I'm using 3d and perspective TikZ libraries for this. I'm also creating three styles:
axis, to change the axis style if you need. The great part of the drawing are axis, so with this we avoid to change them one by one.my view, simply to change the view at the same time in all images.nutation, to change the axis accordingly to the 'nutation' angle.
With this we only need a couple of scopes, one for each image and one more for the central one (the 'nutation' part).
This is my code:
\usetikzlibrary{3d,perspective}
\tikzset
{
axis/.style={thick,-latex},
my view/.style={3d view={65}{20}},
nutation/.style={rotate around x=\nut}
}
\colorlet{my cyan}{cyan!50}
\colorlet{my magenta}{magenta!50!pink}
\begin{document}
\begin{tikzpicture}[line cap=round]
\def\pre{-30} % précession
\def\nut{40} % nutation
\def\rop{30} % rotation propre
% left
\begin{scope}[shift={(-7,0)},my view]
\draw[fill=my cyan] (0,0) circle (2);
\draw[axis] (0,0,0) -- (4,0,0) node[below] {$\vec x_1$};
\draw[axis] (0,0,0) -- (0,4,0) node[right] {$\vec y_1$};
\draw[axis] (0,0,0) -- (0,0,4) node[above] {$\vec z_f=\vec z_1$};
\draw[axis] (0,0,0) -- (\pre:4) node[below] {$\vec x_f$};
\draw[axis] (0,0,0) -- (90+\pre:4) node[below] {$\vec y_f$};
\foreach\i in {0,90}
\draw[-latex] (\i+\pre:2.5) arc (\i+\pre:\i:2.5)
node at (\i+0.5\pre:3) {$\psi$};
\draw[canvas is xy plane at z=2,->] (90:0.5) arc (90:360:0.5) node[pos=0,right] {Précession $\psi$};
\end{scope}
% center
\begin{scope}[my view]
\draw[fill=my magenta,nutation] (2,0) arc (0:-180:2) -- cycle;
\draw[fill=my cyan] (0,0) circle (2);
\draw[axis] (0,0,0) -- (0,4,0) node[right] {$\vec y_1$};
\begin{scope}[nutation]
\draw[fill=my magenta] (2,0) arc (0:180:2) -- cycle;
\draw[axis] (0,0,0) -- (0,4,0) node[above] {$\vec y_2$};
\draw[axis] (0,0,0) -- (0,0,4) node[above] {$\vec z_2$};
\end{scope}
\foreach\i in {0,90}
\draw[my view,canvas is yz plane at x=0,-latex] (\i:2.5) arc (\i:\i+\nut:2.5)
node at (\i+0.5\nut:3) {$\theta$};
\draw[axis] (0,0,0) -- (4,0,0) node[below] {$\vec x_1=\vec x_2$};
\draw[axis] (0,0,0) -- (0,0,4) node[above] {$\vec z_1$};
\draw[canvas is yz plane at x=2,->] (-90:0.5) arc (-90:170:0.5) node[pos=0.3,right] {Nutation $\theta$};
\end{scope}
% right
\begin{scope}[shift={(7,0)},my view,nutation]
\draw[fill=my magenta] (0,0) circle (2);
\draw[axis] (0,0,0) -- (4,0,0) node[right] {$\vec x_2$};
\draw[axis] (0,0,0) -- (0,4,0) node[above] {$\vec y_2$};
\draw[axis] (0,0,0) -- (0,0,4) node[above] {$\vec z_2=\vec z_3$};
\draw[axis] (0,0,0) -- (\rop:4) node[right] {$\vec x_3$};
\draw[axis] (0,0,0) -- (90+\rop:4) node[above] {$\vec y_3$};
\foreach\i in {0,90}
\draw[-latex] (\i:2.5) arc (\i:\i+\rop:2.5)
node at (\i+0.5*\rop:3) {$\varphi$};
\draw[canvas is xy plane at z=2,->] (-90:0.5) arc (-90:160:0.5) node[yshift=-4mm,pos=0,text width=2cm] {Rotation propre $\varphi$};
