Proceedings of the 8th JFPS International
Symposium on Fluid Power, OKINAWA 2011
Oct. 25-28, 2011
2D2-4
DYNAMIC FRICTION CHARACTERISTICS OF PNEUMATIC
ACTUATORS AND THEIR MATHEMATICAL MODEL
Xuan Bo TRAN* and Hideki YANADA**
* Mechanical & Structural Systems Engineering
Toyohashi University of Technology
1-1, Hibarigaoka, Tempaku-cho, Toyohashi 441-8580, Japan
(E-mail: [email protected])
** Department of Mechanical Engineering
Toyohashi University of Technology
1-1, Hibarigaoka, Tempaku-cho, Toyohashi 441-8580, Japan
ABSTRACT
This paper deals with dynamic friction characteristics in the gross-sliding regime of pneumatic cylinders. Using three
pneumatic cylinders, friction characteristics are investigated and modeled under various conditions of velocity and
pressures. It is shown that a hysteresis loop behavior can be seen at low velocities in the friction force-velocity relation
and the friction force varies nearly linearly with the velocity at high velocities. The hysteresis loop is expanded to higher
velocity when the frequency of the velocity variation is increased, and its size increases with increasing driving pressure
and decreases with increasing resistance pressure. It is shown that these behaviors can be relatively accurately modeled
by the modified LuGre model. The pressures have influence on the maximum static friction force, the Coulomb friction
force and the Stribeck velocity of the model.
KEY WORDS
Friction, Dynamic behavior, Pneumatic actuator, Mathematical model
hss
NOMENCLATURE
a
Ai
Fc
FL
Fr
Frss
Fs
g
h
:
:
:
:
:
:
:
:
:
:
acceleration
piston area (i=1,2)
Coulomb friction force
force acting on load cell
friction force
steady-state friction force
maximum static friction force
Stribeck function
dimensionless dynamic
lubricant film thickness
Kf
m
n
pi
ps
v
vb
vs
657
:
:
:
:
:
:
:
:
:
:
:
:
dimensionless steady-state
lubricant film thickness
proportional constant for
lubricant film thickness
mass of the pneumatic piston
exponent for Stribeck curve
pressure (i=1,2)
supply pressure
velocity
velocity at maximum film
thickness
Stribeck velocity
Copyright © 2011 by JFPS, ISBN 4-931070-08-6
z
σ0
σ1
σ2
τh
τhp
τhn
τh0
:
:
:
:
:
:
:
:
:
:
:
:
mean deflection of bristles
stiffness of bristles
micro-viscous friction
coefficient for bristles
viscous coefficient
time constant for lubricant
film dynamics
time constant for acceleration
period
time constant for decelation
period
time constant for dwell time
model by incorporating lubricant film dynamics into the
model and it has been shown that the proposed model,
called the modified LuGre model, can simulate dynamic
behaviors of friction observed in hydraulic cylinders
with a relatively good accuracy [10-12]. However, the
validity of the modified LuGre model in simulating the
dynamic friction characteristics of pneumatic cylinders
has not been investigated.
In this paper, first a new experimental test setup is
developed to examine dynamic friction characteristics
of pneumatic cylinders under various operating
conditions of velocity variation and pressures in the
cylinder chambers. The modified LuGre model is then
used to simulate the dynamic friction characteristics
measured. A parameter investigation is also conducted
to identify the influence of the pressures in the cylinder
chambers on the modified LuGre model’s parameters.
INTRODUCTION
Friction is always present in pneumatic actuator systems
and makes the dynamics of pneumatic position/velocity
control systems rather complex and precise
position/velocity control is usually difficult. It is,
therefore, necessary to investigate the friction
characteristics and develop a suitable friction model for
pneumatic actuators to improve the control performance
of pneumatic actuator systems. However, it is extremely
difficult to obtain friction characteristics of a pneumatic
actuator when the actuator is pneumatically operated
due to the difficulties in controlling the velocity of the
actuator.
