## ANALYSIS ON THE INDEPENDENCE OF EULER ANGLES AND THE COORDINATE TRANSFORMATION MATRIX

ZHANG Yuan, WANG Shimin,1), WANG Qi

Beihang University, Beijing 100083, China

Abstract

The independence of the Eulerian Angles and the rotation sequence in constructing of transformation matrix from body coordinates to fixed coordinate problems are discussed in this paper, by introducing the transition coordinate system. The rigid body position is independent of the rotation sequence corresponding to Eulerian angles. However, there exists a sequence in all of them, in which each of the three rotations is carried out around common coordinate axes of the two coordinate systems associated with each other, and the transformation relation of each rotation is expressed with simple transformation for rotation about a fixed axis. Therefore, this sequence is chosen to construct the transformation matrix from the body coordinate to the fixed coordinate system.

Keywords： Eulerian angles; rotation sequence; transformation matrix

ZHANG Yuan, WANG Shimin, WANG Qi. ANALYSIS ON THE INDEPENDENCE OF EULER ANGLES AND THE COORDINATE TRANSFORMATION MATRIX. Mechanics in Engineering, 2022, 44(2): 385-389 DOI:10.6052/1000-0879-21-288

## 2 中间坐标系与变换矩阵

### 2.2 利用中间坐标系的变换矩阵

${r}_{{k'n}}=A(\varphi ){r}{{'}}$

$\boldsymbol{A}(\varphi)=\left[\begin{array}{ccc}\cos \varphi & \sin \varphi & 0 \\-\sin \varphi & \cos \varphi & 0 \\0 & 0 & 1\end{array}\right]$

${r}_{kn}=A\left( \theta \right){r}_{k'n}$

$\boldsymbol{A}(\theta)=\left[\begin{array}{ccc}1 & 0 & 0 \\0 & \cos \theta & -\sin \theta \\0 & \sin \theta & \cos \theta\end{array}\right]$

${r}=A\left( \psi \right){r}_{kn}$

$A(\psi)=\left[\begin{array}{ccc}\cos \psi & -\sin \psi & 0 \\\sin \psi & \cos \psi & 0 \\0 & 0 & 1\end{array}\right]$

$r=A\left( \psi \right)A\left( \theta \right)A\left( \varphi \right){r'}$

### 图3

$r_{c}=B(\psi) B (\theta) r_{a}$

${r}_{d} =C\left( \psi \right){r}_{a}$

${r}_{c} =B\left( \psi \right)B\left( \theta \right)C^{-1}\left( \psi \right){r}_{d}$

${r}_{c}=A\left( \psi \right)A\left( \theta \right)A{(\psi )}^{-1}{r}_{d}$

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