48 Complementary Filters: Sensor Trust

48.6 Complementary Filter Trust

A complementary filter is a lightweight IMU fusion method that combines gyroscope integration with an accelerometer gravity reference. The gyroscope is responsive over short intervals, but bias and noise accumulate into drift when angular rate is integrated. The accelerometer gives an absolute roll and pitch reference from gravity when the device is not accelerating hard, but it is noisy during vibration and linear motion.

The filter works because those errors are complementary. It trusts the gyroscope for fast changes and uses a small accelerometer correction to pull the estimate back toward the gravity reference over time. In signal terms, the gyroscope path behaves like a high-pass contribution and the accelerometer path behaves like a low-pass correction.

For three-dimensional orientation, name the axis convention before naming the filter. In the usual aircraft convention, roll is rotation about the forward x-axis, pitch is rotation about the lateral y-axis, and yaw is rotation about the vertical z-axis. Wearable systems also need the body-plane mapping: sagittal, coronal, and transverse planes must be tied to the sensor mounting frame. Store that transform with the data; otherwise a correct filter can publish inverted or swapped roll, pitch, and yaw.

Before touching any filter mathematics, pin the vocabulary to a body. Figure 48.1 puts a chest-worn IMU on a person and shows why two of the three angles get a free referee while the third does not.

Walk the three cards in Figure 48.1 downward. Bowing forward is motion in the SAGITTAL plane, so it rotates the trunk about the left–right axis — that is what the filter will call pitch, and the card marks Gravity reference: YES because the tilt swings gravity across the sensor axes. Side-bending does the same job for roll in the CORONAL plane. The TRANSVERSE plane card is the odd one out: twisting about the vertical never moves the gravity vector, so its chip reads Gravity reference: NO. That one amber card is the entire reason this chapter keeps insisting that yaw needs a magnetometer or another heading source — the accelerometer is blind to it by geometry, not by quality.

Phoebe’s Field Notes: Why the Complementary Filter Behaves Like a Physics Ledger

The mathematical gist. The chapter’s 10 ms update first integrates the gyro from 8.00° to 8.12°, then blends 98% of that prediction with 2% of the accelerometer’s 6.00° tilt to obtain 8.0776°. That small correction corresponds to a 0.495 s time constant and leaves 1.82° of an initial 5° drift after 50 equal updates.

Math Bridge · guided foundationsHow does 0.98 turn gyro motion into a stable angle?Let Data Dora integrate one step, blend two sensors, and price the remaining drift. Learn the maths

The fusion diagram in Figure 48.2 turns the worked roll update into a complete evidence path and exposes the separate requirement for yaw.

Read Figure 48.2 from calibrated gyroscope and accelerometer inputs into the roll-and-pitch filter, then inspect the magnetometer branch that supplies heading evidence for yaw. The lower comparison explains the blend: the gyro is responsive but drifts, while gravity supplies a slower tilt reference when linear acceleration is controlled. This connects the worked alpha calculation to the running physics ledger: every output axis must name its reference, calibration, sample period, and conditions of observability.

Overview Knowledge Check

48.7 Tune Alpha by Motion and Rate

The common one-axis complementary update predicts attitude from the gyroscope, then blends that prediction with the accelerometer-derived attitude. The same idea can be applied to roll and pitch separately, or implemented in quaternion form for a full orientation pipeline.

The alpha value is not a universal constant. It should be chosen from the update period, expected vibration, allowed drift, and how often the accelerometer is a valid gravity reference. A higher alpha makes the output smoother during vibration but slower to correct gyro drift. A lower alpha corrects drift faster but can inject acceleration and vibration into the attitude estimate.

Keep raw and calibrated IMU traces separate during review. A stationary calibrated gyroscope should sit near zero angular rate, while the calibrated accelerometer should show two axes near 0 g and the gravity-facing axis near +1 g or -1 g depending on mounting. Magnetometer channels should be stable in the local field but still need hard-iron and soft-iron checks before they are trusted for heading.

What does “raw” actually look like before anyone applies units? Figure 48.3 shows thirty motionless samples exactly as the driver delivers them — as integer codes.

