Yaw, Pitch and Roll composition

Question

I'm trying to understand composition of rotations using eulers angles and rotation matrices. I am facing a counterintuitive situation performing two rotations about different angles. My setting is the following:

• first rotation of an angle $\psi$ about the Z body-axis (yaw)
• second rotation of an angle $\theta$ about the Y body-axis (pitch)
• third rotation of an angle $\varphi$ about the X body-axis (roll)

Denoting with

$$R_X=\begin{bmatrix}1&0&0\\ 0&\cos(\varphi)&-\sin(\varphi)\\ 0&\sin(\varphi)&\cos(\varphi)\end{bmatrix}$$ $$R_Y=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix}$$ $$R_Z=\begin{bmatrix}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{bmatrix}$$

the rotation matrices relative to elementary rotation about the three body axis, I obtain that a general rotation of eulers angles $(yaw,pitch,roll)=(\psi,\theta,\varphi)$ is given by:

$$R=R_X\cdot R_Y\cdot R_Z$$

Suppose now that I would like to perform this 2 consecutive rotations:

1. Rotation of $(yaw,pitch,roll)=(0,\frac{\pi}{4},0)$
2. Rotation of $(yaw,pitch,roll)=(\frac{\pi}{2},0,0)$

The matrices relative to these two rotations are the following:

$$R_1=\begin{bmatrix}\frac{\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\\ 0&1&0\\ -\frac{\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\end{bmatrix}$$ $$R_2=\begin{bmatrix}1&0&0\\ 0&0&-1\\ 0&1&0\end{bmatrix}$$

and composing them I obtain the total rotation matrix

$$R=R_2\cdot R_1 = \begin{bmatrix}0&-1&0\\ \frac{\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\end{bmatrix}$$

If I want to recover the euler angles sequence relative to $R$ I apply these equations:

$$\theta=\arcsin(R_{13})\\ \psi=-\arctan2\left(\frac{R_{12}}{\cos(\theta)},\frac{R_{11}}{\cos(\theta)}\right)\\ \varphi=-\arctan2\left(\frac{R_{23}}{\cos(\theta)},\frac{R_{33}}{\cos(\theta)}\right)$$

obtaining $(yaw,pitch,roll)=(\frac{\pi}{2},0,-\frac{\pi}{4})$.

However applying this sequence of rotation I do not recover the initial frame (sequence of rotations 1-2) unless the rotations 1-2 are performed on different axis respect the body ones.

Probably I miss something and I'm making confusion with these concepts. Please, could you help me to understand where I miss?

Answer 1

I believe the problem you are having is that if you multiply the matrices in the order $R_X\cdot R_Y\cdot R_Z$, then you are rotating first around the global $Z$ axis, then the global $Y$ axis, then the global $X$ axis.

To get the rotations applied to the local (body) axes instead, you simply have to reverse the order of multiplication.

This might seem like magic, so I'll give a short explanation. After applying the $Z$-axis rotation, the body reference frame is not the same as the global reference frame: It has been rotated by $R_Z$. So at this point, if $A$ is a matrix denoting any transformation according to the body frame, then in the global frame that same transformation has the matrix representation $$ R_Z\cdot A\cdot R_Z^{-1} $$ (This will take a vector expressed in the global basis, then $R_Z^{-1}$ is the same vector expressed in the body basis. We apply $A$, then translate back to the global basis with $R_Z$.)

So first applying $R_Z$, then after that applying $R_Y$ in the new local frame has matrix representation $$ \left(R_Z\cdot R_Y\cdot R_Z^{-1}\right)\cdot R_Z = R_Z\cdot R_Y $$ If you want to apply a rotation about the body $X$-axis after this again, then similarily, the global matrix representation of this transformation is $$ (R_Z\cdot R_Y)\cdot R_X\cdot (R_Z\cdot R_Y)^{-1} $$ and if we apply this after our initial rotation of $R_Z\cdot R_Y$, then the result becomes $$ R_Z\cdot R_Y\cdot R_X $$