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bendesarts
Inscrit le: 27 Fév 2009
Messages: 234
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 Posté le: 29 Nov 2009 11:15 Sujet du message: Dérivée d'un vecteur position |
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Bonjour,
Je cherche à dériver un vecteur appelé ici On par rapport à t. Puis, à mettre le système sous la forme "Dérivé du vecteur = J * dq/dt" où q est le vecteur des paramètres.
D'abord, dans mon vecteur On, il faut d'abord que je précise les variables qui dépendent du temps et celle qui ne dépendent pas du temps. Ensui, il faut dériver ce vecteur par rapport au temps. Enfin j'aimerais identifier la matrice jacobienne J sachant que dq/dt = [dtheta1/dt dtheta2/dt dd3/dt dtheta4/dt
Voilà pour le moment l'expression de mon vecteur On que j'aimerais dériver :
| Code: | > restart;
> with(linalg):
f := (d, theta, a, alpha) -> [
> [cos(theta), -cos(alpha) sin(theta), sin(theta) sin(alpha), a cos(theta)],
> [sin(theta), cos(theta) cos(alpha), -cos(theta) sin(alpha), a sin(theta)],
> [0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]];
> T01 := f(d1, theta1, 0, 0);
> T12 := f(d2, theta2, a2, 0);
> T23 := f(d3, 0, a3, 0);
> T34 := f(0, theta4, 0, 0);
> T04 := map(combine, evalm(((T01 &* T12) &* T23) &* T34));
> On := submatrix(T04, 1 .. 3, 4 .. 4);
Warning, the protected names norm and trace have been redefined and unprotected
f := (d, theta, a, alpha) -> [
[cos(theta), -cos(alpha) sin(theta), sin(theta) sin(alpha), a cos(theta)],
[sin(theta), cos(theta) cos(alpha), -cos(theta) sin(alpha), a sin(theta)],
[0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]]
[cos(theta1) -sin(theta1) 0 0 ]
[ ]
[sin(theta1) cos(theta1) 0 0 ]
T01 := [ ]
[ 0 0 1 d1]
[ ]
[ 0 0 0 1 ]
[cos(theta2) -sin(theta2) 0 a2 cos(theta2)]
[ ]
[sin(theta2) cos(theta2) 0 a2 sin(theta2)]
T12 := [ ]
[ 0 0 1 d2 ]
[ ]
[ 0 0 0 1 ]
[1 0 0 a3]
[ ]
[0 1 0 0 ]
T23 := [ ]
[0 0 1 d3]
[ ]
[0 0 0 1 ]
[cos(theta4) -sin(theta4) 0 0]
[ ]
[sin(theta4) cos(theta4) 0 0]
T34 := [ ]
[ 0 0 1 0]
[ ]
[ 0 0 0 1]
T04 := [[cos(theta1 + theta2 + theta4), -sin(theta1 + theta2 + theta4), 0,
cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)], [
sin(theta1 + theta2 + theta4), cos(theta1 + theta2 + theta4), 0,
sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)], [0, 0, 1, d3 + d2 + d1],
[0, 0, 0, 1]]
[cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)]
[ ]
On := [sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)]
[ ]
[ d3 + d2 + d1 ]
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Merci d'avance pour votre aide
Benjamin |
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ALS
Inscrit le: 11 Sep 2006
Messages: 647
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| Posté le: 30 Nov 2009 12:50 Sujet du message: |
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Bonjour, voici ce que je vous propose:
| Code: |
> restart;
> with(linalg):
> f := (d, theta, a, alpha) -> [ [cos(theta), -cos(alpha)* sin(theta), sin(theta) *sin(alpha), a* cos(theta)],
> [sin(theta), cos(theta)* cos(alpha), -cos(theta) *sin(alpha), a* sin(theta)],
> [0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]];
