quaternion.py

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import numpy as np
from dynamic_graph.sot.tools.se3 import SO3
from numpy import linalg
 
 
class Quaternion(object):
    """
    Quaternion class :
    ------------------
 
    A quaternion has a scalar part and a vector part.
    In this class the quaternion is represented as an array of 4 elements :
      - the first element is the scalar part
      - the next 3 elements represents the vector part
 
    One can acces to the array directly with the attribute "array"
      e.g. q1=Quaternion(1,0,0,0) --> q1.array
 
    A quaternion can be instanciated with 1, 2 or 4 elements
      (see : __init__() for more information).
 
    It can also return a rotation vector, a rotation matrix, or a SO3
      (see the methods : to...() for more information).
    """
 
    def __init__(self, *args):
        """
        Instanciation of the quaternion with 1, 2 or 4 arguments  :
        -----------------------------------------------------------
        This creates a 4-sized array (self.array) representing the quaternion
        with the first element representing the scalar part
        and the 3 others the vector part.
 
        With 4 arguments :
        ------------------
          - the first one is used as the scalar part,
            the other three as the vector part.
 
        With 2 arguments :
        ------------------
          - the 1-sized argument is used as the scalar part,
            the 3-sized argument is used as the vector part.
 
        With 1 argument :
        -----------------
          - if it is a quaternion it will create a copy of this quaternion.
          - if it is a scalar, the scalar will be used as the scalar part
            and the vector part will be set at (0,0,0).
          - if it is an array, matrix, tuple or list of 4 elements,
            the first element is used as the scalar part
            and the rest as the vector part.
          - if it is an array, matrix, tuple or list of 3 elements,
            the 3 elements are interpreted as a rotation vector,
            this creates a quaternion representing the same rotation.
          - if it is a to 2 dimension array convertible array, matrix, tuple
            or list with at least (3*3) elements,
            the upper left (3*3) elements are interpreted as a rotation matrix,
            this creates a quaternion representing the same rotation.
 
        With 0 arguments :
        ------------------
        If no argument is given, than the quaternion will be set by default
        to with the scalar part set to 1 and the vector part to (0,0,0).
        (this is the neutral element for multiplication in quaternion space)
 
        To create a quaternion from Roll, Pitch, Yaw angles :
        -----------------------------------------------------
        first instanciate a quaternion and than use the method fromRPY()
        to change the values of it to the dezired ones.
          e.g. : quat().fromRPY(R,P,Y)
        """
 
        error = False
        if len(args) == 0:  # By default, if no argument is given
            self.array = np.array([1.0, 0.0, 0.0, 0.0])
        elif len(args) == 4:  # From 4 elements
            if np.array(args).size == 4:
                self.array = np.double(np.array(args))
            else:
                error = True
        elif len(args) == 1:
            if type(args[0]) == Quaternion:  # From a Quaternion
                self.array = args[0].array.copy()
            elif np.array(args[0]).size == 1:
                # From one sized element, this element will be the scalar part,
                # the vector part will be set at (0,0,0)
                self.array = np.double(
                    np.hstack([np.array(args[0]), np.array([0, 0, 0])])
                )
            elif np.array(args[0]).size == 4 and max(np.array(args[0]).shape) == 4:
                # From an array, matrix, tuple or list of 4 elements
                self.array = np.double(np.array(args[0])).reshape(
                    4,
                )
            elif np.array(args[0]).size == 3 and max(np.array(args[0]).shape) == 3:
                # From an array, matrix, tuple or list of 3 elements interpreted
                # as a rotation vector
                rV = np.double(np.array(args[0])).reshape(
                    3,
                )
                alpha = np.double(linalg.norm(rV))
                if alpha != 0:
                    e = rV / alpha
                else:
                    e = rV
                self.array = np.hstack([np.cos(alpha / 2.0), np.sin(alpha / 2.0) * e])
            elif (
                len(np.array(args[0]).shape) == 2
                and np.array(args[0]).shape[0] >= 3
                and np.array(args[0]).shape[1] >= 3
            ):
                # From a to 2 dimension array convertible array, matrix, tuple or list
                # with at least (3*3) elements interpreted  as a rotation matrix
                rM = np.double(np.array(args[0])[:3, :3])
                selec = np.zeros(4)
                selec[0] = 1 + rM[0, 0] + rM[1, 1] + rM[2, 2]
                selec[1] = 1 + rM[0, 0] - rM[1, 1] - rM[2, 2]
                selec[2] = 1 - rM[0, 0] + rM[1, 1] - rM[2, 2]
                selec[3] = 1 - rM[0, 0] - rM[1, 1] + rM[2, 2]
                param = selec.argmax()
                if selec[param] > 0:
                    q = np.zeros(4)
                    if param == 0:
                        q[0] = np.sqrt(selec[param])
                        q[1] = (rM[2, 1] - rM[1, 2]) / q[0]
                        q[2] = (rM[0, 2] - rM[2, 0]) / q[0]
                        q[3] = (rM[1, 0] - rM[0, 1]) / q[0]
                        self.array = q * 0.5
                        # print '--1--V3'
                    elif param == 1:
                        q[1] = np.sqrt(selec[param])
                        q[0] = (rM[2, 1] - rM[1, 2]) / q[1]
                        q[2] = (rM[1, 0] + rM[0, 1]) / q[1]
                        q[3] = (rM[0, 2] + rM[2, 0]) / q[1]
                        self.array = q * 0.5
                        # print '--2--V3'
                    elif param == 2:
                        q[2] = np.sqrt(selec[param])
                        q[0] = (rM[0, 2] - rM[2, 0]) / q[2]
                        q[1] = (rM[1, 0] + rM[0, 1]) / q[2]
                        q[3] = (rM[2, 1] + rM[1, 2]) / q[2]
                        self.array = q * 0.5
                        # print '--3--V3'
                    elif param == 3:
                        q[3] = np.sqrt(selec[param])
                        q[0] = (rM[1, 0] - rM[0, 1]) / q[3]
                        q[1] = (rM[0, 2] + rM[2, 0]) / q[3]
                        q[2] = (rM[2, 1] + rM[1, 2]) / q[3]
                        self.array = q * 0.5
                        # print '--4--V3'
                else:
                    error = True
            else:
                error = True
        elif (
            len(args) == 2
        ):  # From a scalar part (1 element) and a vector part (3 elements)
            arg0 = np.double(np.array(args[0]))
            arg1 = np.double(np.array(args[1]))
            if arg0.size == 1 and arg1.size == 3:
                self.array = np.zeros(4)
                self.array[0] = arg0
                self.array[1:4] = arg1[:]
            elif arg0.size == 3 and arg1.size == 1:
                self.array = np.zeros(4)
                self.array[0] = arg1
                self.array[1:4] = arg0[:]
            else:
                error = True
 
