如何找到空间中3个正交向量的旋转矩阵我目前的方法将向量旋转到错误的方向

问题描述

我正在寻找一种旋转矩阵，以使三个（几乎）正交向量处于世界坐标系的相同方向。

我的三个（几乎）正交向量可以在python中这样表示：

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，但是方向不正确。运行此代码，

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

给出三个旋转的正交向量：

>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]])

我需要在代码中进行哪些更改以使其正确定位，如下所示：

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

我知道可以通过以正确的顺序将向量旋转90/180度来实现此目的，但是必须有一种更有效的方法来执行上述代码中的其他操作。

感谢您的宝贵时间！

解决方法

弄清楚了。切换为ZYZ旋转模式，并重新设计欧拉角计算方法。希望有一天能对某人有所帮助。

import numpy as np

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

def y_rotation(yrotation):
    yRot = np.array([[np.cos(yrotation),np.sin(yrotation)],1,[-np.sin(yrotation),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,:] = np.linalg.pinv(Rot.transpose()) @ vectors_in[i,:]
    return vectors

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

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,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,new_z1[0,0])
z1rot_2 = z_rotation(z1angle_2)

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