# Quaternions

Note that all these functions work with single quaternions and quaternion vectors, as well as with arrays containing these.

## Quaternion class

- `quat.Quaternion` ... Quaternion class

## Functions for working with quaternions

- `quat.quatconj()` ... Conjugate quaternion
- `quat.quatinv()` ... Quaternion inversion
- `quat.quatmult()` ... Quaternion multiplication

## Conversion routines

- `quat.deg2quat()` ... Convert number or axis angles to quaternion vectors
- `quat.quat2deg()` ... Convert quaternion to corresponding axis angle
- `quat.quat2rotmat()` ... Convert quaternion to corresponding rotation matrix
- `quat.quat2vect()` ... Extract the vector part from a quaternion
- `quat.quat2vel()` ... Calculates the velocity in space from quaternions
- `quat.rotmat2quat()` ... Convert a rotation matrix to the corresponding quaternion
- `quat.vect2quat()` ... Extend a quaternion vector to a unit quaternion.
- `quat.vel2quat()` ... Calculate orientation from a starting orientation and angular velocity.

## Details

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.

A “Quaternion” class is defined, with

- operator overloading for mult, div, and inv.
- indexing

class `quat.``Quaternion`(inData, inType='vector')

  Quaternion class, with multiplocation, division, and inverse. 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 (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’

  `values`

    (4 x n) array – quaternion values

  `inv`()

    Inverse of the quaterion

  `export`(to='rotmat')

    Export to one of the following formats: ‘rotmat’, ‘Euler’, ‘Fick’, ‘Helmholtz’

  Notes

  \[\begin{split}\vec {q}_{Euler} = \left[ {\begin{array}{*{20}{c}} {\cos \frac{\alpha }{2}*\cos \frac{\beta }{2}*\cos \frac{\gamma }{2} - \sin \frac{\alpha }{2}\cos \frac{\beta }{2}\sin \frac{\gamma }{2}} \\ {\cos \frac{\alpha }{2}*\sin \frac{\beta }{2}*\cos \frac{\gamma }{2} + \sin \frac{\alpha }{2}\sin \frac{\beta }{2}\sin \frac{\gamma }{2}} \\ {\cos \frac{\alpha }{2}*\sin \frac{\beta }{2}*\sin \frac{\gamma }{2} - \sin \frac{\alpha }{2}\sin \frac{\beta }{2}\cos \frac{\gamma }{2}} \\ {\cos \frac{\alpha }{2}*\cos \frac{\beta }{2}*\sin \frac{\gamma }{2} + \sin \frac{\alpha }{2}\cos \frac{\beta }{2}\cos \frac{\gamma }{2}} \end{array}} \right]\end{split}\]
  \[\begin{split}\vec {q}_{Fick} = \left[ {\begin{array}{*{20}{c}} {\cos \frac{\psi }{2}*\cos \frac{\phi }{2}*\cos \frac{\theta }{2} + \sin \frac{\psi }{2}\sin \frac{\phi }{2}\sin \frac{\theta }{2}} \\ {\sin \frac{\psi }{2}*\cos \frac{\phi }{2}*\cos \frac{\theta }{2} - \cos \frac{\psi }{2}\sin \frac{\phi }{2}\sin \frac{\theta }{2}} \\ {\cos \frac{\psi }{2}*\sin \frac{\phi }{2}*\cos \frac{\theta }{2} + \sin \frac{\psi }{2}\cos \frac{\phi }{2}\sin \frac{\theta }{2}} \\ {\cos \frac{\psi }{2}*\cos \frac{\phi }{2}*\sin \frac{\theta }{2} - \sin \frac{\psi }{2}\sin \frac{\phi }{2}\cos \frac{\theta }{2}} \end{array}} \right]\end{split}\]
  \[\begin{split}\vec {q}_{Helmholtz} = \left[ {\begin{array}{*{20}{c}} {\cos \frac{\psi }{2}*\cos \frac{\phi }{2}*\cos \frac{\theta }{2} - \sin \frac{\psi }{2}\sin \frac{\phi }{2}\sin \frac{\theta }{2}} \\ {\sin \frac{\psi }{2}*\cos \frac{\phi }{2}*\cos \frac{\theta }{2} + \cos \frac{\psi }{2}\sin \frac{\phi }{2}\sin \frac{\theta }{2}} \\ {\cos \frac{\psi }{2}*\sin \frac{\phi }{2}*\cos \frac{\theta }{2} + \sin \frac{\psi }{2}\cos \frac{\phi }{2}\sin \frac{\theta }{2}} \\ {\cos \frac{\psi }{2}*\cos \frac{\phi }{2}*\sin \frac{\theta }{2} - \sin \frac{\psi }{2}\sin \frac{\phi }{2}\cos \frac{\theta }{2}} \end{array}} \right]\end{split}\]

