【Eigen】【Eigen实践】【Eigen的使用学习记录】
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【Eigen】【Eigen实践】【Eigen的使用学习记录】
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【Eigen】【Eigen實踐】【Eigen的使用學習記錄】
- 0 前言
- 1 Eigen使用
- 1.1 頭文件的使用
- 1.2 定義和初始化
- 1.2.1 定義n*m矩陣
- 1.2.2 定義n*1向量
- 1.2.3 定義n*n矩陣|初始化為0
- 1.2.4 定義動態(tài)矩陣
- 1.3 對Eigen操作
- 1.3.1 輸入數(shù)據(jù)
- 1.3.2 矩陣整個輸出
- 1.3.3 用()訪問矩陣中的元素
- 1.3.4 矩陣和向量相乘|類型轉換
- 1.3.5 得到矩陣的指定塊元素
- 1.4 矩陣運算（隨機數(shù)、轉置、各元素和、跡、數(shù)乘、逆、行列式、特征值、特特征向量、叉乘、點乘、求模、歸一化、“開根號”）
- 1.5 解方程
- 1.5.1 直接求逆
- 1.5.2 QR分解
- 1.5.3 cholesky分解
- 1.5.4 SVD分解
- 1.6 旋轉和平移的表示、轉換和使用
- 1.6.0 數(shù)據(jù)類型匯總
- 1.6.1 旋轉矩陣
- 1.6.1.1 初始化
- 1.6.1.2 坐標變換
- 1.6.2.3 旋轉向量->旋轉矩陣
- 1.6.2 旋轉向量
- 1.6.2.1 初始化
- 1.6.2.3 坐標變換
- 1.6.3 歐拉角
- 1.6.3.1 初始化
- 1.6.3.2 旋轉矩陣->歐拉角
- 1.6.4 歐式變換矩陣se
- 1.6.4.1 初始化
- 1.6.4.2 旋轉向量->se.T|se->旋轉矩陣
- 1.6.4.3 四元數(shù)->se.T
- 1.6.4.4 坐標變換
- 1.6.5 四元數(shù)
- 1.6.5.1 初始化
- 1.6.5.2 旋轉向量->四元數(shù)
- 1.6.5.3 旋轉矩陣->四元數(shù)
- 1.6.5.4 坐標變換
- 1.6.5.5 取虛部`q.vec()`
- 2 slam14的cap3的useigen
- 2.1 eigenMatrix.cpp
- 2.2 CMakeLists.txt
- 2.3 輸出
- 3 slam14的cap3的useigen
- 3.1 useGeometry.cpp
- 3.2 CMakeLists.txt
- 3.3 輸出
- 4 實際的坐標轉換例子
- 4.1 coordinateTransform.cpp
- 4.2 CMakeLists.txt
- 4.3 輸出
- 5 顯示運動軌跡
- 5.1 plotTrajectory.cpp
- 5.2 CMakeLists.txt
- 5.3 輸出
- 6 顯示相機的位姿
- 6.1 visualizeGeometry.cpp
- 6.2 CMakeLists.txt
- 6.3 輸出
- 7 使用QR和Cholesly分解求解示例
- 7.0 說明：自創(chuàng)建100大小的動態(tài)矩陣，并使用QR和Cholesly分解求解
- 7.1 qiujie.cpp
- 7.2 CMakeLists.txt
0 前言
1 Eigen使用
1.1 頭文件的使用
1.2 定義和初始化
- Eigen固定大小矩陣最大支持到50
1.2.1 定義n*m矩陣
// Eigen 中所有向量和矩陣都是Eigen::Matrix，它是一個模板類。它的前三個參數(shù)為：數(shù)據(jù)類型，行，列// 聲明一個2*3的float矩陣 Matrix<float, 2, 3> matrix_23;1.2.2 定義n*1向量
// 同時，Eigen 通過 typedef 提供了許多內(nèi)置類型，不過底層仍是Eigen::Matrix// 例如 Vector3d 實質(zhì)上是 Eigen::Matrix<double, 3, 1>，即三維向量Vector3d v_3d;// 這是一樣的Matrix<float, 3, 1> vd_3d;1.2.3 定義n*n矩陣|初始化為0
// Matrix3d 實質(zhì)上是 Eigen::Matrix<double, 3, 3>Matrix3d matrix_33 = Matrix3d::Zero(); //初始化為零1.2.4 定義動態(tài)矩陣
// 如果不確定矩陣大小，可以使用動態(tài)大小的矩陣Matrix<double, Dynamic, Dynamic> matrix_dynamic;// 更簡單的MatrixXd matrix_x;// 這種類型還有很多，我們不一一列舉1.3 對Eigen操作
1.3.1 輸入數(shù)據(jù)
// 輸入數(shù)據(jù)（初始化）matrix_23 << 1, 2, 3, 4, 5, 6;1.3.2 矩陣整個輸出
// 輸出cout << "matrix 2x3 from 1 to 6: \n" << matrix_23 << endl;1.3.3 用()訪問矩陣中的元素
cout << "print matrix 2x3: " << endl;for (int i = 0; i < 2; i++) {for (int j = 0; j < 3; j++) cout << matrix_23(i, j) << "\t";cout << endl;}1.3.4 矩陣和向量相乘|類型轉換
v_3d << 3, 2, 1;vd_3d << 4, 5, 6;// 但是在Eigen里你不能混合兩種不同類型的矩陣，像這樣是錯的// Matrix<double, 2, 1> result_wrong_type = matrix_23 * v_3d;// 應該顯式轉換Matrix<double, 2, 1> result = matrix_23.cast<double>() * v_3d;cout << "[1,2,3;4,5,6]*[3,2,1]=" << result.transpose() << endl;//隱式轉換Matrix<float, 2, 1> result2 = matrix_23 * vd_3d;cout << "[1,2,3;4,5,6]*[4,5,6]: " << result2.transpose() << endl;1.3.5 得到矩陣的指定塊元素
int n_state = all_frame_count * 3 + 3 + 1;MatrixXd A{n_state, n_state};A.setZero();VectorXd b{n_state};b.setZero();MatrixXd r_A = tmp_A.transpose() * cov_inv * tmp_A;VectorXd r_b = tmp_A.transpose() * cov_inv * tmp_b;A.block<6, 6>(i * 3, i * 3) += r_A.topLeftCorner<6, 6>(); b.segment<6>(i * 3) += r_b.head<6>();A.bottomRightCorner<4, 4>() += r_A.bottomRightCorner<4, 4>();b.tail<4>() += r_b.tail<4>();1.4 矩陣運算（隨機數(shù)、轉置、各元素和、跡、數(shù)乘、逆、行列式、特征值、特特征向量、叉乘、點乘、求模、歸一化、“開根號”）
