別件でまとめてたのが没ったので供養．
自由に使ってもらってかまいませんが自己責任でお願いします．
ジャンクなので動作確認(検算)してません．
可読性のためにコード中に=だけの行をおいてますが，環境によってはここでバグる可能性があります．適宜改行を消して下さい．

ジャンクlatex数式コード置き場

別件でまとめてたのが没ったので供養．自由に使ってもらってかまいませんが自己責任でお願いします．ジャンクなので動作確認(検算)してません．可読性のためにコード中に=だけの行をおいてますが，環境によってはここでバグる可能性があります．適宜改行を消して下さい．

回転行列

x軸回転
$ R_x(\theta_x)= \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right) $

R_x(\theta_x)=
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)

y軸回転
$R_y(\theta_y)= \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right)$

R_y(\theta_y)=
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)

z軸回転
$R_z(\theta_z)= \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right)$

R_z(\theta_z)=
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)

${R'}^{T}R$

$\theta_i=\theta_i(s)$，$s$での微分を'で表記する．
${R}^{T}$は行列$R$の転置を示す．
${R'}_x^{{T}}R_x=$$\theta_x' \left(\begin{matrix} 0 & & \\ & -S\theta_x & C\theta_x \\ & -C\theta_x & -S\theta_x \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right) $$=\theta_x'\left( \begin{matrix} 0 & & \\ & 0 & 1 \\ & -1 & 0 \end{matrix} \right) $

{R'}_x^{T}R_x=
\theta_x'
\left(\begin{matrix} 
0 &           &            \\\\ 
  & -S\theta_x & C\theta_x \\\\
  & -C\theta_x & -S\theta_x
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=
\theta_x'\left(
\begin{matrix} 
0 &    &   \\\\ 
  &  0 & 1 \\\\
  & -1 & 0
\end{matrix} 
\right)

${R'}_y^{T}R_y= \theta_y' \left(\begin{matrix} -S\theta_y & & -C\theta_y \\ & 0 & \\ C\theta_y & & -S\theta_y \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right)= \theta_y' \left(\begin{matrix} 0 & & -1 \\ & 0 & \\ 1 & & 0 \end{matrix}\right)$

{R'}_y^{T}R_y=
\theta_y'
\left(\begin{matrix} 
-S\theta_y &   & -C\theta_y \\\\ 
           & 0 &            \\\\
 C\theta_y &   & -S\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
0 &   & -1 \\\\ 
  & 0 &    \\\\
1 &   &  0
\end{matrix}\right)

${R'}_z^{T}R_z= \theta_z' \left(\begin{matrix} -S\theta_z & C\theta_z & \\ -C\theta_z & -S\theta_z & \\ & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) =\theta_z' \left(\begin{matrix} 0 & 1 & \\ -1 & 0 & \\ & & 0 \end{matrix}\right)$

{R'}_z^{T}R_z=
\theta_z'
\left(\begin{matrix} 
-S\theta_z &  C\theta_z &   \\\\ 
-C\theta_z & -S\theta_z &   \\\\
           &            & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
 0 & 1 &    \\\\ 
-1 & 0 &    \\\\
   &   &  0
\end{matrix}\right)

$R_2^{T}{R_1'}^{T}R_1{R_2}$

1=x,2=y
$R_y^{T}{R_x'}^{T}R_x{R_y}= \theta_x' \left(\begin{matrix} C\theta_y & & -S\theta_y \\ & 1 & \\ S\theta_y & & C\theta_y \end{matrix}\right) \left(\begin{matrix} 0 & & \\ & 0 & 1 \\ & -1 & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) $ $R_y^{T}{R_x'}^{T}R_x{R_y}= \theta_x' \left(\begin{matrix} 0 & S\theta_y & \\ & 0 & 1 \\ & -C\theta_y & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) =\theta_x' \left(\begin{matrix} 0 & S\theta_y & \\ -S\theta_y & 0 & C\theta_y \\ & -C\theta_y & 0 \end{matrix}\right)$

R_y^{T}{R_x'}^{T}R_x{R_y}=
\theta_x'
\left(\begin{matrix} 
 C\theta_y &   & -S\theta_y \\\\ 
           & 1 &           \\\\
 S\theta_y &   &  C\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
0 &    &   \\\\ 
  &  0 & 1 \\\\
  & -1 & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
0 &  S\theta_y &   \\\\ 
  &  0         & 1 \\\\
  & -C\theta_y & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
0          &  S\theta_y &   \\\\ 
-S\theta_y &  0         & C\theta_y \\\\
           & -C\theta_y & 0
\end{matrix}\right)

