Cube around 2 points with correct perspective

Question

I'd like to display a moving rigid body with 2 coordinate frames attached to it. The body should be displayed around every link in the image below, so that the body is parallel to the line connecting the upper and lower frames in the image below. And it appears as if both frames are in/on the body. I know this needs great 3D skills, which I seem to lack. Even though I know how to rotate the coordinate systems, I can't figure out how to rotate the cubes accordingly.

I imagine that one needs to draw the body/cube with the frames, group it somehow and maybe then rotate it into a couple of different positions as a group.

The code of the plot below can be found in my previous question, or alternatively I can provide this:

\documentclass{article}
\usepackage{tikz}
\usepackage[graphics, active, tightpage]{preview}
\PreviewEnvironment{tikzpicture}

\usetikzlibrary{shapes.geometric, arrows.meta, 3d, calc}
\usepackage{tikz-3dplot}
\usetikzlibrary{shapes,positioning}


\begin{document}
\begin{tikzpicture}[scale=2,axis/.style={->,dashed},thick, >=latex]

\tikzset{pics/coordsys/.style n args={4}{
    code = {
        \draw [->, #1] (0,0,0) -- +(1,0,0)[red] node [pos=1.2]{#2};
        \draw [->, #1] (0,0,0) -- +(0,1,0)[green] node [pos=1.2]{#3};
        \draw [->, #1] (0,0,0) -- +(0,0,1)[blue] node [pos=1.2]{#4};
    }
}}

\coordinate (origin) at (0,0,0);
\coordinate (t1M) at (-2,4,0);
\coordinate (t1B) at (-1.2,5.5,-2);
\coordinate (t2M) at (2,5,0);
\coordinate (t2B) at (2,6,-2);
\coordinate (t3M) at (5,6,0);
\coordinate (t3B) at (5,6,-2);
\coordinate (t4M) at (12,6,0);
\coordinate (t4B) at (12,6,-2);

% origin
\draw (origin) pic {coordsys={very thick}{x}{y}{z}};
\node [below right] at (origin.south) {\textit{G}};
\draw [->, dotted] (origin) -- (t1M) node [midway,fill=white] {$q_1, t_1$};
\draw [->, dotted] (origin) -- (t2M) node [midway,fill=white] {$q_2, t_2$};
\draw [->, dotted] (origin) -- (t3M) node [midway,fill=white] {$q_3, t_3$};
\draw [->, dotted] (origin) -- (t4M) node [midway,fill=white] {$q_4, t_4$};

% set fixed rotation of the two frames
\tdplotsetmaincoords{0}{0};
\tdplotsetrotatedcoords{0}{-45}{30};

% Time t1
\draw (t1M) pic {coordsys={}{}{}{}};
\draw [->, thick] (t1M) -- (t1B) node [midway,fill=white] {$q,t$};
\node [above left] at (t1M.north) {$M_1$};
\draw [->, dashed] (t1M) .. controls +(1,-1,0) and +(-1,-1,0) .. (t2M);

\tdplotsetrotatedcoordsorigin{(t1B)};
\draw [tdplot_rotated_coords] (t1B) pic {coordsys={}{}{}{}};
\node [above left] at (t1B.north) {$B_1$};
\draw [->, dashed] (t1B) .. controls +(1,-1,-2) and +(-1,-1,-2) .. (t2B) node [midway,fill=white] {$q_{12}$};

% Time t2
\draw (t2M) pic {coordsys={}{}{}{}};
\draw [->, thick] (t2M) -- (t2B) node [midway,fill=white] {$q,t$};
\node [above left] at (t2M.north) {$M_2$};
\draw [->, dashed] (t2M) .. controls +(1,1,0) and +(-1,0,0) .. (t3M); 

\tdplotsetrotatedcoordsorigin{(t2B)};
\draw [tdplot_rotated_coords] (t2B) pic {coordsys={}{}{}{}};
\node [above left] at (t2B.north) {$B_2$};
\draw [->, dashed] (t2B) .. controls +(1,1,-2) and +(-0.5,0.5,0) .. (t3B) node [midway,fill=white] {$q_{23}$};

% Time t3
\draw (t3M) pic {coordsys={}{}{}{}};
\draw [->, thick] (t3M) -- (t3B) node [midway,fill=white] {$q,t$};
\node [above left] at (t3M.north) {$M_3$};
\draw [->, dashed] (t3M) .. controls +(3,-1,0) and +(-2,-1,0) .. (t4M);

\tdplotsetrotatedcoordsorigin{(t3B)};
\draw [tdplot_rotated_coords] (t3B) pic {coordsys={}{}{}{}};
\node [above right] at (t3B.north) {$B_3$};
\draw [->, dashed] (t3B) .. controls +(1,-1,0) and +(-2,-1,-2) .. (t4B) node [midway,fill=white] {$q_{34}$};

% Time t4
\draw (t4M) pic {coordsys={}{}{}{}};
\draw [->, thick] (t4M) -- (t4B) node [midway,fill=white] {$q,t$};
\node [above left] at (t4M.north) {$M_4$};

\tdplotsetrotatedcoordsorigin{(t4B)};
\draw [tdplot_rotated_coords] (t4B) pic {coordsys={}{}{}{}};
\node [above left] at (t4B.north) {$B_4$};

\end{tikzpicture}
\end{document}

Answer 1

It is fairly easy to add cuboids but they are rotated in one frame. Presumably that's the purpose here, right? (TikZ does not have a 3D engine, so you need to decide yourself which faces are to be drawn, which is why there is an \ifnum. The y' axis is hard coded to be in the background layer...)

