"""
Functions for working with quaternions. Note that all the functions also
work on arrays, and can deal with full quaternions as well as with
quaternion vectors.
"""
# author: Thomas Haslwanter
# date: Feb-2024
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
# The following construct is required since I want to run the module as a script
# inside the skinematics-directory
import os
import sys
file_dir = os.path.dirname(__file__)
if file_dir not in sys.path:
sys.path.insert(0, file_dir)
import vector, rotmat
#import deprecation
#import warnings
#warnings.simplefilter('always', DeprecationWarning)
pi = np.pi
[docs]
class Quaternion():
"""
Quaternion class, with multiplication, division, and inversion.
A Quaternion can be created from vectors, rotation matrices,
or from Fick-angles, Helmholtz-angles, or Euler angles ( in deg).
It provides
* operator overloading for mult, div, and inv.
* indexing
* access to the data, in the attribute * values * .
Parameters
----------
inData: np.ndarray
Contains the data in one of the following formats:
* vector : (3 x n) or (4 x n) array, containing the quaternion values
* rotmat : array, shape (3,3) or (N,9)
single rotation matrix, or matrix with rotation-matrix elements.
* Fick : (3 x n) array, containing (psi, phi, theta) rotations about
the (1,2,3) axes [deg] (Fick sequence)
* Helmholtz : (3 x n) array, containing (psi, phi, theta) rotations about
the (1,2,3) axes [deg] (Helmholtz sequence)
* Euler : (3 x n) array, containing (alpha, beta, gamma) rotations about
the (3,1,3) axes [deg] (Euler sequence)
inType : string
Specifies the type of the input and has to have one of the following values
'vector'[Default], 'rotmat', 'Fick', 'Helmholtz', 'Euler'
Attribute
---------
values : (4 x n) array
quaternion values
Method
------
inv() : Inverse of the quaterion
export(to='rotmat') : Export to one of the following formats: 'rotmat', 'Euler', 'Fick', 'Helmholtz'
Note
----
.. math::
\\vec {q}_{Euler} = \\left[ {\\begin{array}{*{20}{c}}
{\\cos \\frac{\\alpha }{2}*\\cos \\frac{\\beta }{2}*\\cos \\frac{\\gamma }{2} - \\sin \\frac{\\alpha }{2}\\cos \\frac{\\beta }{2}\\sin \\frac{\\gamma }{2}} \\\\
{\\cos \\frac{\\alpha }{2}*\\sin \\frac{\\beta }{2}*\\cos \\frac{\\gamma }{2} + \\sin \\frac{\\alpha }{2}\\sin \\frac{\\beta }{2}\\sin \\frac{\\gamma }{2}} \\\\
{\\cos \\frac{\\alpha }{2}*\\sin \\frac{\\beta }{2}*\\sin \\frac{\\gamma }{2} - \\sin \\frac{\\alpha }{2}\\sin \\frac{\\beta }{2}\\cos \\frac{\\gamma }{2}} \\\\
{\\cos \\frac{\\alpha }{2}*\\cos \\frac{\\beta }{2}*\\sin \\frac{\\gamma }{2} + \\sin \\frac{\\alpha }{2}\\cos \\frac{\\beta }{2}\\cos \\frac{\\gamma }{2}}
\\end{array}} \\right]
.. math::
\\vec {q}_{Fick} = \\left[ {\\begin{array}{*{20}{c}}
{\\cos \\frac{\\psi }{2}*\\cos \\frac{\\phi }{2}*\\cos \\frac{\\theta }{2} + \\sin \\frac{\\psi }{2}\\sin \\frac{\\phi }{2}\\sin \\frac{\\theta }{2}} \\\\
{\\sin \\frac{\\psi }{2}*\\cos \\frac{\\phi }{2}*\\cos \\frac{\\theta }{2} - \\cos \\frac{\\psi }{2}\\sin \\frac{\\phi }{2}\\sin \\frac{\\theta }{2}} \\\\
{\\cos \\frac{\\psi }{2}*\\sin \\frac{\\phi }{2}*\\cos \\frac{\\theta }{2} + \\sin \\frac{\\psi }{2}\\cos \\frac{\\phi }{2}\\sin \\frac{\\theta }{2}} \\\\