1=x,2=z
$R_z^{T}{R_x'}^{T}R_x{R_z}= \theta_x' \left(\begin{matrix} C\theta_z & S\theta_z & \\ -S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) \left(\begin{matrix} 0 & & \\ & 0 & 1 \\ & -1 & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) $ $ R_z^{T}{R_x'}^{T}R_x{R_z}= \theta_x' \left(\begin{matrix} 0 & & S\theta_z \\ & 0 & C\theta_z \\ & -1 & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) =\theta_x'\left(\begin{matrix} 0 & & S\theta_z \\ 0 & 0 & C\theta_z \\ -S\theta_z & -C\theta_z & 0 \end{matrix}\right) $

R_z^{T}{R_x'}^{T}R_x{R_z}=
\theta_x'
\left(\begin{matrix} 
 C\theta_z & S\theta_z &   \\\\ 
-S\theta_z & C\theta_z &   \\\\
           &            &  1
\end{matrix}\right)
\left(\begin{matrix} 
0 &    &   \\\\ 
  &  0 & 1 \\\\
  & -1 & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
0 &    & S\theta_z  \\\\ 
  &  0 & C\theta_z \\\\
  & -1 & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=\theta_x'\left(\begin{matrix} 
 0         &            & S\theta_z  \\\\ 
 0         &  0         & C\theta_z \\\\
-S\theta_z & -C\theta_z & 0 
\end{matrix}\right)

1=y,2=x
$R_x^{T}{R_y'}^{T}R_y{R_x}= \theta_y' \left(\begin{matrix} 1 & & \\ & C\theta_x & S\theta_x \\ & -S\theta_x & C\theta_x \end{matrix}\right) \left(\begin{matrix} 0 & & -1 \\ & 0 & \\ 1 & & 0 \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right)$ $ R_x^{T}{R_y'}^{T}R_y{R_x}= \theta_y' \left(\begin{matrix} 0 & & -1 \\ S\theta_x & 0 & \\ C\theta_x & & 0 \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right) =\theta_y' \left(\begin{matrix} 0 & -S\theta_x & -C\theta_x \\ S\theta_x & 0 & \\ C\theta_x & & 0 \end{matrix}\right)$

R_x^{T}{R_y'}^{T}R_y{R_x}=
\theta_y'
\left(\begin{matrix} 
1 &            &            \\\\ 
  &  C\theta_x &  S\theta_x \\\\
  & -S\theta_x &  C\theta_x
\end{matrix}\right)
\left(\begin{matrix} 
0 &   & -1 \\\\ 
  & 0 &    \\\\
1 &   &  0
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
0         &   & -1 \\\\ 
S\theta_x & 0 &    \\\\
C\theta_x &   &  0
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
0          & -S\theta_x & -C\theta_x \\\\ 
S\theta_x  &  0         &  \\\\
C\theta_x  &            & 0
\end{matrix}\right)

1=y,2=z
$R_z^{T}{R_y'}^{T}R_y{R_z}= \theta_y' \left(\begin{matrix} C\theta_z & S\theta_z & \\ -S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) \left(\begin{matrix} 0 & & -1 \\ & 0 & \\ 1 & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) $ $R_z^{T}{R_y'}^{T}R_y{R_z}= \theta_y' \left(\begin{matrix} 0 & & -C\theta_z \\ & 0 & S\theta_z \\ 1 & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) =\theta_y' \left(\begin{matrix} 0 & & -C\theta_z \\ & 0 & S\theta_z \\ C\theta_z & -S\theta_z & 0 \end{matrix}\right)$

R_z^{T}{R_y'}^{T}R_y{R_z}=
\theta_y'
\left(\begin{matrix} 
 C\theta_z & S\theta_z &   \\\\ 
-S\theta_z & C\theta_z &   \\\\
           &            &  1
\end{matrix}\right)
\left(\begin{matrix} 
0 &    & -1 \\\\ 
  &  0 &    \\\\
1 &    & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
0 &    & -C\theta_z \\\\ 
  &  0 &  S\theta_z \\\\
1 &    & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
0         &            & -C\theta_z \\\\ 
          &  0         &  S\theta_z \\\\
C\theta_z & -S\theta_z & 0
\end{matrix}\right)

