Matlab计算旋转矩阵有两种方法,一种是通过欧拉角,计算yaw,pitch和row轴的旋转角。这里要介绍的是另一种是直接绕一个向量旋转theta角的方法,就是Rodrigues变换,其中的向量就是旋转向量,其得到的矩阵就是旋转矩阵,这个矩阵和欧拉角的方法计算出来是一样的。

        而我们常常将旋转和平移组合在一个3*4的矩阵中,使得物体三维坐标乘以该矩阵,直接得到旋转平移变换之后的坐标。此时就需要将旋转向量转化为一个3×3的旋转矩阵,这就是罗德里格斯(Rodrigues)变换。

一、Rodrigues变换的数学原理

        如题,旋转有两种理解,一种是绕坐标轴依次旋转,另一种是绕任意轴一次直接旋转一个角度得到。将向量B看做是A绕着轴C转过角度theta得到,则:

r8

        同时,我们还要求norm(C)=theta,于是,旋转向量方向与C相同,向量的模大小是弧度制的角度,我们就可以很轻松的求出两个向量的旋转向量和旋转矩阵了。

% 求向量A和向量B的旋转矩阵
clear
A = [-0.9772    0.0953   -0.1896]';
B = [-0.9937    0.0999    0.0501]';
C = cross(B, A);
theta = acos((A'*B) / ( norm(A)*norm(B) ));
Rv = C / norm(C) * theta;
R = rodrigues(Rv);

二、Rodrigues变换的实现

        在OpenCV和Matlab中都用实现Rodrigues变换:

        OpenCV中Calib模块有实现该函数,查看OpenCV文档中的解释,和一篇博客

        正反变换都是同一个函数,输入1*3或3*1,输出3*3,或者输入3*3,输出1*3,第三个参数还可以输出3*9的雅可比矩阵

        cv::Mat om(3, 1, CV_32FC1);
		om.ptr<float>(0)[0] = p[0];
		om.ptr<float>(1)[0] = p[1];
		om.ptr<float>(2)[0] = p[2];
		cv::Mat R(3, 3, CV_32FC1);
		cv::Rodrigues(om, R);
		for (int i = 0; i < 3; ++i){
			for (int j = 0; j < 3; ++j){
				mat[i][j] = R.ptr<float>(i)[j];
			}
		}

        一个工具箱中,Matlab实现rodrigues变换的函数,与上面opencv使用是类似的

function	[out,dout]=rodrigues(in)

% RODRIGUES	Transform rotation matrix into rotation vector and viceversa.
%		
%		Sintax:  [OUT]=RODRIGUES(IN)
% 		If IN is a 3x3 rotation matrix then OUT is the
%		corresponding 3x1 rotation vector
% 		if IN is a rotation 3-vector then OUT is the 
%		corresponding 3x3 rotation matrix
%

%%
%%		Copyright (c) March 1993 -- Pietro Perona
%%		California Institute of Technology
%%

%% ALL CHECKED BY JEAN-YVES BOUGUET, October 1995.
%% FOR ALL JACOBIAN MATRICES !!! LOOK AT THE TEST AT THE END !!

%% BUG when norm(om)=pi fixed -- April 6th, 1997;
%% Jean-Yves Bouguet

%% Add projection of the 3x3 matrix onto the set of special ortogonal matrices SO(3) by SVD -- February 7th, 2003;
%% Jean-Yves Bouguet

% BUG FOR THE CASE norm(om)=pi fixed by Mike Burl on Feb 27, 2007


[m,n] = size(in);
%bigeps = 10e+4*eps;
bigeps = 10e+20*eps;

if ((m==1) & (n==3)) | ((m==3) & (n==1)) %% it is a rotation vector
    theta = norm(in);
    if theta < eps
        R = eye(3);

        %if nargout > 1,

        dRdin = [0 0 0;
            0 0 1;
            0 -1 0;
            0 0 -1;
            0 0 0;
            1 0 0;
            0 1 0;
            -1 0 0;
            0 0 0];

        %end;

    else
        if n==length(in)  in=in'; end; 	%% make it a column vec. if necess.

