在SLAM中常需要估计一个相机的位置和姿态，这是个优化问题，需要对相机位姿求导，而李代数可以方便地表示相机位姿的导数。本笔记记录李群SO(3),SE(3)和李代数so(3),se(3)的对应关系以及李代数的求导表示。最后是李代数的编程练习。

老规矩，手写拍照...

李群与李代数1.jpg

李群与李代数2.jpg

李群与李代数3.jpg

Sophus编程练习

Sophus 可以下载github源码然后cmake编译：

git clone git://github.com/strasdat/Sophus.git
cd Sophus
git checkout a621ff
mkdir svs_build
cd svs_build
cmake .. -DCMAKE_INSTALL_PREFIX:PATH=$HOME/svslocal
make -j4
make install

cpp代码：

#include <iostream>
#include <cmath>


#include <Eigen/Core>
#include <Eigen/Geometry>

#include "sophus/so3.h"
#include "sophus/se3.h"
 
using namespace std; 

int main(int argc , char **argv)
{
  Eigen::Matrix3d R = Eigen::AngleAxisd(M_PI/2,Eigen::Vector3d(0,0,1)).toRotationMatrix();//z轴旋转pi/2
  Sophus::SO3 SO3_R(R);
  Sophus::SO3 SO3_v(0,0,M_PI/2);
  Eigen::Quaterniond q(R);
  Sophus::SO3 SO3_q(q);
  
  cout<<"SO(3) from matrix:"<<SO3_R<<endl;
  cout<<"SO(3) from vector:"<<SO3_v<<endl;
  cout<<"SO(3) from quaternion:"<<SO3_q<<endl;
  
  //使用对数映射获取它的李代数
  Eigen::Vector3d so3 = SO3_R.log();
  cout<<"李代数so3:"<<so3.transpose()<<endl;
  //hat为向量到反对称矩阵
  cout<<"so3 hat:"<<Sophus::SO3::hat(so3)<<endl;
  //vee 为反对称到向量
  cout<<"so3 hat vee:"<<Sophus::SO3::vee(Sophus::SO3::hat(so3)).transpose()<<endl;
  
  //增加绕动模型
  Eigen::Vector3d update_so3(1e-4,0,0);
  Sophus::SO3 SO3_updated = Sophus::SO3::exp(update_so3)*SO3_R;//左乘更新
  cout<<"SO3 updated:"<<SO3_updated<<endl;
  
  //对SE(3)的操作
  
  Eigen::Vector3d t(1,0,0);
  Sophus::SE3 SE3_Rt(R,t);
  Sophus::SE3 SE3_qt(q,t);
  cout<<"SE3 from R,t:"<<endl<<SE3_Rt<<endl;
  cout<<"SE3 from q,t:"<<endl<<SE3_qt<<endl;
  
  typedef Eigen::Matrix<double,6,1> Vector6d;
  Vector6d se3 = SE3_Rt.log();
  cout<<"李代数se3:"<<se3.transpose()<<endl;
  
  cout<<"se3 hat:"<<Sophus::SE3::hat(se3)<<endl;
  cout<<"se3 hat vee:"<<Sophus::SE3::vee(Sophus::SE3::hat(se3)).transpose()<<endl;
  
  
  Vector6d update_se3 ;
  update_se3.setZero();
  update_se3(0,0) = 1e-4d;
  cout<<"update_se3 is:"<<update_se3.transpose()<<endl;
  Sophus::SE3 SE3_updated = Sophus::SE3::exp(update_se3)* SE3_Rt;
  cout<<"SE3 updated:"<<endl<<SE3_updated.matrix()<<endl;
 
  return 0;
  
  
}

CMakeLists.txt

cmake_minimum_required(VERSION 2.6)
project(usesophus)

include_directories( "/usr/include/eigen3" )

find_package( Sophus REQUIRED )
include_directories( ${Sophus_INCLUDE_DIRS} )

add_executable(usesophus main.cpp)
target_link_libraries( usesophus ${Sophus_LIBRARIES} )
install(TARGETS usesophus RUNTIME DESTINATION bin)