1=z,2=x
$R_x^{T}{R_z'}^{T}R_z{R_x}= \theta_z' \left(\begin{matrix} 1 & & \\ & C\theta_x & S\theta_x \\ & -S\theta_x & C\theta_x \end{matrix}\right) \left(\begin{matrix} 0 & 1 & \\ -1 & 0 & \\ & & 0 \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right)$ $ R_x^{T}{R_z'}^{T}R_z{R_x}= \theta_z' \left(\begin{matrix} 0 & 1 & \\ -C\theta_x & 0 & \\ S\theta_x & & 0 \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right) =\theta_z' \left(\begin{matrix} 0 & C\theta_x & -S\theta_x \\ -C\theta_x & 0 & \\ S\theta_x & & 0 \end{matrix}\right)$

R_x^{T}{R_z'}^{T}R_z{R_x}=
\theta_z'
\left(\begin{matrix} 
1 &            &            \\\\ 
  &  C\theta_x &  S\theta_x \\\\
  & -S\theta_x &  C\theta_x
\end{matrix}\right)
\left(\begin{matrix} 
0  & 1 &  \\\\ 
-1 & 0 &  \\\\
   &   & 0
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
 0         & 1 &   \\\\ 
-C\theta_x & 0 &   \\\\
 S\theta_x &   &  0
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=\theta_z'
\left(\begin{matrix} 
0           & C\theta_x & -S\theta_x \\\\ 
-C\theta_x  & 0         &  \\\\
 S\theta_x  &           & 0
\end{matrix}\right)

1=z,2=y
$R_y^{T}{R_z'}^{T}R_z{R_y}= \theta_z' \left(\begin{matrix} C\theta_y & & -S\theta_y \\ & 1 & \\ S\theta_y & & C\theta_y \end{matrix}\right) \left(\begin{matrix} 0 & 1 & \\ -1 & 0 & \\ & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) $ $R_y^{T}{R_x'}^{T}R_x{R_y}= \theta_z' \left(\begin{matrix} 0 & C\theta_y & \\ -1 & 0 & \\ & S\theta_y & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) =\theta_z' \left(\begin{matrix} 0 & C\theta_y & \\ -C\theta_y & 0 & -S\theta_y \\ & S\theta_y & 0 \end{matrix}\right)$

R_y^{T}{R_z'}^{T}R_z{R_y}=
\theta_z'
\left(\begin{matrix} 
 C\theta_y &   & -S\theta_y \\\\ 
           & 1 &           \\\\
 S\theta_y &   &  C\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
 0 &  1 &   \\\\ 
-1 &  0 &   \\\\
   &    & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
0  & C\theta_y &   \\\\ 
-1 & 0         &   \\\\
   & S\theta_y & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
 0         & C\theta_y &   \\\\ 
-C\theta_y & 0         & -S\theta_y \\\\
           & S\theta_y &  0
\end{matrix}\right)

$R_3^{T}R_2^{T}{R_1'}^{T}R_1{R_2}R_3$

1=x,2=y,3=z
$R_z^{T}R_y^{T}{R_x'}^{T}R_x{R_y}R_z= \theta_x' \left(\begin{matrix} C\theta_z & S\theta_z & \\ -S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) \left(\begin{matrix} 0 & S\theta_y & \\ -S\theta_y & 0 & C\theta_y \\ & -C\theta_y & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) $ $R_y^{T}{R_x'}^{T}R_x{R_y}= \theta_x' \left(\begin{matrix} -S\theta_yS\theta_z & S\theta_yC\theta_z & C\theta_yS\theta_z \\ -S\theta_yC\theta_z & -S\theta_yS\theta_z & C\theta_yC\theta_z \\ & -C\theta_y & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) =\theta_x' \left(\begin{matrix} 0 & S\theta_y & C\theta_yS\theta_z \\ -S\theta_y & 0 & C\theta_yC\theta_z \\ -C\theta_yS\theta_z & -C\theta_yC\theta_z & 0 \end{matrix}\right)$