{\\cos \\frac{\\psi }{2}*\\cos \\frac{\\phi }{2}*\\sin \\frac{\\theta }{2} - \\sin \\frac{\\psi }{2}\\sin \\frac{\\phi }{2}\\cos \\frac{\\theta }{2}}
\\end{array}} \\right]
.. math::
\\vec {q}_{Helmholtz} = \\left[ {\\begin{array}{*{20}{c}}
{\\cos \\frac{\\psi }{2}*\\cos \\frac{\\phi }{2}*\\cos \\frac{\\theta }{2} - \\sin \\frac{\\psi }{2}\\sin \\frac{\\phi }{2}\\sin \\frac{\\theta }{2}} \\\\
{\\sin \\frac{\\psi }{2}*\\cos \\frac{\\phi }{2}*\\cos \\frac{\\theta }{2} + \\cos \\frac{\\psi }{2}\\sin \\frac{\\phi }{2}\\sin \\frac{\\theta }{2}} \\\\
{\\cos \\frac{\\psi }{2}*\\sin \\frac{\\phi }{2}*\\cos \\frac{\\theta }{2} + \\sin \\frac{\\psi }{2}\\cos \\frac{\\phi }{2}\\sin \\frac{\\theta }{2}} \\\\
{\\cos \\frac{\\psi }{2}*\\cos \\frac{\\phi }{2}*\\sin \\frac{\\theta }{2} - \\sin \\frac{\\psi }{2}\\sin \\frac{\\phi }{2}\\cos \\frac{\\theta }{2}}
\\end{array}} \\right]
Examples
--------
>>> q = Quaternion(array([[0,0,0.1],
[0,0,0.2],
[0,0,0.5]]))
>>> p = Quaternion(array([0,0,0.2]))
>>> fick = Quaternion( array([[0,0,10],
[0,10,10]]), 'Fick')
>>> combined = p * q
>>> divided = q / p
>>> extracted = q[1:2]
>>> len(q)
>>> data = q.values
>>> 2
>>> inv(q)
"""
def __init__(self, inData, inType='vector'):
""" Initialization """
if inType.lower() == 'vector':
if isinstance(inData, np.ndarray) or isinstance(inData, list):
self.values = unit_q(inData)
elif isinstance(inData, Quaternion):
self.values = inData.values
else:
raise TypeError('Quaternions can only be based on ndarray or Quaternions!')
elif inType.lower() == 'rotmat':
"""Conversion from rotation matrices to quaternions."""
self.values = rotmat2quat(inData)
elif inType.lower() == 'euler':
""" Conversion from Euler angles to quaternions.
(a,b,g) stands for (alpha, beta, gamma) """
inData[inData<0] += 360
inData = np.deg2rad(inData/2)
(ca, cb, cg) = np.cos(inData.T)
(sa, sb, sg) = np.sin(inData.T)
self.values = np.vstack( (ca*cb*cg - sa*cb*sg,
ca*sb*cg + sa*sb*sg,
ca*sb*sg - sa*sb*cg,
ca*cb*sg + sa*cb*cg) ).T
elif inType.lower() == 'fick':
""" Conversion from Fick angles to quaternions.
(p,f,t) stands for (psi, phi, theta) """
inData[inData<0] += 360
inData = np.deg2rad(inData/2)
(cp, cf, ct) = np.cos(inData.T)
(sp, sf, st) = np.sin(inData.T)
self.values = np.vstack( (cp*cf*ct + sp*sf*st,
sp*cf*ct - cp*sf*st,
cp*sf*ct + sp*cf*st,
cp*cf*st - sp*sf*ct) ).T
elif inType.lower() == 'helmholtz':
""" Conversion from Helmholtz angles to quaternions.
(p,f,t) stands for (psi, phi, theta) """
inData[inData<0] += 360
inData = np.deg2rad(inData/2)
(cp, cf, ct) = np.cos(inData.T)
(sp, sf, st) = np.sin(inData.T)
self.values = np.vstack( (cp*cf*ct - sp*sf*st,
sp*cf*ct + cp*sf*st,
cp*sf*ct + sp*cf*st,
cp*cf*st - sp*sf*ct ) ).T
def __len__(self):
"""The "length" is given by the number of quaternions."""
return len(self.values)
def __mul__(self, other):
"""Operator overloading for multiplication."""
if isinstance(other, int) or isinstance(other, float):
return Quaternion(self.values * other)
else:
return Quaternion(q_mult(self.values, other.values))
def __div__(self, other):
"""Operator overloading for division."""
