Xjyuk

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:

documentclassarticle
usepackagetikz
usepackage[graphics, active, tightpage]preview
PreviewEnvironmenttikzpicture

usetikzlibraryshapes.geometric, arrows.meta, 3d, calc
usepackagetikz-3dplot
usetikzlibraryshapes,positioning


begindocument
begintikzpicture[scale=2,axis/.style=->,dashed,thick, >=latex]

tikzsetpics/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 thickxyz;
node [below right] at (origin.south) textitG;
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
tdplotsetmaincoords00;
tdplotsetrotatedcoords0-4530;

% 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) $B_4$;

endtikzpicture
enddocument

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 
usepackagetikz-3dplot 
usetikzlibrarybackgrounds
begindocument 
tdplotsetmaincoords60-15 
begintikzpicture[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 = tikzsetcoordsys/.cd,#1
 draw [->,pic actions] (0,0,0) -- +(1,0,0)[red] node[pos=1.1]
 $pgfkeysvalueof/tikz/coordsys/x$;
 beginscope[on background layer]
 draw [->,pic actions] (0,0,0) -- +(0,1,0)[green!60!black] node[pos=1.1]
 $pgfkeysvalueof/tikz/coordsys/y$;
 endscope
 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 (PY)
 -- (X*3.25/3,0,fB(X*3.25/3)) coordinate (QY);
 tdplotsetrotatedcoords0atan2(fAprime(X*5/3),1)0 
 beginscope[tdplot_rotated_coords]
 path (PY) piccoordsys;
 pgfmathsetmacromyangatan2(fAprime(X*1.25),1)
 pgfmathtruncatemacroitestsign(myang)
 pgfmathsetmacrocuboiddim2/3% 2/3 = 1/scale where scale=1.5
 draw[fill opacity=0.5,fill=gray!70] ($(PY)+(0,0,0)$) 
 -- ($(PY)+(cuboiddim,0,0)$) -- ($(PY)+(cuboiddim,0,cuboiddim)$) 
 -- ($(PY)+(0,0,cuboiddim)$) -- cycle;
 ifnumitest=-1
 draw[fill opacity=0.5,fill=gray] ($(PY)+(cuboiddim,0,0)$) 
 -- ($(PY)+(cuboiddim,0,cuboiddim)$) -- ($(QY)+(cuboiddim,0,cuboiddim)$) 
 -- ($(QY)+(cuboiddim,0,0)$) -- cycle;
 else
 draw[fill opacity=0.5,fill=gray] ($(PY)+(0,0,0)$) 
 -- ($(PY)+(0,0,cuboiddim)$) -- ($(QY)+(0,0,cuboiddim)$) 
 -- ($(QY)+(0,0,0)$) -- cycle;
 fi 
 draw[fill opacity=0.5,fill=gray!70] ($(PY)+(0,0,cuboiddim)$) 
 -- ($(PY)+(cuboiddim,0,cuboiddim)$) -- ($(QY)+(cuboiddim,0,cuboiddim)$) 
 -- ($(QY)+(0,0,cuboiddim)$) -- cycle;
 draw[fill opacity=0.5,fill=gray!50] ($(QY)+(0,0,0)$) 
 -- ($(QY)+(cuboiddim,0,0)$) -- ($(QY)+(cuboiddim,0,cuboiddim)$) 
 -- ($(QY)+(0,0,cuboiddim)$) -- cycle;
 endscope
 tdplotsetrotatedcoords0atan2(fBprime(X*3.25/3),1)0 
 beginscope[tdplot_rotated_coords]
 path (QY) piccoordsys=x=x',y=y',z=z';
 endscope
 
endtikzpicture 
enddocument