\documentclass[border=2mm,tikz]{standalone} 
\usepackage{tikz-3dplot} 
\usetikzlibrary{backgrounds}
\begin{document} 
\tdplotsetmaincoords{60}{-15} 
\begin{tikzpicture}[tdplot_main_coords,scale=1.5,line join=round,>=latex, 
line cap=round,declare function={fA(\t)=-sin(\t*144/(1+\t/5));
fAprime(\t)=pow(60/(5+\t),2)*cos(\t*144/(1+\t/5))*pi/180;
fB(\t)=-sin(\t*216/(1+\t*4/15));
fBprime(\t)=6*pow(90/(15+\t*4),2)*cos(\t*216/(1+\t*4/15))*pi/180;},
pics/coordsys/.style = {
    code = {\tikzset{coordsys/.cd,#1}
        \draw [->,pic actions] (0,0,0) -- +(1,0,0)[red] node[pos=1.1]
        {$\pgfkeysvalueof{/tikz/coordsys/x}$};
        \begin{scope}[on background layer]
        \draw [->,pic actions] (0,0,0) -- +(0,1,0)[green!60!black] node[pos=1.1]
        {$\pgfkeysvalueof{/tikz/coordsys/y}$};
        \end{scope}
        \draw [->,pic actions] (0,0,0) -- +(0,0,1)[blue] node[pos=1.1]
        {$\pgfkeysvalueof{/tikz/coordsys/z}$};
    }
},coordsys/.cd,x/.initial=x,y/.initial=y,z/.initial=z] 
 \draw[dashed] plot[variable=\t,domain=0:5] ({\t},3,{fA(\t)});
 \draw[dashed] plot[variable=\t,domain=0:3.25] ({\t},0,{fB(\t)});
 \foreach \X [count=\Y] in {0,...,3}
 {\draw ({\X*5/3},3,{fA(\X*5/3)}) coordinate (P\Y)
  -- ({\X*3.25/3},0,{fB(\X*3.25/3)}) coordinate (Q\Y);
 \tdplotsetrotatedcoords{0}{atan2(fAprime(\X*5/3),1)}{0} 
 \begin{scope}[tdplot_rotated_coords]
 \path (P\Y) pic{coordsys};
 \pgfmathsetmacro{\myang}{atan2(fAprime(\X*1.25),1)}
 \pgfmathtruncatemacro{\itest}{sign(\myang)}
 \pgfmathsetmacro{\cuboiddim}{2/3}% 2/3 = 1/scale where scale=1.5
 \draw[fill opacity=0.5,fill=gray!70]  ($(P\Y)+(0,0,0)$) 
  --  ($(P\Y)+(\cuboiddim,0,0)$) --  ($(P\Y)+(\cuboiddim,0,\cuboiddim)$) 
  -- ($(P\Y)+(0,0,\cuboiddim)$) -- cycle;
 \ifnum\itest=-1
 \draw[fill opacity=0.5,fill=gray]  ($(P\Y)+(\cuboiddim,0,0)$) 
  --  ($(P\Y)+(\cuboiddim,0,\cuboiddim)$) --  ($(Q\Y)+(\cuboiddim,0,\cuboiddim)$) 
  -- ($(Q\Y)+(\cuboiddim,0,0)$) -- cycle;
 \else
 \draw[fill opacity=0.5,fill=gray]  ($(P\Y)+(0,0,0)$) 
  --  ($(P\Y)+(0,0,\cuboiddim)$) --  ($(Q\Y)+(0,0,\cuboiddim)$) 
  -- ($(Q\Y)+(0,0,0)$) -- cycle;
 \fi 
 \draw[fill opacity=0.5,fill=gray!70]  ($(P\Y)+(0,0,\cuboiddim)$) 
  --  ($(P\Y)+(\cuboiddim,0,\cuboiddim)$) --  ($(Q\Y)+(\cuboiddim,0,\cuboiddim)$) 
  -- ($(Q\Y)+(0,0,\cuboiddim)$) -- cycle;
 \draw[fill opacity=0.5,fill=gray!50]  ($(Q\Y)+(0,0,0)$) 
  --  ($(Q\Y)+(\cuboiddim,0,0)$) --  ($(Q\Y)+(\cuboiddim,0,\cuboiddim)$) 
  -- ($(Q\Y)+(0,0,\cuboiddim)$) -- cycle;
 \end{scope}
 \tdplotsetrotatedcoords{0}{atan2(fBprime(\X*3.25/3),1)}{0} 
 \begin{scope}[tdplot_rotated_coords]
 \path (Q\Y) pic{coordsys={x=x',y=y',z=z'}};
 \end{scope}
 } 
\end{tikzpicture} 
\end{document}