if isinstance(other, int) or isinstance(other, float):
return Quaternion(self.values / other)
else:
return Quaternion(q_mult(self.values, q_inv(other.values)))
def __truediv__(self, other):
"""Operator overloading for division."""
if isinstance(other, int) or isinstance(other, float):
return Quaternion(self.values / other)
else:
return Quaternion(q_mult(self.values, q_inv(other.values)))
def __getitem__(self, select):
return Quaternion(self.values[select])
def __setitem__(self, select, item):
self.values[select] = unit_q(item)
#def __delitem__(self, select):
#np.delete(self.values, select, axis=0)
[docs]
def inv(self):
"""Inverse of a quaternion."""
return Quaternion(q_inv(self.values))
def __repr__(self):
return 'Quaternion ' + str(self.values)
[docs]
def export(self, to='rotmat'):
"""
Conversion to other formats. May be slow for "Fick", "Helmholtz", and "Euler".
Parameters
----------
to : string
content of returned values
* 'rotmat' : rotation matrices (default), each flattened to a 9-dim vector
* 'Euler' : Euler angles
* 'Fick' : Fick angles
* 'Helmholtz' : Helmholtz angles
* 'vector' : vector part of the quaternion
Returns
-------
ndarray, with the specified content
Examples
--------
>>> q = Quaternion([0,0.2,0.1])
>>> rm = q.export()
>>> fick = q.export('Fick')
"""
if to.lower() == 'rotmat' :
return convert(self.values, 'rotmat')
if to.lower() == 'vector' :
return self.values[:,1:]
if to.lower() == 'euler':
Euler = np.zeros((len(self),3))
rm = self.export()
if rm.shape == (3,3):
rm = rm.reshape((1,9))
for ii in range(len(self)):
Euler[ii,:] = rotmat.rotmat2Euler(rm[ii].reshape((3,3)))
return Euler
if to.lower() == 'fick':
Fick = np.zeros((len(self),3))
rm = self.export()
if rm.shape == (3,3):
rm = rm.reshape((1,9))
for ii in range(len(self)):
Fick[ii,:] = rotmat.rotmat2Fick(rm[ii].reshape((3,3)))
return Fick
if to.lower() == 'helmholtz':
Helmholtz = np.zeros((len(self),3))
rm = self.export()
if rm.shape == (3,3):
rm = rm.reshape((1,9))
for ii in range(len(self)):
Helmholtz[ii,:] = rotmat.rotmat2Helmholtz(rm[ii].reshape((3,3)))
return Helmholtz
[docs]
def convert(quat, to='rotmat'):
""" Calculate the rotation matrix corresponding to the quaternion. If
"inQuat" contains more than one quaternion, the matrix is flattened (to
facilitate the work with rows of quaternions), and can be restored to
matrix form by "reshaping" the resulting rows into a (3,3) shape.
Parameters
----------
inQuat : array_like, shape ([3,4],) or (N,[3,4])
quaternions or quaternion vectors
to : string
Has to be one of the following:
- rotmat : rotation matrix
- Gibbs : Gibbs vector
Returns
-------
rotMat : corresponding rotation matrix/matrices (flattened)
Notes
-----
.. math::
{\\bf{R}} = \\left( {\\begin{array}{*{20}{c}}
{q_0^2 + q_1^2 - q_2^2 - q_3^2}&{2({q_1}{q_2} - {q_0}{q_3})}&{2({q_1}{q_3} + {q_0}{q_2})}\\\\
{2({q_1}{q_2} + {q_0}{q_3})}&{q_0^2 - q_1^2 + q_2^2 - q_3^2}&{2({q_2}{q_3} - {q_0}{q_1})}\\\\
{2({q_1}{q_3} - {q_0}{q_2})}&{2({q_2}{q_3} + {q_0}{q_1})}&{q_0^2 - q_1^2 - q_2^2 + q_3^2} \\\\
\\end{array}} \\right)
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> r = quat.convert([0, 0, 0.1], to='rotmat')
>>> r.shape
(1, 9)
>>> r.reshape((3,3))
array([[ 0.98 , -0.19899749, 0. ],
[ 0.19899749, 0.98 , 0. ],
[ 0. , 0. , 1. ]])
"""
if to == 'rotmat':
q = unit_q(quat).T
R = np.zeros((9, q.shape[1]))
R[0] = q[0]**2 + q[1]**2 - q[2]**2 - q[3]**2
R[1] = 2*(q[1]*q[2] - q[0]*q[3])
R[2] = 2*(q[1]*q[3] + q[0]*q[2])
R[3] = 2*(q[1]*q[2] + q[0]*q[3])
R[4] = q[0]**2 - q[1]**2 + q[2]**2 - q[3]**2
R[5] = 2*(q[2]*q[3] - q[0]*q[1])
R[6] = 2*(q[1]*q[3] - q[0]*q[2])
R[7] = 2*(q[2]*q[3] + q[0]*q[1])
R[8] = q[0]**2 - q[1]**2 - q[2]**2 + q[3]**2
if R.shape[1] == 1:
return np.reshape(R, (3,3))
else:
return R.T
elif to == 'Gibbs':
q_0 = q_scalar(quat) # cos(alpha/2)
gibbs = (q_vector(quat).T / q_0).T # tan = sin/cos
return gibbs
[docs]
def deg2quat(inDeg):
"""
Convert axis-angles or plain degree into the corresponding quaternion values.
