Modeling of an air motor servo system and robust sliding ... · PDF fileModeling of an air motor servo system and robust sliding mode controller design ... A fuzzy sliding mode position

Journal of Mechanical Science and Technology 26 (4) (2012) 1161~1169

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0220-2

Modeling of an air motor servo system and robust sliding mode controller design†

Chia-Hua Lu1,* and Yean-Ren Hwang2 1Department of Mechanical Engineering, National Central University, Chung-Li 320, Taiwan

2Department of Mechanical Engineering and the Institute of Opto-Mechatronics Engineering, National Central University, Chung-Li 320, Taiwan

(Manuscript Received November 7, 2010; Revised November 14, 2011; Accepted December 23, 2011)

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Abstract Pneumatic systems offer a number of advantages: typically they are fast, simple to maintain and inexpensive. However, the high com-

pressibility of air makes control of the actuator very difficult. The purpose of this paper is to analyze the behaviors of a ball screw table powered by a vane-type air motor and then to design a robust sliding mode controller for position control. Due to the compressibility of the air and friction in the mechanism, the overall system is nonlinear. A robust sliding mode control is developed to overcome the effects of the uncertainty. Experimental results are given to validate the proposed position control strategy.

Keywords: Pneumatic; Air motor; Robust; Sliding mode control ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

The most commonly used pneumatic actuators are cylinders for translational movements, followed by semi-rotary actua-tors for turning movements. Continuous rotation is typically provided by electric drives. The design of a pneumatic cylin-der is much simpler, and therefore cheaper, than that of an electric linear drive, while there is not much difference with continuously rotating motors. Supplying power to an electric motor is much easier because of the ubiquitous availability of electricity, while compressed air requires additional equip-ment. However, there are many applications where the unique characteristics of pneumatic motors would be advantageous. These advantages include their high power-to-weight ratio, indifference to overload and stalling, coolness of operation, safety in dirty or explosive atmospheres, very high rotational speed and easy energy storage. In particular, their torque vs. speed characteristics make them very well suited for many applications, and they are also easily reversible [1-4].

Unlike electricity or other power sources, air does not gen-erate sparks, poses no health hazard and can be easily stored. Pneumatic actuators can therefore be used in environments where explosions could occur, such as at chemical plants. In addition, if no lubrication is used, air can be vented from the component directly into the atmosphere. A separate return line is not necessary, although some form of silencing is usually applied. A simple bottle is enough to store pneumatic energy

for a long time, even under extreme temperature conditions. For intermittent use, a storage tank can be used and a small compressor will sufficiently fill it.

Electric DC motors with ball screw drivers can be easily controlled and have thus been widely applied in the past. However, due to the compressibility of air, adverse friction characteristics and low damping, pneumatic drives require sophisticated controllers. On the other hand, they also have several advantages. For example, the compressibility of air provides a “cushioning” effect which is important in applica-tions like blow molding or glass forming [5].

Previously, there have been several studies investigating and analyzing the dynamics of air motors. The PI (propor-tional integral) controller was first proposed in Ref. [2], where the all air motor system was considered as a first order linear system. Although the PI control algorithm is simple and highly reliable when the PI values are well adjusted, the PI controller cannot retain high performance at all times because of the nonlinear speed characteristics of the air motor under different operating conditions. In Ref. [6], a nonlinear system was developed by considering all nonlinear phenomena such as un-modeled dynamics. However, these uncertain factors actually affected the performance of the pneumatic devices and the air motor. Hence it may be better to analyze these nonlinear phenomena. A fourth-order nonlinear model was developed in Refs. [7, 8] and its parameters were identified through experimental data. However, its complexity, the num-ber of parameters, and the process of deriving parameters made this method complicated and time consuming. A differ-ent approach, based on neural network techniques, was pro-posed in Ref. [9]. Instead of developing a high order nonlinear

*Corresponding author. Tel.: +886 3 4227151~34342, Fax.: +886 3 4254501 E-mail address: [email protected]

† Recommended by Associate Editor Won Gu Lee © KSME & Springer 2012

1162 C.-H. Lu and Y.-R. Hwang / Journal of Mechanical Science and Technology 26 (4) (2012) 1161~1169

model, a neural-model reference control method, constructed by adopting neural network techniques, was proposed to con-trol the rotational speed of the air motor.

