quaternions.py

'''Functions to operate on, or return, quaternions.

Quaternions here consist of 4 values ``w, x, y, z``, where ``w`` is the
real (scalar) part, and ``x, y, z`` are the complex (vector) part.

Note - rotation matrices here apply to column vectors, that is,
they are applied on the left of the vector.  For example:

>>> import numpy as np
>>> q = [0, 1, 0, 0] # 180 degree rotation around axis 0
>>> M = quat2mat(q) # from this module
>>> vec = np.array([1, 2, 3]).reshape((3,1)) # column vector
>>> tvec = np.dot(M, vec)

Terms used in function names:

* *mat* : array shape (3, 3) (3D non-homogenous coordinates)
* *aff* : affine array shape (4, 4) (3D homogenous coordinates)
* *quat* : quaternion shape (4,)
* *axangle* : rotations encoded by axis vector and angle scalar
'''

import math
import numpy as np

_MAX_FLOAT = np.maximum_sctype(float)
_FLOAT_EPS = np.finfo(float).eps


def fillpositive(xyz, w2_thresh=None):
    ''' Compute unit quaternion from last 3 values

    Parameters
    ----------
    xyz : iterable
       iterable containing 3 values, corresponding to quaternion x, y, z
    w2_thresh : None or float, optional
       threshold to determine if w squared is really negative.
       If None (default) then w2_thresh set equal to
       ``-np.finfo(xyz.dtype).eps``, if possible, otherwise
       ``-np.finfo(np.float).eps``

    Returns
    -------
    wxyz : array shape (4,)
         Full 4 values of quaternion

    Notes
    -----
    If w, x, y, z are the values in the full quaternion, assumes w is
    positive.

    Gives error if w*w is estimated to be negative

    w = 0 corresponds to a 180 degree rotation

    The unit quaternion specifies that np.dot(wxyz, wxyz) == 1.

    If w is positive (assumed here), w is given by:

    w = np.sqrt(1.0-(x*x+y*y+z*z))

    w2 = 1.0-(x*x+y*y+z*z) can be near zero, which will lead to
    numerical instability in sqrt.  Here we use the system maximum
    float type to reduce numerical instability

    Examples
    --------
    >>> import numpy as np
    >>> wxyz = fillpositive([0,0,0])
    >>> np.all(wxyz == [1, 0, 0, 0])
    True
    >>> wxyz = fillpositive([1,0,0]) # Corner case; w is 0
    >>> np.all(wxyz == [0, 1, 0, 0])
    True
    >>> np.dot(wxyz, wxyz)
    1.0
    '''
    # Check inputs (force error if < 3 values)
    if len(xyz) != 3:
        raise ValueError('xyz should have length 3')
    # If necessary, guess precision of input
    if w2_thresh is None:
        try: # trap errors for non-array, integer array
            w2_thresh = -np.finfo(xyz.dtype).eps * 3
        except (AttributeError, ValueError):
            w2_thresh = -_FLOAT_EPS * 3
    # Use maximum precision
    xyz = np.asarray(xyz, dtype=_MAX_FLOAT)
    # Calculate w
    w2 = 1.0 - np.dot(xyz, xyz)
    if w2 < 0:
        if w2 < w2_thresh:
            raise ValueError('w2 should be positive, but is %e' % w2)
        w = 0
    else:
        w = np.sqrt(w2)
    return np.r_[w, xyz]


def quat2mat(q):
    ''' Calculate rotation matrix corresponding to quaternion

    Parameters
    ----------
    q : 4 element array-like

    Returns
    -------
    M : (3,3) array
      Rotation matrix corresponding to input quaternion *q*

    Notes
    -----
    Rotation matrix applies to column vectors, and is applied to the
    left of coordinate vectors.  The algorithm here allows quaternions that
    have not been normalized.

    References
    ----------
    Algorithm from http://en.wikipedia.org/wiki/Rotation_matrix#Quaternion

    Examples
    --------
    >>> import numpy as np
    >>> M = quat2mat([1, 0, 0, 0]) # Identity quaternion
    >>> np.allclose(M, np.eye(3))
    True
    >>> M = quat2mat([0, 1, 0, 0]) # 180 degree rotn around axis 0
    >>> np.allclose(M, np.diag([1, -1, -1]))
    True
    '''
    w, x, y, z = q
    Nq = w*w + x*x + y*y + z*z
    if Nq < _FLOAT_EPS:
        return np.eye(3)
    s = 2.0/Nq
    X = x*s
    Y = y*s
    Z = z*s
    wX = w*X; wY = w*Y; wZ = w*Z
    xX = x*X; xY = x*Y; xZ = x*Z
    yY = y*Y; yZ = y*Z; zZ = z*Z
    return np.array(
           [[ 1.0-(yY+zZ), xY-wZ, xZ+wY ],
            [ xY+wZ, 1.0-(xX+zZ), yZ-wX ],
            [ xZ-wY, yZ+wX, 1.0-(xX+yY) ]])


def mat2quat(M):
    ''' Calculate quaternion corresponding to given rotation matrix

    Parameters
    ----------
    M : array-like
      3x3 rotation matrix

    Returns
    -------
    q : (4,) array
      closest quaternion to input matrix, having positive q[0]

    Notes
    -----
    Method claimed to be robust to numerical errors in M

    Constructs quaternion by calculating maximum eigenvector for matrix
    K (constructed from input `M`).  Although this is not tested, a
    maximum eigenvalue of 1 corresponds to a valid rotation.

