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Magnetometer Calibration Module Documentation

Optional Dependencies Required

Some magnetometer calibration methods (MAGYC and MagFactor3) require jax and gtsam, which are not installed by default. Install the calibration extra to enable them:

pip install navlib[calibration]

On Python >= 3.12, gtsam requires a pre-release build:

pip install navlib[calibration] --pre

If using uv:

[tool.uv]
prerelease = "allow"

This module contains functions to calibrate magnetometers using various methods based on research papers and algorithms.

Functions:

Name Description
cal_mag_ellipsoid_fit

Standard ellipsoid fit method.
cal_mag_ellipsoid_fit_fang

Ellipsoid fit method by Fang et al. [1]
cal_mag_magfactor3

Factor graph based approach to full-magnetometer calibration.
cal_mag_twostep_hi

TWOSTEP method for hard-iron estimation [2].
cal_mag_twostep_hsi

TWOSTEP method for hard-iron and soft-iron estimation [2].
cal_mag_sar_ls

The linear least squares for sensor bias calibration [3].
cal_mag_sar_kf

The Kalman filter for sensor bias calibration [3].
cal_mag_sar_aid

The adaptive identification for sensor bias calibration [3].
cal_mag_sphere_fit

Standard sphere fit method.
cal_mag_magyc_ls

MAGYC-LS method [4].
cal_mag_magyc_nls

MAGYC-NLS method [4].
cal_mag_magyc_bfg

MAGYC-BFG method [4].
cal_mag_magyc_ifg

MAGYC-IFG method [4].
References

[1] Section (III) in J. Fang, H. Sun, J. Cao, X. Zhang, and Y. Tao, “A novel calibration method of magnetic compass based on ellipsoid fitting,” IEEE Transactions on Instrumentation and Measurement, vol. 60, no. 6, pp. 2053--2061, 2011.

[2] Alonso, R. Shuster, M.D. (2002a). TWOSTEP: A fast, robust algorithm for attitude-independent magnetometer-bias determination. Journal of the Astronautical Sciences, 50(4):433-452.

[3] Troni, G. and Whitcomb, L. L. (2019). Field sensor bias calibration with angular-rate sensors: Theory and experimental evaluation with application to magnetometer calibration. IEEE/ASME Transactions on Mechatronics, 24(4):1698--1710.

[4] Rodríguez-Martínez, S., & Troni, G. (2025). Full Magnetometer and Gyroscope Bias Estimation Using Angular Rates: Theory and Experimental Evaluation of a Factor Graph-Based Approach. IEEE Journal of Oceanic Engineering.

The ellipsoid fit method is based on the fact that the error model of a magnetic compass is an ellipsoid, and a constraint least-squares method is adopted to estimate the parameters of an ellipsoid by rotating the magnetic compass in various random orientations.

For further details about the implementation, refer to Aleksandr Bazhin Github repository, where he ports to python [matlab's ellipsoid fit].(http://www.mathworks.com/matlabcentral/fileexchange/24693-ellipsoid-fit)

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required

Returns:

Name Type Description
hard_iron ndarray

Hard iron bias as a (3,) numpy array.
soft_iron ndarray

Soft iron matrix as a (3, 3) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.

Raises:

Type Description
TypeError

If the input is not a numpy array or a list.
ValueError

If the input is not a 3xN or Nx3 numpy array.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_ellipsoid_fit(
    magnetic_field: Union[np.ndarray, list],
) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    The ellipsoid fit method is based on the fact that the error model of a magnetic
    compass is an ellipsoid, and a constraint least-squares method is adopted to
    estimate the parameters of an ellipsoid by rotating the magnetic compass in
    various random orientations.

    For further details about the implementation, refer to Aleksandr Bazhin [Github
    repository](https://github.com/aleksandrbazhin/ellipsoid_fit_python), where he
    ports to python [matlab's ellipsoid fit].(http://www.mathworks.com/matlabcentral/fileexchange/24693-ellipsoid-fit)

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.

    Returns:
        hard_iron (numpy.ndarray): Hard iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Soft iron matrix as a (3, 3) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.

    Raises:
        TypeError: If the input is not a numpy array or a list.
        ValueError: If the input is not a 3xN or Nx3 numpy array.
    """
    # Check if the input is a list and convert it to a numpy array
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)

    # Check if the input is a numpy array
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The input must be a numpy array or a list.")

    # Check if the input is a 3xN or Nx3 numpy array
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The input must be a 3xN or Nx3 numpy array.")

    # Force the array to be a Nx3 numpy array
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T

    # Compute magnetic field calibration
    x, y, z = magnetic_field[:, 0], magnetic_field[:, 1], magnetic_field[:, 2]

    d_matrix = np.array(
        [
            x * x + y * y - 2 * z * z,
            x * x + z * z - 2 * y * y,
            2 * x * y,
            2 * x * z,
            2 * y * z,
            2 * x,
            2 * y,
            2 * z,
            1 - 0 * x,
        ]
    )
    d2 = np.array(x * x + y * y + z * z).T
    u = np.linalg.solve(d_matrix.dot(d_matrix.T), d_matrix.dot(d2))
    a = np.array([u[0] + 1 * u[1] - 1])
    b = np.array([u[0] - 2 * u[1] - 1])
    c = np.array([u[1] - 2 * u[0] - 1])
    v = np.concatenate([a, b, c, u[2:]], axis=0).flatten()
    a_matrix = np.array(
        [
            [v[0], v[3], v[4], v[6]],
            [v[3], v[1], v[5], v[7]],
            [v[4], v[5], v[2], v[8]],
            [v[6], v[7], v[8], v[9]],
        ]
    )

    center = np.linalg.solve(-a_matrix[:3, :3], v[6:9])

    translation_matrix = np.eye(4)
    translation_matrix[3, :3] = center.T

    r_matrix = translation_matrix.dot(a_matrix).dot(translation_matrix.T)

    evals, evecs = np.linalg.eig(r_matrix[:3, :3] / -r_matrix[3, 3])
    evecs = evecs.T

    radii = np.sqrt(1.0 / np.abs(evals))
    radii *= np.sign(evals)

    a, b, c = radii
    r = (a * b * c) ** (1.0 / 3.0)
    D = np.array([[r / a, 0.0, 0.0], [0.0, r / b, 0.0], [0.0, 0.0, r / c]])
    transformation = evecs.dot(D).dot(evecs.T)

    hard_iron = center.reshape(3, 1)
    soft_iron = transformation.reshape(3, 3)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    return hard_iron.flatten(), soft_iron, calibrated_magnetic_field

The ellipsoid fit method is based on the fact that the error model of a magnetic compass is an ellipsoid, and a constraint least-squares method is adopted to estimate the parameters of an ellipsoid by rotating the magnetic compass in various random orientations.

For further details about the implementation, refer to section (III) in J. Fang, H. Sun, J. Cao, X. Zhang, and Y. Tao, “A novel calibration method of magnetic compass based on ellipsoid fitting,” IEEE Transactions on Instrumentation and Measurement, vol. 60, no. 6, pp. 2053--2061, 2011.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required

Returns:

Name Type Description
hard_iron ndarray

Hard iron bias as a (3,) numpy array.
soft_iron ndarray

Soft iron matrix as a (3, 3) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.

Raises:

Type Description
TypeError

If the input is not a numpy array or a list.
ValueError

If the input is not a 3xN or Nx3 numpy array.
RuntimeWarning

If no positive eigenvalues are found.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_ellipsoid_fit_fang(
    magnetic_field: Union[np.ndarray, list],
) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    The ellipsoid fit method is based on the fact that the error model of a magnetic
    compass is an ellipsoid, and a constraint least-squares method is adopted to
    estimate the parameters of an ellipsoid by rotating the magnetic compass in
    various random orientations.

    For further details about the implementation, refer to section (III) in J. Fang,
    H. Sun, J. Cao, X. Zhang, and Y. Tao, “A novel calibration method of magnetic
    compass based on ellipsoid fitting,” IEEE Transactions on Instrumentation
    and Measurement, vol. 60, no. 6, pp. 2053--2061, 2011.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.

    Returns:
        hard_iron (numpy.ndarray): Hard iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Soft iron matrix as a (3, 3) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.

    Raises:
        TypeError: If the input is not a numpy array or a list.
        ValueError: If the input is not a 3xN or Nx3 numpy array.
        RuntimeWarning: If no positive eigenvalues are found.
    """
    # Check if the input is a list and convert it to a numpy array
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)

    # Check if the input is a numpy array
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The input must be a numpy array or a list.")

    # Check if the input is a 3xN or Nx3 numpy array
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The input must be a 3xN or Nx3 numpy array.")

    # Force the array to be a Nx3 numpy array
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T

    # Compute magnetic field calibration
    # Design matrix (S)
    s = np.concatenate(
        [
            np.square(magnetic_field[:, [0]]),
            magnetic_field[:, [0]] * magnetic_field[:, [1]],
            np.square(magnetic_field[:, [1]]),
            magnetic_field[:, [0]] * magnetic_field[:, [2]],
            magnetic_field[:, [1]] * magnetic_field[:, [2]],
            np.square(magnetic_field[:, [2]]),
            magnetic_field[:, :],
            np.ones((magnetic_field.shape[0], 1)),
        ],
        axis=1,
    )

    # Block Matrices: S_11, S_12, S_22
    sTs = s.T @ s
    s_11, s_12, s_22 = sTs[:3, :3], sTs[:3, 3:], sTs[3:, 3:]

    # Constrain matrix C_11
    c_11 = np.array([[0, 0, 2], [0, -1, 0], [2, 0, 0]])

    # Ellipsoid Parameters Estimation
    eigenvals, eigenvecs = np.linalg.eig(
        np.linalg.inv(c_11) @ (s_11 - s_12 @ np.linalg.inv(s_22) @ s_12.T)
    )

    if np.max(eigenvals) < 0:
        warnings.warn(
            "No positive eigenvalues: max eigenvalue = {:.6f}".format(
                np.max(eigenvals)
            ),
            RuntimeWarning,
        )

    a_1 = -eigenvecs[:, [np.argmax(eigenvals)]]
    a_2 = -np.linalg.inv(s_22) @ s_12.T @ a_1
    a = np.concatenate([a_1, a_2], axis=0).flatten()

    # Determine A and b
    a_matrix = np.array(
        [
            [a[0], a[1] / 2, a[3] / 2],
            [a[1] / 2, a[2], a[4] / 2],
            [a[3] / 2, a[4] / 2, a[5]],
        ]
    )
    hard_iron = np.linalg.inv(-2 * a_matrix) @ np.vstack(a[6:9])

    # Determine G and M
    u, s, vh = np.linalg.svd(a_matrix)
    g = u @ np.sqrt(np.diag(s)) @ vh
    soft_iron = np.linalg.inv(g)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    return hard_iron.flatten(), soft_iron, calibrated_magnetic_field

The full-magnetometer calibration least-squares problems can also be modeled as a factor graph. This can be implemented using the GTSAM python wrapper with the magFactor3 factor. This approach allows the user to get the soft-iron (SI) as the identity scaled by a constant and the hard-iron (HI) from the magnetometer bias.

This method assumes that the rotation from the body frame with respect to the world frame and the local magnetic field are known.

