rowan¶
Overview
rowan.conjugate |
Conjugates an array of quaternions |
rowan.inverse |
Computes the inverse of an array of quaternions |
rowan.exp |
Computes the natural exponential function \(e^q\). |
rowan.expb |
Computes the exponential function \(b^q\). |
rowan.exp10 |
Computes the exponential function \(10^q\). |
rowan.log |
Computes the quaternion natural logarithm. |
rowan.logb |
Computes the quaternion logarithm to some base b. |
rowan.log10 |
Computes the quaternion logarithm base 10. |
rowan.multiply |
Multiplies two arrays of quaternions |
rowan.divide |
Divides two arrays of quaternions |
rowan.norm |
Compute the quaternion norm |
rowan.normalize |
Normalize quaternions |
rowan.rotate |
Rotate a list of vectors by a corresponding set of quaternions |
rowan.vector_vector_rotation |
Find the quaternion to rotate one vector onto another |
rowan.from_euler |
Convert Euler angles to quaternions |
rowan.to_euler |
Convert quaternions to Euler angles |
rowan.from_matrix |
Convert the rotation matrices mat to quaternions |
rowan.to_matrix |
Convert quaternions into rotation matrices. |
rowan.from_axis_angle |
Find quaternions to rotate a specified angle about a specified axis |
rowan.to_axis_angle |
Convert the quaternions in q to axis angle representations |
rowan.from_mirror_plane |
Generate quaternions from mirror plane equations. |
rowan.reflect |
Reflect a list of vectors by a corresponding set of quaternions |
rowan.equal |
Check whether two sets of quaternions are equal. |
rowan.not_equal |
Check whether two sets of quaternions are not equal. |
rowan.isfinite |
Test element-wise for finite quaternions. |
rowan.isinf |
Test element-wise for infinite quaternions. |
rowan.isnan |
Test element-wise for NaN quaternions. |
Details
The core rowan
package contains functions for operating on
quaternions. The core package is focused on robust implementations of key
functions like multiplication, exponentiation, norms, and others. Simple
functionality such as addition is inherited directly from numpy due to
the representation of quaternions as numpy arrays. Many core numpy functions
implemented for normal arrays are reimplemented to work on quaternions (
such as allclose()
and isfinite()
). Additionally, numpy
broadcasting
is enabled throughout rowan unless otherwise specified. This means that
any function of 2 (or more) quaternions can take arrays of shapes that do
not match and return results according to numpy’s broadcasting rules.
-
rowan.
allclose
(p, q, **kwargs)¶ Check whether two sets of quaternions are all close.
This is a direct wrapper of the corresponding numpy function.
Parameters: - p ((..,4) np.array) – First set of quaternions
- q ((..,4) np.array) – First set of quaternions
- **kwargs – Keyword arguments to pass to np.allclose
Returns: Whether or not all quaternions are close
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rowan.
conjugate
(q)¶ Conjugates an array of quaternions
Parameters: q ((..,4) np.array) – Array of quaternions Returns: An array containing the conjugates of q Example:
q_star = conjugate(q)
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rowan.
divide
(qi, qj)¶ Divides two arrays of quaternions
Division is non-commutative; this function returns \(q_i q_j^{-1}\).
Parameters: - qi ((..,4) np.array) – Dividend quaternion
- qj ((..,4) np.array) – Divisors quaternions
Returns: An array containing the quotients of row i of qi with column j of qj
Example:
qi = np.array([[1, 0, 0, 0]]) qj = np.array([[1, 0, 0, 0]]) prod = divide(qi, qj)
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rowan.
exp
(q)¶ Computes the natural exponential function \(e^q\).
The exponential of a quaternion in terms of its scalar and vector parts \(q = a + \boldsymbol{v}\) is defined by exponential power series: formula \(e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}\) as follows:
\[\begin{split}\begin{align} e^q &= e^{a+v} \\ &= e^a \left(\sum_{k=0}^{\infty} \frac{v^k}{k!} \right) \\ &= e^a \left(\cos \lvert \lvert \boldsymbol{v} \rvert \rvert + \frac{\boldsymbol{v}}{\lvert \lvert \boldsymbol{v} \rvert \rvert} \sin \lvert \lvert \boldsymbol{v} \rvert \rvert \right) \end{align}\end{split}\]Parameters: q ((..,4) np.array) – Quaternions Returns: Array of shape (…) containing exponentials of q Example:
q_exp = exp(q)
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rowan.
expb
(q, b)¶ Computes the exponential function \(b^q\).
