1 Bonusaufgabe 3

(%i1) ratprint : false ;

\[\operatorname{ }\mbox{%default false}\]

1.1 Angabe

(%i2) phi_0 : %pi / 6 ;

\[\operatorname{ }-\frac{\ensuremath{\pi} }{6}\]

(%i6) d_0_1 : matrix ( [ 2 ] , [ 1 ] , [ 3 ] ) $
d_1_2 : matrix ( [ 4 ] , [ 2 ] , [ 1 ] ) $
d_2_3 : matrix ( [ 3 ] , [ 1 ] , [ 4 ] ) $
d_3_4 : matrix ( [ 3 ] , [ 4 ] , [ 1 ] ) $
(%i11) p_1_s1 : matrix ( [ 2 ] , [ 1 ] , [ 1 / 2 ] ) $
p_2_s2 : matrix ( [ 3 / 2 ] , [ 1 / 2 ] , [ 2 ] ) $
p_3_s3 : matrix ( [ 3 / 2 ] , [ 2 ] , [ 1 / 2 ] ) $
p_4_s4 : matrix ( [ 2 ] , [ 1 / 2 ] , [ 1 ] ) $
p_4_E : matrix ( [ 4 ] , [ 1 ] , [ 2 ] ) $

1.2 Aufgabe 1

(%i17) R_0_1_z : matrix ( [ cos ( phi_0 ) , sin ( phi_0 ) , 0 ] , [ sin ( phi_0 ) , cos ( phi_0 ) , 0 ] , [ 0 , 0 , 1 ] ) $
H_0_1_z : addrow ( addcol ( R_0_1_z , [ 0 , 0 , 0 ] ) , [ 0 , 0 , 0 , 1 ] ) $
R_0_1_x : matrix ( [ 1 , 0 , 0 ] , [ 0 , cos ( q1 ) , sin ( q1 ) ] , [ 0 , sin ( q1 ) , cos ( q1 ) ] ) $
H_0_1_x : addrow ( addcol ( R_0_1_x , [ 0 , 0 , 0 ] ) , [ 0 , 0 , 0 , 1 ] ) $
H_0_1_v : matrix ( [ 0 , 0 , 0 , d_0_1 [ 1 , 1 ] ] , [ 0 , 0 , 0 , d_0_1 [ 2 , 1 ] ] , [ 0 , 0 , 0 , d_0_1 [ 3 , 1 ] ] , [ 0 , 0 , 0 , 0 ] ) $
H_0_1 : H_0_1_z . H_0_1_x + H_0_1_v ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & \frac{\cos{\left( \ensuremath{\mathrm{q1}}\right) }}{2} & -\frac{\sin{\left( \ensuremath{\mathrm{q1}}\right) }}{2} & 2\\ -\frac{1}{2} & \frac{\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{2} & -\frac{\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }}{2} & 1\\ 0 & \sin{\left( \ensuremath{\mathrm{q1}}\right) } & \cos{\left( \ensuremath{\mathrm{q1}}\right) } & 3\\ 0 & 0 & 0 & 1\end{pmatrix}\]

(%i21) R_1_2_x : matrix ( [ 1 , 0 , 0 ] , [ 0 , cos ( q2 ) , sin ( q2 ) ] , [ 0 , sin ( q2 ) , cos ( q2 ) ] ) $
H_1_2_x : addrow ( addcol ( R_1_2_x , [ 0 , 0 , 0 ] ) , [ 0 , 0 , 0 , 1 ] ) $
H_1_2_v : matrix ( [ 0 , 0 , 0 , d_1_2 [ 1 , 1 ] ] , [ 0 , 0 , 0 , d_1_2 [ 2 , 1 ] ] , [ 0 , 0 , 0 , d_1_2 [ 3 , 1 ] ] , [ 0 , 0 , 0 , 0 ] ) $
H_1_2 : H_1_2_x + H_1_2_v ;

\[\operatorname{ }\begin{pmatrix}1 & 0 & 0 & 4\\ 0 & \cos{\left( \ensuremath{\mathrm{q2}}\right) } & -\sin{\left( \ensuremath{\mathrm{q2}}\right) } & 2\\ 0 & \sin{\left( \ensuremath{\mathrm{q2}}\right) } & \cos{\left( \ensuremath{\mathrm{q2}}\right) } & 1\\ 0 & 0 & 0 & 1\end{pmatrix}\]

