Math Bridge: Complementary IMU fusion

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How does 0.98 turn gyro motion into a stable angle?

Integrate, weight, add, and expose what the filter leaves behind.

Data Dora, the guideData Dora guides
The one targetCalculate one complementary-filter update and its drift time scale.
The chapter case8.00°; 12.0°/s; 0.01 s; 6.00°; α=0.98.
What it buys youA filter setting tied to sample time and sensor evidence.

A field team faces an unresolved physical question: How does 0.98 turn gyro motion into a stable angle? They must answer it before changing alpha on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is alpha. The middle card applies this page's relationship. The green card is gyro prediction. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Alpha changes gyro prediction An input card leads through the page relationship to the gyro prediction result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The same α means a different physical filter if Δt changes, so record both together.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for alpha is 0.98.

  2. 2

    Name the relationship. gyro: 8.00 + 12.0x0.01 = 8.12° blend: 0.98x8.12 + 0.02x6.00 = 8.0776° correction = 0.0424° τ = -0.01/ln(0.98) = 0.495 s; fc ≈ 0.32 Hz

  3. 3

    Substitute the chapter fixture. Set alpha to 0.98. The page ledger gives gyro prediction as 8.120 degrees.

  4. 4

    Read the result. Keep degrees beside the value. Use it only inside the technical boundary on this page.

Predict, then change alpha

Try Predict the direction of gyro prediction. Move one control, calculate, then check your prediction.

0.98
Chapter baseline
Gyro prediction

Observe The same α means a different physical filter if Δt changes, so record both together. Reset the control to 0.98 and compare gyro prediction.

Explain Only alpha moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only alpha moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only alpha. It replays the chapter fixture; it is not a field prediction.

1. Ask what each sensor measures

The gyroscope reports turn rate, so time integration makes angle and also accumulates bias. The accelerometer can point to gravity when the device is calm, but motion can fool it. The filter gives fast motion to the gyro and slow correction to gravity.

Data Dora: Alpha is a weight, not a magic confidence score.

2. Name every algebra move

1

Multiply by timeGyro step = ωΔt.

2

Add the old angleθg=θold+ωΔt.

3

Weight both viewsθ=αθg+(1−α)θa.

4

Expose the clockτ=−Δt/ln(α).

3. Reproduce the chapter update

gyro: 8.00 + 12.0×0.01 = 8.12°
blend: 0.98×8.12 + 0.02×6.00 = 8.0776°
correction = 0.0424°
τ = −0.01/ln(0.98) = 0.495 s; fc ≈ 0.32 Hz

After 50 equal updates, 5.0×0.98⁵⁰ = 1.82° of an initial 5° error remains.

4. Try the gyro weight

TryMove α and watch both the immediate correction and the longer drift time change.

Alpha
Gyro prediction
Accel weight
Fused angle
Correction
Time constant
Crossover
Error after 50

ObserveA larger α follows the gyro more closely now but pays down drift more slowly.

ExplainThe same α means a different physical filter if Δt changes, so record both together.

Technical boundaries.

This one-axis blend assumes a trustworthy gravity reference.

Motion
Braking and vibration can corrupt accelerometer tilt
Bias
Gyro calibration still matters
Yaw
Gravity cannot observe heading

Use axis conventions, gravity gates, timestamps, and a heading source where required.

5. Gate the slow correction

Apply accelerometer correction only when measured acceleration is close enough to gravity for the product's motion. Otherwise coast on calibrated gyro evidence and label the estimate degraded.

6. Version the filter

Record sample period, α, gyro bias, axis signs, gravity gate, startup state, yaw source, and what happens after timing gaps or sensor saturation.

7. Check yourself

What does the gyro predict?
Answer: 8.12° after adding 12.0°/s × 0.01 s to 8.00°.
Why is the fused angle lower?
Answer: The 2% accelerometer weight pulls the 8.12° prediction toward 6.00°.
Can this blend recover yaw?
Answer: No. Gravity gives roll and pitch reference, not heading.
Honesty boundary.

All constants and the one-step result come from the chapter's roll example.

8.00°, 12.0°/s, 0.01 s
Chapter gyro step
6.00° and α=0.98
Chapter blend
50 samples and 5°
Chapter drift check

Correct, not complete: one ideal blend does not qualify an IMU attitude estimator.

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