Complementary Filter

Practical

A Complementary Filter is the lightweight, no-math-degree way to fuse two sensors that fail in opposite ways. It matters because the classic robotics problem — estimating a robot’s tilt angle from an IMU — has no single trustworthy sensor: the gyroscope is accurate moment-to-moment but slowly drifts, while the accelerometer is stable over the long run but jittery in the short term. The filter blends them so each covers the other’s weakness, and it does so with a single line of arithmetic.

Prerequisites

State Estimation The broader problem of inferring true state from imperfect sensors.

IMU The inertial measurement unit supplying the gyro and accelerometer signals.

How it works

  gyro rate ──▶ integrate (∫ω·dt) ──▶ HIGH-PASS ──┐
  (drifts slowly)                                  ├──▶ fused tilt angle
  accel ──▶ atan2(gravity) ──▶ LOW-PASS ───────────┘
  (noisy/jittery)

The entire filter is one update per timestep:

angle = α·(angle + gyro·dt) + (1−α)·accel_angle

angle — the current best tilt estimate, carried over between steps.
gyro·dt — the gyroscope’s angular rate multiplied by the timestep dt, integrating rate into an angle increment.
accel_angle — the tilt computed directly from the gravity direction via atan2.
α (alpha) — the blend constant, typically 0.95–0.99.

With α near 1, the filter mostly trusts the integrated gyro (smooth, responsive) and lets the accelerometer term slowly pull the estimate back toward truth — cancelling gyro drift over time. Smaller α leans on the accelerometer and reacts faster but admits more jitter.

Runnable example

import numpy as np

# --- Simulation setup ---
dt = 0.01                      # 100 Hz loop
t = np.arange(0, 10, dt)
true_angle = 0.5 * np.sin(0.5 * t)        # true tilt (radians)
true_rate = np.gradient(true_angle, dt)   # true angular rate

rng = np.random.default_rng(0)
gyro_drift = 0.02              # rad/s constant bias -> integrates into drift
gyro = true_rate + gyro_drift + rng.normal(0, 0.01, t.size)
# accelerometer "sees" gravity tilt, but noisy
accel_x = np.sin(true_angle) + rng.normal(0, 0.08, t.size)
accel_y = np.cos(true_angle) + rng.normal(0, 0.08, t.size)
accel_angle = np.arctan2(accel_x, accel_y)   # tilt from gravity direction

# --- Complementary filter ---
alpha = 0.98
angle = accel_angle[0]         # seed with first accel reading
estimate = np.empty(t.size)
gyro_only = np.empty(t.size)
g = accel_angle[0]
for i in range(t.size):
    angle = alpha * (angle + gyro[i] * dt) + (1 - alpha) * accel_angle[i]
    g += gyro[i] * dt          # naive integration, for comparison
    estimate[i] = angle
    gyro_only[i] = g

rmse = lambda a: np.sqrt(np.mean((a - true_angle) ** 2))
print(f"gyro-only RMSE:     {rmse(gyro_only):.4f} rad  (drifts)")
print(f"accel-only RMSE:    {rmse(accel_angle):.4f} rad  (noisy)")
print(f"complementary RMSE: {rmse(estimate):.4f} rad  (best)")

Run it: the fused estimate beats both the drifting gyro-only integration and the noisy accel-only angle.

Kalman Filter The optimal, covariance-tracking alternative — use it when you need uncertainty estimates or many sensors.

Sensor Fusion The general practice of combining multiple sensors into one estimate.

State Estimation Inferring hidden state from noisy measurements.

IMU The sensor package this filter most often runs on.

Sources

Higgins, W. T. “A Comparison of Complementary and Kalman Filtering.” IEEE Transactions on Aerospace and Electronic Systems, 1975. (The classic equivalence result.)
Colton, S. “The Balance Filter: A Simple Solution for Integrating Accelerometer and Gyroscope Measurements for a Balancing Platform.” MIT, 2007.
Madgwick, S. O. H. “An efficient orientation filter for inertial and inertial/magnetic sensor arrays.” University of Bristol Technical Report, 2010.