Formulas for Euler Axis components

Question

I am trying to derive the formulas given in this book titled Atmoshperic and Spaceflight Dynamics by Ashish Tewari. I found questions related to finding the rotation matrices online but nothing that was meant for deriving these formulas for the euler axis. So, in chapter 2 Tewari writes how any combination of rotations can be expressed as a single rotation about a different axis called the euler axis, and that the euler axis is really just the eigenvector corresponding to the eigenvalue of 1. I followed this, but then he quickly shows that explicit formulas for the three components in R^3 are $$e_1=\frac{c_{23}-c_{32}}{2sin\Theta}$$ $$e_2=\frac{c_{31}-c_{13}}{2sin\Theta}$$ $$e_3=\frac{c_{12}-c_{21}}{2sin\Theta}$$ Tewari says he uses the following two equations can be used to solve for the components: $$Cc_1=c_1$$ $$Trace(C)=\lambda_1+\lambda_2+\lambda_3=1+e^{i\Theta}+e^{-i\Theta}=1+2cos\Theta $$ $$cos\Theta=\frac{1}{2}(Trace(C)-1)$$ Where $C$ is an orthonormal rotation matrix, and $c_1=(e_1,e_2,e_3)$ is the eigenvector with eigenvalue of 1. I was able to follow his explanations to this point, and I tried using the equations he points to, my work looks like the following.

$$Cc_1=1 \ c_1$$ Now solving for the corresponding eigenvector $$(C-I)c_1=0$$ $$\pmatrix{c_{11}-1 & c_{12} & c_{13} \\ c_{21} & c_{22}-1 & c_{23} \\ c_{31} & c_{32} & c_{33}-1 } \pmatrix{e_1 \\ e_2 \\ e_3}=0$$ After some experimentation I found these elimination matrices, soI didn't have to repeat Guassian Elimination every time I restarted from the beginning; they are $$E_1=\pmatrix{1 & 0 & 0 \\ -c_{21}/(c_{11}-1) & 1 & 0 \\ 0 & 0 & 1}$$ $$E_2=\pmatrix{1 & 0 &0 \\ 0 & 1 & 0 \\ -c_{31}/(c_{11}-1) & 0 & 1}$$ then defining $$C_2=E_2 E_1 (C-I) $$ $$C_2= \pmatrix{c_{11}-1 & c_{12} & c_{13} \\ 0 & c_{22}-\frac{c_{12}c_{21}}{c_{11}-1}-1 & c_{23}-\frac{c_{13}c_{21}}{c_{11}-1} \\ 0 & c_{32}-\frac{c_{12}c_{13}}{c_{11}-1} & c_{33}-\frac{c_{13}c_{21}}{c_{11}-1}-1 }$$

and $$E_3= \pmatrix{1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& \frac{ c_{32}-\frac{c_{12}c_{31}}{c_{11}-1} } { \frac{c_{12}c_{21}}{c_{11}-1} -c_{22}+1 } & 1 }$$

finally combining the three elimination matrices into $C_3$

$$ C_3 = \pmatrix{c_{11}-1 & c_{12} & c_{13} \\ 0 & c_{22}-\frac{c_{12}c_{21}}{c_{11}-1}-1 & c_{23}-\frac{c_{13}c_{21}}{c_{11}-1} \\ 0 & 0 & c_{33}+ \frac{(c_{23}-\frac{c_{13}c_{21}}{c_{11}-1})( c_{32}-\frac{c_{12}c_{31}}{c_{11}-1}) } {1-c_{22}-\frac{c_{12}c_{21}}{c_{11}-1}} -\frac{c_{13}c_{31}}{c_{11}-1} -1 }$$ and the following still holds $$C_3 c_1=E_3 C_2 = E_3 E_2 E_1 (C-I)c_1 =0$$ which is the same as $$\pmatrix{c_{11}-1 & c_{12} & c_{13} \\ 0 & c_{22}-\frac{c_{12}c_{21}}{c_{11}-1}-1 & c_{23}-\frac{c_{13}c_{21}}{c_{11}-1} \\ 0 & 0 & c_{33}+ \frac{(c_{23}-\frac{c_{13}c_{21}}{c_{11}-1})( c_{32}-\frac{c_{12}c_{31}}{c_{11}-1}) } {1-c_{22}-\frac{c_{12}c_{21}}{c_{11}-1}} -\frac{c_{13}c_{31}}{c_{11}-1} -1 } \pmatrix{e_1\\e_2\\e_3}=0$$

This is where I get stuck. Normally when solving for the eigenvectors we could set the third unknown variable ($e_3$ in this case) equal to $1$ and there would be some kind of relationship we could make between $e_3$ and $e_2$ and $e_1$. But here I'm not sure what relationship we can find. If we set $e_3=1$ it would mean $e_3$ would NOT be equal to the formula $e_3=\frac{c_{12}-c_{21}}{2sin\Theta}$

I also tried to use related information not mentioned by Tewari, such as the fact that the eigenvector's magnitude should be equal to 1, giving $$e_3 = (1-e_2^2-e_1^2)^{\frac{1}{2}}$$ and that $sin\Theta$ might be related to the $Trace(C)$ by $$sin\Theta = \sqrt{1-cos^2\Theta} = \sqrt{1- (\frac{1}{2}(Trace(C)-1))^{2}}=\\ = \sqrt{1-\frac{1}{4}(Trace(C)^2-2Trace(C)+1)} =\\ = \frac{1}{2}\sqrt{-Trace(C)^2+2Trace(C)+3}$$ But I am still stumped.

Answer 1

The simplest way to derive the mentioned formula for components of axis is to use directly Rodrigues formula.

$$C(e,\theta) =I+\sin(\theta)S(e)+(1-\cos(\theta))S^2(e)$$

where $S(e)$ is a skew-symmetric matrix corresponding to the axis vector $e$

$$S(e)=\begin{bmatrix} 0 & -e_3 & e_2 \\ e_3 & 0 & -e_1 \\ -e_2 & e_1 & 0 \end{bmatrix}$$

Additionally we have $$S^2(e)=ee^T-I$$ ($e$ has to be unit vector i.e $e_1^2+e_2^2+e_3^2=1$)

The formula for $e_1,e_2,e_3$ is derived from skew-symmmetrical part of Rodrigues formula i.e.

$$\dfrac{1}{2}(C -C^T)= \sin(\theta)S(e)$$,

btw the formula for cosine of rotation angle $\text{tr}(C)=1+2\cos(\theta)$ can be derived from the rest (symmetrical) part of rotation matrix i.e.

$$I+ (1-\cos(\theta))(ee^T-I)= (1-\cos(\theta)) ee^T +I\cos(\theta) $$

(trace of skew-symmetric part is equal to $0$ so we can neglect this part).