虽迟但到，手眼标定代码实现篇

你好，我是小鱼。今天周末，在小仙女带领下，剪了个帅气发型。今天说说手眼标定的代码实现。

之前介绍过手眼标定算法Tsai的原理，今天介绍算法的代码实现，分别有Python、C++、Matlab版本的算法实现方式。

该算法适用于将相机装在手抓上和将相机装在外部两种情况
论文已经传到git上，地址：https://gitee.com/ohhuo/handeye-tsai

Python版本

使用前需要安装库：

pip3 install transforms3dpip3 install numpy

#!/usr/bin/env python
# coding: utf-8
import transforms3d as tfs
import numpy as np
import math

def get_matrix_eular_radu(x,y,z,rx,ry,rz):rmat = tfs.euler.euler2mat(math.radians(rx),math.radians(ry),math.radians(rz))rmat = tfs.affines.compose(np.squeeze(np.asarray((x,y,z))), rmat, [1, 1, 1])return rmat

def skew(v):return np.array([[0,-v[2],v[1]],[v[2],0,-v[0]],[-v[1],v[0],0]])

def rot2quat_minimal(m):quat =  tfs.quaternions.mat2quat(m[0:3,0:3])return quat[1:]

def quatMinimal2rot(q):p = np.dot(q.T,q)w = np.sqrt(np.subtract(1,p[0][0]))return tfs.quaternions.quat2mat([w,q[0],q[1],q[2]])

hand = [1.1988093940033604, -0.42405585264804424, 0.18828251788562061, 151.3390418721659, -18.612399542280507, 153.05074895025035,1.1684831621733476, -0.183273375514656, 0.12744868246620855, -161.57083804238462, 9.07159838346732, 89.1641128844487,1.1508343174145468, -0.22694301453461405, 0.26625166858469146, 177.8815855486261, 0.8991159570568988, 77.67286224959672]
camera = [-0.16249272227287292, -0.047310635447502136, 0.4077761471271515, -56.98037030812389, -6.16739631361851, -115.84333735802369,0.03955405578017235, -0.013497642241418362, 0.33975949883461, -100.87129330834215, -17.192685528625265, -173.07354634882094,-0.08517949283123016, 0.00957852229475975, 0.46546608209609985, -90.85270962096058, 0.9315977976503153, 175.2059707654342]


Hgs,Hcs = [],[]
for i in range(0,len(hand),6):Hgs.append(get_matrix_eular_radu(hand[i],hand[i+1],hand[i+2],hand[i+3],hand[i+4],hand[i+5]))    Hcs.append(get_matrix_eular_radu(camera[i],camera[i+1],camera[i+2],camera[i+3],camera[i+4],camera[i+5]))

Hgijs = []
Hcijs = []
A = []
B = []
size = 0
for i in range(len(Hgs)):for j in range(i+1,len(Hgs)):size += 1Hgij = np.dot(np.linalg.inv(Hgs[j]),Hgs[i])Hgijs.append(Hgij)Pgij = np.dot(2,rot2quat_minimal(Hgij))
Hcij = np.dot(Hcs[j],np.linalg.inv(Hcs[i]))Hcijs.append(Hcij)Pcij = np.dot(2,rot2quat_minimal(Hcij))
A.append(skew(np.add(Pgij,Pcij)))B.append(np.subtract(Pcij,Pgij))
MA = np.asarray(A).reshape(size*3,3)
MB = np.asarray(B).reshape(size*3,1)
Pcg_  =  np.dot(np.linalg.pinv(MA),MB)
pcg_norm = np.dot(np.conjugate(Pcg_).T,Pcg_)
Pcg = np.sqrt(np.add(1,np.dot(Pcg_.T,Pcg_)))
Pcg = np.dot(np.dot(2,Pcg_),np.linalg.inv(Pcg))
Rcg = quatMinimal2rot(np.divide(Pcg,2)).reshape(3,3)


A = []
B = []
id = 0
for i in range(len(Hgs)):for j in range(i+1,len(Hgs)):Hgij = Hgijs[id]Hcij = Hcijs[id]A.append(np.subtract(Hgij[0:3,0:3],np.eye(3,3)))B.append(np.subtract(np.dot(Rcg,Hcij[0:3,3:4]),Hgij[0:3,3:4]))id += 1

