Calibration, filtering and sensor fusion basics

Raw sensor readings are never quite right. Earlier lessons showed that every sensor has biases, scale errors, misalignments, noise and drift, and that no single sensor covers every situation. This lesson brings those threads together into three practical skills. Calibration removes the systematic errors that can be measured and modeled. Filtering reduces random noise at the cost of delay. And sensor fusion combines sensors whose errors differ, so that each covers the other's weaknesses. The complementary filter developed here is the simplest useful fusion method and runs on countless drones and balancing robots; the Kalman filter, introduced briefly at the end, is developed properly in ROB-310.

Calibration: removing systematic error

Calibration compares a sensor's output with known references and fits a model that corrects it. The simplest model is linear, with an offset b and a scale factor s:

Two reference points determine both parameters; more points, fitted by least squares, average out noise and reveal nonlinearity. Multi-axis sensors need more. A three-axis accelerometer or magnetometer is modeled with a bias vector and a 3 × 3 matrix that captures scale factors, cross-axis coupling and misalignment. A classic procedure for accelerometers places the sensor in six orientations, each axis pointing up and then down, so that each axis sees +1 g and −1 g; the average of each pair gives the bias, and half the difference gives the scale. Magnetometers are calibrated by rotating the robot through as many orientations as possible: ideally the readings lie on a sphere, but hard-iron errors shift its center and soft-iron errors stretch it into an ellipsoid, and fitting the ellipsoid gives the correction.

Calibration also covers the geometry between sensors, called extrinsic calibration: the exact transform from a camera, lidar or IMU to the robot's base frame (ROB-210). An error of one degree in a lidar's mounting angle places an obstacle 20 m away about 35 cm from where it really is. Camera-lidar, camera-IMU and hand-eye calibration procedures estimate these transforms from data, and the results are stored in the robot's transform tree. Calibration is never finished: temperature, vibration, wear and knocks change sensors over time, so robots need routine checks and often online calibration that estimates biases continuously during operation.

Filtering noise: moving averages and low-pass filters

Random noise cannot be calibrated away, but it can be reduced by filtering. A moving average of the last N samples cuts white noise by \( \sqrt{N} \), as lesson 1 showed, but it delays the signal by about \( (N-1)/2 \) samples. A more economical alternative is the first-order low-pass filter, also called an exponential moving average, which needs only the previous output:

The parameter \( \alpha \) between 0 and 1 sets the smoothing: values near 1 give heavy smoothing and slow response, values near 0 follow the raw signal. It corresponds to a time constant \( \tau \) and a cutoff frequency \( f_c = 1/(2\pi\tau) \). Every filter of this kind trades noise for lag, and lag in a feedback loop reduces stability margins (ROB-230 and ROB-300), so filters inside control loops must be chosen carefully. Outliers, such as a single wild lidar return or an encoder glitch, are better handled with a median filter or a validity check than with averaging, which smears them into neighboring samples.

Complementary sensors and the complementary filter

The real power of fusion comes from combining sensors whose errors live at different frequencies. Lesson 3 gave the classic example. A gyroscope measures rotation rate precisely and responds instantly, but its integrated angle drifts slowly: its errors are low-frequency. An accelerometer gives tilt from gravity with no drift, but is noisy and disturbed by every bump and acceleration: its errors are high-frequency. A complementary filter takes the high-frequency part of the gyroscope angle and the low-frequency part of the accelerometer angle and adds them, so each sensor contributes where it is good.

The weights sum to one, which is where the name comes from. The crossover time constant \( \tau = \alpha\Delta t/(1 - \alpha) \) decides which sensor dominates at which timescale: for changes faster than \( \tau \) the gyroscope wins, for slower changes the accelerometer corrects the drift. Typical values are \( \alpha = 0.98 \) at 100 Hz, giving a crossover of about half a second. The filter is a few lines of code, runs on the smallest microcontroller and works remarkably well, which is why it is found in self-balancing robots, gimbals and hobby drones.

import numpy as np
rng = np.random.default_rng(3)
dt, n = 0.01, 3000                       # 100 Hz for 30 s
t = np.arange(n) * dt
true = 10 * np.sin(0.5 * t)              # true pitch, degrees
rate = np.gradient(true, dt)
gyro = rate + 0.8 + rng.normal(0, 0.3, n)              # 0.8 deg/s bias plus noise
acc = true + rng.normal(0, 3.0, n)                     # tilt from accelerometer, noisy
gyro_only = np.cumsum(gyro) * dt
alpha, est = 0.98, np.zeros(n)
for k in range(1, n):
    est[k] = alpha * (est[k-1] + gyro[k] * dt) + (1 - alpha) * acc[k]
def rms(e): return np.sqrt(np.mean(e**2))
print(f"accelerometer only : RMS error {rms(acc - true):5.2f} deg")
print(f"gyroscope only     : RMS error {rms(gyro_only - true):5.2f} deg (drifts)")
print(f"complementary 0.98 : RMS error {rms(est - true):5.2f} deg")

accelerometer only : RMS error  3.01 deg
gyroscope only     : RMS error 14.03 deg (drifts)
complementary 0.98 : RMS error  0.46 deg

The output shows the pattern clearly. Each sensor alone is poor, one noisy and one drifting, but the fused estimate is far better than either. This is the central idea of sensor fusion, and every more sophisticated method builds on it.

Toward the Kalman filter

The complementary filter uses a fixed blend. The Kalman filter, developed by Rudolf Kálmán in 1960 and used to navigate the Apollo spacecraft, generalizes the idea: it keeps an estimate of the state and of its uncertainty, predicts both forward with a motion model, and when a measurement arrives, blends prediction and measurement with a gain that depends on their relative uncertainties. A precise measurement gets a large weight; a noisy one a small weight. For a single quantity, the update looks like the complementary filter with a gain that adapts automatically:

Here \( P^{-} \) is the variance of the prediction and R the variance of the measurement noise, which is exactly what datasheet noise figures and the Allan analysis of lesson 3 provide. The extended and unscented Kalman filters, and particle filters, handle nonlinear robot models; they are the subject of ROB-310, and they underpin the localization and SLAM systems of ROB-340.

Time synchronization and practical fusion

Fusion only works if the measurements being combined refer to the same moment and the same frame. A camera image that is 60 ms old fused with a fresh IMU reading will place features in the wrong position whenever the robot moves. Good systems timestamp every measurement at acquisition, preferably in hardware, synchronize clocks between computers (for example with the Precision Time Protocol), trigger sensors together where possible and interpolate measurements to common times. In ROS 2, message headers carry timestamps and frame IDs for this reason, and tf2 interpolates transforms in time (ROB-110). Many baffling fusion bugs turn out to be a timestamp in the wrong clock, a missing extrinsic calibration or a frame mix-up rather than a problem with the filter.