Journal (~f Man~ facturing Systems , ~ Vol. 24/No. 4

20O5

TECHNICAL NOTE Illll

A Novel, Low-Cost Pneumatic Positioning System M. Br ianThomas , Gary P. Maul, and Enrico Jayawiyanto, Manufactur ing Automat ion Laboratory, The Ohio State University, Columbus, Ohio, USA

Abstract Applications requiring accurate position control are in increasing

use in industry. Pneumatic servo systems can provide a clean, accurate, robust positioning system; however, the proportional valves used in such systems are relatively expensive. The use of solenoid valves to replace a proportional flow control valve can significantly lower the price of a positioning system.This substitution is possible if the solenoid valves are operated using a pulse-width- modulated (PWM) control scheme. This work presents a design in which position control is realized on a single-rod, double-acting cyl- inder.Two solenoid valves operate conventionally to fill either cham- ber of the cylinder, while a third valve uses pulse-width modulation in metering the exhaust flow.The experimental apparatus is capable of positioning a load along a horizontal axis to within _+0.10 mm.

Keywords: Pneumatic Cylinder, Solenoid Valve, Position Control

Introduction Position control applications have typically used one

of two actuator technologies. Hydraulic actuators have speed and force profiles compatible with many indus- trial processes but can present a number of workplace hazards to personnel. Electromagnetic actuators, on the other hand, are clean and reliable in their operation but often require a mechanical transmission, both to con- vert high speed and low torque to a more useful combi- nation and to convert rotary motion to linear motion. While linear motors can overcome the need for a trans- mission, they can be expensive. Pneumatic actuators af- ford the opportunity to design a positioning system that may be directly coupled with a load like a hydraulic actuator; is clean and reliable like electric motors; and is inexpensive. The challenge to the use of pneumatics is the highly nonlinear dynamics that makes conven- tional control strategies such as PID (proportional-inte- gral-derivative) ineffective.

This nonlinear behavior has relegated the use of pneu- matic cylinders in automated equipment to applications in which positioning accuracy is only required at the end of the actuator's stroke. In such applications, the nonlin- ear dynamics of the pneumatic cylinder are not impor- tant, as positioning accuracy is obtained by moving to and against a hard stop. When an arbitrary positioning

capability is required, systems using DC servomotors or hydraulic cylinders are usually selected over pneumatics, as these technologies may employ linear control strate- gies. With a nonlinear controller, though, pneumatic ac- tuators may replace their more expensive counterparts in some positioning applications.

The first significant analysis of the dynamics of a pneu- matic cylinder was performed by Shearer (1956). Using principles of thermodynamics, he developed a linear model valid for small motions about the midstroke posi- tion of a symmetric pneumatic cylinder. While this linear model has limited utility, Shearer's methodology toward developing the model has been used in most subsequent works. Liu and Bobrow (1988) expand the linear model to apply to any initial position for the cylinder. Their linear model is used in conjunction with a PD (propor- tional-derivative) controller to situate the poles of the closed-loop system. Shih and Tseng (1995) use a differ- ent approach, using system identification techniques to develop an empirical linear model for a specific pneu- matic positioning system. The authors then employ clas- sical methods in designing a PID controller.

With the widespread availability of programmable com- puters, simulation and control of highly nonlinear sys- tems is now practical. A thorough nonlinear model is developed by Richer and Hunnuzlu (2000a), taking into account the effects of flow through a proportional valve, leakage, and propagation losses and delays in the air lines. A subsequent paper from the same authors applies the nonlinear model to a sliding mode control scheme for force control (Richer and Hurmuzlu 2000b). Another analysis of the nonlinear model is conducted by Kawakami et al. (1988). In this work, the authors find little differ- ence between models assuming adiabatic thermodynamic processes and those assuming isothermal processes. The nonlinear models compare favorably to the experimental data; however, the linear dynamic model produces sig- nificant errors in simulation. Pu and Weston (1990) con- duct an analysis of the steady-state speed of pneumatic cylinders and rotary vane motors, considering the non- linear flow of air through the valve and plumbing.