Others have applied control schemes to guarantee accurate position control of pneumatic cylinders [10-12] which are subject to uncertain pneumatic dynamics. A fuzzy sliding mode position control method incorporating a ball screw driven by an air motor was proposed in Ref. [13], in which fuzzy logic control was applied to compensate for the disad-vantages of disturbances and chattering. In the last decade, various backstepping nonlinear design methodologies have been proposed [14, 15] for use in pneumatic nonlinear systems.

The contribution of this current study is to investigate the behavior of a vane-type air motor system which takes advan-tage of robust sliding mode control as a means of tracking position. With it, we should be able to decrease chattering while preserving strong robustness with different initial loads on a ball screw table. The experimental results confirm the good performance of the control system and perfect removal of the steady state position error.

2. Air motor ball screw table system

A schematic diagram of the air motor system and photo-graph of the experimental system are shown in Fig. 1 and Fig.

3, respectively. The system consists of an air motor (GAST 1AM), an air tank, an electronic proportional control valve (FESTO), a filter/regulator with lubricant (SHAKO FRL-600), an optical linear scale (FUTABA FJH5515IDR) with accu-racy of 0.5 mμ , and a digital signal processor (DSP, TI C240). The path of the airflow starts from the air tank and moves through the filter, then through the control valve, and finally enters the air motor. The responses for the output voltages are shown in Fig. 2.

3. Robust sliding mode controller design

The air flow is assumed to enter the air motor from chamber A and to exhaust from chamber B, as shown in Fig. 1. In the diagram r is the inner rotor radius, R is the radius of the motor body, d is the difference of R and r ( d R r= − ), φ is the motor rotating angle, and rx is the vane’s working radius measured from the rotor center.

The vane’s working radius measured from the rotor center, rx is given by

2 2 2cos sinrx d R dφ φ= + − . (1)

Considering friction and different payloads, using Newton’s

second law of angular motion we get

(a) Air motor system

(b) Air motor inner structure

Fig. 1. Schematic diagram of the air motor system.

Fig. 2. Position responses with different output voltages.

Fig. 3. Photograph of the experimental air motor system.

C.-H. Lu and Y.-R. Hwang / Journal of Mechanical Science and Technology 26 (4) (2012) 1161~1169 1163

( )c fM M S M Jφ φ φ− − = (2)

where φ is the angular velocity; φ represents the angular acceleration; cM is the stiction coefficient; fM is the fric-tion coefficient; and ( )S φ is described as follows:

( ) ( )0 , 0

, 0 S

sign

φφ

φ φ

⎧ =⎪= ⎨≠⎪⎩

. (3)

The drive torque is determined by the difference of the

torque acting on the vane between the drive and exhaust chambers, and (assuming 2 2 2sind Rφ << ) is given by

( )( ) ( )( )( )2 2 2

/ 2

cos2 2 cos / 2a b r r

a b

M P P x r L x r

P P d Rd R r Lφ φ

= − − −

= − + + − (4)

where L is the depth of the inner air motor.

The dynamics of the air motor system can then be described by

( ) ( )

( ) ( ) ( )

( ) ( )

1 2

2 2 2

2 2 21 1

1

1 cos2 2 cos2

fc

x t x tM

x t x t M S x tJ J

L d x t Rd x t R r uJ

=

= − − ⎡ ⎤ +⎣ ⎦

⎡ ⎤+ + −⎣ ⎦

(5)

where the input u is ( )a bP P− ; 1( )x t φ= ; 2 ( )x t φ= .

Assuming that x is the displacement of the ball screw table and that sρ is the screw pitch. The displacement sx ρ φ= and the velocity sx ρ φ= .

Eq. (2) can be rearranged as follows:

( )c f l mM M S M T Jφ φ φ− − − = (6)

where mJ is the total system inertia (motor and ball screw), and lT is the load torque.

The dynamics of the system considering friction can be de-scribed by the following equation:

( ) ( )( )( )2 2 21 1cos2 2 cos

f

m

s

m

s sc l

m s m

Mx x

J

L d x t Rd x t R r PJ

xM S TJ P J

ρ

ρ ρ

= − +

+ + − Δ

⎛ ⎞− −⎜ ⎟

⎝ ⎠

(7)

where PΔ is the ( a bP P− ).