\end{scope}
\end{tikzpicture}
\end{document}
And the output:
Update 1: The second figure:
\documentclass[tikz,border=2.718mm]{standalone}
\begin{document}
\begin{tikzpicture}[line cap=round]
% axis
\foreach\i in {1,2,3}
{
\begin{scope}[shift={(6\i,0)}]
\foreach\j in {0,1}
{
\draw[thick,latex-latex,rotate=20\j] (0,4) coordinate (Y\i\j) |- (4,0) coordinate (X\i\j);
\draw[->] (90\j:3) arc (90\j:90*\j+20:3) coordinate[midway] (A\i\j);
}
\draw[fill=white] (0,0) circle (0.2);
\fill (0,0) coordinate (O\i) circle (0.1);
\coordinate (P\i) at (2,3);
\end{scope}
}
% left
\node[yshift=-5mm] at (O1) {\strut$\vec z_f=\vec z_1$};
\node[right] at (X10) {\strut$\vec x_f$};
\node[above] at (Y10) {\strut$\vec y_f$};
\node[right] at (X11) {\strut$\vec x_1$};
\node[above] at (Y11) {\strut$\vec y_1$};
\node[right] at (A10) {\strut$\psi$};
\node[above] at (A11) {\strut$\psi$};
\node at (P1) {\strut Précession};
% center
\node[yshift=-5mm] at (O2) {\strut$\vec x_1=\vec x_2$};
\node[right] at (X20) {\strut$\vec y_1$};
\node[above] at (Y20) {\strut$\vec z_1$};
\node[right] at (X21) {\strut$\vec y_2$};
\node[above] at (Y21) {\strut$\vec z_2$};
\node[right] at (A20) {\strut$\theta$};
\node[above] at (A21) {\strut$\theta$};
\node at (P2) {\strut Nutation};
% right
\node[yshift=-5mm] at (O3) {\strut$\vec z_2=\vec z_1$};
\node[right] at (X30) {\strut$\vec x_2$};
\node[above] at (Y30) {\strut$\vec y_2$};
\node[right] at (X31) {\strut$\vec x_3$};
\node[above] at (Y31) {\strut$\vec y_3$};
\node[right] at (A30) {\strut$\varphi$};
\node[above] at (A31) {\strut$\varphi$};
\node at (P3) {\strut Rotation propre};
\end{tikzpicture}
\end{document}
Update 2: a second version for the second figure, using a \pic (almost the same output):
\tikzset
{%
pics/axis/.style n args={7}{
% #1 = x axis label (without subscript): x,y,z
% #2 = y axis label (without subscript): x,y,z
% #3 = z axis label (without subscript): x,y,z
% #4 = original axis label (subscript)
% #5 = rotated axis label (subscript)
% #6 = angle
% #7 = figure label
code={%
\draw[thick,latex-latex] (0,4) node[above] {\strut$\vec#2_#4$} |- (4,0) node[right] {\strut$\vec#1_#4$};
\draw[thick,latex-latex,rotate=20] (0,4) node[above] {\strut$\vec#2_#5$} |- (4,0) node[right] {\strut$\vec#1_#5$};
\foreach\j in {0,1}
\draw[->] (90*\j:3) arc (90*\j:90*\j+20:3) node at (90*\j+10:3.25) {$#6$};
\draw[fill=white] (0,0) circle (0.2);
\fill (0,0) circle (0.1);
\node at (0,-0.5) {\strut$\vec#3_#4=\vec#3_#5$};
\node at (2,3) {\strut#7};
}}
}
\begin{document}
\begin{tikzpicture}[line cap=round]
\pic at (0,0) {axis={x}{y}{z}{f}{1}{\psi}{Precéssion}};
\pic at (6,0) {axis={y}{z}{x}{1}{2}{\theta}{Nutation}};
\pic at (12,0) {axis={x}{y}{z}{2}{3}{\varphi}{Rotation propre}};
\end{tikzpicture}
\end{document}
