Quaternions, Transformation Matrix

Question

hknatas 2018-11-11

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https://ww2.mathworks.cn/matlabcentral/answers/429216-quaternions-transformation-matrix

编辑： James Tursa 2020-5-8

Hello.

As you see in the figure, my C matrix depends on q1, q2, q3 and q4 quaternions. I caculated all the q1(i),q2(i),q3(i),q4(i) values when i = 0:1:54000. Namely for 54000 iteration, I have 54000 different q1, q2, q3, q4 values. Right now, I want to calculate C matrix for i = 0:1:54000. As a result, I need 54000 different C matrices. For i = 1, q1(1), q2(1), q3(1) and q4(1) should be used and so on i need to reach i = 54000. As i say above, i have the code to calculate all q1,q2,q3 and q4. Just i need to insert these values to matrix respectively. I guess for matrices, i can't use the same plan as I calculated quternions.

Thanks for the help already.

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Answer 1

Bruno Luong 2018-11-11

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https://ww2.mathworks.cn/matlabcentral/answers/429216-quaternions-transformation-matrix#answer_346341

编辑：Bruno Luong 2018-11-11

在 MATLAB Online 中打开

Such thing is straight forward in MATLAB

q1 = reshape(q1,1,1,[]);
q2 = reshape(q2,1,1,[]);
q3 = reshape(q3,1,1,[]);
q4 = reshape(q4,1,1,[]);
% The matrix is i C(:,:,i) for i=1,..., 54000  
C = [q1.^2-q2.^2-q3.^2+q4.^2, 2*(q1.*q2+q3.*q4), 2*(q1.*q3-q2.*q4);
     2*(q1.*q2-q3.*q4), -q1.^2+q2.^2-q3.^2+q4.^2, 2*(q2.*q3-q1.*q4);
    2*(q1.*q3+q2.*q4), 2*(q2.*q3-q1.*q4), -q1.^2-q2.^2+q3.^2+q4.^2]

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hknatas 2018-11-11

Thanks for your effort, I guess I understood what we did finally.

James Tursa 2020-5-8

编辑：James Tursa 2020-5-8

@Bruno: Correct on that error catch. The C(2,3) term should have a + instead of a -.

Also, for the benefit of other readers, note that the quaternion to direction cosine matrix formula above (with the correction) assumes the following:

Quaternion is vector-scalar order. I.e., the vector is [q1;q2;q3] and the scalar is q4. Note that this does not match either the Aerospace Toolbox or the Robotics Toolbox, which have the scalar first and vector last.

Quaternion is either Right Chain right-handed Hamilton convention or Left Chain left-handed JPL convention. I.e., it must be one of the following:

v_body = q^-1 * v_ref * q with right-handed Hamilton convention (ij=k, jk=i, ki=j)

or

v_body = q * v_ref * q^-1 with left-handed JPL convention (ij=-k, jk=-i, ki=-j)

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