// 一些矩陣運算// 四則運算就不演示了，直接用+-*/即可。matrix_33 = Matrix3d::Random(); // 隨機數(shù)矩陣cout << "random matrix: \n" << matrix_33 << endl;cout << "transpose: \n" << matrix_33.transpose() << endl; // 轉置cout << "sum: " << matrix_33.sum() << endl; // 各元素和cout << "trace: " << matrix_33.trace() << endl; // 跡cout << "times 10: \n" << 10 * matrix_33 << endl; // 數(shù)乘cout << "inverse: \n" << matrix_33.inverse() << endl; // 逆cout << "det: " << matrix_33.determinant() << endl; // 行列式 // 特征值// 實對稱矩陣可以保證對角化成功SelfAdjointEigenSolver<Matrix3d> eigen_solver(matrix_33.transpose() * matrix_33);cout << "Eigen values = \n" << eigen_solver.eigenvalues() << endl;cout << "Eigen vectors = \n" << eigen_solver.eigenvectors() << endl; matrix1_33 = Matrix3d::Random(); // 隨機數(shù)矩陣matrix2_33 = Matrix3d::Random(); // 隨機數(shù)矩陣matrix3_33 = matrix1_33.cross(matrix2_33);//叉乘matrix4_33 = matrix1_33.dot(matrix2_33);//叉乘double mo = matrix3_33.norm();//求模 Eigen::Vector3d ljm_norm; ljm_norm.normalize();//向量的歸一化 //開根號Eigen::VectorXd S ;Eigen::VectorXd S_sqrt = S.cwiseSqrt(); // 這個求得就是 S^(1/2)，不過這里是向量還不是矩陣1.5 解方程
初始化求解方程：
// 解方程// 我們求解 matrix_NN * x = v_Nd 這個方程// N的大小在前邊的宏里定義，它由隨機數(shù)生成// 直接求逆自然是最直接的，但是求逆運算量大Matrix<double, MATRIX_SIZE, MATRIX_SIZE> matrix_NN= MatrixXd::Random(MATRIX_SIZE, MATRIX_SIZE);matrix_NN = matrix_NN * matrix_NN.transpose(); // 保證半正定Matrix<double, MATRIX_SIZE, 1> v_Nd = MatrixXd::Random(MATRIX_SIZE, 1);1.5.1 直接求逆
clock_t time_stt = clock(); // 計時// 直接求逆Matrix<double, MATRIX_SIZE, 1> x = matrix_NN.inverse() * v_Nd;cout << "time of normal inverse is "<< 1000 * (clock() - time_stt) / (double) CLOCKS_PER_SEC << "ms" << endl;cout << "x = " << x.transpose() << endl;1.5.2 QR分解
// 通常用矩陣分解來求，例如QR分解，速度會快很多time_stt = clock();x = matrix_NN.colPivHouseholderQr().solve(v_Nd);cout << "time of Qr decomposition is "<< 1000 * (clock() - time_stt) / (double) CLOCKS_PER_SEC << "ms" << endl;cout << "x = " << x.transpose() << endl;1.5.3 cholesky分解
- cholesky分解需要是正定矩陣
1.5.4 SVD分解
Eigen::Matrix3d W = Eigen::Matrix3d::Zero();//初始化...Eigen::JacobiSVD<Eigen::Matrix3d> svd(W,Eigen::ComputeFullU | Eigen::ComputeFullV);Eigen::Matrix3d U = svd.matrixU();//得到U矩陣Eigen::Matrix3d V = svd.matrixV();//得到V矩陣1.6 旋轉和平移的表示、轉換和使用
1.6.0 數(shù)據(jù)類型匯總
- 每種都有double 和float類型，把后面的d改為f就可以了
1.6.1 旋轉矩陣
1.6.1.1 初始化
// 3D 旋轉矩陣直接使用 Matrix3d 或 Matrix3fMatrix3d rotation_matrix = Matrix3d::Identity();1.6.1.2 坐標變換
Vector3d v(1, 0, 0);// 或者用旋轉矩陣v_rotated = rotation_matrix * v;cout << "(1,0,0) after rotation (by matrix) = " << v_rotated.transpose() << endl;輸出：
(1,0,0) after rotation (by matrix) = 0.707 0.707 01.6.2.3 旋轉向量->旋轉矩陣
cout.precision(3);cout << "rotation matrix =\n" << rotation_vector.matrix() << endl; //用matrix()轉換成矩陣輸出：
rotation matrix =0.707 -0.707 00.707 0.707 00 0 1 // 也可以直接賦值rotation_matrix = rotation_vector.toRotationMatrix();同1
1.6.2 旋轉向量
1.6.2.1 初始化
// 旋轉向量使用 AngleAxis, 它底層不直接是Matrix，//但運算可以當作矩陣（因為重載了運算符）//沿 Z 軸旋轉 45 度AngleAxisd rotation_vector(M_PI / 4, Vector3d(0, 0, 1));1.6.2.3 坐標變換