Parameters:

Name Type Description Default
magnetic_field Union[ndarray, list]

Magnetic field raw data

 required
rph Union[ndarray, list]

Roll, pitch and heading data

 required
magnetic_declination float

Magnetic declination in degrees

 required
reference_magnetic_field Union[ndarray, list]

Reference magnetic field

 required
optimizer str

Optimization algorithm to use. Options are "dogleg" or "lm" for the Dogleg and Levenberg-Marquardt optimizers respectively.

 'dogleg'
relative_error_tol float

Relative error tolerance for the optimizer. Default is 1.00e-12

 1e-12
absolute_error_tol float

Absolute error tolerance for the optimizer. Default is 1.00e-12

 1e-12
max_iter int

Maximum number of iterations for the optimizer. Default is 1000

 1000

Returns:

Name Type Description
hard_iron ndarray

Hard-iron offset in as a (3.) numpy array
soft_iron ndarray

Soft-iron scaling matrix
corrected_magnetic_field ndarray

Corrected magnetic field data
optimization_errors list

List of optimization errors in each iteration

Raises:

Type Description
TypeError

If the magnetic field input is not a numpy array or a list
TypeError

If the reference magnetic field input is not a numpy array or a list
TypeError

If the rph input is not a numpy array or a list
ValueError

If the magnetic field input is not a 3xN or Nx3 numpy array
ValueError

If the reference magnetic field input is not a 3, numpy array
ValueError

If the rph input is not a 3xN or Nx3 numpy array
TypeError

If the magnetic declination is not a float
ValueError

If the optimizer is not a string or not "dogleg" or "lm"
TypeError

If the relative error tolerance is not a float
TypeError

If the absolute error tolerance is not a float
ValueError

If the maximum number of iterations is not a positive integer
Source code in navlib/cal/cal_mag.py 
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def cal_mag_magfactor3(
    magnetic_field: Union[np.ndarray, list],
    rph: Union[np.ndarray, list],
    magnetic_declination: float,
    reference_magnetic_field: Union[np.ndarray, list],
    optimizer: str = "dogleg",
    relative_error_tol: float = 1.00e-12,
    absolute_error_tol: float = 1.00e-12,
    max_iter: int = 1000,
) -> Tuple[np.ndarray, np.ndarray, np.ndarray, list]:
    """
    The full-magnetometer calibration least-squares problems can also be modeled
    as a factor graph. This can be implemented using the [GTSAM](https://github.com/borglab/gtsam)
    python wrapper with the magFactor3 factor. This approach allows the user to get the soft-iron
    (SI) as the identity scaled by a constant and the hard-iron (HI) from the
    magnetometer bias.

    This method assumes that the rotation from the body frame with respect to
    the world frame and the local magnetic field are known.

    Args:
        magnetic_field (Union[np.ndarray, list]): Magnetic field raw data
        rph (Union[np.ndarray, list]): Roll, pitch and heading data
        magnetic_declination (float): Magnetic declination in degrees
        reference_magnetic_field (Union[np.ndarray, list]): Reference magnetic field
        optimizer (str): Optimization algorithm to use. Options are "dogleg" or "lm"
            for the Dogleg and Levenberg-Marquardt optimizers respectively.
        relative_error_tol (float): Relative error tolerance for the optimizer. Default is 1.00e-12
        absolute_error_tol (float): Absolute error tolerance for the optimizer. Default is 1.00e-12
        max_iter (int): Maximum number of iterations for the optimizer. Default is 1000

    Returns:
        hard_iron (np.ndarray): Hard-iron offset in as a (3.) numpy array
        soft_iron (np.ndarray): Soft-iron scaling matrix
        corrected_magnetic_field (np.ndarray): Corrected magnetic field data
        optimization_errors (list): List of optimization errors in each iteration

    Raises:
        TypeError: If the magnetic field input is not a numpy array or a list
        TypeError: If the reference magnetic field input is not a numpy array or a list
        TypeError: If the rph input is not a numpy array or a list
        ValueError: If the magnetic field input is not a 3xN or Nx3 numpy array
        ValueError: If the reference magnetic field input is not a 3, numpy array
        ValueError: If the rph input is not a 3xN or Nx3 numpy array
        TypeError: If the magnetic declination is not a float
        ValueError: If the optimizer is not a string or not "dogleg" or "lm"
        TypeError: If the relative error tolerance is not a float
        TypeError: If the absolute error tolerance is not a float
        ValueError: If the maximum number of iterations is not a positive integer
    """
    try:
        import gtsam
    except ImportError:
        raise ImportError(
            "gtsam is required for this calibration method. "
            "Install with: pip install navlib[calibration] (requires Python >= 3.12)."
        )

    # Check if the magnetic field input is a list and convert it to a numpy array
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)

    # Check if the reference magnetic field input is a list and convert it to a numpy array
    if isinstance(reference_magnetic_field, list):
        reference_magnetic_field = np.array(reference_magnetic_field).flatten()

    # Check if the rph input is a list and convert it to a numpy array
    if isinstance(rph, list):
        rph = np.array(rph)

    # Check if the magnetic field input is a numpy array
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field input must be a numpy array or a list.")

    # Check if the reference magnetic field input is a numpy array
    if not isinstance(reference_magnetic_field, np.ndarray):
        raise TypeError(
            "The reference magnetic field input must be a numpy array or a list."
        )

    # Check if the rph input is a numpy array
    if not isinstance(rph, np.ndarray):
        raise TypeError("The rph input must be a numpy array or a list.")

    # Check if the magnetic field input is a 3xN or Nx3 numpy array
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field input must be a 3xN or Nx3 numpy array.")

    # Check if the reference magnetic field input is a 3, numpy array
    reference_magnetic_field = reference_magnetic_field.flatten()
    if reference_magnetic_field.shape[0] != 3:
        raise ValueError(
            "The reference magnetic field input must be a 3, or 1x3, or 3x1 numpy array."
        )

    # Check if the rph input is is a 3xN or Nx3 numpy array
    if rph.ndim != 2 or (rph.shape[0] != 3 and rph.shape[1] != 3):
        raise ValueError("The rph input must be a 3xN or Nx3 numpy array.")

    # Force the magnetic field array to be a Nx3 numpy array
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T

    # Force the rph array to be a Nx3 numpy array
    if rph.shape[0] == 3:
        rph = rph.T

    # Check that the magnetic declination is a float
    if not isinstance(magnetic_declination, float):
        raise TypeError("The magnetic declination must be a float.")

    # Check that the optimizer is a string and is either "dogleg" or "lm"
    if not isinstance(optimizer, str) or optimizer not in ["dogleg", "lm"]:
        raise ValueError("The optimizer must be a string and either 'dogleg' or 'lm'.")

    # Check that the relative error tolerance is a float
    if not isinstance(relative_error_tol, float) or relative_error_tol <= 0:
        raise TypeError("The relative error tolerance must be a float.")

    # Check that the absolute error tolerance is a float
    if not isinstance(absolute_error_tol, float) or absolute_error_tol <= 0:
        raise TypeError("The absolute error tolerance must be a float.")

    # Check that the maximum number of iterations is a positive integer
    if not isinstance(max_iter, int) or max_iter <= 0:
        raise ValueError("The maximum number of iterations must be a positive integer.")

    # Compute attitude based on magnetic heading
    magnetic_hdg = ahrs_raw_hdg(magnetic_field, rph) - np.deg2rad(magnetic_declination)
    magnetic_rph = np.concatenate([rph[:, :2], magnetic_hdg.reshape(-1, 1)], axis=1)

    # Compute calibration
    # Smoothing and Mapping Factor Graph
    # 1. Create the non-linear graph
    graph = gtsam.NonlinearFactorGraph()

    # 2. noise model for each factor.
    residual_noise = gtsam.noiseModel.Isotropic.Sigma(3, 0.001)

    # 3. Creates values structure with initial values: S -> Scale, D -> Direction, B -> Bias
    initial = gtsam.Values()
    initial.insert(S(0), 1.0)
    initial.insert(B(0), gtsam.Point3(0, 0, 0))
    keys = [S(0), B(0)]

    # 4. Add factor for each measurement into a single node
    h0 = gtsam.Point3(reference_magnetic_field.flatten())

    for i in range(magnetic_field.shape[0]):
        mi = gtsam.Point3(magnetic_field[i, :])
        bRw = gtsam.Rot3(rph2rot(magnetic_rph[i, :]).T)

        # 5.1 magFactor3
        rf = gtsam.CustomFactor(
            residual_noise,
            keys,
            partial(_cal_mag_magfactor3_residual_factor, mi, h0, bRw),
        )
        graph.add(rf)

    # 5. If not online optimize the full batch
    # 5.1 Create optimizer parameters
    params = (
        gtsam.DoglegParams()
        if optimizer == "dogleg"
        else gtsam.LevenbergMarquardtParams()
    )
    params.setRelativeErrorTol(relative_error_tol)
    params.setAbsoluteErrorTol(absolute_error_tol)
    params.setMaxIterations(max_iter)
    params.setLinearSolverType("MULTIFRONTAL_CHOLESKY")

    # 5.2 Create optimizer
    if optimizer == "dogleg":
        optimizer = gtsam.DoglegOptimizer(graph, initial, params)
    else:
        optimizer = gtsam.LevenbergMarquardtOptimizer(graph, initial, params)

    # 5.3 Optimize
    result, optimization_errors = _cal_mag_magfactor3_gtsam_optimize(optimizer, params)

    # 7. Process Results
    hard_iron = np.vstack(result.atPoint3(B(0)))
    soft_iron = result.atDouble(S(0)) * np.eye(3)

    # Correct the magnetic field
    corrected_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    corrected_magnetic_field = corrected_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    corrected_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ corrected_magnetic_field
    ).squeeze()

    return hard_iron.flatten(), soft_iron, corrected_magnetic_field, optimization_errors

Proposed method for the full calibration of a three-axis magnetometer using magnetic field and angular rate measurements. This particular approach is based on a factor graph processing all the data in a batch manner.

In particular MAGYC-BFG embeds the volume constraint for the soft-iron into a reparametrization for the Cholesky decomposition of the soft-iron matrix, allowing for the use of half the factors.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time stamps of the measurements.

 required
measurements_window int

Window size for the measurements.

 25
optimizer str

Optimization algorithm to use. Options are "dogleg" or "lm" for the Dogleg and Levenberg-Marquardt optimizers respectively.

 'dogleg'
relative_error_tol float

Relative error tolerance for the optimizer. Default is 1.00e-07

 1e-07
absolute_error_tol float

Absolute error tolerance for the optimizer. Default is 1.00e-07

 1e-07
max_iter int

Maximum number of iterations for the optimizer. Default is 1000

 1000

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
soft_iron ndarray

Estimated soft-iron matrix as a (3, 3) numpy array.
gyro_bias ndarray

Estimated gyroscope bias as a (3,) numpy array in rad/s.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.
calibrated_angular_rate ndarray

Calibrated angular rate measurements in rad/s.
optimization_status Dict[str, Union[List[float], int]]

Dictionary with the optimization status. The keys are "error" and "iterations".

