I want to convert a 4x1 vector column to skew symmetric matrix in matlab

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Waleed new 2018-7-19

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/411234-i-want-to-convert-a-4x1-vector-column-to-skew-symmetric-matrix-in-matlab

评论： Waleed new 2018-7-21

采纳的回答： James Tursa

在 MATLAB Online 中打开

for example

Q=[a;b;c;d]

S is skew symmetric which satisfies the condition -S= S transpose is that true the

S(Q) =[0   -a   d   -c
       a    0   c    b
      -d   -c   0   -a
       c   -b   a    0]   ?

and how do it in matlab directly ?

采纳的回答

James Tursa 2018-7-19

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/411234-i-want-to-convert-a-4x1-vector-column-to-skew-symmetric-matrix-in-matlab#answer_329544

在 MATLAB Online 中打开

You could just use the code you have already typed above. E.g.,

a = Q(1); b = Q(2); c = Q(3); d = Q(4);
S = [0   -a   d   -c
     a    0   c    b
    -d   -c   0   -a
     c   -b   a    0];

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James Tursa 2018-7-20

编辑：James Tursa 2018-7-20

在 MATLAB Online 中打开

If you have the Aerospace Toolbox installed that has the quaternion routines, you can play around with the following code. It first demonstrates the quaternion convention that MATLAB uses in these routines. Then it constructs the associated Rodrigues formulation. For some reason, the Rodrigues construction only works at replicating the quat2dcm stuff if the scalar element is negative (hence the qc stuff). Here it is:

% Demonstrates quaternion convention, by James Tursa
function quat_convention
vecpart = @(q) reshape(q(2:4),[],1);
disp('------------------------------------------------------------------');
disp('------------------------------------------------------------------');
disp(' ');
disp('The following code demonstrates the MATLAB quaternion convention');
disp(' ');
disp('Create an arbitrary unit quaternion');
q = randn(1,4); q = q/norm(q)
disp(' ');
disp('Create an arbitrary vector');
v = randn(3,1)
disp(' ');
disp('Get corresponding direction cosine matrix');
disp('dc = quat2dcm(q)');
dc = quat2dcm(q)
disp(' ');
disp('Rotate the vector using the direction cosine matrix');
disp('dc*v')
disp(dc*v);
disp(' ');
disp('Rotate the vector using the quaternion');
disp('conj(q)*v*q')
disp(vecpart(quatmultiply(quatconj(q),quatmultiply([0 v'],q))));
disp(' ');
disp('Differrence in rotated vectors (should be small)');
dcv = dc*v;
qcvq = vecpart(quatmultiply(quatconj(q),quatmultiply([0 v'],q)));
disp(max(abs(dcv(:)-qcvq(:))));
disp(' ');
disp('Quaternion convention is scalar first and successive rotations are RIGHT multiplies');
disp(' ');
disp('------------------------------------------------------------------');
disp(' ');
disp('Rodrigues Rotation Formula');
disp(' ');
if( q(1) < 0 )
    qc = q;
else
    qc = -q;
end
disp('cos(theta)');
c = cos(acos(qc(1))*2)
disp(' ');
disp('sin(theta)');
s = sin(asin(norm(qc(2:4)))*2)
disp(' ');
disp('theta (deg)');
disp(atan2(s,c)*180/pi);
disp(' ');
disp('Rotation axis');
k = qc(2:4)/norm(qc(2:4))
disp(' ');
disp('Skew symmetric matrix');
K = [0 -k(3)  k(2); k(3) 0 -k(1); -k(2) k(1) 0]
disp(' ');
disp('R = eye(3) + s*K + (1-c)*K^2');
R = eye(3) + s*K + (1-c)*K^2
disp(' ');
disp('Differrence in dc matrices (should be small)');
disp(max(abs(R(:)-dc(:))));
end