Force Control of a Flexible Arm via Parallel Compensation

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Force

Control of a Flexible Arm

via Parallel Compensation


Li
ang
-
Yih Liu

Department of
Automation

Enginee
ring

&

Institute of Mechatronoptic Systems
,

Ch
ien Kuo Technology

U
niversity



Abstract

A contact

force control
scheme is proposed in this
paper for
a constrained one
-
link flexible arm based on a
linear distributed parameter model.
This model is then
used to design a integral
controller
by parallel
compensation. The control offers a good
regulation of
the contact force

exerted by a flexible arm while i
n
contact with a surface
. In order to obtain exact solutions
of the infinite dimensional system, infinite product
formulation is used throughout the paper. Numerical
simulations are
given
to
demonstrate

the effectiveness
of the proposed
method
.


Keywords

c
ontact f
orce control
,
flexible arm
, parallel
compensation
.



1. Introduction

F
or

a
pplication of
robot manipulators to space
robots used for satellite ca
p
turing and large space
structure construction, and light weight industrial robots
used for assembly, d
e
burring and grinding tasks.

It is
necessary to
control

the
contact force
exerted by
manipulators

hands
.

Several methods have been
proposed for force control of flexible arms. Most of the
works on force control are b
ased on finite
-
dimensional
approximate m
o
d
els
, and some works b
ased on
distributed parameter
mo
d
els

for a constrained flexible
manipulator.
Based on finite
-
dimensional approximate
mo
d
els, Chiou and Shahinpoor [1, 2] studied a
si
n
gle
-
link and a two
-
link constrained flexible
m
a
nipulators, and poin
ted out that the link flexibility is
the main source of dynamic inst
a
bility of the force
controlled systems. Later, in [3], Li used the distributed
parameter model of a single
-
link co
n
strained flexible
manipulator to demonstrate that an inherent limit
a
tion

on the achievable bandwidth of force control is due to
the presence of infinitely many non
-
minimum phase
zeros of the transfer function from the actuator torque to
the tip contact force. However, no controller design was
consi
d
ered in [3]. In [4, 5], Mats
uno et al. proposed
h
y
brid position
-
force controllers
using
the distri
b
uted
parameter models for a rigid
-
flexible and a
flexible
-
flexible constrained planar two
-
link
m
a
nipulators. However, the e
x
perimental results in [5]
showed that the controllers could n
ot ensure global
st
a
bility of the original systems when there were large
initial tracking errors and the desired tip speed was too
fast. On the basis of nonlinear f
i
nite
-
dimensional
dynamic models, var
i
ous hybrid force
-
position
controllers were also propos
ed in [6
-
10] for some two
and three dimensional co
n
strained flexible robots.
Because of spillover pro
b
lems the controllers proposed
in [6
-
10] do not guarantee the stability of the original
di
s
tributed parameter systems. More recently, Matsuno
and Kasai [11
] derived the di
s
tributed parameter model
for a constrained one
-
link flex
i
ble arm with a
concentrated tip mass, and obtained a f
i
nite
-
dimensional
model for force feedback and co
m
pliance control.
Matsuno and Coworkers [12, 13] also extended their
works to t
he constrained one
-
link arm with a symmetric
rigid tip body and a nonsymmetric rigid tip body.
Distributed p
a
rameter models were derived in [12, 13],
but f
i
nite
-
dimensional modal models were still used for
controller designs. It can be seen from the
aforem
e
n
tioned researches that no attempt has been
made to design a f
i
nite
-
dimensional controller for the
constrained one
-
link flexible arm based on the original
distri
b
uted parameter model of the arm.
In a recent work,
Liu [14]
derived a
linear distributed p
a
ra
meter model

of
the
constrained one
-
link flexible arm, and obtained a
in
f
i
nite
-
dimensional modal model for

integral
control.

Thus, t
he objective of this paper is

to
extend
his

works
to show that perfect asymptotic tracking of a desired
contact force tr
a
ject
ory can be achieved using a

linear
distributed parameter model with internal material
damping of the constrained one
-
link flexible arm.

To
remove the nonminimum phase obstacle relating the
joint torque input and the tip contact force output, a new
input an
d a new output are generated by using the
measurements of contact force, bending moment and
shear force at the root of the beam. It will be shown that
the transfer function from the new input to the new
output is
minimum
. Then, a
simple

integral

controller

is
shown to
be able to improve the pe
r
formance of the
infinite dimensional closed loop system. To pr
e
serve the
exact poles and zeros of the system, the infinite product
representations of transfer functions are employed
throughout the paper. Numerical sim
ulations are
given

to demo
n
strate the excellent performance of the
proposed
method
.



