PracticasSistemas

❯◆■❱❊❘❙■❉❆❉ P❖▲■❚➱❈◆■❈❆ ❉❊ ❈❆❘❚❆●❊◆❆

P❘➪❈❚■❈❆❙ ❉❊ ❙■❙❚❊▼❆❙ ❘❖❇❖❚■❩❆❉❖❙
●r❛❞♦ ❡♥ ■♥❣❡♥✐❡rí❛ ❊❧❡❝tró♥✐❝❛ ■♥❞✉str✐❛❧ ② ❆✉t♦♠át✐❝❛

❆♥t♦♥✐♦ ❏❛✈✐❡r Pér❡③ ❘♦❞rí❣✉❡③
❉❛♥✐❡❧ ●❛r❝í❛ ▼❛rtí♥❡③

❮♥❞✐❝❡ ❣❡♥❡r❛❧
Prá❝t✐❝❛ ✶

✶

Prá❝t✐❝❛ ✷

✷

Prá❝t✐❝❛ ✸

✹

Prá❝t✐❝❛ ✹

✽

Prá❝t✐❝❛ ✺

✶✶

Prá❝t✐❝❛ ✻

✶✸

Prá❝t✐❝❛ ✼

✶✼

Prá❝t✐❝❛ ✽

✷✶

Prá❝t✐❝❛ ✶
❊❧ ♦❜❥❡t✐✈♦ ❞❡ ❡st❛ ♣rá❝t✐❝❛ ❡s r❡❛❧✐③❛r ✉♥❛ s❡r✐❡ ❞❡❡❥❡r❝✐❝✐♦s ❝♦♥ ❧❛
▼❛t▲❛❜✳

❙②♠❜♦❧✐❝ ▼❛t❤ ❚♦♦❧❜♦① ❞❡

▲♦ ♣r✐♠❡r♦ q✉❡ s❡ ❤❛rá ❡s ❝r❡❛r ✉♥❛ ♠❛tr✐③ A ❝✉❛❞r❛❞❛ ❞❡ ✉♥❛ ❞✐♠❡♥s✐ó♥ ❞❡s❡❛❞❛✱ ❡♥ ❧❛ q✉❡
❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ ❞✐❝❤❛ ♠❛tr✐③ ❝♦♥t❡♥❣❛ ❡❧ tér♠✐♥♦ ✱ ❛❞❡♠ás ❞❡ ✐♥❞✐❝❛r ❡♥ q✉❡ ♣♦s✐❝✐ó♥ s❡ ❡♥✲
❝✉❡♥tr❛ ❞❡♥tr♦ ❞❡ ❧❛ ♠❛tr✐③✱ ❡s ❞❡❝✐r✱ an×m ✱ s✐❡♥❞♦ n ❡❧ ♥ú♠❡r♦ ❞❡ ✜❧❛s ② m ❡❧ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♠♥❛s✳

❛

❆ ❝♦♥t✐♥✉❛❝✐ó♥ s❡ ❝r❡❛rá ✉♥❛ ♠❛tr✐③ B ❞❡ ❧❛ ♠✐s♠❛ ❞✐♠❡♥s✐ó♥ q✉❡ ❧❛ ❛♥t❡r✐♦r ② ❡♥❝❛❞❛
❡❧❡♠❡♥t♦ ❞❡ ést❛ ❝♦♥t❡♥❞rá ❛ x ❡❧❡✈❛❞♦ ❛ ❧❛ s✉♠❛ ❞❡ ❧❛ ✜❧❛ ② ❝♦❧✉♠♥❛ ❡♥ ❧❛ q✉❡ s❡ ❡♥❝✉❡♥tr❛ ❞✐❝❤♦
❡❧❡♠❡♥t♦✱ ❡s ❞❡❝✐r✱ xi+j s✐❡♥❞♦ i ❡❧ ♥ú♠❡r♦ ❞❡ ❧❛ ✜❧❛ ② j ❡❧ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♠♥❛✳
P♦r ú❧t✐♠♦✱ s❡ ❤❛rá q✉❡ x = 2 ❡♥ B ② s❡ ♠♦str❛rá ❧❛ ♠❛tr✐③ ❝♦♥ ❧♦s ❝á❧❝✉❧♦s r❡❛❧✐③❛❞♦s✳
❆q✉í s❡ ❛❞❥✉♥t❛ ❡❧ ❝ó❞✐❣♦ ❞❡❧

