AHRS: Attitude and Heading Reference Systems

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AHRS is a collection of functions and algorithms in pure Python used to estimate the orientation of mobile systems.

Orginally, an AHRS is a set of orthogonal sensors providing attitude information about an aircraft. This field has now expanded to smaller devices, like wearables, automated transportation and all kinds of systems in motion.

This package’s focus is fast prototyping, education, testing and modularity. Performance is NOT the main goal. For optimized implementations there are endless resources in C/C++ or Fortran.

AHRS is compatible with Python 3.6 and newer.

Installation

The most recommended method is to install AHRS directly from this repository to get the latest version:

git clone https://github.com/Mayitzin/ahrs.git
cd ahrs
python -m pip install .

Or using pip for the stable releases:

pip install ahrs

AHRS depends merely on NumPy. More packages are avoided, to reduce its third-party dependency.

Versioning

In order to update the version, use hatch and adjust it automatically

hatch version <major, minor, patch>

Important novelties in 0.3

(Click on each topic to see more details.)

The World Magnetic Model (WMM) is fully implemented.

It is a re-implementation of the Spherical Harmonics approximation used by the United States’ National Geopatial-Intelligence Agency. It can be used to estimate all magnetic field elements on any given place of Earth for dates between 2015 and 2025.

python
>>> from ahrs.utils import WMM
>>> wmm = WMM(latitude=10.0, longitude=-20.0, height=10.5)
>>> wmm.magnetic_elements
{'X': 30499.640469609083, 'Y': -5230.267158472566, 'Z': -1716.633311360368,
'H': 30944.850352270452, 'F': 30992.427998627096, 'I': -3.1751692563622993,
'D': -9.73078560629778, 'GV': -9.73078560629778}

The Ellipsoid model of the World Geodetic System (WGS84) is also included.

The estimation of the main and derived parameters of the WGS84 using the ellipsoid model are implemented:

python
>>> from ahrs.utils import WGS
>>> wgs = WGS()      # Creates an ellipsoid model, using Earth's characteristics by default
>>> wgs_properties = [x for x in dir(wgs) if not (hasattr(wgs.__getattribute__(x), '__call__') or x.startswith('__'))]
>>> for p in wgs_properties:
...     print('{:<{w}}  {}'.format(p, wgs.__getattribute__(p), w=len(max(wgs_properties, key=len))))
...
a                                          6378137.0
arithmetic_mean_radius                     6371008.771415059
aspect_ratio                               0.9966471893352525
atmosphere_gravitational_constant          343591934.4
authalic_sphere_radius                     6371007.1809182055
b                                          6356752.314245179
curvature_polar_radius                     6399593.625758493
dynamic_inertial_moment_about_X            8.007921777277886e+37
dynamic_inertial_moment_about_Y            8.008074799852911e+37
dynamic_inertial_moment_about_Z            8.03430094201443e+37
dynamical_form_factor                      0.0010826298213129219
equatorial_normal_gravity                  9.78032533590406
equivolumetric_sphere_radius               6371000.790009159
f                                          0.0033528106647474805
first_eccentricity_squared                 0.0066943799901413165
geometric_dynamic_ellipticity              0.003258100628533992
geometric_inertial_moment                  8.046726628049449e+37
geometric_inertial_moment_about_Z          8.073029370114392e+37
gm                                         398600441800000.0
gravitational_constant_without_atmosphere  398600098208065.6
is_geodetic                                True
linear_eccentricity                        521854.00842338527
mass                                       5.972186390142457e+24
mean_normal_gravity                        9.797643222256516
normal_gravity_constant                    0.0034497865068408447
normal_gravity_potential                   62636851.71456948
polar_normal_gravity                       9.832184937863065
second_degree_zonal_harmonic               -0.00048416677498482876
second_eccentricity_squared                0.006739496742276434
w                                          7.292115e-05

It can be used, for example, to estimate the normal gravity acceleration (in m/s^2) at any location on Earth.

python
>>> wgs.normal_gravity(50.0, 1000.0)    # Normal gravity at latitude = 50.0 °, 1000 m above surface
9.807617683884756

Setting the fundamental parameters (a, f, GM, w) yields a different ellipsoid. For the moon, for instance, we build a new model:

python
>>> moon_a = ahrs.MOON_EQUATOR_RADIUS
>>> moon_f = (ahrs.MOON_EQUATOR_RADIUS-ahrs.MOON_POLAR_RADIUS)/ahrs.MOON_EQUATOR_RADIUS
>>> moon_gm = ahrs.MOON_GM
>>> moon_w = ahrs.MOON_ROTATION
>>> moon = WGS(a=moon_a, f=moon_f, GM=moon_gm, w=moon_w)
>>> moon.normal_gravity(10.0, h=500.0)  # Gravity on moon at 10° N and 500 m above surface
1.6239259827292798
>>> moon.is_geodetic     # Only the Earth is geodetic
False

A full implementation of the Earth Gravitational Model (EGM2008) using Spherical Harmonics is NOT available here.

The International Gravity Formula and the EU’s WELMEC normal gravity reference system are also implemented.

python
>>> ahrs.utils.international_gravity(50.0)       # Latitude = 50° N
9.810786421572386
>>> ahrs.utils.welmec_gravity(50.0, 500.0)       # Latitude = 50° N,   height above sea = 500 m
9.809152687885897

New class DCM (derived from numpy.ndarray).

