apuntes de robótica - universitat de valencia

6/2/2018 apuntes de rob tica - universitat de valencia

1/177

30


2/177


3/177


4/177


5/177


6/177

y/p/r

TM


7/177

TM

Kg = 0

Kg= 0


8/177


9/177


10/177


11/177


12/177


13/177


14/177


15/177

Cilindrico Esferico

Artic. de paralelogramo Mixto (SCARA)

Cartesiano tipo portico

Eje vertical usual

Articulacion cilindrica

zy

Cartesiano tipo cantilever

x


16/177


17/177

Union

miento

Union

rotula

Union

apoyo

plano

Union

pivote

deslizante

Union

deslizante

helicoidal

Union

pivote

(rotacional)

Uniondeslizante

(traslacional)

Nombre de

empotra

Union

lineal

deslizante

Union

lineal

anular

Union

puntual

Union

libre

0 traslac.

0 rotac.

1 rotac.

0 traslac.

0 rotac.1 traslac.

1 rotac. y

1 traslac.

conjugadas

1rotac.

1 traslac.

1 rotac.

2 traslac.

3 rotac.

0 traslac.

2 rotac.

2 traslac.

3 rotac.

1 traslac.

3 rotac.

2 traslac.

3 rotac.

3 traslac.

0

1

1

1

2

3

3

4

4

5

6No hay simbolo

No hay contacto entre los dos cuerpos

la union

Movimien.

relativos

Num.

de g.l.Simbolos

(vistas lateral y frontal)

C1

C2

C1

C2

C1

C2

C2

C1

C2

C1

C2C1

C2

C2

C1

C2

C1

C2

C1

C1C2

C1

C2

C1

C2

C2

C1

C2

C1

C1 C2

C1

C1: cuerpo 1

C2: cuerpo 2


18/177

(x,y,z)


19/177

y x

z

xy

z

x

y

z

Sistema no

ortogonalSistema ortonormal Sistema ortonormal

dextrogiro: x y = z levogiro: x y = z

x

(x,y,z)

x

y

z

w

x = xwy = ywz = zw

w

w

x

v

x

x =x+v

x

y

z

=

xy

z

+

vxvy

vz

x

y

z

1

=

1 0 0 vx0 1 0 vy0 0 1 vz0 0 0 1

x

yz1


20/177

x =H x

x= (H)1 x

(H)1 =

1 0 0 vx0 1 0 vy0 0 1

vz

0 0 0 1

x

x

v

x

y

x

y

i

j

j i

i

j

i

j

x= xi + yj =xi + yj i

= cos i + sin jj = sin i + cos j

xi + yj =x(cos i + sin j) + y( sin i + cos j)

xy

=

cos sin sin cos

x

y

z

xyz1

=

cos sin 0 0sin cos 0 0

0 0 1 00 0 0 1

x

y

z

1

R

R

x

yz1

=

cos(

)

sin(

) 0 0

sin() cos () 0 00 0 1 00 0 0 1

x

y

z

1

=

cos sin 0 0

sin cos 0 00 0 1 00 0 0 1

x

y

z

1


21/177

Rot(x, ) =

1 0 0 00 cos

sin 0

0 sin cos 00 0 0 1

Rot(y, ) =

cos 0 sin 00 1 0 0

sin 0 cos 00 0 0 1

Rot(z, ) =

cos sin 0 0sin cos 0 0

0 0 1 00 0 0 1

y

900

x

z

900

y

Y = T2R3T1R2R1X

X

R1 R2 T1 R3

T2

r = (rx, ry, rz)

Rr,

P P

P P

Rr, =Rx,Ry,Rz,Ry,Rx,

Rr, =

1 0 0 00 c s 00 s c 00 0 0 1

c 0 s 00 1 0 0s 0 c 0

0 0 0 1

c s 0 0s c 0 00 0 1 00 0 0 1

c 0

s 0

0 1 0 0s 0 c 00 0 0 1

1 0 0 0

0 c s 00 s c 00 0 0 1


22/177

ry + rz22

ry + rz22

Y

x

z

P

P

P

r

o

ry

rx

rx

rz

1

2

4

5

3

P es el trasladado de P por la rotacion 1 (angulo )

P es el trasladado de P por la rotacion 2 (angulo )

s =

ryr2y+ r

2z

c = rz

r2y+ r2z

s = rx| r | =rx

c =

r2y+ r

2z

| r | =

r2y+ r2z

Rr, =

r2xv + c rxryv rzs rxrzv+ ryc 0rxryv + rzs r

2yv + c ryrzv rxs 0

rxrzv rys ryrzv + rxs r2xv+ c 00 0 0 1

s= sin

c = cos

v = 1 cos

rx = ry = 0


23/177

(rx, ry, rz)

Z=Z

Z=Z

Y=Y

Y

X X

XX

Y Y

Z

X=X

XX

Y

Z

Y

Y

Z

r

p

y

r esta en el plano Z/Z

p esta en el plano X/X/X

y esta en el plano X/X

Angulos de EulerAngulos y/p/r

y/p/r

z

y

y

z

z

R,,=

c s 0 0s c 0 00 0 1 00 0 0 1

c 0 s 00 1 0 0s 0 c 0

0 0 0 1

c s 0 0s c 0 00 0 1 00 0 0 1

=

ccc ss ccs sc cc 0scc+ cs scs+ cc sc 0

sc ss c 00 0 0 1


24/177

y/p/r

r

p

y

Ry,p,r =

cy sy 0 0sy cy 0 00 0 1 00 0 0 1

cp 0 sp 00 1 0 0sp 0 cp 0

0 0 0 1

1 0 0 00 cr sr 00 sr cr 00 0 0 1

=

cycp cyspsr sycr cyspcr+ sysr 0sycp syspsr+ cycr syspcr cysr 0sp cpsr cpcr 00 0 0 1

xy=z

X OX = OP + P X

xi+ yj+ zk =pxi+pyj+pzk+ xi + yj + zk

i = cos(ix)i+ cos(iy)j+ cos(iz)k

j = cos(jx)i+ cos(jy)j+ cos(jz)k

k = cos(kx)i+ cos(ky)j+ cos(kz)k

x y = z


25/177

O

X

k

ji

k

j

i P

X

Y

Z

X

Z

Y

x= px+ x cos(ix) + y

cos(jx) + z cos(kx)

y=py+ x cos(iy) + y

cos(jy) + z cos(ky)

z=pz+ x cos(iz) + y

cos(jz) + z cos(kz)

xyz1

= cos(ix) cos(jx) cos(kx) pxcos(iy) cos(jy) cos(ky) pycos(iz) cos(jz) cos(kz) pz

0 0 0 1

x

y

z

1

T=

R33 p31

f13 1 1

=

Rot. T ras.