❙❝r✐♣t ❝r❡❛❞♦ ❡♥ ▼❛t▲❛❜✿

n=input('Cual es el orden de la matriz?: ')
A=sym('A[]');
for i=1 : n
for j=1 : nA(i,j) = sym(['a',num2str(i),'-',num2str(j)]);
end
end
A
B=sym('B[]');
for i=1 : n
for j=1 : n
B(i,j) = sym(['x','^',num2str(i+j)]);
end
end
B
B=subs(B,'x',num2str(2))

✶

Prá❝t✐❝❛ ✷
❙❝r✐♣t ❡♥ ▼❛t▲❛❜ q✉❡ ♥♦s s❡r✈✐rá ♣❛r❛ ❛❣✐❧✐③❛r ❡❧ ❝á❧❝✉❧♦ ❞❡

❊♥ ❡st❛ ♣rá❝t✐❝❛ r❡❛❧✐③❛♠♦s ✉♥
♠✉❧t✐♣❧✐❝❛❝✐ó♥ ❞❡ ♠❛tr✐❝❡s ❞❡ r♦t❛❝✐ó♥✳

❋✐❣✉r❛ ✶
❈ó♠♦ s❛❜❡♠♦s ❞❡ ❧❛ t❡♦rí❛✱ ❧❛s ♠❛tr✐❝❡s ❞❡ r♦t❛❝✐ó♥ ♥♦s ♣❡r♠✐t❡♥ ❝♦♥♦❝❡r ❧❛♦r✐❡♥t❛❝✐ó♥ ❞❡ ✉♥
s✐st❡♠❛ ❞❡ ❡❥❡s ❝♦♦r❞❡♥❛❞♦s r❡❧❛t✐✈♦s r❡s♣❡❝t♦ ❛ ♦tr♦ ✜❥♦ ❝♦♥♦❝✐❡♥❞♦ ❧♦s á♥❣✉❧♦s ❞❡ ❣✐r♦ r❡❛❧✐③❛❞♦s
r❡s♣❡❝t♦ ❛ ❝❛❞❛ ❡❥❡✳ ❙í ❧♦s ❣✐r♦s s❡ ❤❛❝❡♥ s♦❜r❡ ❧♦s ❡❥❡s ♠ó✈✐❧❡s✱ ❧❛s ♠❛tr✐❝❡s s❡ ♣♦s✲♠✉❧t✐♣❧✐❝❛♥✳ ❙í
❧♦s ❣✐r♦s s❡ ❤❛❝❡♥ s♦❜r❡ ❧♦s ❡❥❡s ✜❥♦s✱ ❧❛s ♠❛tr✐❝❡s s❡ ♣r❡✲♠✉❧t✐♣❧✐❝❛♥✳




1
0
0
R(x, α) =  0 cos(α) − sin(α) 
0 sin(α) cos(α)



cos(φ)
0
R(y, φ) = 
− sin(φ)


cos(θ) −sin(θ) 0
R(z, θ) =  sin(θ) cos(θ) 0 
0
0
1


0 sin(φ)

1
0
0 cos(φ)



✭✶✮

❊♥ ✭✶✮ ♣♦❞❡♠♦s ✈❡r ❧❛s ♠❛tr✐❝❡s ❝♦rr❡s♣♦♥❞✐❡♥t❡s ❛ ❧♦s ❣✐r♦s ❡♥ ❝❛❞❛ ❡❥❡✳ ❆ ❝♦♥t✐♥✉❛❝✐ó♥ s❡
❛❞❥✉♥t❛ ❡❧
q✉❡ ❝❛❧❝✉❧❛ s✐♠❜ó❧✐❝❛♠❡♥t❡ ❧❛ ♠❛tr✐③ ❞❡ r♦t❛❝✐ó♥ ❝♦rr❡s♣♦♥❞✐❡♥t❡ s♦❜r❡ ❧♦s ❡❥❡s
♠ó✈✐❧❡s✱ ✐♥❞✐❝❛♥❞♦ ❡❧ ♥ú♠❡r♦ ❞❡ ❣✐r♦s ❛ r❡❛❧✐③❛r ② s♦❜r❡ q✉❡ ❡❥❡ s❡ ❧❧❡✈❛ ❛ ❝❛❜♦ ❝❛❞❛ ❣✐r♦✳

❙❝r✐♣t

alfa=1;