Can be used with a plain number or with an axis angle.
Parameters
----------
inDeg : float or (N,3)
quaternion magnitude or quaternion vectors.
Returns
-------
outQuat : float or array (N,3)
number or quaternion vector.
Notes
-----
.. math::
| \\vec{q} | = sin(\\theta/2)
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> quat.deg2quat(array([[10,20,30], [20,30,40]]))
array([[ 0.08715574, 0.17364818, 0.25881905],
[ 0.17364818, 0.25881905, 0.34202014]])
>>> quat.deg2quat(10)
0.087155742747658166
"""
deg = (inDeg+180)%360-180
return np.sin(0.5 * deg * pi/180)
[docs]
def q_conj(q):
""" Conjugate quaternion
Parameters
----------
q: array_like, shape ([3,4],) or (N,[3/4])
quaternion or quaternion vectors
Returns
-------
qconj : conjugate quaternion(s)
Examples
--------
>>> quat.q_conj([0,0,0.1])
array([ 0.99498744, -0. , -0. , -0.1 ])
>>> quat.q_conj([[cos(0.1),0,0,sin(0.1)],
>>> [cos(0.2), 0, sin(0.2), 0]])
array([[ 0.99500417, -0. , -0. , -0.09983342],
[ 0.98006658, -0. , -0.19866933, -0. ]])
"""
q = np.atleast_2d(q)
if q.shape[1]==3:
q = unit_q(q)
qConj = q * np.r_[1, -1,-1,-1]
if q.shape[0]==1:
qConj=qConj.ravel()
return qConj
[docs]
def q_inv(q):
""" Quaternion inversion
Parameters
----------
q: array_like, shape ([3,4],) or (N,[3/4])
quaternion or quaternion vectors
Returns
-------
qinv : inverse quaternion(s)
Notes
-----
.. math::
q^{-1} = \\frac{q_0 - \\vec{q}}{|q|^2}
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> quat.q_inv([0,0,0.1])
array([-0., -0., -0.1])
>>> quat.q_inv([[cos(0.1),0,0,sin(0.1)],
>>> [cos(0.2),0,sin(0.2),0]])
array([[ 0.99500417, -0. , -0. , -0.09983342],
[ 0.98006658, -0. , -0.19866933, -0. ]])
"""
q = np.atleast_2d(q)
if q.shape[1]==3:
return -q
else:
qLength = np.sum(q**2, 1)
qConj = q * np.r_[1, -1,-1,-1]
return (qConj.T / qLength).T
[docs]
def q_mult(p,q):
"""
Quaternion multiplication: Calculates the product of two quaternions r = p * q
If one of both of the quaterions have only three columns,
the scalar component is calculated such that the length
of the quaternion is one.
The lengths of the quaternions have to match, or one of
the two quaternions has to have the length one.
If both p and q only have 3 components, the returned quaternion
also only has 3 components (i.e. the quaternion vector)
Parameters
----------
p,q : array_like, shape ([3,4],) or (N,[3,4])
quaternions or quaternion vectors
Returns
-------
r : quaternion or quaternion vector (if both
p and q are contain quaternion vectors).