P ersp. Esc.global

3 3

det(R) = 1

det(T) = 1

R1 =RT

f

f= (0, 0, 0)


26/177

T =

nx ox ax px

ny oy ay pynz oz az pz0 0 0 1

n o a p0 0 0 1

z

a

y

o

x n

a

o

n

yaw

z z

(y,p,r)

x z y x

z y

Ry,p,r

Ry,p,r=

cyspsr sycr cyspcr+ sysr cycp 0syspsr+ cycr syspcr cysr sycp 0

cpsr cpcr sp 00 0 0 1


27/177

M

M=

nx ox ax pxny oy ay pynz oz az pz0 0 0 1

m1(i, j)

(1)i+jdet(A(j, i))

A(j, i)

(i, j)

M

i

j

m111 =det(A(nx)) =

oy ay pyoz az pz0 0 1

=oyaz ozay = (o a)x = nx

m114 =

det(A(041)) =

ox ax pxoy ay py

oz az pz

=

ox oy ozax ay az

px py pz

=

(o

a)

p=

n

p

M1 =

nx ny nz npox oy oz opax ay az ap0 0 0 1

f : J R6

(ax, ay, az)

y/p/r

R6

T

(ax, ay, az, ox, oy, oz, nx, ny, nz, px, py, pz)

T

T (0, 0, 0, 1)


28/177

T

T

T

L

L L L

L

JJ

J

J0

1

1 22

3

3

4

4

Fijo

ai i

i + 1

ai

i i i+ 1

ai

di i/i + 1

i i 1/i i


29/177

i i 1/i i/i + 1

i

i

i

a

i

i

di

Eje i+1

a i

Enlace i

Eje i

Eje i1

Eje i

Eje i+1

a

a

d

Linea paralela

a la normal

eje i / eje i+1

i1

i

i

Enlace i1

Enlace i

i

i

Enlace i aislado Enlaces i e i1

zi1 i zi i+ 1

z

xi z zi z (zi1)

ai

yi

xi yi=zi

ai1

i

i 1 i

x0 z0

z1

z0

z

y


30/177

xz

yy

x

z

Linea paralela

al eje z

i1

i1

i1

i

i

i

i

Eje i

Eje i+1

a

a

d

Linea paralela

a la normal

eje i / eje i+1

i1

i

i

Enlace i1

Enlace i

i

Eje i1

i1

z

x

z

ai x

z

z

z

x

z

x

x

ai

i di ai i

di i i

ai

i di

i 0

0

900

1800

2700

i 1

i

zi1 i xi1 xi ai1 ai zi1

zi1 zi di xi1 xi zi1 zi


31/177

xi ai

i xi1 x

i

xi1 xi i zi

1

zi y

i

1

yi

i 1

i

i1Ai = Rotzi1,i T r(0,0,di) T r(ai,0,0) Rotxi,i

i1Ai=

ci si 0 0si ci 0 00 0 1 0

0 0 0 1

1 0 0 00 1 0 00 0 1 di0 0 0 1

1 0 0 ai0 1 0 00 0 1 0

0 0 0 1

1 0 0 00 ci si 00 si ci 0

0 0 0 1

=

=

ci si 0 0si ci 0 00 0 1 di0 0 0 1

1 0 0 ai0 ci si 00 si ci 00 0 0 1

=

ci sici sisi aicisi cici cisi aisi0 si ci di0 0 0 1

i

i

i 1

i

i

1

4a

ai zi1 zi

xi i zi1 zi xi xi

i xi1 xi zi1 zi1

di i 1 i

i

ai

zi1 zi1

n

0An=0A1 1A2 . . . n1An

i1Ai i, ai, i di i ai

di i

0An i, ai, i, di i= 1 . . . n


32/177

n

n

z

zi1 i

x

xi zi zi1

y

xi yi=zi

(i, ai, i, di)

i1Ai

0An

R2T R3

L1 L2

L5

L6

z

z0

z1

z1z2 L3

z3 z2 z2

z4


33/177

xy

z

xy

z

yxz

x

z

yx y

z

y

x

z

y =o

x =n

z =a

6

66

3

3

3

4 4

4

55

5

2 2

2

1

1

1

0

0

0

d

d

d

d

1

1

2

3

4

5

6

6

Origenes coincidentes

(d = d = 0)4 5

Sistema

del mundo

2

z5

z6 z5

x6 y6

x

x0 z0 z

x1 z z1 z z0

x1

x2 z1 z2

x2

x3 z2 z3

x4 z3 z4

z2 x4

x5

z5 z3 x5 x3

z3 z4 z3

z5


34/177

x6 z5 z6

y

xi yi= zi

z0 z1 z1 z2

z

z

y

x

n

o

a

1 z0 z1 x1 x1

z0 z1

2

a1 z0 z1

d1 z0

d1

1 x0 x1 z0 z0

x0

x1 2

Articulacion i i ai di1 (

2)

2 0 d1= 412mm.

2 (2

) 2

0 d2= 154mm.3

2

0 0 (d3)

4 (0) 2 0 05 (0)

2 0 0

6 (0) 0 0 d6= 263mm.


35/177

J=

12d3

456

i1Ai i

0A1 =

c1 0 s1 0s1 0 c1 0

0 1 0 d10 0 0 1

1A2 =

c2 0 s2 0s2 0 c2 0

0 1 0 d20 0 0 1

2A3 =

0 1 0 01 0 0 00 0 1 d30 0 0 1

3A4 =

c4 0 s4 0s4 0 c4 0

0 1 0 00 0 0 1

4A5 =

c5 0 s5 0s5 0 c5 0

0 1 0 00 0 0 1

5A6 =

c6 s6 0 0s6 c6 0 0

0 0 1 d60 0 0 1

0A3 =0A1

1A22A3 =

c1 0 s1 0s1 0 c1 0

0 1 0 d10 0 0 1

c2 0 s2 0s2 0 c2 0

0 1 0 d20 0 0 1

0 1 0 01 0 0 00 0 1 d20 0 0 1

=

c1 0 s1 0s1 0 c1 0

0 1 0 d10 0 0 1

0 c2 s2 d3s20 s2 c2 d3c21 0 0 d20 0 0 1

=

s1 c1c2 c1s2 d3c1s2 d2s1c1 s1c2 s1s2 d3s1s2+d2c1

0 s2 c2 d3c2+d10 0 0 1

1 =2 2 =2 d3 = l

l

0A3 =

1 0 0 d20 0 1 l0 1 0 d10 0 0 1

(d2, l , d1) x3

(1, 0, 0) x0

y3 z3 z0 y0

(0, 0, 1) (0, 1, 0)