        else:
            error = True
 
        if not error and self.array.shape != (4,):
            del self.array
            error = True
        if error:
            raise TypeError(
                "Impossible to instanciate the Quaternion object "
                "with the given arguments"
            )
 
    def __str__(self):
        """
        String representation of the quaternion.
        """
        aff = "[ "
        aff += str(self.array[0]) + "  +  "
        aff += str(self.array[1]) + " i  +  "
        aff += str(self.array[2]) + " j  +  "
        aff += str(self.array[3]) + " k ]"
        return aff
 
    def __neg__(self):
        """
        Returns a quaternion which elements are the opposite of the original,
        (this opposite quaternion represents the same rotation).
        """
        return Quaternion(-self.array)
 
    def __add__(self, other):
        """
        If other is not a quaternion it is casted to a quaternion,
        the elements are added one to one.
        """
        if type(other) != Quaternion:
            q2 = Quaternion(other)
        else:
            q2 = other
        return Quaternion(self.array + q2.array)
 
    def __sub__(self, other):
        """
        If other is not a quaternion it is casted to a quaternion,
        the elements are substracted one to one.
        """
        if type(other) != Quaternion:
            q2 = Quaternion(other)
        else:
            q2 = other
        return Quaternion(self.array - q2.array)
 
    def __mul__(self, other):
        """
        If other is not a quaternion it is casted to a quaternion,
        the result of the quaternion multiplication is returned.
        """
        if type(other) != Quaternion:
            q2 = Quaternion(other)
        else:
            q2 = other
        qr = np.zeros(4)
        qr[0] = self.array[0] * q2.array[0] - np.vdot(self.array[1:], q2.array[1:])
        qr[1:4] = (
            np.cross(self.array[1:4], q2.array[1:4])
            + self.array[0] * q2.array[1:4]
            + q2.array[0] * self.array[1:4]
        )
        return Quaternion(qr)
 
    def __rmul__(self, other):
        """
        other is casted to a quaternion,
        the result of the quaternion multiplication is returned.
        """
        return Quaternion(other) * self
 
    def __abs__(self):
        """
        Returns the norm of the quaternion.
        """
        return np.double(linalg.norm(self.array))
 
    def conjugate(self):
        """
        Returns the conjugate of the quaternion.
        """
        return Quaternion(self.array[0], -self.array[1:4])
 
    def inv(self):
        """
        Returns the inverse of the quaternion.
        """
        return Quaternion(self.conjugate().array / (abs(self) ** 2))
 
    def __div__(self, other):
        """
        If other is not a quaternion it is casted to a quaternion,
        the result of the quaternion multiplication with the inverse of other
        is returned.
        """
        if type(other) != Quaternion:
            q2 = Quaternion(other)
        else:
            q2 = other
        return self * q2.inv()
 
