the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Effects of friction models on simulation of pneumatic cylinder

### Van Lai Nguyen

### Khanh Duong Tran

This study examines effects of three friction models: a steady-state friction model (SS model), the LuGre model (LG model), and the revised LuGre model (RLG model) on the motion simulation accuracy of a pneumatic cylinder. An experimental set-up of an electro-pneumatic servo system is built, and characteristics of the piston position, the pressures in the two-cylinder chambers and the friction force are measured and calculated under different control inputs to the proportional flow control valves. Mathematical model of the electro-pneumatic servo system is derived, and simulations are carried out under the same conditions as the experiments. Comparisons between measured characteristics and simulated ones show that the RLG model can give the best agreement among the three friction models while the LG model can only simulate partly the stick-slip motion of the piston at low velocities. The comparison results also show that the SS model used in this study is unable to simulate the stick-slip motion as well as creates much oscillations in the friction force characteristics at low velocities.

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Friction usually exists between piston/rod seals and contacting surfaces in fluid power cylinders and has an important aspect in fluid power control systems. Friction may occur in nonlinear manner and cause limit cycles and unexpected stick-slip oscillation at low operating velocities. These nonlinear characteristics of the friction make accurate simulation and position control of the fluid power cylinders difficult to achieve. In order to overcome these difficulties, it is, therefore, necessary to develop an accurate friction model for the fluid power cylinders.

Classical friction models that describe the steady-state relation between velocity and friction force, which can be characterized by the viscous and Coulomb friction with Stribeck effect combination have been proposed (Hibi and Ichikawa, 1977; Armstrong, 1991; Armstrong et al., 1994; Pennestrì et al., 2016; Marques et al., 2016, 2019; Brown and McPhee, 2016). However, some friction behaviours cannot be captured by these classical friction models, as for example, hysteretic behaviour with oscillating velocity, stiction behaviour and breakaway-force variations (Armstrong et al., 1994). In addition, in the mechanics-related controller design, simple classical models are not enough to address applications with high precision positioning requirements and low velocity tracking. Thus, in order to obtain accurate friction compensation and best control performance, a friction model with dynamic behaviours is necessary.

Several friction models that describe the dynamic behaviours of friction have been proposed so far (Haessig and Friedland, 1991; Canudas et al., 1995; Dupont, 1995; Swevers et al., 2000; Dupont et al., 2002), and among them, the LG model (Canudas et al., 1995) is most widely utilized in control applications (Lu et al., 2009; Freidovich et al., 2010; Hoshino et al., 2012; Green et al., 2013; Ahmed et al., 2015; Wojtyra, 2017; Piatkowski and Wolski, 2018). The model can simulate arbitrary steady-state friction characteristics and it can capture hysteretic behaviour due to frictional lag, spring-like behaviour in stiction and give a varying break-away force depending on the rate of change of the applied force. However, Yanada and Sekikawa (2008) have shown that the LG model cannot simulate a decrease of the maximum friction force observed after one cycle of the velocity variation in a hydraulic cylinder when the piston velocity varies sinusoidally with velocity reversals. In order to overcome this limitation of the LG model, they have modified the LG model by incorporating lubricant film dynamics into the model to obtain a new friction model called the modified LuGre model (MLG model).

Next, Tran et al. (2012) have pointed out that the MLG model cannot simulate the real hysteretic behaviours of the friction force–velocity curve in the fluid lubrication regime of hydraulic cylinders. The MLG model was then improved by replacing the usual fluid friction term, which is proportional to velocity, with a first-order lead dynamic. It has been verified that the improved model, called the new modified LuGre model (NMLG model), can capture accurately most of the friction behaviours observed in the hydraulic cylinders (Tran et al., 2012) and in the pneumatic cylinders (Tran and Yanada, 2013) in entire sliding regime. In addition, the usefulness of the NMLG model in simulating the operating characteristics of a hydraulic servo system have been verified by Tran et al. (2014). In a recent study, Tran et al. (2016) have shown that the NMLG model cannot capture the friction characteristics observed experimentally in pneumatic cylinders when the pneumatic cylinders operated in pre-sliding regime and they have proposed a new friction model by incorporating a hysteresis function into the NMLG model. Although the usefulness of the new friction model, called the revised LuGre model (RLG model) in this study has been verified, the validity of this model in simulating the motion of electro-pneumatic servo systems has not been investigated.