(%i25) R_2_3_x : matrix ( [ 1 , 0 , 0 ] , [ 0 , cos ( q3 ) , sin ( q3 ) ] , [ 0 , sin ( q3 ) , cos ( q3 ) ] ) $
H_2_3_x : addrow ( addcol ( R_2_3_x , [ 0 , 0 , 0 ] ) , [ 0 , 0 , 0 , 1 ] ) $
H_2_3_v : matrix ( [ 0 , 0 , 0 , d_2_3 [ 1 , 1 ] ] , [ 0 , 0 , 0 , d_2_3 [ 2 , 1 ] ] , [ 0 , 0 , 0 , d_2_3 [ 3 , 1 ] ] , [ 0 , 0 , 0 , 0 ] ) $
H_2_3 : H_2_3_x + H_2_3_v ;

\[\operatorname{ }\begin{pmatrix}1 & 0 & 0 & 3\\ 0 & \cos{\left( \ensuremath{\mathrm{q3}}\right) } & -\sin{\left( \ensuremath{\mathrm{q3}}\right) } & 1\\ 0 & \sin{\left( \ensuremath{\mathrm{q3}}\right) } & \cos{\left( \ensuremath{\mathrm{q3}}\right) } & 4\\ 0 & 0 & 0 & 1\end{pmatrix}\]

(%i29) R_3_4 : matrix ( [ 1 , 0 , 0 ] , [ 0 , 1 , 0 ] , [ 0 , 0 , 1 ] ) $
H_3_4_tx : matrix ( [ 1 , 0 , 0 , q4 ] , [ 0 , 1 , 0 , 0 ] , [ 0 , 0 , 1 , 0 ] , [ 0 , 0 , 0 , 1 ] ) $
H_3_4_v : matrix ( [ 0 , 0 , 0 , d_3_4 [ 1 , 1 ] ] , [ 0 , 0 , 0 , d_3_4 [ 2 , 1 ] ] , [ 0 , 0 , 0 , d_3_4 [ 3 , 1 ] ] , [ 0 , 0 , 0 , 0 ] ) $
H_3_4 : H_3_4_tx + H_3_4_v ;

\[\operatorname{ }\begin{pmatrix}1 & 0 & 0 & \ensuremath{\mathrm{q4}}+3\\ 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\end{pmatrix}\]

(%i31) H_0_2 : H_0_1 . H_1_2 $
trigsimp ( % ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & -\frac{\sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{2} & -\frac{\cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{2} & -\frac{\sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sqrt{3}-4}{2}\\ -\frac{1}{2} & -\frac{\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{2} & -\frac{\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{2} & -\frac{\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }+2}{2}\\ 0 & \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) } & \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) } & 2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }+3\\ 0 & 0 & 0 & 1\end{pmatrix}\]

(%i33) H_0_3 : H_0_2 . H_2_3 $
trigsimp ( % ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & -\frac{\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & \frac{\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -\cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & -\frac{\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-7 \sqrt{3}-4}{2}\\ -\frac{1}{2} & -\frac{\left( \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & \frac{\left( \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & -\frac{\left( \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }+5}{2}\\ 0 & \left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) } & \left( -\cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) } & \left( \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }+3\\ 0 & 0 & 0 & 1\end{pmatrix}\]

(%i35) H_0_4 : H_0_3 . H_3_4 $
trigsimp ( % ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & -\frac{\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & \frac{\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -\cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & \\ -\frac{1}{2} & -\frac{\left( \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & \frac{\left( \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & -\\ 0 & \left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) } & \left( -\cos{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) } \cos{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) } \sin{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) } & \left( \left( -4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( 4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }+3\\ 0 & 0 & 0 & 1\end{pmatrix}\]

1.3 Aufgabe 2

Wir müssen eine Reihe zum Vektor hinzufügen, damit die Multiplikation erfolgen kann. Danach kann diese Reihe wieder entfernt werden.
(%i45) p_0_s1 : submatrix ( 4 , H_0_1 . addrow ( p_1_s1 , matrix ( [ 1 ] ) ) ) $
trigsimp ( % ) ;
p_0_s2 : submatrix ( 4 , H_0_2 . addrow ( p_2_s2 , matrix ( [ 1 ] ) ) ) $
trigsimp ( % ) ;
p_0_s3 : submatrix ( 4 , H_0_3 . addrow ( p_3_s3 , matrix ( [ 1 ] ) ) ) $
trigsimp ( % ) ;
p_0_s4 : submatrix ( 4 , H_0_4 . addrow ( p_4_s4 , matrix ( [ 1 ] ) ) ) $
trigsimp ( % ) ;
p_0_E : submatrix ( 4 , H_0_4 . addrow ( p_4_E , matrix ( [ 1 ] ) ) ) $
trigsimp ( % ) ;

