IMU Accelerometer + Gyroscope Complementary Filter Calculator

Calculates the fused angle estimate using a complementary filter that combines high-frequency gyroscope integration with low-frequency accelerometer angle measurement. The filter balances gyroscope drift correction against accelerometer noise.

Accelerometer Inputs

Gyroscope Inputs

Filter Parameter

α close to 1 → trust gyroscope more; α close to 0 → trust accelerometer more
Results will appear here.

Formulas Used

1. Accelerometer Angle

Roll:   θ_accel = atan2(ay, az)
Pitch: θ_accel = atan2(−ax, √(ay² + az²))

2. Gyroscope Integration

θ_gyro = θ_prev + ω · Δt

where ω is the gyroscope angular rate (°/s) and Δt is the time step (s).

3. Complementary Filter

θ_fused = α · θ_gyro + (1 − α) · θ_accel

α = filter coefficient (0 ≤ α ≤ 1)
High α → trusts gyroscope (good for fast dynamics, prone to drift)
Low α → trusts accelerometer (good for slow dynamics, noisy for vibration)

4. Equivalent Time Constant & Cutoff Frequency

τ = α · Δt / (1 − α)
fc = 1 / (2π · τ)

The filter acts as a high-pass filter on the gyroscope and a low-pass filter on the accelerometer, with crossover at fc.

Assumptions & References

  • The accelerometer measures gravity plus linear acceleration noise; best results when the sensor is quasi-static or slowly moving.
  • The gyroscope rate is in degrees per second (°/s); convert from rad/s by multiplying by 180/π if needed.
  • Δt should match the actual sensor sampling interval for accurate integration.
  • Typical α values: 0.95–0.99 for most IMU applications at 100 Hz sampling.
  • The complementary filter does not correct for gyroscope bias drift over long periods; a Kalman or Mahony filter is preferred for that.
  • Accelerometer angle assumes gravity dominates; large linear accelerations will corrupt the angle estimate.
  • Reference: Mahony et al. (2008), "Nonlinear Complementary Filters on the Special Orthogonal Group"; Madgwick (2010), "An efficient orientation filter for inertial and inertial/magnetic sensor arrays."
  • Formula derivation: Euston et al. (2008), "A complementary filter for attitude estimation of a fixed-wing UAV."

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