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I'm disassembling a rotation matrix to Euler angles (Tait-Bryan angles more specifically in the order x-y-z, that is rotation around x axis first) and back to a rotation matrix. I used the transforms3d python library (https://github.com/matthew-brett/transforms3d) and also followed this tutorial www.gregslabaugh.net/publications/euler.pdf Both give the same result.
The problem is that the reassambled rotation matrix doesn't match the one that I started with.
The matrix I'm working with was created by the "decomposeHomographyMat" function from openCV, so I expect it to be a valid rotation matrix. Maybe it is a special case? The matrix is
The three angles are [-1.8710997 , 0.04623301, -0.03679793]. If I convert them back to a rotation matrix I get
where R_23 cannot be a rounding error.
Following the paper above, rotation around the y axis (beta) can be calculated by asin(-R_31). Another valid angle would be pi-asin(-R_31). The angle around the x axis (alpha) can be calculated by atan2(R_32,R_33). I could also get alpha by asin(R_32/cos(beta)) or by acos(R_33/cos(beta)). If I use the latter two equations I only get the same result for alpha if I use beta=pi-arcsin(-R_31), which implies that there is only one valid solution for beta. atan2(R_32,R_33) gives a different result from both.
Anyway something seems to be wrong with my matrix or I cannot figure out why the disassambly doesn't work.
import numpy as np
def rot2eul(R):
beta = -np.arcsin(R[2,0])
alpha = np.arctan2(R[2,1]/np.cos(beta),R[2,2]/np.cos(beta))
gamma = np.arctan2(R[1,0]/np.cos(beta),R[0,0]/np.cos(beta))
return np.array((alpha, beta, gamma))
def eul2rot(theta) :
R = np.array([[np.cos(theta[1])*np.cos(theta[2]), np.sin(theta[0])*np.sin(theta[1])*np.cos(theta[2]) - np.sin(theta[2])*np.cos(theta[0]), np.sin(theta[1])*np.cos(theta[0])*np.cos(theta[2]) + np.sin(theta[0])*np.sin(theta[2])],
[np.sin(theta[2])*np.cos(theta[1]), np.sin(theta[0])*np.sin(theta[1])*np.sin(theta[2]) + np.cos(theta[0])*np.cos(theta[2]), np.sin(theta[1])*np.sin(theta[2])*np.cos(theta[0]) - np.sin(theta[0])*np.cos(theta[2])],
[-np.sin(theta[1]), np.sin(theta[0])*np.cos(theta[1]), np.cos(theta[0])*np.cos(theta[1])]])
return R
R = np.array([[ 0.9982552 , -0.03323557, -0.04880523],
[-0.03675031, 0.29723396, -0.95409716],
[-0.04621654, -0.95422606, -0.29549393]])
ang = rot2eul(R)
eul2rot(ang)
import transforms3d.euler as eul
ang = eul.mat2euler(R, axes='sxyz')
eul.euler2mat(ang[0], ang[1], ang[2], axes='sxyz')
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Given a rotation matrix R, we can compute the Euler angles, ψ, θ, and φ by equating each element in R with the corresponding element in the matrix product Rz(φ)Ry(θ)Rx(ψ). This results in nine equations that can be used to find the Euler angles. Starting with R31, we find R31 = − sin θ.
Description. eul = rotm2eul( rotm ) converts a rotation matrix, rotm , to the corresponding Euler angles, eul . The input rotation matrix must be in the premultiply form for rotations. The default order for Euler angle rotations is "ZYX" .
Euler angles are specified by the three angles, viz., ψ , θ , ϕ . Euler angles represent three consecutive rotations in the order of ψ , θ , ϕ so that one coordinate axes system is made to coincide with another system.
quat = eul2quat( eul ) converts a given set of Euler angles, eul , to the corresponding quaternion, quat . The default order for Euler angle rotations is "ZYX" . quat = eul2quat( eul , sequence ) converts a set of Euler angles into a quaternion. The Euler angles are specified in the axis rotation sequence, sequence .
It turns out the rotation matrix has a negative determinant, which makes it an improper rotation matrix. The openCV function "decomposeHomographyMat" has a bug: https://github.com/opencv/opencv/issues/4978
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