how to achieve and draw the final rotation of a frame with Euler angles.

Question

I would like to do the final rotation using Euler angles. In this link Euler Angles, the final rotation undergoes three successive rotations, starting from frame (x,y,z), going through (x',y',z') and then (x'',y'',z'') and finally ending with frame (x''',y''',z'''). I've seen this post and it helped me to some extend. In my case, the first rotation yields

The second rotation yields

Now the third rotation doesn't yield what in the aforementioned link.

x2, x3 should point in the same direction.

This is the code

\documentclass[preview,varwidth ,multi,border=3pt]{standalone}

\usepackage{tikz}  
\usepackage{tikz-3dplot} 
\usepackage[active,tightpage]{preview}  
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{1cm}


\begin{document}

\tdplotsetmaincoords{70}{100}

\begin{tikzpicture}[scale=5,tdplot_main_coords]

\coordinate (O) at (0,0,0);

% main Frame (x,y,z)
\draw[thick,->,line width=0.55mm] (O) -- (2.0,0,0) node[anchor=north west]{$X$};
\draw[thick,->,line width=0.55mm] (O) -- (0,2.0,0) node[anchor=west]      {$Y$};
\draw[thick,->,line width=0.55mm] (O) -- (0,0,2.0) node[anchor=south]     {$Z$};


% define yaw, pitch, and roll
\def \yaw   {30}
\def \roll  {30}
\def \pitch {30}

%_____________________________________________________________________
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%( Yaw )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% First rotation yields Frame 1 (about z axis main)
% Frame 1 (x1,y1, z1)
\tdplotsetrotatedcoords{\yaw}{0}{0}


\draw[thick,->,line width=0.65mm, color=red, tdplot_rotated_coords] 
(O) -- (1.5,0,0) node[xshift=0mm,anchor=north west] {$x^{1}$};
\draw[thick,->,line width=0.65mm, color=red, tdplot_rotated_coords]  
(O) -- (0,1.5,0) node[anchor=west]{$y^{1}$};
\draw[thick,->,line width=0.65mm, color=red, tdplot_rotated_coords]  
(O) -- (0,0,1.5)  node[anchor=south,xshift=3mm, yshift=-1mm]{$z^{1}$};

%_____________________________________________________________________
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%( Pitch )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Second rotation yields Frame 2 (about y1 axis)
% Frame 2 (x2,y2, z2)
\tdplotsetrotatedcoords{\yaw}{\pitch}{0}


\draw[thick,->,line width=0.65mm, color=blue, tdplot_rotated_coords] 
(O) -- (1.0,0,0) node[xshift=0mm,anchor=north west]{$x^{2}$};
\draw[thick,->,line width=0.65mm, color=blue,tdplot_rotated_coords]  
(O) -- (0,1.0,0) node[anchor=west]{$y^{2}$};
\draw[thick,->,line width=0.65mm, color=blue,tdplot_rotated_coords]  
(O) -- (0,0,1.0)  node[anchor=south,xshift=3mm, yshift=-1mm]{$z^{2}$};


%_____________________________________________________________________
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%( Roll )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Third rotation yields Frame 3 (about x2 axis) <-- (it should be )
% Frame 3 (x3,y3,z3)
\tdplotsetrotatedcoords{\yaw}{\pitch}{\roll}


\draw[thick,->,line width=0.65mm, color=green, tdplot_rotated_coords] 
(O) -- (0.5,0,0) node[xshift=0mm,anchor=north west]{$x^{3}$};
\draw[thick,->,line width=0.65mm, color=green,tdplot_rotated_coords]  
(O) -- (0,0.5,0) node[anchor=west]{$y^{3}$};
\draw[thick,->,line width=0.65mm, color=green,tdplot_rotated_coords]  
(O) -- (0,0,.5)  node[anchor=south]{$z^{3}$};


\end{tikzpicture}


\end{document}

Answer 1

I think this may be a duplicate of Navigation system coordinates in tikz-3dplot: By default, tikz-3dplot uses the rotation sequence ZYZ (that is it rotates around the Z axis first, then around the new Y axis Y', then around the new Z axis Z''), while you want the rotation sequence ZYX. For this, you have to redefine the transformation parameters:

\newcommand{\tdseteulerxyz}{
\renewcommand{\tdplotcalctransformrotmain}{%
%perform some trig for the Euler transformation
\tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
\tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
\tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
%
\tdplotmult{\sasb}{\sinalpha}{\sinbeta}
\tdplotmult{\sasg}{\sinalpha}{\singamma}
\tdplotmult{\sasbsg}{\sasb}{\singamma}
%
\tdplotmult{\sacb}{\sinalpha}{\cosbeta}
\tdplotmult{\sacg}{\sinalpha}{\cosgamma}
\tdplotmult{\sasbcg}{\sasb}{\cosgamma}
%
\tdplotmult{\casb}{\cosalpha}{\sinbeta}
\tdplotmult{\cacb}{\cosalpha}{\cosbeta}
\tdplotmult{\cacg}{\cosalpha}{\cosgamma}
\tdplotmult{\casg}{\cosalpha}{\singamma}
%
\tdplotmult{\cbsg}{\cosbeta}{\singamma}
\tdplotmult{\cbcg}{\cosbeta}{\cosgamma}
%
\tdplotmult{\casbsg}{\casb}{\singamma}
\tdplotmult{\casbcg}{\casb}{\cosgamma}
%
%determine rotation matrix elements for Euler transformation
\pgfmathsetmacro{\raaeul}{\cacb}
\pgfmathsetmacro{\rabeul}{\casbsg - \sacg}
\pgfmathsetmacro{\raceul}{\sasg + \casbcg}
\pgfmathsetmacro{\rbaeul}{\sacb}
\pgfmathsetmacro{\rbbeul}{\sasbsg + \cacg}
\pgfmathsetmacro{\rbceul}{\sasbcg - \casg}
\pgfmathsetmacro{\rcaeul}{-\sinbeta}
\pgfmathsetmacro{\rcbeul}{\cbsg}
\pgfmathsetmacro{\rcceul}{\cbcg}
}
}

Then you can activate the new rotation sequence using \tdseteulerxyz.

