archimedes.spatial.Quaternion

class archimedes.spatial.Quaternion(array: ndarray)

Quaternion representation of a rotation in 3 dimensions.

This class is closely modeled after [scipy.spatial.transform.Rotation]( https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.transform.Rotation.html) with a few differences:

• The quaternion is always represented in scalar-first format (i.e. [w, x, y, z]) instead of scalar-last ([x, y, z, w]).

• This class is designed for symbolic computation, so some checks (e.g. for valid rotation matrices) are omitted, since these cannot be done symbolically.

• The class does not support multiple rotations in a single object

• This implementation supports kinematic calculations

The following operations on quaternions are supported:

• Application on vectors (rotations of vectors)

• Quaternion Composition

• Quaternion Inversion

• Kinematic time derivative given angular velocity

Parameters:

quat (array_like, shape (4,)) – Quaternion representing the rotation in scalar-first format (w, x, y, z).

array

Underlying numpy array representing the quaternion.

Type:

np.ndarray, shape (4,)

Examples

>>> from archimedes.spatial import Quaternion
>>> import numpy as np

Consider a counter-clockwise rotation of 90 degrees about the z-axis. This corresponds to the following quaternion (in scalar-first format):

>>> q = Quaternion([np.cos(np.pi/4), 0, 0, np.sin(np.pi/4)])

The quaternion can be expressed in any of the other formats:

>>> q.as_matrix()
array([[ 2.22044605e-16,  1.00000000e+00,  0.00000000e+00],
   [-1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
   [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
>>> np.rad2deg(q.as_euler('zyx'))
array([90.,  0.,  0.])

The same quaternion can be initialized using a rotation matrix:

>>> q = Quaternion.from_matrix([[0, 1, 0],
...                    [-1, 0, 0],
...                    [0, 0, 1]])

Representation in other formats:

>>> np.rad2deg(q.as_euler('zyx'))
array([90.,  0.,  0.])

The from_euler method is flexible in the range of input formats it supports. Here we initialize a quaternion about a single axis:

>>> q = Quaternion.from_euler(np.deg2rad(90), 'z')

The associated rotation matrix can be used to change coordinate systems. If the quaternion represents the orientation of a frame B relative to a frame A, then the rotation matrix transforms vectors from frame A to frame B:

>>> v_A = np.array([1, 0, 0])  # Vector in frame A
>>> R_BA = q.as_matrix()
>>> R_BA @ v_A  # Vector in frame B
[6.12323e-17, -1, 0]

The kinematics method can be used to compute the time derivative of the quaternion as an attitude representation given the angular velocity in the body frame using quaternion kinematics:

>>> w_B = np.array([0, 0, np.pi/2])  # 90 deg/s about z-axis
>>> q.kinematics(w_B)
array([-0.55536037,  0.        ,  0.        ,  0.55536037])

See also

scipy.spatial.transform.Rotation

Similar class in SciPy

RigidBody

Rigid body dynamics supporting Quaternion attitude representation

euler_to_dcm

Directly calculate rotation matrix from roll-pitch-yaw angles

euler_kinematics

Transform roll-pitch-yaw rates to body-frame angular velocity

quaternion_kinematics

Low-level quaternion kinematics function

__init__(array: ndarray) → None

Methods

__init__(array)
as_euler(seq)	Return the Euler angles from the quaternion
as_matrix()	Return the quaternion as a rotation matrix.
as_quat()	Return the same object - dummy method for API consistency.
from_euler(euler[, seq])	Create a Quaternion from Euler angles.
from_matrix(matrix)	Create a Quaternion from a rotation matrix.
from_quat(quat)	Returns a copy of the Quaternion object - dummy method for API consistency.
identity()	Return a quaternion representing the identity rotation.
inv()	Return the inverse of the quaternion
kinematics(w[, baumgarte])	Return the time derivative of the quaternion given angular velocity w.
mul(other[, normalize])	Compose (multiply) this quaternion with another
normalize()	Return a normalized version of this quaternion.
replace(**updates)	Returns a new object replacing the specified fields with new values.

Attributes

array