// 用 AngleAxis 可以進行坐標變換Vector3d v(1, 0, 0);Vector3d v_rotated = rotation_vector * v;cout << "(1,0,0) after rotation (by angle axis) = " << v_rotated.transpose() << endl;輸出：
(1,0,0) after rotation (by angle axis) = 0.707 0.707 01.6.3 歐拉角
1.6.3.1 初始化
1.6.3.2 旋轉矩陣->歐拉角
// 歐拉角: 可以將旋轉矩陣直接轉換成歐拉角// ZYX順序，即roll pitch yaw順序Vector3d euler_angles = rotation_matrix.eulerAngles(2, 1, 0); cout << "yaw pitch roll = " << euler_angles.transpose() << endl;輸出：
yaw pitch roll = 0.785 -0 01.6.4 歐式變換矩陣se
1.6.4.1 初始化
// 歐氏變換矩陣使用 Eigen::IsometryIsometry3d T = Isometry3d::Identity(); // 雖然稱為3d，實質(zhì)上是4＊4的矩陣對于仿射和射影變換，使用 Eigen::Affine3d 和 Eigen::Projective3d 即可，略
1.6.4.2 旋轉向量->se.T|se->旋轉矩陣
T.rotate(rotation_vector); // 按照rotation_vector進行旋轉T.pretranslate(Vector3d(1, 3, 4)); // 把平移向量設成(1,3,4)cout << "Transform matrix = \n" << T.matrix() << endl;輸出：
Transform matrix = 0.707 -0.707 0 10.707 0.707 0 30 0 1 40 0 0 11.6.4.3 四元數(shù)->se.T
Isometry3d T1w(q1), T2w(q2);1.6.4.4 坐標變換
// 用變換矩陣進行坐標變換Vector3d v_transformed = T * v; // 相當于R*v+tcout << "v tranformed = " << v_transformed.transpose() << endl;輸出：
v tranformed = 1.71 3.71 41.6.5 四元數(shù)
1.6.5.1 初始化
- 使用四元數(shù)進行坐標變換之前要進行歸一化處理q.normalize()
初始化是w在前
1.6.5.2 旋轉向量->四元數(shù)
// 四元數(shù)// 可以直接把AngleAxis賦值給四元數(shù)，反之亦然Quaterniond q = Quaterniond(rotation_vector);cout << "quaternion from rotation vector = " << q.coeffs().transpose()<< endl; // 請注意coeffs的順序是(x,y,z,w),w為實部，前三者為虛部輸出：
quaternion from rotation vector = 0 0 0.383 0.9241.6.5.3 旋轉矩陣->四元數(shù)
- q = rotation_matrix;好像直接這樣也可以，不加Quaterniond()，加了更加清晰
輸出：
quaternion from rotation matrix = 0 0 0.383 0.9241.6.5.4 坐標變換
// 使用四元數(shù)旋轉一個向量，使用重載的乘法即可v_rotated = q * v; // 注意數(shù)學上是qvq^{-1}cout << "(1,0,0) after rotation = " << v_rotated.transpose() << endl;// 用常規(guī)向量乘法表示，則應該如下計算cout << "should be equal to " << (q * Quaterniond(0, 1, 0, 0) * q.inverse()).coeffs().transpose() << endl;輸出：
(1,0,0) after rotation = 0.707 0.707 0 should be equal to 0.707 0.707 0 01.6.5.5 取虛部q.vec()
Eigen::Quaterniond corrected_delta_q ,Qi,Qj;Eigen::Matrix<double, 15, 1> residuals;residuals.block<3, 1>(O_R, 0) = 2 * (corrected_delta_q.inverse() * (Qi.inverse() * Qj)).vec();2 slam14的cap3的useigen
2.1 eigenMatrix.cpp
#include <iostream>using namespace std;#include <ctime> // Eigen 核心部分 #include <Eigen/Core> // 稠密矩陣的代數(shù)運算（逆，特征值等） #include <Eigen/Dense>using namespace Eigen;#define MATRIX_SIZE 50/**************************** * 本程序演示了 Eigen 基本類型的使用 ****************************/int main(int argc, char **argv) {// Eigen 中所有向量和矩陣都是Eigen::Matrix，它是一個模板類。它的前三個參數(shù)為：數(shù)據(jù)類型，行，列// 聲明一個2*3的float矩陣Matrix<float, 2, 3> matrix_23;// 同時，Eigen 通過 typedef 提供了許多內(nèi)置類型，不過底層仍是Eigen::Matrix// 例如 Vector3d 實質(zhì)上是 Eigen::Matrix<double, 3, 1>，即三維向量Vector3d v_3d;// 這是一樣的Matrix<float, 3, 1> vd_3d;// Matrix3d 實質(zhì)上是 Eigen::Matrix<double, 3, 3>Matrix3d matrix_33 = Matrix3d::Zero(); //初始化為零// 如果不確定矩陣大小，可以使用動態(tài)大小的矩陣Matrix<double, Dynamic, Dynamic> matrix_dynamic;// 更簡單的MatrixXd matrix_x;// 這種類型還有很多，我們不一一列舉// 下面是對Eigen陣的操作// 輸入數(shù)據(jù)（初始化）matrix_23 << 1, 2, 3, 4, 5, 6;// 輸出cout << "matrix 2x3 from 1 to 6: \n" << matrix_23 << endl;// 用()訪問矩陣中的元素cout << "print matrix 