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technical note

Pneumatic servo motion applications, such as that in Liu and Bobrow (1988), typically use five-port propor- tional flow control valves. Wang, Pu, and Moore (1999) use a five-port proportional valve to control a single-rod cylinder. The control scheme uses PID control with ve- locity and acceleration feedforward gains. To overcome the effects of stiction, a switching scheme saturates the valve when motion is initiated. Drakunov et al. (1997) apply techniques of feedback linearization toward a slid- ing mode controller for a rodless pneumatic cylinder. An exception to this use of proportional valves in servo ap- plications is the work by Kobayashi, Cotsaftis, and Takamore (1995). Here, the authors use a single-acting cylinder with a spring return. Two PWM-driven solenoid valves control the fill and exhaust flows.

While servo applications typically use proportional flow control valves, pneumatic positioning applications more often use a small number of solenoid valves in conjunc- tion with a PWM control scheme. Lai, Menq, and Singh (1990) use a pair of two-position, three-way valves to meter flow" to each chamber a pneumatic cylinder, as does Shih and Ma (1998). A paper by van Varseveld and Bone (1997) uses a similar physical arrangement, but applies PWM to only one valve at a time. Chillari, Guccione, and Muscato (2001) use a four-valve arrangement in com- paring control strategies for pneumatic actuators. Noritsugu and Takaiwa (1995) use two solenoid valves with a pulse-code modulated (PCM) valve to meter the exhaust. A proportional valve is used by Fok and Ong (1999) in their study of the repeatability of a pneumatic positioning axis.

Positioning accuracy of pneumatic positioning cylin- ders is typically less than 0.25 mm. Using PID control, van Varseveld and Bone (1997) find a worst-case posi- tioning error of 0.21 ram. Fuzzy control provides a low- load positioning accuracy of _+0.075 mm under no-load conditions, and _+0.1 man when an external load is ap- plied (Shih and Ma 1998). Position and pressure obsel~v- ers are used in the control system of Noritsugu and Takaiwa (1995), resulting in positioning accuracies of _+0.1 mm and _+0.2 mm under no-load and load condi- tions, respectively. The study of Fok and Ong (1999), using a proportional valve, resulted in repeatabilities of 0.1 nun to 0.3 ram, depending on the starting and stop- ping position. Root mean square (RMS) trajectory track- ing errors are calculated in Chillari, Guccione, and Muscato (2001), which compares the ability of six control strate- gies to track representative command trajectories. The RMS tracking errors fell between 0.2 mm and 2.8 mm, depending on the strategy and trajectory. The authors

found that their fuzzy controller, with pressure feedback, generally provided the smallest tracking error of the six control schemes.

Apparatus and Control Philosophy The purpose of this study is to determine the accuracy

of a pneumatic positioning system, using low-cost com- ponents, in reaching a fixed command position. A target accuracy of _+0.10mm is established. The experimental apparatus is shown schematically in Figure 1. The air cylinder, mounted horizontally, has a 27.0 mm (1-1/16 in.) bore and a 101.6 mm (4 in.) stroke and is mounted horizontally. A linear potentiometer provides position feedback signal to the controller. An inertial load of 1.64 kg may be attached to the cylinder rod. With the hard- ware configuration of Figure 1, the system is realistically limited to moving loads of 10 kg (22 lb) or smaller at speeds of 75 mm/s (3 in./s) or less. This configuration is scalable for larger loads or faster positioning.

Two regulators allow for independent pressure con- trol to both chambers of the cylinder. The pressures are set as to balance the pressure forces on the piston. Sole- noid-actuated fill valves control the extension and re- traction of the cylinder. These valves operate in a conventional manner, based on the desired direction of travel. Positioning control is obtained by metering the exhaust through a third, PWM-driven solenoid valve. Table l describes the motion control strategy for the experimental apparatus.

In practice, it is preferable that the pulse-width modu- lator run at a moderate frequency, between 10 and 100 Hz. When the PWM control signal operates faster than the mechanical response time of the exhaust valve, the distinct on-off periods of the command signal are attenu- ated in the mechanical action of the valve, resulting in an "average" flow rate based on the duty cycle of the com- mand signal. For the valve used in this study, it was ob- served that the mechanical position of the valve did not allow air flow when the duty cycle fell below 25% at 50 Hz. This effect is attributed to the mechanical design of the valve. To ensure air flow through the valve, a mini- nmm duty cycle is enforced in the command signal.