None of the numbers in Figure 48.3 is a physical unit yet. The gyroscope panel wobbles inside a ±50 LSB band, and its card gives the decoder ring: 131 LSB per °/s on a ±250 °/s range. The accelerometer’s most important teaching is the z trace pinned near −16,384 LSB — that is not a fault, it is gravity expressed at 16,384 LSB per g, and it will still be there in every “stationary” recording you ever review. The magnetometer’s three levels differ from each other because the local field plus hard-iron bias set a private offset per axis. Flat and boring is the healthy signature here; the next two figures show what turns these codes into checkable physics.

Alpha tuning is meaningless until the inputs have a healthy stationary baseline. Inspect the three charts in Figure 48.4 to establish that baseline before comparing drift under motion.

The top chart in Figure 48.4 shows Gyroscope — near-zero angular rate, with x, y, and z close to zero. In Accelerometer — gravity on one axis only, z holds about -1g while x and y remain near 0g. The Compass — stable local field, no drift panel keeps three separate offsets without a trend. These are calibration signatures, not interchangeable zero targets; the later complementary filter can be reviewed only after each sensor matches its own physical reference.

The step between those two figures — raw codes in, calibrated traces out — deserves its own arithmetic. Figure 48.5 walks one sample per sensor through both corrections so the calibrated traces stop being magic.

Read one row of Figure 48.5 end to end. The accelerometer’s −16,650 LSB becomes −1.016 g at step 1 · APPLY SCALE, and only the second step, 2 · REMOVE BIAS, brings it to the clean −1.000 g the calibrated chart displayed. The order matters for review: scale converts codes to units, bias centres the units on the physics, and the STATIONARY TARGET column — 0 °/s, −1 g, steady field — is what a reviewer can actually check. If a “calibrated” trace misses its target at rest, one of these two numbers was wrong, and no amount of filter tuning downstream will repair it.

Gyroscope angle is obtained by integrating angular rate, commonly with a rectangle or trapezoidal numerical rule. That integration also accumulates bias and white noise: a class trace can show tens of degrees of apparent angle drift over about 30 seconds even though the gyroscope itself is unaffected by earth gravity. Accelerometer tilt estimates avoid long-term drift but are noisy and sluggish when filtered, so they are best treated as roll and pitch correction evidence rather than a complete attitude solution.

“Integrating angular rate” sounds abstract until you see that each 10 ms interval simply contributes one slice of area under the rate curve. Figure 48.6 zooms into a single slice below and prices the difference between the two common rules.

Read the shaded slice in Figure 48.6: it sits between two real samples, ω[k−1] = 10.0 °/s and ω[k] = 12.0 °/s, over one Δt = 0.01 s interval. The trapezoid averages the two rates and books 0.11° of angle; the dashed rectangle at the newer sample’s height would book 0.12°, overpaying by the red sliver — which is exactly why the chapter’s quick 8.00 → 8.12 update is the rectangle answer while a trapezoidal integrator lands on 8.11. Note what the side card refuses to promise: a finer Δt shrinks that geometric sliver, but any rule faithfully integrates whatever bias hides in ω. Sharper integration buys motion accuracy, never drift immunity.

So how bad can that integrated bias get? Figure 48.7 runs the classic classroom experiment: rock a board back and forth between 0° and 90° for half a minute and watch the integrated angle in the lower panel walk away.

The top panel of Figure 48.7 looks perfectly healthy — the rocking is symmetric and no single sample is suspicious — yet the dashed line marks its mean = −1.7 °/s, not 0. The integral sign between the panels is doing the damage: every biased sample is added and never forgotten, so the red card’s bill, error = bias × time, reaches −1.7 °/s × 30 s ≈ −51° by the right edge. The card also separates the two failure currencies: noise integrates into a slow random walk that grows like √t, while bias buys a straight ramp that grows like t. Calibration shrinks the ramp’s slope; only an external reference — gravity, in this chapter — can keep paying it off.