Several experimental methods have been proposed to
investigate the friction characteristics of pneumatic
actuators [1-3]. Schoroeder and Singh [1] proposed an
experimental test setup in which the friction force was
calculated by detecting the force exchanged by the rods
of the tested pneumatic cylinder and of a load
pneumatic cylinder assembled with a reversed working
direction. Belforte et al. [2] proposed an experimental
test setup in which the velocity of the test pneumatic
cylinder was controlled by a driving hydraulic cylinder
and the pressures of the chambers were controlled by
pressure-proportional valves in order to measure the
friction force under a broad range of operating
conditions of velocity and pressures. Nouri [3] proposed
an experimental test setup to identify the friction force
in both the pre-sliding and gross-sliding regimes of a
rodless cylinder. However, these experimental methods
mainly focused on investigating the friction
characteristics under steady-state conditions. To the best
of the authors' knowledge, the friction characteristics of
pneumatic actuators under dynamic conditions have not
fully been investigated.
Several mathematical models that describe the dynamic
behaviors of friction have been proposed so far [4-9]
and among them, the LuGre model [6] is most widely
utilized. However, all these models cannot simulate well
the friction behaviors of a hydraulic cylinder in the
gross-sliding regime as shown in [10]. Yanada and
Sekikawa [10] have made a modification to the LuGre
TEST SETUP AND EXPERIMENTS
The test setup used in this investigation is shown in
Figure 1. It consists of a pneumatic cylinder under test
and an electro-hydraulic servo cylinder system. The
pneumatic piston was driven by the hydraulic piston in
order to precisely control the velocity of the pneumatic
piston. Pressures in two chambers of the pneumatic
cylinder were independently controlled by using two
proportional valves (pressure type). The valves provide
air flow up to 0.025 m3/s and allow controlling the
pressures up to 0.65 MPa. The motion of the hydraulic
cylinder was controlled by a computer through an
amplifier and a servovalve. The supply pressure of the
servovalve was set at 2 MPa, providing enough force to
drive the pneumatic piston.
A load cell with a rated output of 500 N and with an
accuracy less than 0.15% R.O., which was set between
the rod of the pneumatic cylinder and the rod of the
hydraulic cylinder, was used to measure the force acting
on the pneumatic piston. Two pressure sensors with an
accuracy less than 2% F.S. were used to measure the
pressures, p1, p2, in the cylinder chambers, and the
piston velocity, v, was measured using a tachometer
generator with a ripple of less than 2% by converting
linear motion of the piston to rotational motion through
a pulley and belt system.
The values of velocity, pressures, and force from the
sensors were read into a computer through an A/D
converter and the computer provided the control signals
to the proportional valves and servovalve though a D/A
converter. The acceleration, a, of the piston was
calculated by an approximate differentiation of the
piston velocity accompanied by a first-order low pass
filter with 50 Hz cutoff frequency. The velocity of the
pneumatic cylinder, v, and the pressures, p1, p2, were
controlled by using PID control laws. Experimental data,
i.e., velocity, v, pressures, p1, p2, and acceleration, a,
were recorded at the interval of 0.5 ms (2 kHz).
658
Copyright © 2011 by JFPS, ISBN 4-931070-08-6
Figure 1 Schema of experimental test setup
The friction force, Fr, is obtained from the equation of
motion of the pneumatic piston using the measured
values of the pressures in the cylinder chambers, the
inertia force and the force acting on the load cell as
follows:
Fr = p1 A1 − p 2 A2 − ma + FL
the experimental result.
MODIFIED LUGRE MODEL
Yanada and Sekikawa [10] have extended the LuGre
model [6], called the modified LuGre model, for
simulating the dynamic friction behaviors of hydraulic
cylinders by incorporating a dimensionless lubricant film
thickness parameter, h, into the Stribeck function. The
model is described by
(1)
where m is the mass of the pneumatic piston, A1, A2 are
the piston areas, and FL is the force acting on the load
cell.
Three different pneumatic cylinders were used for the
experiments: standard, smooth, and low speed cylinders.