\[\operatorname{ }\begin{pmatrix}-\frac{\sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sqrt{3}-8}{4}\\ -\frac{\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{4}\\ \frac{2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }+6}{2}\end{pmatrix}\]

\[\operatorname{ }\begin{pmatrix}-\frac{\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-11 \sqrt{3}-8}{4}\\ -\frac{\left( \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+2 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }+7}{4}\\ -\frac{\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -\sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-6}{2}\end{pmatrix}\]

\[\operatorname{ }\begin{pmatrix}\\ \\ -\end{pmatrix}\]

\[\operatorname{ }\begin{pmatrix}\\ -\\ -\end{pmatrix}\]

\[\operatorname{ }\begin{pmatrix}\\ -\\ \left( \left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( 5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }+3\end{pmatrix}\]

(%i46) j_p_s : jacobian ( [ p_0_s1 , p_0_s2 , p_0_s3 , p_0_s4 , p_0_E ] , [ q1 , q2 , q3 , q4 ] ) $
(%i47) j_v_0_s1 : trigsimp ( addcol ( j_p_s [ 1 , 1 ] , j_p_s [ 1 , 2 ] , j_p_s [ 1 , 3 ] , j_p_s [ 1 , 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}-\frac{2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }}{4} & 0 & 0 & 0\\ -\frac{2 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{4} & 0 & 0 & 0\\ -\frac{\sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{2} & 0 & 0 & 0\end{pmatrix}\]

(%i48) j_v_0_s2 : trigsimp ( addcol ( j_p_s [ 2 , 1 ] , j_p_s [ 2 , 2 ] , j_p_s [ 2 , 3 ] , j_p_s [ 2 , 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -\sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{4} & \frac{\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -\sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{4} & 0 & 0\\ \frac{\left( 4 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }-4 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-2 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{4} & \frac{\left( 4 \sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -\sqrt{3} \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sqrt{3} \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{4} & 0 & 0\\ -\frac{\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }+2 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }}{2} & -\frac{\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }}{2} & 0 & 0\end{pmatrix}\]

(%i49) j_v_0_s3 : trigsimp ( addcol ( j_p_s [ 3 , 1 ] , j_p_s [ 3 , 2 ] , j_p_s [ 3 , 3 ] , j_p_s [ 3 , 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix} & & \frac{\left( \left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{4} & 0\\ & & & 0\\ & & \frac{\left( \left( \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( -4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-\cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & 0\end{pmatrix}\]

(%i50) j_v_0_s4 : trigsimp ( addcol ( j_p_s [ 4 , 1 ] , j_p_s [ 4 , 2 ] , j_p_s [ 4 , 3 ] , j_p_s [ 4 , 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix} & & \frac{\left( \left( 9 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-9 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-9 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -9 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{4} & \frac{\sqrt{3}}{2}\\ & & & -\frac{1}{2}\\ & & \frac{\left( \left( 4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-9 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -9 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( -9 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 9 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & 0\end{pmatrix}\]

(%i51) j_v_0_E : trigsimp ( addcol ( j_p_s [ 5 , 1 ] , j_p_s [ 5 , 2 ] , j_p_s [ 5 , 3 ] , j_p_s [ 5 , 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix} & & \frac{\left( \left( 5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }+3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( 3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }}{2} & \frac{\sqrt{3}}{2}\\ & & & -\frac{1}{2}\\ \left( \left( 3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -\sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }-\sin{\left( \ensuremath{\mathrm{q1}}\right) }+2 \cos{\left( \ensuremath{\mathrm{q1}}\right) } & \left( \left( 3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -\sin{\left( \ensuremath{\mathrm{q1}}\right) }-4 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( \cos{\left( \ensuremath{\mathrm{q1}}\right) }-4 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) } & \left( \left( 3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( \left( -5 \sin{\left( \ensuremath{\mathrm{q1}}\right) }-3 \cos{\left( \ensuremath{\mathrm{q1}}\right) }\right) \sin{\left( \ensuremath{\mathrm{q2}}\right) }+\left( 5 \cos{\left( \ensuremath{\mathrm{q1}}\right) }-3 \sin{\left( \ensuremath{\mathrm{q1}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q2}}\right) }\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) } & 0\end{pmatrix}\]