In this paper, the effects of the RLG model on the simulation accuracy of an electro-pneumatic servo system are examined in comparation with the LG model and a SS model (static + Coulomb + viscous friction). For this purpose, an experimental setup of the electro-pneumatic servo system using a pneumatic cylinder is proposed. Characteristics of the piston position, the pressures in the cylinder chambers and the friction force of the piston are measured and analysed under various operating conditions of control inputs to proportional flow control valves. Mathematical model of the electro-pneumatic system is developed by incorporating one of the three friction models into the entire system model, and the pneumatic cylinder's characteristics are simulated using MATLAB/Simulink under the same conditions as the experiments. Comparisons of simulation and experimental results are carried out to show the effects of each friction model and to verify the validity of the RLG model for pneumatic cylinders.

The organization of this paper is as follows: Brief descriptions of the SS model, the LG model, and the RLG model are given in Sect. 2. Section 3 describes the electro-pneumatic servo system and its mathematical model. Experimental and simulation results are presented and discussed in Sect. 4. Finally, main conclusions are drawn in Sect. 5.

In this section, the three friction models: the SS model, the LG model and the RLG model are described in short.

## 2.1 Steady-state friction model

The SS model used in this study is a combination of static friction, Coloumb
friction and viscous friction (Armstrong, 1991). The model characteristics
are presented by a Stribeck curve as shown in Fig. 1. In this friction
model, the friction force *F*_{r} depends on the velocity input and is
calculated by the following equation:

where *F*_{s} is the static friction force, *F*_{c} is the Coulomb friction
force, *v*_{s} is the Stribeck velocity, *n* is the exponent that affects the
slope of the Stribeck curve, *σ*_{2} is the viscous friction
coefficient and *v* is the relatively tangential velocity between two
contacting surfaces.

## 2.2 LuGre model

The LG model (Canudas et al., 1995) is a combination of a stiction force with
an arbitrary steady-state friction force which can include the Stribeck
effect. It is assumed in this model that two matting surfaces make contact
at several asperities through elastic bristles as shown in Fig. 2. When a
tangential force is applied to a surface, the bristles will deflect like
springs; and when the force is sufficiently large, some of the bristles will
break and then slip. The mean deflection of the elastic bristle is denoted
as *z* and is defined as:

where *σ*_{0} is the stiffness of the elastic bristle and g(*v*) is the
Stribeck function and is defined as:

The friction force is given by

where *σ*_{1} is the micro-viscous friction coefficient.

In Eq. (4), the first two terms represent the friction force generated from the bending of the elastic bristles and the third term stands for the viscous friction. In steady-state condition, the friction force is given by Eq. (1).

## 2.3 Revised LuGre model

The RLG model (Tran et al., 2016) is a model where three modifications were
made from the LG model. Firstly, a lubricant film dynamic has been
incorporated into the function *g*(*v*) in Eqs. (2) and (3) to obtain a new
Stribeck function *g*(*v*,*h*) as shown in Eqs. (5) and (6). Secondly, the friction force
term *σ*_{0}*z* in Eqs. (2) and (4) of the LG model has been replaced
by a hysteresis function *F*(*z*) as shown in Eqs. (5) and (7). And thirdly, the
usual fluid friction term *σ*_{2}*v* in Eq. (4) has been replaced by a
first order lead dynamics ${\mathit{\sigma}}_{\mathrm{2}}(v+T\mathrm{d}v/\mathrm{d}t)$ as shown in Eq. (7).