They are of the same size but have different operating
conditions of the velocity and pressure as shown in Table
1. In this experiment, dynamic friction force-velocity
characteristic was measured under different conditions of
the velocity and pressures. The input velocity of the
pneumatic piston was varied sinusoidally in both the
extending and retracting strokes of the cylinder between
0.005 and 0.12 m/s at three different frequencies of 0.5, 2,
and 4 Hz. Pressures, p1 and p2, in the cylinder chambers
were varied between 0 and 0.6 MPa. Every experiment
was conducted three times to verify the repeatability of
σ 0z
dz
=v−
v
dt
g (v, h )
Fr = σ 0 z + σ 1
(2)
dz
+ σ 2v
dt
(3)
where z is the mean deflection of the elastic bristles, v is
the velocity between the two surfaces in contact, Fr is the
friction force, σ0 is the stiffness of the elastic bristles, σ1
is the micro-viscous friction coefficient, and σ2 is the
viscous friction coefficient. g(v,h) is a Stribeck function
that expresses the Coulomb friction and the Stribeck
Table 1 Specifications of pneumatic cylinders tested
Specifications
Bore diameter (mm)
Rod diameter (mm)
Stroke (mm)
Operating velocity (mm/s)
Operating pressure (MPa)
Standard cylinder
50 - 750
0.05 - 1
659
Smooth cylinder
25
10
300
5 - 500
0.02 - 1
Low speed cylinder
0.5 - 300
0.005 - 1
Copyright © 2011 by JFPS, ISBN 4-931070-08-6
effect and is given by
g ( v , h ) = Fc + [(1 − h ) F s − Fc ]e − ( v
vs ) n
(4)
where Fc is the Coulomb friction force, Fs corresponds to
the maximum static friction force, vs is the Stribeck
velocity, and n is an appropriate exponent. The lubricant
film dynamics can be given by
dh
1
=
( hss − h )
dt τ h
(5)
⎧τ hp (v ≠ 0, h ≤ hss )
⎪
τ h = ⎨τ hn (v ≠ 0, h ≥ hss )
⎪τ
(v = 0)
⎩ h0
(6)
⎧⎪ K | v | 2 / 3 ( v ≤ vb )
hss = ⎨ f
2/3
( v > vb )
⎪⎩K f | vb |
(7)
K f = (1 − Fc Fs )|vb |−2 3
(8)
a) Velocity variation (f=0.5 Hz)
where hss is the dimensionless steady-state lubricant film
thickness parameter, Kf is the proportional constant for
lubricant film thickness, vb is the velocity at which the
steady-state friction force becomes minimum, and τhp, τhn,
and τh0 are the time constants for acceleration,
deceleration, and dwell periods, respectively. In Eq. (6),
h<hss corresponds to the acceleration period, h>hss to the
deceleration period.
For steady-state, friction force is given by
Frss = Fc + [(1 − h ss ) Fs − Fc ]e − ( v v s ) + σ 2 v
n
b) Friction force vs. velocity
Figure 2 Dynamic friction force-velocity characteristics
for three pneumatic cylinders in extending stroke
(p1=0.3 MPa, p2=0 MPa)
pneumatic cylinders. The hysteresis loops of the smooth
and low speed cylinders are relatively small as compared
to the one of the standard cylinder. At high velocities, the
friction forces are increased nearly linearly with the
velocity.
Figure 3 shows dynamic friction force-velocity
characteristics measured in the retracting stroke of three
pneumatic cylinders. In this case, the pressures, p1 and p2,
in the cylinder chambers were kept constant at 0 and 0.3
MPa, respectively. As can be seen from Figure 3(b), the
hysteresis behavior can be obtained only for the standard
cylinder at small velocities. For the smooth and low
speed cylinders, the friction force varies almost linearly
with the velocity in the whole velocity range.
Figure 4 shows dynamic friction force-velocity
characteristics of the standard cylinder measured at three
different frequencies of velocity variation: 0.5, 2, and 4
Hz, for both the extending and retracting strokes. It is
shown in Figure 4(a) for the case of extending stroke that
when the frequency is increased, the hysteresis loop is
expanded to higher velocities and becomes larger. In
addition, a reduction of the friction force at small
velocities can be seen when the frequency is increased.