We define the exponential of a quaternion to an arbitrary base relative to the exponential function \(e^q\) using the change of base formula as follows:
\[\begin{split}\begin{align} b^q &= y \\ q &= \log_b y = \frac{\ln y}{\ln b}\\ y &= e^{q\ln b} \end{align}\end{split}\]Parameters: q ((..,4) np.array) – Quaternions Returns: Array of shape (…) containing exponentials of q Example:
q_exp = exp(q, 2)
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rowan.
exp10
(q)¶ Computes the exponential function \(10^q\).
Wrapper around
expb()
.Parameters: q ((..,4) np.array) – Quaternions Returns: Array of shape (…) containing exponentials of q Example:
q_exp = exp(q, 2)
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rowan.
equal
(p, q)¶ Check whether two sets of quaternions are equal.
This function is a simple wrapper that checks array equality and then aggregates along the quaternion axis.
Parameters: - p ((..,4) np.array) – First set of quaternions
- q ((..,4) np.array) – First set of quaternions
Returns: A boolean array of shape (…) indicating equality.
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rowan.
from_axis_angle
(axes, angles)¶ Find quaternions to rotate a specified angle about a specified axis
Parameters: - axes ((..,3) np.array) – An array of vectors (the axes)
- angles (float or (..,1) np.array) – An array of angles in radians. Will be broadcast to match shape of v as needed
Returns: An array of the desired rotation quaternions
Example:
import numpy as np axis = np.array([[1, 0, 0]]) ang = np.pi/3 quat = about_axis(axis, ang)
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rowan.
from_euler
(alpha, beta, gamma, convention='zyx', axis_type='intrinsic')¶ Convert Euler angles to quaternions
For generality, the rotations are computed by composing a sequence of quaternions corresponding to axis-angle rotations. While more efficient implementations are possible, this method was chosen to prioritize flexibility since it works for essentially arbitrary Euler angles as long as intrinsic and extrinsic rotations are not intermixed.
Parameters: - alpha ((..) np.array) – Array of \(\alpha\) values in radians.
- beta ((..) np.array) – Array of \(\beta\) values in radians.
- gamma ((..) np.array) – Array of \(\gamma\) values in radians.
- convention (str) – One of the 12 valid conventions xzx, xyx, yxy, yzy, zyz, zxz, xzy, xyz, yxz, yzx, zyx, zxy
- axes (str) – Whether to use extrinsic or intrinsic rotations
Returns: An array containing the converted quaternions
Example:
rands = np.random.rand(100, 3) alpha, beta, gamma = rands.T ql.from_euler(alpha, beta, gamma)
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rowan.
from_matrix
(mat, require_orthogonal=True)¶ Convert the rotation matrices mat to quaternions
Thhis method uses the algorithm described by Bar-Itzhack in [Itzhack00]. The idea is to construct a matrix K whose largest eigenvalue corresponds to the desired quaternion. One of the strengths of the algorithm is that for nonorthogonal matrices it gives the closest quaternion representation rather than failing outright.
[Itzhack00] Itzhack Y. Bar-Itzhack. “New Method for Extracting the Quaternion from a Rotation Matrix”, Journal of Guidance, Control, and Dynamics, Vol. 23, No. 6 (2000), pp. 1085-1087 https://doi.org/10.2514/2.4654 Parameters: mat ((..,3,3) np.array) – An array of rotation matrices Returns: An array containing the quaternion representations of the elements of mat (i.e. the same elements of SO(3))
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rowan.
from_mirror_plane
(x, y, z)¶ Generate quaternions from mirror plane equations.
Reflection quaternions can be constructed of the from \((0, x, y, z)\), i.e. with zero real component. The vector \((x, y, z)\) is the normal to the mirror plane.
Parameters: - x ((..) np.array) – First planar component
- y ((..) np.array) – Second planar component
- z ((..) np.array) – Third planar component
Returns: An array of quaternions corresponding to the provided reflections.
Example:
plane = (1, 2, 3) quat_ref = from_mirror_plane(*plane)
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rowan.
inverse
(q)¶ Computes the inverse of an array of quaternions
Parameters: q ((..,4) np.array) – Array of quaternions Returns: An array containing the inverses of q Example:
q_inv = inverse(q)
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rowan.
isclose
(p, q, **kwargs)¶ Element-wise check of whether two sets of quaternions close.
