你好,我是小智。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 transforms3d
pip3 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 += 1
        Hgij = 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 pairs
    int 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 - Pgij
    Mat 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 6
            vec_Hgij.push_back(Hgij);
            //Rotation axis for Rgij which is the 3D rotation from gripper coordinate frame Gi to Gj
            Mat 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 7
            vec_Hcij.push_back(Hcij);
            //Rotation axis for Rcij
            Mat 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 - Pgij
            Mat 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 rotation
    Mat 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 - Tgij
            diff = 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 build
cmake ..   
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 pose
        Pgij = 2*rot2quat(Hgij);            % ... and the corresponding quaternion

        Hcij = Hc(:,:,j)*inv(Hc(:,:,i));    % Transformation from i-th to j-th camera pose
        Pcij = 2*rot2quat(Hcij);            % ... and the corresponding quaternion

      k = k+1;                            % Form linear system of equations
      A((3*k-3)+(1:3), 1:3) = skew(Pgij+Pcij); % left-hand side
      B((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 pose
        Hcij = Hc(:,:,j)*inv(Hc(:,:,i));    % Transformation from i-th to j-th camera pose

      k = k+1;                            % Form linear system of equations
      A((3*k-3)+(1:3), 1:3) = Hgij(1:3,1:3)-eye(3); % left-hand side
      B((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,小智会第一时间改正~

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