Notes
-----
.. math::
q \\circ p = \\sum\\limits_{i=0}^3 {q_i I_i} * \\sum\\limits_{j=0}^3 \\
{p_j I_j} = (q_0 p_0 - \\vec{q} \\cdot \\vec{p}) + (q_0 \\vec{p} + p_0 \\
\\vec{q} + \\vec{q} \\times \\vec{p}) \\cdot \\vec{I}
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> p = [cos(0.2), 0, 0, sin(0.2)]
>>> q = [[0, 0, 0.1],
>>> [0, 0.1, 0]]
>>> r = quat.q_mult(p,q)
"""
flag3D = False
p = np.atleast_2d(p)
q = np.atleast_2d(q)
if p.shape[1]==3 & q.shape[1]==3:
flag3D = True
if len(p) != len(q):
assert (len(p)==1 or len(q)==1), \
'Both arguments in the quaternion multiplication must have the same number of rows, unless one has only one row.'
p = unit_q(p).T
q = unit_q(q).T
if np.prod(np.shape(p)) > np.prod(np.shape(q)):
r=np.zeros(np.shape(p))
else:
r=np.zeros(np.shape(q))
r[0] = p[0]*q[0] - p[1]*q[1] - p[2]*q[2] - p[3]*q[3]
r[1] = p[1]*q[0] + p[0]*q[1] + p[2]*q[3] - p[3]*q[2]
r[2] = p[2]*q[0] + p[0]*q[2] + p[3]*q[1] - p[1]*q[3]
r[3] = p[3]*q[0] + p[0]*q[3] + p[1]*q[2] - p[2]*q[1]
if flag3D:
# for rotations > 180 deg
r[:,r[0]<0] = -r[:,r[0]<0]
r = r[1:]
r = r.T
return r
[docs]
def quat2deg(inQuat):
"""Calculate the axis-angle corresponding to a given quaternion.
Parameters
----------
inQuat: float, or array_like, shape ([3/4],) or (N,[3/4])
quaternion(s) or quaternion vector(s)
Returns
-------
axAng : corresponding axis angle(s)
float, or shape (3,) or (N,3)
Notes
-----
.. math::
| \\vec{q} | = sin(\\theta/2)
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> quat.quat2deg(0.1)
array([ 11.47834095])
>>> quat.quat2deg([0.1, 0.1, 0])
array([ 11.47834095, 11.47834095, 0. ])
>>> quat.quat2deg([cos(0.1), 0, sin(0.1), 0])
array([ 0. , 11.4591559, 0. ])
"""
return 2 * np.arcsin(q_vector(inQuat)) * 180 / pi
[docs]
def quat2seq(quats, seq='nautical'):
"""
This function takes a quaternion, and calculates the corresponding
angles for sequenctial rotations.
Parameters
----------
quats : ndarray, nx4
input quaternions
seq : string
Has to be one the following:
- Euler ... Rz * Rx * Rz
- Fick ... Rz * Ry * Rx
- nautical ... same as "Fick"
- Helmholtz ... Ry * Rz * Rx
Returns
-------
sequence : ndarray, nx3
corresponding angles [deg]
same sequence as in the rotation matrices
Examples
--------
>>> quat.quat2seq([0,0,0.1])
array([[ 11.47834095, -0. , 0. ]])
>>> quaternions = [[0,0,0.1], [0,0.2,0]]
skin.quat.quat2seq(quaternions)
array([[ 11.47834095, -0. , 0. ],
[ 0. , 23.07391807, 0. ]])
>>> skin.quat.quat2seq(quaternions, 'nautical')
array([[ 11.47834095, -0. , 0. ],
[ 0. , 23.07391807, 0. ]])
>>> skin.quat.quat2seq(quaternions, 'Euler')
array([[ 11.47834095, 0. , 0. ],
[ 90. , 23.07391807, -90. ]])
"""
# Ensure that it also works for a single quaternion
quats = np.atleast_2d(quats)
# If only the quaternion vector is entered, extend it to a full unit quaternion
if quats.shape[1] == 3:
quats = unit_q(quats)
if seq =='Fick' or seq =='nautical':
R_zx = 2 * (quats[:,1]*quats[:,3] - quats[:,0]*quats[:,2])
R_yx = 2 * (quats[:,1]*quats[:,2] + quats[:,0]*quats[:,3])
R_zy = 2 * (quats[:,2]*quats[:,3] + quats[:,0]*quats[:,1])
phi = -np.arcsin(R_zx)
theta = np.arcsin(R_yx / np.cos(phi))
psi = np.arcsin(R_zy / np.cos(phi))
sequence = np.column_stack((theta, phi, psi))
elif seq == 'Helmholtz':
R_yx = 2 * (quats[:,1]*quats[:,2] + quats[:,0]*quats[:,3])
R_zx = 2 * (quats[:,1]*quats[:,3] - quats[:,0]*quats[:,2])