For the case of retracting stroke in Figure 4(b), it is
shown that the hysteresis loop is also expanded to higher
velocities but becomes smaller when the frequency is
increased. The effects of the frequency of velocity
(9)
The static parameters of the modified LuGre model, Fs,
Fc, vs, vb, n, and σ2, can be identified experimentally
from steady-state friction force-velocity characteristic
and the dynamic parameters, σ0, σ1 and τh, can be
identified experimentally by the methods proposed in
[11].
RESULTS AND DISCUSSION
Figure 2 shows dynamic friction force-velocity
characteristics measured in the extending stroke of three
pneumatic cylinders. Figure 2(a) shows the sinusoidal
velocity variation of the pneumatic piston and Figure
2(b) shows the friction force versus velocity curves. The
pressures, p1 and p2, in the cylinder chambers were
controlled and kept constant at 0.3 and 0 MPa,
respectively. It is shown in Figure 2(b) that a hysteresis
behavior can be obtained from the friction force-velocity
curves at small velocities ( v ≤ 0.02 m/s ) for all the three
660
Copyright © 2011 by JFPS, ISBN 4-931070-08-6
a) Velocity variation (f=0.5 Hz)
a) Effect of p1 (p2=0 MPa, f=0.5 Hz)
b) Friction force vs. velocity
Figure 3 Dynamic friction force-velocity characteristics
for three pneumatic cylinders in retracting stroke
(p1=0 MPa, p2=0.3 MPa)
b) Effect of p2 (p1=0.6 MPa, f=0.5 Hz)
Figure 5 Dynamic friction force-velocity characteristics
under different pressures in extending stroke
(standard cylinder)
variation on the dynamic friction force-velocity
characteristics for the smooth and low speed cylinders
are similar to those for the standard cylinder.
Figure 5 shows the effects of pressures, p1 and p2, on the
dynamic friction force-velocity characteristics for the
standard cylinder in the extending stroke. In this case, p1
is the driving pressure and p2 is the resistance pressure.
Figure 5(a) shows the effect of pressure, p1, when the
pressure, p2, is kept constant at 0 MPa, and Figure 5(b)
shows the effect of pressure, p2, when the pressure, p1, is
kept constant at 0.6 MPa. As can be seen from Figures
5(a) and 5(b) that the hysteresis loop becomes larger with
increasing pressure, p1, and becomes smaller with
increasing pressure, p2.
Figure 6 shows the effects of pressures, p1 and p2, on the
dynamic friction force-velocity characteristics for the
standard cylinder in the retracing stroke. In this case, p2
is the driving pressure and p1 is the resistance pressure.
Figure 6(a) shows the effect of pressure, p2, when the
pressure, p1, is kept constant at 0 MPa, and Figure 6(b)
shows the effect of pressure, p1, when the pressure, p2, is
kept constant at 0.6 MPa. As can be seen from Figures
6(a) and 6(b) that the hysteresis loop becomes larger with
increasing pressure, p 2 , and becomes smaller with
increasing pressure, p1. From the results obtained in
Figures 5 and 6, it can conclude that the driving pressure
makes hysteresis loop larger and the resistance pressure
a) Extension (p1=0.3 MPa, p2=0 MPa)
b) Retraction (p1=0 MPa, p2=0.3 MPa)
Figure 4 Dynamic friction force-velocity characteristics
under different frequencies (standard cylinder)
661
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a) Effect of p2 (p1=0 MPa, f=0.5 Hz)
a) Effect of p1 (p2=0 MPa)
b) Effect of p1 (p2=0.6 MPa, f=0.5 Hz)
Figure 6 Dynamic friction force-velocity characteristics
under different pressures in retracting stoke
(standard cylinder)
b) Effect of p2 (p1=0.6 MPa)
Figure 7 Relations between the pressures and the model
parameters in extending stroke (standard cylinder)
makes hysteresis loop smaller. The effects of the
pressures on the dynamic friction force-velocity
characteristics for the smooth and low speed cylinders
are similar to those for the standard cylinder.