36/177

3A6 = 3A4

4A55A6 =

c4 0 s4 0s4 0 c4 0

0 1 0 00 0 0 1

c5 0 s5 0s5 0 c5 0

0 1 0 00 0 0 1

c6 s6 0 0s6 c6 0 0

0 0 1 d60 0 0 1

=

c4 0 s4 0s4 0 c4 0

0 1 0 00 0 0 1

c5c6 c5s6 s5 d6s5s5c6 s5s6 c5 d6c5

s6 c6 0 00 0 0 1

=

c4c5c6 s4s6 c4c5s6 s4s6 c4s5 d6c4s5s4c5c6+ c4s6 s4c5s6+ c4s6 s4s5 d6s4s5

s5c6 s5s6 c5 d6c50 0 0 1

0A3 3A6

T R5

Hombro

Motor del

hombro

Parte superior

del brazo

Tubo para

cables

Motoreductor (eje z)

Placa de

conexiones

Base

Eje de la

Mueca y mano

mueca

Antebrazo

Polea intermedia

Motor del yaw

Motor del antebrazo

Correa del eje z


37/177

254 mm 254 mm

694 mm

915 mm

33 mm

98o

Eje del shoulder

Eje del elbow

Eje del yaw

Desplazamiento

en zed

220

180

151

180

90 mm

o

o

0

0

(shoulder)

(elbow)

(yaw)

Vista lateral Vista superior

X0

Y2

Y3

X5

a2

a3

d6

d

5

6

2

3

4

d4

Z0

Y0

Z2

X2

Y4X3

Z3

Z4,Y5

X4

X6

Y6

Z6

X1

Z1

Y1

d3

Z5

z

z0

z1

z0z2 z1

z3 z2


38/177

z4 z3

z5

z6

z5

x6 y6

x

x0 z0 z

x1 z z1 z z0

x

x2 z1 z2

x3 x2

x4 z3 z4

y0 x4

x5 z4 z5

x5 900 x5

x6 z5 z6

x5

y

xi yi= zi

z3

z4

z

y

x

n

o

a

1 z0 z1 a1 z0 z1 d1 z0


39/177

d1

d

1 x0 x1 z0 z0

x0

x1

1

2 z1 z2 a2 z1 z2

d2

a2

d2 2 x1 x2 z1 z1

x1

x2

2

x2 x3 3

a3

d3

4 z3 z4 x4 x4

z3 z4

4 2 a4 z3 z4 4 x3 x4 z3 z3

x3

x4

d4

z3

5 z4 z5 x5 x5

z3 z5 5

2

a5 z4 z5 d5

6 z5 z6 a6 6 x5 x6 z5 z5

x 6 d6


40/177

z5

Articulacion i i ai di1 0 0 0 (d)2 (

2) 0 a2= 254mm. 0

3 (0) 0 a3= 254mm. d3 = 75mm.4 ()

2 0 d4 = 85mm.

5 (2

) 2

0 06 (0) 0 0 d6 = 176mm.

J=

d

23456

i1Ai i

0A1 =

1 0 0 00 1 0 0

0 0 1 d10 0 0 1

1A2 =

c2 s2 0 a2c2s2 c2 0 a2s2

0 0 1 00 0 0 1

2A3 =

c3 s3 0 a3c3s3 c3 0 a3s3

0 0 1 d30 0 0 1

3A4 =

c4 0 s4 0s4 0 c4 0

0 1 0 d40 0 0 1

4A5 =

c5 0 s5 0s5 0 c5 0

0 1 0 00 0 0 1

5A6 =

c6 s6 0 0s6 c6 0 0

0 0 1 d60 0 0 1

0A3 = 0A1

1A22A3 =

1 0 0 00 1 0 00 0 1 d10 0 0 1

c2 s2 0 a2c2s2 c2 0 a2s2

0 0 1 00 0 0 1

c3 s3 0 a3c3s3 c3 0 a3s3

0 0 1 d30 0 0 1

=

cos(2+3) sin(2+3) 0 a3cos(2+3) +a2cos(2)sin(2+3) cos(2+3) 0 a3sin(2+3) +a2sin(2)

0 0 1 d2+d30 0 0 1

3A6=3A4

4A55A6=

c4c5c6+s4s6 c4c5s6+s4c6 c4s5 c4s5d6s4c5c6 c4s6 s4c5s6 c4c6 s4s5 s4s5d6

s5c6 s5s6 c5 c6d6 d4


41/177

0A6

A11= c(2+3+4)c5c6+s(2+3+4)s6A12= s(2+3+4)c6 c(2+3+4)c5s6A13= c(2+3+4)s5A14

= d6s

5c(

2+

3+

4) +a

3c(

2+

3) +a

2c(

2)

A21= s(2+3+4)c5c6 c(2+3+4)s6A22= c(2+3+4)c6 s(2+3+4)c5s6A23= s(2+3+4)s5A24= d6s5s(2+3+4) +a3s(2+3) +a2s(2)

A31= s5c6A32= s5s6A33= c5A34= d1 d3 d4 d6c5A41= 0

A42= 0

A43= 0A44= 1


42/177

T

Ji

T

T =

nx ox ax px


(X,Y,Z,y,p,r)

X=px(J1 . . . J 6) y= arctan(ayax

)

Y =py(J1 . . . J 6) p= arctan( azax/ cos(y)

)

Z=pz(J1 . . . J 6) r= arctan(

nzoz )

y/p/r


43/177

n a

o

X

Y

Z

R

P

S

R

S

P

(X, Y , Z)

a

o

n

P a

S a

a

S=| S| a

| S|

S

R

R= P S

R

R

(px, py, pz) 0A3

0A6

(J1, J2, J3) Ji i di

R3 R2T RT R T R2

RT2

T RT

T2R

T3

i 0

2

2

i= 1 . . . 3

8 27 = 216

0A3

(1, 2, 3)

