ComplementaryFilter

A complementary filter is a simple sensor fusion method used to combine measurements from two sensors with complementary frequency characteristics to estimate a quantity such as orientation or angle. It leverages the strengths of each sensor: one provides accurate low-frequency information, while the other supplies high-frequency information, producing a more robust estimate than either sensor alone.

In attitude estimation for inertial measurement units, a common use is to fuse gyroscope data with accelerometer data. The standard form for a one-dimensional angle estimate is: angle_hat_k = alpha * (angle_hat_{k-1} + gyro_rate_k * dt) + (1 - alpha) * angle_accel_k. Here angle_accel_k is derived from accelerometer measurements (for example, pitch or roll from arctan relationships), gyro_rate_k is the angular rate from the gyroscope, dt is the time step, and alpha (0 < alpha < 1) weights the contribution of the integrated gyro signal versus the accelerometer measurement. A high alpha favors the gyro (good for fast motion but prone to drift), while a low alpha favors the accelerometer (stable orientation reference but noisy during dynamics).

Variants may implement the filter in Euler angles or quaternions, and can be extended to 2D or 3D but retain the same principle of combining a high-pass (gyroscope) and a low-pass (accelerometer) term. Complementary filters are valued for their simplicity and low computational cost but can be sensitive to non-gravitational accelerations and require careful tuning of alpha.