taitbryan
¶
Euler angle rotations and their conversions for TaitBryan zyx convention
See euler
for general discussion of Euler angles and conventions.
This module has specialized implementations of the extrinsic Z axis, Y axis, X axis rotation convention.
The conventions in this module are therefore:
axes \(i, j, k\) are the \(z, y, x\) axes respectively. Thus an Euler angle vector \([ \alpha, \beta, \gamma ]\) in our convention implies a \(\alpha\) radian rotation around the \(z\) axis, followed by a \(\beta\) rotation around the \(y\) axis, followed by a \(\gamma\) rotation around the \(x\) axis.
the rotation matrix applies on the left, to column vectors on the right, so if
R
is the rotation matrix, andv
is a 3 x N matrix with N column vectors, the transformed vector setvdash
is given byvdash = np.dot(R, v)
.extrinsic rotations  the axes are fixed, and do not move with the rotations.
a righthanded coordinate system
The convention of rotation around z
, followed by rotation around
y
, followed by rotation around x
, is known (confusingly) as
“xyz”, pitchrollyaw, Cardan angles, or TaitBryan angles.
Terms used in function names:
mat : array shape (3, 3) (3D nonhomogenous coordinates)
euler : (sequence of) rotation angles about the z, y, x axes (in that order)
axangle : rotations encoded by axis vector and angle scalar
quat : quaternion shape (4,)

Convert axis, angle pair to Euler angles 

Return angle, axis corresponding to these Euler angles 

Return matrix for rotations around z, y and x axes 

Return quaternion corresponding to these Euler angles 

Discover Euler angle vector from 3x3 matrix 

Return Euler angles corresponding to quaternion q 
axangle2euler¶

transforms3d.taitbryan.
axangle2euler
(vector, theta)¶ Convert axis, angle pair to Euler angles
 Parameters
 vector3 element sequence
vector specifying axis for rotation.
 thetascalar
angle of rotation
 Returns
 zscalar
 yscalar
 xscalar
Rotations in radians around z, y, x axes, respectively
Notes
It’s possible to reduce the amount of calculation a little, by combining parts of the
angle_axis2mat
andmat2euler
functions, but the reduction in computation is small, and the code repetition is large.Examples
>>> z, y, x = axangle2euler([1, 0, 0], 0) >>> np.allclose((z, y, x), 0) True
euler2axangle¶

transforms3d.taitbryan.
euler2axangle
(z, y, x)¶ Return angle, axis corresponding to these Euler angles
Uses the z, then y, then x convention above
 Parameters
 zscalar
Rotation angle in radians around zaxis (performed first)
 yscalar
Rotation angle in radians around yaxis
 xscalar
Rotation angle in radians around xaxis (performed last)
 Returns
 vectorarray shape (3,)
axis around which rotation occurs
 thetascalar
angle of rotation
Examples
>>> vec, theta = euler2axangle(0, 1.5, 0) >>> np.allclose(vec, [0, 1, 0]) True >>> theta 1.5
euler2mat¶