T R2 1 = 0

x0 x1 x0 z0


44/177

i

0A11A2

2A3

i = 0

1 0 0 00 1 0 00 0 1 d0 0 0 1

c2 s2 0 a2c2s2 c2 0 a2s2

0 0 1 d20 0 0 1

c3 s3 0 a3c3s3 c3 0 a3s3

0 0 1 d30 0 0 1

i= 2

1 0 0 00 0 1 00 1 0 d0 0 0 1

c2 0 s2 a2c2s2 0 c2 a2s2

0 1 0 d20 0 0 1

c3 0 s3 a3c3s3 0 c3 a3s3

0 1 0 d30 0 0 1

i= 2

1 0 0 00 0 1 00 1 0 d0 0 0 1

c2 0 s2 a2c2s2 0 c2 a2s2

0 1 0 d20 0 0 1

c3 0 s3 a3c3s3 0 c3 a3s3

0 1 0 d30 0 0 1

1=2 = 3= 0

0A3=

1 0 0 00 1 0 00 0 1 d0 0 0 1

c2 s2 0 a2c2s2 c2 0 a2s2

0 0 1 d20 0 0 1

c3 s3 0 a3c3s3 c3 0 a3s3

0 0 1 d30 0 0 1

=

c(2+ 3) s(2+ 3) 0 a3c(2+ 3) + a2c2s(2+ 3) c(2+ 3) 0 a3s(2+ 3) + a2s2

0 0 1 d2+ d

3+ d

0 0 0 1

R

d

2 3

Rx = a3c(2+ 3) + a2c2

Ry = a3s(2+ 3) + a2s2

Rz = d2+ d3+ d

d d= Rz d2 d3

a2 a3

a

Rx = a(c(2+ 3) + c2)

Ry = a(s(2+ 3) + s2)

Rx = 2a cos(2+3

2) cos3

2

Ry = 2a sin(2+3

2) cos

32


45/177

g =2+ 3/2

3 = 2 arc cos

R2x+ R

2y

2a

2 = atan2(Ry, Rx) arccos

R2x+ R2y

2a

R= P S0

1, 2 3

0A3

3A6

3A6=

c4c5c6+ s4s6 c4c5s6+ s4s6 c4s5 c4s5d6s4c5c6 c4s6 s4c5s6 c4c6 s4s5 s4s5d6

s4c6

s4s6

c5

c5d6

d40 0 0 1

S

S3x = c4s5d6

S3y = s4s5d6

S3z = c5d6 d4

S3

S

S0

0A3

S3 = 3R0 S03R0= (

0R3)T

4 = atan2(S3y , S

3x)

5

= arc cos(

S3z + d4

d6)

6

o

T

(nx, ny, nz, ox, . . . e t c)


46/177

3

A6 =

c4 0 s4 0s4 0

c4 0

0 1 0 d40 0 0 1

c5 0 s5 0s5 0

c5 0

0 1 0 00 0 0 1

c6 s6 0 0s6 c6 0 0

0 0 1 d60 0 0 1

=

nx ox ax pxny oy ay py

nz oz az pz0 0 0 1

3A6 =

c4c5c6+ s4s6 c4c5s6+ s4s6 c4s5 c4s5d6s4c5c6 c4s6 s4c5s6 c4c6 s4s5 s4s5d6

s4c6 s4s6 c5 c5d6 d40 0 0 1

=

nx ox ax pxny oy ay pynz oz az pz0 0 0 1

4 T

3A4

(3A4)1

5

6

c4s5 = ax

s4s5 = ay

4=atan2(ay, ax)

cos 4 = 11 + tan2 4

= axa2x+ a

2y

sin 4 = 11 + cotan24

= aya2x+ a

2y

3A4

(3A4)1 =

c4 s4 0 0

0 0 1 d4s4 c4 0 0

0 0 0 1

=

axa2x+a

2y

aya2x+a

2y

0 0

0 0 1 d4aya2x+a

2y

axa2x+a

2y

0 0

0 0 0 1

T = 3A4 4A5 5A6 = (3A4)1 T = 4A5 5A6

axa2x+a

2y

aya2x+a

2y

0 0

0 0 1 d4aya2x+a

2y

axa2x+a

2y

0 0

0 0 0 1

nx ox ax px


= c5 0 s5 0

s5 0 c5 00 1 0 00 0 0 1

c6 s6 0 0s5 c5 0 0

0 0 1 d60 0 0 1


47/177

axnx+aynya2x+a

2y

axox+ayoya2x+a

2y

a2x+ a

2y

axpx+aypya2x+a

2y

nz oz az pz+ d4aynxaxny

a2x+a2yayoxaxoy

a2x+a2y 0 aypxaxpy

a2x+a2y0 0 0 1

=

c5c6 c5s6 s5 d6s5s5c6 s5s6 c5 d6c5

s6 c6 0 00 0 0 1

5

(1, 3)

(2, 3)

tan 5=

a2x+ a2y

az= 5 = atan2(

a2x+ a

2y,az)

(4A5)1 =

c5 s5 0 00 0 1 0

s5 c5 0 00 0 0 1

=

az a2x+ a2y 0 0

0 0 1 0a2x+ a

2y az 0 0

0 0 0 1

(4A5)1 (3A4)1 T = 5A6

6

6 = arc cos(axnx ayny)aza2x+ a2y + nz

1 a2z

A31 A32

atan2

a2x+ a

2y

z

f C1

x0

x0


48/177

x x + x

y

y + y

0 0 0

0

0 0

f

x0

R R

x0 f y0 = f(x0)

f(x0+ x0) f(x0) + (tan())x0=y0+ dfdx

x=x0

x0

Rnj

R6

nj

i

X

X = F()

X0+ X X0

F(X0+ X) F(X0) + JfX0

JfX0 f X0= ( y,p,r)

JfX0 =

x1

. . . xnj

y

1. . . y

njz1

. . . znj

y1 . . . ynjp

1. . . p

njr1

. . . rnj

X

X=JfX


49/177

X

X=JfX

vxvyvzypr

=

Jf

X

123456

123456

=

Jf

1

X

vxvyvzypr

X

J

v

J1X = 1

det(JX)Ad(JX)

Ad(JX)

JX det(JX)

J1X

X

m(X) =

det(JX JTX)

det

JTX X

m(X) =det(J)