    A quaternion q*-1 corresponds to the same rotation as q; thus the
    sign of the reconstructed quaternion is arbitrary, and we return
    quaternions with positive w (q[0]).

    References
    ----------
    * http://en.wikipedia.org/wiki/Rotation_matrix#Quaternion
    * Bar-Itzhack, Itzhack Y. (2000), "New method for extracting the
      quaternion from a rotation matrix", AIAA Journal of Guidance,
      Control and Dynamics 23(6):1085-1087 (Engineering Note), ISSN
      0731-5090

    Examples
    --------
    >>> import numpy as np
    >>> q = mat2quat(np.eye(3)) # Identity rotation
    >>> np.allclose(q, [1, 0, 0, 0])
    True
    >>> q = mat2quat(np.diag([1, -1, -1]))
    >>> np.allclose(q, [0, 1, 0, 0]) # 180 degree rotn around axis 0
    True

    Notes
    -----
    http://en.wikipedia.org/wiki/Rotation_matrix#Quaternion

    Bar-Itzhack, Itzhack Y. (2000), "New method for extracting the
    quaternion from a rotation matrix", AIAA Journal of Guidance,
    Control and Dynamics 23(6):1085-1087 (Engineering Note), ISSN
    0731-5090

    '''
    # Qyx refers to the contribution of the y input vector component to
    # the x output vector component.  Qyx is therefore the same as
    # M[0,1].  The notation is from the Wikipedia article.
    Qxx, Qyx, Qzx, Qxy, Qyy, Qzy, Qxz, Qyz, Qzz = M.flat
    # Fill only lower half of symmetric matrix
    K = np.array([
        [Qxx - Qyy - Qzz, 0,               0,               0              ],
        [Qyx + Qxy,       Qyy - Qxx - Qzz, 0,               0              ],
        [Qzx + Qxz,       Qzy + Qyz,       Qzz - Qxx - Qyy, 0              ],
        [Qyz - Qzy,       Qzx - Qxz,       Qxy - Qyx,       Qxx + Qyy + Qzz]]
        ) / 3.0
    # Use Hermitian eigenvectors, values for speed
    vals, vecs = np.linalg.eigh(K)
    # Select largest eigenvector, reorder to w,x,y,z quaternion
    q = vecs[[3, 0, 1, 2], np.argmax(vals)]
    # Prefer quaternion with positive w
    # (q * -1 corresponds to same rotation as q)
    if q[0] < 0:
        q *= -1
    return q


def qmult(q1, q2):
    ''' Multiply two quaternions

    Parameters
    ----------
    q1 : 4 element sequence
    q2 : 4 element sequence

    Returns
    -------
    q12 : shape (4,) array

    Notes
    -----
    See : http://en.wikipedia.org/wiki/Quaternions#Hamilton_product
    '''
    w1, x1, y1, z1 = q1
    w2, x2, y2, z2 = q2
    w = w1*w2 - x1*x2 - y1*y2 - z1*z2
    x = w1*x2 + x1*w2 + y1*z2 - z1*y2
    y = w1*y2 + y1*w2 + z1*x2 - x1*z2
    z = w1*z2 + z1*w2 + x1*y2 - y1*x2
    return np.array([w, x, y, z])


def qconjugate(q):
    ''' Conjugate of quaternion

    Parameters
    ----------
    q : 4 element sequence
       w, i, j, k of quaternion

    Returns
    -------
    conjq : array shape (4,)
       w, i, j, k of conjugate of `q`
    '''
    return np.array(q) * np.array([1.0, -1, -1, -1])


def qnorm(q):
    ''' Return norm of quaternion

    Parameters
    ----------
    q : 4 element sequence
       w, i, j, k of quaternion

    Returns
    -------
    n : scalar
       quaternion norm
    '''
    return np.dot(q, q)


def qisunit(q):
    ''' Return True is this is very nearly a unit quaternion '''
    return np.allclose(qnorm(q), 1)


def qinverse(q):
    ''' Return multiplicative inverse of quaternion `q`

    Parameters
    ----------
    q : 4 element sequence
       w, i, j, k of quaternion

    Returns
    -------
    invq : array shape (4,)
       w, i, j, k of quaternion inverse
    '''
    return qconjugate(q) / qnorm(q)


def qeye():
    ''' Return identity quaternion '''
    return np.array([1.0,0,0,0])


def rotate_vector(v, q):
    ''' Apply transformation in quaternion `q` to vector `v`

    Parameters
    ----------
    v : 3 element sequence
       3 dimensional vector
    q : 4 element sequence
       w, i, j, k of quaternion

    Returns
    -------
    vdash : array shape (3,)
       `v` rotated by quaternion `q`

    Notes
    -----
    See: http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Describing_rotations_with_quaternions
    '''
    varr = np.zeros((4,))
    varr[1:] = v
    return qmult(q, qmult(varr, qconjugate(q)))[1:]


def nearly_equivalent(q1, q2, rtol=1e-5, atol=1e-8):
    ''' Returns True if `q1` and `q2` give near equivalent transforms

    `q1` may be nearly numerically equal to `q2`, or nearly equal to `q2` * -1
    (because a quaternion multiplied by -1 gives the same transform).