(%i54) R_0_1 : R_0_1_z . R_0_1_x $
R_1_2 : R_1_2_x $
R_2_3 : R_2_3_x $
(%i58) R_0_2 : R_0_1 . R_1_2 $
R_0_3 : R_0_2 . R_2_3 $
R_0_4 : R_0_3 . R_3_4 $
R_0_E : R_0_4 $
(%i61) R : [ R_0_1 , R_0_2 , R_0_3 , R_0_4 , R_0_E ] $
q : [ q1 , q2 , q3 , q4 ] $
q_d : [ qd1 , qd2 , qd3 , qd4 ] $
Figure 1:
Diagram
(%i68) sum : 0 $
calc : R_0_1 $
for y : 1 while y < = 4 do ( sum : sum + ( diff ( calc , q [ y ] , 1 ) . transpose ( calc ) · q_d [ y ] ) ) $
trigsimp ( sum ) $
matrix ( [ sum [ 3 ] [ 2 ] ] , [ sum [ 1 ] [ 3 ] ] , [ sum [ 2 ] [ 1 ] ] ) $
J_sum : jacobian ( [ % ] , q_d ) $
J_w_0_1 : trigsimp ( addcol ( J_sum [ 1 ] [ 1 ] , J_sum [ 1 ] [ 2 ] , J_sum [ 1 ] [ 3 ] , J_sum [ 1 ] [ 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & 0 & 0 & 0\\ -\frac{1}{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

(%i75) sum : 0 $
calc : R_0_2 $
for y : 1 while y < = 4 do ( sum : sum + ( diff ( calc , q [ y ] , 1 ) . transpose ( calc ) · q_d [ y ] ) ) $
trigsimp ( sum ) $
matrix ( [ sum [ 3 ] [ 2 ] ] , [ sum [ 1 ] [ 3 ] ] , [ sum [ 2 ] [ 1 ] ] ) $
J_sum : jacobian ( [ % ] , q_d ) $
J_w_0_2 : trigsimp ( addcol ( J_sum [ 1 ] [ 1 ] , J_sum [ 1 ] [ 2 ] , J_sum [ 1 ] [ 3 ] , J_sum [ 1 ] [ 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 & 0\\ -\frac{1}{2} & -\frac{1}{2} & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

(%i82) sum : 0 $
calc : R_0_3 $
for y : 1 while y < = 4 do ( sum : sum + ( diff ( calc , q [ y ] , 1 ) . transpose ( calc ) · q_d [ y ] ) ) $
trigsimp ( sum ) $
matrix ( [ sum [ 3 ] [ 2 ] ] , [ sum [ 1 ] [ 3 ] ] , [ sum [ 2 ] [ 1 ] ] ) $
J_sum : jacobian ( [ % ] , q_d ) $
J_w_0_3 : trigsimp ( addcol ( J_sum [ 1 ] [ 1 ] , J_sum [ 1 ] [ 2 ] , J_sum [ 1 ] [ 3 ] , J_sum [ 1 ] [ 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0\\ -\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

(%i89) sum : 0 $
calc : R_0_4 $
for y : 1 while y < = 4 do ( sum : sum + ( diff ( calc , q [ y ] , 1 ) . transpose ( calc ) · q_d [ y ] ) ) $
trigsimp ( sum ) $
matrix ( [ sum [ 3 ] [ 2 ] ] , [ sum [ 1 ] [ 3 ] ] , [ sum [ 2 ] [ 1 ] ] ) $
J_sum : jacobian ( [ % ] , q_d ) $
J_w_0_4 : trigsimp ( addcol ( J_sum [ 1 ] [ 1 ] , J_sum [ 1 ] [ 2 ] , J_sum [ 1 ] [ 3 ] , J_sum [ 1 ] [ 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0\\ -\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