The position signal from the linear potentiometer is sent to an analog-to-digital signal conditioner. PD control is realized using LabView software, which converts the con- tinuous command signal to a PWM signal operating at 50 Hz. While the dynamics of the pneumatic cylinder are non- linear and vary with the cylinder's position, a lxajectory- following capability is not required in this application. PD

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Journal of Manufacturing Systems Wol. 24/No. 4

2005

technical note

y SUPPLY

GU~]'OeS

JST !

FILL VALVE~

C~'L~NDER a~d LOAD

LINEAR POTENTIOMETER

Figure 1 Experimental Apparatus

Table 1 Control Strategy

Desired Extend Retract EMaaust Motion Fill Valve Fill Valve Valve

Extend cylinder On Off PWM Retract cylinder Off On PWM Stop cylinder On On OFF

control is judged to be suitable for this position-control application. Table 2 lists the parameters used in the PD controller. Only one parameter, the minimum exhaust valve's duty cycle, DCMm, is adjusted in the LabView pro- gram to account for changes in pressure and load.

Figure 2 shows the exhaust valve duty cycle as a func- tion of the error signal. The enforcement of a minimum duty cycle has the effect of increasing the effective gain for small displacements outside the positioning tolerance band. This allows the system to more quickly overcome static friction for short moves.

The dynamics of a pneumatic cylinder may be described by a fourth-order set of nonlinear differential equations. The governing equations were first derived by Shearer (1956); later authors have applied his methodology in deriving variations of these equations. Assuming an adia- batic process, the differential equations describing the dynamics of a pneumatic cylinder can be expressed as

kRT rh, kPl 2 = Alx -7 (1)

Table 2 Control Parameters

Parameter Symbol %Salue PWM drive frequency - 50 Hz Controller update rate - 12 Hz Proportional gain k~, 12.5% per mm Differential gain k. 0.8% per mm.sec -~ Tolerance band - __0.1 mm Minimum duty cycle DCMLv Varies by pressure

and load

100% - k duty cycle

er ror

- ~-- tolerance bund Figure 2

Exhaust Valve Duty Cycle as a Function of the Error Signal

kRT + kP2 Jc P2 =a2-~-x)rh2 (L -x ) (2)

~(Mg+PjA _PzA2_ParMARoD_b2_Fmc) (3)

where x, 2 , P], and P2 are the system state variables-- position, velocity, pressure in the blind end, and pressure in the rod end, respectively.

This formulation uses state variables that are easy to measure experimentally, but the nonlinear terms--particu- larly the i/x and 1/(L - x) terms--prohibit an analytical solution. In addition, the gas mass flow inputs to the sys- tem, rh 1 and rh z , are nonlinear functions of the command signal and the pressttre in each side of the cylinder, and could be considered as additional states of the system.

Recognizing the mass flow inputs, rn I and rh 2 , are not strictly independent of each other in this system, a reduced- order model of the system may be developed. Moving the pneumatic cylinder of Figure 1 from one position to an- other requires the removal of gas from one chamber of the cylinder and addition of gas to the other. "Hae ideal gas law

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Journal o[" Mant(facturing Systems Vol. 24/No. 4 2005

technical note

shows that for an ideal, frictionless cylinder, the mass of gas removed from one side of the cylinder is equal to the gas added to the second, regardless of the design of the cylinder. In effect, the function of the valves is to change the equilibrium position of the system by means of the flow of gas. This may be expressed mathematically as

2co = kVALvEU (4)

In this expression, u is the command signal sent to PWM-controlled exhaust valve. The sign of the control signal determines which fill valves in F~'gure 1 are open or closed (Table 1). The control gain, kVALVE, in Eq. (4) is not strictly constant, but can be considered the math- ematical representation of the pressure dynamics in Eqs. (1) and (2). Considering it a bounded constant enables treatment of the system dynamics using linear analysis tools, with the understanding that some detail of the sys- tems dynamics has been lost.