2.
Mathematical Model



2.1
Dynamics of a Constrained Flexible Arm


In this paper, a

co
n
strained one
-
link flexible

arm

is
shown
in Fig. 1
. The flexible arm

is assumed to
be a
uniform, homogeneous, Euler
-
Bernoulli beam of length

, mass per unit l
ength

,

internal material damping
d
c

and flexural rigidity
EI
. The hub is modelled by a


si
n
gle
-
m
ass moment of inertia
h
I
, where the driving
torque
)
(
t


is applied. The end
-
effector has a
concentrated mass
p
m
, where the contact force exerted
by the smooth rigid constraint su
r
face is
)
(
t

.

Let
)
(
t


designate the hub rotation angle, and
)
,
(
t
x
v

denote the
small elastic deflection of the link. Since
)
,
(
t
v


is
a
s
sumed small,
)
(
t


must also be sma
ll. The equations
of mo
t
ion and the corresponding boundary conditions
are

well
-
established

[1
5
]
:












)
(
)
(
)
,
(
)
(
)
(
0
t
t
dx
t
x
v
t
x
x
t
I
h











(
1
)


0
)
,
(
)
,
(
)]
,
(
)
(
[





t
x
v
EI
c
t
x
EIv
t
x
v
t
x
xxxx
d
xxxx








(
2
)


0
)
,
0
(

t
v

(
3
)


0
)
,
0
(

t
v
x

(
4
)


0
)
,
(
)
,
(


t
v
EI
c
t
EIv
xx
d
xx




(
5
)


)
(
)
,
(
)
,
(
t
t
v
EI
c
t
EIv
xxx
d
xxx








(
6
)


Substituting Eqs. (
2
) and

(
6
) into (
1
), pe
r
forming
int
e
gration by parts and making use of Eq. (
5
), an
altern
a
tive form for Eq. (
1
) is obtained:



)
,
0
(
)
,
0
(
)
(
)
(
t
v
EI
c
t
EIv
t
t
I
xx
d
xx
h









(
7
)


Now, introduce a new joint input variable
)
(
t
u

such
that



)
(
)
(
)
(
t
I
t
t
u
h







(
8
)


2.
2

Tr
ansfer Function

The transfer function can be derived by taking the
Laplace transform of Eqs. (
2
)
-
(
8
) a
s
suming zero initial
conditions. Let
s

be the Laplace transform variable,
and define the
following
dimensionless p
a
rameters



2
2
4
ˆ
s
s
EI



,


s
c
s
d



1
ˆ
2
4

,

3



h
I


(
9
)


After tedious algebraic manipulations
[15]

and

the application of infinite product
represe
n
tation of transcendental functions given in
the Appendix,
we

o
b
tains
































N
s
s
c
s
s
c
s
s
s
G
N
n
n
d
N
n
n
z
d
,
ˆ
1
ˆ
1
1
)
ˆ
(
)
ˆ
(
:
)
ˆ
(
1
1
2
2
1
2
2









(
10
)































1
4
4
2
1
2
2
ˆ
1
ˆ
1
1
)
ˆ
(
)
ˆ
(
:
)
ˆ
(
n
d
n
n
z
d
u
n
s
s
c
s
s
c
s
u
s
s
G






(
11
)






























1
4
4
2
1
2
2
ˆ
1
ˆ
1
)
ˆ
(
)
ˆ
(
:
)
ˆ
(
n
d
n
n
d
u
n
s
s
c
s
s
c
s
u
s
s
G






(
12
)
































1
4
4
2
1
2
2
ˆ
1
ˆ
1
)
1
(
3
)
ˆ
(
)
ˆ
(
:
)
ˆ
(
n
d
n
n
d
d
u
n
s
s
c
s
s
c
s
c
EI
s
u
s
s
G







(
13
)


)
1
(
1
)
ˆ
(
)
ˆ
,
0
(
:
)
ˆ
(
s
c
EI
s
u
s
v
s
G
d
xx
u
v
xx




(
14
)

where
n
z


and
n



are defined in the Appendix.
The numerical values of
)
(



n

can be computed
usin
g
2
n
n




, where
)
(


n
,

2
,
1

n

are the
real positive roots of the denominator of Eq. (
10
),
namely


0
)
cos
sinh
sin
(cosh
sin
sinh
2
3











(
1
5
)


Selected values of
n



thus obtained are listed
in Table 1.