R_z^{T}R_y^{T}{R_x'}^{T}R_x{R_y}R_z=
\theta_x'
\left(\begin{matrix} 
 C\theta_z & S\theta_z &   \\\\ 
-S\theta_z & C\theta_z &   \\\\
           &            &  1
\end{matrix}\right)
\left(\begin{matrix} 
0          &  S\theta_y &   \\\\ 
-S\theta_y &  0         & C\theta_y \\\\
           & -C\theta_y & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_z & -S\theta_z &   \\\\ 
 S\theta_z &  C\theta_z &   \\\\
           &            &  1
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
-S\theta_yS\theta_z &  S\theta_yC\theta_z & C\theta_yS\theta_z  \\\\ 
-S\theta_yC\theta_z & -S\theta_yS\theta_z & C\theta_yC\theta_z \\\\
                    & -C\theta_y          & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_z & -S\theta_z &   \\\\ 
 S\theta_z &  C\theta_z &   \\\\
           &            &  1
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
0                   &  S\theta_y          & C\theta_yS\theta_z  \\\\ 
-S\theta_y          &  0                  & C\theta_yC\theta_z \\\\
-C\theta_yS\theta_z & -C\theta_yC\theta_z & 0
\end{matrix}\right)

1=x,2=z,3=y
$R_y^{T}R_z^{T}{R_x'}^{T}R_x{R_z}R_y= \theta_x' \left(\begin{matrix} C\theta_y & & -S\theta_y \\ & 1 & \\ S\theta_y & & C\theta_y \end{matrix}\right) \left(\begin{matrix} 0 & & S\theta_z \\ 0 & 0 & C\theta_z \\ -S\theta_z & -C\theta_z & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) $ $ R_z^{T}{R_x'}^{T}R_x{R_z}= \theta_x' \left(\begin{matrix} S\theta_yS\theta_z & S\theta_yC\theta_z & C\theta_yS\theta_z \\ & 0 & C\theta_z \\ -C\theta_yS\theta_z & -C\theta_yC\theta_z & S\theta_yS\theta_z \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) =\theta_x' \left(\begin{matrix} 0 & S\theta_yC\theta_z & S\theta_z \\ -S\theta_yC\theta_z & 0 & C\theta_yC\theta_z \\ -S\theta_z & -C\theta_yC\theta_z & 0 \end{matrix}\right) $

R_y^{T}R_z^{T}{R_x'}^{T}R_x{R_z}R_y=
\theta_x'
\left(\begin{matrix} 
 C\theta_y &   & -S\theta_y \\\\ 
           & 1 &           \\\\
 S\theta_y &   &  C\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
 0         &            & S\theta_z  \\\\ 
 0         &  0         & C\theta_z \\\\
-S\theta_z & -C\theta_z & 0 
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
 S\theta_yS\theta_z &  S\theta_yC\theta_z & C\theta_yS\theta_z  \\\\ 
                    &  0                  & C\theta_z \\\\
-C\theta_yS\theta_z & -C\theta_yC\theta_z & S\theta_yS\theta_z
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_x'
\left(\begin{matrix} 
  0                  &  S\theta_yC\theta_z & S\theta_z  \\\\ 
 -S\theta_yC\theta_z &  0                  & C\theta_yC\theta_z \\\\
 -S\theta_z          & -C\theta_yC\theta_z & 0 
\end{matrix}\right)

1=y,2=x,3=z
$R_z^{T}R_x^{T}{R_y'}^{T}R_y{R_x}R_z= \theta_y' \left(\begin{matrix} C\theta_z & S\theta_z & \\ -S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) \left(\begin{matrix} 0 & -S\theta_x & -C\theta_x \\ S\theta_x & 0 & \\ C\theta_x & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right)$ $R_z^{T}R_x^{T}{R_y'}^{T}R_y{R_x}R_z= \theta_y' \left(\begin{matrix} S\theta_xS\theta_z & -S\theta_xC\theta_z & -C\theta_xC\theta_z \\ S\theta_xC\theta_z & S\theta_xS\theta_z & C\theta_xS\theta_z\\ C\theta_x & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_z & -S\theta_z & \\ S\theta_z & C\theta_z & \\ & & 1 \end{matrix}\right) =\theta_y' \left(\begin{matrix} 0 & -S\theta_x & -C\theta_xC\theta_z \\ S\theta_x & 0 & C\theta_xS\theta_z\\ C\theta_xC\theta_z & -C\theta_xS\theta_z & 0 \end{matrix}\right) $