2x3: " << endl;for (int i = 0; i < 2; i++) {for (int j = 0; j < 3; j++) cout << matrix_23(i, j) << "\t";cout << endl;}// 矩陣和向量相乘（實際上仍是矩陣和矩陣）v_3d << 3, 2, 1;vd_3d << 4, 5, 6;// 但是在Eigen里你不能混合兩種不同類型的矩陣，像這樣是錯的// Matrix<double, 2, 1> result_wrong_type = matrix_23 * v_3d;// 應該顯式轉換Matrix<double, 2, 1> result = matrix_23.cast<double>() * v_3d;cout << "[1,2,3;4,5,6]*[3,2,1]=" << result.transpose() << endl;Matrix<float, 2, 1> result2 = matrix_23 * vd_3d;cout << "[1,2,3;4,5,6]*[4,5,6]: " << result2.transpose() << endl;// 同樣你不能搞錯矩陣的維度// 試著取消下面的注釋，看看Eigen會報什么錯// Eigen::Matrix<double, 2, 3> result_wrong_dimension = matrix_23.cast<double>() * v_3d;// 一些矩陣運算// 四則運算就不演示了，直接用+-*/即可。matrix_33 = Matrix3d::Random(); // 隨機數(shù)矩陣cout << "random matrix: \n" << matrix_33 << endl;cout << "transpose: \n" << matrix_33.transpose() << endl; // 轉置cout << "sum: " << matrix_33.sum() << endl; // 各元素和cout << "trace: " << matrix_33.trace() << endl; // 跡cout << "times 10: \n" << 10 * matrix_33 << endl; // 數(shù)乘cout << "inverse: \n" << matrix_33.inverse() << endl; // 逆cout << "det: " << matrix_33.determinant() << endl; // 行列式// 特征值// 實對稱矩陣可以保證對角化成功SelfAdjointEigenSolver<Matrix3d> eigen_solver(matrix_33.transpose() * matrix_33);cout << "Eigen values = \n" << eigen_solver.eigenvalues() << endl;cout << "Eigen vectors = \n" << eigen_solver.eigenvectors() << endl;// 解方程// 我們求解 matrix_NN * x = v_Nd 這個方程// N的大小在前邊的宏里定義，它由隨機數(shù)生成// 直接求逆自然是最直接的，但是求逆運算量大Matrix<double, MATRIX_SIZE, MATRIX_SIZE> matrix_NN= MatrixXd::Random(MATRIX_SIZE, MATRIX_SIZE);matrix_NN = matrix_NN * matrix_NN.transpose(); // 保證半正定Matrix<double, MATRIX_SIZE, 1> v_Nd = MatrixXd::Random(MATRIX_SIZE, 1);clock_t time_stt = clock(); // 計時// 直接求逆Matrix<double, MATRIX_SIZE, 1> x = matrix_NN.inverse() * v_Nd;cout << "time of normal inverse is "<< 1000 * (clock() - time_stt) / (double) CLOCKS_PER_SEC << "ms" << endl;cout << "x = " << x.transpose() << endl;// 通常用矩陣分解來求，例如QR分解，速度會快很多time_stt = clock();x = matrix_NN.colPivHouseholderQr().solve(v_Nd);cout << "time of Qr decomposition is "<< 1000 * (clock() - time_stt) / (double) CLOCKS_PER_SEC << "ms" << endl;cout << "x = " << x.transpose() << endl;// 對于正定矩陣，還可以用cholesky分解來解方程time_stt = clock();x = matrix_NN.ldlt().solve(v_Nd);cout << "time of ldlt decomposition is "<< 1000 * (clock() - time_stt) / (double) CLOCKS_PER_SEC << "ms" << endl;cout << "x = " << x.transpose() << endl;return 0; }2.2 CMakeLists.txt
cmake_minimum_required(VERSION 2.8)project(useEigen)set(CMAKE_BUILD_TYPE "Release") set(CMAKE_CXX_STANDARD 11)include_directories("/usr/local/include/eigen3") add_executable(eigenMatrix src/eigenMatrix.cpp)2.3 輸出
matrix 2x3 from 1 to 6: 1 2 3 4 5 6 print matrix 2x3: 1 2 3 4 5 6 [1,2,3;4,5,6]*[3,2,1]=10 28 [1,2,3;4,5,6]*[4,5,6]: 32 77 random matrix: 0.680375 0.59688 -0.329554 -0.211234 0.823295 0.5364590.566198 -0.604897 -0.444451 transpose: 0.680375 -0.211234 0.5661980.59688 0.823295 -0.604897 -0.329554 0.536459 -0.444451 sum: 1.61307 trace: 1.05922 times 10: 6.80375 5.9688 -3.29554 -2.11234 8.23295 5.364595.66198 -6.04897 -4.44451 inverse: -0.198521 2.22739 2.83571.00605 -0.555135 -1.41603-1.62213 3.59308 3.28973 det: 