Here's the complete code:

\documentclass[border=3pt]{standalone}

\usepackage{tikz}  
\usepackage{tikz-3dplot}


\newcommand{\tdseteulerxyz}{
\renewcommand{\tdplotcalctransformrotmain}{%
%perform some trig for the Euler transformation
\tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
\tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
\tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
%
\tdplotmult{\sasb}{\sinalpha}{\sinbeta}
\tdplotmult{\sasg}{\sinalpha}{\singamma}
\tdplotmult{\sasbsg}{\sasb}{\singamma}
%
\tdplotmult{\sacb}{\sinalpha}{\cosbeta}
\tdplotmult{\sacg}{\sinalpha}{\cosgamma}
\tdplotmult{\sasbcg}{\sasb}{\cosgamma}
%
\tdplotmult{\casb}{\cosalpha}{\sinbeta}
\tdplotmult{\cacb}{\cosalpha}{\cosbeta}
\tdplotmult{\cacg}{\cosalpha}{\cosgamma}
\tdplotmult{\casg}{\cosalpha}{\singamma}
%
\tdplotmult{\cbsg}{\cosbeta}{\singamma}
\tdplotmult{\cbcg}{\cosbeta}{\cosgamma}
%
\tdplotmult{\casbsg}{\casb}{\singamma}
\tdplotmult{\casbcg}{\casb}{\cosgamma}
%
%determine rotation matrix elements for Euler transformation
\pgfmathsetmacro{\raaeul}{\cacb}
\pgfmathsetmacro{\rabeul}{\casbsg - \sacg}
\pgfmathsetmacro{\raceul}{\sasg + \casbcg}
\pgfmathsetmacro{\rbaeul}{\sacb}
\pgfmathsetmacro{\rbbeul}{\sasbsg + \cacg}
\pgfmathsetmacro{\rbceul}{\sasbcg - \casg}
\pgfmathsetmacro{\rcaeul}{-\sinbeta}
\pgfmathsetmacro{\rcbeul}{\cbsg}
\pgfmathsetmacro{\rcceul}{\cbcg}
}
}

\begin{document}
\tdseteulerxyz
\tdplotsetmaincoords{70}{100}

\begin{tikzpicture}[scale=5,tdplot_main_coords]

\coordinate (O) at (0,0,0);

% main Frame (x,y,z)
\draw[thick,->,line width=0.55mm] (O) -- (2.0,0,0) node[anchor=north west]{$X$};
\draw[thick,->,line width=0.55mm] (O) -- (0,2.0,0) node[anchor=west]      {$Y$};
\draw[thick,->,line width=0.55mm] (O) -- (0,0,2.0) node[anchor=south]     {$Z$};


% define yaw, pitch, and roll
\def \yaw   {30}
\def \roll  {30}
\def \pitch {30}

%_____________________________________________________________________
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%( Yaw )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% First rotation yields Frame 1 (about z axis main)
% Frame 1 (x1,y1, z1)
\tdplotsetrotatedcoords{\yaw}{0}{0}


\draw[thick,->,line width=0.65mm, color=red, tdplot_rotated_coords] 
(O) -- (1.5,0,0) node[xshift=0mm,anchor=north west] {$x^{1}$};
\draw[thick,->,line width=0.65mm, color=red, tdplot_rotated_coords]  
(O) -- (0,1.5,0) node[anchor=west]{$y^{1}$};
\draw[thick,->,line width=0.65mm, color=red, tdplot_rotated_coords]  
(O) -- (0,0,1.5)  node[anchor=south,xshift=3mm, yshift=-1mm]{$z^{1}$};

%_____________________________________________________________________
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%( Pitch )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Second rotation yields Frame 2 (about y1 axis)
% Frame 2 (x2,y2, z2)
\tdplotsetrotatedcoords{\yaw}{\pitch}{0}


\draw[thick,->,line width=0.65mm, color=blue, tdplot_rotated_coords] 
(O) -- (1.0,0,0) node[xshift=0mm,anchor=north west]{$x^{2}$};
\draw[thick,->,line width=0.65mm, color=blue,tdplot_rotated_coords]  
(O) -- (0,1.0,0) node[anchor=west]{$y^{2}$};
\draw[thick,->,line width=0.65mm, color=blue,tdplot_rotated_coords]  
(O) -- (0,0,1.0)  node[anchor=south,xshift=3mm, yshift=-1mm]{$z^{2}$};


%_____________________________________________________________________
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%( Roll )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Third rotation yields Frame 3 (about x2 axis) <-- (it should be )
% Frame 3 (x3,y3,z3)
\tdplotsetrotatedcoords{\yaw}{\pitch}{\roll}


\draw[thick,->,line width=0.65mm, color=green, tdplot_rotated_coords] 
(O) -- (0.5,0,0) node[xshift=0mm,anchor=north west]{$x^{3}$};
\draw[thick,->,line width=0.65mm, color=green,tdplot_rotated_coords]  
(O) -- (0,0.5,0) node[anchor=west]{$y^{3}$};
\draw[thick,->,line width=0.65mm, color=green,tdplot_rotated_coords]  
(O) -- (0,0,.5)  node[anchor=south]{$z^{3}$};


\end{tikzpicture}


\end{document}