Raises:

Type Description
TypeError

If the magnetic field, angular rate, and time are not numpy arrays or lists.
ValueError

If the magnetic field and angular rate are not 3xN or Nx3 numpy arrays, or if the time is not a 1D numpy array.
ValueError

If the optimizer is not a string or not "dogleg" or "lm"
TypeError

If the relative error tolerance is not a float
TypeError

If the absolute error tolerance is not a float
ValueError

If the maximum number of iterations is not a positive integer
ValueError

If the measurements window is not a positive integer
Source code in navlib/cal/cal_mag.py 
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def cal_mag_magyc_bfg(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
    measurements_window: int = 25,
    optimizer: str = "dogleg",
    relative_error_tol: float = 1.00e-07,
    absolute_error_tol: float = 1.00e-07,
    max_iter: int = 1000,
) -> Tuple[
    np.ndarray,
    np.ndarray,
    np.ndarray,
    np.ndarray,
    np.ndarray,
    Dict[str, Union[List[float], int]],
]:
    """
    Proposed method for the full calibration of a three-axis magnetometer
    using magnetic field and angular rate measurements. This particular approach
    is based on a factor graph processing all the data in a batch manner.

    In particular MAGYC-BFG embeds the volume constraint for the soft-iron into
    a reparametrization for the Cholesky decomposition of the soft-iron matrix,
    allowing for the use of half the factors.


    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time stamps of the measurements.
        measurements_window (int): Window size for the measurements.
        optimizer (str): Optimization algorithm to use. Options are "dogleg" or "lm"
            for the Dogleg and Levenberg-Marquardt optimizers respectively.
        relative_error_tol (float): Relative error tolerance for the optimizer. Default is 1.00e-07
        absolute_error_tol (float): Absolute error tolerance for the optimizer. Default is 1.00e-07
        max_iter (int): Maximum number of iterations for the optimizer. Default is 1000

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Estimated soft-iron matrix as a (3, 3) numpy array.
        gyro_bias (numpy.ndarray): Estimated gyroscope bias as a (3,) numpy array in rad/s.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.
        calibrated_angular_rate (numpy.ndarray): Calibrated angular rate measurements in rad/s.
        optimization_status (Dict[str, Union[List[float], int]]): Dictionary with
            the optimization status. The keys are "error" and "iterations".

    Raises:
        TypeError: If the magnetic field, angular rate, and time are not numpy arrays or lists.
        ValueError: If the magnetic field and angular rate are not 3xN or Nx3 numpy
            arrays, or if the time is not a 1D numpy array.
        ValueError: If the optimizer is not a string or not "dogleg" or "lm"
        TypeError: If the relative error tolerance is not a float
        TypeError: If the absolute error tolerance is not a float
        ValueError: If the maximum number of iterations is not a positive integer
        ValueError: If the measurements window is not a positive integer
    """
    try:
        import gtsam
    except ImportError:
        raise ImportError(
            "gtsam is required for this calibration method. "
            "Install with: pip install navlib[calibration] (requires Python >= 3.12)."
        )

    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Check that the optimizer is a string and is either "dogleg" or "lm"
    if not isinstance(optimizer, str) or optimizer not in ["dogleg", "lm"]:
        raise ValueError("The optimizer must be a string and either 'dogleg' or 'lm'.")

    # Check that the relative error tolerance is a float
    if not isinstance(relative_error_tol, float) or relative_error_tol <= 0:
        raise TypeError("The relative error tolerance must be a float.")

    # Check that the absolute error tolerance is a float
    if not isinstance(absolute_error_tol, float) or absolute_error_tol <= 0:
        raise TypeError("The absolute error tolerance must be a float.")

    # Check that the maximum number of iterations is a positive integer
    if not isinstance(max_iter, int) or max_iter <= 0:
        raise ValueError("The maximum number of iterations must be a positive integer.")

    # Check that the measurements window is a positive integer
    if not isinstance(measurements_window, int) or measurements_window <= 0:
        raise ValueError("The measurements window must be a positive integer.")

    # Compute the magnetic field derivative
    magnetic_field_derivative = np.diff(magnetic_field, axis=0) / np.diff(time).reshape(
        -1, 1
    )
    magnetic_field_derivative = np.concatenate(
        [np.zeros((1, 3)), magnetic_field_derivative], axis=0
    )

    # Compute the magnetic calibration
    # Smoothing and Mapping Factor Graph
    # 1. Create the non-linear graph
    graph = gtsam.NonlinearFactorGraph()

    # 2. noise model for each factor.
    residual_noise = gtsam.noiseModel.Isotropic.Sigma(3, 1e-6)

    # 3. Creates values structure with initial values
    initial = gtsam.Values()
    initial.insert(S(0), np.array([0.0, 0.0, 0.0, 0.0, 0.0]))
    initial.insert(B(0), gtsam.Point3(0, 0, 0))
    initial.insert(W(0), gtsam.Point3(0, 0, 0))
    keys = [S(0), B(0), W(0)]

    # 4. Add factor for each measurement accumulates in the measurements window into a single node
    measurements_window = int(measurements_window)
    m_dot_window = np.empty((measurements_window, 3))
    m_window = np.empty((measurements_window, 3))
    w_window = np.empty((measurements_window, 3))

    # 5. Add factors to the graph
    for i in range(magnetic_field.shape[0]):
        # Get sensor measurements and estimated magnetic field derivative
        m_dot_window[i % measurements_window, :] = magnetic_field_derivative[i, :]
        m_window[i % measurements_window, :] = magnetic_field[i, :]
        w_window[i % measurements_window, :] = angular_rate[i, :]

        if i % measurements_window == 0 and i != 0:
            # Average measurements by the measurements window size.
            m_dot_meadian = np.median(m_dot_window, axis=0).reshape(3, 1)
            m_median = np.median(m_window, axis=0).reshape(3, 1)
            w_median = np.median(w_window, axis=0).reshape(3, 1)

            # 5.1 Residual factor
            rf = gtsam.CustomFactor(
                residual_noise,
                keys,
                partial(
                    _cal_mag_magyc_residual_factor, m_dot_meadian, m_median, w_median
                ),
            )
            graph.push_back(rf)

            # 5.2 Reset the measurements window
            m_dot_window = np.empty((measurements_window, 3))
            m_window = np.empty((measurements_window, 3))
            w_window = np.empty((measurements_window, 3))

    # 6. Optimize the graph
    # 6.1 Create optimizer parameters
    params = (
        gtsam.DoglegParams()
        if optimizer == "dogleg"
        else gtsam.LevenbergMarquardtParams()
    )
    params.setRelativeErrorTol(relative_error_tol)
    params.setAbsoluteErrorTol(absolute_error_tol)
    params.setMaxIterations(max_iter)
    params.setLinearSolverType("MULTIFRONTAL_CHOLESKY")

    # For dogleg method set the trust region. For good estimations, it ranges between 0.1 and 1.0
    if optimizer == "dogleg":
        params.setDeltaInitial(0.5)

    # 6.2 Create optimizer
    if optimizer == "dogleg":
        optimizer = gtsam.DoglegOptimizer(graph, initial, params)
    else:
        optimizer = gtsam.LevenbergMarquardtOptimizer(graph, initial, params)

    # 6.3 Optimize
    result, optimization_status = _cal_mag_magyc_gtsam_optimize(optimizer, params)

    # 7. Process Results
    l_params = result.atVector(S(0))
    b = result.atVector(B(0))
    d = result.atVector(W(0))

    lower_triangular_matrix = np.array(
        [
            [exp(l_params[0]), 0, 0],
            [l_params[1], exp(l_params[2]), 0],
            [l_params[3], l_params[4], 1 / exp(l_params[0] + l_params[2])],
        ]
    )
    soft_iron = np.linalg.inv(lower_triangular_matrix @ lower_triangular_matrix.T)
    hard_iron = soft_iron @ np.vstack(b)
    gyro_bias = np.vstack(d)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    # Calibrated gyroscope measurements
    calibrated_angular_rate = angular_rate - gyro_bias.flatten()

    return (
        hard_iron.flatten(),
        soft_iron,
        gyro_bias.flatten(),
        calibrated_magnetic_field,
        calibrated_angular_rate,
        optimization_status,
    )

Proposed method for the full calibration of a three-axis magnetometer using magnetic field and angular rate measurements. This particular approach is based on a factor graph processing all the data in an incremental manner.

In particular MAGYC-IFG embeds the volume constraint for the soft-iron into a reparametrization for the Cholesky decomposition of the soft-iron matrix, allowing for the use of half the factors.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time stamps of the measurements.

 required
measurements_window int

Window size for the measurements.

 25

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
soft_iron ndarray

Estimated soft-iron matrix as a (3, 3) numpy array.
gyro_bias ndarray

Estimated gyroscope bias as a (3,) numpy array in rad/s.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.
calibrated_angular_rate ndarray

Calibrated angular rate measurements in rad/s.
optimization_status Dict[str, ndarray]

Dictionary with the SI, HI and Wb for each iterations. The keys are: "soft_iron", "hard_iron", "gyro_bias", and "time".

Raises:

Type Description
TypeError

If the magnetic field, angular rate, and time are not numpy arrays or lists.
ValueError

If the magnetic field and angular rate are not 3xN or Nx3 numpy arrays, or if the time is not a 1D numpy array.
ValueError

If the measurements window is not a positive integer
Source code in navlib/cal/cal_mag.py 
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def cal_mag_magyc_ifg(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
    measurements_window: int = 25,
) -> Tuple[
    np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray, Dict[str, np.ndarray]
]:
    """
    Proposed method for the full calibration of a three-axis magnetometer
    using magnetic field and angular rate measurements. This particular approach
    is based on a factor graph processing all the data in an incremental manner.

    In particular MAGYC-IFG embeds the volume constraint for the soft-iron into
    a reparametrization for the Cholesky decomposition of the soft-iron matrix,
    allowing for the use of half the factors.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time stamps of the measurements.
        measurements_window (int): Window size for the measurements.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Estimated soft-iron matrix as a (3, 3) numpy array.
        gyro_bias (numpy.ndarray): Estimated gyroscope bias as a (3,) numpy array in rad/s.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.
        calibrated_angular_rate (numpy.ndarray): Calibrated angular rate measurements in rad/s.
        optimization_status (Dict[str, np.ndarray]): Dictionary with the SI, HI
            and Wb for each iterations. The keys are: "soft_iron", "hard_iron",
            "gyro_bias", and "time".

    Raises:
        TypeError: If the magnetic field, angular rate, and time are not numpy arrays or lists.
        ValueError: If the magnetic field and angular rate are not 3xN or Nx3 numpy
            arrays, or if the time is not a 1D numpy array.
        ValueError: If the measurements window is not a positive integer
    """
    try:
        import gtsam
    except ImportError:
        raise ImportError(
            "gtsam is required for this calibration method. "
            "Install with: pip install navlib[calibration] (requires Python >= 3.12)."
        )

    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Check that the measurements window is a positive integer
    if not isinstance(measurements_window, int) or measurements_window <= 0:
        raise ValueError("The measurements window must be a positive integer.")

