r3f 2.0.10

A library for three-dimensional, reference-frame conversions

Rotation of 3-dimensional Frames

Functions

This library includes four sets of functions: general array checks, attitude-representation conversions, reference-frame conversions, and rotation matrix (direction cosine matrix) utilities.

All twenty possible conversions among the following five attitude representations are provided: rotation vector, rotation axis and angle, roll and pitch and yaw (RPY) Euler angles, direction cosine matrix (DCM), and quaternion. However, some of the conversions are built using other conversions. In the following table, the low-level conversions are marked with an x and the conversions build using the other conversions are marked with an o:

	vector	ax, ang	RPY	DCM	quat
vector	-	x	o	o	o
ax, ang	x	-	o	x	x
RPY	o	x	-	x	x
DCM	o	x	x	-	x
quat	o	x	x	x	-

The roll, pitch, and yaw angles are applied in a zyx sequence of passive rotations. The quaternions follow a Hamilton convention. Here is an example:

roll = 20*np.pi/180
pitch = 45*np.pi/180
yaw = 10*np.pi/180
C = r3f.rpy_to_dcm([roll, pitch, yaw])
>> C = [[ 0.69636424  0.1227878  -0.70710678]
    [ 0.07499469  0.96741248  0.24184476]
    [ 0.71375951 -0.2214413   0.66446302]]

In addition to the conversion from the z, y, x sequence of Euler angles to a DCM, the function rot is also provided for creating a DCM from a generic set of Euler angles in any desired sequence of axes. Although this rot function could be used, two additional functions are provided for generating rotation matrices: dcm_inertial_to_ecef and dcm_ecef_to_navigation. By default, all angles are treated as being in radians, but if the degs parameter is set to True, then they are treated as being in degrees.

This library includes all twelve possible conversions among the following four frames: ECEF (Earth-centered, Earth-fixed), geodetic (latitude, longitude, and height above ellipsoid), local-level tangent, and local-level curvilinear. By default, all local-level coordinates are interpreted as having a North, East, Down (NED) orientation, but if the ned parameter is set to False, the coordinates are interpreted as having an East, North, Up (ENU) orientation. Here is an example:

lat = 45*np.pi/180
lon = 0.0
hae = 1000.0
[xe, ye, ze] = r3f.geodetic_to_ecef([lat, lon, hae])
>> xe = 4518297.985630118
>> ye = 0.0
>> ze = 4488055.515647106

The rotation matrix utility functions are an orthonormalize_dcm function, a rodrigues_rotation function, and an inverse_rodrigues_rotation function. The orthonormalize_dcm function will work to make a rotation matrix normalized and orthogonal, a proper rotation matrix. The two Rodrigues's rotation functions are meant for converting a vector to the matrix exponential of the skew-symmetric matrix of that vector and back again.

Passive Rotations

Unless specifically otherwise stated, all rotations are interpreted as passive. This means they represent rotations of reference frames, not of vectors.

Vectorization

When possible, the functions are vectorized in order to handle processing batches of values. A set of scalars is a 1D array. A set of vectors is a 2D array, with each vector in a column. So, a (3, 7) array is a set of seven vectors, each with 3 elements. If the axis parameter is set to 0, the transpose is true. A set of matrices is a 3D array with each matrix in a stack. The first index is the stack number. So, a (5, 3, 3) array is a stack of five 3x3 matrices. Roll, pitch, and yaw are not treated as a vector but as three separate quantities. The same is true for latitude, longitude, and height above ellipsoid. A quaternion is passed around as an array.

Robustness

In general, the functions in this library check that the inputs are of the correct type and shape. They do not generally handle converting inputs which do not conform to the ideal type and shape. Generally, the allowed types are int, float, list, and np.ndarray.

Constants

The defined constants are

Name	Value	Definition
A_E	6378137.0	Earth's semi-major axis (m)
F_E	298.257223563	Earth's flattening constant
B_E	6356752.314245	Earth's semi-minor axis (m)
E2	6.694379990141317e-3	Earth's eccentricity squared (ND)
W_EI	7.2921151467e-5	sidereal Earth rate (rad/s)

Functions

The defined functions for attitude-representation conversion are

• axis_angle_to_vector
• vector_to_axis_angle
• rpy_to_vector
• vector_to_rpy
• dcm_to_vector
• vector_to_dcm
• quat_to_vector
• vector_to_quat
• rpy_to_axis_angle
• axis_angle_to_rpy
• dcm_to_axis_angle
• axis_angle_to_dcm
• quat_to_axis_angle
• axis_angle_to_quat
• dcm_to_rpy
• rpy_to_dcm
• rot
• rotate
• quat_to_rpy
• rpy_to_quat
• quat_to_dcm
• dcm_to_quat
• dcm_inertial_to_ecef
• dcm_ecef_to_navigation

The functions for reference-frame conversions are

• geodetic_to_ecef
• ecef_to_geodetic
• tangent_to_ecef
• ecef_to_tangent
• curvilinear_to_ecef
• ecef_to_curvilinear
• tangent_to_geodetic
• geodetic_to_tangent
• curvilinear_to_geodetic
• geodetic_to_curvilinear
• curvilinear_to_tangent
• tangent_to_curvilinear

A few additional rotation matrix utility functions are

• is_ortho
• orthonormalize_dcm
• rodrigues_rotation
• inverse_rodrigues_rotation