Similarly, Eq. (3) may be expressed in terms of the cylinder's equilibrium position, XEQ, instead of pressure. The mass of gas in the chambers determines the equilib- rium position of the cylinder. Displacement from equi- librium changes the pressures in the cylinder, creating a restorative force imbalance on the piston. In the reduced- order model, the restorative force may be treated as an equivalent spring. Equation (3) is now

1 )~='~(Mg-fCcrL (X-XeQ)-b.t-Fve,c ) (5)

where kCyL is the cylinder's spring stiffness coefficient. Like the control gain, kvacw, this coefficient is not con- stant but varies as a function of both stroke and displace- ment from equilibrium. Assuming displacement from equilibrium is sufficiently small and ignoring the volume of gas in the supply lines leading to the cylinder, the stiff- ness coefficient is expressed as

lCcyL = RT I ~ m2 (6)

This equation is derived in Thomas (2003). Equations (4) and (5) describe the open-loop dynam-

ics of the reduced-order system model. Assuming the ex- perimental apparatus is operating in the horizontal plane, the closed-loop system dynamics with a PD controller may be expressed in matrix form as

i ] [ 0 : , 71i 1 [--[,VALVEk,--kvALVEAD 0 jkxEo/

0 0

LkvALwkeJ L o I

(7)

This expression has the appearance of a linear, third- order differential equation with a command input, Xcog, and disturbance term, FFR1O The terms kcr L and kVALV f are time-variant and nonlinear functions of the state vari- ables, though, and prevent a purely linear analysis of the system. Having bounds of the nonlinear terms, though, allows for some understanding of the system dynanfics. The characteristic equation, X, of the closed-loop "lin- ear" system may be expressed as

(~l(lq-kvaLvEkD)S-l-(kcYLk-K~vEkp) (8)

For the pneumatic cylinder in this experiment, Table 3 lists the nominal values of the system variables for the cylinder at the mid-stroke position, having a mass load and a pressure of 414 kPa in the blind end. Under these conditions, the roots of the closed-loop system are s = - 5.88 and s = -3.96 + 95.11i. The real root has a time constant of 0.17 s, and the lightly damped complex roots have a natural frequency of 15.1 Hz.

Moving the cylinder away from its mid-stroke posi- tion increases the effective stiffness, /~crL, which in turn moves the complex roots of Eq. (8) away from the real axis without significantly changing the real root. Uncer- tainty in the control gain, kVALVE, moves all three roots, with an increase in kVALV E moving the real root away from, and the complex roots toward, the imaginary axis. Friction, though, is the strongest nonlinear term in Eq. (7), having two significant effects. First, it is expected to diminish high-frequency oscillation associated with the complex roots of Eq. (8), allowing the single real root to dominate the system response. Second, friction will cause the c~ hnder s position to lag behind the equilibrium po- sition. Both phenomena are observed in comparing Figure

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Journal of ManuJ~wturing Systems Vol. 24/No. 4

2005

I I I I I

technical note

Table 3 Nominal Values of System Parameters

Method of Variable Meaning Value Determination

M Mass load 1.64 kg Measurement b Viscous dmaaping 0.02 N-sec/mm Empirical

coefficient ]~crL Equivalent spring 9.32 N/ram Eq. (8)

stift'ness kVALVE Control valve 0.75 mm/%-sec Empirical

gain k Proportional 12.5 %/mm Control

feedback gain parameter k D Derivative 0.8 %-sec/mm Control

feedback gain parameter F~.~c Coulomb friction 45 N Empirical

4 with Figure 3. Figure 3 shows the linear response of the system, without friction, for extension to 50ram using the values listed in Table 3. Figure. 4 includes the effect of Coulomb friction in the response, and clearly exhibits both the lack of oscillation and the lag between the equi- librium position and the cylinder's position.