    def __pow__(self, n):
        """
        Returns quaternion**n with quaternion**0 = Quaternion(1,0,0,0).
        """
        r = Quaternion()
        for i in range(n):
            r = r * self
        return r
 
    def normalize(self):
        """
        Changes the values of the quaternion to make it a unit quaternion
        representing the same rotation as the original one
        and returns the updated version.
        """
        self.array /= abs(self)
        return self
 
    def normalized(self):
        """
        Returns the unit quaternion representation of the quaternion
        without changing the original.
        """
        qr = Quaternion(self)
        qr.normalize()
        return qr
 
    def toSO3(self):
        """
        Returns the corresponding SO3.
        """
        return SO3(self.toRotationMatrix())
 
    def toRotationMatrix(self):
        """
        Returns a (3*3) array (rotation matrix)
        representing the same rotation as the (normalized) quaternion.
        """
        q = self.normalized().array
        rm = np.zeros((3, 3))
        rm[0, 0] = 1 - 2 * (q[2] ** 2 + q[3] ** 2)
        rm[0, 1] = 2 * q[1] * q[2] - 2 * q[0] * q[3]
        rm[0, 2] = 2 * q[1] * q[3] + 2 * q[0] * q[2]
        rm[1, 0] = 2 * q[1] * q[2] + 2 * q[0] * q[3]
        rm[1, 1] = 1 - 2 * (q[1] ** 2 + q[3] ** 2)
        rm[1, 2] = 2 * q[2] * q[3] - 2 * q[0] * q[1]
        rm[2, 0] = 2 * q[1] * q[3] - 2 * q[0] * q[2]
        rm[2, 1] = 2 * q[2] * q[3] + 2 * q[0] * q[1]
        rm[2, 2] = 1 - 2 * (q[1] ** 2 + q[2] ** 2)
        return rm
 
    def toRotationVector(self):
        """
        Returns a 3-sized array (rotation vector)
        representing the same rotation as the (normalized) quaternion.
        """
        q = self.normalized().array
        rV = np.zeros(3)
        alpha = 2 * np.arccos(q[0])
        if linalg.norm(q[1:4]) != 0:
            rV = alpha * q[1:4] / linalg.norm(q[1:4])
        return rV
 
    def copy(self):
        """
        Returns a copy of the quaternion.
        """
        return Quaternion(self)
 
    def toRPY(self):
        """
        Returns a 3-sized array with representing the same rotation
        as the (normalized) quaternion. With :
          - the first element representing the Roll,
          - the second the Pitch
          - the third the Yaw
        Where Roll Pitch and Yaw are the angles so that the rotation
        with the quaternion represents the same rotation as :
          - A rotation of R (Roll) about the original x-axis,
            followed by a rotation of P (Pitch) about the original y-axis,
            followed by a rotation of Y (Yaw) about the original z-axis.
          - Or otherwise a rotation of Y about the original z-axis,
            followed by a rotation of P about the new y-axis,
            followed by a rotation of R about the new x-axis.
        """
        q = self.normalized().array
        r = np.arctan2(2 * (q[0] * q[1] + q[2] * q[3]), 1 - 2 * (q[1] ** 2 + q[2] ** 2))
        p = np.arctan2(
            2 * (q[0] * q[2] - q[3] * q[1]),
            np.sqrt(
                (2 * (q[0] * q[1] + q[2] * q[3])) ** 2
                + (1 - 2 * (q[1] ** 2 + q[2] ** 2)) ** 2
            ),
        )  # We cas use arcsin but arctan2 is more robust
        y = np.arctan2(2 * (q[0] * q[3] + q[1] * q[2]), 1 - 2 * (q[2] ** 2 + q[3] ** 2))
        return np.array([r, p, y])
 
    def fromRPY(self, R, P, Y):
        """
        Set the values of the quaternion to the values of a unit quaternion
        representing the same rotation as the one performed by Roll Pitch Yaw :
          - A rotation of R (Roll) about the original x-axis,
            followed by a rotation of P (Pitch) about the original y-axis,
            followed by a rotation of Y (Yaw) about the original z-axis.
          - Or otherwise a rotation of Y about the original z-axis,
            followed by a rotation of P about the new y-axis,
            followed by a rotation of R about the new x-axis.
        """
        r = R / 2.0
        p = P / 2.0
        y = Y / 2.0
        self.array[0] = np.cos(r) * np.cos(p) * np.cos(y) + np.sin(r) * np.sin(
            p
        ) * np.sin(y)
        self.array[1] = np.sin(r) * np.cos(p) * np.cos(y) - np.cos(r) * np.sin(
            p
        ) * np.sin(y)
        self.array[2] = np.cos(r) * np.sin(p) * np.cos(y) + np.sin(r) * np.cos(
            p
        ) * np.sin(y)
        self.array[3] = np.cos(r) * np.cos(p) * np.sin(y) - np.sin(r) * np.sin(
            p
        ) * np.cos(y)
        return self.normalize()