R_z^{T}R_x^{T}{R_y'}^{T}R_y{R_x}R_z=
\theta_y'
\left(\begin{matrix} 
 C\theta_z & S\theta_z &   \\\\ 
-S\theta_z & C\theta_z &   \\\\
           &           &  1
\end{matrix}\right)
\left(\begin{matrix} 
0          & -S\theta_x & -C\theta_x \\\\ 
S\theta_x  &  0         &  \\\\
C\theta_x  &            & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
S\theta_xS\theta_z & -S\theta_xC\theta_z & -C\theta_xC\theta_z \\\\ 
S\theta_xC\theta_z &  S\theta_xS\theta_z &  C\theta_xS\theta_z\\\\
C\theta_x          &                     & 0
\end{matrix}\right)
\left(\begin{matrix} 
C\theta_z & -S\theta_z &   \\\\ 
S\theta_z &  C\theta_z &   \\\\
          &            &  1
\end{matrix}\right)
=
\theta_y'
\left(\begin{matrix} 
0                   & -S\theta_x          & -C\theta_xC\theta_z \\\\ 
S\theta_x           &  0                  &  C\theta_xS\theta_z\\\\
C\theta_xC\theta_z  & -C\theta_xS\theta_z &  0
\end{matrix}\right)

1=y,2=z,3=x
$R_x^{T}{R_z}^{T}{R_y'}^{T}R_y{R_z}{R_x}= \theta_y' \left(\begin{matrix} 1 & & \\ & C\theta_x & S\theta_x \\ & -S\theta_x & C\theta_x \end{matrix}\right) \left(\begin{matrix} 0 & & -C\theta_z \\ & 0 & S\theta_z \\ C\theta_z & -S\theta_z & 0 \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right)$ $ R_x^{T}{R_z}^{T}{R_y'}^{T}R_y{R_z}{R_x}= \theta_y' \left(\begin{matrix} 0 & & -C\theta_z \\ S\theta_xC\theta_z & -S\theta_xS\theta_z & C\theta_xS\theta_z \\ C\theta_xC\theta_z & -C\theta_xS\theta_z & -S\theta_xS\theta_z \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right) =\theta_y' \left(\begin{matrix} 0 & -S\theta_xC\theta_z & -C\theta_xC\theta_z \\ S\theta_xC\theta_z & 0 & S\theta_z \\ C\theta_xC\theta_z & -S\theta_z & 0 \end{matrix}\right)$

R_x^{T}{R_z}^{T}{R_y'}^{T}R_y{R_z}{R_x}=
\theta_y'
\left(\begin{matrix} 
1 &            &            \\\\ 
  &  C\theta_x &  S\theta_x \\\\
  & -S\theta_x &  C\theta_x
\end{matrix}\right)
\left(\begin{matrix} 
0         &            & -C\theta_z \\\\ 
          &  0         &  S\theta_z \\\\
C\theta_z & -S\theta_z & 0
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)$
$
R_x^{T}{R_z}^{T}{R_y'}^{T}R_y{R_z}{R_x}=
\theta_y'
\left(\begin{matrix} 
0                  &                     & -C\theta_z \\\\ 
S\theta_xC\theta_z & -S\theta_xS\theta_z &  C\theta_xS\theta_z \\\\
C\theta_xC\theta_z & -C\theta_xS\theta_z & -S\theta_xS\theta_z
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=\theta_y'
\left(\begin{matrix} 
0                  & -S\theta_xC\theta_z & -C\theta_xC\theta_z \\\\ 
S\theta_xC\theta_z &  0                  & S\theta_z \\\\
C\theta_xC\theta_z & -S\theta_z          & 0
\end{matrix}\right)

1=z,2=x,3=y
$R_y^{T}R_x^{T}{R_z'}^{T}R_z{R_x}R_y= \theta_z' \left(\begin{matrix} C\theta_y & & -S\theta_y \\ & 1 & \\ S\theta_y & & C\theta_y \end{matrix}\right) \left(\begin{matrix} 0 & C\theta_x & -S\theta_x \\ -C\theta_x & 0 & \\ S\theta_x & & 0 \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) $ $ R_y^{T}R_x^{T}{R_z'}^{T}R_z{R_x}R_y= \theta_z' \left(\begin{matrix} -S\theta_xS\theta_y & C\theta_xC\theta_y & -S\theta_xC\theta_y \\ -C\theta_x & 0 & \\ S\theta_xC\theta_y & C\theta_xS\theta_y & -S\theta_xS\theta_y \end{matrix}\right) \left(\begin{matrix} C\theta_y & & S\theta_y \\ & 1 & \\ -S\theta_y & & C\theta_y \end{matrix}\right) =\theta_z' \left(\begin{matrix} 0 & C\theta_xC\theta_y & -S\theta_x \\ -C\theta_xC\theta_y & 0 & -C\theta_xS\theta_y \\ S\theta_x & C\theta_xS\theta_y & 0 \end{matrix}\right) $