A proportional valve is adopted in the system to control pressure. The PΔ is proportional to the control input

pu . p pP k uΔ = , where pk is the coefficient. Eq. (7) can be derived as follows:

( ) ( )( )2 2 21 1cos2 2 cos

f

m

p sp

m

df

Mx x

Jk

L d x t Rd x t R r uJD

ρ

= − +

+ + −

+

(8)

where s sdf c l

m s m

xD M S TJ Jρ ρ

ρ⎡ ⎤⎛ ⎞

= − −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

.

In order to facilitate the controller design, the over-all sys-tem can be described by

( ) ( ) ( )px a t b t u d t= − + + (9)

where

( )

( ) ( ) ( )( )

( )

2 2 21 1

cos2 2 cos

f

m

p s

m

df

Ma t

Jk

b t L d x t Rd x t R rJ

d t D

ρ

=

= + + −

=

；

.

It is assumed that

( )( ) ( )

( ) ( )

1min max

1min max

1

b t

b t t

b t d t D

β β

α α α

−

−

−

≤ ≤

≤ ≤

≤

. (10)

The inlet pressure is restricted to 6 bar so that minβ , maxβ , minα and maxα are all positive numbers; D 0> . The reference input is r and the tracking error can be de-

fined as

.e x r= − (11) The sliding mode surface is chosen as follows: s e ce= + (12)

where c is the slope of the surface, the system convergence speed. We now define a Lyapunov function as follows:

212

V s= . (13)

Eq. (14) can now be derived from Eq. (12) by

( )s e ce x cx r cr x cx R= + = + − + = + + (14)

where ( )R r cr= − − . Now, substituting Eq. (9) into Eq. (14), we get

( )1 1 1 1

s x cx bu bd R

b b x b cx b R u b d

α

α− − − −

= − + + + +

= − + + + + (15)


1 1 1 1 1pb s b x b cx b R u b dα− − − − −= − + + + + . (16)

The designed control input is

p eq vessu u u= + (17)

where equ is the equivalent control input for no disturbance;

vessu is the robust control input for a disturbance. These two control inputs are designed to make the system converge to the original point. The equ

and vessu are designed as follows:

{ } ( )

ˆ ˆˆ

sgneq

vss

u x cx R ks

u cx x D R s

α β β

β α β

= − − −

= − Δ + Δ + + Δ (18)

where k is a positive constant,

max min max min

max min max min

ˆ ,2 2

ˆ ,2 2

α α α αα α

β β β ββ β

+ −= Δ =

+ −= Δ =

.

(19)

Substituting Eq. (17) into Eq. (16) we get

{ } ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 1 1

-1

-1 -1

1 1

ˆ ˆˆ

cx sgn s b d

ˆ ˆ b cx- cx sgn s b R- R sgn s

ˆ - sgn sgn

b s b ax b cx b R x cx R ks

x D R

b a x x s b d D s ks

α β β

β α β

β β β β

α α

− − − −

− −

= − + + + − − −

− Δ + Δ + + Δ +

= − Δ + − Δ

+ − Δ + − −

(20)

and the following equations can be obtained:

( ) ( )( )

1 1 1

1 1 2

ˆ ˆ

ˆ

b ss b cxs cx s b Rs R s

b a xs x s b ds D s ks

β β β β

α α

− − −

− −

= − − Δ + − − Δ

+ − − Δ + − −

(21)

and

1

1

ˆ

ˆb

b aβ β

α α

−

−

− ≤ Δ

− ≤ Δ. (22)

Then, it can be proven that 2V ks≤ − and the system will

be asymptotically stable. In order to improve the chattering, the ( )sgn s function can be changed to the ( )sat s function (saturation function). Fig. 4 shows a block diagram of the robust sliding mode controller.

The mathematical model can be validated by comparing the open loop step dynamics of the ball screw table system re-sponses obtained both experimentally and in the simulation. Specifications of the air motor used for the experiment and the

simulation are listed in Table 1. The system responses for an open-loop air motor system are shown in Fig. 5 and the simu-lation results are based on Eq. (12). The simulated and ex-perimental results are consistent.

The results of the simulation of nonlinearity problems using the PID control design without friction and disturbance com-pensation are shown in Fig. 6. The nonlinearity included fric-tion and disturbances, and a loading mass on the ball screw table of 5 gk . It is obvious that for step position control, the nonlinearity would cause the system to vibrate, as shown in Fig. 6(b). The system cannot effectively stay at the desired final position. The different experimental output responses depend on various pressures and mechanical problems, such as the compressibility of air, pressure leakage, artificial factors (mechanism assembly), and so on. The ball screw table per-formance with an inlet pressure of 4 bar is shown in Fig. 7. Each experimental result shows a slightly different output response, but the variation is small. The main causes of this phenomenon are nonlinearities and uncertainties in the system.