Since
)
ˆ
(
s
G



has infinitely many zeros in
0
)
ˆ
Re(

s
,
)
ˆ
(
s
G



is non
-
minimum phase. Similarly,
)
ˆ
(
s
G
u


is also non
-
minimum phase
, due to t
he effect of
introducing the new input
)
(
t
u

via angular acceleration
measur
ement is merely to change the poles
)
ˆ
(
s
G



to
the poles of
)
ˆ
(
s
G
u

.



3.
Construction of
Controller


3.1
Achieving
M
inimum

Phase
by
Using Parallel
Compensation

I
n

order to
alleviate the non
-
minimum phase
problem, the right half
-
plane zeros can be replaced by
the left half
-
plane zeros by the method of redefinition of
output. Define a virtual contact force
)
,
(
k
t
f

such that









sinh
sin
)
cos
cosh
1
)(
1
(
)
sinh
(sin
2
)
ˆ
(
)
,
ˆ
(
:
)
,
ˆ
(







k
k
s
u
k
s
f
k
s
G
fu

(
16
)


where
k

is a real constant. It was shown in
[1
6
]

that for
758
.
0

k
, one can write
























1
2
2
ˆ
1
)
c o s
c o s h
1
)(
1
(
)
sinh
(sin
n
n
d
s
s
c
k
k








(
17
)


The numerical values can be computed using
2
n
n




, where
)
(
k
n

,

2
,
1

n

are the real positive
roots of the numerator of Eq. (
16
). Selected values of
n



are listed in Table 1. Thus, one has a minimum
phase stable transfer function






























1
4
4
2
1
2
2
ˆ
1
ˆ
1
)
,
ˆ
(
n
d
n
n
d
fu
n
s
s
c
s
s
c
k
s
G






(
18
)


Note that the above redefinition of output is equivalent
to the parallel compensation
[1
4
]

as shown in Fig.
2
. It
can be show
n that the parallel compensator
)
,
ˆ
(
k
s
T

has
the form
































1
4
4
2
1
2
2
2
ˆ
1
)
1
(
ˆ
1
ˆ
120
)
1
(
11
)
,
ˆ
(
n
d
d
n
n
d
n
s
s
c
s
c
s
s
c
s
k
k
s
T





(
19
)


where
2
n
n





and
)
(
k
n

,

2
,
1

n

are the real
positive roots of the equation



0
)
sinh
(sin
)
cos
cosh
1
(












(
2
0
)


Selected
values of
n



are computed and listed in
Table 1.


3
.
2


Integral
Controller

For the control structure as shown in Fig.
2
, the
objective is to make
)
(
t


to track asymptotically a
desired contact force trajectory
)
(
t
d


using an integral
control, where
I
k

is a real positive constant and
758
.
0

k
. Obviously,


)
ˆ
(
)
,
ˆ
(
ˆ
1
ˆ
)
,
ˆ
(
s
k
s
G
s
k
s
k
k
s
u
d
u
f
I
I









(
2
1
)


The poles of the closed
-
loop system are given
by the roots of the characteristic equation
and

)
ˆ
(
s
G
u
y

given by Eq. (
18
),
then


1
ˆ
1
ˆ
1
ˆ
1
4
4
2
1
2
2





























n
d
n
n
d
I
n
s
s
c
s
s
c
s
k








(
22
)


Since the poles and zeros of the left hand side of
Eq. (
2
2
) located on the imaginary
s
ˆ
-
axis are di
s
tinct
and alternate each other

as
0

d
c

(see the first co
l
umn
of Table 1), it can be easily shown by the method of root
locus that all roots of Eq. (
2
2
) lie in the open left half
s
ˆ
-
plane. It is reasonable to co
n
jecture that the damping
effect should be enhanced when i
n
terna
l damping are
considered. Let the roots of Eq. (
2
2
) be written as




,
2
,
1
,
1
ˆ
2
4






n
j
s
EI
s
n
n
n
n
n
n







(
23
)


where
)
(
1


n
n
n




and
n


are the natural
frequency and damping ratio of the

n
-
th

closed
-
loop
pole.

