I am writing a research paper for an engineering journal and I am having difficulty writing a couple of simple matrix equations.
For equation 1, I have a $4 \times 4$ transformation matrix $T$. I want to define $\delta_x$ where $\delta_x$ is the norm of the translation vector of $T$.
For equation 2, I have the same transformation matrix $T$. I want to define $\delta_\theta$ where $\delta_\theta$ is the norm of the vector $<\alpha, \beta, \gamma >$, and $\alpha, \beta, \gamma$ are the Euler angles of the $3 \times 3$ rotation matrix of $T$.
Is there a correct way to write these equations?
Best Answer
For problem 1) I would do the following.
Let $T$ be the following $4\times4$ transformation matrix
$$ T=\left(\begin{array}{cc} R & x\\ 0 & 1 \end{array}\right) $$
where $R$ is a $3\times3$ rotation (i.e. orthogonal) matrix and $x$ a $3\times 1$ (shift) vector.
Then we define
$$ \delta_x = \parallel x \parallel. $$
For part 2 I would suggest:
Let now decompose $R$ into three elementary rotations defining Euler angles, i.e.
$$ R = X(\alpha) Y(\beta) Z(\gamma) \tag{1} $$
where $X(\phi),Y(\phi),Z(\phi)$ represent rotations around their respective axis by an angle $\phi$. Let $\theta$ be the $3\times 1$ vector of angles $\theta=(\alpha,\beta,\gamma)$.
Then
$$\delta_\theta = \parallel \theta \parallel $$
PS
Note that I'm not sure that Eq. (1) is the definition you use for Euler angles. Another possibly more standard one is $R= XZX$.