Experimental Conditions The PD control algorithm was tested using different

configurations of the experimental apparatus to evalu- ate the positioning accuracy. Three nominal gauge pres- sure settings were applied to the blind end of the cylinder: 207 kPa (30 psi), 310 kPa (45 psi), and 414 kPa (60 psi). The pressure in the rod end of the cylinder was set proportionately higher (227 kPa, 339 kPa, and 453 kPa, respectively) to account for the difference in piston ar- eas to achieve balance in the pressure forces. Tests were conducted with no external load on the cylinder, and with a mass load rigidly attached to the cylinder. When the load is attached to the cylinder, an unknown fric- tional force is generated, acting between the mass and its supporting surface. The mass load is supported in a manner that does not generate side loads on the cylinder rod. With LabView, the dynamic responses of the sys- tem were recorded for position step commands to the 10 ram, 50 mm, and 90 lnm positions, in both extension and retraction. Positioning error was calculated by av- eraging the last 12 data points in a record, ending ap- proximately 8 seconds after the step command. The positioning en'or in a single record was typically either always positive or always negative.

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Figure 3 Linear Simulation without Friction, Extension to 50 mm

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[~cy l inder position - - equilibrium I

Figure 4 Linear Simulation ~th Friction, Extension to 50 mm

Results No-Load Cond i t ions

Figures 5, 6, and 7 show the transient response of the positioning system in extension. Table 4 shows the steady- state positioning errors, calculated using the last 12 data points in each data record. Figure 8 shows the transient response for positioning from 100 mna to 10 ram, with Table 5 showing the positioning errors in retraction. The maximum positioning error in Tables 4 and 5 is 0.19 ram, though most positioning errors fall within the _+0.10 mm en'or tolerance band.

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Journal o[ Manufacturing Systems Vol. 24/No. 4 20O5

technical note

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t ime (see)

Command ~ - 2 0 7 k P a ~ 3 1 0 k P a m - -414kPa

Figure 5 No-Load Extension to 10 mm

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Command I I 2 0 7 k P a - - 3 1 0 k P a - - - -414kPa

Figure 7 No-Load Extension to 90 mm

40 [ j l~[ - i - -

!3 - I-i . . . . "~ r I~ 20 "~- -- ~ --I

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. . . . . . T . . . . . . . .

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t ime (see)

Con l~nd ~ , 2 0 7 k P a ~ 3 1 0 k P a m . -414kPa

Figure 6 No-Load Extension to 50 min

Table 4 No-Load Pos i t ion ing Er ror on Extens ion

103 kPa 207 kPa 310 kPa 10mm -0 .01mm 0.04mm 0.10mm 50mm -0.09mm 0.03mm -0.12mm 90mm 0.01mm 0.11mm -0.19mm

110

100

90

80

70

60

50

40

30

20

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t ime (see)

Command I - 2 0 7 k F a ~ 3 1 0 1 d z a ~ - i414kPa

Figure 8 No-Load Retraction to 10 mm

Table 5 No-Load Pos i t ion ing Er ror on Ret ract ion

103 kPa 207 kPa 3111 kPa 10 mm -0.05 mm 0.00 mm 0.15 mm 50 mm -0.04 nma 0.06 mm -0.03 mm 90 mm -0.07 mm 0.03 mm 0.05 mm

Inert ia l Load

An external mass adds a 1.64 kg external load with an unknown fi icf ion component to the actuator. Figures 9, 10, and 11 show the system response to step command inputs in extension. Table 6 gives the posit ioning errors for the inertial load in both extension and retraction. The range o f posit ioning errors in Table 6 is comparable to those in Tables 4 and 5.

Grav i tat iona l (Constant) Load

The experimental apparatus was reoriented for verti- cal motion. The mass load, suspended be low the pneu- matic cylinder, applies a constant force to the cylinder. The pressure in the blind end of the cyl inder was set at 207 kPa. The pressure in the rod end was increased from 340 kPa to 370 k_Pa to account for the added weight of the payload.

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Journal o . I " Manufacturing Systems Vo l . 24 /No . 4

2005

technical note

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Command i "207kPa ,310kPa ~ - -414kPa

bYgure 9 Mass Load Extens ion to 10 mm

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Figure 10 Mass Load Extens ion to 50 mm

It was observed that the system response to a step in- put would oscillate under the command position for some period of time before motion stopped. Closer inspection of the data revealed that the cylinder moves sufficiently fast as to cross the tolerance band around the conm]and position in less than one sampling period. During upward motion (reta'action), the weight of the external payload adds to the restorative force in the actuator. The com- bined acceleration from the air pressure and gravity forces the payload down faster than when only an inertial load is present. The system will oscillate indefinitely until a sample happens to fall within the tolerance band.