0.208598 Eigen values = 0.02428990.9921541.80558 Eigen vectors = -0.549013 -0.735943 0.3961980.253452 -0.598296 -0.760134 -0.796459 0.316906 -0.514998 time of normal inverse is 0.073ms x = -55.7896 -298.793 130.113 -388.455 -159.312 160.654 -40.0416 -193.561 155.844 181.144 185.125 -62.7786 19.8333 -30.8772 -200.746 55.8385 -206.604 26.3559 -14.6789 122.719 -221.449 26.233 -318.95 -78.6931 50.1446 87.1986 -194.922 132.319 -171.78 -4.19736 11.876 -171.779 48.3047 84.1812 -104.958 -47.2103 -57.4502 -48.9477 -19.4237 28.9419 111.421 92.1237 -288.248 -23.3478 -275.22 -292.062 -92.698 5.96847 -93.6244 109.734 time of Qr decomposition is 0.043ms x = -55.7896 -298.793 130.113 -388.455 -159.312 160.654 -40.0416 -193.561 155.844 181.144 185.125 -62.7786 19.8333 -30.8772 -200.746 55.8385 -206.604 26.3559 -14.6789 122.719 -221.449 26.233 -318.95 -78.6931 50.1446 87.1986 -194.922 132.319 -171.78 -4.19736 11.876 -171.779 48.3047 84.1812 -104.958 -47.2103 -57.4502 -48.9477 -19.4237 28.9419 111.421 92.1237 -288.248 -23.3478 -275.22 -292.062 -92.698 5.96847 -93.6244 109.734 time of ldlt decomposition is 0.018ms x = -55.7896 -298.793 130.113 -388.455 -159.312 160.654 -40.0416 -193.561 155.844 181.144 185.125 -62.7786 19.8333 -30.8772 -200.746 55.8385 -206.604 26.3559 -14.6789 122.719 -221.449 26.233 -318.95 -78.6931 50.1446 87.1986 -194.922 132.319 -171.78 -4.19736 11.876 -171.779 48.3047 84.1812 -104.958 -47.2103 -57.4502 -48.9477 -19.4237 28.9419 111.421 92.1237 -288.248 -23.3478 -275.22 -292.062 -92.698 5.96847 -93.6244 109.734進程已結束,退出代碼03 slam14的cap3的useigen
3.1 useGeometry.cpp
#include <iostream> #include <cmath>using namespace std;#include <Eigen/Core> #include <Eigen/Geometry>using namespace Eigen;// 本程序演示了 Eigen 幾何模塊的使用方法int main(int argc, char **argv) {// Eigen/Geometry 模塊提供了各種旋轉和平移的表示// 3D 旋轉矩陣直接使用 Matrix3d 或 Matrix3fMatrix3d rotation_matrix = Matrix3d::Identity();// 旋轉向量使用 AngleAxis, 它底層不直接是Matrix，但運算可以當作矩陣（因為重載了運算符）AngleAxisd rotation_vector(M_PI / 4, Vector3d(0, 0, 1)); //沿 Z 軸旋轉 45 度cout.precision(3);cout << "rotation matrix =\n" << rotation_vector.matrix() << endl; //用matrix()轉換成矩陣// 也可以直接賦值rotation_matrix = rotation_vector.toRotationMatrix();//cout << "rotation matrix =\n" << rotation_matrix << endl;// 用 AngleAxis 可以進行坐標變換Vector3d v(1, 0, 0);Vector3d v_rotated = rotation_vector * v;cout << "(1,0,0) after rotation (by angle axis) = " << v_rotated.transpose() << endl;// 或者用旋轉矩陣v_rotated = rotation_matrix * v;cout << "(1,0,0) after rotation (by matrix) = " << v_rotated.transpose() << endl;// 歐拉角: 可以將旋轉矩陣直接轉換成歐拉角Vector3d euler_angles = rotation_matrix.eulerAngles(2, 1, 0); // ZYX順序，即roll pitch yaw順序cout << "yaw pitch roll = " << euler_angles.transpose() << endl;// 歐氏變換矩陣使用 Eigen::IsometryIsometry3d T = Isometry3d::Identity(); // 雖然稱為3d，實質(zhì)上是4＊4的矩陣T.rotate(rotation_vector); // 按照rotation_vector進行旋轉T.pretranslate(Vector3d(1, 3, 4)); // 把平移向量設成(1,3,4)cout << "Transform matrix = \n" << T.matrix() << endl;// 用變換矩陣進行坐標變換Vector3d v_transformed = T * v; // 相當于R*v+tcout << "v tranformed = " << v_transformed.transpose() << endl;// 對于仿射和射影變換，使用 Eigen::Affine3d 和 Eigen::Projective3d 即可，略// 四元數(shù)// 