Euler Angles from Rotation Matrix

Hello!

I'm trying to get symbolic form of alpha, beta, gamma angles from a rotation matrix R in the sequence 3-1-3 (i.e. first rotation of gamma about Z axis, then a rotation of alpha about X axis and finally a rotation of gamma about Z axis) but I don't know how to do so.

Could someone can help me?

Thank you!

syms alpha beta gamma

R1 = [1, 0, 0; 0, cos(alpha), -sin(alpha); 0, sin(alpha), cos(alpha)]; % Rotation matrix about X axis of an angle alpha                 
R3 = [cos(gamma), -sin(gamma), 0; sin(gamma), cos(gamma), 0; 0, 0, 1]; % Rotation matrix about Z axis of an angle gamma
        
R313 = R3*R1*R3; % Final rotation matrix in the sequence 3-1-3

Answer 1

[David Goodmanson]

Hi Gregory,

If you mean you want the form of the rotation martrix in terms of alpha,beta,gamma, that's just

syms a b g
R1 = [1, 0, 0; 0, cos(b), -sin(b); 0, sin(b), cos(b)];                 
R3m = [cos(a), -sin(a), 0; sin(a), cos(a), 0; 0, 0, 1];
R3n = [cos(g), -sin(g), 0; sin(g), cos(g), 0; 0, 0, 1];
Rtot = R3m*R1*R3n        
Rtot =
 
[cos(a)*cos(g) -cos(b)*sin(a)*sin(g), -cos(a)*sin(g) -cos(b)*cos(g)*sin(a),  sin(a)*sin(b)]
[cos(g)*sin(a) +cos(a)*cos(b)*sin(g),  cos(a)*cos(b)*cos(g) -sin(a)*sin(g), -cos(a)*sin(b)]
[                      sin(b)*sin(g),                        cos(g)*sin(b),         cos(b)]

If you mean you have a rotation matrix M and need expressions for a,b,g then

b = acos(M33)    % M33 = M(3,3) etc.

Two choices here; b and (2*pi-b) are possible since cos is the same either way. That essentially means you get to pick the sign of sin(b). Denote that sign by S. Then

g = atan2(S*M31,S*M32)
a = atan2(S*M13,-S*M23)

Generally one of the euler angles is restricted to 0,theta<pi. That might or might not determine the angle b uniquely.

[Gregory Cottone]

Sorry I have made a mistake in my question, I meant first a rotation of alpha about Z axis, then a rotation of beta about X axis and finally a rotation of gamma about Z axis, therefore I have a sequence of three different angles (alpha, beta and gamma) about Z-X-Z axis.

Sorry I'm new in using Matlab but I don't know how to pick the sign of sin(beta).

I would also know if is it possible to write a script that can give the three angles for the all range

0 < alpha,beta,gamma < 2*pi and not for the restriction of 0 < theta < pi.

Really thank you sir.

Gregory

[David Goodmanson]

Depending on your convention, you might have to use

R3(g)R1(b)R3(a) instead of R3(a)R1(b)R3(g). a,b,g = alpha,beta,gamma

For euler angles, if all three of them have the domain 0<=angle<=2*pi, then for every physical rotation there are two different sets of angles that give the same 3d rotation. This is not a good situation. Restricting one of the angles to 0<=angle<=pi gets rid of that problem, and the restriced set of angles still cover every possible 3d rotation.

[Gregory Cottone]

Perfect, I understand, therefore this is for example one of the reasons why I should not use this parametrization to design a flight simulator for acrobatic airplane.

Thank you sir.

Gregory