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I'd like to visualize elemental rotations using Z-Y-X convention in Matlab. If you'd like to follow me exactly, I'll be refering to a paper Quadrotor Dynamics and Control by Randal Beard.

The base coordinate frame, $F^v$, is called the vehicle frame (page 6). It is a right-handed cartesian frame of reference. The unit vectors in $F^v$ are $x^v, y^v, z^v$ which can be visualized in Matlab using the below script:

x_v = [1;0;0]; %Define unit vectors in vehicle frame
y_v = [0;1;0];
z_v = [0;0;1];
o_v = [x_v'; y_v'; z_v']; %Convert to quiver3 compliant form
quiver3(zeros(3,1), zeros(3,1), zeros(3,1), o_v(:,1), o_v(:,2), o_v(:,3)) %Draw the frame

I have written a function to create a Direction Cosine Matrix from Euler angles, as defined on page 9 of the aforementioned paper:

function dcm = RPY_2_DCM(roll, pitch, yaw)
  cR = cos(roll);  sR = sin(roll);
  cP = cos(pitch); sP = sin(pitch);
  cY = cos(yaw);   sY = sin(yaw);

  dcm = [cP*cY          cP*sY          -sP;
         sR*sP*cY-cR*sY sR*sP*sY+cR*cY sR*cP;
         cR*sP*cY+sR*sY cR*sP*sY-sR*cY cR*cP];

Drawing the vehicle-1 $F^{v1}$ frame as a rotation around $z^v$ by yaw angle with the below script works fine (vehicle frame in blue, vehicle-1 frame in green):

yaw = pi/6;
x_v1 = RPY_2_DCM(0,0,yaw)*x_v; %Draw vehicle-1 coordinate frame obtained by right-handed rotation of F_v around z_v
y_v1 = RPY_2_DCM(0,0,yaw)*y_v;
z_v1 = RPY_2_DCM(0,0,yaw)*z_v;
o_v1 = [x_v1'; y_v1'; z_v1'];
quiver3(zeros(3,1), zeros(3,1), zeros(3,1), o_v1(:,1), o_v1(:,2), o_v1(:,3))

Vehicle-1 frame

Now, I'd like to rotate vectors in $F^{v1}$ around $y^{v1}$ axis of vehicle-1 by adding:

pitch = pi/6;
x_v2 = RPY_2_DCM(0,pitch,0)*x_v1;
y_v2 = RPY_2_DCM(0,pitch,0)*y_v1;
z_v2 = RPY_2_DCM(0,pitch,0)*z_v1;
o_v2 = [x_v2'; y_v2'; z_v2'];
quiver3(zeros(3,1), zeros(3,1), zeros(3,1), o_v2(:,1), o_v2(:,2), o_v2(:,3))

But instead of rotating around $y^{v1}$ I obtain a rotation of $F^{v1}$ around $y^v$ (see the picture below, $F^{v2}$ in red). What am I doing wrong?

Vehicle-2 frame

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Your rotation matrices will be in the wrong order the way you do it! Mathematically its M1*M2*M3*V with M1,M2,M3 being the FIRST second third rotation matrix and V a vector. So you see the first transformation is the last multiplication on the vector! In your example of your second transformation you do M2*M1*V ... wrong order... wrong result...

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A bit late, but anyway maybe my answer can help other people.

You want to perform rotations about rotating axes (intrinsic rotations), you can get the same result by inverting the order of rotations and perform the rotations about the fixed axes (extrinsic rotations), so if you want to perform intrinsic rotations about the Z axis, then about the new Y axis and finally about the new X axis you can rotate about the x axis, then about the y axis and finally about the z axis.

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