    # Compute the magnetic calibration
    # Smoothing and Mapping Factor Graph
    # 1. Create the non-linear graph
    graph = gtsam.NonlinearFactorGraph()

    # 2. Set iSAM2 parameters and create iSAM2 object
    isam_parameters = gtsam.ISAM2Params()
    dogleg_parameters = gtsam.ISAM2DoglegParams()
    dogleg_parameters.setInitialDelta(0.5)
    dogleg_parameters.setAdaptationMode("ONE_STEP_PER_ITERATION")
    isam_parameters.setOptimizationParams(dogleg_parameters)
    isam = gtsam.ISAM2(isam_parameters)

    # 3. noise model for each factor.
    residual_noise = gtsam.noiseModel.Isotropic.Sigma(3, 1e-6)

    # 4. Creates values structure with initial values
    initial = gtsam.Values()
    initial.insert(S(0), np.array([0.0, 0.0, 0.0, 0.0, 0.0]))
    initial.insert(B(0), gtsam.Point3(0, 0, 0))
    initial.insert(W(0), gtsam.Point3(0, 0, 0))
    keys = [S(0), B(0), W(0)]

    # Dictionary to save the progress of parameters during optimization
    optimization_status = {"S": [], "B": [], "W": [], "T": []}

    # 5. Add factor for each measurement accumulates in the measurements window into a single node
    measurements_window = int(measurements_window)
    m_window = np.empty((measurements_window, 3))
    w_window = np.empty((measurements_window, 3))
    t_window = np.empty((measurements_window,))

    # 6. Add factors to the graph
    for i in range(magnetic_field.shape[0]):
        # Get sensor measurements and estimated magnetic field derivative
        t_window[i % measurements_window] = time[i]
        m_window[i % measurements_window, :] = magnetic_field[i, :]
        w_window[i % measurements_window, :] = angular_rate[i, :]

        if i % measurements_window == 0 and i != 0:
            # Compute the derivative of the magnetic field for the window
            m_dot_window = np.diff(m_window, axis=0) / np.diff(t_window).reshape(-1, 1)

            # Average measurements by the measurements window size.
            m_dot_meadian = np.median(m_dot_window, axis=0).reshape(3, 1)
            m_median = np.median(m_window, axis=0).reshape(3, 1)
            w_median = np.median(w_window, axis=0).reshape(3, 1)

            # 6.1 Residual factor
            rf = gtsam.CustomFactor(
                residual_noise,
                keys,
                partial(
                    _cal_mag_magyc_residual_factor, m_dot_meadian, m_median, w_median
                ),
            )
            graph.push_back(rf)

            # 6.2 Perform incremental update to iSAM2's internal Bayes tree, optimizing only the affected variables.
            # Set iterations to start optimization, otherwise the optimizations starts as a ill-posed problem.
            # CHECK: If the measurement window is set to small values, the optimiaztion will raise RuntimeError
            # try:
            if (i // measurements_window) % 10 == 0:
                isam.update(graph, initial)
                current = isam.calculateEstimate()

                # Save the current parameters
                for key, variable in zip([S(0), B(0), W(0)], "SBW"):
                    vector = current.atVector(key).reshape(1, -1)
                    optimization_status[variable].append(vector)
                # Save the time as a unix timestamp in microseconds
                optimization_status["T"].append(int(datetime.now().timestamp() * 1e6))

                # except RuntimeError:
                #     warnings.warn("Skipping graph optimization due to indetermined system.")
                # finally:
                graph = gtsam.NonlinearFactorGraph()
                initial = gtsam.Values()

            # 6.5 Reset the measurements window
            t_window = np.empty((measurements_window,))
            m_window = np.empty((measurements_window, 3))
            w_window = np.empty((measurements_window, 3))

    # 7. Process Results
    # Update optimization status to have the actual matrices instead of the keys
    optimization_steps = len(optimization_status["S"])
    optimization_status_final = {
        "soft_iron": np.empty((optimization_steps, 9)),
        "hard_iron": np.empty((optimization_steps, 3)),
        "gyro_bias": np.empty((optimization_steps, 3)),
        "time": np.empty((optimization_steps,)),
    }

    for i in range(optimization_steps):
        # Get parameters
        l_params = optimization_status["S"][i].flatten()
        b = optimization_status["B"][i]
        d = optimization_status["W"][i]

        # Compute soft-iron, hard-iron and gyroscope bias
        lower_triangular_matrix = np.array(
            [
                [exp(l_params[0]), 0, 0],
                [l_params[1], exp(l_params[2]), 0],
                [l_params[3], l_params[4], 1 / exp(l_params[0] + l_params[2])],
            ]
        )
        soft_iron_i = np.linalg.inv(lower_triangular_matrix @ lower_triangular_matrix.T)
        hard_iron_i = soft_iron_i @ b.reshape(3, 1)
        gyro_bias_i = d.reshape(3, 1)

        # Fill the new optimization status dictionary
        optimization_status_final["soft_iron"][i, :] = soft_iron_i.flatten()
        optimization_status_final["hard_iron"][i, :] = hard_iron_i.flatten()
        optimization_status_final["gyro_bias"][i, :] = gyro_bias_i.flatten()
        optimization_status_final["time"][i] = optimization_status["T"][i]

    # Average the last 20% of the optimization steps to get the final calibration
    optimization_steps = int(0.2 * optimization_steps)
    soft_iron = np.mean(
        optimization_status_final["soft_iron"][-optimization_steps:], axis=0
    ).reshape(3, 3)
    hard_iron = np.mean(
        optimization_status_final["hard_iron"][-optimization_steps:], axis=0
    )
    gyro_bias = np.mean(
        optimization_status_final["gyro_bias"][-optimization_steps:], axis=0
    )

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    # Calibrated gyroscope measurements
    calibrated_angular_rate = angular_rate - gyro_bias.flatten()

    return (
        hard_iron.flatten(),
        soft_iron,
        gyro_bias.flatten(),
        calibrated_magnetic_field,
        calibrated_angular_rate,
        optimization_status_final,
    )

Proposed method for the full calibration of a three-axis magnetometer using magnetic field and angular rate measurements. This particular approach is based on a least squares optimization and poses the probems as a linear least squares optimization problem.

Even though a closed solution can be computed, it is an ill-conditioned problem and the optimization is preferred.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time stamps of the measurements.

 required

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
soft_iron ndarray

Estimated soft-iron matrix as a (3, 3) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.

Raises:

Type Description
TypeError

If the magnetic field, angular rate, and time are not numpy arrays or lists.
ValueError

If the magnetic field and angular rate are not 3xN or Nx3 numpy arrays, or if the time is not a 1D numpy array.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_magyc_ls(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    Proposed method for the full calibration of a three-axis magnetometer
    using magnetic field and angular rate measurements. This particular approach
    is based on a least squares optimization and poses the probems as a linear
    least squares optimization problem.

    Even though a closed solution can be computed, it is an ill-conditioned problem
    and the optimization is preferred.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time stamps of the measurements.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Estimated soft-iron matrix as a (3, 3) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.

    Raises:
        TypeError: If the magnetic field, angular rate, and time are not numpy arrays or lists.
        ValueError: If the magnetic field and angular rate are not 3xN or Nx3 numpy
            arrays, or if the time is not a 1D numpy array.
    """
    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Compute the skew symmetric matrix of the angular rate
    skew_symmetric_angular_rate = np.apply_along_axis(
        _cal_mag_magyc_vec_to_so3_jax, 1, angular_rate
    )

    # Compute the magnetic field derivative
    magnetic_field_derivative = np.diff(magnetic_field, axis=0) / np.diff(time).reshape(
        -1, 1
    )
    magnetic_field_derivative = np.concatenate(
        [np.zeros((1, 3)), magnetic_field_derivative], axis=0
    ).reshape(-1, 3, 1)

    # Reshape magnetic field
    magnetic_field_3d = magnetic_field.reshape(-1, 3, 1)

    # Compute the magnetic calibration
    # Least Squares Initial Guess and Constraints
    x0 = np.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])

    # Optimization
    res = least_squares(
        _cal_mag_magyc_ls_cost_function,
        x0,
        jac=_cal_mag_magyc_ls_jacobian,
        method="dogbox",
        verbose=0,
        loss="linear",
        max_nfev=1000,
        ftol=1.00e-06,
        gtol=None,
        xtol=None,
        x_scale="jac",
        args=(
            magnetic_field_3d,
            magnetic_field_derivative,
            skew_symmetric_angular_rate,
        ),
    )

    # Compute SI and HI
    x = res["x"]
    lower_triangular_matrix = np.array(
        [[exp(x[0]), 0, 0], [x[1], exp(x[2]), 0], [x[3], x[4], 1 / exp(x[0] + x[2])]]
    )
    soft_iron = np.linalg.inv(lower_triangular_matrix @ lower_triangular_matrix.T)
    hard_iron = soft_iron @ x[5:].reshape(3, 1)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    return hard_iron.flatten(), soft_iron, calibrated_magnetic_field

Proposed method for the full calibration of a three-axis magnetometer and a three-axis gyroscope using magnetic field and angular rate measurements. This particular approach is based on a least squares optimization and poses the probems as a non-linear least squares optimization problem.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time stamps of the measurements.

 required

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
soft_iron ndarray

Estimated soft-iron matrix as a (3, 3) numpy array.
gyro_bias ndarray

Estimated gyroscope bias as a (3,) numpy array in rad/s.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.
calibrated_angular_rate ndarray

Calibrated angular rate measurements in rad/s.

Raises:

Type Description
TypeError

If the magnetic field, angular rate, and time are not numpy arrays or lists.
ValueError

If the magnetic field and angular rate are not 3xN or Nx3 numpy arrays, or if the time is not a 1D numpy array.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_magyc_nls(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
    """
    Proposed method for the full calibration of a three-axis magnetometer
    and a three-axis gyroscope using magnetic field and angular rate measurements.
    This particular approach is based on a least squares optimization and poses
    the probems as a non-linear least squares optimization problem.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time stamps of the measurements.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Estimated soft-iron matrix as a (3, 3) numpy array.
        gyro_bias (numpy.ndarray): Estimated gyroscope bias as a (3,) numpy array in rad/s.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.
        calibrated_angular_rate (numpy.ndarray): Calibrated angular rate measurements in rad/s.

    Raises:
        TypeError: If the magnetic field, angular rate, and time are not numpy arrays or lists.
        ValueError: If the magnetic field and angular rate are not 3xN or Nx3 numpy
            arrays, or if the time is not a 1D numpy array.
    """
    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Compute the skew symmetric matrix of the angular rate
    skew_symmetric_angular_rate = np.apply_along_axis(
        _cal_mag_magyc_vec_to_so3_jax, 1, angular_rate
    )

    # Compute the magnetic field derivative
    magnetic_field_derivative = np.diff(magnetic_field, axis=0) / np.diff(time).reshape(
        -1, 1
    )
    magnetic_field_derivative = np.vstack(
        [np.zeros((1, 3)), magnetic_field_derivative]
    ).reshape(-1, 3, 1)

    # Reshape magnetic field
    magnetic_field_3d = magnetic_field.reshape(-1, 3, 1)

    # Compute the magnetic calibration
    # Least Squares Initial Guess and Constraints
    x0 = np.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])

    # Optimization
    res = least_squares(
        _cal_mag_magyc_nls_cost_function,
        x0,
        method="dogbox",
        jac=(
            _cal_mag_magyc_compute_jacobian_nls_jax
            if callable(_cal_mag_magyc_nls_jacobian)
            else "2-point"
        ),
        verbose=0,
        loss="linear",
        max_nfev=1000,
        ftol=1.00e-06,
        gtol=None,
        xtol=None,
        x_scale="jac",
        args=(
            magnetic_field_3d,
            magnetic_field_derivative,
            skew_symmetric_angular_rate,
        ),
    )

    # Compute SI, HI and Wb
    x = res["x"]
    lower_triangular_matrix = np.array(
        [[exp(x[0]), 0, 0], [x[1], exp(x[2]), 0], [x[3], x[4], 1 / exp(x[0] + x[2])]]
    )
    soft_iron = np.linalg.inv(lower_triangular_matrix @ lower_triangular_matrix.T)
    hard_iron = soft_iron @ x[5:8].reshape(3, 1)
    gyro_bias = x[8:].reshape(3, 1)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    # Calibrated gyroscope measurements
    calibrated_angular_rate = angular_rate - gyro_bias.flatten()

    return (
        hard_iron.flatten(),
        soft_iron,
        gyro_bias.flatten(),
        calibrated_magnetic_field,
        calibrated_angular_rate,
    )

The adaptive identification for sensor bias calibration proposes that the unknown sensor bias, \(b\), can be estimated on-line with a novel adaptive identification algorithm. The possible advantages of this adaptive approach are that (i) it does not require numerical differentiation of the sensor measurement \(x(t)\), (ii) it is less computationally expensive than the SAR-KF, and (iii) it could be combined with other nonlinear observer methods.

