使用eigen库进行空间变换

在三维空间中，常常需要变换当前机器人的位姿计算定义的绝对坐标系和当前机器人所处相对坐标系之间的关系。而主要的变换则是平移和旋转，有时候可能需要尺度变换，那么就可以描述为：

Transform<float,3,Affine> T = Translation3f(p) * AngleAxisf(a,axis) * Scaling(s);

而Affine3d T是一个4*4齐次矩阵变换。

旋转

Eigen::Quaternionf quater;

可以采用沿着某一个轴进行计算，

Eigen::AngleAxisd(M_PI/2.0，Eigen::Vector3d::UnitZ());

也可以直接给定参数

quater.x() = 0;
quater.y() = 0;
quater.z() = sin(M_PI/2.0 / 2.0);
quater.w() = cos(M_PI/2.0 / 2.0);

平移

Eigen::Translation3f translation(x,y,z);

尺度缩放

Scaling(sx, sy)
Scaling(sx, sy, sz)
Scaling(s)
Scaling(vecN)

仿射变换

//仿射变换矩阵
Eigen::Affine3f affine3f = translation*quater.toRotationMatrix();

//求逆矩阵
affine3f = affine3f.inverse();

对三维点进行变换

//变换前的点
Eigen::Vector3f v3f_a(x_m, y_m, 0.0);

//变换后的点
Eigen::Vector3f v3f_b = affine3f*v3f_a;

eigen空间变换详细文档，以便查找其他函数的使用方法。有时候需要知道affine3f的四维矩阵Matrix4f，采用如下装换

Eigen::Matrix4f a;
Eigen::Affine3f b;
b.matrix() = a;

Eigen库使用教程

Multiplication:

Eigen::Matrix3d A = Eigen::Matrix3d::Identity();

Eigen::Vector3d a(0.5, 3, -0.4);

Eigen::Vector3d Aa = A * a;

std::cout << "The multiplication of A * a is " << std::endl << Aa << std::endl;

Eigen::MatrixXd B = Eigen::MatrixXd::Identity(6, 5);

Eigen::VectorXd b(5);

b << 1, 4, 6, -2, 0.4;

Eigen::VectorXd Bb = B * b;

std::cout << "The multiplication of B * b is " << std::endl << Bb << std::endl;

Transpose and inverse:

Eigen::MatrixXd A(3, 2);

A << 1, 2,

2, 3,

3, 4;

Eigen::MatrixXd B = A.transpose();// the transpose of A is a 2x3 matrix

Eigen::MatrixXd C = (B * A).inverse();// computer the inverse of BA, which is a 2x2 matrix

Dot product and cross product:

Eigen::Vector3d v(1, 2, 3);

Eigen::Vector3d w(0, 1, 2);

double vDotw = v.dot(w); // dot product of two vectors

Eigen::Vector3d vCrossw = v.cross(w); // cross product of two vectors

Accessing matrix:

Eigen::MatrixXd A = Eigen::MatrixXd::Random(7, 9);

std::cout << "The element at fourth row and 7the column is " << A(3, 6) << std::endl;

Eigen::MatrixXd B = A.block(1, 2, 3, 3);

std::cout << "Take sub-matrix whose upper left corner is A(1, 2)" << std::endl << B << std::endl;

Eigen::VectorXd a = A.col(1); // take the second column of A

Eigen::VectorXd b = B.row(0); // take the first row of B

Eigen::VectorXd c = a.head(3);// take the first three elements of a

Eigen::VectorXd d = b.tail(2);// take the last two elements of b

Quaternion:

Eigen::Quaterniond q(2, 0, 1, -3); 

std::cout << "This quaternion consists of a scalar " << q.w() << " and a vector " << std::endl << q.vec() << std::endl;

q.normalize();

std::cout << "To represent rotation, we need to normalize it such that its length is " << q.norm() << std::endl;

Eigen::Vector3d v(1, 2, -1);

Eigen::Quaterniond p;

p.w() = 0;

p.vec() = v;

Eigen::Quaterniond rotatedP = q * p * q.inverse(); 

Eigen::Vector3d rotatedV = rotatedP.vec();

std::cout << "We can now use it to rotate a vector " << std::endl << v << " to " << std::endl << rotatedV << std::endl;

Eigen::Matrix3d R = q.toRotationMatrix(); // convert a quaternion to a 3x3 rotation matrix

std::cout << "Compare with the result using an rotation matrix " << std::endl << R * v << std::endl;

Eigen::Quaterniond a = Eigen::Quterniond::Identity();

Eigen::Quaterniond b = Eigen::Quterniond::Identity();

Eigen::Quaterniond c; // Adding two quaternion as two 4x1 vectors is not supported by the EIgen API. That is, c = a + b is not allowed. We have to do this in a hard way

c.w() = a.w() + b.w();

c.x() = a.x() + b.x();

c.y() = a.y() + b.y();

c.z() = a.z() + b.z();

cs4496 Computer Animation 使用eigen 教程。