where *T* is the time constant for fluid friction dynamics. *h* is the
dimensionless lubricant film thickness and is given by

where *h*_{ss} is the dimensionless steady-state lubricant film thickness
parameter, *K*_{f} is the proportional constant for lubricant film thickness,
*v*_{b} is the velocity within which the lubricant film thickness is varied,
and *τ*_{hp},*τ*_{hn} and *τ*_{h0} are the time constants for
acceleration, deceleration, and dwelling periods, respectively. In Eq. (9),
*h*≤*h*_{ss} corresponds to the acceleration periods and *h*>*h*_{ss}
corresponds to the deceleration periods. It is noted that the lubricant used
for the packing of pneumatic cylinders is grease and is not oil. Regarding
the behaviour of film formation of grease between contact surfaces, it has
been shown by Li et al. (2009) that the film thickness becomes thinner
during acceleration and thicker during deceleration than the steady-state
film thickness. This behaviour of grease film is the same as that of oil
film in Sugimura et al. (1998). Therefore, it is believed that the lubricant
film dynamics described by Eqs. (8) to (11) hold also for grease and can be
applied to pneumatic cylinders.

The hysteresis function *F*(*z*) in Eq. (5) is a function that simulates the
hysteresis behaviour with nonlocal memory in pre-sliding regime. *F*(*z*) consists
of many functions *f*_{i}(*z*) (*i*=1, 2…) in which each function
*f*_{i}(*z*) models a segments of transition curve. The friction
force-deflection curve in pre-sliding regime of the pneumatic cylinder
consists of transition curves, i.e. curves between velocity reversal points
(Tran et al., 2016). Each velocity reversal starts a new transition curve
and each transition curve can be divided into some segments depending on its
shape. Each segment can be approximated by a function *f*_{i}(*z*) as follows:

where *c*_{i} and *k*_{i} are the segment parameters that can be identified
from experimental friction force-displacement characteristics,
*c*_{i}*k*_{i} indicates the stiffness of the bristles at *z*=*z*_{i}, *z*_{i} is
the initial deflection on the *i*th segment, and *f*_{i}(*z*_{i}) is the
initial friction force of the *i*th segment and equals to the final friction
force of the (*i*−1)th segment.

Implementation of the function *F*(*z*) in pre-sliding regime requires two memory
sets for the functions *f*_{i}(*z*): one for ascending curves (*v*>0)
and one for descending curves (*v*<0). The sets begin at velocity
reversal and remove when a hysteresis loop is closed. At each inverse point
of velocity, the deflection *z* takes a maximum value *z*_{m} (Fig. 3a) or a
minimum value *z*_{n} (Fig. 3b). At these points, a new transition curve
begins and the function *f*_{i+1}(*z*) is calculated by resetting *z*_{i} in Eq. (12) to *z*_{m} or *z*_{n} and *f*_{i}(*z*_{i}) will take the value *F*_{m} or
*F*_{n}, respectively.

For internal loops created on an ascending curve (Fig. 4a) and on a
descending curve of external loop (Fig. 4b), the internal loop is formed by
two curves 2 and 3 between two velocity reversal points at *z*_{n} and
*z*_{m}. When the velocity reverses at *z*_{n}, the numerical program will
judge the state to be on the internal loop by checking the variation of
velocity sign from positive to negative. The state is calculated by a
function *f*_{i+1}(*z*) using the parameters *c*_{i+1}, *k*_{i+1} and the values
of *z*_{n} and *F*_{n}. When the velocity reverses at *z*_{m}, the numerical
program will remain the state on the internal loop and the state is
calculated by a function *f*_{i+2}(*z*) using the parameters *c*_{i+2},
*k*_{i+2} and the values of *z*_{m} and *F*_{m}. After the velocity reversal
point *z*_{m}, a function *f*_{i}(*z*) of the curve 1 on the external loop is
calculated together with the function *f*_{i+2}(*z*) of the curve 3. When the
value of *f*_{i+2}(*z*) reaches the value of *f*_{i}(*z*) at the point in the
vicinity of *z*_{n} and when there is no change in velocity sign, the
friction state has to follow the curve 1 or curve 4 on the external loop
after the intersection point *z*_{n}. The values of *z*_{n}, *F*_{n}, *z*_{m}
and *F*_{m} of the internal loop are automatically cleared from the program
after the intersection point *z*_{n}.

