pneumatic system model with friction phenomena for modern system ...

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I
NTERNATIONAL

C
ARPATHIAN
C
ONTROL

C
ONFERENCE

I CCC’ 2010

Eger, Hungary

May 26
-
29, 2010











PNEUMATIC SYSTEM MODEL WITH FRICTION PHENOMENA
FOR MODERN SYSTEM CONTROL DESIGN


Patrícia Gróf

1

Béla Takarics
2

András Czmerk
3

1

Computer and Automation Research Institute/Cognitive Informatics

Research Group,

Budapest, Hungary, grof @sztaki.hu

2

Budapest
University of Technology and Economics/Department of Mechatronics, Optics and
Engineering Informatics, Budapest Hungary,
bela.takarics@gmail.com

3

Budapest University of Technology and Economics/Department of

Mechatronics, Optics and
Engineering Informatics, Budapest Hungary,
czmerk@mogi.bme.hu


Abstract:

The main
goal

of this paper to develop a mathematical model which describes the
movement of a piston of pneumatic
cylinder controlled by a proportional valve, keeping the non
linear

behavior of pneumatic system. In further researches t
he Linear Parameter Varying (LPV)
representations and Linear Matrix Inequality (LMI) based analysis and system control design are in
th
e focus of modern control theories
.

Key words:

pneumatic cylinder, positioning, system model, friction

1

Introduction

As an important driver element, the
pneumatic cylinder is widely used in industrial
applications for many automation purposes
thanks to
their variety of advantages, such as:
simple, clean, low cost, high speed, high power
to weight ratio, easy maintenance and inherent
compliance. Traditionally, they are used for
motion between two hard stop. The design of a
stable robust position controlle
r for a pneumatic
servo
-
system is difficult since it is a very
nonlinear time
-
variant controlled plant because
of the compressibility of air, the friction force
between the piston and the cylinder, air mass
flow rate through the servo
-
valve, etc. In most
c
ases, applications of pneumatic actuators use
servo valves.

2

Friction model

Friction is a physical phenomenon and
expressed in quantitative terms as a force
F
f
,
being the force exerted by either of two
contacting bodies tending to oppose relative
tangent
i
al displacement of the other
.

The easiest and probably the most well
known model is the so
-
called
Coulomb friction

model. Though it greatly over simplifies the
frictional phenomena it is widely used in the
motion control problems, when dynamic effects
are

not concerned. Also, the Coulomb model is
a common piece of all more developed models
(see Figure 1a). The Coulomb friction force
F
c

is a force of constant magnitude, acting in the
direction opposite to motion

v
(
t
)
.

When

v
(
t
)

0
:

))
(
(
sign
)
(
t
v
F
t
F
c
f




N
c
F
F


,


where
F
N

is the normal component of the
force pressing surfaces together and


is the
frictional factor.


is determined by
measurements under certain conditions. One of


the biggest problems of the

Coulomb model is,
that it cannot handle the vicinity of zero
velocity, hence the properties of motion at
starting or zero velocity crossing, i.e. static and
rising static friction
F
s
. To apply the model for
those cases a

0
>


factor has been introduced.

When

v
(
t
)

0
:

N
s
f
F
F
t
F
0
)
(



.


At motion start, it replaces



until the
process arrives to steady state. The values of


and

0

can be found in any major physics or
engineering tabulations for different material
pairs in both dry a
nd lubricated conditions. The
first tabulations of those kinds date back to the
beginning of 18
th

century.


The
viscous friction

element models the
friction force as a force proportional to the
sliding velocity:

When

v
(
t
)

0
:

)
(
)
(
t
v
F
t
F
v
f




where
F
v

is the coefficient of viscous
friction.

The model is used for the friction caused by
the viscosity of the fluids, specifically
lubricants. A combination with Coulomb
friction yields (see Figure 1b):

When

v
(
t
)

0
:

)),
(
(
sign
)
(
t
v
v
F
t
F
v
v
f






where
δ
v

is a geometry
-
dependent
parameter. The model can be refined by adding
the influence of an external force for the friction
at rest. This, however, leads to a discontinuous
function (see Figure 1c). Here, an important
contribution has been made by Stribec
k.
Armstrong
-
Hélouvry proposed a mod
el which
involves a nonlinear
, but continuous function
(see Figure 1d):

When

v
(
t
)

0
:



v
F
t
v
e
F
F
F
t
F
v
v
v
C
s
C
f
S


















))
(
(
sign
)
(

where
v
s

is the Stribeck velocity,


is an
empirical parameter,
F
S

is the static friction
force
. A similar model
was employed by
Hess and Soom [9
].

When

v
(
t
)

0
:

v
F
t
v
v
v
F
F
F
t
F
v
s
C
s
C
f














))
(
(
sign
)
/
(
1
)
(
)
(
2






(7)

The Stribeck curve is an advanced model of
friction as a function of velocity (see Figure 1d).
Although it is still valid only in steady stat
e, it
includes the model of Coulomb, static and
viscous friction as built
-
in elements. There are
several more advanced models in the technical
literature.

1.1 Friction of the pneumatic cylinder

One can find in the technical literature that
pneumatic cylin
ders have Stribeck fri
ction
without the viscous term
.

The friction model can be described with the
following model:






















erm
Stribeck t
2
)
1000
)
term
Coulomb
1000
)
)
/
(
1
(
(
1
(
2
)
1
(
2






























s
C
s
v
C
s
c
v
c
v
v
J
F
F
e
F
F
J
F
e
J
F
v

(1
)

where the signum friction is approximated
as


1
)
1
(
2
)
(
sign
1000




v
e
v
.



