关于ZYZ旋转和XYZ旋转

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概述

1. 以下公式默认为右手坐标系；
2. ZYZ通常可以避免死解情况，因此在六轴末端解算时常被用到；
3. 参考文章

1、XYZ旋转

function R = RotationZYX(xyz) X = [1, 0, 0; ... 0, cos(xyz(1)), -sin(xyz(1)); ... 0, sin(xyz(1)), cos(xyz(1))]; Y = [cos(xyz(2)), 0, sin(xyz(2)); ... 0, 1, 0; ... -sin(xyz(2)), 0, cos(xyz(2))]; Z = [cos(xyz(3)), -sin(xyz(3)), 0; ... sin(xyz(3)), cos(xyz(3)), 0; ... 0, 0, 1]; R = Z * Y * X; end 

C++代码：

cv::Mat EulerToMat(cv::Point3f Euler) { 
    float alpha = Euler.x / 180 * CV_PI; float belta = Euler.y / 180 * CV_PI; float gamma = Euler.z / 180 * CV_PI; cv::Mat Rx = (cv::Mat_<float>(3, 3) << 1, 0, 0, 0, cos(alpha), -sin(alpha), 0, sin(alpha), cos(alpha)); cv::Mat Ry = (cv::Mat_<float>(3, 3) << cos(belta), 0, sin(belta), 0, 1, 0, -sin(belta), 0, cos(belta) ); cv::Mat Rz = (cv::Mat_<float>(3, 3) << cos(gamma), -sin(gamma), 0, sin(gamma), cos(gamma), 0, 0, 0, 1 ); return Rz * Ry * Rx; } 

2、ZYZ旋转

XYZ旋转一般是绕自身轴旋转(内旋)，旋转矩阵的构成为：RxRyRz。

MATLAB代码：

syms a b c Ra=[cos(a) -sin(a) 0; sin(a) cos(a) 0; 0 0 1]; Rb=[cos(b) 0 sin(b); 0 1 0;-sin(b) 0 cos(b)]; Rc=[cos(c) -sin(c) 0;sin(c) cos(c) 0;0 0 1]; Rzyz=Ra*Rb*Rc % 定义姿态旋转角度 angle1 = deg2rad(-8.668); % 绕z轴旋转角度 angle2 = deg2rad(146.282); % 绕y轴旋转角度 angle3 = deg2rad(175.557); % 绕新z轴旋转角度 r11 = cos(angle1) * cos(angle2) * cos(angle3) - sin(angle1) * sin(angle3); r12 = -cos(angle1) * cos(angle2) * sin(angle3) - sin(angle1) * cos(angle3); r13 = cos(angle1) * sin(angle2); r21 = sin(angle1) * cos(angle2) * cos(angle3) + cos(angle1) * sin(angle3); r22 = -sin(angle1) * cos(angle2) * sin(angle3) + cos(angle1) * cos(angle3); r23 = sin(angle1) * sin(angle2); r31 = -sin(angle2) * cos(angle3); r32 = sin(angle2) * sin(angle3); r33 = cos(angle2); % 姿态旋转矩阵 ZYZ_matrix = [r11 r12 r13; r21 r22 r23; r31 r32 r33]; 

C++代码：

float angle1 = config.ReadFloat("Parmeter", "angleA", 0.0); float angle2 = config.ReadFloat("Parmeter", "angleE", 0.0); float angle3 = config.ReadFloat("Parmeter", "angleR", 0.0); angle1 = angle1 * CV_PI / 180; angle2 = angle2 * CV_PI / 180; angle3 = angle3 * CV_PI / 180; float r11 = cos(angle1) * cos(angle2) * cos(angle3) - sin(angle1) * sin(angle3); float r12 = -cos(angle1) * cos(angle2) * sin(angle3) - sin(angle1) * cos(angle3); float r13 = cos(angle1) * sin(angle2); float r21 = sin(angle1) * cos(angle2) * cos(angle3) + cos(angle1) * sin(angle3); float r22 = -sin(angle1) * cos(angle2) * sin(angle3) + cos(angle1) * cos(angle3); float r23 = sin(angle1) * sin(angle2); float r31 = -sin(angle2) * cos(angle3); float r32 = sin(angle2) * sin(angle3); float r33 = cos(angle2); cv::Mat ZYZ_matrix = (cv::Mat_<float>(3, 3) << r11, r12, r13, r21, r22, r23, r31, r32, r33);