Table 1. Parameters of the experimental system. Parameterr Value Units Parameterr Values Unit

L 44.5 mm mJ 8.7×10-4 2kgm

r 32.5 mm dR 15 mm

R 36.5 mm ck 0.42 bar / v

d 4 mm sρ 0.795 mm / rad

fM 0.34 N cM 0.06 Ns

Fig. 4. Block diagram of the robust sliding mode controller.

Fig. 5. System responses for experimental and simulation results with the step input open-loop.


4. Experimental results

The experiments were divided into two parts: under no-loading and loading conditions. Figs. 8-10 show the results of position control, where ssP (position at a steady state) is 10 and 20 mm . There was no serious overshoot with our system and the steady state error was 0.003 mm ( ssP =9.997 mm ), 0.002 mm ( ssP =19.998 mm ), respectively. The S function ( s e ce= + ) converged to zero. The output voltage is shown in Figs. 9 and 10. Fig. 11 shows the tracking sinusoidal wave results for an amplitude of 10 mm and a frequency of 0.05 zH .

Fig. 12 shows the tracking error. The maximum error is about 0.04 mm . According to the experimental results under no-loading conditions, the robust sliding mode can achieve a minimum steady state error of 0.002 mm for position control and a maximum error of 0.04 mm for the tracking sinusoidal wave. The chattering phenomenon is very small. The above experimental results are achieved under no-loading conditions. For real system control, however, the system may carry some load and also require accurate position control. Fig. 13 shows

(a)

(b)

Fig. 6. Simulation results of nonlinearity problems for the PID control design.

Fig. 7. Experimental results with the same inlet pressure.

Fig. 8. Position control results at 10 mm and 20 mm under no-loading conditions.

Fig. 9. S function ( s e ce= + ) results at 10 mm and 20 mm under no-loading conditions.

Fig. 10. Output voltage results at 10 mm and 20 mm under no-loading conditions.


the position results under a loading mass of 5 gk for two steps: one is 10 mm and the other is 20 mm . The steady state errors are 0.001 mm and 0.002 ,mm respectively. It is obvi-ous that the control voltage will increase to propel the ball-screw table and maintain the steady state error within 2 mμ , as shown in Fig. 13. The loading conditions for tracking the si-nusoidal wave and tracking error are shown in Figs. 15 and 16. The maximum tracking error is 0.05 ,mm compared to 0.04 mm under the no-loading conditions. The proposed ro-bust sliding mode control can maintain the steady state error

of the ball-screw table system to within 2 mμ and the track-ing error within 0.05 mm when tracking sinusoidal waves, regardless of being under no-loading or loading conditions.

The experimental performance of the two proposed control-lers, a PID controller and the robust sliding mode controller, are compared. The gains uK , and uT are used to set the PK ,

IK , and DK gains depending on the Ziegler–Nichols tuning method ( 0.6P uK K= ; 2 /I p uK K T= ; 0.125D P uK K T= ). Fi-nally, the variable parameters uK and uT are adjusted to be approximately 25.7 and 3.3, respectively.

Fig. 11. Experimental results showing the tracked sinusoidal wave under no-loading conditions.

Fig. 12. Error in the tracking sinusoidal wave under no-loading condi-tions.

Fig. 13. Experimental results for two step position control under a loading mass of 5 .gk

Fig. 14. Output voltage for two step position control under a loading mass of 5 .gk

Fig. 15. Experimental results for the tracking sinusoidal wave under a loading mass of 5 .gk

Fig. 16. Error in the tracking sinusoidal wave under a loading mass of 5 .gk


Figs. 17-19 show the results obtained for the two controllers with various loading conditions. Fig. 17 shows the results of step position control with a loading mass of 5 gk . The steady state errors for the two controllers are shown in Table 2. The maximum steady state error is 5 mμ compared to 15 mμ . The results of the tracking sinusoidal wave (amplitude 15 mm and frequency 0.025 zH ) and loading mass increased gradually from 5 to 15 kg during the experiments, as shown in Figs. 18 and 19. At 10 to 20 seconds, the loading mass was 5 kg ; at 20 to 30 seconds, the loading mass was 10 kg ; at 30 to 40 sec-onds, the loading mass was 15 kg . Figs. 18 and 19 show the results for the PID and robust sliding mode controller, respec-tively. Figs. 18(b) and 19(b) show that the error increases as the loading mass gets heavier. Figs. 18(c) and 19(c) show the output voltage. It can be seen that even the PID controller can catch cosine waves resulting in successful tracking. However, as indicated in Table 3, the error for the maximum axis is ap-proximately 0.15 mm with the PID control and 0.06 mm with the proposed controller. The aim of these experiments was to demonstrate the proposed control. There was a greater reduction in the chattering phenomenon than with the PID controller allowing for the achievement of robust performance