Then
)
,
ˆ
(
k
s
G
u
f
,
)
ˆ
(
s
G
u

,
)
ˆ
(
s
G
u

,
)
ˆ
(
s
G
u
v
xx
,
)
ˆ
(
s
G
u


given by Eqs. (
18
) and (
1
1
)
-
(
14
), the
closed
-
loop responses of some relevant variables can be
computed as follows































N
s
s
s
k
s
s
s
c
k
k
s
f
d
N
n
n
n
n
I
N
n
n
d
I
),
ˆ
(
ˆ
ˆ
2
1
)
ˆ
(
ˆ
1
)
,
ˆ
(
1
1
2
2
1
2
2






(
24
)






























N
s
s
s
k
s
s
s
c
k
s
d
N
n
n
n
n
I
N
n
n
z
d
I
),
ˆ
(
ˆ
ˆ
2
1
)
ˆ
(
ˆ
1
)
ˆ
(
1
1
2
2
1
2
2







(
25
)

































N
s
s
s
k
s
s
c
s
s
c
EI
k
s
d
N
n
n
n
n
I
d
N
n
n
d
I
),
ˆ
(
ˆ
ˆ
2
1
)
ˆ
)(
1
(
ˆ
1
3
)
ˆ
(
1
1
2
2
1
2
2
2








(
26
)































N
s
s
s
k
s
s
c
n
s
s
c
EI
k
s
v
d
N
n
n
n
n
I
d
N
n
d
I
xx
),
ˆ
(
ˆ
ˆ
2
1
)
ˆ
)(
1
(
ˆ
1
)
ˆ
,
0
(
1
1
2
2
1
4
4
2






(
27
)






























N
s
s
s
k
s
s
s
c
k
s
d
N
n
n
n
n
I
N
n
n
d
I
),
ˆ
(
ˆ
ˆ
2
1
)
ˆ
(
ˆ
1
)
ˆ
(
1
1
2
2
1
2
2










(
28
)



4.
Numerical Simulation

In this
section
,

in order to demonstrate the
effectiveness of the proposed control approach
. Some
results

is evaluated here through numerical simulation
using the
parameters of an experimental apparatus
described in
[12]
.
The physical
parameters
are
114
.
0


m
/
kg
,
2
.
23

EI
2
m
N

,
7
.
0


m
,
4
10
17
.
1



d
c
s
and
01
.
0

h
I
2
m
kg


(
1
10
557
.
2




).

The virtual contact force parameter
and controller gain that resulted in good performance
were selected as
7
.
0

k

and
533
.
7

I
k
. The desired
contact force was

se
t
as


t
t
d
e
e
t
19
18
18
19
1
)
(










(
2
9
)


The simulation results which are shown in Fig. 3.

which results in
5
)
7
.
0
,
(
lim
)
(
lim






t
f
t
t
t

N
,
2
10
52
.
3
)
(
lim





t
t

rad, and
5
.
3
)
(
lim



t
t

m
N

.


5. Conclusions

T
he contact force

control
method for
a
constrained one
-
link flexible arm has been
developed

i
n
this paper
. It is

based on a linear distributed

parameter
model

with internal material damping
.

A

minimum
phase stable transfer function is obtained for the
noncollocated system

by using
the feedback of joint
angular acceleration to generate a new input and the
redefinition of output via parallel compensation
.
On the
basis of the distributed parameter model, a simple

integral control
ler has been constructed. The controller
was pro
ved to ensure the
perfect asymptotic tracking of
a desired contact force trajectory with internal stability.
Numerical simulations
results demonstrated the
excellent performance of the proposed
integral
controller
.



6
. References

1.

Chiou
,

B.C. and
Shahinpo
or,

M.,
Dynamic Stability
Analysis of a One
-
Link Force
-
Controlled Flexible
Manipulator,

Journal of Robotic Systems, Vol. 5, No.
5
,
pp. 443
-
451, 1988.

2.

Chiou
,
B
.
C.

and

Shahinpoor,

M.,
Dynamic Stability
Analysis of a Two
-
Link Force
-
Controlled Flexible
Manipu
lator,

ASME Journal of Dynamic Systems,
Measurement, and Control, Vol. 112, No. 6, pp.
661
-
666, 1990.

3.

