Calibration Engineering - Ch.05: IMU Calibration [EN]

STATERA Library · Series 07 · Volume 10 · Chapter 05 · EN

Source: https://github.com/Statera-Library/07_Vol_10_Calibration_Engineering/blob/main/Chapter_05_IMU_Calibration/Chapter_05_IMU_Calibration_EN.md

Volume 10 Calibration Engineering Chapter 05. IMU Calibration

05.01. 6 Axis Bias Calibration

A six-axis inertial measurement unit combines a three-axis accelerometer and a three-axis gyroscope to measure linear acceleration and angular velocity along orthogonal X, Y, and Z axes.

Even when the sensor is perfectly stationary, its outputs rarely equal their theoretical values because electronic offsets, mechanical stresses, manufacturing tolerances, and internal signal-conditioning errors introduce bias. Six-axis bias calibration estimates these persistent offsets so that subsequent inertial measurements more accurately represent the physical motion of the robot or vehicle.

Accelerometer bias is the additive error present in each acceleration channel. For a stationary IMU, the measured acceleration should contain only the gravitational acceleration projected onto the sensor axes. If an axis is aligned upward or downward with gravity, its ideal magnitude is approximately one g, while the two perpendicular axes should ideally measure zero. Deviations from these expected values provide information about the constant offsets affecting the accelerometer channels.

Gyroscope bias is commonly observed as a nonzero angular-rate output while the IMU is completely stationary. Ideally, all three gyroscope channels should report zero angular velocity under static conditions, but practical sensors exhibit small offsets caused by electronic imbalance, temperature, mechanical stress, and internal noise. Because gyroscope measurements are integrated to estimate orientation, even a small constant bias can accumulate into substantial attitude drift during extended operation.

A basic bias model represents the measured inertial signal as the sum of the true physical quantity, a bias term, and measurement noise. For acceleration, this can be expressed conceptually as a_m = a_true + b_a + n_a, while angular velocity follows ω_m = ω_true + b_g + n_g. The calibration objective is to estimate the three-element accelerometer bias vector b_a and the three-element gyroscope bias vector b_g and subtract them from subsequent measurements.

The IMU should remain mechanically stationary during bias data acquisition because vibration or unintended rotation can be interpreted as sensor offset. The mounting fixture should be rigid, the supporting surface should be stable, and sufficient settling time should be provided after repositioning. Motors, cooling fans, manipulators, or other vibration sources near the calibration fixture should be controlled where practical because low-frequency motion can distort the estimated mean values.

Gyroscope bias can be estimated by collecting a sufficiently long sequence of stationary samples and calculating the average output of each axis. Random zero-mean noise tends to decrease through averaging, whereas the systematic offset remains. The resulting three mean values become the initial gyroscope bias estimate. The observation interval must be long enough to produce a stable estimate without allowing significant temperature variation or environmental disturbance to dominate the measurement period.

Accelerometer calibration requires special treatment because gravity is always present. A common static procedure places the IMU in multiple known orientations so that each sensor axis is alternately aligned with positive and negative gravity. Measurements obtained from opposing orientations separate the constant bias from the gravitational component. For an individual axis, averaging the outputs measured under +g and −g orientations provides an estimate of its additive offset while reducing dependence on the gravity term.

A practical six-position procedure therefore places the sensor in approximately:

• +X, −X,
• +Y, −Y,
• +Z, and −Z gravity orientations.

At each position, the IMU is allowed to settle before a block of samples is recorded and averaged. The six datasets provide symmetrical observations of gravitational loading across all accelerometer axes while simultaneously supplying stationary measurements that can be used to characterize gyroscope zero-rate bias.

The quality of the calibration depends strongly on data acquisition discipline. Sampling frequency, acquisition duration, sensor operating mode, digital filtering, and full-scale range should remain consistent throughout the procedure. Raw or minimally processed measurements are generally preferable when estimating fundamental sensor errors because undocumented filtering or automatic correction may hide the underlying bias. Calibration metadata should therefore record the configuration used to generate each dataset.

Bias should not be confused with random noise. Increasing the number of samples reduces uncertainty in the estimated mean but does not eliminate a true systematic offset. Statistical examination of the stationary data can distinguish a stable mean displacement from short-term fluctuations. The sample mean represents the primary bias estimate, while standard deviation and related stability metrics indicate the uncertainty and repeatability associated with that estimate.

Outliers caused by shocks, fixture movement, communication errors, or transient disturbances should be identified before final parameter estimation. However, aggressive rejection of samples can also create an unrealistically clean calibration dataset. A robust procedure examines time histories and statistical distributions, removes samples only when a defensible disturbance criterion is satisfied, and preserves enough representative data to characterize the actual behavior of the IMU under normal operating conditions.

Temperature is particularly important because inertial sensor bias is rarely constant across the complete operating range. A calibration performed immediately after power-up may differ from one performed after the IMU reaches thermal equilibrium. For a basic six-axis bias calibration, the sensor should normally be stabilized at a defined temperature before data collection. More extensive temperature-dependent behavior belongs to temperature compensation, which is treated separately within the IMU calibration structure.

After the six bias values have been estimated, corrected measurements are obtained by subtracting the corresponding bias from each raw channel. The corrected accelerometer vector becomes a_corr = a_meas − b_a, and the corrected gyroscope vector becomes ω_corr = ω_meas − b_g. These corrections should be applied consistently at the appropriate sensor-processing layer before the measurements are consumed by attitude estimation, localization, navigation, stabilization, or sensor-fusion algorithms.