(%i96) sum : 0 $
calc : R_0_E $
for y : 1 while y < = 4 do ( sum : sum + ( diff ( calc , q [ y ] , 1 ) . transpose ( calc ) · q_d [ y ] ) ) $
trigsimp ( sum ) $
matrix ( [ sum [ 3 ] [ 2 ] ] , [ sum [ 1 ] [ 3 ] ] , [ sum [ 2 ] [ 1 ] ] ) $
J_sum : jacobian ( [ % ] , q_d ) $
J_w_0_E : trigsimp ( addcol ( J_sum [ 1 ] [ 1 ] , J_sum [ 1 ] [ 2 ] , J_sum [ 1 ] [ 3 ] , J_sum [ 1 ] [ 4 ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0\\ -\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

1.4 Aufgabe 3

(%i99) m : [ m1 , m2 , m3 , m4 ] $
J_w : [ J_w_0_1 , J_w_0_2 , J_w_0_3 , J_w_0_4 ] $
J_v : [ j_v_0_s1 , j_v_0_s2 , j_v_0_s3 , j_v_0_s4 ] $

1.4.1 Massematrix

1.4.1.1 Translatorischer Anteil

(%i102) sum : 0 $
for i : 1 while i < = 4 do ( sum : sum + ( ( transpose ( J_v [ i ] ) · m [ i ] ) . J_v [ i ] ) ) $
MM_v : trigsimp ( sum ) ;

\[\operatorname{ }\begin{pmatrix}- & - & - & 0\\ - & \frac{\left( 128 \ensuremath{\mathrm{m4}}+60 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( 100 \ensuremath{\mathrm{m4}}+32 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+165 \ensuremath{\mathrm{m4}}+85 \ensuremath{\mathrm{m3}}+17 \ensuremath{\mathrm{m2}}}{4} & \frac{\left( 64 \ensuremath{\mathrm{m4}}+30 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( 50 \ensuremath{\mathrm{m4}}+16 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+97 \ensuremath{\mathrm{m4}}+17 \ensuremath{\mathrm{m3}}}{4} & 0\\ - & \frac{\left( 64 \ensuremath{\mathrm{m4}}+30 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( 50 \ensuremath{\mathrm{m4}}+16 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+97 \ensuremath{\mathrm{m4}}+17 \ensuremath{\mathrm{m3}}}{4} & \frac{97 \ensuremath{\mathrm{m4}}+17 \ensuremath{\mathrm{m3}}}{4} & 0\\ 0 & 0 & 0 & \ensuremath{\mathrm{m4}}\end{pmatrix}\]

1.4.1.2 Rotatorischer Anteil

(%i106) I1 : matrix ( [ I1xx , 0 , 0 ] , [ 0 , I1yy , 0 ] , [ 0 , 0 , I1zz ] ) $
I2 : matrix ( [ I2xx , 0 , 0 ] , [ 0 , I2yy , 0 ] , [ 0 , 0 , I2zz ] ) $
I3 : matrix ( [ I3xx , 0 , 0 ] , [ 0 , I3yy , 0 ] , [ 0 , 0 , I3zz ] ) $
I4 : matrix ( [ I4xx , 0 , 0 ] , [ 0 , I4yy , 0 ] , [ 0 , 0 , I4zz ] ) $
(%i111) I_0_s1 : trigsimp ( R_0_1 . I1 . transpose ( R_0_1 ) ) $
I_0_s2 : trigsimp ( R_0_2 . I2 . transpose ( R_0_2 ) ) $
I_0_s3 : trigsimp ( R_0_3 . I3 . transpose ( R_0_3 ) ) $
I_0_s4 : trigsimp ( R_0_4 . I4 . transpose ( R_0_4 ) ) $
I_0_s : [ I_0_s1 , I_0_s2 , I_0_s3 , I_0_s4 ] $
(%i114) sum : 0 $
for i : 1 while i < = 4 do ( sum : sum + ( ( transpose ( J_w [ i ] ) . I_0_s [ i ] ) . J_w [ i ] ) ) $
MM_w : trigsimp ( sum ) ;

\[\operatorname{ }\begin{pmatrix}\ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}}+\ensuremath{\mathrm{I2xx}}+\ensuremath{\mathrm{I1xx}} & \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}}+\ensuremath{\mathrm{I2xx}} & \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}} & 0\\ \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}}+\ensuremath{\mathrm{I2xx}} & \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}}+\ensuremath{\mathrm{I2xx}} & \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}} & 0\\ \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}} & \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}} & \ensuremath{\mathrm{I4xx}}+\ensuremath{\mathrm{I3xx}} & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