Li,

D., Tip
-
Contact Force Control of One
-
Link
Flexible Manip
u
lator: An Inherent Performance
Limitation
,

Proceedings of 1990 American Control
Conference, S
an Diego, CA.,

pp. 697
-
701,
1
990.

4.

Matsuno,

F.
,

Sakawa
, Y.,

Asano,

T.,

Quasi
-
Static
Hybrid Pos
i
tion/Force Control of a Flexible
Manipulator
,

Pr
o
ceedings of 1991 IEEE
International Conference on Robotics and
Autom
a
tion, Sacr
a
mento, CA.,

pp. 2838
-
2842, 1991.

5.

Matsuno,

F.,
Asano
, T.,

Sakawa,

Y.,

Quasi
-
Static
Hybrid Pos
i
tion/Force Control of Constrained Planar
Two
-
Link Flexible M
a
nipulators
,

IEEE Trans.
Robotics and Automation, Vol. 10, No. 3, pp.
287
-
297, 1994.

6.

Hu
,

F.L.

and Ulsoy,
A.G.
,

Force and Motion C
ontrol
of a Co
n
strained Flexible Arm,

ASME Journal of
Dynamic Systems, Measurement, and Control, Vol.
116, No. 3,

pp. 336
-
343, 1994.

7.

Matsuno
, F.

and Yamamoto,
K.,
Dynamic Hybrid
Pos
i
tion/Force Control of a Two Degree
-
of
-
Freedom
Flexible M
a
nipulator
,

Journal of Ro
botic Systems,
Vol. 11, No. 5,

pp. 355
-
366, 1994.

8.

Yim
, W.

and Singh,

S.N.
,

Inverse Force and Motion
Control of Constrained Elastic Robots,

ASME
Journal of D
y
namic Systems, Measurement, and
Control, Vol. 117, No. 3,

pp. 374
-
383, 1995a.

9.

Yim
, W.

and Singh, S.
N.
,
Sliding Mode Force,
Motion Co
n
trol, and Stabilization of Elastic
Manipulator in the Presence of Uncertai
n
ties,
Journal of Robotic Systems, Vol.12, No. 5,

pp.
315
-
330, 1995b.

10.

Yoshihawa,
T.,

Harada
, H.,

Matsumoto,

A.,
Hybrid
Pos
i
tion/Force Control of Fle
xible
-
Macro/

Rigid
-
Micro Manipul
a
tor Systems,
IEEE Trans.
Robotics and Automation, Vol. 12, No. 4,

pp.
663
-
640, 1996.

11.

Matsuno
, F.

and Kasai,

S.,
Modeling and Robust
Force Control of Constrained One
-
Link Flexible
Arms,
Journal of Robotic Sy
s
tems, Vol. 15, N
o. 8,

pp. 447
-
464, 1998.

12.

Matsuno,

F.,
Umeyama
, S.,

Kasai,

S.,
Exper
i
mental
Study on Robust For
ce Control of a Flexible Arm
with a Sy
m
metric Rigid Tip Body,
Procee
d
ings of
1997 IEEE International Conference on Robotics
and Automation, Albuquerque, New Mexic
o, pp.
3136
-
3141, 1997.

13.

Morita,
Y., K
obayashi,

Y.,
Kando,

H.,
Matsuno,

F.,

Ka
n
zawa
T.,
Ukai,

H.,
Robust Force Control of a
Flexible Arm with a No
n
symmetric Rigid Body,
Journal of Robotic Systems, Vol. 18, No. 5,

pp.
221
-
235, 2001.

14.

Liu
,
L.Y.
,

Noncollocated
Integral
Control of a
Co
n
straint One
-
Link Flexible Arm,
Proceedings of
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The 14
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Conference on Automation
Technology Conference
, Changhua, Taiwan, June
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-
3, pp. c67
-
c72, 2006
.


15.

Liu
,
L.Y.
,

PD
Control of a Co
n
straint Flexible Arm

with Internal Damping
,
P
roceedings of 2007 CACS
International Automatic Control Conference,
Taichung, Taiwan, Nov. 9
-
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-
1033, 2007
.

16.

Y
uan
, K.

and Liu,

L.Y.,
Achieving Minimum Phase
Transfer Function for a Noncollocated Single
-
Link
Flexible M
a
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Asian Journal of C
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2, No. 3,

pp. 179
-
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17.