For further information refer to section IV.C in Troni, G. and Whitcomb, L. L. (2019). Field sensor bias calibration with angular-rate sensors: Theory and experimental evaluation with application to magnetometer calibration. IEEE/ASME Transactions on Mechatronics, 24(4):1698--1710.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time measurements in a 1D numpy array or list.

 required
gains tuple

Gains defined in the set of equations (5) of the proposed method as a tuple of floats, by default (1.0, 1.0)

 (1.0, 1.0)
f_normalize bool

Whether the k2 gain should be scaled by and adaptive constant computed as the reciprocal of the norm of the gyroscope measurement for that step, by default False.

 False

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.
calibrated_filtered_magnetic_field ndarray

Calibrated and filtered magnetic field measurements.

Raises:

Type Description
TypeError

If the magnetic field, angular rate, or time are not numpy arrays or lists.
ValueError

If the magnetic field, angular rate, or time are not 3xN or Nx3 numpy arrays.
ValueError

If the magnetic field, angular rate, and time do not have the same number of samples.
TypeError

If the gains are not a tuple of floats.
TypeError

If the f_normalize is not a boolean.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_sar_aid(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
    gains: Tuple[float, float] = (1.0, 1.0),
    f_normalize: bool = False,
) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    The adaptive identification for sensor bias calibration proposes that the
    unknown sensor bias, $b$, can be estimated on-line with a novel adaptive
    identification algorithm. The possible advantages of this adaptive approach
    are that (i) it does not require numerical differentiation of the sensor
    measurement $x(t)$, (ii) it is less computationally expensive than the SAR-KF,
    and (iii) it could be combined with other nonlinear observer methods.

    For further information refer to section IV.C in Troni, G. and Whitcomb, L. L.
    (2019). Field sensor bias calibration with angular-rate sensors: Theory and
    experimental evaluation with application to magnetometer calibration. IEEE/ASME
    Transactions on Mechatronics, 24(4):1698--1710.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time measurements in a 1D numpy array or list.
        gains (tuple): Gains defined in the set of equations (5) of the proposed method as
            a tuple of floats, by default (1.0, 1.0)
        f_normalize (bool): Whether the k2 gain should be scaled by and adaptive
            constant computed as the reciprocal of the norm of the gyroscope measurement
            for that step, by default False.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.
        calibrated_filtered_magnetic_field (numpy.ndarray): Calibrated and filtered magnetic field measurements.

    Raises:
        TypeError: If the magnetic field, angular rate, or time are not numpy arrays or lists.
        ValueError: If the magnetic field, angular rate, or time are not 3xN or Nx3 numpy arrays.
        ValueError: If the magnetic field, angular rate, and time do not have the same number of samples.
        TypeError: If the gains are not a tuple of floats.
        TypeError: If the f_normalize is not a boolean.
    """
    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Check that the gains are a tuple of floats
    if not isinstance(gains, tuple):
        raise TypeError("The gains must be a tuple of floats.")
    if not all(isinstance(gain, float) for gain in gains):
        raise TypeError("The gains must be a tuple of floats.")

    # Check that the f_normalize is a boolean
    if not isinstance(f_normalize, bool):
        raise TypeError("The f_normalize must be a boolean.")

    # Compute the magnetic calibration
    # Initial parameters
    b0 = np.zeros((3,))
    k1 = gains[0]
    k2 = gains[1]
    mf = magnetic_field.reshape(3, -1)
    w = angular_rate.reshape(3, -1)
    dt = np.diff(time)
    dt_vec = np.concatenate([np.array([dt[0]]), dt])

    # Compute the skew-symmetric matrix of the angular rate.
    skew_symmetric_angular_rate = np.apply_along_axis(vec_to_so3, 1, angular_rate)

    # Adaptive ID system
    mh = np.zeros((3, mf.shape[1] + 1))
    mhd = np.zeros((3, mf.shape[1]))
    bh = np.zeros((3, mf.shape[1] + 1))
    bhd = np.zeros((3, mf.shape[1]))
    mh[:, 0] = magnetic_field[0, :]
    bh[:, 0] = b0

    for ix in range(mf.shape[1]):
        mhd[:, ix] = (
            -skew_symmetric_angular_rate[ix, :, :] @ mh[:, ix]
            + skew_symmetric_angular_rate[ix, :, :] @ bh[:, ix]
            - k1 * (mh[:, ix] - mf[:, ix])
        )

        if (np.linalg.norm(w[:, ix]) > 0.01) and f_normalize:
            k_adap = 1 / np.linalg.norm(w[:, ix])
            bhd[:, ix] = (
                -k_adap
                * k2
                * skew_symmetric_angular_rate[ix, :, :]
                @ (mh[:, ix] - mf[:, ix])
            )
        else:
            bhd[:, ix] = (
                -k2 * skew_symmetric_angular_rate[ix, :, :].T @ (mh[:, ix] - mf[:, ix])
            )

        mh[:, ix + 1] = mh[:, ix] + dt_vec[ix] * mhd[:, ix]
        bh[:, ix + 1] = bh[:, ix] + dt_vec[ix] * bhd[:, ix]

    # Final Bias averaging last 20%
    hard_iron = np.mean(
        bh[:, -int(np.round(mf.shape[1] * 0.2)) :], axis=1, keepdims=True
    )

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field - hard_iron.flatten()
    calibrated_filtered_magnetic_field = mh[:, :-1].T

    return (
        hard_iron.flatten(),
        calibrated_magnetic_field,
        calibrated_filtered_magnetic_field,
    )

The Kalman filter for sensor bias calibration uses the system model with a discretization of the continuous-time system the sensor bias estimation can be solved with a standard discrete-time Kalman filter implementation that does not require differentiation.

\[\dot{x}_i = -\omega_i \times (x_i - b)\]

Where \(x(t)\) is the measured magnetic field, \(\dot{x(t)}\) is the measured magnetic field differentiated with respect to time, \(\omega(t)\) is the measured angular-rate in instrument coordinates, and \(b\) is the hard-iron.

For further information refer to section IV.B in Troni, G. and Whitcomb, L. L. (2019). Field sensor bias calibration with angular-rate sensors: Theory and experimental evaluation with application to magnetometer calibration. IEEE/ASME Transactions on Mechatronics, 24(4):1698--1710.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time measurements in a 1D numpy array or list.

 required
gains tuple

Kalman filter gains.

 (1.0, 1.0)
f_normalize bool

Whether the k2 gain should be scaled by and adaptive constant computed as the reciprocal of the norm of the gyroscope measurement for that step, by default False.

 False

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.
calibrated_filtered_magnetic_field ndarray

Calibrated and filtered magnetic field measurements.

Raises:

Type Description
TypeError

If the magnetic field, angular rate, or time are not numpy arrays or lists.
ValueError

If the magnetic field, angular rate, or time are not 3xN or Nx3 numpy arrays.
ValueError

If the magnetic field, angular rate, and time do not have the same number of samples.
TypeError

If the gains are not a tuple of floats.
TypeError

If the f_normalize is not a boolean.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_sar_kf(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
    gains: Tuple[float, float] = (1.0, 1.0),
    f_normalize: bool = False,
) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    The Kalman filter for sensor bias calibration uses the system model with  a
    discretization of the continuous-time system the sensor bias estimation can
    be solved with a standard discrete-time Kalman filter implementation that
    does not require differentiation.

    $$\\dot{x}_i = -\\omega_i \\times (x_i - b)$$

    Where $x(t)$ is the measured magnetic field, $\\dot{x(t)}$ is the measured
    magnetic field differentiated with respect to time, $\\omega(t)$ is the
    measured angular-rate in instrument coordinates, and $b$ is the hard-iron.

    For further information refer to section IV.B in Troni, G. and Whitcomb, L. L.
    (2019). Field sensor bias calibration with angular-rate sensors: Theory and
    experimental evaluation with application to magnetometer calibration. IEEE/ASME
    Transactions on Mechatronics, 24(4):1698--1710.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time measurements in a 1D numpy array or list.
        gains (tuple): Kalman filter gains.
        f_normalize (bool): Whether the k2 gain should be scaled by and adaptive
            constant computed as the reciprocal of the norm of the gyroscope measurement
            for that step, by default False.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.
        calibrated_filtered_magnetic_field (numpy.ndarray): Calibrated and filtered magnetic field measurements.

    Raises:
        TypeError: If the magnetic field, angular rate, or time are not numpy arrays or lists.
        ValueError: If the magnetic field, angular rate, or time are not 3xN or Nx3 numpy arrays.
        ValueError: If the magnetic field, angular rate, and time do not have the same number of samples.
        TypeError: If the gains are not a tuple of floats.
        TypeError: If the f_normalize is not a boolean.
    """
    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Check that the gains are a tuple of floats
    if not isinstance(gains, tuple):
        raise TypeError("The gains must be a tuple of floats.")
    if not all(isinstance(gain, float) for gain in gains):
        raise TypeError("The gains must be a tuple of floats.")

    # Check that the f_normalize is a boolean
    if not isinstance(f_normalize, bool):
        raise TypeError("The f_normalize must be a boolean.")