This function is a simple wrapper that checks using the corresponding numpy function and then aggregates along the quaternion axis.
Parameters: - p ((..,4) np.array) – First set of quaternions
- q ((..,4) np.array) – First set of quaternions
- **kwargs – Keyword arguments to pass to np.isclose
Returns: A boolean array of shape (…)
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rowan.
isinf
(q)¶ Test element-wise for infinite quaternions.
A quaternion is defined as infinite if any elements are infinite.
Parameters: q ((..,4) np.array) – Quaternions to check Returns: A boolean array of shape (…) indicating infinite quaternions.
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rowan.
isfinite
(q)¶ Test element-wise for finite quaternions.
A quaternion is defined as finite if all elements are finite.
Parameters: q ((..,4) np.array) – Quaternions to check Returns: A boolean array of shape (…) indicating finite quaternions.
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rowan.
isnan
(q)¶ Test element-wise for NaN quaternions.
A quaternion is defined as NaN if any elements are NaN.
Parameters: q ((..,4) np.array) – Quaternions to check Returns: A boolean array of shape (…) indicating NaN.
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rowan.
is_unit
(q)¶ Check if all input quaternions have unit norm
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rowan.
log
(q)¶ Computes the quaternion natural logarithm.
The natural of a quaternion in terms of its scalar and vector parts \(q = a + \boldsymbol{v}\) is defined by inverting the exponential formula (see
exp()
), and is defined by the formula \(\frac{x^k}{k!}\) as follows:\[\begin{equation} \ln(q) = \ln\lvert\lvert q \rvert\rvert + \frac{\boldsymbol{v}}{\lvert\lvert \boldsymbol{v} \rvert\rvert} \arccos\left(\frac{a}{q}\right) \end{equation}\]Parameters: q ((..,4) np.array) – Quaternions Returns: Array of shape (…) containing logarithms of q Example:
ln_q = log(q)
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rowan.
logb
(q, b)¶ Computes the quaternion logarithm to some base b.
The quaternion logarithm for arbitrary bases is defined using the standard change of basis formula relative to the natural logarithm.
\[\begin{split}\begin{align} \log_b q &= y \\ q &= b^y \\ \ln q &= y \ln b \\ y &= \log_b q = \frac{\ln q}{\ln b} \end{align}\end{split}\]Parameters: - q ((..,4) np.array) – Quaternions
- n ((..) np.array) – Scalars to use as log bases
Returns: Array of shape (…) containing logarithms of q
Example:
log_q = log(q, 2)
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rowan.
log10
(q)¶ Computes the quaternion logarithm base 10.
Wrapper around
logb()
.Parameters: q ((..,4) np.array) – Quaternions Returns: Array of shape (…) containing logarithms of q Example:
log_q = log(q, 2)
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rowan.
multiply
(qi, qj)¶ Multiplies two arrays of quaternions
Note that quaternion multiplication is generally non-commutative.
Parameters: - qi ((..,4) np.array) – First set of quaternions
- qj ((..,4) np.array) – Second set of quaternions
Returns: An array containing the products of row i of qi with column j of qj
Example:
qi = np.array([[1, 0, 0, 0]]) qj = np.array([[1, 0, 0, 0]]) prod = multiply(qi, qj)
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rowan.
norm
(q)¶ Compute the quaternion norm
Parameters: q ((..,4) np.array) – Quaternions to find norms for Returns: An array containing the norms for qi in q Example:
q = np.random.rand(10, 4) norms = norm(q)
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rowan.
normalize
(q)¶ Normalize quaternions
Parameters: q ((..,4) np.array) – Array of quaternions to normalize Returns: An array containing the unit quaternions q/norm(q) Example:
q = np.random.rand(10, 4) u = normalize(q)
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rowan.
not_equal
(p, q)¶ Check whether two sets of quaternions are not equal.
This function is a simple wrapper that checks array equality and then aggregates along the quaternion axis.
Parameters: - p ((..,4) np.array) – First set of quaternions
- q ((..,4) np.array) – First set of quaternions
Returns: A boolean array of shape (…) indicating inequality.
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rowan.
power
(q, n)¶ Computes the power of a quaternion \(q^n\).
Quaternions raised to a scalar power are defined according to the polar decomposition angle \(\theta\) and vector \(\hat{u}\): \(q^n = \lvert\lvert q \rvert\rvert^n \left( \cos(n\theta) + \hat{u} \sin(n\theta)\right)\). However, this can be computed more efficiently by noting that \(q^n = \exp(n \ln(q))\).