MA = np.asarray(A).reshape(size*3,3)
MB = np.asarray(B).reshape(size*3,1)
Tcg = np.dot(np.linalg.pinv(MA),MB).reshape(3,)
print(tfs.affines.compose(Tcg,np.squeeze(Rcg),[1,1,1]))

运行结果：

python3 tsai.py                             [[-0.01522186 -0.99983174 -0.01023609 -0.02079774] [ 0.99976822 -0.01506342 -0.01538198  0.00889827] [ 0.0152252  -0.01046786  0.99982929  0.08324514] [ 0.          0.          0.          1.        ]]

C++版本：

//Reference:
//R. Y. Tsai and R. K. Lenz, "A new technique for fully autonomous and efficient 3D robotics hand/eye calibration."
//In IEEE Transactions on Robotics and Automation, vol. 5, no. 3, pp. 345-358, June 1989.
//C++ code converted from Zoran Lazarevic's Matlab code:
//http://lazax.com/www.cs.columbia.edu/~laza/html/Stewart/matlab/handEye.m
static void calibrateHandEyeTsai(const std::vector<Mat>& Hg, const std::vector<Mat>& Hc,Mat& R_cam2gripper, Mat& t_cam2gripper)
{//Number of unique camera position pairsint K = static_cast<int>((Hg.size()*Hg.size() - Hg.size()) / 2.0);//Will store: skew(Pgij+Pcij)Mat A(3*K, 3, CV_64FC1);//Will store: Pcij - PgijMat B(3*K, 1, CV_64FC1);
std::vector<Mat> vec_Hgij, vec_Hcij;vec_Hgij.reserve(static_cast<size_t>(K));vec_Hcij.reserve(static_cast<size_t>(K));
int idx = 0;for (size_t i = 0; i < Hg.size(); i++){for (size_t j = i+1; j < Hg.size(); j++, idx++){//Defines coordinate transformation from Gi to Gj//Hgi is from Gi (gripper) to RW (robot base)//Hgj is from Gj (gripper) to RW (robot base)Mat Hgij = homogeneousInverse(Hg[j]) * Hg[i]; //eq 6vec_Hgij.push_back(Hgij);//Rotation axis for Rgij which is the 3D rotation from gripper coordinate frame Gi to GjMat Pgij = 2*rot2quatMinimal(Hgij);
//Defines coordinate transformation from Ci to Cj//Hci is from CW (calibration target) to Ci (camera)//Hcj is from CW (calibration target) to Cj (camera)Mat Hcij = Hc[j] * homogeneousInverse(Hc[i]); //eq 7vec_Hcij.push_back(Hcij);//Rotation axis for RcijMat Pcij = 2*rot2quatMinimal(Hcij);
//Left-hand side: skew(Pgij+Pcij)skew(Pgij+Pcij).copyTo(A(Rect(0, idx*3, 3, 3)));//Right-hand side: Pcij - PgijMat diff = Pcij - Pgij;diff.copyTo(B(Rect(0, idx*3, 1, 3)));}}


Mat Pcg_;//Rotation from camera to gripper is obtained from the set of equations://    skew(Pgij+Pcij) * Pcg_ = Pcij - Pgij    (eq 12)solve(A, B, Pcg_, DECOMP_SVD);
Mat Pcg_norm = Pcg_.t() * Pcg_;//Obtained non-unit quaternion is scaled back to unit value that//designates camera-gripper rotationMat Pcg = 2 * Pcg_ / sqrt(1 + Pcg_norm.at<double>(0,0)); //eq 14
Mat Rcg = quatMinimal2rot(Pcg/2.0);
idx = 0;for (size_t i = 0; i < Hg.size(); i++){for (size_t j = i+1; j < Hg.size(); j++, idx++){//Defines coordinate transformation from Gi to Gj//Hgi is from Gi (gripper) to RW (robot base)//Hgj is from Gj (gripper) to RW (robot base)Mat Hgij = vec_Hgij[static_cast<size_t>(idx)];//Defines coordinate transformation from Ci to Cj//Hci is from CW (calibration target) to Ci (camera)//Hcj is from CW (calibration target) to Cj (camera)Mat Hcij = vec_Hcij[static_cast<size_t>(idx)];
//Left-hand side: (Rgij - I)Mat diff = Hgij(Rect(0,0,3,3)) - Mat::eye(3,3,CV_64FC1);diff.copyTo(A(Rect(0, idx*3, 3, 3)));
//Right-hand side: Rcg*Tcij - Tgijdiff = Rcg*Hcij(Rect(3, 0, 1, 3)) - Hgij(Rect(3, 0, 1, 3));diff.copyTo(B(Rect(0, idx*3, 1, 3)));}}
Mat Tcg;//Translation from camera to gripper is obtained from the set of equations://    (Rgij - I) * Tcg = Rcg*Tcij - Tgij    (eq 15)solve(A, B, Tcg, DECOMP_SVD);
R_cam2gripper = Rcg;t_cam2gripper = Tcg;
}