R_yz = 2 * (quats[:,2]*quats[:,3] - quats[:,0]*quats[:,1])
theta = np.arcsin(R_yx)
phi = -np.arcsin(R_zx / np.cos(theta))
psi = -np.arcsin(R_yz / np.cos(theta))
sequence = np.column_stack((phi, theta, psi))
elif seq == 'Euler':
Rs = convert(quats, to='rotmat').reshape((-1,3,3))
beta = np.arccos(Rs[:,2,2])
# special handling for (beta == 0)
bz = beta == 0
# there the gamma-values are set to zero, since alpha/gamma is degenerated
alpha = np.nan * np.ones_like(beta)
gamma = np.nan * np.ones_like(beta)
alpha[bz] = np.arcsin(Rs[bz,1,0])
gamma[bz] = 0
alpha[~bz] = np.arctan2(Rs[~bz,0,2], Rs[~bz,1,2])
gamma[~bz] = np.arctan2(Rs[~bz,2,0], Rs[~bz,2,1])
sequence = np.column_stack((alpha, beta, gamma))
else:
raise ValueError('Input parameter {0} not known'.format(seq))
return np.rad2deg(sequence)
[docs]
def q_vector(inQuat):
"""
Extract the quaternion vector from a full quaternion.
Parameters
----------
inQuat : array_like, shape ([3,4],) or (N,[3,4])
quaternions or quaternion vectors.
Returns
-------
vect : array, shape (3,) or (N,3)
corresponding quaternion vectors
Notes
-----
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> quat.q_vector([[np.cos(0.2), 0, 0, np.sin(0.2)],[cos(0.1), 0, np.sin(0.1), 0]])
array([[ 0. , 0. , 0.19866933],
[ 0. , 0.09983342, 0. ]])
"""
inQuat = np.atleast_2d(inQuat)
if inQuat.shape[1] == 4:
vect = inQuat[:,1:]
else:
vect = inQuat
if np.min(vect.shape)==1:
vect = vect.ravel()
return vect
[docs]
def q_scalar(inQuat):
"""
Extract the quaternion scalar from a full quaternion.
Parameters
----------
inQuat : array_like, shape ([3,4],) or (N,[3,4])
quaternions or quaternion vectors.
Returns
-------
vect : array, shape (1,) or (N,1)
Corresponding quaternion scalar.
If the input is only the quaternion-vector, the scalar part for a unit
quaternion is calculated and returned.
Notes
-----
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> quat.q_scalar([[np.cos(0.2), 0, 0, np.sin(0.2)],[np.cos(0.1), 0, np.sin(0.1), 0]])
array([ 0.98006658, 0.99500417])
"""
inQuat = np.atleast_2d(inQuat)
if inQuat.shape[1] == 4:
scalar = inQuat[:,0]
else:
scalar = np.sqrt(1-np.linalg.norm(inQuat, axis=1))
if np.min(scalar.shape)==1:
scalar = scalar.ravel()
return scalar
[docs]
def unit_q(inData):
""" Utility function, which turns a quaternion vector into a unit quaternion.
If the input is already a full quaternion, the output equals the input.
Parameters
----------
inData : array_like, shape (3,) or (N,3)
quaternions or quaternion vectors
Returns
-------
quats : array, shape (4,) or (N,4)
corresponding unit quaternions.
Notes
-----
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> quats = array([[0,0, sin(0.1)],[0, sin(0.2), 0]])
>>> quat.unit_q(quats)
array([[ 0.99500417, 0. , 0. , 0.09983342],
[ 0.98006658, 0. , 0.19866933, 0. ]])
"""
inData = np.atleast_2d(inData)
(m,n) = inData.shape
if (n!=3)&(n!=4):
raise ValueError('Quaternion must have 3 or 4 columns')
if n == 3:
qLength = 1-np.sum(inData**2,1)
numLimit = 1e-12
# Check for numerical problems
if np.min(qLength) < -numLimit:
raise ValueError('Quaternion is too long!')
else:
# Correct for numerical problems
qLength[qLength<0] = 0
outData = np.hstack((np.c_[np.sqrt(qLength)], inData))
else:
outData = inData
return outData
[docs]
def calc_quat(omega, q0, rate, CStype):
"""
Take an angular velocity (in rad/s), and convert it into the
corresponding orientation quaternion.