>
>
> T01 := f(d1, theta1, 0, 0);
> T12 := f(d2, theta2, a2, 0);
> T23 := f(d3, 0, a3, 0);
> T34 := f(0, theta4, 0, 0);
> T04 := map(combine, evalm(((T01 &* T12) &* T23) &* T34));
>
> On := submatrix(T04, 1 .. 4, 4 .. 4);
>
f := (d, theta, a, alpha) -> [[cos(theta), -cos(alpha) sin(theta),
sin(theta) sin(alpha), a cos(theta)], [sin(theta),
cos(theta) cos(alpha), -cos(theta) sin(alpha), a sin(theta)],
[0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]]
T01 := [[cos(theta1), -sin(theta1), 0, 0],
[sin(theta1), cos(theta1), 0, 0], [0, 0, 1, d1], [0, 0, 0, 1]
]
T12 := [[cos(theta2), -sin(theta2), 0, a2 cos(theta2)],
[sin(theta2), cos(theta2), 0, a2 sin(theta2)], [0, 0, 1, d2],
[0, 0, 0, 1]]
T23 := [[1, 0, 0, a3], [0, 1, 0, 0], [0, 0, 1, d3], [0, 0, 0, 1]]
T34 := [[cos(theta4), -sin(theta4), 0, 0],
[sin(theta4), cos(theta4), 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
T04 :=
[cos(%1) , -sin(%1) , 0 ,
cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)]
[sin(%1) , cos(%1) , 0 ,
sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)]
[0 , 0 , 1 , d3 + d2 + d1]
[0 , 0 , 0 , 1]
%1 := theta1 + theta2 + theta4
[cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)]
[ ]
[sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)]
On := [ ]
[ d3 + d2 + d1 ]
[ ]
[ 1 ]
> On:=subs(theta1=theta1(t),theta2=theta2(t),d3=d3(t),theta4=theta4(t),evalm(On));
[cos(%1) a3 + a2 cos(%1)]
[ ]
[sin(%1) a3 + a2 sin(%1)]
On := [ ]
[ d3(t) + d2 + d1 ]
[ ]
[ 1 ]
%1 := theta1(t) + theta2(t)
> dOn:=map(x->diff(x,t),On);
[-sin(%2) %1 a3 - a2 sin(%2) %1]
[ ]
[cos(%2) %1 a3 + a2 cos(%2) %1 ]
[ ]
dOn := [ d ]
[ -- d3(t) ]
[ dt ]
[ ]
[ 0 ]
/d \ /d \
%1 := |-- theta1(t)| + |-- theta2(t)|
\dt / \dt /
%2 := theta1(t) + theta2(t)
> J:=matrix([[diff(theta1(t),t)],[diff(theta2(t),t)],[diff(d3(t),t)],[diff(theta4(t),t)]]);
>
[d ]
[-- theta1(t)]
[dt ]
[ ]
[d ]
[-- theta2(t)]
[dt ]
J := [ ]
[ d ]
[ -- d3(t) ]
[ dt ]
[ ]
[d ]
[-- theta4(t)]
[dt ]
> A:=matrix([[-(a2+a3)*sin(theta1(t)+theta2(t)),-(a2+a3)*sin(theta1(t)+theta2(t)),0,0],[(a2+a3)*cos(theta1(t)+theta2(t)),(a2+a3)*cos(theta1(t)+theta2(t)),0,0],[0,0,1,0],[0,0,0,0]]);
[-(a3 + a2) sin(%1) -(a3 + a2) sin(%1) 0 0]
[ ]
[(a3 + a2) cos(%1) (a3 + a2) cos(%1) 0 0]
A := [ ]
[ 0 0 1 0]
[ ]
[ 0 0 0 0]
%1 := theta1(t) + theta2(t)
> simplify(evalm(dOn-evalm(A &* J)));
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
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Donc dOn = AJ.
A+ |
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bendesarts
Inscrit le: 27 Fév 2009
Messages: 234
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| Posté le: 01 Déc 2009 8:19 Sujet du message: regroupement de termes ou factorisation dans une expression |
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Ok merci beaucoup pour votre aide
Vous êtes d'une grande aide.
Encore une petite question:
Après m'être inspiré fortement de ce que vous m'avez proposé, j'en suis là. J'aimerais regrouper dans mon expression finale dOn en fonction des composantes de ce qu'on appellait J. En d'autres termes, regrouper dans l'expression dOn les termes en dérivées dtheta1/dt dtheta2/dt ...