To prevent this undesirable behavior, the minimum duty cycle function of Figure 2 was made asymmetric. A lower minimmn duty cycle for downward motion com- pensates for the additional gravitational load acting on the cylinder. Figure 12 shows the step response of the

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. . . . . Corm~nd i -207kPa , .310kPa~- .414kPa

Figure 11 Mass Load Extens ion to 90 mm

1~ble 6 Mass Load Pos i t ion ing Er rors

103 kPa 207 kPa 310 kPa 10 mm, extending 0.06 mm -0.14 mm -0.18 mm 50 ram, extending -0.10 mm -0.06 mm -0.02 mm 911 ntm, extending -0.02 mm -0.10 mm -0.07 mm

10 mm, retracting -0.09 mm -0.17 mm 0.05 mm 50 rnm, retracting 0.09 mm 0.03 wan 0.05 mm 90 nun, retracting 0.02 mm 0.00 mm -0.08 mm

vertically-loaded system moving to 50 mm, from both the fully extended and fully retracted positions. The ver- tical scale in Figure 12 is reversed so that the ordinate matches the physical arrangement of the apparatus. Table 7 shows the positioning errors for the vertical load ar- rangement. Except for one datum, the positioning errors in the vertical orientation are similar to those found in the horizontal.

Increasing the sampling period of the controller, so that the cylinder could not cross the tolerance band be- tween samples, would allow the controller to stop the cylinder while in the control band. This approach was not pursued due to time constraints.

Discussion The second section of this paper discussed the control

philosophy for the pneumatic positioning system. A mini- mum duty cycle (DCMIN) in the exhaust valve is enforced, as the average mechanical position valve spool does not allow flow at duty cycles of less than 25%. Without a minimum control effort, the system would not be able to compensate for small positioning errors. The width of

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Journal c~f Manufacturing Systems Vol. 24/No. 4 2005

technical note

1

40

60

80

100

120

time (see)

0.0 0 .5 1 ,0 1 .5 2 .0 2.5

0~ . t 1

n i r 20~ . . . . . . ~ . . . . . T . . . . . . r . . . . . . . . . . . .

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40 . . . . . . . . . ~ - - j . . . . . . r- . . . . . . . . . . . .

t . . . . . . . . . :S . . . . . . . . ' ,

. . . . . . . . . . T . . . . . . ~- . . . . . . p . . . . . . r ! I

I I I I - - I . . . . . . - . . . . . . F . . . . . . I . . . . . .

I I r I I I

Command ~ - o Extend ing Ret rac t ing 1

Figure 12 "Vertical Mass Load Motion to 50 mm

Table 7 Vertical Load Positioning Errors

Extending (down) Retracting (up) 10 mm -0.10 mm -0.15 mm 5(1 mm -0.36 mm -0.00 mm 90 nun 0.10 mm 0.03 mm

this dead band, without DCMIN, may be estimated. By divid- ing the 25% minimum duty cycle for air flow by the 12.5% per mm proportional error gain, the width of the dead band is +2 mm about the command position.

Through the course of this work, the minimum duty cycle, DCMm , is the controller parameter from Table 2 that is adjusted for variations in pressure and external loading. This suggests that, once the proportional and derivative gains are determined for a particular hardware configuration, adjustment of DC~,N alone is sufficient to compensate for variations in the supply pressures, loads, or the plant. For a vertical load, two values of DCM1N are required. Proper adjustment of DC,~IN is necessary for good performance of the controller. If this parameter is too small, the response time and accuracy of the control- ler will suffer. If it is too large, the system can limit cycle about the desired position. Table 8 gives the minimum duty cycles used in this paper.

