How a complementary filter blends a gyro and accelerometer

A complementary filter blends two sensors that fail in opposite ways. A gyro measures how fast a board is turning; added up over time it gives a smooth angle that slowly drifts. An accelerometer measures which way gravity points; it gives an angle that never drifts but jitters. Trust the gyro for the short term, pull gently toward the accelerometer for the long term, and you get a tilt that's both steady and correct.

Two sensors, two flaws

A gyro reports a rate, degrees per second. To get an angle, add up rate × time on every sample. But no gyro is perfect: one that reads 2 °/s high when still adds a phantom 2° every second, and after 10 seconds the angle is 20° off. The error never stops growing.

An accelerometer, sitting still, measures gravity, so the tilt comes straight from its readings, with no adding up and no drift:

But any vibration or acceleration adds to gravity, so the angle jumps several degrees from sample to sample, and during a fast move it's confused by the motion itself.

One line blends them

Each step, the first part moves the angle by the gyro's rate, smooth and quick. The second part pulls it 2% of the way toward the accelerometer's angle. Fast changes come from the gyro, because 2% a step is too little to pass the jitter. Slow errors, like drift, get pulled out by that steady 2% before they can build up.

float angle = 0;

void loop() {
  float gyroRate = readGyro();      // degrees per second
  float accelAngle = readAccel();   // degrees, from atan2
  angle = 0.98 * (angle + gyroRate * 0.01) + 0.02 * accelAngle;
  delay(10);                        // 100 samples a second
}

Choosing the weight

The weight sets a time constant: errors slower than it are corrected by the accelerometer, changes faster than it come from the gyro.

With α = 0.98 at 100 samples a second, τ ≈ 0.5 s. At 0.5, half and half, the accelerometer's jitter comes straight through. At 0.99 the angle is calm, but the pull is too weak to hold a 2 °/s drift near zero, and it settles a couple of degrees off.

Where you'll meet it

Self-balancing robots, quadcopter flight controllers, camera gimbals and phones' motion sensing all fuse a gyro with an accelerometer. A Kalman filter does the same job with more maths and an estimate of each sensor's noise. On a microcontroller, this one line is usually enough.