with different load disturbances. The system will have differ-ent responses even with the same inlet pressure as shown in Fig. 7. This phenomenon was caused by the nonlinearities and uncertainties of the system. According to the experimental results, the robust sliding mode controller can maintain system

Table 2. Experimental results for step position control.

Step position control

Robust sliding mode control PID control

Position ssP Error Position ssP Error

10 mm 9.995 mm 5 um 10 mm 9.985 mm 15 mμ15 mm 14.995 mm 5 um 15 mm 14.985 mm 15 mμ

Table 3. Experimental results for tracking position control.

Tracking position control (amplitude 15 mm and frequency 0.025 zH )

Robust sliding mode control PID control

Tracking error Tracking error

Cosine wave ± 0.04 mm Cosine wave ± 0.15 mm

Sine wave ± 0.06 mm Sine wave ± 0.15 mm

Fig. 17. Experimental results of position control for robust and PID control under a loading mass of 5 .gk

(a) Experimental results for tracking two different waves

(b) Errors for tracking two different waves

(c) Output voltage for tracking two different waves

Fig. 18. Experimental results for the PID control under various loading mass conditions (0-10 secs, loading 0 kg ; 10-20 secs, loading 5 kg ;20-30 secs, loading 10 kg ; 30-40 secs, loading 15 kg ).


track trajectories more robustly than the PID controller can. The maximum error was reduced by half a degree.

The experimental results illustrate the excellent perform-ance of the robust sliding mode controller. Clearly, even under loading conditions, the system produces accurate results.

5. Conclusion

Although the PID controller is highly reliable when the PID values are well adjusted, this performance cannot be retained at all times with an air motor because of nonlinear uncertainties. Thus, the conventional PID feedback controllers are not effective for position control of pneumatic systems. In this study, we devel-oped a robust sliding mode controller for an air motor ball screw table designed to overcome the problems of nonlinearity that occur due to variations in the compressibility of air as well as motor and ball screw table friction, which cause the system to yield different responses even with the same inlet pressure. Our robust sliding controller design provides very good traction per-formance. The step tracking position results show that with our system, the number of steady state errors can be maintained within 5, even under mass loading conditions, and the sinusoidal wave tracking results show that the air motor system can be stable. The experimental results all clearly show that our system has very strong robustness under loading conditions and the stability of the proposed robust sliding mode control system can be guaranteed.

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(a) Experimental results for tracking two different waves

(b) Errors for tracking two different waves

(c) Output voltage for tracking two different waves

Fig. 19. Experimental results for the robust sliding mode control under various loading mass conditions (0-10 secs, loading 0 kg ; 10-20 secs, loading 5 kg ; 20-30 secs, loading 10 kg ; 30-40 secs, loading 15 kg ).


cost solenoid valves, IEEE/ASME Trans. on Mechatronics, 12 (2) (2007) 216-219.

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[14] C. H. Lu, Y. R. Hwang and Y. T. Shen, Backstepping slid-ing-mode control for a pneumatic control system, Journal of Systems and Control Engineering, 224 (6) (2010) 763- 770.

[15] C. H. Lu, Y. R. Hwang and Y. T. Shen, Backstepping slid-ing mode tracking control of a vane-type air motor X-Y table motion system, ISA Transactions, 50 (2011) 278-286.

Lu, Chia-Hua was born in Taiwan. She received her B.S. degree in the depart-ment of Mechanical Engineering from Ming Chi University of Technology in 1999 and M.S. degree in the department of Mechanical Engineering from Na-tional Yunlin University of Science and Technology in 2008. She has been pur-

suing her Ph.D in National Central University from 2008 to now. Her current research topics include backstepping sliding model control and fuzzy design on pneumatic devices.

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Modeling of an air motor servo system and robust sliding ... · PDF fileModeling of an air motor servo system and robust sliding mode controller design ... A fuzzy sliding mode position