✷

❆♥t♦♥✐♦ ❏❛✈✐❡r Pér❡③❘♦❞rí❣✉❡③
❉❛♥✐❡❧ ●❛r❝í❛ ▼❛rtí♥❡③

❙✐st❡♠❛s ❘♦❜♦t✐③❛❞♦s

phi=1;
theta=1;
Ralfa=sym('[1,0,0;0,cos(alfa_1),-sin(alfa_1);0,sin(alfa_1),cos(alfa_1)]');
Rphi=sym('[cos(phi_1),0,sin(phi_1);0,1,0;-sin(phi_1),0,cos(phi_1)]');
Rtheta=sym('[cos(theta_1),-sin(theta_1),0;sin(theta_1),cos(theta_1),0;0,0,1]');
R=eye(3);
n=input('Numero de giros: ');
for k=1 : n
o=menu(['Giro ' num2str(k)],'sobre eje X', 'Sobre ejeY', 'Sobre eje Z');
if o==1
if alfa==1
R1=Ralfa;
alfa=alfa+1;
else
R1=subs(Ralfa,'alfa_1',['alfa_' num2str(alfa)]);
alfa=alfa+1;
end
else if o==2
if phi==1
R1=Rphi;
phi=phi+1;
else
R1=subs(Rphi,'phi_1',['phi_' num2str(phi)]);
phi=phi+1;
end
else
if theta==1
R1=Rtheta;
theta=theta+1;
else
R1=subs(Rtheta,'theta_1',['theta_' num2str(theta)]);
theta=theta+1;
end
end
end
R=R*R1;
end
Rdisp('Simplificando...')
R=simple(R)

✸

Prá❝t✐❝❛ ✸
❊♥ ❡st❛ ♥✉❡✈❛ ♣rá❝t✐❝❛ s❡ ❤❛rá ✉s♦ ❞❡ ❧❛s ▼❛tr✐❝❡s ❞❡ ❘♦t❛❝✐ó♥ ❍♦♠♦❣é♥❡❛✳ ❉✐❝❤❛s ♠❛tr✐❝❡s
♥♦s ♣❡r♠✐t❡♥ ❝♦♥♦❝❡r ❧❛ ♣♦s✐❝✐ó♥ ❞❡ ✉♥ ♣✉♥t♦ s♦❜r❡ ❧♦s ❡❥❡s ✜❥♦s q✉❡ s❡ ❡♥❝✉❡♥tr❛ r❡♣r❡s❡♥t❛❞♦ ❡♥
❡❧ s✐st❡♠❛ ❞❡ ❡❥❡s ♠ó✈✐❧❡s✱ ❝♦♥♦❝✐❡♥❞♦ ❞❡ ést❡ ❧❛ ♣♦s✐❝✐ó♥ ❞❡ s✉ ♦r✐❣❡♥ r❡s♣❡❝t♦ ❛ ❧♦s ❡❥❡s ✜❥♦s ② s✉
♦r✐❡♥t❛❝✐ó♥ r❡s♣❡❝t♦ ❛❧ ♠✐s♠♦✳

 
rX0
nx
 rY 0  ny

 
 rZ0  =nz
1
0

ox
oy
oz
0

ax
ay
az
0


 
Px
rXn


Py 
 ·  rY n 
Pz   rZn 
1
1

✭✷✮

❊♥ ❧❛ ❡❝✉❛❝✐ó♥ ✭✷✮ ✈❡♠♦s q✉❡ ❡❧ ✈❡❝t♦r r r❡♣r❡s❡♥t❛ ❧❛ ♣♦s✐❝✐ó♥ ❞❡ ✉♥ ♣✉♥t♦ r❡s♣❡❝t♦ ❛ ❧♦s ❡❥❡s
♠ó✈✐❧❡s✳ ❊❧ ✈❡❝t♦r P ✐♥❞✐❝❛ ❧❛ ♣♦s✐❝✐ó♥ ❞❡❧ ❡❥❡ ♠ó✈✐❧ r❡s♣❡❝t♦ ❛❧ ✜❥♦✳ ❉♦♥❞❡ n o a ❡s ❧❛ s✉❜♠❛tr✐③
❞❡ r♦t❛❝✐ó♥ q✉❡ ♦r✐❡♥t❛ ❛❧ ❡❥❡ ♠ó✈✐❧ r❡s♣❡❝t♦ ❛❧ ✜❥♦✳ ❨ ♣♦r ú❧t✐♠♦✱ ❡❧ ✈❡❝t♦r r ♥♦s...