        %m3 = [in,theta]

        dm3din = [eye(3);in'/theta];

        omega = in/theta;

        %m2 = [omega;theta]

        dm2dm3 = [eye(3)/theta -in/theta^2; zeros(1,3) 1];

        alpha = cos(theta);
        beta = sin(theta);
        gamma = 1-cos(theta);
        omegav=[[0 -omega(3) omega(2)];[omega(3) 0 -omega(1)];[-omega(2) omega(1) 0 ]];
        A = omega*omega';

        %m1 = [alpha;beta;gamma;omegav;A];

        dm1dm2 = zeros(21,4);
        dm1dm2(1,4) = -sin(theta);
        dm1dm2(2,4) = cos(theta);
        dm1dm2(3,4) = sin(theta);
        dm1dm2(4:12,1:3) = [0 0 0 0 0 1 0 -1 0;
            0 0 -1 0 0 0 1 0 0;
            0 1 0 -1 0 0 0 0 0]';

        w1 = omega(1);
        w2 = omega(2);
        w3 = omega(3);

        dm1dm2(13:21,1) = [2*w1;w2;w3;w2;0;0;w3;0;0];
        dm1dm2(13: 21,2) = [0;w1;0;w1;2*w2;w3;0;w3;0];
        dm1dm2(13:21,3) = [0;0;w1;0;0;w2;w1;w2;2*w3];

        R = eye(3)*alpha + omegav*beta + A*gamma;

        dRdm1 = zeros(9,21);

        dRdm1([1 5 9],1) = ones(3,1);
        dRdm1(:,2) = omegav(:);
        dRdm1(:,4:12) = beta*eye(9);
        dRdm1(:,3) = A(:);
        dRdm1(:,13:21) = gamma*eye(9);

        dRdin = dRdm1 * dm1dm2 * dm2dm3 * dm3din;


    end;
    out = R;
    dout = dRdin;

    %% it is prob. a rot matr.
elseif ((m==n) & (m==3) & (norm(in' * in - eye(3)) < bigeps)...
        & (abs(det(in)-1) < bigeps))
    R = in;

    % project the rotation matrix to SO(3);
    [U,S,V] = svd(R);
    R = U*V';

    tr = (trace(R)-1)/2;
    dtrdR = [1 0 0 0 1 0 0 0 1]/2;
    theta = real(acos(tr));


    if sin(theta) >= 1e-4,

        dthetadtr = -1/sqrt(1-tr^2);

        dthetadR = dthetadtr * dtrdR;
        % var1 = [vth;theta];
        vth = 1/(2*sin(theta));
        dvthdtheta = -vth*cos(theta)/sin(theta);
        dvar1dtheta = [dvthdtheta;1];

        dvar1dR =  dvar1dtheta * dthetadR;


        om1 = [R(3,2)-R(2,3), R(1,3)-R(3,1), R(2,1)-R(1,2)]';

        dom1dR = [0 0 0 0 0 1 0 -1 0;
            0 0 -1 0 0 0 1 0 0;
            0 1 0 -1 0 0 0 0 0];

        % var = [om1;vth;theta];
        dvardR = [dom1dR;dvar1dR];

        % var2 = [om;theta];
        om = vth*om1;
        domdvar = [vth*eye(3) om1 zeros(3,1)];
        dthetadvar = [0 0 0 0 1];
        dvar2dvar = [domdvar;dthetadvar];


        out = om*theta;
        domegadvar2 = [theta*eye(3) om];

        dout = domegadvar2 * dvar2dvar * dvardR;


    else
        if tr > 0; 			% case norm(om)=0;

            out = [0 0 0]';

            dout = [0 0 0 0 0 1/2 0 -1/2 0;
                0 0 -1/2 0 0 0 1/2 0 0;
                0 1/2 0 -1/2 0 0 0 0 0];
        else