In each experiment, the pressure in both chambers of the cylinder was adjusted to establish force equilibrium over both faces of the cylinder piston. In practice, a per- fect force balance is not obtainable. Measurement errors exist in pressure gauges and sensors, and the internal mechanisms of regulators suffer from drift and resolu- tion limitations. As a result, small force imbalances will induce drift in the cylinder away from the command po-

Table 8 Minimum Duty Cycles

Cylinder Blind End Pressure Payload DCw~,

207 kPa None 50% 310 kPa None 37.5 % 414 kPa None 30% 207 kPa 1.64 kg 50% 310 kPa 1.64 kg 41% 414 kPa 1.64 kg 32t, 310 kPa 1.64 kg 36% (up)

(vertical) 25% (down)

50,2

50 .1

EE 50,0 g '~ 49 .9

o .

49 .8

49 .7

i i i i i i i | i i i i i F i J i i i i t J

. - ,= , ~- D- =- , - ,~=- - -L - - - - ~ , - - t i I I i i , , J , ,

; _ _ i I I ~ 1 I I

i I i I I I

4.0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0 7 .5 8 .0

time ( see)

o 310 kPa]

Figure 13 Position l toiding Detail

sition. Evidence for this was observed in the experimen- tal data, as most experiments had a positioning error that was either always positive or always negative. The force imbalance is responsible for many of the results outside the _+0.10 mm tolerance band in Tables 4, 5, and 6. The minimum duty cycle acts to increase the effective feed- back gain for small displacements. With this, the control- ler is able to return the cylinder to within the tolerance band in one sample period.

Figure 13 shows the positioning detail for extension to 50 mm with 414 kPa applied to the blind end of the cylin- der and an external mass load. In this test, the pressure force imbalance induces a drift toward the lower tolerance bound (retraction). From Figure 13, it is apparent that in- creased resolution in the analog-to-digital conversion of the sensor and a faster sampling rate would allow a smaller positioning tolerance to be applied to the system.

The stiffness of the pneumatic cylinder may be calcu- lated using the ideal gas law. This analysis, supported by

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technical note

previous works (Fok and Ong 1999), assumes displace- ments are small enough to allow ibr an isothermal analy- sis. Neglecting the volume of air in the external tubing, the cylinder stiffness is expressed as

Note that Eq. (6) could be manipulated, using the ideal gas law equations, to find Eq. (9). Both equations are singular at the limits of travel, when x = 0 or x = L. In the real system, the compliance of the air in the air lines connected to the cylinder will prevent an infinite stiff- ness. The minimum stiffness, however, is of more inter- est to the design engineer. Differentiation shows Eq. (9) has a minimum value at the mid-stroke position, when x = L/2. Here, the stiffness is

kcrL MIN -- 4PjAI (10) ' L

For the system used in this paper, the stiffness at mid- stroke is 4.66 N/mm (26.6 lb/in.) for the system with a blind-end pressure of 207 kPa (30 psi). At 310 kPa (45 psi), the stiffness is 6.98 N/ram (39.9 lb/in.), and at 414 kPa (60 psi), the stiffness is 9.32 N/mm (53.2 lb/in.).

Qualitatively, the experimental data agree with the es- timated pole locations in the linearized analysis of Eq. (8). The time rise is consistent with a first-order linear system with a time constant of 0.17 s, and the oscillatory behavior generated by the complex roots is suppressed by the Coulomb friction in the cylinder. Figure 14 shows the experimental data from Figure 10 overlaid with the match- ing simulation data from Eq. (7) and Figure 4. As with previous works such as Kawakami et al. (1988), discrep- ancies exist between the simulated and actual dynamic response in Figure 14. Many of these discrepancies can be attributed to the simplifying assumptions made for the simulation. The simulation model treats many of the nonlinear elements in Eq_(7) as linear functions, such as the cylinder stiffness, kcrL, and the flow valve coef- ficient, kj. The model also ignores the effects of stiction. Stiction accounts for the delay in the experimental re- sponse seen in Figure 14, and plays a significant role in the final positioning due to stick-slip motion over small motion increments.