The blend factor only makes sense after seeing what each IMU channel gets wrong over the same motion. The paired chart in Figure 48.8 holds the time window constant so the tuning discussion can separate accumulated bias from sample-by-sample noise.

Compare the two traces in the chart Figure 48.8 at 0s and 30s. The Gyro-integrated angle accumulates small angular-rate errors until the annotation marks roughly 50 deg drift by t = 30s. The Accelerometer-derived angle remains noisy on each sample, yet gravity resets its reference so it has no matching long-term drift. A larger alpha trusts the fast gyro branch for short motion; the complementary term must still pull the estimate toward the driftless gravity reference. The chosen alpha and sample interval therefore belong in the review record because motion spectrum and update rate set how these two labelled failure modes trade off.

Common accelerometer tilt estimates
pitch = atan2(-ax, sqrt(ay^2 + az^2))
roll  = atan2( ay, az )

Axis signs vary by board and mounting. Record the coordinate frame,
unit conversion, and body transform before comparing values across devices.

Those two atan2 lines have a picture behind them, and it is worth seeing once properly. Figure 48.9 draws the 20°-rolled device and shows where the numbers inside atan2 come from.

In Figure 48.9 the device’s blue z — device up axis leans 20° away from the dashed true vertical, and gravity — which never moves — casts two shadows onto the body frame: ay = 0.34 g on the lateral axis and az = 0.94 g along the device’s vertical. atan2 of those two shadows returns exactly the roll = 20° the geometry started with, and because the answer is a ratio, the strength of gravity cancels out — nothing is integrated, so nothing can drift. The amber card carries the licence condition the formulas hide: the magnitude check √(0.34² + 0.94²) = 1.00 g is what certifies the shadows as gravity’s. When that magnitude leaves 1 g, the same trigonometry runs happily on the wrong force — which is precisely the failure the validity-gate section below is built to catch.

Worked example: one roll update at 100 Hz
previous roll estimate: 8.00 deg
gyro roll rate: 12.0 deg/s
sample interval dt: 0.01 s
accelerometer roll estimate: 6.00 deg
alpha: 0.98

gyro prediction:
roll_gyro = 8.00 + 12.0 * 0.01 = 8.12 deg

complementary update:
roll_new = alpha * roll_gyro + (1 - alpha) * roll_accel
roll_new = 0.98 * 8.12 + 0.02 * 6.00
roll_new = 7.9576 + 0.1200
roll_new = 8.0776 deg

Interpretation:
The estimate follows the gyro's fast motion but is nudged 0.0424 deg toward
the accelerometer gravity estimate during this 10 ms update.

The arithmetic above is correct but linear text hides the architecture. Figure 48.10 lays the same update out as the signal loop it really is, with every intermediate value on the wire.

Two bands in Figure 48.10 carry the whole philosophy. The FAST PATH does almost all the work — its prediction arrives at the Σ node carrying 7.9576° of the answer — while the SLOW CORRECTION contributes a humble 0.1200°. The part the worked example cannot show is the feedback edge across the top: the fresh 8.08° becomes next cycle’s “previous θ”, so the 0.02 correction is applied again 100 times every second. That repetition is what the left chip prices as τ = Δt/(1−α) = 0.50 s, and the middle chip translates into a crossover near 0.32 Hz — below it the gravity reference wins, above it the gyro does. The amber chip repeats this chapter’s standing warning at the exact place designers forget it: yaw never enters this loop, so publish its drift label with the output.

Correction time-scale example
sample interval dt: 0.01 s
alpha: 0.98

approximate time constant:
tau = dt / (1 - alpha)
tau = 0.01 / 0.02 = 0.50 s

If gyro integration starts 5.0 deg away from the gravity reference:
after 0.50 s, remaining error is about 5.0 * exp(-1) = 1.84 deg
after 1.00 s, remaining error is about 5.0 * exp(-2) = 0.68 deg
after 2.50 s, remaining error is about 5.0 * exp(-5) = 0.03 deg

Interpretation:
At 100 Hz and alpha 0.98, drift correction is visible within seconds,
but each individual update still heavily favors the gyro prediction.

Practitioner Knowledge Check