            % case norm(om)=pi;
            if(0)

                %% fixed April 6th by Bouguet -- not working in all cases!
                out = theta * (sqrt((diag(R)+1)/2).*[1;2*(R(1,2:3)>=0)'-1]);
                %keyboard;

            else

                % Solution by Mike Burl on Feb 27, 2007
                % This is a better way to determine the signs of the
                % entries of the rotation vector using a hash table on all
                % the combinations of signs of a pairs of products (in the
                % rotation matrix)

                % Define hashvec and Smat
                hashvec = [0; -1; -3; -9; 9; 3; 1; 13; 5; -7; -11];
                Smat = [1,1,1; 1,0,-1; 0,1,-1; 1,-1,0; 1,1,0; 0,1,1; 1,0,1; 1,1,1; 1,1,-1;
                    1,-1,-1; 1,-1,1];

                M = (R+eye(3,3))/2;
                uabs = sqrt(M(1,1));
                vabs = sqrt(M(2,2));
                wabs = sqrt(M(3,3));

                mvec = ([M(1,2), M(2,3), M(1,3)] + [M(2,1), M(3,2), M(3,1)])/2;
                syn  = ((mvec > eps) - (mvec < -eps)); % robust sign() function
                hash = syn * [9; 3; 1];
                idx = find(hash == hashvec);
                svec = Smat(idx,:)';

                out = theta * [uabs; vabs; wabs] .* svec;

            end;

            if nargout > 1,
                fprintf(1,'WARNING!!!! Jacobian domdR undefined!!!\n');
                dout = NaN*ones(3,9);
            end;
        end;
    end;

else
    error('Neither a rotation matrix nor a rotation vector were provided');
end;

return;

%% test of the Jacobians:

%% TEST OF dRdom:
om = randn(3,1);
dom = randn(3,1)/1000000;
[R1,dR1] = rodrigues(om);
R2 = rodrigues(om+dom);
R2a = R1 + reshape(dR1 * dom,3,3);
gain = norm(R2 - R1)/norm(R2 - R2a)

%% TEST OF dOmdR:
om = randn(3,1);
R = rodrigues(om);
dom = randn(3,1)/10000;
dR = rodrigues(om+dom) - R;

[omc,domdR] = rodrigues(R);
[om2] = rodrigues(R+dR);
om_app = omc + domdR*dR(:);
gain = norm(om2 - omc)/norm(om2 - om_app)


%% OTHER BUG: (FIXED NOW!!!)
omu = randn(3,1);   
omu = omu/norm(omu)
om = pi*omu;        
[R,dR]= rodrigues(om);
[om2] = rodrigues(R);
[om om2]

%% NORMAL OPERATION
om = randn(3,1);         
[R,dR]= rodrigues(om);
[om2] = rodrigues(R);
[om om2]

%% Test: norm(om) = pi
u = randn(3,1);
u = u / sqrt(sum(u.^2));
om = pi*u;
R = rodrigues(om);
R2 = rodrigues(rodrigues(R));
norm(R - R2)

%% Another test case where norm(om)=pi from Chen Feng (June 27th, 2014)
R = [-0.950146567583153 -6.41765854280073e-05 0.311803617668748; ...
     -6.41765854277654e-05 -0.999999917385145 -0.000401386434914383; ...
      0.311803617668748 -0.000401386434914345 0.950146484968298];
om = rodrigues(R)
norm(om) - pi

%% Another test case where norm(om)=pi from 浣欐垚涔�(July 1st, 2014)
R = [-0.999920129411407	-6.68593208347372e-05	-0.0126384464118876; ...
     9.53007036072085e-05	-0.999997464662094	-0.00224979713751896; ...
    -0.0126382639492467	-0.00225082189773293	0.999917600647740];
om = rodrigues(R)
norm(om) - pi



        上面的matlab函数来自加州理工的张正友标定工具箱。

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