Modellbildung | Koordinatensysteme
Funktionsdefinitionen
Drehmatrizen als Homogene RotationenDie Matrizen drehen jeweils nur um eine Achse und m\303\274ssen hintereinander geschaltet werden um allgemeine Rotationen im Raum beschreiebn zu k\303\266nnen. Siehe Skriptum Fachvertiefung
HR_x := theta -> <<1,0,0,0>|<0,cos(theta),sin(theta),0>|<0,-sin(theta),cos(theta),0>|<0,0,0,1>>;
HR_y := theta -> <<cos(theta),0,-sin(theta),0>|<0,1,0,0>|<sin(theta),0,cos(theta),0>|<0,0,0,1>>:
HR_z := theta -> <<cos(theta),sin(theta),0,0>|<-sin(theta),cos(theta),0,0>|<0,0,1,0>|<0,0,0,1>>:
Homogene Translationen mit dem allgemeinen Verschiebungs-Vektor d
HT := d -> <<1,0,0,0>|<0,1,0,0>|<0,0,1,0>|<d,1>>;
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Kinematik
Die Kinematik beschreibt die geometrischen Parameter.
kin_par := {d = 2.5}
Homogene Transformationen zwischen den Koordinatensystemen an jedem Gelenk
H_01 := HT(<0,d,0>).HR_y(-Pi/2).HR_x(-Pi/2);
H_02 := HT(<d/2,-d/2,0>).HR_z(-Pi/4);
H_23 := HT(<sqrt(2)/2*d,0,0>).HR_y(Pi/4);
H_03 := simplify(H_02.H_23);
Berechnung der Punkte im Basiskoordinatensystem
P1 := H_01.<0,1,1,1>;
P3 := H_03.<1,1,2,1>;
Visualisierung der Angabe
Plots der Koordinatensystemerefresh durch Klicken auf das Symbol
# visualisation
r:=0.015*6: # radii of balls
l:=0.2*6: # length of arrows
listi := a -> convert(a,list):
plotPoint := (T,col) -> plottools[sphere](listi(T[1..3]),r,color=col,style=patchnogrid):
plotOrigin := T -> plotPoint(T[1..4,4],'black'):
plotArrow := (T,ax,col) -> plots[arrow](listi((T.<0,0,0,1>)[1..3]),listi((T.<l*ax,0>)[1..3]),color=col):
plotCoords := (T) -> [plotOrigin(T),
plotArrow(T,<1,0,0>,'red'),
plotArrow(T,<0,1,0>,'green'),
plotArrow(T,<0,0,1>,'blue')]:
Tlist := map(x->eval(x,kin_par),[IdentityMatrix(4,4),H_01,H_02,H_03]):
plots:-display((
plotCoords(Tlist[1]),
plotCoords(Tlist[2]),
plotCoords(Tlist[3]),
plotCoords(Tlist[4]),
plotPoint(eval(P1,kin_par),'red'),
plotPoint(eval(P3,kin_par),'red')
),
scaling=constrained,
orientation = [25, 60],
axes=Framed
);
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