; if the motor frequency drifts from the centre, the depth falls quickly
• ArduPilot’s harmonic notch can track motor frequency from throttle, ESC telemetry or FFT
• Low-pass filters add phase lag even below their cutoff, so lowering the cutoff risks stability
• Tune in order: fix vibration, set filters, then tune the loops, comparing logs each time

Check your understanding

1. A notch with $f_0$ 100 Hz and 50 Hz bandwidth has what $Q$?
2. A motor at 5,400 rpm has what fundamental frequency?
3. What is the phase lag of a second-order low-pass at its cutoff?
4. Why does a notch affect stability less than lowering a low-pass cutoff?
5. Which INS_HNTCH_MODE does the documentation call generally the best?