    # Compute the magnetic calibration
    # Initial parameters
    b0 = np.zeros((3,))
    k1a = gains[0]
    k1b = gains[1] if len(gains) >= 2 else gains[0]
    k2 = gains[2] if len(gains) >= 3 else gains[1]
    mf = magnetic_field.reshape(3, -1)
    w = angular_rate.reshape(3, -1)
    dt = np.diff(time)
    dt_vec = np.concatenate([np.array([dt[0]]), dt])

    # Kalman Model
    Bc = np.zeros([6, 1])
    # Measurement model
    H1 = np.hstack([np.eye(3), np.zeros([3, 3])])
    # Process noise covariance
    Qc = np.diag([k1a, k1a, k1a, k1b, k1b, k1b])
    # Variance in the measurements
    R = np.diag([k2, k2, k2])

    # KF
    F1 = _cal_mag_sar_kf_transition_matrix([0, 0, 0])
    n = F1.shape[0]
    m = F1.shape[1]
    MM = np.zeros([n, mf.shape[1]])
    PP = np.zeros([n, m, mf.shape[1]])
    AA = np.zeros([n, m, mf.shape[1]])
    QQ = np.zeros([n, m, mf.shape[1]])
    KK = np.zeros([n, H1.shape[0], mf.shape[1]])

    # Initial guesses for the state mean and covariance.
    x = np.hstack([mf[:, 0], b0])
    p01 = 0.001  # P0 gyro
    p02 = 0.001  # P0 bias
    P0 = np.diag([p01, p01, p01, p02, p02, p02])
    P = P0

    # Filtering steps.
    for ix in range(mf.shape[1]):
        # Discretization of the continous-time system (dtk)
        dtk = dt_vec[ix]
        u = w[:, ix]

        [Ak, Bk, Qk] = _cal_mag_sar_kf_lti_discretize(
            _cal_mag_sar_kf_transition_matrix(u), Bc, Qc, dtk
        )

        AA[:, :, ix] = Ak
        QQ[:, :, ix] = Qk

        # Prediction
        [x, P] = _cal_mag_sar_kf_predict(x, P, Ak, Qk)
        [x, P, K, dy, S] = _cal_mag_sar_kf_update(x, P, mf[:, ix], H1, R)

        MM[:, ix] = x
        PP[:, :, ix] = P
        KK[:, :, ix] = K

    # Final Bias averaging last 20%
    hard_iron = np.mean(
        MM[3:, -int(np.round(mf.shape[1] * 0.2)) :], axis=1, keepdims=True
    )

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field - hard_iron.flatten()
    calibrated_filtered_magnetic_field = MM[:3, :].T

    return (
        hard_iron.flatten(),
        calibrated_magnetic_field,
        calibrated_filtered_magnetic_field,
    )

The linear least squares for sensor bias calibration is seeks to minimize the sum of squared residuals

\[\sum_{i=1}^{n} \frac{1}{\sigma_i^2} ||\dot{x}_i + \omega_i \times (x_i - b)||^2\]

Where \(x(t)\) is the measured magnetic field, \(\dot{x(t)}\) is the measured magnetic field differentiated with respect to time, \(\omega(t)\) is the measured angular-rate in instrument coordinates, \(b\) is the hard-iron, and \(\times\) is the standard cross product operator.

This optimization problem can be solved in an analytical way. For further information refer to section IV.A in Troni, G. and Whitcomb, L. L. (2019). Field sensor bias calibration with angular-rate sensors: Theory and experimental evaluation with application to magnetometer calibration. IEEE/ASME Transactions on Mechatronics, 24(4):1698--1710.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
angular_rate ndarray or list

Angular rate measurements in a 3xN or Nx3 numpy array or list.

 required
time ndarray or list

Time measurements in a 1D numpy array or list.

 required

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements.

Raises:

Type Description
TypeError

If the magnetic field, angular rate, or time are not numpy arrays or lists.
ValueError

If the magnetic field, angular rate, or time are not 3xN or Nx3 numpy arrays.
ValueError

If the magnetic field, angular rate, and time do not have the same number of samples.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_sar_ls(
    magnetic_field: Union[np.ndarray, list],
    angular_rate: Union[np.ndarray, list],
    time: Union[np.ndarray, list],
) -> Tuple[np.ndarray, np.ndarray]:
    """
    The linear least squares for sensor bias calibration is seeks to minimize the
    sum of squared residuals

    $$\\sum_{i=1}^{n} \\frac{1}{\\sigma_i^2} ||\\dot{x}_i + \\omega_i \\times (x_i - b)||^2$$

    Where $x(t)$ is the measured magnetic field, $\\dot{x(t)}$ is the measured
    magnetic field differentiated with respect to time, $\\omega(t)$ is the
    measured angular-rate in instrument coordinates, $b$ is the hard-iron, and
    $\\times$ is the standard cross product operator.

    This optimization problem can be solved in an analytical way. For further
    information refer to section IV.A in Troni, G. and Whitcomb, L. L. (2019).
    Field sensor bias calibration with angular-rate sensors: Theory and experimental
    evaluation with application to magnetometer calibration. IEEE/ASME Transactions
    on Mechatronics, 24(4):1698--1710.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        angular_rate (numpy.ndarray or list): Angular rate measurements in a 3xN or
            Nx3 numpy array or list.
        time (numpy.ndarray or list): Time measurements in a 1D numpy array or list.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements.

    Raises:
        TypeError: If the magnetic field, angular rate, or time are not numpy arrays or lists.
        ValueError: If the magnetic field, angular rate, or time are not 3xN or Nx3 numpy arrays.
        ValueError: If the magnetic field, angular rate, and time do not have the same number of samples.
    """
    # Check if the magnetic_field, angular_rate, and time are lists and convert them to numpy arrays
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)
    if isinstance(angular_rate, list):
        angular_rate = np.array(angular_rate)
    if isinstance(time, list):
        time = np.array(time)

    # Check if the magnetic_field, angular_rate, and time are numpy arrays
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field must be a numpy array or a list.")
    if not isinstance(angular_rate, np.ndarray):
        raise TypeError("The angular rate must be a numpy array or a list.")
    if not isinstance(time, np.ndarray):
        raise TypeError("The time must be a numpy array or a list.")

    # Check if the magnetic_field and angular_rate are 3xN or Nx3 numpy arrays
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field must be a 3xN or Nx3 numpy array.")
    if angular_rate.ndim != 2 or (
        angular_rate.shape[0] != 3 and angular_rate.shape[1] != 3
    ):
        raise ValueError("The angular rate must be a 3xN or Nx3 numpy array.")

    # Check if the time is a 1D numpy array
    time = time.flatten()
    if time.ndim != 1:
        raise ValueError("The time must be a (n, ), (n, 1) or (1, n) numpy array.")

    # Force the magnetic_field and angular_rate to be Nx3 numpy arrays
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T
    if angular_rate.shape[0] == 3:
        angular_rate = angular_rate.T

    # Check if the magnetic_field, angular_rate, and time have the same number of samples
    if (
        magnetic_field.shape[0] != angular_rate.shape[0]
        or magnetic_field.shape[0] != time.shape[0]
    ):
        raise ValueError(
            "The magnetic field, angular rate, and time must have the same number of samples."
        )

    # Compute the magnetic calibration
    # Get the data variance
    magnetic_field_variance = _cal_mag_sar_ls_get_sigma_noise(magnetic_field)
    sigma_i = np.linalg.norm(magnetic_field_variance).reshape(-1, 1, 1)

    # Compute the skew-symmetric matrix of the angular rate.
    skew_symmetric_angular_rate = np.apply_along_axis(vec_to_so3, 1, angular_rate)

    # Compute the magnetic field derivative
    magnetic_field_derivative = np.diff(magnetic_field, axis=0) / np.diff(time).reshape(
        -1, 1
    )
    magnetic_field_derivative = np.concatenate(
        [np.zeros((1, 3)), magnetic_field_derivative], axis=0
    )

    # Estimate b
    b1_inv = np.linalg.inv(
        np.einsum("ijk->jk", (skew_symmetric_angular_rate**2) * (1 / sigma_i))
    )

    yi = np.einsum(
        "ijk->ikj",
        np.cross(angular_rate.reshape(-1, 1, 3), magnetic_field.reshape(-1, 1, 3))
        + magnetic_field_derivative.reshape(-1, 1, 3),
    )
    b2 = np.einsum("ijk->jk", (skew_symmetric_angular_rate @ yi) * (1 / sigma_i))

    hard_iron = b1_inv @ b2

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field - hard_iron.flatten()

    return hard_iron.flatten(), calibrated_magnetic_field

The sphere fit method fits a sphere to a collection of data using a closed form for the solution. With this purpose, propose an optimization problem that seeks to minimize the sum:

\[\sum_i ((x_i-x_c)^2+(y_i-y_c)^2+(z_i-z_c)^2-r^2)^2\]

Where x, y, and z is the data; \(x_c\), \(y_c\), and \(z_c\) are the sphere center; and r is the radius.

The method assumes that points are not in a singular configuration and are real numbers to solve this problem. If you have coplanar data, use a circle fit with svd for determining the plane, recommended Circle Fit (Pratt method), by Nikolai Chernov

Inspired by Alan Jennings, University of Dayton, implementation (source)

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required

Returns:

Name Type Description
hard_iron ndarray

Hard iron bias as a (3,) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field measurements

Raises:

Type Description
TypeError

If the input is not a numpy array or a list.
ValueError

If the input is not a 3xN or Nx3 numpy array.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_sphere_fit(
    magnetic_field: Union[np.ndarray, list],
) -> Tuple[np.ndarray, np.ndarray]:
    """
    The sphere fit method fits a sphere to a collection of data using a closed
    form for the solution. With this purpose, propose an optimization problem that
    seeks to minimize the sum:

    $$\\sum_i ((x_i-x_c)^2+(y_i-y_c)^2+(z_i-z_c)^2-r^2)^2$$

    Where x, y, and z is the data; $x_c$, $y_c$, and $z_c$ are the sphere center;
    and r is the radius.

    The method assumes that points are not in a singular configuration and are
    real numbers to solve this problem. If you have coplanar data, use a circle
    fit with svd for determining the plane, recommended [Circle Fit (Pratt method),
    by Nikolai Chernov](http://www.mathworks.com/matlabcentral/fileexchange/22643)

    Inspired by Alan Jennings, University of Dayton, implementation ([source](
    https://www.mathworks.com/matlabcentral/fileexchange/34129-sphere-fit-least-squared))

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.

    Returns:
        hard_iron (numpy.ndarray): Hard iron bias as a (3,) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field measurements

    Raises:
        TypeError: If the input is not a numpy array or a list.
        ValueError: If the input is not a 3xN or Nx3 numpy array.
    """
    # Check if the input is a list and convert it to a numpy array
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)

    # Check if the input is a numpy array
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The input must be a numpy array or a list.")

    # Check if the input is a 3xN or Nx3 numpy array
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The input must be a 3xN or Nx3 numpy array.")

    # Force the array to be a Nx3 numpy array
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T

    # Compute magnetic field calibration
    mf = magnetic_field
    a_matrix = np.array(
        [
            [
                np.mean(mf[:, 0] * (mf[:, 0] - np.mean(mf[:, 0]))),
                2 * np.mean(mf[:, 0] * (mf[:, 1] - np.mean(mf[:, 1]))),
                2 * np.mean(mf[:, 0] * (mf[:, 2] - np.mean(mf[:, 2]))),
            ],
            [
                0,
                np.mean(mf[:, 1] * (mf[:, 1] - np.mean(mf[:, 1]))),
                2 * np.mean(mf[:, 1] * (mf[:, 2] - np.mean(mf[:, 2]))),
            ],
            [0, 0, np.mean(mf[:, 2] * (mf[:, 2] - np.mean(mf[:, 2])))],
        ]
    )

    a_matrix = a_matrix + a_matrix.T
    b_matrix = np.array(
        [
            [
                np.mean(
                    (mf[:, 0] ** 2 + mf[:, 1] ** 2 + mf[:, 2] ** 2)
                    * (mf[:, 0] - np.mean(mf[:, 0]))
                )
            ],
            [
                np.mean(
                    (mf[:, 0] ** 2 + mf[:, 1] ** 2 + mf[:, 2] ** 2)
                    * (mf[:, 1] - np.mean(mf[:, 1]))
                )
            ],
            [
                np.mean(
                    (mf[:, 0] ** 2 + mf[:, 1] ** 2 + mf[:, 2] ** 2)
                    * (mf[:, 2] - np.mean(mf[:, 2]))
                )
            ],
        ]
    )

    hard_iron = np.array(np.linalg.lstsq(a_matrix, b_matrix, rcond=None)[0])

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field - hard_iron.flatten()

    return hard_iron.flatten(), calibrated_magnetic_field

The TWOSTEP method proposes a fast, robust algorithm for estimating magnetometer biases when the attitude is unknown. This algorithm combines the convergence in a single step of a heuristic algorithm currently in use with the correct treatment of the statistics of the measurements and does without discarding data.

