quaternions.py

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

The module also includes functions for the closely related angle, axis
pair as a specification for rotation.

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)
'''

import math
import numpy as np

_MAX_FLOAT = np.maximum_sctype(np.float)
_FLOAT_EPS = np.finfo(np.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
        except (AttributeError, ValueError):
            w2_thresh = -_FLOAT_EPS
    # 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 %f' % 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 non-unit
    quaternions.

    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 == 0.0:
        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]).
    
    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
    # 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 mult(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 conjugate(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 norm(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 isunit(q):
    ''' Return True is this is very nearly a unit quaternion '''
    return np.allclose(norm(q),1)


def inverse(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 conjugate(q) / norm(q)


def eye():
    ''' 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 mult(q, mult(varr, conjugate(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
    (becuase 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
    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:
        vector /= math.sqrt(np.dot(vector, vector))
    t2 = theta / 2.0
    st2 = math.sin(t2)
    return np.concatenate(([math.cos(t2)],
                           vector * st2))


def axangle2rmat(vector, theta):
    ''' Rotation matrix of angle `theta` around `vector`

    Parameters
    ----------
    vector : 3 element sequence
       vector specifying axis for rotation.
    theta : scalar
       angle of rotation

    Returns
    -------
    mat : array shape (3,3)
       rotation matrix specified rotation

    Notes
    -----
    From: http://en.wikipedia.org/wiki/Rotation_matrix#Axis_and_angle
    '''
    x, y, z = vector[:3]
    n = math.sqrt(x*x + y*y + z*z)
    x = x/n
    y = y/n
    z = z/n
    c = math.cos(theta); s = math.sin(theta); C = 1-c
    xs = x*s;   ys = y*s;   zs = z*s
    xC = x*C;   yC = y*C;   zC = z*C
    xyC = x*yC; yzC = y*zC; zxC = z*xC
    return np.array([
            [ x*xC+c,   xyC-zs,   zxC+ys ],
            [ xyC+zs,   y*yC+c,   yzC-xs ],
            [ zxC-ys,   yzC+xs,   z*zC+c ]])


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)

    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]
    '''
    w, x, y, z = quat
    vec = np.asarray([x, y, z])
    if identity_thresh is None:
        try:
            identity_thresh = np.finfo(vec.dtype).eps * 3
        except ValueError: # integer type
            identity_thresh = _FLOAT_EPS * 3
    n = math.sqrt(x*x + y*y + z*z)
    if n < identity_thresh: 
        # if vec is nearly 0,0,0, this is an identity rotation
        return np.array([1.0, 0, 0]), 0.0
    return  vec / n, 2 * math.acos(max(min(w,1),-1))