可以直接把AngleAxis賦值給四元數(shù)，反之亦然Quaterniond q = Quaterniond(rotation_vector);cout << "quaternion from rotation vector = " << q.coeffs().transpose()<< endl; // 請注意coeffs的順序是(x,y,z,w),w為實部，前三者為虛部// 也可以把旋轉矩陣賦給它q = Quaterniond(rotation_matrix);cout << "quaternion from rotation matrix = " << q.coeffs().transpose() << endl;// 使用四元數(shù)旋轉一個向量，使用重載的乘法即可v_rotated = q * v; // 注意數(shù)學上是qvq^{-1}cout << "(1,0,0) after rotation = " << v_rotated.transpose() << endl;// 用常規(guī)向量乘法表示，則應該如下計算cout << "should be equal to " << (q * Quaterniond(0, 1, 0, 0) * q.inverse()).coeffs().transpose() << endl;return 0; }3.2 CMakeLists.txt
cmake_minimum_required(VERSION 2.8) project(geometry)include_directories("/usr/include/eigen3")add_executable(eigenFeometry src/useGeometry.cpp)3.3 輸出
rotation matrix =0.707 -0.707 00.707 0.707 00 0 1 (1,0,0) after rotation (by angle axis) = 0.707 0.707 0 (1,0,0) after rotation (by matrix) = 0.707 0.707 0 yaw pitch roll = 0.785 -0 0 Transform matrix = 0.707 -0.707 0 10.707 0.707 0 30 0 1 40 0 0 1 v tranformed = 1.71 3.71 4 quaternion from rotation vector = 0 0 0.383 0.924 quaternion from rotation matrix = 0 0 0.383 0.924 (1,0,0) after rotation = 0.707 0.707 0 should be equal to 0.707 0.707 0 0進程已結束,退出代碼04 實際的坐標轉換例子
4.1 coordinateTransform.cpp
#include <iostream> #include <vector> #include <algorithm> #include <Eigen/Core> #include <Eigen/Geometry>using namespace std; using namespace Eigen;int main(int argc, char ** argv) {Quaterniond q1(0.35, 0.2,0.3,0.1), q2(-0.5,0.4,-0.1,0.2);q1.normalize();q2.normalize();Vector3d t1(0.3,0.1,0.1),t2(-0.1,0.5,0.3);Vector3d p1(0.5,0,0.2);Isometry3d T1w(q1), T2w(q2);T1w.pretranslate(t1);T2w.pretranslate(t2);Vector3d p2 = T2w*T1w.inverse()*p1;cout << endl << p2.transpose() << endl;return 0; }4.2 CMakeLists.txt
cmake_minimum_required(VERSION 2.8) project(coordinateTransform)include_directories("/usr/include/eigen3")add_executable(coordinatetransform src/coordinateTransform.cpp)4.3 輸出
-0.0309731 0.73499 0.2961085 顯示運動軌跡
- 該歷程需要一個trajectory.txt的位姿數(shù)據(jù)，自取：鏈接: https://pan.baidu.com/s/1ITzeIl_DsG2ckMU-eSEnhg 提取碼: m35v
5.1 plotTrajectory.cpp
#include <pangolin/pangolin.h> #include <Eigen/Core> #include <unistd.h>// 本例演示了如何畫出一個預先存儲的軌跡using namespace std; using namespace Eigen;// path to trajectory file string trajectory_file = "../src/trajectory.txt";void DrawTrajectory(vector<Isometry3d, Eigen::aligned_allocator<Isometry3d>>);int main(int argc, char **argv) {vector<Isometry3d, Eigen::aligned_allocator<Isometry3d>> poses;ifstream fin(trajectory_file);if (!fin) {cout << "cannot find trajectory file at " << trajectory_file << endl;return 1;}while (!fin.eof()) {double time, tx, ty, tz, qx, qy, qz, qw;fin >> time >> tx >> ty >> tz >> qx >> qy >> qz >> qw;Isometry3d Twr(Quaterniond(qw, qx, qy, qz));Twr.pretranslate(Vector3d(tx, ty, tz));poses.push_back(Twr);}cout << "read total " << poses.size() << " pose entries" << endl;// draw trajectory in pangolinDrawTrajectory(poses);return 0; }/*******************************************************************************************/ void DrawTrajectory(vector<Isometry3d, Eigen::aligned_allocator<Isometry3d>> poses) {// create