| Code: | with(linalg):
f := (d, theta, a, alpha) -> [
> [cos(theta), -cos(alpha) sin(theta), sin(theta) sin(alpha), a cos(theta)],
> [sin(theta), cos(theta) cos(alpha), -cos(theta) sin(alpha), a sin(theta)],
> [0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]];
> T01 := f(d1, theta1, 0, 0);
> T12 := f(d2, theta2, a2, 0);
> T23 := f(d3, 0, a3, 0);
> T34 := f(0, theta4, 0, 0);
> T4h := f(0, 0, a4, 0);
> T0h := map(combine, evalm((((T01 &* T12) &* T23) &* T34) &* T4h));
> On := submatrix(T0h, 1 .. 3, 4 .. 4);
> On := subs(theta1 = theta1(t), theta2 = theta2(t), d3 = d3(t),
> theta4 = theta4(t), evalm(On));
> / d \
> dOn := map|x -> --- x, On|;
> \ dt /
Warning, the protected names norm and trace have been redefined and unprotected
f := (d, theta, a, alpha) -> [
[cos(theta), -cos(alpha) sin(theta), sin(theta) sin(alpha), a cos(theta)],
[sin(theta), cos(theta) cos(alpha), -cos(theta) sin(alpha), a sin(theta)],
[0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]]
[cos(theta1) -sin(theta1) 0 0 ]
[ ]
[sin(theta1) cos(theta1) 0 0 ]
T01 := [ ]
[ 0 0 1 d1]
[ ]
[ 0 0 0 1 ]
[cos(theta2) -sin(theta2) 0 a2 cos(theta2)]
[ ]
[sin(theta2) cos(theta2) 0 a2 sin(theta2)]
T12 := [ ]
[ 0 0 1 d2 ]
[ ]
[ 0 0 0 1 ]
[1 0 0 a3]
[ ]
[0 1 0 0 ]
T23 := [ ]
[0 0 1 d3]
[ ]
[0 0 0 1 ]
[cos(theta4) -sin(theta4) 0 0]
[ ]
[sin(theta4) cos(theta4) 0 0]
T34 := [ ]
[ 0 0 1 0]
[ ]
[ 0 0 0 1]
[1 0 0 a4]
[ ]
[0 1 0 0 ]
T4h := [ ]
[0 0 1 0 ]
[ ]
[0 0 0 1 ]
T0h := [[cos(theta1 + theta2 + theta4), -sin(theta1 + theta2 + theta4), 0, cos(theta1
+ theta2 + theta4) a4 + cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)],
[sin(theta1 + theta2 + theta4), cos(theta1 + theta2 + theta4), 0, sin(theta1
+ theta2 + theta4) a4 + sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)],
[0, 0, 1, d3 + d2 + d1], [0, 0, 0, 1]]
On := [[cos(theta1 + theta2 + theta4) a4 + cos(theta1 + theta2) a3
+ a2 cos(theta1 + theta2)], [sin(theta1 + theta2 + theta4) a4
+ sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)], [d3 + d2 + d1]]
On := [[cos(theta1(t) + theta2(t) + theta4(t)) a4
+ cos(theta1(t) + theta2(t)) a3 + a2 cos(theta1(t) + theta2(t))], [sin(
theta1(t) + theta2(t) + theta4(t)) a4 + sin(theta1(t) + theta2(t)) a3
+ a2 sin(theta1(t) + theta2(t))], [d3(t) + d2 + d1]]
[[
dOn := [[
[[
// d \ / d \
-sin(theta1(t) + theta2(t) + theta4(t)) ||--- theta1(t)| + |--- theta2(t)|
\\ dt / \ dt /
/ d \\
+ |--- theta4(t)|| a4
\ dt //
// d \ / d \\
- sin(theta1(t) + theta2(t)) ||--- theta1(t)| + |--- theta2(t)|| a3
\\ dt / \ dt //
// d \ / d \\] [
- a2 sin(theta1(t) + theta2(t)) ||--- theta1(t)| + |--- theta2(t)||], [cos(
\\ dt / \ dt //] [
// d \ / d \
theta1(t) + theta2(t) + theta4(t)) ||--- theta1(t)| + |--- theta2(t)|
\\ dt / \ dt /
/ d \\
+ |--- theta4(t)|| a4
\ dt //
// d \ / d \\
+ cos(theta1(t) + theta2(t)) ||--- theta1(t)| + |--- theta2(t)|| a3
\\ dt / \ dt //
// d \ / d \\]
+ a2 cos(theta1(t) + theta2(t)) ||--- theta1(t)| + |--- theta2(t)||],
\\ dt / \ dt //]
[ d ]]
[--- d3(t)]]
[ dt ]]
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ALS
Inscrit le: 11 Sep 2006
Messages: 647
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| Posté le: 02 Déc 2009 8:05 Sujet du message: |
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Bonjour,
voici une solution en jonglant un peu avec les substitutions, factorisations et en utilisant la fonction collect.