transforms3d.taitbryan.
euler2mat
(z, y, x)¶ Return matrix for rotations around z, y and x axes
Uses the convention of staticframe rotation around the z, then y, then x axis.
 Parameters
 zscalar
Rotation angle in radians around zaxis (performed first)
 yscalar
Rotation angle in radians around yaxis
 xscalar
Rotation angle in radians around xaxis (performed last)
 Returns
 Marray shape (3,3)
Rotation matrix giving same rotation as for given angles
Notes
The direction of rotation is given by the righthand rule. Orient the thumb of the right hand along the axis around which the rotation occurs, with the end of the thumb at the positive end of the axis; curl your fingers; the direction your fingers curl is the direction of rotation. Therefore, the rotations are counterclockwise if looking along the axis of rotation from positive to negative.
Examples
>>> zrot = 1.3 # radians >>> yrot = 0.1 >>> xrot = 0.2 >>> M = euler2mat(zrot, yrot, xrot) >>> M.shape == (3, 3) True
The output rotation matrix is equal to the composition of the individual rotations
>>> M1 = euler2mat(zrot, 0, 0) >>> M2 = euler2mat(0, yrot, 0) >>> M3 = euler2mat(0, 0, xrot) >>> composed_M = np.dot(M3, np.dot(M2, M1)) >>> np.allclose(M, composed_M) True
When applying M to a vector, the vector should column vector to the right of M. If the right hand side is a 2D array rather than a vector, then each column of the 2D array represents a vector.
>>> vec = np.array([1, 0, 0]).reshape((3,1)) >>> v2 = np.dot(M, vec) >>> vecs = np.array([[1, 0, 0],[0, 1, 0]]).T # giving 3x2 array >>> vecs2 = np.dot(M, vecs)
Rotations are counterclockwise.
>>> zred = np.dot(euler2mat(np.pi/2, 0, 0), np.eye(3)) >>> np.allclose(zred, [[0, 1, 0],[1, 0, 0], [0, 0, 1]]) True >>> yred = np.dot(euler2mat(0, np.pi/2, 0), np.eye(3)) >>> np.allclose(yred, [[0, 0, 1],[0, 1, 0], [1, 0, 0]]) True >>> xred = np.dot(euler2mat(0, 0, np.pi/2), np.eye(3)) >>> np.allclose(xred, [[1, 0, 0],[0, 0, 1], [0, 1, 0]]) True
euler2quat¶

transforms3d.taitbryan.
euler2quat
(z, y, x)¶ Return quaternion corresponding to these Euler angles
Uses the z, then y, then x convention above
 Parameters
 zscalar
Rotation angle in radians around zaxis (performed first)
 yscalar
Rotation angle in radians around yaxis
 xscalar
Rotation angle in radians around xaxis (performed last)
 Returns
 quatarray shape (4,)
Quaternion in w, x, y z (real, then vector) format
Notes
Formula from Sympy  see
eulerangles.py
inderivations
subdirectory
mat2euler¶

transforms3d.taitbryan.
mat2euler
(M, cy_thresh=None)¶ Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
 Parameters
 Marraylike, shape (3,3)
 cy_threshNone or scalar, optional
threshold below which to give up on straightforward arctan for estimating x rotation. If None (default), estimate from precision of input.
 Returns
 zscalar
 yscalar
 xscalar
Rotations in radians around z, y, x axes, respectively
Notes
If there was no numerical error, the routine could be derived using Sympy expression for z then y then x rotation matrix, (see
eulerangles.py
inderivations
subdirectory):[ cos(y)*cos(z), cos(y)*sin(z), sin(y)], [cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z)  sin(x)*sin(y)*sin(z), cos(y)*sin(x)], [sin(x)*sin(z)  cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
This gives the following solutions for
[z, y, x]
:z = atan2(r12, r11) y = asin(r13) x = atan2(r23, r33)
Problems arise when
cos(y)
is close to zero, because both of:z = atan2(cos(y)*sin(z), cos(y)*cos(z)) x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to
atan2(0, 0)
, and highly unstable.The
cy
fix for numerical instability in this code is from: Euler Angle Conversion by Ken Shoemake, p2229 ; in: Graphics Gems IV, Paul Heckbert (editor), Academic Press, 1994, ISBN: 0123361559. Specifically it comes fromEulerAngles.c
and deals with the case where cos(y) is close to zero:The code appears to be licensed (from the website) as “can be used without restrictions”.
quat2euler¶

transforms3d.taitbryan.
quat2euler
(q)¶ Return Euler angles corresponding to quaternion q
 Parameters
 q4 element sequence
w, x, y, z of quaternion
 Returns
 zscalar
Rotation angle in radians around zaxis (performed first)
 yscalar
Rotation angle in radians around yaxis
 xscalar
Rotation angle in radians around xaxis (performed last)
Notes
It’s possible to reduce the amount of calculation a little, by combining parts of the
quat2mat
andmat2euler
functions, but the reduction in computation is small, and the code repetition is large.