Skip to content

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 
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
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 
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
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 
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
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 
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
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 
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
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 
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
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 
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
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 
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
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 
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
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 
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
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 
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
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 
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
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 
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
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