50/177

T

T = 0A1 1A2 . . . 5A6

i= 1 . . . 6

Ui

Ui= i1Ai iAi+1 . . . 5A6

i

Ui=

nxi oxi axi pxi

nyi oyi ayi pyinzi ozi azi pzi0 0 0 1

pxinyi pyinxipxioyi pyioxipxiayi pyiaxi

nxioxiaxi

nzioziazi000

i

i

Ui


51/177

z

x

yO

ss+ s

C

BA

dsdt

d

s

s

| s |=| AB|=| CA | d=| OA | sin() | | dt| s |

dt

=

| OA

|sin()

|

|

s

s

OC

s

ds

dt = s

R(ds

dt) = RTN

N(ds

dt) + RTNs


52/177

z

x

y

z

x

y

x

y

zO

O

O

0

0

0

i

i+1

i+1

i+1

i+1

i

i

ir

r

r

O,i+1

i,i+1

O,i

ro,i+1=ro,i+ ri,i+1

dro,i+1dt

R

= dro,i

dt

R

+ dri,i+1

dt

R

R

vi+1= R

vi+ dri,i+1

dtRi

+ R

i+1 R

ri,i+1

ri,i+1=di+1ki

dri,i+1dt

Ri

= di+1ki

i+1=i

ri,i+1

dri,i+1dt

Ri

= 0

i+1=i+ i+1ki


53/177

Rai+1= Rai+

d2ri,i+1

dt

2 Ri

+ Ri+1

dri,i+1

dtRi

+ Ri+1

ri,i+1+

Ri+1

dri,i+1

dtRi

+ Ri+1

ri,i+1

Rai+1= Rai+

d2ri,i+1dt2

Ri

+ 2 Ri+1 dri,i+1dt

Ri

+ Ri+1 ri,i+1+ Ri+1 Ri+1 ri,i+1

Ri+1 dri,i+1

dt

Ri

d2ri,i+1

dt2Ri

= di+1ki

0

i+1=di+1

dt =

didt

= i d(i+i+1ki)

dt = i+i+1ki+i i+1ki

R

Articulacion i es traslacional Articulacion i es rotacional

vi+1 = vi+ di+1ki+i+1 di+1ki (T1a) vi+1= vi+i+1 ri,i+1 (T1b)ai+1= ai+ di+1ki+ 2i+1 di+1ki+ ai+1= ai+ i+1 ri,i+1+

+i+1 di+1ki+i+1 (i+1 di+1ki) (T2a) +i+1 (i+1 ri,i+1) (T2b)i+1= i (T3a) i+1 = i+ i+1ki (T3b)

i+1= i (T4a) i+1= i+i+1ki+i i+1ki (T4b)

i, i,i di, di,di

(L0) v0 = 0, a0 =g, 0 =

0, 0 = 0 g

1, 1, v1 a1 (1, 1, 1)

d1, d1, d1 (2, 2, 2) 2, 2, v2 a2


54/177

mi Fi i

i

Fi=mia i

i=dL

dt

F

=r F

r

L

L=V

ri pidVi

ri

pi pi=mivi

w

L= I w

I

I= mi

(r2iy+ r2iz)dVi

mirixriydVi

mirixrizdVi

mirixriydVi mi(r2ix+ r2iz)dVi miriyrizdVi mirixriydVi miriyrizdVi mi(r2ix+ r2iy)dVi

ri= (rix, riy, riz) mi

dL

dt

R

=I+ I

I

R(I) = RRotTN N(I) RRotN RRotN


55/177

vv

c

cii

i

ii

CMO i

vci=vi+i cii

cii dciidt

= 0

aci= ai+ i cii+i (i cii)

vci aci

i

Fi=miaci

fi1,i fi,i+1+ mig=miaci

fkl k l

fi+1,i fi,i+1

i i=

dL

dt

i1,i i,i+1 ci1,i fi1,i+ ci,i fi,i+1= Iii+i Iii

ci1,i=ri1,i+ cii

fi1,i= fi,i+1 mig+ miaci

i1,i=i,i+1+ (ri1,i cii) fi1,i+ cii fi,i+1+ Iii+i Iii


56/177

R

fn,n+1= Mg

n,n+1 = R (Mg)

fn1,n = Mg mng+ mnacn

n1,n= R (Mg) + (rn1 cnn) fn1,n+ cnn (Mg) + Inn+n Inn

acn rn1

acn acn= an+ n cnn+n (n cnn)

fn,n+1= 0 n,n+1 = 0

Ji di i


57/177

v a

J J J

f y

Se obtienen

f y

Se obtienen

J J J

v a

J J J

v a

v a

v a

v a

v a

Se conocen Se conocen

Se obtienen

usando

idem, usando

Se conocen

usando

Se obtienen

y se conocen

y se conocen

0 0 0 0 111 0,1 0,1

1 1 1 1

2 2 2

1,2 1,2

n n

n n n n

Se obtienen

f y

f y

n1

cn1 cn1

cn cn

n,n+1 n,n+1

n1,n n1,n

c1 c1

n1

n

n1 n1

usando T3a/b, T4a/b T1a/b, T2a/b

usando 2.5 y 2.6

usando 2.7 y 2.8

T3a/b, T4a/bT1a/b, T2a/b,

usando 2.5 y 2.6

usando 2.7 y 2.8

T1a/b, T2a/b

T3a/b, T4a/busando 2.7 y 2.8

usando 2.5 y 2.6


58/177

R

R

6

Rnj

f : R R6t [0 . . . t0] (x,y,z,y,p,r)

C1

f

C1

C1

: R Rt (t) con t [0 . . . t0]

(0) =0 (0) = 0

(tf) =f (tf) = 0


59/177

0 f

t

(t) =a0+ a1t + a2t2 + a3t

3

ai

a0 = 0

a0+ a1tf+ a2t2f+ a3t3f = f

a1 = 0

a1+ 2a2tf+ 3a3t2f = 0

ai tf

a0 = 0

a1 = 0

a2 = 3

t2f(f 0)

a3 = 2t3f

(f 0)

0

f i i = 1 . . . n

(ti1) = i1 a0+ a1ti1+ a2t2i1+ a3t

3i1 = i1

(ti) = i a0+ a1ti+ a2t2i + a3t

3i = i

(ti1) = i1 a1+ 2a2ti1+ 3a3t2i1 =

i1(ti) = i a1+ 2a2ti+ 3a3t

2i =

i

ti

(ti)


