1、本文内容
四元数、罗德里格斯公式、欧拉角、旋转矩阵推导和资料
2、转载请注明出处：
https://blog.csdn.net/qq_41102371/article/details/126002167

资料

四元数
 Understanding Quaternions 中文翻译《理解四元数》 https://www.qiujiawei.com/understanding-quaternions/
http://mars.cs.umn.edu/tr/reports/Trawny05b.pdf
几种三维空间旋转的表达方式转换
三维旋转：欧拉角、四元数、旋转矩阵、轴角之间的转换 https://zhuanlan.zhihu.com/p/45404840
彻底搞懂“旋转矩阵/欧拉角/四元数”，让你体会三维旋转之美 https://blog.csdn.net/weixin_45590473/article/details/122884112
展示欧拉角与四元数动态变换关系的网站
https://quaternions.online/

罗德里格斯公式

https://en.wikipedia.org/wiki/Rodrigues’_rotation_formula
罗德里格斯公式（Rodrigues Formula） https://blog.csdn.net/weixin_40215443/article/details/123950141
二维xy坐标旋转 https://blog.csdn.net/qq_41102371/article/details/116245483#t4

几种表达方式

$K K 是 k k 的叉乘矩阵，即 K = k ∧ \mathbf{K}=\mathbf{k}^{\land}$

四元数转旋转矩阵

根据 http://mars.cs.umn.edu/tr/reports/Trawny05b.pdf，四元数转旋转矩阵为：
在这里插入图片描述

$\begin{aligned} {}_G^L\mathbf{C}(\bar{q})&=(2q_4^2-1)\mathbf{I}_{3\times3}-2q_4[\mathbf{q}_{\times}]+2\mathbf{q}\mathbf{q}^T\\ &=(2q_4^2-1)\mathbf{I}_{3\times3}-2q_4[\mathbf{q}_{\times}]+2[\mathbf{q}_{\times}]^2+2|\mathbf{q}|^2\cdot \mathbf{I}_{3\times3}\\ &=(2q_4^2+2|\mathbf{q}|^2-1)\mathbf{I}_{3\times3}-2q_4[\mathbf{q}_{\times}]+2[\mathbf{q}_{\times}]^2\\ &=(2q_4^2+2(q_1^2+q_2^2+q_3^2)-1)\mathbf{I}_{3\times3}-2q_4[\mathbf{q}_{\times}]+2[\mathbf{q}_{\times}]^2\\ &=(2-1)\mathbf{I}_{3\times3}-2q_4[\mathbf{q}_{\times}]+2[\mathbf{q}_{\times}]^2\\ &=\mathbf{I}_{3\times3}-2q_4[\mathbf{q}_{\times}]+2[\mathbf{q}_{\times}]^2 \end{aligned}$

近似

根据 http://mars.cs.umn.edu/tr/reports/Trawny05b.pdf原文当 $\delta\bar{q}$

四元数与轴角

还是参考 http://mars.cs.umn.edu/tr/reports/Trawny05b.pdf，

$\bar{q}=\begin{bmatrix}\mathbf{q}\\q_4\end{bmatrix}=\begin{bmatrix}q_1\\q_2\\q_3\\q_4\end{bmatrix}=\begin{bmatrix}k_x\sin(\theta/2)\\k_y\sin(\theta/2)\\k_z\sin(\theta/2)\\\cos(\theta/2)\end{bmatrix}=\begin{bmatrix}\hat{\mathbf{k}}\sin(\theta/2)\\\cos(\theta/2)\end{bmatrix},\hat{\mathbf{k}}=[k_x,k_y,k_z]^T,q_4=\cos(\theta/2)$

转换代码

欧拉角转旋转矩阵

#include <Eigen/Core>
#include <Eigen/Dense>
#include <iostream>
#define PI 3.1415926

int main(int argc, char* argv[]){
    std::cout<<PI<<std::endl;
    if(argc<4){
        std::cout<<"please input a 3x1 vector,for example:\neuler2rt 45 30 60"<<std::endl;
        return 0;  
    }
    
    Eigen::Vector3d eulerAngle(atof(argv[1]),atof(argv[2]),atof(argv[3]));
    std::cout<<"eulerAngle:\nx: "<<eulerAngle[0]<<"    y: "<<eulerAngle[1]<<"    z: "<<eulerAngle[2]<<std::endl;
    eulerAngle=eulerAngle/180*PI;


    Eigen::Matrix3d rotation_matrix = Eigen::Matrix3d::Identity();
        Eigen::AngleAxisd rollAngle(Eigen::AngleAxisd(eulerAngle[0],Eigen::Vector3d::UnitX()));
    Eigen::AngleAxisd pitchAngle(Eigen::AngleAxisd(eulerAngle[1],Eigen::Vector3d::UnitY()));
    Eigen::AngleAxisd yawAngle(Eigen::AngleAxisd(eulerAngle[2],Eigen::Vector3d::UnitZ()));


    rotation_matrix=rollAngle*pitchAngle*yawAngle;

    std::cout<<"rotation_matrix:\n"<<rotation_matrix<<std::endl;
    Eigen::Vector3d eulerAngle2=rotation_matrix.eulerAngles(0,1,2);
     std::cout<<"eulerAngle:\nx: "<<eulerAngle2[0]/PI*180<<"    y: "<<eulerAngle2[1]/PI*180<<"    z: "<<eulerAngle2[2]/PI*180<<std::endl;

    return 0;
}

使用

旋转矩阵转欧拉角和四元数

#include <Eigen/Core>
#include <Eigen/Dense>
#include <iostream>
#include <fstream>
#define PI 3.1415926

int main(int argc, char* argv[]){
    if(argc<2){
        std::cout<<"please input a file including a 3x3 matrix"<<std::endl;
        return 0;  
    }
    std::string filename;
    filename=argv[1];  
    std::ifstream fin;
    fin.open(filename,std::ios::in);
    if(!fin.is_open()){
        std::cout<<"read file "<<filename<<" failed."<<std::endl;
        return 0;
    }
    Eigen::Matrix3d rotation_matrix = Eigen::Matrix3d::Identity();

    for (std::size_t i = 0; i < 3; ++i) {
        for (std::size_t j = 0; j < 3; ++j) {
            double value;
            fin >> value;
            // std::cout<<value<<" ";
            rotation_matrix(i, j) = value;
        }
    }
    std::cout<<std::endl;
	fin.close();

    std::cout<<"rotation_matrix:\n"<<rotation_matrix<<std::endl;
    Eigen::Vector3d eulerAngle=rotation_matrix.eulerAngles(0,1,2);
    // std::cout<<eulerAngle<<std::endl;
    std::cout<<"eulerAngle:\nx: "<<eulerAngle[0]/PI*180<<"    y: "<<eulerAngle[1]/PI*180<<"    z: "<<eulerAngle[2]/PI*180<<std::endl;

	//转四元数
    Eigen::Quaterniond rotation_q(rotation_matrix);
    std::cout<<"rotation_q:\nx: "<<rotation_q.x()<<" y:"<<rotation_q.y()<<" z:"<<rotation_q.z()<<" w:"<<rotation_q.w()<<" "<<std::endl;


    return 0;
}

使用

rt2euler r_3x3.txt

其中r_3x3.txt格式如下

1 0 0
0 1 0
0 0 1

参考

使用Eigen实现四元数、欧拉角、旋转矩阵、旋转向量之间的转换
 http://t.zoukankan.com/long5683-p-14373627.html

完

如有错漏，敬请指正
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四元数、罗德里格斯公式、欧拉角、旋转矩阵推导和资料

目录

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资料