This new class represents 3x3 Direction Cosine Matrices used to describe orientations / rotations operations.

python
>>> from ahrs import DCM
>>> R = DCM(x=10.0, y=20.0, z=30.0)
>>> type(R)
<class 'ahrs.common.dcm.DCM'>
>>> R.view()
DCM([[ 0.81379768 -0.46984631  0.34202014],
     [ 0.54383814  0.82317294 -0.16317591],
     [-0.20487413  0.31879578  0.92541658]])
>>> R.inv     # or R.I
array([[ 0.81379768  0.54383814 -0.20487413]
       [-0.46984631  0.82317294  0.31879578]
       [ 0.34202014 -0.16317591  0.92541658]])
>>> R.log
array([0.26026043, 0.29531805, 0.5473806 ])
>>> R.to_axisangle()        # Axis in 3D NumPy array, and angle as radians
(array([0.38601658, 0.43801381, 0.81187135]), 0.6742208510527136)
>>> R.to_quaternion()
array([0.94371436, 0.12767944, 0.14487813, 0.26853582])
>>> R.to_quaternion(method='itzhack', version=2)
array([ 0.94371436, -0.12767944, -0.14487813, -0.26853582])

New class QuaternionArray (derived from numpy.ndarray).

This class can be used to simultaneously handle an array with several quaternions at once.

python
>>> Q = QuaternionArray(np.random.random((3, 4))-0.5)
>>> Q.view()
QuaternionArray([[ 0.31638467,  0.59313477, -0.62538687, -0.39621099],
                 [ 0.24973118, -0.37958194, -0.67851278, -0.57721079],
                 [-0.44643469,  0.17200957, -0.72678553,  0.49284031]])
>>> Q.w
array([ 0.31638467,  0.24973118, -0.44643469])
>>> Q.to_DCM()
array([[[-0.09618377, -0.49116723, -0.86573866],
        [-0.99258756, -0.017584  ,  0.1202528 ],
        [-0.07428738,  0.8708878 , -0.48583519]],

       [[-0.58710377,  0.80339746,  0.09930598],
        [ 0.22680733,  0.04549051,  0.97287669],
        [ 0.77708918,  0.5937029 , -0.20892408]],

       [[-0.54221755,  0.19001389,  0.81847104],
        [-0.69007015,  0.45504228, -0.56279633],
        [-0.47937805, -0.86996048, -0.115609  ]]])
>>> Q.conjugate()
array([[ 0.31638467, -0.59313477,  0.62538687,  0.39621099],
       [ 0.24973118,  0.37958194,  0.67851278,  0.57721079],
       [-0.44643469, -0.17200957,  0.72678553, -0.49284031]])
>>> Q.average()
array([ 0.19537239,  0.17826049, -0.87872408, -0.39736232])

Type hints are added.
NumPy is now the only third-party dependency.
New submodule frames to represent the position of an object in different reference frames.
Metrics for rotations in 3D spaces using quaternions and direction cosine matrices.
New operations, properties and methods for class Quaternion (now also derived from numpy.ndarray)
A whole bunch of new constant values (mainly for Geodesy) accessed from the top level of the package.
Docstrings are improved with further explanations, references and equations whenever possible.

More Attitude Estimators

One of the biggest improvements in this version is the addition of many new attitude estimation algorithms.

All estimators are refactored to be consistent with the corresponding articles describing them. They have in-code references to the equations, so that you can follow the original articles along with the code.

These estimators are based on two main solutions:

Wahba’s Problem (WP), which finds a rotation matrix between two coordinate systems. This means we compare measurement vectors against reference vectors. Their difference is the rotation. The solution to Wahba’s problem mainly compares accelerometers and magnetometers against the gravitational and geomagnetic vectors, correspondingly.
Dead Reckoning (DR) integrating the measured local angular velocity to increasingly estimate the angular position of the sensor.

Implemented attitude estimators are:

Algorithm	Gyroscope	Accelerometer	Magnetometer
AQUA	YES	YES	Optional
Complementary	YES	YES	Optional
Davenport’s	NO	YES	YES
EKF	YES	YES	YES
FAMC	NO	YES	YES
FLAE	NO	YES	YES
Fourati	YES	YES	YES
FQA	NO	YES	Optional
Integration	YES	NO	NO
Madgwick	YES	YES	Optional
Mahony	YES	YES	Optional
OLEQ	NO	YES	YES
QUEST	NO	YES	YES
ROLEQ	YES	YES	YES
SAAM	NO	YES	YES
Tilt	NO	YES	Optional
TRIAD	NO	YES	YES

To use the sensor data to estimate the attitude simply pass the data to a desired estimator, and it will automatically estimate the quaternions with the given parameters.

>>> attitude = ahrs.filters.Madgwick(acc=acc_data, gyr=gyro_data)
>>> attitude.Q.shape
(6959, 4)

Some algorithms allow a finer tuning of its estimation with different parameters. Check their documentation to see what can be tuned.

>>> attitude = ahrs.filters.Madgwick(acc=acc_data, gyr=gyro_data, mag=mag_data, gain=0.1, frequency=100.0)

Speaking of documentation…

Documentation

A comprehensive documentation, with examples, is now available in
Read the Docs.

Note for future versions

ahrs moves away from plotting and data handling submodules to better focus in the algorithmic parts. Submodules io and plot are not built in the package anymore, and will be entirely removed from the base code in the next release.

This way you can also choose your favorite libraries for data loading and visualization. This also means, getting rid of its dependency on matplotlib too.