Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 2 docx
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 2 docx
... q
1
2
+ l
2
2
cos(q
1
2
+ q
2
2
)
+ l
3
2
cos(q
1
2
+ q
2
2
+ q
3
2
) + l
4
2
cos(q
1
2
+ q
2
2
+ q
3
2
+ q
4
2
)
z
o2
+ l
1
2
sin q
1
2
+ l
2
2
sin(q
1
2
+ q
2
2
)
+ l
3
2
sin(q
1
2
+ q
2
2
+ ... l
1
2
cos q
1
2
+ l
2
2
cos(q
1
2
+ q
2
2
)
+ l
3
2
cos(q
1
2
+ q
2
2
+ q
3
2
) + l
4
2
cos(q
1
2
+ q
2
2
+ q
3
2
+ q
4
2
)
z
o2...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 3 pot
... =
1
2
F
12
y
12
+
1
2
F
23
y
23
=
1
2
c
p
y
2
12
+
1
2
c
k
y
2
23
=
1
2
c
p
(y
1
− y
2
)
2
+
1
2
c
k
(y
2
− y
3
)
2
=
1
2
c
p
(Y
1
− Y
2
+ s
1
)
2
+
1
2
c
k
(Y
2
− Y
3
+ s
3
)
2
=
1
2
π
12
(Y
1
− ... defined by
π
12
= π
21
= c
p
1 +
s
1
||Y
1
− Y
2
||
2
= c
p
1 −
Y
10
− Y
20
||Y
1
− Y
2
||
2
,
π
23
= π
23
= c
k
1...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 4 ppt
... is
L = T − =
1
2
m(
˙
Y
10
+˙y
2
)
2
−
1
2
c
p
(y
1
− y
2
)
2
−
1
2
c
k
(y
2
− y
3
)
2
(38)
=
1
2
m(
˙
Y
10
+˙y
2
)
2
−
1
2
c
p
(Y
1
− Y
2
+ s
1
)
2
−
1
2
c
k
(Y
2
− Y
3
+ s
3
)
2
,
so that the ... −c
k
c
k
⎤
⎦
·
⎡
⎣
y
1
y
2
y
3
⎤
⎦
=
1
2
y
T
Ky,
y = col(y
1
,y
2
,y
3
), ( 32)
=
1
2
⎡
⎣
Y
1
Y
2
Y
3
⎤
⎦
T
⎡
⎣
π
12
−π
12
0
−π
12
π
12
+...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 5 pdf
... quantities (see relations
(27 ), (28 ), (29 ) and (31)).
A
d
=
1
2
T
F
int
=
1
2
T
π
, (68)
where F
int
= F
int
(Y ) and = (Y) are the vectors of internal forces and relative
displacements ... diag(K
e1
,K
e2
,K
e3
, ),andex-
panded vectors of the force F
e
= column(F
e1
,F
e2
,F
e3
, ) and displacement
e
= column(
e1
,
e2
,
e3
, ). If the equations of stiffness of...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 6 ppsx
... 0
r
ia
+ˆη − (r
ja
+ˆη)
2
I
3×3
,c
ψ
ij
,c
θ
ij
,c
ϕ
ij
= diag
c
ij
1 −
ρ
ij 0
r
ia
− r
ja
2
I
3×3
,c
ψ
ij
,c
θ
ij
,c
ϕ
ij
= π
ij
(Y
i
,Y
j
) ( 129 )
92 Multi-Arm Cooperating Robots
of ... R
6m+6
. (114)
88 Multi-Arm Cooperating Robots
Y
Ii
=
r
ia
A
ia
+
ˆη
0
= Y
i
+ η
i
= Y
i
+¯η, i = 0, ,m. ( 123 )
The overall potential energy at that point is
2...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 8 doc
... Figure 22 d. Simulation results for τ
1
1
= 50 [Nm] and τ
1
2
=−50 [Nm]
1 32 Multi-Arm Cooperating Robots
part referring to the physical quantity used in modeling and the number indicat-
ing the ... 21 b. Simulation results for τ
j
i
= 0, i, j = 1, 2, 3
128 Multi-Arm Cooperating Robots
133
Figure 22 e. Simulation results for τ
1
1
= 50 [Nm] and τ
1
2
=−50 [Nm]
Mathem...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 9 ppt
... R
6×1
,
W
s
(Y
s
) = diag(W
2
(Y
2
), ,W
m
(Y
m
)) ∈ R
(6m−6)×(6m−6)
,
F
bs
(Y
s
,
˙
Y
s
) = col(F
b2
(Y
2
,
˙
Y
2
), ,F
bm
(Y
m
,
˙
Y
m
))
= col
˙
W
2
(Y
2
)
˙
Y
2
−
∂T
2
(Y
2
,
˙
Y
2
)
∂Y
2
, ,
˙
W
m
(Y
m
)
˙
Y
m
−
∂T
m
(Y
m
,
˙
Y
m
)
∂Y
m
∈ ... equations (21 7) and (21 8), differential equations should be solved.
Using the indexing system defined in (20 6), (20 7...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 10 pps
... c
k
)Y
2
− c
k
Y
3
+ mg and F
c
= (F
c1
,F
c2
)
T
. Models of
the manipulators are taken in the form
m
1
¨q
1
+ m
1
g = τ
1
+ f
c1
,f
c1
=−F
c1
,
m
2
¨q
2
+ m
2
g = τ
2
+ f
c2
,f
c2
=−F
c2
. (26 2)
Kinematic ... determined.
Synthesis of Nominals
Figure 28 . Nominal input to a closed-loop cooperative system for gripping
1 82 Multi-Arm Cooperating Robots
169
ping F
s
e0
is det...
Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 11 pptx
... of time for the
non-controlled object and of time and state for the controlled object. For the con-
20 0 Multi-Arm Cooperating Robots
For a linear system of n
x
ordinary first-order differential ... torques τ ,andthe
20 4 Multi-Arm Cooperating Robots
20 3
– elastic;
• in view of their redundanc y
– non-redundant, and
– redundant;
• in view of joint compliance
– with non-compli...
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