MATLAB: Euler 3D rotation between two vectors

Question

Hello everyone,

I would like to obtain the Euler angles needed to rotate a vector u = (0,0,1) to a vector v, defined between an arbitrary point (x,y,z) and the origin (0,0,0). So v = (x,y,z). I am currently working with a right-handed 3d coordinate system, where X is pointing to the right side of the current reader, Y is pointing upwards and therefore Z is pointing outside the screen of the computer.

I calculate the angles needed to rotate by taking as a reference axis Y -> X -> Z. I have tried to use Euler angles directly, but somehow the objects that I rotate end up in a different location that they should be. Searching on the internet, I have found loads of information about "rotation matrixes" and "quartenions", but I am lost about how to apply them to my case. Any help would be appreciated 🙂

EDIT: After some tinkering and searching on internet I managed to obtain an answer for a XYZ coordinated system. Still not useful for me, but maybe other people can profit from it.

a = [0 0 1].';
b = [0 5 0].';
% Method described in
% https://math.stackexchange.com/questions/180418/calculate-rotation-matrix-to-align-vector-a-to-vector-b-in-3d
a = a./norm(a);
b = b./norm(b);
v = cross(a,b);
vx = [0 -v(3) v(2) ; v(3) 0 -v(1); -v(2) v(1) 0 ];
c = dot(a,b);
I = eye(3);
R=I+vx+vx^2*(1/(1+c));
R = round(R,5);
% From Rot matrix to euler coordinates, follows XYZ, described in:
% http://www.gregslabaugh.net/publications/euler.pdf
if (R(3,1) ~=1) && (R(3,1) ~=-1)
    theta_1 = -asin(R(3,1));
    theta_2 = pi-theta_1;
    chi_1 = atan2((R(3,2)/cos(theta_1)),(R(3,3)/cos(theta_1)));
    chi_2 = atan2((R(3,2)/cos(theta_2)),(R(3,3)/cos(theta_2)));
    phi_1 = atan2((R(2,1)/cos(theta_1)),(R(1,1)/cos(theta_1)));
    phi_2 = atan2((R(2,1)/cos(theta_2)),(R(1,1)/cos(theta_2)));
    theta = min(theta_1,theta_2);
    chi = min(chi_1,chi_2);
    phi = min(phi_1,phi_2);
else
    phi = 0;
    if R(3,1) == -1
        theta = pi/2;
        chi = phi+atan2(R(1,2),R(1,3));
    else
        theta = -pi/2;
        chi = -phi+atan2(-R(1,2),-R(1,3));
    end
end
theta = rad2deg(theta)
chi = rad2deg(chi)
phi = rad2deg(phi)

Answer 1

Here is rotation using Rodriguez matrix

Here is rotation about Y axis and Z.

Here is the difference between them

See attached script