When the piston movement enters its sliding regime, i.e., when the
deflection reaches *F*_{s}∕*σ*_{0}, the model is then switched to the
NMLG model. At this condition, the hysteresis function *F*(*z*) is set equally to
*σ*_{0}*z*.

In steady-state condition, friction force is described by

The static parameters *F*_{s}, *F*_{c}, *v*_{s}, *v*_{b}, *n*, and *σ*_{2}
of the three models are identified from measured steady-state friction
characteristics using the least-squares method and the dynamic parameters
*σ*_{0}, *σ*_{1}, *τ*_{h}, and *T* are identified from
measured dynamic friction characteristics by the methods proposed in Tran et
al. (2012). The function *f*_{i}(*z*) is identified from measured friction
force-displacement characteristics by the methods proposed in Tran et al. (2016).

In this section, an experimental test setup of the electro-pneumatic servo system is firstly introduced, and its mathematical model is then developed.

## 3.1 Experimental test setup

Figures 5 and 6 show the experimental test setup used in this investigation.
The system consists of a pneumatic cylinder (Eq. 1) (SMC, CM2L25-300) fixed
horizontally on a flat plate made of steel. The cylinder has internal
diameter of 0.025 m, rod diameter of 0.01 m and piston stroke of 0.3 m,
respectively. The piston end was connected to a load mass (Eq. 4) which can
slide on a guiding bar (Eq. 5). The load mass was varied from 0.5 to 5 kg. The
piston motion was controlled by two flow proportional control valves (Eq. 6)
(SMC, VEF3121). The two valves can supply a flow rate up to 720 L min^{−1} with a
rated voltage of 5 VDC. According to the valve characteristic, if the valve
control inputs *u*_{1} or *u*_{2} vary from 2.5 to 5 VDC, the valves will
provide air into the cylinder chamber (the valves are operated at left
position); and if the valve control inputs *u*_{1} or *u*_{2} vary from 0 to 2.5 VDC, the valves will release air into the atmosphere (the valves are
operated at right position). Therefore, by combining signals between
*u*_{1} and *u*_{2} of the two valves, the extending and retracting motions of
the piston can be obtained.

The position of the piston was measured by a position sensor (Eq. 2) with a
measurement range of 300 mm (Novotechnik, LWH0300). The pressures in the
two-cylinder chambers were measured by two pressure sensors (Eq. 3) with a
measurement range of 1 MPa (SMC, PSE540). Measuring accuracies of the
position sensor and the pressure sensors are less than 0.5 % F.S and 1 % F.S, respectively. The source pressure was set at 0.5 MPa. The position
signal and the pressure signals were read via a personal computer through a
12 bits analog to digital converter (ADC). The computer sent the control
signals *u*_{1} and *u*_{2} to the two valves through a 12 bits digital to
analog converter (DAC) (Eq. 8) (Advantech, USB4711). Two amplifiers (Eq. 7) (SMC,
VEA250) were used to convert the voltage signals to the current signals of
the valves. The program for data acquisition was done by using Microsoft
visual C$++$ software. The signals were recorded at the interval of 1.16 ms.