1.4.1.3 Gesamt

(%i115) MM : trigsimp ( MM_v + MM_w ) ;

\[\operatorname{ }\begin{pmatrix}- & - & - & 0\\ - & \frac{\left( 128 \ensuremath{\mathrm{m4}}+60 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( 100 \ensuremath{\mathrm{m4}}+32 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+165 \ensuremath{\mathrm{m4}}+85 \ensuremath{\mathrm{m3}}+17 \ensuremath{\mathrm{m2}}+4 \ensuremath{\mathrm{I4xx}}+4 \ensuremath{\mathrm{I3xx}}+4 \ensuremath{\mathrm{I2xx}}}{4} & \frac{\left( 64 \ensuremath{\mathrm{m4}}+30 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( 50 \ensuremath{\mathrm{m4}}+16 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+97 \ensuremath{\mathrm{m4}}+17 \ensuremath{\mathrm{m3}}+4 \ensuremath{\mathrm{I4xx}}+4 \ensuremath{\mathrm{I3xx}}}{4} & 0\\ - & \frac{\left( 64 \ensuremath{\mathrm{m4}}+30 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( 50 \ensuremath{\mathrm{m4}}+16 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }+97 \ensuremath{\mathrm{m4}}+17 \ensuremath{\mathrm{m3}}+4 \ensuremath{\mathrm{I4xx}}+4 \ensuremath{\mathrm{I3xx}}}{4} & \frac{97 \ensuremath{\mathrm{m4}}+17 \ensuremath{\mathrm{m3}}+4 \ensuremath{\mathrm{I4xx}}+4 \ensuremath{\mathrm{I3xx}}}{4} & 0\\ 0 & 0 & 0 & \ensuremath{\mathrm{m4}}\end{pmatrix}\]

1.4.2 Coriolismatrix

(%i118) CM : zeromatrix ( 4 , 4 ) $
for j : 1 while j < = 4 do (
   for k : 1 while k < = 4 do (
       for i : 1 while i < = 4 do (
           cc_ijk : 1 / 2 · ( diff ( MM [ k ] [ j ] , q [ i ] ) + diff ( MM [ k ] [ i ] , q [ j ] ) diff ( MM [ i ] [ j ] , q [ k ] ) ) ,
           CM [ k ] [ j ] : CM [ k ] [ j ] + cc_ijk · q_d [ i ]
       ) ) ) $
trigsimp ( CM ) ;

\[\operatorname{ }\begin{pmatrix}- & - & - & 0\\ - & -\frac{\left( \left( 25 \ensuremath{\mathrm{m4}}+8 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -32 \ensuremath{\mathrm{m4}}-15 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }\right) \, \ensuremath{\mathrm{qd3}}}{2} & - & 0\\ & \frac{\left( \left( 25 \ensuremath{\mathrm{m4}}+8 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -32 \ensuremath{\mathrm{m4}}-15 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }\right) \, \ensuremath{\mathrm{qd2}}+\left( \left( 25 \ensuremath{\mathrm{m4}}+8 \ensuremath{\mathrm{m3}}\right) \sin{\left( \ensuremath{\mathrm{q3}}\right) }+\left( -32 \ensuremath{\mathrm{m4}}-15 \ensuremath{\mathrm{m3}}\right) \cos{\left( \ensuremath{\mathrm{q3}}\right) }\right) \, \ensuremath{\mathrm{qd1}}}{2} & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\]

1.4.3 Vektor der verallgemeinerten Kräfte

(%i119) vk : trigsimp ( transpose ( j_v_0_E ) . matrix ( [ fxe ] , [ fye ] , [ fze ] ) ) ;

\[\operatorname{ }\begin{pmatrix}\\ \\ \\ -\frac{\ensuremath{\mathrm{fye}}-\sqrt{3} \ensuremath{\mathrm{fxe}}}{2}\end{pmatrix}\]

1.4.4 Vektor der Potentialkräfte

(%i126) p : [ p_0_s1 , p_0_s2 , p_0_s3 , p_0_s4 , p_0_E ] $
(%i168) V : 0 $
for i : 1 while i < = 4 do ( V : V + ( ( m [ i ] · g ) · matrix ( [ 0 ] , [ 0 ] , [ 1 ] ) . p [ i ] ) ) $
vp : trigsimp ( transpose ( jacobian ( [ V ] , q ) ) ) ;

\[\operatorname{ }\begin{pmatrix}\\ \\ \\ 0\end{pmatrix}\]


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