pangolin window and plot the trajectorypangolin::CreateWindowAndBind("Trajectory Viewer", 1024, 768);glEnable(GL_DEPTH_TEST);glEnable(GL_BLEND);glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA);pangolin::OpenGlRenderState s_cam(pangolin::ProjectionMatrix(1024, 768, 500, 500, 512, 389, 0.1, 1000),pangolin::ModelViewLookAt(0, -0.1, -1.8, 0, 0, 0, 0.0, -1.0, 0.0));pangolin::View &d_cam = pangolin::CreateDisplay().SetBounds(0.0, 1.0, 0.0, 1.0, -1024.0f / 768.0f).SetHandler(new pangolin::Handler3D(s_cam));while (pangolin::ShouldQuit() == false) {glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);d_cam.Activate(s_cam);glClearColor(1.0f, 1.0f, 1.0f, 1.0f);glLineWidth(2);for (size_t i = 0; i < poses.size(); i++) {// 畫每個位姿的三個坐標軸Vector3d Ow = poses[i].translation();Vector3d Xw = poses[i] * (0.1 * Vector3d(1, 0, 0));Vector3d Yw = poses[i] * (0.1 * Vector3d(0, 1, 0));Vector3d Zw = poses[i] * (0.1 * Vector3d(0, 0, 1));glBegin(GL_LINES);glColor3f(1.0, 0.0, 0.0);glVertex3d(Ow[0], Ow[1], Ow[2]);glVertex3d(Xw[0], Xw[1], Xw[2]);glColor3f(0.0, 1.0, 0.0);glVertex3d(Ow[0], Ow[1], Ow[2]);glVertex3d(Yw[0], Yw[1], Yw[2]);glColor3f(0.0, 0.0, 1.0);glVertex3d(Ow[0], Ow[1], Ow[2]);glVertex3d(Zw[0], Zw[1], Zw[2]);glEnd();}// 畫出連線for (size_t i = 0; i < poses.size(); i++) {glColor3f(0.0, 0.0, 0.0);glBegin(GL_LINES);auto p1 = poses[i], p2 = poses[i + 1];glVertex3d(p1.translation()[0], p1.translation()[1], p1.translation()[2]);glVertex3d(p2.translation()[0], p2.translation()[1], p2.translation()[2]);glEnd();}pangolin::FinishFrame();usleep(5000); // sleep 5 ms} }5.2 CMakeLists.txt
cmake_minimum_required(VERSION 2.8)project(plotTrajectory)include_directories("/usr/include/eigen3") find_package(Pangolin REQUIRED) include_directories(${Pangolin_INCLUDE_DIRS})add_executable(plottrajectory src/plotTrajectory.cpp) target_link_libraries(plottrajectory ${Pangolin_LIBRARIES})5.3 輸出
6 顯示相機的位姿
6.1 visualizeGeometry.cpp
#include <iostream> #include <iomanip>using namespace std;#include <Eigen/Core> #include <Eigen/Geometry>using namespace Eigen;#include <pangolin/pangolin.h>struct RotationMatrix {Matrix3d matrix = Matrix3d::Identity(); };ostream &operator<<(ostream &out, const RotationMatrix &r) {out.setf(ios::fixed);Matrix3d matrix = r.matrix;out << '=';out << "[" << setprecision(2) << matrix(0, 0) << "," << matrix(0, 1) << "," << matrix(0, 2) << "],"<< "[" << matrix(1, 0) << "," << matrix(1, 1) << "," << matrix(1, 2) << "],"<< "[" << matrix(2, 0) << "," << matrix(2, 1) << "," << matrix(2, 2) << "]";return out; }istream &operator>>(istream &in, RotationMatrix &r) {return in; }struct TranslationVector {Vector3d trans = Vector3d(0, 0, 0); };ostream &operator<<(ostream &out, const TranslationVector &t) {out << "=[" << t.trans(0) << ',' << t.trans(1) << ',' << t.trans(2) << "]";return out; }istream &operator>>(istream &in, TranslationVector &t) {return in; }struct QuaternionDraw {Quaterniond q; };ostream &operator<<(ostream &out, const QuaternionDraw quat) {auto c = quat.q.coeffs();out << "=[" << c[0] << "," << c[1] << "," << c[2] << "," << c[3] << "]";return out; }istream &operator>>(istream &in, const QuaternionDraw