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updated orientation/quaternion class to follow Rowenhorst_etal2015

This commit is contained in:
Philip Eisenlohr 2018-09-06 14:19:09 -04:00
parent eae6dc8e67
commit 6c3b5f17eb
1 changed files with 53 additions and 107 deletions
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160
lib/damask/orientation.py
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@ -30,12 +30,19 @@ class Quaternion:
"""
Orientation represented as unit quaternion.
All methods and naming conventions based on http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions.
All methods and naming conventions based on Rowenhorst_etal2015
Convention 1: coordinate frames are right-handed
Convention 2: a rotation angle ω is taken to be positive for a counterclockwise rotation
when viewing from the end point of the rotation axis unit vector towards the origin
Convention 3: rotations will be interpreted in the passive sense
Convention 4: Euler angle triplets are implemented using the Bunge convention,
with the angular ranges as [0, 2π],[0, π],[0, 2π]
Convention 5: the rotation angle ω is limited to the interval [0, π]
w is the real part, (x, y, z) are the imaginary parts.
Representation of rotation is in ACTIVE form!
(Derived directly or through angleAxis, Euler angles, or active matrix)
Vector "a" (defined in coordinate system "A") is actively rotated to new coordinates "b".
Vector "a" (defined in coordinate system "A") is passively rotated
resulting in new coordinates "b" when expressed in system "B".
b = Q * a
b = np.dot(Q.asMatrix(),a)
"""
@ -309,10 +316,12 @@ class Quaternion:
return np.outer([i for i in self],[i for i in self])
def asMatrix(self):
return np.array(
[[1.0-2.0*(self.y*self.y+self.z*self.z), 2.0*(self.x*self.y-self.z*self.w), 2.0*(self.x*self.z+self.y*self.w)],
[ 2.0*(self.x*self.y+self.z*self.w), 1.0-2.0*(self.x*self.x+self.z*self.z), 2.0*(self.y*self.z-self.x*self.w)],
[ 2.0*(self.x*self.z-self.y*self.w), 2.0*(self.x*self.w+self.y*self.z), 1.0-2.0*(self.x*self.x+self.y*self.y)]])
qbarhalf = 0.5*(self.w**2 - self.x**2 - self.y**2 - self.z**2)
return 2.0*np.array(
[[ qbarhalf + self.x**2 , self.x*self.y - self.w*self.z, self.x*self.z + self.w*self.y],
[ self.x*self.y + self.w*self.z, qbarhalf + self.y**2 , self.y*self.z - self.w*self.x],
[ self.x*self.z - self.w*self.y, self.y*self.z + self.w*self.x, qbarhalf + self.z**2 ],
])
def asAngleAxis(self,
degrees = False):
@ -335,52 +344,28 @@ class Quaternion:
return np.inf*np.ones(3) if self.w == 0.0 else np.array([self.x, self.y, self.z])/self.w
def asEulers(self,
type = "bunge",
degrees = False,
standardRange = False):
):
"""
Orientation as Bunge-Euler angles.
Conversion of ACTIVE rotation to Euler angles taken from:
Melcher, A.; Unser, A.; Reichhardt, M.; Nestler, B.; Poetschke, M.; Selzer, M.
Conversion of EBSD data by a quaternion based algorithm to be used for grain structure simulations
Technische Mechanik 30 (2010) pp 401--413.
"""
angles = [0.0,0.0,0.0]
if type.lower() == 'bunge' or type.lower() == 'zxz':
if abs(self.x) < 1e-4 and abs(self.y) < 1e-4:
x = self.w**2 - self.z**2
y = 2.*self.w*self.z
angles[0] = math.atan2(y,x)
elif abs(self.w) < 1e-4 and abs(self.z) < 1e-4:
x = self.x**2 - self.y**2
y = 2.*self.x*self.y
angles[0] = math.atan2(y,x)
angles[1] = math.pi
else:
chi = math.sqrt((self.w**2 + self.z**2)*(self.x**2 + self.y**2))
q03 = self.w**2+self.z**2
q12 = self.x**2+self.y**2
chi = np.sqrt(q03*q12)
if abs(chi) < 1e-10 and abs(q12) < 1e-10:
eulers = np.array([math.atan2(-2*self.w*self.z,self.w**2-self.z**2),0,0])
elif abs(chi) < 1e-10 and abs(q03) < 1e-10:
eulers = np.array([math.atan2( 2*self.x*self.y,self.x**2-self.y**2),np.pi,0])
else:
eulers = np.array([math.atan2((self.x*self.z-self.w*self.y)/chi,(-self.w*self.x-self.y*self.z)/chi),
math.atan2(2*chi,q03-q12),
math.atan2((self.w*self.y+self.x*self.z)/chi,( self.y*self.z-self.w*self.x)/chi),
])
x = (self.w * self.x - self.y * self.z)/2./chi