    Parameters
    ----------
    q1 : 4 element sequence
       w, x, y, z of first quaternion
    q2 : 4 element sequence
       w, x, y, z of second quaternion

    Returns
    -------
    equiv : bool
       True if `q1` and `q2` are nearly equivalent, False otherwise

    Examples
    --------
    >>> q1 = [1, 0, 0, 0]
    >>> nearly_equivalent(q1, [0, 1, 0, 0])
    False
    >>> nearly_equivalent(q1, [1, 0, 0, 0])
    True
    >>> nearly_equivalent(q1, [-1, 0, 0, 0])
    True
    '''
    q1 = np.array(q1)
    q2 = np.array(q2)
    if np.allclose(q1, q2, rtol, atol):
        return True
    return np.allclose(q1 * -1, q2, rtol, atol)


def axangle2quat(vector, theta, is_normalized=False):
    ''' Quaternion for rotation of angle `theta` around `vector`

    Parameters
    ----------
    vector : 3 element sequence
       vector specifying axis for rotation.
    theta : scalar
       angle of rotation in radians.
    is_normalized : bool, optional
       True if vector is already normalized (has norm of 1).  Default
       False.

    Returns
    -------
    quat : 4 element sequence of symbols
       quaternion giving specified rotation

    Examples
    --------
    >>> q = axangle2quat([1, 0, 0], np.pi)
    >>> np.allclose(q, [0, 1, 0,  0])
    True

    Notes
    -----
    Formula from http://mathworld.wolfram.com/EulerParameters.html
    '''
    vector = np.array(vector)
    if not is_normalized:
        # Cannot divide in-place because input vector may be integer type,
        # whereas output will be float type; this may raise an error in versions
        # of numpy > 1.6.1
        vector = vector / math.sqrt(np.dot(vector, vector))
    t2 = theta / 2.0
    st2 = math.sin(t2)
    return np.concatenate(([math.cos(t2)],
                           vector * st2))


def quat2axangle(quat, identity_thresh=None):
    ''' Convert quaternion to rotation of angle around axis

    Parameters
    ----------
    quat : 4 element sequence
       w, x, y, z forming quaternion.
    identity_thresh : None or scalar, optional
       Threshold below which the norm of the vector part of the quaternion (x,
       y, z) is deemed to be 0, leading to the identity rotation.  None (the
       default) leads to a threshold estimated based on the precision of the
       input.

    Returns
    -------
    theta : scalar
       angle of rotation.
    vector : array shape (3,)
       axis around which rotation occurs.

    Examples
    --------
    >>> vec, theta = quat2axangle([0, 1, 0, 0])
    >>> vec
    array([ 1.,  0.,  0.])
    >>> np.allclose(theta, np.pi)
    True

    If this is an identity rotation, we return a zero angle and an arbitrary
    vector:

    >>> quat2axangle([1, 0, 0, 0])
    (array([ 1.,  0.,  0.]), 0.0)

    If any of the quaternion values are not finite, we return a NaN in the
    angle, and an arbitrary vector:

    >>> quat2axangle([1, np.inf, 0, 0])
    (array([ 1.,  0.,  0.]), nan)

    Notes
    -----
    A quaternion for which x, y, z are all equal to 0, is an identity rotation.
    In this case we return a 0 angle and an arbitrary vector, here [1, 0, 0].

    The algorithm allows for quaternions that have not been normalized.
    '''
    w, x, y, z = quat
    Nq = w * w + x * x + y * y + z * z
    if not np.isfinite(Nq):
        return np.array([1.0, 0, 0]), float('nan')
    if identity_thresh is None:
        try:
            identity_thresh = np.finfo(Nq.type).eps * 3
        except (AttributeError, ValueError): # Not a numpy type or not float
            identity_thresh = _FLOAT_EPS * 3
    if Nq < _FLOAT_EPS ** 2:  # Results unreliable after normalization
        return np.array([1.0, 0, 0]), 0.0
    if Nq != 1:  # Normalize if not normalized
        s = math.sqrt(Nq)
        w, x, y, z = w / s, x / s, y / s, z / s
    len2 = x * x + y * y + z * z
    if len2 < identity_thresh ** 2:
        # if vec is nearly 0,0,0, this is an identity rotation
        return np.array([1.0, 0, 0]), 0.0
    # Make sure w is not slightly above 1 or below -1
    theta = 2 * math.acos(max(min(w, 1), -1))
    return  np.array([x, y, z]) / math.sqrt(len2), theta