  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)

  `export`(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: |
    Return type: | ndarray, with the specified content

    Examples

        >>> q = Quaternion([0,0.2,0.1])
        >>> rm = q.export()
        >>> fick = q.export('Fick')

  `inv`()

    Inverse of a quaternion.

`quat.``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 – number or quaternion vector.
  Return type: | float or array (N,3)

  Notes

  \[| \vec{q} | = sin(\theta/2)\]

  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

`quat.``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 – float, or shape (3,) or (N,3)
  Return type: | corresponding axis angle(s)

  Notes

  \[| \vec{q} | = sin(\theta/2)\]

  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.       ])

`quat.``quat2rotmat`(inQuat)

  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
  Returns: | rotMat
  Return type: | corresponding rotation matrix/matrices (flattened)

  Notes

  \[\begin{split}{\bf{R}} = \left( {\begin{array}{*{20}{c}} {q_0^2 + q_1^2 - q_2^2 - q_3^2}&{2({q_1}{q_2} - {q_0}{q_3})}&{2({q_1}{q_3} + {q_0}{q_2})}\\ {2({q_1}{q_2} + {q_0}{q_3})}&{q_0^2 - q_1^2 + q_2^2 - q_3^2}&{2({q_2}{q_3} - {q_0}{q_1})}\\ {2({q_1}{q_3} - {q_0}{q_2})}&{2({q_2}{q_3} + {q_0}{q_1})}&{q_0^2 - q_1^2 - q_2^2 + q_3^2} \\ \end{array}} \right)\end{split}\]

  More info under http://en.wikipedia.org/wiki/Quaternion

  Examples

      >>> r = quat.quat2rotmat([0, 0, 0.1])
      >>> r.shape
      (1, 9)
      >>> r.reshape((3,3))
      array([[ 0.98      , -0.19899749,  0.        ],
          [ 0.19899749,  0.98      ,  0.        ],
          [ 0.        ,  0.        ,  1.        ]])

`quat.``quat2vect`(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 – corresponding quaternion vectors
  Return type: | array, shape (3,) or (N,3)

  Notes

  More info under http://en.wikipedia.org/wiki/Quaternion

  Examples

      >>> quat.quat2vect([[cos(0.2), 0, 0, sin(0.2)],[cos(0.1), 0, sin(0.1), 0]])
      array([[ 0.        ,  0.        ,  0.19866933],
             [ 0.        ,  0.09983342,  0.        ]])

`quat.``quat2vel`(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: | vel – angular velocity [rad/s].
  Return type: | array, shape (3,) or (N,3)

  Notes

  The angular velocity is given by

  > \[\omega = 2 * \frac{dq}{dt} \circ q^{-1}\]

  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.        ]])
              .......

`quat.``quatconj`(q)

  Conjugate quaternion

  Parameters: | q (array_like, shape ([3,4],) or (N,[3/4])) – quaternion or quaternion vectors
  Returns: | qconj
  Return type: | conjugate quaternion(s)

  Examples

      >>>  quat.quatconj([0,0,0.1])
      array([ 0., -0., -0., -1.])