quat) {return in; }int main(int argc, char **argv) {pangolin::CreateWindowAndBind("visualize geometry", 1000, 600);glEnable(GL_DEPTH_TEST);pangolin::OpenGlRenderState s_cam(pangolin::ProjectionMatrix(1000, 600, 420, 420, 500, 300, 0.1, 1000),pangolin::ModelViewLookAt(3, 3, 3, 0, 0, 0, pangolin::AxisY));const int UI_WIDTH = 500;pangolin::View &d_cam = pangolin::CreateDisplay().SetBounds(0.0, 1.0, pangolin::Attach::Pix(UI_WIDTH), 1.0, -1000.0f / 600.0f).SetHandler(new pangolin::Handler3D(s_cam));// uipangolin::Var<RotationMatrix> rotation_matrix("ui.R", RotationMatrix());pangolin::Var<TranslationVector> translation_vector("ui.t", TranslationVector());pangolin::Var<TranslationVector> euler_angles("ui.rpy", TranslationVector());pangolin::Var<QuaternionDraw> quaternion("ui.q", QuaternionDraw());pangolin::CreatePanel("ui").SetBounds(0.0, 1.0, 0.0, pangolin::Attach::Pix(UI_WIDTH));while (!pangolin::ShouldQuit()) {glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);d_cam.Activate(s_cam);pangolin::OpenGlMatrix matrix = s_cam.GetModelViewMatrix();Matrix<double, 4, 4> m = matrix;RotationMatrix R;for (int i = 0; i < 3; i++)for (int j = 0; j < 3; j++)R.matrix(i, j) = m(j, i);rotation_matrix = R;TranslationVector t;t.trans = Vector3d(m(0, 3), m(1, 3), m(2, 3));t.trans = -R.matrix * t.trans;translation_vector = t;TranslationVector euler;euler.trans = R.matrix.eulerAngles(2, 1, 0);euler_angles = euler;QuaternionDraw quat;quat.q = Quaterniond(R.matrix);quaternion = quat;glColor3f(1.0, 1.0, 1.0);pangolin::glDrawColouredCube();// draw the original axisglLineWidth(3);glColor3f(0.8f, 0.f, 0.f);glBegin(GL_LINES);glVertex3f(0, 0, 0);glVertex3f(10, 0, 0);glColor3f(0.f, 0.8f, 0.f);glVertex3f(0, 0, 0);glVertex3f(0, 10, 0);glColor3f(0.2f, 0.2f, 1.f);glVertex3f(0, 0, 0);glVertex3f(0, 0, 10);glEnd();pangolin::FinishFrame();} }6.2 CMakeLists.txt
cmake_minimum_required( VERSION 2.8 ) project( visualizeGeometry )set(CMAKE_CXX_STANDRAD 14)# 添加Eigen頭文件 include_directories( "/usr/include/eigen3" )# 添加Pangolin依賴 find_package( Pangolin ) include_directories( ${Pangolin_INCLUDE_DIRS} )add_executable( visualizeGeometry src/visualizeGeometry.cpp ) target_link_libraries( visualizeGeometry ${Pangolin_LIBRARIES} )6.3 輸出
7 使用QR和Cholesly分解求解示例
7.0 說明：自創(chuàng)建100大小的動態(tài)矩陣，并使用QR和Cholesly分解求解
- 這是salm14的第第二節(jié)課習題的2.5題
7.1 qiujie.cpp
#include <iostream> using namespace std;#include <ctime> #include <Eigen/Core> #include <Eigen/Dense>#define MATRIX_SIZE 100int main(int argc, char** argv) {Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic> A;A = Eigen::MatrixXd::Random(MATRIX_SIZE,MATRIX_SIZE);A = A.transpose() * A;//喬利斯基分解需要正定矩陣Eigen::Matrix<double, Eigen::Dynamic,1> B;B = Eigen::MatrixXd::Random(MATRIX_SIZE, 1);Eigen::Matrix<double, Eigen::Dynamic,1> X;X = A.llt().solve(B);cout << "Cholesly's = " << X << endl;X = A.colPivHouseholderQr().solve(B);cout << "QR's = " << X << endl;return 0;}7.2 CMakeLists.txt
CMAKE_MINIMUM_REQUIRED( VERSION 2.8) PROJECT( qiujie )set( CMAKE_BUILD_TYPE "Release" )INCLUDE_DIRECTORIES( "/usr/include/eigen3" )ADD_EXECUTABLE( qiujie src/qiujie.cpp )總結
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