The friction force, *F*_{r}, was obtained from the equation of motion of the
pneumatic piston using the measured values of the pressures in the cylinder
chambers, the acceleration of the piston and the weight of the load mass as
follows:

where *A*_{1} and *A*_{2} are the piston areas, *M* is the total mass of the
piston, piston rod and the external load. *a* is the piston acceleration and
was calculated by an approximation of second differentiation of the measured
piston position. The noise in the calculated acceleration signal was filtered
by an acausal first order low-pass filter with a bandwidth of 32 Hz.

## 3.2 Modelling of the electro-pneumatic servo system

The objective of this section is to derive the dynamic equations of the entire electro-pneumatic servo system. In order to obtain the air flow dynamics in the pneumatic cylinder, the following assumptions are used:

- a.
The used air is an ideal gas and its kinetic energy is negligible in the cylinder chamber.

- b.
The leakages of the cylinder are negligible.

- c.
The temperature variation in cylinder chambers is negligible with respect to the supply temperature.

- d.
The pressure and the temperature in the cylinder chamber are homogeneous.

- e.
The evolution of the gas in each chamber is polytropic process.

- f.
The supply and ambient pressures are constant.

As mentioned in Sect. 3.1, if the supplied voltage to the proportional
valve varies from 2.5 to 5 VDC, the valve will provide air into the cylinder
chamber (the building pressure case) and if the supplied voltage varies from
0 to 2.5 VDC, the valve will release air into the atmosphere (the exhausting
pressure case). In addition, it is noted that the proportional valves are
overlap and there exists a dead zone in relation between the mass flow rate
and the voltage signal of the valves. Therefore, the mass flow
rates ${\dot{m}}_{j}$ (*j*=1 and 2) that flow into or out from the
chambers of the pneumatic cylinder can be derived in terms of the voltage
inputs *u*_{j} of the two valves as follows:

where *u*_{m} and *u*_{n} are respectively the upper and lower voltage limits of
the dead-zone, *p*_{s} and *p*_{j} respectively the source air pressure and
the pressure in the chamber *j* of the cylinder, *R* is the gas constant, *k* is the
specific heat ratio, *T*_{s} is the temperature of the supply source, and
*K*_{V1} and *K*_{V2} are respectively the valve gains for the
building pressure case and the exhausting pressure case. The operating
condition ${u}_{m}\le {u}_{j}\le \mathrm{5}$ corresponds to the case when the
pressure in the chamber *j* is the building pressure, $\mathrm{0}\le {u}_{j}\le {u}_{n}$ to the case when the pressure in the chamber *j* is
the exhausting pressure, and ${u}_{n}<{u}_{j}<{u}_{m}$ to the
case when all the valve ports are closed (the dead-zone condition of the
valve). *γ*_{jb} and *γ*_{je} are respectively the
modifying factors when the pressure in the chamber *j* is building pressure
and exhausting pressure. These factors are given by Tressler et al. (2002)
as follows:

where *p*_{atm} is the atmosphere pressure.

The dynamic relationships between the mass flow rates ${\dot{m}}_{\mathrm{1}}$,
${\dot{m}}_{\mathrm{2}}$ and the pressures *p*_{1}, *p*_{2} in the cylinder chambers can
be obtained with basis on energy conversation arguments in a pneumatic
cylinder and are given by Hodgson et al. (2012) as follows:

where *v* is the piston velocity. *V*_{1} and *V*_{2} refer to the volumes of the
cylinder chambers 1 and 2, respectively and are calculated as:

where *L* is the piston stroke, *x* is the piston position, and *V*_{10} and
*V*_{20} are the dead volumes in the cylinder chambers 1 and 2 respectively.

Motion equation of the cylinder piston according to Newton's second law is given by

where *F*_{r} is the friction force which has been described by one of the
three friction models in Sect. 2. The system parameters used in simulation
are shown in Table 1. The parameters of *u*_{m} and *u*_{n} were determined
from the measured characteristics of the pressures *p*_{1} and *p*_{2} at
constant voltage inputs of the valves. *u*_{m} and *u*_{n} were taken at the
voltage values at which the pressure *p*_{1} or *p*_{2} starts to increase,
i.e. the air starts to flow into the cylinder chamber.