Goodson,

R.E.
,

Distributed System Simulation
Using Inf
i
nite Product Expansions
,
Simulation, pp.


255
-
263,

1970.


7
.
Appendix


I
n this paper
, t
he infinite product expansions
for transcendental functions are summarize
d as
follows
[1
4
, 1
7
]
.

A1.

,
ˆ
1
1
1
2
sin
sinh
1
2
2
















n
n
z
d
s
s
c




,
2
2
n
n
z
a




,
0
tan
tanh


n
n
a
a

,
0
(

n
a
)
r e a l











4
1
n
a
n

as


n



2
1
365
.
2
2

z

,


2
2
498
.
5
2

z

,


2
3
639
.
8
2

z

,



2
4
781
.
11
2

z

,



2
5
923
.
14
2

z

,


2
6
064
.
18
2

z


A2.
















1
4
4
2
2
ˆ
1
1
1
sinh
sin
n
d
n
s
s
c






A3.




cosh
sin
sinh
cos


















1
2
2
3
ˆ
1
1
1
3
2
n
n
d
s
s
c



,

2
n
n
c





,
0
tan
tanh


n
n
c
c

,
0
(

n
c
)
r e a l











4
1
n
c
n

as



n


2
1
927
.
3



,
2
2
069
.
7



,
2
3
210
.
10



,


2
4
352
.
13



,

2
5
493
.
16



,
2
6
635
.
19




A4.

















1
2
2
ˆ
1
1
1
2
cos
cosh
1
n
pn
d
s
s
c



,
.
2
n
pn
b




,
0
1
cos
cosh


n
n
b
b

,
0
(

n
b
).
real











2
1
n
b
n

as
.


n


2
1
875
.
1

p

,
2
2
694
.
4

p

,
2
3
855
.
7

p

,


2
4
996
.
10

p

,
2
5
137
.
14

p

,
2
6
279
.
17

p


Note that the asymptotic expressions are quite
accurate (to thr
ee decimal places) for
5

n
.


8
.
Figures and Tables



)
,
(
t
x
v
h
I

)
(
t

p
m
)
(
t

)
(
t

X
x
)
,
(
t
x
u
x

Fig.1

C
onstrained
one
-
link
flexible arm


)
(
t
d

s
k
I
ˆ


)
(
t
u
)
ˆ
(
s
G
u

)
,
ˆ
(
k
s
T
)
1
,
(
)
(
t
f
t


)
,
ˆ
(
k
s
G
u
f
)
,
(
k
t
f


Fig.
2

A block diagram of an
Integral control
via
parallel
compensat
ion



Fig.
3


)
(
t




Fig.
3

)
(
t







0

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

0

2

4

6

8

x 10

-
3

C
ontact forces, N



time(sec)

Joint angle, rad

time (sec)








Fig.
3


)
(
t







Table 1
Values of roots of associated transc
endental
equations


n

)
7
.
0
(

k
n



n



)
10
557
.
2
(
1






n

1

2
847
.
2

2
900
.
4

2
816
.
1

2

2
082
.
4

2
725
.
7

2
992
.
3

3

2
145
.
8

2
086
.
11

2
080
.
7

4

2
777
.
10

2
066
.
14

2
214
.
10

5

2
301
.
14

2
336
.
17

2
353
.
13

6

2
142
.
17

2
371
.
20

2
494
.
16






n

(odd)



n

(even
)

2
)
2
1
(
3
7
)
2
1
(



















n
n
2
)
2
1
(
3
7
)
2
1
(



















n
n

2
)
2
1
(
1
)
2
1
(



















n
n

2
)
2
1
(
1
)
2
1
(



















n
n



2
3
3
)
4
3
(
9108
.
3
)
4
3
(



















n
n




受拘束撓性臂之加速度回饋控制

柳良義

建國科技


自動化系暨機電光所



摘要



本文利用線性分佈參數模式探討受拘束
有阻
尼單桿撓性臂端點之接觸力控制問題,
進而
藉由平行
補償設計一
積分控制器
使接觸力漸近達到其預期


此外,
為求得此一無限維系統之正合解,本文採
用了無窮乘積表示法。實例之數值模擬驗證了本文所
提出方法之優異性。




關鍵字

接觸


控制,
撓性臂

平行補償


0

0.5

1

1.5

0

0.2

0.4

0.6

0.8

Control torque , N
-
m


time (sec)

….

….

….

….