The Stribe
ck curve
in this case
has
3
parameters
, the static friction
F
S
, the Coulomb



friction
F
C

and the Stribeck velocity,
v
S
. The
3
parameters were measured and have the
following values:

F
s

= 160
N
;

F
c

= 130
N
;

v
s

= 0.05
m/s
;


2

Pneumatic system model

The scheme

of the pneumatic system is
shown in Fig. 1. The valve is denoted by V, the
cylinder is denoted by C, and TD assigns the
transducer. The numbers 1
-
5 assign the
connecting ports of the valve. Port 1 is the
connecting point of compressed air supplier
unit. P
ort 2 and 4 are connected with chamber B
and A in order. Finally, port 5 and 3 are
connected with the atmosphere. The valve has
two main states according to the ports. In state
1
-
4 (see Fig. 1.) chamber A is charging, and the
piston is moving right, while
in state 1
-
2
chamber B is charging, and the piston is moving
left.


Fig. 3. Scheme of the system

While we try to model the system, it can be
recognized that it contains several nonlinear
phenomenon, the compressibility of the air, the
nonlinear flow thro
ugh the valves, which
control the motion of the piston. For formulate
the motion equation of the piston, and equation
of the pressure deviation of the two different
chambers, four basic physical laws can be
applied: the universal gas law, the Bernoulli
equ
ation for co
mpressible mediums,
the Euler
equation

and the continuity equation.

]
. The
speciality

here is that the input
voltage signal is replaced by the effective area
of the proportional valve’s orifice:

)
(
)
(
)
(
x
F
x
k
x
d
A
A
p
A
A
p
x
M
f
eff
b
eff
a















(2)

where
M
is the load mass,

x



is the acceleration of the piston,

a
p

is the pressure in chamber A,

b
p

is the pressure in chamber B,

eff
A

is the input signal, the effective orif
ice
area of the valve

A

is the area of the piston, the two side are
equal, because this is a rodless cylinder

d
is the damping factor, and its value is
approximately zero,

k
is the stiffness
, and its value is
approximately zero,

x


is the velocity of the piston,

x

is the displacement of the piston,

f
F

is the friction force between the wall of
the cylinder and the piston, in this

model it is
neglected.

The deviation of the pressure is in the two
chambers is described by the following
equations:





x
A
p
m
T
R
V
x
A
V
p
m
T
R
V
p
a
a
in
DA
a
a
a
in
a
a


















1
1

(3)








x
A
p
m
T
R
V
x
L
A
V
p
m
T
R
V
p
b
b
bki
DB
b
b
b
bki
b
b




















1
1
.

(4)

The mass flow rate can be formulated as a
function of the effective or
ifice area,
eff
A
. The
pressure in the chambers depends on the flow
rate, that we have to determine
in
m


and
out
m

;
see paper [2]

In the present case the state variables of the
pneumatic system are th
e followings:



1
x
= the displacement of the piston

2
x
= the velocity of the piston

3
x
= the acceleration of the piston

a
p

= the pressure in the A chamber

b
p

= the pressure in the B chamber

The A and B matrices according to the
equations of motion (1
-
4
):

eff
b
a
b
a
A
B
B
B
p
p
x
x
x
A
A
A
A
p
p
x
x
x


































































5
4
3
3
2
1
5
,
5
4
,
4
3
,
3
2
,
3
3
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0





.

(
5
)

where











M
k
A
p
V
x
L
A
M
A
A
p
V
x
A
M
A
p
p
x
A
b
db
a
da
b
a















1
1
1
2
,
3
,
,
,

(6)




























erm
Stribeck t
2
2
)
1000
)
term
Coulomb
1000
2
3
,
3
)
)
/
(
1
(
(
1
(
2
)
1
(
2
2
2


































s
C
s
x
C
s
c
x
c
v
x
J
F
F
e
F
F
J
F
e
J
F
M
d
x
A
,

(7)






db
V
x
L
A
x
A
x
x
A





1
2
2
1
5
,
5
,
,

(8)










Ψ
T
R
2
p
Ψ
T
R
2
p
,
,
b
1
i
s
1
1
3























T
R
V
x
L
A
M
A
T
R
V
x
A
M
A
p
p
x
B
db
da
b
a
,

(9)




Ψ
T
R
2
p
,
s
1
1
4







x
A
T
R
p
x
B
a
,

(10)








Ψ
T
R
2
p
x
-
L
A
,
b
1
1
5








T
R
p
x
B
b
.

(11)

Here






t
x
t
y
1

,


(12)



0
0
0
0
1

C
,

and D=0.

This is

a new representation of the existing
models is proposed which is suitable for
controller design.



Acknowledgement

The resear
ch was supported by HUNOROB
project (HU0045), a grant from Iceland,
Liechtenstein and Norway through the EEA
Financial Mechanism and the Hungarian
National Development Agency and the National
Science Research Fund (OTKA K62836)
.


References

[1.]

P

G
RÓF
:
Identifi
cation and control of
pneumatic positioning systems Final
Project, Budapest University of
Technology and Economics, 2006,
Budapest.

[2.]

J.

G
YEVIKI
,

I.T.

T
ÓTH
,

K.

R
ÓZSAHEGYI
,

Sliding Mode Control and its
Application on a Servopneumatic

Positioning System
,

Transactions on
automatic control and computer
science Vol.49 (63), 2004, ISSN
1224
-
600X. pp. 99
-
104.

[3.]

P.

K
ORONDI
,

J.

G
YEVIKI
,

Robust
Position C
ontrol for a Pneumatic
Cylinder
, EPE
-
PEMC 06