as_davenport#
- Rotation.as_davenport(self, axes, order, degrees=False)#
Represent as Davenport angles.
Any orientation can be expressed as a composition of 3 elementary rotations.
For both Euler angles and Davenport angles, consecutive axes must be are orthogonal (
axis2
is orthogonal to bothaxis1
andaxis3
). For Euler angles, there is an additional relationship betweenaxis1
oraxis3
, with two possibilities:axis1
andaxis3
are also orthogonal (asymmetric sequence)axis1 == axis3
(symmetric sequence)
For Davenport angles, this last relationship is relaxed [1], and only the consecutive orthogonal axes requirement is maintained.
A slightly modified version of the algorithm from [2] has been used to calculate Davenport angles for the rotation about a given sequence of axes.
Davenport angles, just like Euler angles, suffer from the problem of gimbal lock [3], where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation.
- Parameters:
- axesarray_like, shape (3,) or ([1 or 2 or 3], 3)
Axis of rotation, if one dimensional. If two dimensional, describes the sequence of axes for rotations, where each axes[i, :] is the ith axis. If more than one axis is given, then the second axis must be orthogonal to both the first and third axes.
- orderstring
If it belongs to the set {‘e’, ‘extrinsic’}, the sequence will be extrinsic. If if belongs to the set {‘i’, ‘intrinsic’}, sequence will be treated as intrinsic.
- degreesboolean, optional
Returned angles are in degrees if this flag is True, else they are in radians. Default is False.
- Returns:
- anglesndarray, shape (3,) or (N, 3)
Shape depends on shape of inputs used to initialize object. The returned angles are in the range:
First angle belongs to [-180, 180] degrees (both inclusive)
Third angle belongs to [-180, 180] degrees (both inclusive)
Second angle belongs to a set of size 180 degrees, given by:
[-abs(lambda), 180 - abs(lambda)]
, wherelambda
is the angle between the first and third axes.
References
[1]Shuster, Malcolm & Markley, Landis. (2003). Generalization of the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304.
[2]Bernardes E, Viollet S (2022) Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE 17(11): e0276302. 10.1371/journal.pone.0276302
Examples
>>> from scipy.spatial.transform import Rotation as R >>> import numpy as np
Davenport angles are a generalization of Euler angles, when we use the canonical basis axes:
>>> ex = [1, 0, 0] >>> ey = [0, 1, 0] >>> ez = [0, 0, 1]
Represent a single rotation:
>>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([90., 0., 0.]) >>> r.as_euler('zxy', degrees=True) array([90., 0., 0.]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (3,)
Represent a stack of single rotation:
>>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([[90., 0., 0.]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (1, 3)
Represent multiple rotations in a single object:
>>> r = R.from_rotvec([ ... [0, 0, 90], ... [45, 0, 0]], degrees=True) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([[90., 0., 0.], [ 0., 45., 0.]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (2, 3)