| Code: |
> restart;
> with(linalg):
> f := (d, theta, a, alpha) -> [ [cos(theta), -cos(alpha)* sin(theta), sin(theta) *sin(alpha), a* cos(theta)],
> [sin(theta), cos(theta)* cos(alpha), -cos(theta) *sin(alpha), a* sin(theta)],
> [0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]];
> T01 := f(d1, theta1, 0, 0);
> T12 := f(d2, theta2, a2, 0);
> T23 := f(d3, 0, a3, 0);
> T34 := f(0, theta4, 0, 0);
> T04 := map(combine, evalm(((T01 &* T12) &* T23) &* T34));
>
> On := submatrix(T04, 1 .. 4, 4 .. 4);
> On:=subs(theta1=theta1(t),theta2=theta2(t),d3=d3(t),theta4=theta4(t),evalm(On));
> dOn:=map(x->diff(x,t),On);
f := (d, theta, a, alpha) -> [[cos(theta), -cos(alpha) sin(theta),
sin(theta) sin(alpha), a cos(theta)], [sin(theta),
cos(theta) cos(alpha), -cos(theta) sin(alpha), a sin(theta)],
[0, sin(alpha), cos(alpha), d], [0, 0, 0, 1]]
T01 := [[cos(theta1), -sin(theta1), 0, 0],
[sin(theta1), cos(theta1), 0, 0], [0, 0, 1, d1], [0, 0, 0, 1]
]
T12 := [[cos(theta2), -sin(theta2), 0, a2 cos(theta2)],
[sin(theta2), cos(theta2), 0, a2 sin(theta2)], [0, 0, 1, d2],
[0, 0, 0, 1]]
T23 := [[1, 0, 0, a3], [0, 1, 0, 0], [0, 0, 1, d3], [0, 0, 0, 1]]
T34 := [[cos(theta4), -sin(theta4), 0, 0],
[sin(theta4), cos(theta4), 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
T04 :=
[cos(%1) , -sin(%1) , 0 ,
cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)]
[sin(%1) , cos(%1) , 0 ,
sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)]
[0 , 0 , 1 , d3 + d2 + d1]
[0 , 0 , 0 , 1]
%1 := theta1 + theta2 + theta4
[cos(theta1 + theta2) a3 + a2 cos(theta1 + theta2)]
[ ]
[sin(theta1 + theta2) a3 + a2 sin(theta1 + theta2)]
On := [ ]
[ d3 + d2 + d1 ]
[ ]
[ 1 ]
[cos(%1) a3 + a2 cos(%1)]
[ ]
[sin(%1) a3 + a2 sin(%1)]
On := [ ]
[ d3(t) + d2 + d1 ]
[ ]
[ 1 ]
%1 := theta1(t) + theta2(t)
[-sin(%2) %1 a3 - a2 sin(%2) %1]
[ ]
[cos(%2) %1 a3 + a2 cos(%2) %1 ]
[ ]
dOn := [ d ]
[ -- d3(t) ]
[ dt ]
[ ]
[ 0 ]
/d \ /d \
%1 := |-- theta1(t)| + |-- theta2(t)|
\dt / \dt /
%2 := theta1(t) + theta2(t)
//d \ /d \\
-sin(theta1(t) + theta2(t)) ||-- theta1(t)| + |-- theta2(t)|| a3 -
\\dt / \dt //
a2 sin(theta1(t) + theta2(t))
//d \ /d \\
||-- theta1(t)| + |-- theta2(t)||
\\dt / \dt //
> m:=subs(diff(theta1(t),t)=dth1,diff(theta2(t),t)=dth2,evalm(dOn));
[-sin(%1) (dth1 + dth2) a3 - a2 sin(%1) (dth1 + dth2)]
[ ]
[cos(%1) (dth1 + dth2) a3 + a2 cos(%1) (dth1 + dth2) ]
[ ]
m := [ d ]
[ -- d3(t) ]
[ dt ]
[ ]
[ 0 ]
%1 := theta1(t) + theta2(t)
> m:=map( x->factor( collect(x,[dth1,dth2])),evalm(m));
[-sin(theta1(t) + theta2(t)) (dth1 + dth2) (a3 + a2)]
[ ]
[cos(theta1(t) + theta2(t)) (dth1 + dth2) (a3 + a2) ]
[ ]
m := [ d ]
[ -- d3(t) ]
[ dt ]
[ ]
[ 0 ]
> dOn:=subs(dth1=diff(theta1(t),t),dth2=diff(theta2(t),t),evalm(m));
dOn :=
[
[-sin(theta1(t) + theta2(t))
[
//d \ /d \\ ]
||-- theta1(t)| + |-- theta2(t)|| (a3 + a2)]
\\dt / \dt // ]
[ //d \ /d \\
[cos(theta1(t) + theta2(t)) ||-- theta1(t)| + |-- theta2(t)||
[ \\dt / \dt //
]
(a3 + a2)]
]
[d ]
[-- d3(t)]
[dt ]
[0]
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