Based on the experimental results, all the parameters of
the the modified LuGre model were identified at
different conditions of the pressures, p1 and p2, for three
cylinders. In this paper, only the identification results for
the case of extending stroke are shown. The results
identified for three cylinders at a condition of p1=p2=0
MPa are shown in Table 2. For other conditions of
pressures, the identification results show that the values
of the maximum static friction force (Fs), the Coulomb
friction force (Fc) and the Stribeck velocity (vs) of the
modified LuGre model are changed with the pressures
while other parameters are unchanged. Figure 7 shows
the relations between the values of the parameters, Fs, Fc,
vs, and the pressures, p1, p2 for the standard cylinder. It is
shown in Figure 7 that the value of Fc increases with
both p1 and p2, while the values of Fs and vs increase with
p1 and decrease with p2.
Figure 8 shows comparisons between the dynamic
friction characteristics measured and the ones simulated
by the modified LuGre model for the standard, smooth
and low speed cylinders. The simulations were done
using MATLAB/Simulink. As can be seen from Figure 8,
all the simulated results are in good overall agreement
with the measured results. However, the hysteresis
behaviors of the smooth and low speed cylinders at small
velocities are hardly simulated by the modified LuGre
model.
Figures 9 shows the simulation results obtained by the
modified LuGre model and needs to be compared with
the experimental results shown in Figure 4(a) for the
effect of the frequency. The comparison shows that the
modified LuGre model cannot predict the expansion of
the hysteresis loop in the measured result when the
Table 2 Values of parameters of the modified LuGre
model for three cylinders at p1=0 MPa, p2=0 MPa
Parameters
Fs0 [N]
Fc0 [N]
vs0 [m/s]
vb [m/s]
n
σ2 [Ns/m]
σ0 [N/m]
σ1 [Ns/m]
τhp [s]
τhn [s]
τh0 [s]
Standard
cylinder
16
5
0.005
0.025
2.5
25
1.5×104
0.1
0.02
0.15
20
Smooth
cylinder
3.6
3.6
0.005
0.025
0.5
72
1.5×104
0.1
0.01
0.2
20
Low speed
cylinder
4.5
4.5
0.005
0.025
0.5
53
1.5×104
0.1
0.01
0.2
20
662
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a) Standard cylinder
a) Effect of p1 (p2=0 MPa)
b) Effect of p2 (p1=0.6 MPa)
Figure 10 Simulation results corresponding to Figure 5
b) Smooth cylinder
frequency is increased; as shown in Figure 9, the
hysteresis loop predicted by the modified LuGre model
becomes smaller by increasing the frequency. However,
the model can predict the reduction of the friction force
at small velocities when the frequency is increased. The
reason for the discrepancy between the simulated results
and measured ones obtained in Figures 8 and 9 may be
due to the effects of other factors of mechanism, which is
not clear at present and are not taken into account in the
modified LuGre model.
Figure 10 shows the simulation results obtained by the
modified LuGre model and needs to be compared with
the experimental results shown in Figure 5. The
comparison shows that the modified LuGre model can
predict accurately the variation of the hysteresis loop
when the pressures are varied.
c) Low speed cylinder
Figure 8 Comparison between measured and simulated
results for three pneumatic cylinders (p1=0.3 MPa,
p2=0.0 MPa, f=0.5 Hz)
CONCLUSION
In this paper, dynamic behaviors of friction of pneumatic
cylinders are experimentally investigated at various
conditions of velocity and pressures. The experimental
results show that a hysteresis behavior can be obtained at
low velocities in the dynamic friction force-velocity
relation and the friction force varies nearly linearly with
the velocity at high velocities. The hysteresis loop is
expanded to higher velocity when the frequency of the
velocity variation is increased, and its size increases with
Figure 9 Simulation results corresponding to Figure 4(a)
663
Copyright © 2011 by JFPS, ISBN 4-931070-08-6
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the driving pressure and decreases with the resistance
pressure. It is shown that these behaviors can be
relatively accurately modeled by the modified LuGre
model. The pressures have influence only on the
maximum static friction force, the Coulomb friction
force and the Stribeck velocity.
ACKNOWLEDGEMENT
The authors would like to express their gratitude to Prof.
Y. Wakasawa for his help in part of the experiments and
to SMC Corporation for donating the major pneumatic
components used in the investigation.
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664
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