C++版本食用方法：

终端指令

git clone https://gitee.com/ohhuo/handeye-tsai.git   cd handeye-tsai/cpp     mkdir build   cd buildcmake ..   make./opencv_example

示例：

sangxin@sangxin-ubu~ git clone https://gitee.com/ohhuo/handeye-tsai.git

正克隆到 'handeye-tsai'...
remote: Enumerating objects: 60, done.
remote: Counting objects: 100% (60/60), done.
remote: Compressing objects: 100% (57/57), done.
remote: Total 60 (delta 9), reused 0 (delta 0), pack-reused 0
展开对象中: 100% (60/60), 完成.

sangxin@sangxin-ubu~ cd handeye-tsai/cpp
sangxin@sangxin-ubu~ mkdir build
sangxin@sangxin-ubu~ cd build
sangxin@sangxin-ubu~ cmake ..

-- The C compiler identification is GNU 7.5.0
-- The CXX compiler identification is GNU 7.5.0
-- Check for working C compiler: /usr/bin/cc
-- Check for working C compiler: /usr/bin/cc -- works
-- Detecting C compiler ABI info
-- Detecting C compiler ABI info - done
-- Detecting C compile features
-- Detecting C compile features - done
-- Check for working CXX compiler: /usr/bin/c++
-- Check for working CXX compiler: /usr/bin/c++ -- works
-- Detecting CXX compiler ABI info
-- Detecting CXX compiler ABI info - done
-- Detecting CXX compile features
-- Detecting CXX compile features - done
-- Found OpenCV: /usr/local (found version "4.5.1")
-- OpenCV library status:
--     config: /usr/local/lib/cmake/opencv4
--     version: 4.5.1
--     libraries: opencv_calib3d;opencv_core;opencv_dnn;opencv_features2d;opencv_flann;opencv_gapi;opencv_highgui;opencv_imgcodecs;opencv_imgproc;opencv_ml;opencv_objdetect;opencv_photo;opencv_stitching;opencv_video;opencv_videoio
--     include path: /usr/local/include/opencv4
-- Configuring done
-- Generating done
-- Build files have been written to: /home/sangxin/code/ramp/other/handeye-tsai/cpp/build

sangxin@sangxin-ubu~ make

Scanning dependencies of target opencv_example
[ 33%] Building CXX object CMakeFiles/opencv_example.dir/example.cpp.o
[ 66%] Building CXX object CMakeFiles/opencv_example.dir/calibration_handeye.cpp.o
[100%] Linking CXX executable opencv_example
[100%] Built target opencv_example

sangxin@sangxin-ubu~ ./opencv_example

Hand eye calibration
[0.02534592279128711, -0.999507800830298, -0.01848621857599331, 0.03902588103574497;0.99953544041497, 0.02502485833258339, 0.01739712102291752, 0.002933439485668206;-0.01692594317342544, -0.01891857671220042, 0.9996777480282706, -0.01033683416650518;0, 0, 0, 1]
Homo_cam2gripper 是否包含旋转矩阵：1