In this section, experimental characteristics of the piston position, the
pressures in the cylinder chambers, the inertial force and the friction
force of the piston under different operating conditions of the voltage
signals *u*_{1} and *u*_{2} are firstly presented and analysed. Comparisons
between the simulation results of the three friction models and the
experimental results are then presented and discussed to show effects of
each friction model.

## 4.1 Experimental results

Figure 7 shows the measured characteristics of the piston position, the
pressures *p*_{1} an *p*_{2} in the cylinder chambers, the inertial force and
the friction force when the valves were supplied by constant voltage values.
The control inputs *u*_{1} and *u*_{2} were given by 2.875 and 2.19 VDC,
respectively. For this case, the air from air tank was supplied to the
cylinder chamber 1 through the valve 1 and the air in the cylinder chamber 2
was exhausted to the atmosphere through the valve 2. Flow rates of the air
supplied to and exhausted from the cylinder chambers in this case are
relatively small. As can be seen in Fig. 7a for the position characteristic,
the piston firstly remains at an initial position of 0.035 m for 1.8 s
then moves a small distance to a new position of 0.045 m. After that the
piston suddenly stops and remains at the new position for 0.5 s then
the piston moves again. This movement process of the piston is continued
until the stroke end of the piston. This characteristic is called
“stick-slip” motion and has been observed in pneumatic cylinders (Sakiichi
et al., 1988; Peng et al., 2012) and in other mechanisms (Mate et al., 1987;
Lampaert et al., 2004; Landolsi et al., 2009). This motion of the piston can
be explained that when air is supplied to the chamber 1, air is compressed
and the pressure *p*_{1} is increased (in Fig. 7b) while the pressure
*p*_{2} in the chamber 2 is remained at 0 MPa. In the first 1.8 s, the
increase in pressure *p*_{1} is not large enough to overcome the friction
force so that the piston remains stationary. As the pressure *p*_{1} rises to
a value large enough of about 0.022 MPa, creating an enough force to
overcome the friction force, then the piston starts moving (slip). However,
as the piston moves, the volume of the chamber 1 expands and the pressure
*p*_{1} decreases and thus the piston stops moving (stick). Air continues to
be fed into the chamber 1 and, after a period, the pressure *p*_{1} is
increased again to an enough value to overcome the friction force then the
piston moves again. The process is then repeated. It is further noted in
Fig. 7b that the maximum pressure *p*_{1} obtained from the second moving
onwards are less than the maximum pressure *p*_{1} in the first ones. In Fig. 7c, the inertial force obtained is small and therefore variation of the
friction force (Fig. 7d) is similar to that of the pressure *p*_{1}. Value of
the friction force is maximum (10.5 N) at the time when the piston starts
moving. When the piston has moved, the friction force decreases. In the next
cycles, the value of the friction force varies between 4.5 to 8.4 N.

When the signal *u*_{1} of the valve 1 was given by a higher value (*u*_{1}=2.99 V) and *u*_{2} of the valve 2 was given by a lower value (*u*_{2}=2.09 V) than those in case of Fig. 7, the piston remains at the initial
position 0.03 m for 0.45 s then moves smoothly to a new position of
0.24 m at 1.4 s as shown in Fig. 8a. Stick-slip motion of the piston
cannot be observed. It is noted that the air flow rate supplied to the
cylinder chamber 1 is relatively large in this case. This means that if the
air is provided largely enough to the cylinder chamber in a short time, a
continuous motion of the cylinder piston can be obtained. In addition, it is
shown in Fig. 8b that due to a large amount of the air flow rate supplied to
the cylinder chamber 1, the pressure of *p*_{1} continues increasing after
the piston has moved. The pressure is increased to the maximum value of
about 0.03 MPa at 0.58 s and then slightly decreased. Although the
maximum inertial force in Fig. 8c is larger than that in Fig. 7d, the
variation of the friction force in Fig. 8d is also mainly depend on the
variation of the pressure *p*_{1}.