60/177

t t t t

0

1

2

3

4

t0 1 2 3 4

a(t)

b(t)

a(t) = a10+ a11t + a12t2 + a13t

3 a(t) = a11+ 2a12t + 3a13t2 a(t) = 2a12+ 6a13t

b(t) = a20+ a21t + a22t2 + a23t

3 b(t) = a21+ 2a22t + 3a23t2 b(t) = 2a22+ 6a23t

tf1

tf2

0 = a(0) = a10 (1)v = a(tf1) = a10+ a11tf1+ a12t

2f1+ a13t

3f1 (2)

v = b(0) = a20 (3)g = b(tf2) = a20+ a21tf2+ a22t

2f2+ a23t

3f2 (4)

a(tf1) = b(0) = a11+ 2a12tf1+ 3a13t2f1 = a21 (5)a(tf1) = b(0) = 2a12+ 6a13tf1 = 2a22 (6)

a(0) = 0 = a11 = 0 (7)b(tf2) = 0 = a21+ 2a22tf2+ 3a23t2f2 = 0 (8)

v

g


61/177

a10=0 a20 = v a11= 0

a12t2f1+ a13t

3f1 = v 0 (2)

a21tf2+ a22t2f2+ a23t

3f2 = g v (4)

2a12tf1+ 3a13t2f1 a21t3f1 = 0 (5)

2a12+ 6a13tf1 2a22 = 0 (6)a21+ 2a22tf2+ 3a23t

2f2 = 0 (8)

t2f1 t3f1 0 0 0

0 0 tf2 t2f2 t

3f2

2tf1 3t3f1 1 0 0

2 6tf1 0 2 00 0 1 2tf2 3t

2f2

a12a13a21a22a23

=

v 0g 0

000

c

a(tf1) = c

a11+ 2a12tf1+ 3a13t2f1 =

c a21= c

n

4n

4n

tb tf

0

tb tb tftb

tf tb tf

0

tb

(t) =0+1

2t2

b=0+1

2t2b


62/177

0 tb

t t t th f b f

b

h

f

0

(t) =t

tb (t) =tb

tb=h bth tb

th =tf/2

h= (0+ f)/2

tb=0+f

2 (0+ 12 t2b)

tf2 tb

t2b tftb+ (f 0) = 0

tb

tb

tb= tf

2

2t2f 4(f 0)

2

2t2f 4(f 0)> 0

4(f 0)t2f


63/177

t

t

t

tj tk

j k tdjk

tjk

j

t

t

t

jk

djk

k

k

l

tj

j


64/177

k

jk =k j

tdjk

k

k =sig(kl jk) | k|

tk =kl jk

k

kl = jk + ktk

tjk =tdjk 12

tj 12

tk

i

i 1

i+ 1

i

i 1

i

i

i+ 1

i

ti

n

o

a

y/p/r


65/177

X

Y

r

Vista superior

a

Recta x=a vy

r

X

Y

Z

Vista en perspectiva

z

T RT

x = r cos()

y = r sin()

z = z

= atan2(y, x)

r =

x2 + y2

z = z

J=

xr

x

xz

y

r

y

y

z

zr z zz

=

cos() r sin() 0sin() r cos() 0

0 0 1

vxvy

vz

=

cos() r sin() 0sin() r cos() 0

0 0 1

r

z

rz

= 1r

r cos() r sin() 0 sin() cos() 00 0 r

vxvy

vz

=

x

x2+y2y

x2+y2 0y

x2+y2x

x2+y2 0

0 0 1

vxvy

vz


66/177

r = 1

x2 + y2(xvx+ yvy)

=

1

x2 + y2 (yvx+ xvy)z = z

x = a

a

vx= 0 vy =cte

r = y

a2 + y2vy

= a

a2 + y2vy

x = a

y = 0

r = 0

= vya

a

(0, 0)


67/177


68/177


69/177


70/177

Lamina de contacto

Devanado de hilo

d

V

V

r

R

R

r

Vs

s

0

V0= r

RVs

r

d

100 Vs

V

d

Tension

obtenida

Tension

esperada

%(res) = 100Vs/N

Vs=

100

N


71/177

V(t) =V0sin (t)

V(t) = V0

cos

sin(t)

Tension

de ref.

(alterna)

S S

S S S S

R

R

1

2 3

1

2 3

1

2

120o

TM

V(t) =Vsin(t)

V1=Vsin(2XS

)sin(t)

V2=V cos(2XS

)sin(t)

X

S

X


72/177

Pistas en la parte movil

Pistas en la regleta fija

Regleta fija

Parte movil

TM


73/177

360/No sectores

2N

opistas = Nosectores

0

1

01

0

1

01

0

1

01

0

1

01

0

1

01

0

1

01

0

1

01

0

1

01

SH

SA

a


74/177

A

BC

D

E

FG

H

I

10

11 01

00

01

11 10

00

0110

01

10

00

1101 10

00

10

10

10

10

01

01

01

01

Entradas sobre las flechas de transicion: canal1/canal2Salida (en negrilla) bajo los estados: (UP counter,DOWN counter)


75/177

Escobilla

Iman permanente

Devanado

Onda alterna Onda rectificada

Varios devanados

V resultante

V resultante

da

da

V = ddt

= d(BS)dt

= B dS0cos dt

=BS0sin d

dt =BS0 sin(t)

BS0


76/177

n

2n

v=dx

dt x

t

x t

ets

P(kT)

v(k)

k k+ 1

T t

T

P

k

kT

t

T

v(k)

T

a

F = ma

F = kx


77/177

x

k

Resorte

Encoder

lineal

Masa m k x

m

x es leida por el encoder

k es la cte. del sistema equivalente de los dos resortes

F = m a = k x a =

(vista lateral)

F

Superficie inferior de material conductor

Base de silicio


78/177

Puntos focales

Objeto

8/10cm

I

d


79/177

Int. emitida

Int. reflejada


80/177

1

1

2

23

3

ObjetoObjeto

Catadioptrico

Sensor de corte de haz Sensor de retrorreflector

A B

Laser Lente cilindrica

Cinta transportadora

CamarasObjeto

SA SB

Laser

Espejos moviles

Colimador

A

B

A

B

450

200C

TM


81/177

Lamina de plastico con

recubrimiento de oro

0 10 30 50

Rejilla protectora

Lamina que actua

como muelle

Anillo de

aislamiento

Lamina de aluminio

con orificios

Condensador

tierra

300 Vfrec. de ultrasonidos)

50 30 10 0

022.5

45

67.5

90 90

67.5

45

22.5

Respuesta en dB en funcion del angulo,

normalizada a la respuesta para angulo 0

TM

C= S

d

S

d


82/177


83/177

Lamina de plasticoLamina de elastomero

Electrodos

Senal a la

circuiteria de

desacoplo

Cavidad

Tubo de aire

comprimido Contactos

Capa

plastica

Lamina

metalica

T

Ia T =KTIa Ia


84/177

R

T =RF

= F = KIa

R

0,9

Ia

Ia

KT

R= L

S

L

S

R

L

L

Vs

V

Flexion

Extension

F = (Fx, Fy, Fz) M = (Mx, My, Mz)


