field guide

Complementary filter for attitude estimation

Fusing a drifting gyro and a noisy accelerometer into one roll/pitch estimate by splitting the frequency domain - the simplest useful attitude filter.

updated 2026-06-17 state-estimation sensor-fusion

The raw MPU-6050 gives accelerometer (m/s²) and gyroscope (rad/s) readings. Neither alone is sufficient for attitude estimation — the complementary filter fuses them into a single roll/pitch estimate.

The problem

Sensor	Strength	Weakness
Gyro	Fast, clean angular velocity	Integrates to angle, but biases accumulate → drift
Accel	Absolute roll/pitch reference at rest (gravity direction)	Noisy under vibration, corrupted by linear acceleration

Each sensor covers what the other lacks. The complementary filter exploits this by splitting the frequency domain: gyro dominates at high frequencies, accel at low frequencies.

The filter (roll and pitch)

One line per axis:

roll  = (1 - α) · (roll_prev  + p · dt) + α · roll_accel
pitch = (1 - α) · (pitch_prev + q · dt) + α · pitch_accel

Where roll_accel and pitch_accel are computed from the accelerometer (see accel-roll-pitch-derivation for the full derivation):

float roll_accel  = atan2f(ay, az);
float pitch_accel = atan2f(-ax, sqrtf(ay*ay + az*az));

These use the fact that at rest, the accelerometer measures only gravity — so the direction of the gravity vector in the sensor frame directly gives the tilt angles.

Choosing alpha

α is the cross-fade frequency. The crossover point is:

$f_c = \frac{\alpha}{2\pi \cdot dt}$

α (at 200 Hz)	Crossover	Behavior
0.5	~16 Hz	Heavy accel blend — jittery
0.02	~0.3 Hz	Good for hover, moderate flight
0.01	~0.16 Hz	Better for aggressive flight, slower drift recovery
0.001	~0.016 Hz	Almost pure gyro — minutes of drift

For a quadcopter in attitude-hold or gentle flight, α = 0.02–0.05 works well. For acro/rate mode where sustained acceleration is common (the accel no longer points at gravity), drop to α = 0.005–0.01.

What it doesn’t do

Yaw. There’s no magnetic reference in a 6-DOF IMU. Yaw is pure gyro integration and will drift. This is acceptable for rate mode — the pilot commands yaw rate, and relative heading is all that matters.
Absolute position. The complementary filter estimates attitude only. Position requires additional sensors (barometer, GPS, optical flow).

On hardware

The filter runs inside the control task, after mpu6050_read() and before control_update(). The gyro integration step (p · dt) uses the same dt as the control loop. The roll_accel and pitch_accel terms can be computed at a lower rate (every 4th tick, 50 Hz) since the accel is slow-changing — saves CPU cycles on the microcontroller.

The pitch formula sign assumes the sensor is mounted with +x forward, +y right, +z down (NED). If the sensor mount flips any axis, adjust the sign in the atan2 calls — not in the control code.