y = (self.w * self.y + self.x * self.z)/2./chi
angles[0] = math.atan2(y,x)
x = self.w**2 + self.z**2 - (self.x**2 + self.y**2)
y = 2.*chi
angles[1] = math.atan2(y,x)
x = (self.w * self.x + self.y * self.z)/2./chi
y = (self.z * self.x - self.y * self.w)/2./chi
angles[2] = math.atan2(y,x)
if standardRange:
angles[0] %= 2*math.pi
if angles[1] < 0.0:
angles[1] += math.pi
angles[2] *= -1.0
angles[2] %= 2*math.pi
return np.degrees(angles) if degrees else angles
return np.degrees(eulers) if degrees else eulers
# # Static constructors
@ -408,7 +393,7 @@ class Quaternion:
halfangle = math.atan(np.linalg.norm(rodrigues))
c = math.cos(halfangle)
w = c
x,y,z = c*rodrigues
x,y,z = rodrigues/c
return cls([w,x,y,z])
@ -431,24 +416,19 @@ class Quaternion:
@classmethod
def fromEulers(cls,
eulers,
type = 'Bunge',
degrees = False):
if not isinstance(eulers, np.ndarray): eulers = np.array(eulers,dtype='d')
eulers = np.radians(eulers) if degrees else eulers
c = np.cos(0.5 * eulers)
s = np.sin(0.5 * eulers)
sigma = 0.5*(eulers[0]+eulers[2])
delta = 0.5*(eulers[0]-eulers[2])
c = np.cos(0.5*eulers[1])
s = np.sin(0.5*eulers[1])
if type.lower() == 'bunge' or type.lower() == 'zxz':
w = c[0] * c[1] * c[2] - s[0] * c[1] * s[2]
x = c[0] * s[1] * c[2] + s[0] * s[1] * s[2]
y = - c[0] * s[1] * s[2] + s[0] * s[1] * c[2]
z = c[0] * c[1] * s[2] + s[0] * c[1] * c[2]
else:
w = c[0] * c[1] * c[2] - s[0] * s[1] * s[2]
x = s[0] * s[1] * c[2] + c[0] * c[1] * s[2]
y = s[0] * c[1] * c[2] + c[0] * s[1] * s[2]
z = c[0] * s[1] * c[2] - s[0] * c[1] * s[2]
w = c * np.cos(sigma)
x = -s * np.cos(delta)
y = -s * np.sin(delta)
z = -c * np.sin(sigma)
return cls([w,x,y,z])
@ -460,49 +440,16 @@ class Quaternion:
if m.shape != (3,3) and np.prod(m.shape) == 9:
m = m.reshape(3,3)
tr = np.trace(m)
if tr > 1e-8:
s = math.sqrt(tr + 1.0)*2.0
w = 0.5*math.sqrt(1.+m[0,0]+m[1,1]+m[2,2])
x = 0.5*math.sqrt(1.+m[0,0]-m[1,1]-m[2,2])
y = 0.5*math.sqrt(1.-m[0,0]+m[1,1]-m[2,2])
z = 0.5*math.sqrt(1.-m[0,0]-m[1,1]+m[2,2])
return cls(
[ s*0.25,
(m[2,1] - m[1,2])/s,
(m[0,2] - m[2,0])/s,
(m[1,0] - m[0,1])/s,
])
x *= -1 if m[2,1] < m[1,2] else 1
y *= -1 if m[0,2] < m[2,0] else 1
z *= -1 if m[1,0] < m[0,1] else 1
elif m[0,0] > m[1,1] and m[0,0] > m[2,2]:
t = m[0,0] - m[1,1] - m[2,2] + 1.0
s = 2.0*math.sqrt(t)
return cls(
[ (m[2,1] - m[1,2])/s,
s*0.25,
(m[0,1] + m[1,0])/s,
(m[2,0] + m[0,2])/s,
])
elif m[1,1] > m[2,2]:
t = -m[0,0] + m[1,1] - m[2,2] + 1.0
s = 2.0*math.sqrt(t)
return cls(
[ (m[0,2] - m[2,0])/s,
(m[0,1] + m[1,0])/s,
s*0.25,
(m[1,2] + m[2,1])/s,
])
else:
t = -m[0,0] - m[1,1] + m[2,2] + 1.0
s = 2.0*math.sqrt(t)
return cls(
[ (m[1,0] - m[0,1])/s,
(m[2,0] + m[0,2])/s,
(m[1,2] + m[2,1])/s,
s*0.25,
])
return cls( np.array([w,x,y,z])/math.sqrt(w**2 + x**2 + y**2 + z**2))
@classmethod
@ -829,7 +776,7 @@ class Orientation:
else:
self.quaternion = Quaternion.fromRandom(randomSeed=random)
elif isinstance(Eulers, np.ndarray) and Eulers.shape == (3,): # based on given Euler angles
self.quaternion = Quaternion.fromEulers(Eulers,type='bunge',degrees=degrees)
self.quaternion = Quaternion.fromEulers(Eulers,degrees=degrees)
elif isinstance(matrix, np.ndarray) : # based on given rotation matrix
self.quaternion = Quaternion.fromMatrix(matrix)
elif isinstance(angleAxis, np.ndarray) and angleAxis.shape == (4,): # based on given angle and rotation axis
@ -855,16 +802,15 @@ class Orientation:
return 'Symmetry: %s\n' % (self.symmetry) + \
'Quaternion: %s\n' % (self.quaternion) + \
'Matrix:\n%s\n' % ( '\n'.join(['\t'.join(map(str,self.asMatrix()[i,:])) for i in range(3)]) ) + \
'Bunge Eulers / deg: %s' % ('\t'.join(map(str,self.asEulers('bunge',degrees=True))) )
'Bunge Eulers / deg: %s' % ('\t'.join(map(str,self.asEulers(degrees=True))) )
def asQuaternion(self):
return self.quaternion.asList()
def asEulers(self,
type = 'bunge',
degrees = False,
standardRange = False):
return self.quaternion.asEulers(type, degrees, standardRange)
):
return self.quaternion.asEulers(degrees)
eulers = property(asEulers)
def asRodrigues(self):