Sorry for boring you during summer vacation, my friends. I am haunted by the approximated expression of Euler angle rotation matrix found in this textbook. In the appendix, the author declares that the conventional Euler (Tait–Bryan) angle matrix could be approximated into the following second order form as $$ \begin{bmatrix} 1 & & \\ & 1 & \\ & & 1 \end{bmatrix} + \begin{bmatrix} 0 & -\theta_z & \theta_y \\ \theta_z & 0 & -\theta_x\\ -\theta_y& \theta_x& 0 \end{bmatrix} -\frac{1}{2} \begin{bmatrix} \theta_y^2+\theta_z^2 & & \\ & \theta_x^2+\theta_z^2 & \\ & & \theta_x^2+\theta_y^2 \end{bmatrix} + \begin{bmatrix} 0 & \theta_x\theta_y & \theta_x\theta_z \\ \theta_x\theta_y & 0 & \theta_y\theta_z\\ \theta_x\theta_z& \theta_y\theta_z& 0 \end{bmatrix} $$ Normally, the first two matrices are used for approximating the Euler angles following small rotation hypothesis. And in my mind, the second matrix represents physically the rotational motion of small rotations. However, no further explication and analysis are presented in the textbook for all the other matrices. Thus, I wonder if someone could share some insight with this form of approximation (I cannot find a similar approximation in anywhere else but in the textbook). Thanks a lot in advance.

  • $\begingroup$ I believe the fourth matrix, like the third matrix, should be multiplied by a factor of 1/2. $\endgroup$ – David Hammen Jul 23 '18 at 10:32

This follows from Rodrigues' rotation formula.

Define $\theta$ as the magnitude of the rotation, $\theta=\sqrt{\theta_x^2+\theta_y^2+\theta_z^2}$, and the matrix $\mathbf K$ as $$\mathbf K = \frac1{\theta}\,\begin{bmatrix} 0&-\theta_z &\phantom{-}\theta_y \\ \phantom{-}\theta_z & 0 & -\theta_x \\ -\theta_y & \phantom{-}\theta_x & 0 \end{bmatrix}$$ Note that the square of this matrix is $$\mathbf K^2 = \frac1{\theta^2}\,\begin{bmatrix} -\theta_y^2-\theta_z^2&\theta_x\theta_y &\theta_x\theta_z \\ \theta_x\theta_y & -\theta_x^2-\theta_z^2 & \theta_y\theta_z \\ \theta_x\theta_z & \theta_y\theta_z & -\theta_x^2-\theta_y^2 \end{bmatrix}$$

By Rodriques' rotation formula, the rotation matrix is $$\mathbf R = \mathbf I + \sin\theta\,\mathbf K + (1-\cos\theta)\,\mathbf K^2$$ To second order, this becomes $$\begin{aligned} \mathbf R &\approx \mathbf I + \theta\,\mathbf K + \frac12\theta^2\,\mathbf K^2 \\ &= \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} + \begin{bmatrix} 0&-\theta_z &\phantom{-}\theta_y \\ \phantom{-}\theta_z & 0 & -\theta_x \\ -\theta_y & \phantom{-}\theta_x & 0 \end{bmatrix} + \frac12\begin{bmatrix} -\theta_y^2-\theta_z^2&\theta_x\theta_y &\theta_x\theta_z \\ \theta_x\theta_y & -\theta_x^2-\theta_z^2 & \theta_y\theta_z \\ \theta_x\theta_z & \theta_y\theta_z & -\theta_x^2-\theta_y^2 \end{bmatrix} \end{aligned}$$

  • $\begingroup$ Thank you for your immediate and very helpful answer. The Rodrigues's rotation formula extend my understanding of the rotation matrix. You are definitely right that the coefficient before the fourth matrix should be 0.5, if the rotation matrix is constructed by Rodrigues's formula. However, it is found that coefficient becomes 1, when Taylor expansion is applied on a rotation by Euler angles. It is very bizarre. Thanks again. $\endgroup$ – Zihan Shen Jul 23 '18 at 12:12

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