Problem with Euler angles from YZX Rotation Matrix

Question

I've gotten stuck getting my euler angles out my rotation matrix.

My conventions are:

Left-handed (x right, z back, y up)
YZX
Left handed angle rotation

My rotation matrix is built up from Euler angles like (from my code):

    var xRotationMatrix = $M([
        [1,  0,   0, 0], 
        [0, cx, -sx, 0], 
        [0, sx,  cx, 0], 
        [0,  0,   0, 1]
    ]);

    var yRotationMatrix = $M([
        [ cy, 0, sy, 0], 
        [  0, 1,  0, 0], 
        [-sy, 0, cy, 0], 
        [  0, 0,  0, 1]
    ]);
    var zRotationMatrix = $M([
        [cz, -sz, 0, 0], 
        [sz,  cz, 0, 0], 
        [ 0,   0, 1, 0], 
        [ 0,   0, 0, 1]
    ]);

Which results in a final rotation matrix as:

R(YZX) = | cy.cz, -cy.sz.cx + sy.sx,  cy.sz.sx + sy.cx, 0|
         |    sz,             cz.cx,            -cz.sx, 0|
         |-sy.cz,  sy.sz.cx + cy.sx, -sy.sz.sx + cy.cx, 0|
         |     0,                 0,                 0, 1|

I'm calculating my euler angles back from this matrix using this code:

this.anglesFromMatrix = function(m) {
    var y = 0, x = 0, z = 0;

    if (m.e(2, 1) > 0.999) {
        y = Math.atan2(m.e(1, 3), m.e(3, 3));
        z = Math.PI / 2;
        x = 0;
    } else if (m.e(2, 1) < -0.999) {
        y = Math.atan2(m.e(1, 3), m.e(3, 3));
        z = -Math.PI / 2;
        x = 0;
    } else {
        y = Math.atan2(-m.e(3, 1), -m.e(1, 1));
        x = Math.atan2(-m.e(2, 3), m.e(2, 2));
        z = Math.asin(m.e(2, 1));
    }
    return {theta: this.deg(x), phi: this.deg(y), psi: this.deg(z)};
};

I've done the maths backwards and forwards a few times, but I can't see what's wrong. Any help would hugely appreciated.

I would have, but there is far more material on 3D rotation on stackoverflow. And math.stackexchange claims to be the site "for people studying math or in math related professions". That excludes me. — Brendon McLean
@Scorpi0: This is an appropriate question here. @Brendon: Your code looks OK. I would try the trivial cases: rotate +90 deg around the X axis and then check the result. Then around the Y axis and so on. Only one rotation at a time, and always start from the original state, no consecutive rotations. — Ali
Thanks @Ali - yeah I tried that. The results were wrong - one rotation affecting two angles. — Brendon McLean
I am really disappointed, this question is appropriate here. @Brendon I am glad you finally found the typo. — Ali

David Hammen David Hammen · Accepted Answer · 2011-06-28T11:55:24

Your matrix and euler angles aren't consistent. It looks like you should be using

y = Math.atan2(-m.e(3, 1), m.e(1, 1));

instead of

y = Math.atan2(-m.e(3, 1), -m.e(1, 1));

for the general case (the else branch).

I said "looks like" because -- what language is this? I'm assuming you have the indexing correct for this language. Are you sure about atan2? There is no single convention for atan2. In some programming languages the sine term is the first argument, in others, the cosine term is the first argument.

Problem with Euler angles from YZX Rotation Matrix

3 Answers