Such similar above behaviours can be also observed for the cases when the piston retracts, i.e. for case when the valve 2 supplies air to the cylinder chamber 2 and the valve 1 exhausts air from the cylinder chamber 1 to the atmosphere.

Figure 9 shows the experimental characteristics of the pneumatic cylinder
when the valve signals *u*_{1} and *u*_{2} were varied with sinusoidal waves.
The signals of the valves were given by ${u}_{\mathrm{1}}=\mathrm{2.5}+\mathrm{0.5}\mathrm{sin}\left(\mathrm{2}\mathit{\pi}ft\right)$ and
${u}_{\mathrm{2}}=\mathrm{2.5}-\mathrm{0.4}\mathrm{sin}\left(\mathrm{2}\mathit{\pi}ft\right)$ with a low frequency of 0.2 Hz. It can be seen
in Fig. 9a that the piston moves in a trapezoidal form with the
corresponding frequency of the valve signals and the position amplitude
tends to increase slightly after each cycle. Like the results in Figs. 7 and
8, the piston moves only when the pressure *p*_{1} in the extending stroke or
*p*_{2} in the retracting stroke is increased to a value large enough,
corresponding to the appropriately increased and decreased values of the
valve signals *u*_{1} and *u*_{2}. In the extending stroke of the piston, the
increase of pressure *p*_{1} is relatively large, to a maximum value of 0.04 MPa, while the increase of the pressure *p*_{2} is very small, near the
atmosphere pressure. In contrast, the increase of pressure *p*_{2} is
relatively large, to a maximum value of 0.0506 MPa, while the increase of
the pressure *p*_{1} is very small in the retracting stroke (Fig. 9b). The
peaks of the pressure *p*_{2} in the retracting stroke are higher than those
of the pressure *p*_{1} in the extending stroke. This result is due to a
difference in the piston areas. The friction force observed in Fig. 9d
varied in a sinusoidal form and the variation of the friction force is
repeated in each cycle. It can be realized that this variation of the
friction force in the pneumatic cylinder is different from that observed in
hydraulic cylinders (Tran et al., 2012). In hydraulic cylinders, the
friction force characteristic hydraulic cylinders showed a decrease of the
maximum friction force observed after one cycle of the velocity variation.

## 4.2 Simulation results

This section shows comparisons between the simulated results of the three
friction models (the RLG model, the LG model, and the SS model) and the
measured results. Simulations were done using MATLAB/Simulink. An *ode3* solver
with a fixed-time step of $\mathrm{1.16}\times {\mathrm{10}}^{-\mathrm{3}}$ s was chosen for
numerical integration algorithm. The parameter values of the three friction
models used in these simulations are shown in Table 2. These values were
taken from Table 2 in Tran et al. (2016) in which the static parameters of
the models, *F*_{s}, *F*_{c}, *v*_{s}, *v*_{b}, *n*, and *σ*_{2}, were identified
experimentally from the steady-state friction characteristics using the
least-squares method and the dynamic parameters, *σ*_{0}, *σ*_{1}, *τ*_{h}, and *T* were identified experimentally by the method
proposed by Tran et al. (2012).

Figure 10 shows comparisons between the measured characteristics of the
piston position, the pressures *p*_{1} in the cylinder chambers, the friction
force and those simulated by the three friction models when low and constant
voltage signals of the two proportional valves are supplied. It can realize
in Fig. 10a that the stick-slip motion of the piston position can be
relatively well predicted by the RLG model while the stick-slip motion can
be partly predicted by the LG model; the number of stick-slip cycles of the
piston motion predicted by the LG model are less than those by the RLG
model. In addition, it is shown that the SS model cannot predict the
stick-slip motion of the piston; the SS model can only create continue
moving curves of the piston position and the pressures. In addition, much
oscillation is observed in the friction force characteristic simulated by
the SS model in Fig. 10c. Such these limitations of the SS model may be due
to lack of the stiction characteristic combined in the model. These results
obtained in Fig. 10 verify a better simulation capacity of the RLG model
comparing to the LG model and the SS model when the piston is operated at
low velocity range.