Parameters
----------
omega : array, shape (3,) or (N,3)
angular velocity [rad/s].
q0 : array (3,)
vector-part of quaternion (!!)
rate : float
sampling rate (in [Hz])
CStype: string
coordinate_system, space-fixed ("sf") or body_fixed ("bf")
Returns
-------
quats : array, shape (N,4)
unit quaternion vectors.
Notes
-----
1) The output has the same length as the input. As a consequence, the last velocity vector is ignored.
2) For angular velocity with respect to space ("sf"), the orientation is given by
.. math::
q(t) = \\Delta q(t_n) \\circ \\Delta q(t_{n-1}) \\circ ... \\circ \\Delta q(t_2) \\circ \\Delta q(t_1) \\circ q(t_0)
.. math::
\\Delta \\vec{q_i} = \\vec{n(t)}\\sin (\\frac{\\Delta \\phi (t_i)}{2}) = \\frac{\\vec \\omega (t_i)}{\\left| {\\vec \\omega (t_i)} \\right|}\\sin \\left( \\frac{\\left| {\\vec \\omega ({t_i})} \\right|\\Delta t}{2} \\right)
3) For angular velocity with respect to the body ("bf"), the sequence of quaternions is inverted.
4) Take care that you choose a high enough sampling rate!
Examples
--------
>>> v0 = np.r_[0., 0., 100.] * np.pi/180.
>>> omega = np.tile(v0, (1000,1))
>>> rate = 100
>>> out = quat.calc_quat(omega, [0., 0., 0.], rate, 'sf')
array([[ 1. , 0. , 0. , 0. ],
[ 0.99996192, 0. , 0. , 0.00872654],
[ 0.9998477 , 0. , 0. , 0.01745241],
...,
[-0.74895572, 0. , 0. , 0.66262005],
[-0.75470958, 0. , 0. , 0.65605903],
[-0.76040597, 0. , 0. , 0.64944805]])
"""
omega_05 = np.atleast_2d(omega).copy()
# The following is (approximately) the quaternion-equivalent of the trapezoidal integration (cumtrapz)
if omega_05.shape[1]>1:
omega_05[:-1] = 0.5*(omega_05[:-1] + omega_05[1:])
omega_t = np.sqrt(np.sum(omega_05**2, 1))
omega_nonZero = omega_t>0
# initialize the quaternion
q_delta = np.zeros(omega_05.shape)
q_pos = np.zeros((len(omega_05),4))
q_pos[0,:] = unit_q(q0)
# magnitude of position steps
dq_total = np.sin(omega_t[omega_nonZero]/(2.*rate))
q_delta[omega_nonZero,:] = omega_05[omega_nonZero,:] * np.tile(dq_total/omega_t[omega_nonZero], (3,1)).T
for ii in range(len(omega_05)-1):
q1 = unit_q(q_delta[ii,:])
q2 = q_pos[ii,:]
if CStype == 'sf':
qm = q_mult(q1,q2)
elif CStype == 'bf':
qm = q_mult(q2,q1)
else:
print('I don''t know this type of coordinate system!')
q_pos[ii+1,:] = qm
return q_pos
[docs]
def calc_angvel(q, rate=1, winSize=5, order=2):
"""
Take a quaternion, and convert it into the
corresponding angular velocity
Parameters
----------
q : array, shape (N,[3,4])
unit quaternion vectors.
rate : float
sampling rate (in [Hz])
winSize : integer
window size for the calculation of the velocity.
Has to be odd.
order : integer
Order of polynomial used by savgol to calculate the first derivative
Returns
-------
angvel : array, shape (3,) or (N,3)
angular velocity [rad/s].
Notes
-----
The angular velocity is given by
.. math::
\\omega = 2 * \\frac{dq}{dt} \\circ q^{-1}
Examples
--------
>>> rate = 1000
>>> t = np.arange(0,10,1/rate)
>>> x = 0.1 * np.sin(t)
>>> y = 0.2 * np.sin(t)
>>> z = np.zeros_like(t)
array([[ 0.20000029, 0.40000057, 0. ],
[ 0.19999989, 0.39999978, 0. ],
[ 0.19999951, 0.39999901, 0. ]])
.......