85/177

x

y

z

f

f

f

6

5

7

8

2

1

3

4

y

x

z

M

M

M

z

x

y

(w1, . . . , w8)

FxFyFzMxMy

Mz

=

0 0 K13 0 0 0 K17 0K21 0 0 0 K25 0 0 0

0 K32 0 K34 0 K36 0 K380 0 0 K44 0 0 0 K480 K52 0 0 0 K56 0 0

K61 0 K63 0 K65 0 K67 0

w1w2w3w4w5w6

w7w8


86/177

C1 C2

M

Fd


87/177

(b)

(a)

c

C1 C2

P R P

P

C1 C2 C1 C2

P1 P1P2 P2

Masa Masa

Cilindro

Piston

Cilindro

Piston

Desp. izq. Desp. der.P1 < P2 P1 > P2

P1

P0 P0


88/177

F K

A

++ K K

K

1

A Vdt

E

E

E i Q Q

Servo

amplif.

Servovalvula

Actuador Integrador

Transductor

de posicion

Servo

valvula

Actuador

Carga

F

K K

F

x,v

d

a s

d

pq

c

f

esa q

0

fx

E

E

i

v x

c

f

Servoamplif.+

Transductor

P2 A2

P0 P1

Af

P0

P2 A2

P0 P1

P0


89/177

Compresor

P P

PP

P

1 2

0

0

1

e

x

Hacia el sistema

a controlar

Atmosfera

A

A

2

f

Escobillas

B

B1

B1 B


90/177

Torque

I

Torque

Vel()

R

L

R

E

V

Res. arm. Res. per. mag.

Induc. arm.

a

a

a

L

g

+

Ra La

RL

Eg Va

RL

Va(t) =RaIa(t) + LadIa(t)

dt + Eg(t)

Eg(t) =KE(t)

T(t) =KTIa(t)


91/177

JM

I

JL B

Tf Tg

T =J = J

T(t) = (JM+ JL)(t) + B(t) + Tf(t) + Tg(t)

Eg

T

(s)

(t)

JT =JM+JL

Va(s) =RaIa(s) + LasIa(s) + KE(s)

T(s) =KTIa(s) =JTs(s) + B(s) + Tf(s) + Tg(s)

Ia(s) =Va(s) KE(s)

Ra+ Las

(s) =T(s) (Tf(s) + Tg(s))

JTs + B

V

E

K

K

T

T + T

1

R + L s1

J s+B

E

f g

T

+ +

VE I

g

a aT

g a

Tg = 0

Tf = 0

Va

Gm(s) = (s)

Va(s)=

KT(Ra+Las)(JTs+B)

1 + KEKT

(Ra+Las)(JTs+B)

=KTLaJT

s2 + ( LaRa

+ BJT

)s+ ( BRaLaJT

+ KTKELAJT

)


92/177

Gm(s) = (s)

Va(s) Km

1 + ms

(t) = d(t)

dt

(s) =s(s)

1,80


93/177

Nr

Ns

1

2

3

4

5

6

A

B

C

D

D

C

B

A


94/177

600

rad/paso= (2

Nr 2

Ns) = 2

Ns NrNsNr

Ns Nr

pasos/rev =

2 rad/rev

2 NsNrNsNr

rad/paso =

NsNr

Ns Nrpasos/rev

2 8686

= 12

= 150

8686

= 24

. . . 1000 0100 0010 0001 1000 . . .

3,60

ABCDA

AABBBCCCDDDA

. . . 1000 1100 0100 0110 0010 0011 0001 1001 . . .


95/177

S NS N

Rotor

Iman

permanente

A

A

BB B

A

B

A

S

S

SS

SN N

NN

N

Vista superior Vista inferior

Vista lateral

3 4

S

1 2

N

1 2

N

3 4

S

I = 0 I = +I1 2

3 4

S

1 2

N

I = I I = 01 2

1 2

N

3 4

S

I = +I I = 0

I = 0 I = I

1 2

1 2

Paso 1 Paso 2

Paso 3 Paso 4


96/177

Paso 0 Paso 1

Paso 0 Paso 1

sin(x+ ) = sin x cos + cos x sin cos(x+ ) = cos x cos sin x sin si+1=sic+ cisci+1=cic+ sis

si+1

ci+1=

c s

s c si

ci


97/177

Ic=IB

M

M

C1

C2

Q

Q Q

Q Q

Q

V

V

V

V

1 2

3 4

1

2

CCCC

CC

Tipo H Tipo T

I

I

1

2

i

C1

C2

Q1 Q4

I1 C1

C2

Q2 Q3

I2

IB

Vi

Wdis=VCEIC VCE


98/177

IC VCE

VCC IC IC I

VCE

T T

T

1 2

Ideal Real

T

T1 T2

T1 T2

Vef= 1

T

T0

V(t)dt= 1

T

T10

(VCC)dt + TT1

VCCdt

=

1

T [VCCT1+ VCC(T T1)] = VCC

T [T2 T1]

T1 = T2

T1 =T T2 = 0 T1 = 0 T2 =T

TM

TM


99/177

T

Tr

r

1

2

1

22

1

r1 N1 r2

N2 T1 T2

1 2

N1r1

=N2

r2

N1 N2 N1/N2 N1 : N2

r11 = r22

T11 = T22

N1N2

= r1r2

=21

=T1T2

=21

=21

J2

2

T2 = J22=T1N2N1

T1=N1

N2J22

1 =

N2N1

2


100/177

1

T1 Jeq

T1=Jeq1

Jeq

Jeq1=JeqN2N1

2=N1N2

J22

Jeq =

N1N2

2J2

J2 2< 1 T2> T1

100 : 1

n

p

np : 1

Carga

Rodamiento

fijo (tope)