Matlab版本：

% handEye - performs hand/eye calibration
%
%     gHc = handEye(bHg, wHc)
%
%     bHg - pose of gripper relative to the robot base..
%           (Gripper center is at: g0 = Hbg * [0;0;0;1] )
%           Matrix dimensions are 4x4xM, where M is ..
%           .. number of camera positions.
%           Algorithm gives a non-singular solution when ..
%           .. at least 3 positions are given
%           Hbg(:,:,i) is i-th homogeneous transformation matrix
%     wHc - pose of camera relative to the world ..
%           (relative to the calibration block)
%           Dimension: size(Hwc) = size(Hbg)
%     gHc - 4x4 homogeneous transformation from gripper to camera
%           , that is the camera position relative to the gripper.
%           Focal point of the camera is positioned, ..
%           .. relative to the gripper, at
%                 f = gHc*[0;0;0;1];
%
% References: R.Tsai, R.K.Lenz "A new Technique for Fully Autonomous
%           and Efficient 3D Robotics Hand/Eye calibration", IEEE
%           trans. on robotics and Automaion, Vol.5, No.3, June 1989
%
% Notation: wHc - pose of camera frame (c) in the world (w) coordinate system
%                 .. If a point coordinates in camera frame (cP) are known
%                 ..     wP = wHc * cP
%                 .. we get the point coordinates (wP) in world coord.sys.
%                 .. Also refered to as transformation from camera to world
%

function gHc = handEye(bHg, wHc)

M = size(bHg,3);

K = (M*M-M)/2;               % Number of unique camera position pairs
A = zeros(3*K,3);            % will store: skew(Pgij+Pcij)
B = zeros(3*K,1);            % will store: Pcij - Pgij
k = 0;

% Now convert from wHc notation to Hc notation used in Tsai paper.
Hg = bHg;
% Hc = cHw = inv(wHc); We do it in a loop because wHc is given, not cHw
Hc = zeros(4,4,M); for i = 1:M, Hc(:,:,i) = inv(wHc(:,:,i)); end;

for i = 1:M,for j = i+1:M;Hgij = inv(Hg(:,:,j))*Hg(:,:,i);    % Transformation from i-th to j-th gripper posePgij = 2*rot2quat(Hgij);            % ... and the corresponding quaternion
Hcij = Hc(:,:,j)*inv(Hc(:,:,i));    % Transformation from i-th to j-th camera posePcij = 2*rot2quat(Hcij);            % ... and the corresponding quaternion
k = k+1;                            % Form linear system of equationsA((3*k-3)+(1:3), 1:3) = skew(Pgij+Pcij); % left-hand sideB((3*k-3)+(1:3))      = Pcij - Pgij;     % right-hand side
end;
end;

% Rotation from camera to gripper is obtained from the set of equations:
%    skew(Pgij+Pcij) * Pcg_ = Pcij - Pgij
% Gripper with camera is first moved to M different poses, then the gripper
% .. and camera poses are obtained for all poses. The above equation uses
% .. invariances present between each pair of i-th and j-th pose.

Pcg_ = A \ B;                % Solve the equation A*Pcg_ = B

% Obtained non-unit quaternin is scaled back to unit value that
% .. designates camera-gripper rotation
Pcg = 2 * Pcg_ / sqrt(1 + Pcg_'*Pcg_);

Rcg = quat2rot(Pcg/2);         % Rotation matrix


% Calculate translational component
k = 0;
for i = 1:M,for j = i+1:M;Hgij = inv(Hg(:,:,j))*Hg(:,:,i);    % Transformation from i-th to j-th gripper poseHcij = Hc(:,:,j)*inv(Hc(:,:,i));    % Transformation from i-th to j-th camera pose
k = k+1;                            % Form linear system of equationsA((3*k-3)+(1:3), 1:3) = Hgij(1:3,1:3)-eye(3); % left-hand sideB((3*k-3)+(1:3))      = Rcg(1:3,1:3)*Hcij(1:3,4) - Hgij(1:3,4);     % right-hand side
end;
end;

Tcg = A \ B;

gHc = transl(Tcg) * Rcg;    % incorporate translation with rotation


return

如果有错误的地方，还请各位指出，小鱼会第一时间改正~

作者介绍：

我是小鱼，机器人领域资深玩家，现深圳某独脚兽机器人算法工程师一枚

初中学习编程，高中开始学习机器人，大学期间打机器人相关比赛实现月入2W+（比赛奖金）

目前在输出机器人学习指南、论文注解、工作经验，欢迎大家关注小鱼，一起交流技术，学习机器人

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