如何求空间中3个正交向量的旋转矩阵。我当前的方法将向量旋转到错误的方向

Question

我希望找到旋转矩阵，以获得三个（几乎）正交向量，它们位于世界坐标系的同一方向

我的三个（几乎）正交向量可以用python表示如下：

vectors = np.array([[ 0.43187079,  0.90161148,  0.02417362],
   [-0.46076794,  0.19750816,  0.86526495],
   [ 0.77535832, -0.38482109,  0.50073167]])

我当前使用的代码可以使向量与世界坐标平行，但方向不正确。运行此代码

xrotation = np.arctan2(vectors[2, 1], vectors[2, 2])
xRot = np.array([[1, 0, 0],
                 [0, np.cos(xrotation), -np.sin(xrotation)],
                 [0, np.sin(xrotation), np.cos(xrotation)]])
vectors_x = np.zeros((3, 3))
for i in range(3):
    vectors_x[i, :] = np.linalg.inv(xRot.transpose()) @ vectors[i, :]
yrotation = np.arctan2(vectors_x[1, 2], vectors_x[1, 0])
yRot = np.array([[np.cos(yrotation), 0, np.sin(yrotation)],
                 [0, 1, 0],
                 [-np.sin(yrotation), 0, np.cos(yrotation)]])
vectors_y = np.zeros((3, 3))
for i in range(3):
    vectors_y[i, :] = np.linalg.pinv(yRot.transpose()) @ vectors_x[i, :]

zrotation = np.arctan2(vectors_y[0, 0], vectors_y[0, 1])
zRot = np.array([[np.cos(zrotation), -np.sin(zrotation), 0],
                 [np.sin(zrotation), np.cos(zrotation), 0],
                 [0, 0, 1]])
vectors_z = np.zeros((3, 3))
for i in range(3):
    vectors_z[i, :] = np.linalg.pinv(zRot.transpose()) @ vectors_y[i, :]

给出三个旋转正交向量：

>vectors_z
>array([[-1.11022302e-16,  1.00000000e+00,  3.19660393e-09],
       [ 1.00000000e+00, -3.70417658e-09, -2.77555756e-16],
       [ 2.12261116e-09, -1.98949113e-09, -1.00000000e+00]])

我需要在代码中更改什么才能使其处于正确的方向，如下所示：

array([[ 1, 0, 0],
       [ 0, 1, 0],
       [ 0, 0, 1]])

我知道可以通过以正确的顺序旋转向量90/180度来实现这一点，但必须有一种更有效的方法，通过在上面的代码中执行其他操作来实现这一点

谢谢你的时间

Answer 1

我明白了。切换到ZYZ旋转模式并重新定义欧拉角计算方法。希望有一天这能帮助别人

import numpy as np

def z_rotation(zrotation):
    z1Rot = np.array([[np.cos(zrotation), -np.sin(zrotation), 0],
                      [np.sin(zrotation), np.cos(zrotation), 0],
                      [0, 0, 1]])
    return z1Rot

def y_rotation(yrotation):
    yRot = np.array([[np.cos(yrotation), 0, np.sin(yrotation)],
                     [0, 1, 0],
                     [-np.sin(yrotation), 0, np.cos(yrotation)]])
    return yRot

def forward_rotation(Rot,vectors_in):
    vectors = np.zeros((3, 3))
    for i in range(3):
        vectors[i, :] = vectors_in[i, :] @ Rot
    return vectors


def reverse_rotation(Rot, vectors_in):
    vectors = np.zeros((3, 3))
    for i in range(3):
        vectors[i, :] = np.linalg.pinv(Rot.transpose()) @ vectors_in[i, :]
    return vectors

org_vectors = np.array([[1,0,0],[0,1,0],[0,0,1]])

z1_angle = (-.5 + np.random.random()) * 1800
y_angle = (-.5 + np.random.random()) * 1800
z2_angle = (-.5 + np.random.random()) * 1800

z1 = z1_angle*np.pi/180
y = y_angle*np.pi/180
z2 = z2_angle*np.pi/180

z1Rot = z_rotation(z1)
z1vectors = forward_rotation(z1Rot, org_vectors)


yRot = y_rotation(y)
yvectors = forward_rotation(yRot, z1vectors)

z2Rot = z_rotation(z2)
z2vectors = forward_rotation(z2Rot, yvectors)

z2angle_calc = np.arctan2(z2vectors[2,1],z2vectors[2,0])
z2rot_2 = z_rotation(z2angle_calc)

new_y = forward_rotation(z2rot_2, z2vectors)

yangle_2 = np.arctan2(new_y[2,0],new_y[2,2])
yrot_2 = y_rotation(yangle_2)

new_z1 = forward_rotation(yrot_2, new_y)


z1angle_2 = yangle_2 = np.arctan2(new_z1[0,1],new_z1[0, 0])
z1rot_2 = z_rotation(z1angle_2)

new_org_vectors = forward_rotation(z1rot_2, new_z1)
print(new_org_vectors)