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.

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

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.

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

Steady-state friction 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

Bristle model.

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).

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

The hysteresis function

Implementation of the function

Numerical implementation of the function

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

Numerical implementation of internal hysteresis loop:

When the piston movement enters its sliding regime, i.e., when the
deflection reaches

In steady-state condition, friction force is described by

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

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

Schematic of the experimental test setup.

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

The friction force,

Photo of the experimental test setup.

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:

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

The leakages of the cylinder are negligible.

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

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

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

The supply and ambient pressures are constant.

The dynamic relationships between the mass flow rates

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

System parameters.

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

Figure 7 shows the measured characteristics of the piston position, the
pressures

Experimental characteristics at operating conditions of

When the signal

Experimental characteristics at operating conditions of

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

Experimental characteristics at operating conditions of

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

Values of the parameters of the three models used in simulation.

Figure 10 shows comparisons between the measured characteristics of the
piston position, the pressures

Comparison between measured characteristics of the
piston position, pressures and friction force and those simulated using the
three friction models:

Figure 11 compares the measured characteristics of the piston position, the
pressures

Comparison between measured characteristics of the
piston position, pressures and friction force and those simulated using the
three friction models:

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

Comparison between measured characteristics of the
piston position, pressures and friction force and those simulated using the
three friction models:

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

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

The supplement related to this article is available online at:

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.