Bias calibration should also remain distinct from scale-factor and cross-coupling calibration. Bias represents an additive zero offset, whereas scale-factor error changes measurement magnitude and cross-coupling allows motion along one physical axis to influence another measurement axis. The calibration structure therefore treats six-axis bias first and addresses scale factor and cross-coupling separately, allowing each error source to be identified, estimated, stored, and validated without unnecessarily mixing parameter effects.

Corrected stationary data provide the first verification of calibration quality. Gyroscope outputs should cluster close to zero when no rotation is present, while the accelerometer magnitude should remain close to the expected gravitational acceleration. Repeating the measurements in several orientations helps reveal residual offsets or inconsistent corrections. The objective is not to force every instantaneous sample to an ideal value, but to remove systematic mean error while preserving legitimate sensor noise and physical motion.

For mobile robots and autonomous vehicles, residual IMU bias directly affects downstream state estimation. Accelerometer bias can create false velocity and position changes after integration, while gyroscope bias produces progressive orientation error that can rotate acceleration, odometry, LiDAR, camera, or GNSS information into incorrect directions. Reliable bias calibration therefore improves not only the IMU output itself but also the consistency of localization, mapping, navigation, motion control, and multi-sensor fusion.

Production systems should store calibrated bias parameters together with sensor identity, calibration time, firmware or configuration information, operating temperature, and relevant quality indicators. Parameters should be associated with the individual IMU rather than treated as universally interchangeable values. Replacing a sensor, changing its mechanical installation, updating internal firmware, or detecting abnormal drift may justify recalibration so that the active parameter set remains traceable to the installed hardware.

Six-axis bias calibration ultimately establishes the zero-reference foundation for the broader IMU calibration chain. Once accelerometer and gyroscope offsets have been reliably estimated, subsequent procedures can characterize scale factors, cross-axis coupling, temperature dependence, and alignment between the IMU frame and the robot body frame. The resulting calibration sequence transforms raw inertial outputs into measurements suitable for dependable state estimation and provides a controlled basis for final IMU calibration validation.

05.02. Scale Factor and Cross Coupling

Zhang's checkerboard method is a widely used camera calibration technique for estimating the intrinsic parameters of a camera from multiple observations of a planar calibration target. The target normally consists of a checkerboard with accurately defined square geometry. Because all reference points lie on one plane, calibration can be performed without constructing a complex three-dimensional calibration fixture, making the method practical for laboratory, production, and robotics applications.

The fundamental principle is to observe the same planar checkerboard from several different orientations and positions. Each image establishes a geometric relationship between known checkerboard coordinates and their corresponding image coordinates. Since the physical points can be represented on a plane with Z = 0, this relationship can be expressed using a planar projective transformation called a homography. Multiple homographies provide sufficient constraints to recover the camera's intrinsic parameters.

The camera projection model relates a three-dimensional point in the camera coordinate system to a two-dimensional image point through intrinsic and extrinsic transformations. The intrinsic matrix K typically contains the focal lengths fx and fy, the principal point coordinates cx and cy, and potentially a skew parameter between the image axes. The extrinsic parameters describe the rotation and translation between the checkerboard coordinate frame and the camera coordinate frame for each captured image.

For a planar checkerboard, the mapping between target coordinates and image coordinates can be written conceptually as s[u v 1]ᵀ = H[X Y 1]ᵀ, where H is the homography matrix and s is an arbitrary scale factor. The homography contains information derived from both the camera intrinsic matrix and the target pose. By estimating H independently from several checkerboard images, constraints on the shared intrinsic parameters can be assembled while allowing each image to have a different target pose.

Checkerboard corner detection is therefore a critical stage of the calibration procedure. The internal intersections between adjacent black and white squares provide repeatable reference features whose physical spacing is known. Initial corner locations are normally detected from image intensity patterns and then refined to subpixel accuracy. Accurate localization is important because even small corner errors can propagate through homography estimation and influence the final focal length, principal point, distortion, and pose estimates.

The physical checkerboard geometry must be defined consistently. Calibration software generally requires the number of internal corners along the horizontal and vertical directions together with the physical square size. Confusing the number of squares with the number of internal corners is a common setup error. The square dimensions establish the physical scale of translation estimates, while the planar geometric relationships used to determine camera intrinsic parameters depend primarily on accurate and consistent corner coordinates.

Image diversity strongly affects parameter observability and calibration quality. Capturing many nearly identical front-facing images provides less useful geometric information than observing the checkerboard with substantial changes in orientation. The target should be tilted around different axes, translated across the image, and observed at different distances while remaining sufficiently sharp and visible. Such variation supplies independent constraints and reduces numerical ambiguity in the intrinsic parameter estimation process.

The checkerboard should also cover different regions of the image sensor. Measurements concentrated near the optical center provide limited information about behavior toward the image boundaries, particularly when lens distortion becomes significant. Useful calibration datasets therefore include checkerboard observations near the center, sides, corners, upper and lower regions of the image. At the same time, the complete set of required internal corners should remain detectable and sufficiently large for reliable subpixel localization.

After homographies have been estimated, Zhang's method constructs algebraic constraints based on the orthogonality and equal-scale properties of the rotation vectors associated with each checkerboar