This algorithm was the in a first publication developed for the estimation of the hard-iron (Alonso, R. Shuster, M.D. (2002a). TWOSTEP: A fast, robust algorithm for attitude-independent magnetometer-bias determination. Journal of the Astronautical Sciences, 50(4):433-452.

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
reference_magnetic_field ndarray or list

Reference magnetic field measurements in a 3, or 1x3, or 3x1 numpy array or list.

 required
max_iterations int

Maximum number of iterations for the second step.

 2000
measurement_noise_std float

Standard deviation that characterizes the measurements' noise, by default 0.001 G.

 0.001

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field as a 3xN numpy array.

Raises:

Type Description
TypeError

If the magnetic field or reference magnetic field inputs are not numpy arrays or lists.
ValueError

If the magnetic field input is not a 3xN or Nx3 numpy array, if the reference magnetic field input is not a 3, or 1x3, or 3x1 numpy array, if the maximum number of iterations is not a positive integer, or if the measurement noise standard deviation is not a positive float.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_twostep_hi(
    magnetic_field: Union[np.ndarray, list],
    reference_magnetic_field: Union[np.ndarray, list],
    max_iterations: int = 2000,
    measurement_noise_std: float = 1e-3,
) -> Tuple[np.ndarray, np.ndarray]:
    """
    The TWOSTEP method proposes a fast, robust algorithm for estimating magnetometer
    biases when the attitude is unknown. This algorithm combines the convergence
    in a single step of a heuristic algorithm currently in use with the correct
    treatment of the statistics of the measurements and does without discarding
    data.

    This algorithm was the in a first publication developed for the estimation of
    the hard-iron (Alonso, R. Shuster, M.D. (2002a). TWOSTEP: A fast, robust
    algorithm for attitude-independent magnetometer-bias determination. Journal
    of the Astronautical Sciences, 50(4):433-452.

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        reference_magnetic_field (numpy.ndarray or list): Reference magnetic field
            measurements in a 3, or 1x3, or 3x1 numpy array or list.
        max_iterations (int): Maximum number of iterations for the second step.
        measurement_noise_std (float): Standard deviation that characterizes the
            measurements' noise, by default 0.001 G.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field as a
                3xN numpy array.

    Raises:
        TypeError: If the magnetic field or reference magnetic field inputs are not
            numpy arrays or lists.
        ValueError: If the magnetic field input is not a 3xN or Nx3 numpy array, if
            the reference magnetic field input is not a 3, or 1x3, or 3x1 numpy array,
            if the maximum number of iterations is not a positive integer, or if the
            measurement noise standard deviation is not a positive float.
    """
    # Check if the magnetic field input is a list and convert it to a numpy array
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)

    # Check if the reference magnetic field input is a list and convert it to a numpy array
    if isinstance(reference_magnetic_field, list):
        reference_magnetic_field = np.array(reference_magnetic_field).flatten()

    # Check if the magnetic field input is a numpy array
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field input must be a numpy array or a list.")

    # Check if the reference magnetic field input is a numpy array
    if not isinstance(reference_magnetic_field, np.ndarray):
        raise TypeError(
            "The reference magnetic field input must be a numpy array or a list."
        )

    # Check if the magnetic field input is a 3xN or Nx3 numpy array
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field input must be a 3xN or Nx3 numpy array.")

    # Check if the reference magnetic field input is a 3, numpy array
    if reference_magnetic_field.ndim != 1 and reference_magnetic_field.size != 3:
        raise ValueError(
            "The reference magnetic field input must be a 3, or 1x3, or 3x1 numpy array."
        )

    # Force the magnetic field array to be a Nx3 numpy array
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T

    # Check that the maximum number of iterations is a positive integer
    if not isinstance(max_iterations, int) or max_iterations <= 0:
        raise ValueError("The maximum number of iterations must be a positive integer.")

    # Check that the measurement noise standard deviation is a positive float
    if not isinstance(measurement_noise_std, float) or measurement_noise_std <= 0:
        raise ValueError(
            "The measurement noise standard deviation must be a positive float."
        )

    # Compute magnetic field calibration
    mf = magnetic_field

    # First step
    b0 = np.zeros((3, 1))

    # Effective measurement from paper equation (3a)
    b_matrix = np.ones((mf.shape)) * reference_magnetic_field
    z_k = (
        np.square(np.linalg.norm(mf, axis=1))
        - np.square(np.linalg.norm(b_matrix, axis=1))
    ).reshape(-1, 1)

    # Sensor measurements noise modeled as white gaussian with standard deviation epsilon_k
    epsilon_sq_k = np.ones(mf.shape) * (measurement_noise_std**2)

    # Sensor error scalar measurement noise characterization as gaussian.
    # Gaussian distribution mean, equation (7a)
    mu_k = -np.sum(epsilon_sq_k, axis=1, keepdims=True)

    # Gaussian distribution variance, equation (5.15)
    sigma_sq_k = (
        4
        * (
            (mf.reshape(-1, 1, 3) - b0.reshape(1, 3))
            @ np.apply_along_axis(np.diag, 1, epsilon_sq_k)
            @ (mf.reshape(-1, 3, 1) - b0)
        )
        + 2
        * np.apply_along_axis(
            lambda x: np.square(np.trace(np.diag(x))), 1, epsilon_sq_k
        ).reshape(-1, 1, 1)
    ).reshape(-1, 1)

    # Calculate  centered sigma squared, equation (14)
    sigma_sq_bar = 1 / np.sum(1 / sigma_sq_k)

    # Center  the  data
    mu_bar, mu_k_tilde = _cal_mag_twostep_center_data(mu_k, sigma_sq_k, sigma_sq_bar)
    z_bar, z_k_tilde = _cal_mag_twostep_center_data(z_k, sigma_sq_k, sigma_sq_bar)
    b_bar, b_k_tilde = _cal_mag_twostep_center_data(b_matrix, sigma_sq_k, sigma_sq_bar)

    # Offset and error covariance matrix calculation from paper equations (33) and (34)
    F_bb_tilde = np.einsum(
        "ijk->jk",
        (4 / sigma_sq_k.reshape(-1, 1, 1))
        * (b_k_tilde.reshape(-1, 3, 1) @ b_k_tilde.reshape(-1, 1, 3)),
    )
    F_zb = np.einsum(
        "ijk->jk",
        ((z_k_tilde - mu_k_tilde) * (2 / sigma_sq_k)).reshape(-1, 1, 1)
        * b_k_tilde.reshape(-1, 3, 1),
    )
    b = np.linalg.inv(F_bb_tilde) @ F_zb

    # Second Step: Iterative
    F_bb_bar = (
        (4 / sigma_sq_bar) * (b_bar.reshape(-1, 1) - b) @ (b_bar.reshape(-1, 1) - b).T
    )
    b_asterisk = np.copy(b)

    if np.max(np.diag(F_bb_bar) / np.diag(F_bb_tilde)) > 0.001:
        F_bb = F_bb_tilde + F_bb_bar
        gg = (F_bb_tilde @ (b - b_asterisk)) - (1 / sigma_sq_bar) * (
            z_bar - 2 * (b_bar @ b) + np.linalg.norm(b) ** 2 - mu_bar
        ) * 2 * (b_bar.reshape(-1, 1) - b)
        bn = b - np.linalg.inv(F_bb) @ gg

        iter = 1
        while ((bn - b).T @ F_bb @ (bn - b)) > 0.001:
            b = np.copy(bn)
            gg = (F_bb_tilde @ (b - b_asterisk)) - (1 / sigma_sq_bar) * (
                z_bar - 2 * (b_bar @ b) + np.linalg.norm(b) ** 2 - mu_bar
            ) * 2 * (b_bar.reshape(-1, 1) - b)
            F_bb_bar = (
                (4 / sigma_sq_bar)
                * (b_bar.reshape(-1, 1) - b)
                @ (b_bar.reshape(-1, 1) - b).T
            )
            F_bb = F_bb_tilde + F_bb_bar
            bn = b - np.linalg.inv(F_bb) @ gg

            iter += 1
            if iter > max_iterations:
                warnings.warn(
                    "Second step: Maximum number of iterations reached.", RuntimeWarning
                )
                break

    hard_iron = bn.reshape(3, 1)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field - hard_iron.flatten()

    return hard_iron.flatten(), calibrated_magnetic_field

The TWOSTEP method proposes a fast, robust algorithm for estimating magnetometer biases when the attitude is unknown. This algorithm combines the convergence in a single step of a heuristic algorithm currently in use with the correct treatment of the statistics of the measurements and does without discarding data.

This algorithm was extended in a second iteration to compute also the soft-iron (Alonso, R. Shuster, M.D. (2002b). Complete linear attitude-independent magnetometer calibration. Journal of the Astronautical Science, 50(4):477-490).

Parameters:

Name Type Description Default
magnetic_field ndarray or list

Magnetic field measurements in a 3xN or Nx3 numpy array or list.

 required
reference_magnetic_field ndarray or list

Reference magnetic field measurements in a 3, or 1x3, or 3x1 numpy array or list.

 required
max_iterations int

Maximum number of iterations for the second step.

 2000
measurement_noise_std float

Standard deviation that characterizes the measurements' noise, by default 0.001 G.

 0.001

Returns:

Name Type Description
hard_iron ndarray

Estimated hard-iron bias as a (3,) numpy array.
soft_iron ndarray

Estimated soft-iron matrix as a (3,3) numpy array.
calibrated_magnetic_field ndarray

Calibrated magnetic field as a 3xN numpy array.

Raises:

Type Description
TypeError

If the magnetic field or reference magnetic field inputs are not numpy arrays or lists.
ValueError

If the magnetic field input is not a 3xN or Nx3 numpy array, if the reference magnetic field input is not a 3, or 1x3, or 3x1 numpy array, if the maximum number of iterations is not a positive integer, or if the measurement noise standard deviation is not a positive float.
Source code in navlib/cal/cal_mag.py 
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def cal_mag_twostep_hsi(
    magnetic_field: Union[np.ndarray, list],
    reference_magnetic_field: Union[np.ndarray, list],
    max_iterations: int = 2000,
    measurement_noise_std: float = 1e-3,
) -> Tuple[np.ndarray, np.ndarray]:
    """
    The TWOSTEP method proposes a fast, robust algorithm for estimating magnetometer
    biases when the attitude is unknown. This algorithm combines the convergence
    in a single step of a heuristic algorithm currently in use with the correct
    treatment of the statistics of the measurements and does without discarding
    data.

    This algorithm was extended in a second iteration to compute also the soft-iron
    (Alonso, R. Shuster, M.D. (2002b). Complete linear attitude-independent
    magnetometer calibration. Journal of the Astronautical Science, 50(4):477-490).

    Args:
        magnetic_field (numpy.ndarray or list): Magnetic field measurements in a
            3xN or Nx3 numpy array or list.
        reference_magnetic_field (numpy.ndarray or list): Reference magnetic field
            measurements in a 3, or 1x3, or 3x1 numpy array or list.
        max_iterations (int): Maximum number of iterations for the second step.
        measurement_noise_std (float): Standard deviation that characterizes the
            measurements' noise, by default 0.001 G.