R_y^{T}R_x^{T}{R_z'}^{T}R_z{R_x}R_y=
\theta_z'
\left(\begin{matrix} 
 C\theta_y &   & -S\theta_y \\\\ 
           & 1 &           \\\\
 S\theta_y &   & C\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
0           & C\theta_x & -S\theta_x \\\\ 
-C\theta_x  & 0         &  \\\\
 S\theta_x  &           & 0
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
-S\theta_xS\theta_y &  C\theta_xC\theta_y & -S\theta_xC\theta_y  \\\\ 
-C\theta_x          &  0                  &  \\\\
 S\theta_xC\theta_y &  C\theta_xS\theta_y & -S\theta_xS\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
 C\theta_y &   & S\theta_y \\\\ 
           & 1 &           \\\\
-S\theta_y &   & C\theta_y
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
  0                  &  C\theta_xC\theta_y & -S\theta_x  \\\\ 
 -C\theta_xC\theta_y &  0                  & -C\theta_xS\theta_y \\\\
  S\theta_x          &  C\theta_xS\theta_y &  0 
\end{matrix}\right)

1=z,2=y,3=x
$R_x^{T}{R_y}^{T}{R_z'}^{T}{R_y}R_z{R_x}= \theta_z' \left(\begin{matrix} 1 & & \\ & C\theta_x & S\theta_x \\ & -S\theta_x & C\theta_x \end{matrix}\right) \left(\begin{matrix} 0 & C\theta_y & \\ -C\theta_y & 0 & -S\theta_y \\ & S\theta_y & 0 \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right)$ $ R_x^{T}{R_y}^{T}{R_z'}^{T}{R_y}R_z{R_x}= \theta_z' \left(\begin{matrix} 0 & C\theta_y & \\ -C\theta_xC\theta_y & S\theta_xS\theta_y & -C\theta_xS\theta_y \\ S\theta_xC\theta_y & C\theta_xS\theta_y & S\theta_xS\theta_y \end{matrix}\right) \left(\begin{matrix} 1 & & \\ & C\theta_x & -S\theta_x \\ & S\theta_x & C\theta_x \end{matrix}\right) =\theta_z' \left(\begin{matrix} 0 & C\theta_xC\theta_y & -S\theta_xC\theta_y \\ -C\theta_xC\theta_y & 0 & -S\theta_y \\ S\theta_xC\theta_y & S\theta_y & 0 \end{matrix}\right)$

R_x^{T}{R_y}^{T}{R_z'}^{T}{R_y}R_z{R_x}=
\theta_z'
\left(\begin{matrix} 
1 &            &            \\\\ 
  &  C\theta_x &  S\theta_x \\\\
  & -S\theta_x &  C\theta_x
\end{matrix}\right)
\left(\begin{matrix} 
 0         & C\theta_y &   \\\\ 
-C\theta_y & 0         & -S\theta_y \\\\
           & S\theta_y &  0
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=
\theta_z'
\left(\begin{matrix} 
 0                  & C\theta_y          &   \\\\ 
-C\theta_xC\theta_y & S\theta_xS\theta_y & -C\theta_xS\theta_y  \\\\
 S\theta_xC\theta_y & C\theta_xS\theta_y &  S\theta_xS\theta_y
\end{matrix}\right)
\left(\begin{matrix} 
1 &           &            \\\\ 
  & C\theta_x & -S\theta_x \\\\
  & S\theta_x &  C\theta_x
\end{matrix}\right)
=\theta_z'
\left(\begin{matrix} 
 0                  & C\theta_xC\theta_y & -S\theta_xC\theta_y  \\\\ 
-C\theta_xC\theta_y & 0                  & -S\theta_y  \\\\
 S\theta_xC\theta_y & S\theta_y          &  0
\end{matrix}\right)

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