Eje


101/177

P

P = 4

L= 1/P

x

v

a

= P x

=P v

= P a

1

2Mv

2

=

1

2Jeq

2

=

1

2JeqP

2

v

2

Jeq =M/P

2

CargaCarga

Carga

Jeq =

N1N2

2J2


102/177

N1N2

=

J1J2

Jeq =J1

J22

J2=J1

=P ot. suministrada

P ot. consumida =

T rab. suministrado

T rab. realizado

T

=Tsal(real) s

Tent e =Tsal(real) N1N2

Tent=

Tsal(real)

Tent rel acoplo

100%

20 %


103/177

= error en el angulo

= angulo girado

di ai i


104/177

Desp. desde

la izquierda

Desp. desde

la derecha

Res. en la posicionRes. en control

Puntos consecutivos


105/177


106/177

A

Gamp(s) = A

1 + as

Gm(s) = Km1 + ms

Kg

Kp

1s

V =


107/177

(t)

K

K

A K1 + s 1 + s

1s

+ +

d

a m

g

p

m

(t)

(s) (s)

Kg = 0

G(s) = AKms(1 + sm)(1 + sa)

H = Kp

1 + GH= 0 = 1 + Ks(1 + sm)(1 + sa)

= 0

K = AKmKp

s= 0

s=

1/m s=

1/a

a m

| 1a|>| 1

m|

0

3

3

= a+m3am

a

m

1 1

Kg = 0

a m K

A

Kp


108/177

K

K

A

Kp Km

Kp A

Kg

K

A K1 + s 1 + s

1s

+

d

a m

g

p

m

++

K

s

K gK

A K1 + s 1 + s

1s

+

d

a m

p

m

+ s

equivalente a

1 + GH= 0 = 1 + AKm (Kp+ Kgs)s(1 + sm)(1 + sa)

GH

s =KpKg

2

= 12

(KpKg 1

m 1

g)

Kp

Kg

Kp Kg

KpKg

> 1m

0

1

m


109/177

a

1m

1 K

Kg

p

m

1K

Kg

p

a

1

Kg= 0

1/a

A

Gin(s) =AKm1+sm

1 + KgAKm1+sm

= AKmms + AKgKm+ 1

G(s) =

AKmms+AKgKm+1

1s

1 + AKmms+AKgKm+1

1s

Kp=

1

Kp

AKmKpm

s2 + AKgKm+1m

s + AKpKmm

= 1

Kp

2ns2 + 2ns+ 2n

n =

AKmKpm

=12

1 + AKmKgmAKpKm

Kp Kg

Kg

Kp


110/177

+

K

1 + s

m

K

R + L s

1

B + J s

K

T

a a T

E

T + Tf gr

++

sin suponer

T + T =0

f gr

m

K g

+

1s

K

A1 + s

+ +

d

a

p

K

R + L s

1

B + J s

K

T

a a T

E

T + Tf gr

++ (t)(t)

(s)

(s)

(s) =1s

(s)

(s) = L{(t)}

(s) = L{(t)} = L

d(t)dt

(s) = 1

B+ JTs

KT

Ra+ sLa

KE(s) + A

sa+ 1(Kg(s) Kp(s) + d(s))

(Tf(s) + Tgr(s))

(s) =Geq(s)

d(s)[Tf(s) + Tgr(s)] (sa+ 1)(Ra+ sLa)

AKT

Geq(s) = AKT

s(sa+ 1) [KEKT+ (sJT+ B)(Ra+ sLa)] + AKT(sKg+ Kp)

Tf+Tgr d

d0

(s) =Geq(s) d0s


111/177

lmt

(t) = lms0

s(s) = lms0

sGeq(s)d0s

= AKTAKTKp

d0=d0Kp

Kp Kp

d

TL(s) = Tf(s) +Tgr(s)

TL(s) =TL0/s

lmt

(t) = lms0

s(s) = lms0

sGeq(s)(TL0)(1 + sa)(Ra+ sLa)

AKT

s =

RaTL0AK

TK

p

Kp

A

Kp

Kg

K g

+

1s

+A

1 + s+

d

a

K

R + L s

1

B + J s

K

T

a a T

E

T + Tf gr

++

K

K s

Ks

+++

p

d

I

Kp

Kp


112/177

Kp1

Kp2

Kp3

Kd2

Kd3

Kp1

Kp2

Kp3

K d1K

d1K

d2 K

d3

(a)

PPD

PI PID

< < > >

(b)

(c) (d)

d


113/177

Amp.D/APos. des.

(digital) Proces. Motor Tac. Enc Carga

(kT)

Pos. des.

(digital) Proces. D/A

A/D

Amp. Motor Tac. Enc CargaSenal

control

(kT)

(kT)

+Vel.

dig.

(kT)

T

T

Senal

analogica

Valor digital

obtenido

Aprox. digital

correcta

Ampliacion

T

Kg


114/177

Kg

Kg

Kp Kg

Kp

KI

Kd

+Pos.

des.

errorControlador Motor en

una artic.

Modulo de

adaptacionpar.

Pos. presente


115/177


116/177


117/177


118/177

T


119/177


120/177

Programa

a nivel robot

Usuario

Sistema op.

de tiempo real

Sistema op.

multitarea

Herramientas

de medicion

y test

Instrumentos

de medida

Controladores

de articulacion

Modelador

de solidos

Base de datos

CAD

geometrico

Modelador

Proceso

Control

del robot

Definicion

de la tarea

(offline)

(online)

Herramientas

de programacion a nivel tareaVisualizacion

Modelado

geom. de la tarea

Modelos

cinematico y

dinamico

Sensores

internosexternos

internaexterna

Programa

RobotObjetos

a manipular

Actuadores

Simulacion

Sensorizacion


121/177


122/177


123/177


124/177


125/177

Mundo

y

z

x

bloque

taladrobloque

D B

A

C

z

x

x

yz

alimentador

a

o

n

y

cabeza_tornillo


126/177

>

900

Z

Z


127/177

X

Y

Z

Z


128/177

75 mm

1

2

3

4

Origen del

robot (0,0,0)

Punto de deposit

Origen del plano xy

(0,0,448)


129/177

Modelo

del mundoPlanificacion de las subtareas

Programa a nivel robot

Conocimiento

externo

Especificaciones de la tarea

Descripcion

de la tarea


130/177


131/177

r

H

r

x= y = 0

z

z= 0

z= H

apuntes de robótica - universitat de valencia

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