Parameters: - q ((..,4) np.array) – Quaternions.
- n ((..) np.arrray) – Scalars to exponentiate quaternions with.
Returns: Array of shape (…) containing of q
Example:
q_n = pow(q^n)
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rowan.
reflect
(q, v)¶ Reflect a list of vectors by a corresponding set of quaternions
For help constructing a mirror plane, see
from_mirror_plane()
.Parameters: - q ((..,4) np.array) – Quaternions to use for reflection
- v ((..,3) np.array) – Vectors to reflect.
Returns: An array of the vectors in v reflected by q
Example:
q = np.random.rand(1, 4) v = np.random.rand(1, 3) v_rot = rotate(q, v)
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rowan.
rotate
(q, v)¶ Rotate a list of vectors by a corresponding set of quaternions
Parameters: - q ((..,4) np.array) – Quaternions to rotate by.
- v ((..,3) np.array) – Vectors to rotate.
Returns: An array of the vectors in v rotated by q
Example:
q = np.random.rand(1, 4) v = np.random.rand(1, 3) v_rot = rotate(q, v)
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rowan.
to_axis_angle
(q)¶ Convert the quaternions in q to axis angle representations
Parameters: q ((..,4) np.array) – An array of quaternions Returns: A tuple of np.arrays (axes, angles) where axes has shape (…,3) and angles has shape (…,1). The angles are in radians
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rowan.
to_euler
(q, convention='zyx', axis_type='intrinsic')¶ Convert quaternions to Euler angles
Euler angles are returned in the sequence provided, so in, e.g., the default case (‘zyx’), the angles returned are for a rotation \(Z(\alpha) Y(\beta) X(\gamma)\).
Note
In all cases, the \(\alpha\) and \(\gamma\) angles are between \(\pm \pi\). For proper Euler angles, \(\beta\) is between \(0\) and \(pi\) degrees. For Tait-Bryan angles, \(\beta\) lies between \(\pm\pi/2\).
For simplicity, quaternions are converted to matrices, which are then converted to their Euler angle representations. All equations for rotations are derived by considering compositions of the three elemental rotations about the three Cartesian axes:
\begin{eqnarray*} R_x(\theta) =& \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \\ \end{array}\right)\\ R_y(\theta) =& \left(\begin{array}{ccc} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0\\ -\sin \theta & 1 & \cos \theta \\ \end{array}\right)\\ R_z(\theta) =& \left(\begin{array}{ccc} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \\ \end{array}\right)\\ \end{eqnarray*}Extrinsic rotations are represented by matrix multiplications in the proper order, so \(z-y-x\) is represented by the multiplication \(XYZ\) so that the system is rotated first about \(Z\), then about \(y\), then finally \(X\). For intrinsic rotations, the order of rotations is reversed, meaning that it matches the order in which the matrices actually appear i.e. the \(z-y'-x''\) convention (yaw, pitch, roll) corresponds to the multiplication of matrices \(ZYX\). For proof of the relationship between intrinsic and extrinsic rotations, see the Wikipedia page on Davenport chained rotations.
For more information, see the Wikipedia page for Euler angles (specifically the section on converting between representations).
Parameters: - q ((..,4) np.array) – Quaternions to transform
- convention (str) – One of the 6 valid conventions zxz, xyx, yzy, zyz, xzx, yxy
- axes (str) – Whether to use extrinsic or intrinsic
Returns: An array with Euler angles \((\alpha, \beta, \gamma)\) as the last dimension (in radians)
Example:
rands = np.random.rand(100, 3) alpha, beta, gamma = rands.T ql.from_euler(alpha, beta, gamma) alpha_return, beta_return, gamma_return = ql.to_euler(full)
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rowan.
to_matrix
(q, require_unit=True)¶ Convert quaternions into rotation matrices.
Uses the conversion described on Wikipedia.
Parameters: q ((..,4) np.array) – An array of quaternions Returns: The array containing the matrix representations of the elements of q (i.e. the same elements of \(SO(3))\)
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rowan.
vector_vector_rotation
(v1, v2)¶ Find the quaternion to rotate one vector onto another
Parameters: - v1 ((..,3) np.array) – Vector to rotate
- v2 ((..,3) np.array) – Desired vector
Returns: Array (…, 4) of quaternions that rotate v1 onto v2.