Figure 7 shows an apparent anomaly in its 414 kPa no- load response. Immediately after the cylinder passes its command position, the cylinder jumps 5 nun away from its command position. This behavior is attributed to the fact that the equilibrium position, xEQ, leads the cylinder's

60., ,, , ,,

40

g g 30

o. 20

l0

0 0.0 0.5 1.0 1.5 2.0

Time (see)

-o - -Data ~S imula t ion

Figure 14 Mass Load Extension to 50 mm

actual position, as seen in Figure 4. When enough lead exists, the restorative force generated by the difference between the equilibrium and actual position is enough to break the stiction that might otherwise prevent the system from moving away from its command position. A contrib- uting factor to this is resonance in the system. At 90 nun and 414 kPa with no load, the resonant frequency of the cylinder is estimated to be 54 Hz, very close to the 50 Hz PWM frequency of the exahaust valve. This same frequency match occurs around 10 nun for the same pressure and loading conditions, but does not appear elsewhere in the experimental conditions. Figure 5 shows a similar anomaly for the movement to 10 iron, but the scale of Figure 5 exaggerates the magnitude of this dynanfic.

A statistical analysis of the data in Tables 4 through 7 was performed to determine if any correlation between the positioning error and the control parameters exists. In testing for differences in the mean, no significant rela- tionships were found to exist in the positioning error with respect to the cylinder pressures, loading conditions, di- rection of travel, or command position. Further analysis examined a set of 25 repeated runs under the same nomi- nal operating conditions. The repeated positioning error behaves like a random variable with a normal distribu- tion. A Fourier analysis of the error as a function of time shows no significant frequency information.

The cost of the hardware used in the experimental appa- ratus (Figure 1), discounting the controller, is approximately $530. For comparison, a similar pneumatic system using a proportional air valve wotfld cost about $760, due to the higher cost of the valve. Replicating the experimental ap- paratus with a hydraulic system would cost $1,300, not including the reservoir and pump unit. A linear motor sys-

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technical note Illl

tern, with integated position feedback, power supply, and servo controller, costs more than $2,200. An open-loop stepper motor-timing belt system can be realized for less than $530, but such a system would be susceptible to posi- tioning errors from missed steps. System costs were ob- tained with reference to McMaster-Carr (2004).

While the system has been successfully demonstrated in both the vertical and horizontal configurations, it is expected that a similar pneumatic positioning system would find more practical application in the horizontal configu- ration. In the vertical configuration, the rod end and blind end pressures are set for a specific loading condition. Any changes in the external load would upset the force bal- ance created by these specific pressures. Therefore, the practical application of this positioning system is restricted to those applications in which external loads act perpen- dicularly to the cylinder's axis of motion. An example of such an application is a gantry-style robot, in which the positioning cylinders move the payload in the horizontal plmae. A pick-and-place end effector, realized with con- ventional pneumatic cylinders, can act in the vertical axis to place or retrieve parts fi'om multiple positions in the working space of the robot.

Conclusions A low-cost position control system for a pneumatic

actuator is presented. The system can achieve positioning accuracies of under 0.2 ram, similar to pneumatic posi- tioning systems presented in previous works. The control system uses a modified PD controller to determine the duty cycle on a PWM-driven solenoid valve controlling the exhaust flow from the cylinder. The control algo- rithm has been shown to be effective in both horizontal and vertical configurations, though it is expected to be applied in horizontal applications where there are no sig- nificant variations to the load applied to the cylinder. This system provides a significant reduction in the cost of the hardware over competitive technologies.

Nomenclature Symbols

A b DCMl, v F g k k

area

coefficient of viscous friction minimum duty cycle of exhaust valve force acceleration of gravity ratio of specific heats gain (/~ indicates a time-variant gain)

L M

P R s

T x

Subscripts

1 2 ATM CYL EQ FRIC ROD E4LVE

cylinder stroke payload mass mass flow rate of gas pressure gas constant LaPlace operator temperature position

blind end of cylinder rod end of cylinder atmospheric cylinder equilibrium friction rod valve

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Authors' Biographies Dr. M. Brian Thomas is an assistant professor of industrial engi-

neering at Cleveland State University and is a 2003 PhD graduate of The Ohio State University.

Dr. Gary P. Maul is an associate professor of industrial engineer- ing at The Ohio State University. He has more than 30 years of experience in industrial automation and material handling,

Mr. Enrico Jayawiyanto earned his master's degree in industrial engineering from The Ohio State University in 2003. He is working as an automated systems engineer.

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