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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?

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  • $\begingroup$ You say difference but you write a product. Also the first notation $[X]_4$ seems different from the second $[X]_{a,b,c}$. Why don't you explain exactly what you want to say so that people can help you better $\endgroup$
    – lcv
    May 26, 2020 at 19:12
  • $\begingroup$ @Icv I edited the question to be more precise $\endgroup$
    – brohan322
    May 26, 2020 at 20:29
  • $\begingroup$ Maybe you could add what you mean with "transformation matrix". $\endgroup$
    – md2perpe
    May 26, 2020 at 21:55
  • $\begingroup$ It's a 4x4 homogenous transformation matrix, consisting of a 3x3 rotation matrix in the upper left corner and a 3x1 translation vector in the last column. $\endgroup$
    – brohan322
    May 26, 2020 at 21:59
  • $\begingroup$ Like in math.stackexchange.com/questions/2230290/… ? $\endgroup$
    – md2perpe
    May 26, 2020 at 22:19

1 Answer 1

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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 $R= XZX$.

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