Figure 11 compares the measured characteristics of the piston position, the
pressures *p*_{1} and *p*_{2} in the cylinder chambers, the friction force
with those simulated by the three friction models at the following operating
conditions of the valve signals: ${u}_{\mathrm{1}}=\mathrm{2.5}+\mathrm{0.5}\mathrm{sin}\left(\mathrm{2}\mathit{\pi}ft\right)$ (V) and
${u}_{\mathrm{2}}=\mathrm{2.5}-\mathrm{0.4}\mathrm{sin}\left(\mathrm{2}\mathit{\pi}ft\right)$ (V). The varying frequency of the valve signals
was 0.2 Hz and the load mass was 0.5 kg. The comparison results show that
both the RLG model and the LG model give the same simulation results and can
simulate the measured characteristics with a relatively good accuracy. It is
noted that in the RLG model the lubricant film dynamics is added into the LG
model in order to simulate a decrease of the maximum friction force observed
after one cycle of the velocity variation in a hydraulic cylinder. However,
such this decrease of the maximum friction force cannot be observed
experimentally in the pneumatic cylinder as shown in Figs. 9d and 11d.
Therefore, the usefulness of the RLG model comparing to the LG model cannot
be realized for the pneumatic cylinder when the piston is operated under
sliding regime and under varied velocity conditions with reversal. For the
simulated results of the SS model, it can realize that the SS model can also
track well the measured position of the piston (Fig. 11a). However, the
pressures and the friction force predicted by the SS model are much smaller
than those of the measured ones. In addition, the SS model causes much
oscillations at the stop intervals of the piston position in the friction
force characteristic as shown in Fig. 11d.

Figure 12 show comparisons between the measured characteristics of the
piston and those simulated by the three friction models when the voltage
signals of the two proportional valves were varied by sinusoidal waves with
a high frequency *f*=1 Hz. It shows that the same simulation results as
obtained in Fig. 11 can be also achieved by the three friction models in
Fig. 12. Therefore, the simulation results obtained from Figs. 10 to 12
verify that the RLG model is the best for the pneumatic cylinder in the
three friction models considered in this study.

In this study, both experiments and simulations were conducted to examine the effects of friction models on the simulation accuracy of an electro-pneumatic servo system. Three friction models: a SS model (static + Coulomb + viscous friction), the LG model and the RLG model were examined. The characteristics of the piston position, the piston velocity, the pressures in the cylinder chambers and the friction force under different operating conditions of the valve inputs were measured, analysed and used for verification of the friction models. It has been verified that the RLG model can give the best simulation results among the three friction models. The LG model give relatively good results except for a limitation in simulating the stick-slip motion of the piston at low velocity condition. It has been also verified that the steady-state fiction model used in this study is unable to simulate the stick-slip motion as well as causes much oscillation in the friction force characteristics. The application of the RLG model or the LG model to control of a servo pneumatic system with friction compensation will be the subject for a future research.

The data in this study can be requested from the corresponding author.

The supplement related to this article is available online at: https://doi.org/10.5194/ms-10-517-2019-supplement.

XBT and VLN designed the experiment and VLN carried it out. XBT and KDT developed the mathematical model of the system. VLN built the simulation code and performed the simulations. XBT prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

The authors would like to thank Huy Thuong Dao for his assistance in collecting the experimental data.

This research has been supported by the Hanoi University of Science and Technology (grant no. T2018-PC-042).

This paper was edited by Marek Wojtyra and reviewed by two anonymous referees.

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