    Returns:
        hard_iron (numpy.ndarray): Estimated hard-iron bias as a (3,) numpy array.
        soft_iron (numpy.ndarray): Estimated soft-iron matrix as a (3,3) numpy array.
        calibrated_magnetic_field (numpy.ndarray): Calibrated magnetic field as a
            3xN numpy array.

    Raises:
        TypeError: If the magnetic field or reference magnetic field inputs are not
            numpy arrays or lists.
        ValueError: If the magnetic field input is not a 3xN or Nx3 numpy array, if
            the reference magnetic field input is not a 3, or 1x3, or 3x1 numpy array,
            if the maximum number of iterations is not a positive integer, or if the
            measurement noise standard deviation is not a positive float.
    """
    # Check if the magnetic field input is a list and convert it to a numpy array
    if isinstance(magnetic_field, list):
        magnetic_field = np.array(magnetic_field)

    # Check if the reference magnetic field input is a list and convert it to a numpy array
    if isinstance(reference_magnetic_field, list):
        reference_magnetic_field = np.array(reference_magnetic_field).flatten()

    # Check if the magnetic field input is a numpy array
    if not isinstance(magnetic_field, np.ndarray):
        raise TypeError("The magnetic field input must be a numpy array or a list.")

    # Check if the reference magnetic field input is a numpy array
    if not isinstance(reference_magnetic_field, np.ndarray):
        raise TypeError(
            "The reference magnetic field input must be a numpy array or a list."
        )

    # Check if the magnetic field input is a 3xN or Nx3 numpy array
    if magnetic_field.ndim != 2 or (
        magnetic_field.shape[0] != 3 and magnetic_field.shape[1] != 3
    ):
        raise ValueError("The magnetic field input must be a 3xN or Nx3 numpy array.")

    # Check if the reference magnetic field input is a 3, numpy array
    if reference_magnetic_field.ndim != 1 and reference_magnetic_field.size != 3:
        raise ValueError(
            "The reference magnetic field input must be a 3, or 1x3, or 3x1 numpy array."
        )

    # Force the magnetic field array to be a Nx3 numpy array
    if magnetic_field.shape[0] == 3:
        magnetic_field = magnetic_field.T

    # Check that the maximum number of iterations is a positive integer
    if not isinstance(max_iterations, int) or max_iterations <= 0:
        raise ValueError("The maximum number of iterations must be a positive integer.")

    # Check that the measurement noise standard deviation is a positive float
    if not isinstance(measurement_noise_std, float) or measurement_noise_std <= 0:
        raise ValueError(
            "The measurement noise standard deviation must be a positive float."
        )

    # Compute magnetic field calibration
    mf = magnetic_field

    stop_tol = 1e-24  # Stop Condition from Alonso paper
    I3 = np.eye(3, dtype=np.float64)

    # TWOSTEP Centered estimate
    # Set initial guess for b and D.
    b0 = np.zeros((3, 1))
    d0 = np.zeros((3, 3))

    # Form L matrix, equations (5.10b) and (5.12a)
    l1 = 2 * mf
    l2 = -np.square(mf)
    l3 = -2 * mf[:, [0]] * mf[:, [1]]
    l4 = -2 * mf[:, [0]] * mf[:, [2]]
    l5 = -2 * mf[:, [1]] * mf[:, [2]]
    L_k = np.concatenate([l1, l2, l3, l4, l5], axis=1)

    # Compute sensor error as scalar measurement, equation (5.7a)
    h_matrix = np.ones((mf.shape)) * reference_magnetic_field
    z_k = (
        np.square(np.linalg.norm(mf, axis=1))
        - np.square(np.linalg.norm(h_matrix, axis=1))
    ).reshape(-1, 1)

    # Sensor measurements noise modeled as white gaussian with standard deviation epsilon_k
    epsilon_sq_k = np.ones(mf.shape) * (measurement_noise_std**2)

    # Sensor error scalar measurement noise characterization as gaussian.
    # Gaussian distribution mean, equation (5.14)
    mu_k = -np.sum(epsilon_sq_k, axis=1, keepdims=True)

    # Gaussian distribution variance, equation (5.15)
    sigma_sq_k = (
        4
        * np.einsum(
            "ijk->ikj",
            np.tile(I3 + d0, (mf.shape[0], 1, 1)) @ mf.reshape(-1, 3, 1)
            - np.tile(b0, (mf.shape[0], 1, 1)),
        )
        @ np.apply_along_axis(np.diag, 1, epsilon_sq_k)
        @ (
            np.tile(I3 + d0, (mf.shape[0], 1, 1)) @ mf.reshape(-1, 3, 1)
            - np.tile(b0, (mf.shape[0], 1, 1))
        )
        + 2
        * np.apply_along_axis(
            lambda x: np.square(np.trace(np.diag(x))), 1, epsilon_sq_k
        ).reshape(-1, 1, 1)
    ).reshape(-1, 1)

    # Calculate centered sigma squared, equation (5.18)
    sigma_sq_bar = 1 / np.sum(1 / sigma_sq_k)

    # Center the data, equation (5.19)
    mu_bar, mu_k_tilde = _cal_mag_twostep_center_data(mu_k, sigma_sq_k, sigma_sq_bar)
    z_bar, z_k_tilde = _cal_mag_twostep_center_data(z_k, sigma_sq_k, sigma_sq_bar)
    L_bar, L_k_tilde = _cal_mag_twostep_center_data(L_k, sigma_sq_k, sigma_sq_bar)

    # Compute fisher information matrix
    I_fisher_tilde, I_fishinv_tilde = _cal_mag_twostep_TS_fisher_centered(
        sigma_sq_k, L_k_tilde
    )

    # Compute centered estimate, equation (5.24)
    f_matrix = np.einsum(
        "ijk->jk",
        (
            (1 / sigma_sq_k).reshape(-1, 1, 1)
            * ((z_k_tilde - mu_k_tilde).reshape(-1, 1, 1) * L_k_tilde.reshape(-1, 9, 1))
        ),
    )
    theta_0_tilde = I_fishinv_tilde @ f_matrix

    # TWOSTEP Center correction
    theta_n, theta_np1 = (
        theta_0_tilde,
        theta_0_tilde,
    )  # Initiate theta for  first  iteration
    n = 0  # Initialise  iteration counter
    TS_err = np.inf  # Initial  condition  for  error.

    # ABC is used to remove intensive calculations out of for loop
    abc = -np.einsum(
        "ijk->jk",
        (
            (1 / sigma_sq_k).reshape(-1, 1, 1)
            * ((z_k_tilde - mu_k_tilde).reshape(-1, 1, 1) * L_k_tilde.reshape(-1, 9, 1))
        ),
    )

    while TS_err > stop_tol and n < max_iterations:
        if n != 0:  # If  we are not  in the first	iteration
            theta_n = theta_np1

        # Extract  c  and  E  components
        c, e_matrix = _cal_mag_twostep_theta_to_c_E(theta_n)

        # Compute  second  derivative  of  b^2  wrt theta
        tmp = (
            np.linalg.solve((np.eye(3) + e_matrix), c)
            @ np.linalg.solve((np.eye(3) + e_matrix), c).T
        )
        dbsqdtheta_p = np.concatenate(
            [
                2 * np.linalg.solve((np.eye(3) + e_matrix), c),
                -np.diag(tmp).reshape(3, 1),
                np.vstack([-2 * tmp[0, 1], -2 * tmp[0, 2], -2 * tmp[1, 2]]),
            ]
        )
        # Compute gradient of J
        dJdThetap_tilde = abc + I_fisher_tilde @ theta_n
        dJdThetap_bar = (
            -(1 / sigma_sq_bar)
            * (L_bar.reshape(-1, 1) - dbsqdtheta_p)
            * (
                z_bar
                - (L_bar.reshape(1, -1) @ theta_n)
                + (c.T @ np.linalg.solve((np.eye(3) + e_matrix), c))
                - mu_bar
            )
        )
        dJdTheta = dJdThetap_tilde + dJdThetap_bar

        # Calculate Fisher matrix
        I_fisher_bar = _cal_mag_twostep_TS_fisher_center(
            sigma_sq_bar, L_bar, dbsqdtheta_p
        )

        # Update theta
        theta_np1 = theta_n - np.linalg.solve((I_fisher_tilde + I_fisher_bar), dJdTheta)

        # Compute error
        TS_err = ((theta_np1 - theta_n).T @ (I_fisher_tilde + I_fisher_bar)) @ (
            theta_np1 - theta_n
        )
        n += 1

    b, d_matrix = _cal_mag_twostep_theta_to_b_D(theta_np1)

    # Extract covariance matrix
    m_cd = np.array(
        [
            [b[0, 0], 0, 0, b[1, 0], b[2, 0], 0],
            [0, b[1, 0], 0, b[0, 0], 0, b[2, 0]],
            [0, 0, b[2, 0], 0, b[0, 0], b[1, 0]],
        ]
    )
    m_ed = np.array(
        [
            [2 * d_matrix[0, 0], 0, 0, 2 * d_matrix[0, 1], 2 * d_matrix[0, 2], 0],
            [0, 2 * d_matrix[1, 1], 0, 2 * d_matrix[0, 1], 0, 2 * d_matrix[1, 2]],
            [0, 0, 2 * d_matrix[2, 2], 0, 2 * d_matrix[0, 2], 2 * d_matrix[1, 2]],
            [
                d_matrix[0, 1],
                d_matrix[0, 1],
                0,
                d_matrix[0, 0] + d_matrix[1, 1],
                d_matrix[1, 2],
                d_matrix[0, 2],
            ],
            [
                d_matrix[0, 2],
                0,
                d_matrix[0, 2],
                d_matrix[1, 2],
                d_matrix[0, 0] + d_matrix[2, 2],
                d_matrix[0, 1],
            ],
            [
                0,
                d_matrix[1, 2],
                d_matrix[1, 2],
                d_matrix[0, 2],
                d_matrix[0, 1],
                d_matrix[1, 1] + d_matrix[2, 2],
            ],
        ]
    )
    dbD_dcE = np.eye(9)
    dbD_dcE[:3, :3], dbD_dcE[:3, 3:] = np.eye(3) + d_matrix, m_cd
    dbD_dcE[3:, :3], dbD_dcE[3:, 3:] = np.zeros((6, 3)), 2 * np.eye(6) @ m_ed
    dbD_dcE = np.linalg.inv(dbD_dcE)
    # Cov_est = dbD_dcE @ np.linalg.solve((I_fisher_tilde + I_fisher_bar), dbD_dcE.T)

    # END   TWOSTEP
    hard_iron = (np.linalg.inv(np.eye(3) + d_matrix)) @ b
    soft_iron = np.linalg.inv(np.eye(3) + d_matrix)

    # Calibrate magnetic field
    calibrated_magnetic_field = magnetic_field.copy()[..., np.newaxis]
    N = magnetic_field.shape[0]
    calibrated_magnetic_field = calibrated_magnetic_field - np.tile(
        hard_iron.reshape(3, 1), (N, 1, 1)
    )
    calibrated_magnetic_field = (
        np.tile(np.linalg.inv(soft_iron), (N, 1, 1)) @ calibrated_magnetic_field
    ).squeeze()

    return hard_iron.flatten(), soft_iron, calibrated_magnetic_field