"""
if np.mod(winSize, 2) != 1:
raise ValueError('Window size must be odd!')
numCols = q.shape[1]
if numCols < 3 or numCols > 4:
raise TypeError('quaternions must have 3 or 4 columns')
# This has to be done: otherwise q_mult will "complete" dq_dt to be a unit
# quaternion, resulting in wrong value
if numCols == 3:
q = unit_q(q)
dq_dt = signal.savgol_filter(q, window_length=winSize, polyorder=order, deriv=1, delta=1./rate, axis=0)
angVel = 2 * q_mult(dq_dt, q_inv(q))
return angVel[:,1:]
if __name__=='__main__':
"""These are some simple tests to see if the functions produce the
proper output.
More extensive tests are found in tests/test_quat.py"""
a = np.r_[np.cos(0.1), 0,0,np.sin(0.1)]
b = np.r_[np.cos(0.2), 0,np.sin(0.2),0]
seq = quat2seq(np.vstack((a,b)), seq='Euler')
print(seq)
"""
from skinematics.vector import rotate_vector
v0 = np.r_[0., 0., 100.] * np.pi/180.
vel = np.tile(v0, (1000,1))
rate = 100
out = calc_quat(vel, [0., 0., 0.], rate, 'sf')
rate = 1000
t = np.arange(0,10,1./rate)
x = 0.1 * np.sin(t)
y = 0.2 * np.sin(t)
z = np.zeros_like(t)
q = np.column_stack( (x,y,z) )
vel = calc_angvel(q, rate, 5, 2)
qReturn = vel2quat(vel, q[0], rate, 'sf' )
plt.plot(q)
plt.plot(qReturn[:,1:],'--')
plt.show()
q = Quaternion(np.array([0,0,10]), 'Fick')
print(q)
rm = q.export(to='rotmat')
print(rm)
q2 = Quaternion(rm, inType='rotmat')
print(q2)
a = np.r_[np.cos(0.1), 0,0,np.sin(0.1)]
b = np.r_[np.cos(0.1),0,np.sin(0.1), 0]
c = np.vstack((a,b))
d = np.r_[np.sin(0.1), 0, 0]
e = np.r_[2, 0, np.sin(0.1), 0]
print(q_mult(a,a))
print(q_mult(a,b))
print(q_mult(c,c))
print(q_mult(c,a))
print(q_mult(d,d))
print('The inverse of {0} is {1}'.format(a, q_inv(a)))
print('The inverse of {0} is {1}'.format(d, q_inv(d)))
print('The inverse of {0} is {1}'.format(e, q_inv(e)))
print(q_mult(e, q_inv(e)))
print(q_vector(a))
print('{0} is {1} degree'.format(a, quat2deg(a)))
print('{0} is {1} degree'.format(c, quat2deg(c)))
print(quat2deg(0.2))
x = np.r_[1,0,0]
vNull = np.r_[0,0,0]
print(rotate_vector(x, a))
v0 = [0., 0., 100.]
vel = np.tile(v0, (1000,1))
rate = 100
out = vel2quat(vel, [0., 0., 0.], rate, 'sf')
print(out[-1:])
plt.plot(out[:,1:4])
plt.show()
print(deg2quat(15))
print(deg2quat(quat2deg(a)))
q = np.array([[0, 0, np.sin(0.1)],
[0, np.sin(0.01), 0]])
rMat = convert(q, to='rotmat)
print(rMat[1].reshape((3,3)))
qNew = rotmat2quat(rMat)
print(qNew)
q = Quaternion(np.array([0,0,0.5]))
p = Quaternion(np.array([[0,0,0.5], [0,0,0.1]]))
r = Quaternion([0,0,0.5])
print(p*q)
print(q*3)
print(q*pi)
print(q/p)
print(q/5)
Fick = p.export('Fick')
Q_fick = q.export('Fick')
import pprint
pprint.pprint(Fick)
pprint.pprint(np.rad2deg(Fick))
p = Quaternion(np.array([[0,0,0.5], [0,0,0.1], [0,0,0.1]]))
p[1:] = [[0,0,0],[0,0,0.01]]
print(p)
"""
