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

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 ?

Answer 1

[James Tursa]

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];

[Waleed new]

is there any command to do it automatically if the number of lines of vector is big ? thank you

[James Tursa]

There are many different skew symmetric matrices you could form from the elements of the vector. Which one is appropriate would depend on your particular application. What is this for and how do you intend to use it? How large is your vector?

[Waleed new]

actually i'am looking for the skew symmetric matrix of quaternion vector

Q=[q0; q1; q2; q3]

and i want to do it directly in matlab without creating function so i think may be there is a command to do it automatically

[James Tursa]

For what purpose? What will this matrix be used for downstream? Does this quaternion Q represent a small rotation and you want the corresponding small rotation skew symmetric matrix approximation to it? Or what ...?

[Waleed new]

i want to use Rodriguez rotation formula to define the rotation matrix in quaternion representation using this expression :

     R(Q(t))= I+ S(Q(t))+S^2(Q(t))

      Q(t)= [q0(t);q1(t);q2(t);q3(t)]

I is matrix identity

S is skew symmetric matrix

R rotation matrix in quaternion representation___

[James Tursa]

Assuming that R is the 3x3 rotation matrix corresponding to Q, and S is the 3x3 skew symmetric matrix corresponding to the Rodriguez formula, this expression doesn't quite match the matrix representation found here (K is used instead of S in the "Matrix notation" section):

https://en.wikipedia.org/wiki/Rodrigues%2527_rotation_formula

Is this the same thing you are trying to do? Or something else?

[Waleed new]

yes I made a mistake it does not match the matrix representation found

R= I +sin(teta)*S + (1-cos(teta))*S^2

the rotation matrix in euler angles representation :

the rotation matrix in quaternions representation :

thank you James Tursa . I think now it's complicated to me but i will try to understand what i'am doing. those images are from "A Survey of Attitude Representations Malcolm D. Shuster The Journal of the Astronautical Sciences, Vol. 41, No. 4, October–December 1993, pp. 439–517"

[James Tursa]

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

[Waleed new]

Thank you very much I appreciate your consideration time.