      >>> quat.quatconj([[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.        ]])

`quat.``quatinv`(q)

  Quaternion inversion

  Parameters: | q (array_like, shape ([3,4],) or (N,[3/4])) – quaternion or quaternion vectors
  Returns: | qinv
  Return type: | inverse quaternion(s)

  Notes

  \[q^{-1} = \frac{q_0 - \vec{q}}{|q|^2}\]

  More info under http://en.wikipedia.org/wiki/Quaternion

  Examples

      >>>  quat.quatinv([0,0,0.1])
      array([[-0. , -0. , -0.1]])

      >>> quat.quatinv([[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.        ]])

`quat.``quatmult`(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 – p and q are contain quaternion vectors).
  Return type: | quaternion or quaternion vector (if both

  Notes

  \[q \circ p = \sum\limits_{i=0}^3 {q_i I_i} * \sum\limits_{j=0}^3 \ {p_j I_j} = (q_0 p_0 - \vec{q} \cdot \vec{p}) + (q_0 \vec{p} + p_0 \ \vec{q} + \vec{q} \times \vec{p}) \cdot \vec{I}\]

  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.quatmult(p,q)

`quat.``rotmat2quat`(rMat)

  Assumes that R has the shape (3,3), or the matrix elements in columns

  Parameters: | rMat (array, shape (3,3) or (N,9)) – single rotation matrix, or matrix with rotation-matrix elements.
  Returns: | outQuat – corresponding quaternion vector(s)
  Return type: | array, shape (4,) or (N,4)

  Notes

  \[\begin{split}\vec q = 0.5*copysign\left( {\begin{array}{*{20}{c}} {\sqrt {1 + {R_{11}} - {R_{22}} - {R_{33}}} ,}\\ {\sqrt {1 - {R_{11}} + {R_{22}} - {R_{33}}} ,}\\ {\sqrt {1 - {R_{11}} - {R_{22}} + {R_{33}}} ,} \end{array}\begin{array}{*{20}{c}} {{R_{32}} - {R_{23}}}\\ {{R_{13}} - {R_{31}}}\\ {{R_{21}} - {R_{12}}} \end{array}} \right)\end{split}\]

  More info under http://en.wikipedia.org/wiki/Quaternion

  Examples

      >>> rotMat = array([[cos(alpha), -sin(alpha), 0],
      >>>    [sin(alpha), cos(alpha), 0],
      >>>    [0, 0, 1]])
      >>> quat.rotmat2quat(rotMat)
      array([[ 0.99500417,  0.        ,  0.        ,  0.09983342]])

`quat.``vect2quat`(inData)

  Utility function, which turns a quaternion vector into a unit quaternion.

  Parameters: | inData (array_like, shape (3,) or (N,3)) – quaternions or quaternion vectors
  Returns: | quats – corresponding unit quaternions.
  Return type: | array, shape (4,) or (N,4)

  Notes

  More info under http://en.wikipedia.org/wiki/Quaternion

  Examples

      >>> quats = array([[0,0, sin(0.1)],[0, sin(0.2), 0]])
      >>> quat.vect2quat(quats)
      array([[ 0.99500417,  0.        ,  0.        ,  0.09983342],
             [ 0.98006658,  0.        ,  0.19866933,  0.        ]])

`quat.``vel2quat`(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 – unit quaternion vectors.
  Return type: | array, shape (N,4)

  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

  > \[q(t) = \Delta q(t_n) \circ \Delta q(t_{n-1}) \circ ... \circ \Delta q(t_2) \circ \Delta q(t_1) \circ q(t_0)\]
  > \[\Delta \vec{q_i} = \vec{n(t)}\sin (\frac{\Delta \phi (t_i)}{2}) = \frac{\vec \omega (t_i)}{\left| {\vec \omega (t_i)} \right|}\sin \left( \frac{\left| {\vec \omega ({t_i})} \right|\Delta t}{2} \right)\]

  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.vel2quat(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]])