A High Performance Pneumatic Force Actuator SystemPart 1 - Nonlinear Mathematical Model∗

Edmond Richerand

Yildirim HurmuzluSouthern Methodist University

School of Engineering and Applied ScienceMechanical Engineering Department

Dallas, TX 75275

February 12, 2001

Abstract

In this paper, we developed a detailed mathematical model of dual action pneumaticactuators controlled with proportional spool valves. Effects of nonlinear flow throughthe valve, air compressibility in cylinder chambers, leakage between chambers, end ofstroke inactive volume, and time delay and attenuation in the pneumatic lines werecarefully considered. System identification, numerical simulation and model validationexperiments were conducted for two types of air cylinders and different connectingtubes length, showing very good agreement. This mathematical model will be used inthe development of high performance nonlinear force controllers, with applications inteleoperation, haptic interfaces, and robotics.

1 Introduction

Modern force-reflecting teleoperation, haptic interfaces, and other applications in roboticsrequire high performance force actuators, with high force output per unit weight. It is alsoimportant to have linear, fast and accurate response, as well as low friction and mechan-ical impedance. Traditional geared electrical motors can not provide these characteristics.Few newly designed motors may have special direct-drive actuators, with no intermediate

∗This article has appeared in the September 2000 issue of ASME Journal of Dynamic Systems Measure-ment and Control, Vol. 122, No.3, pp. 416-425

1

mechanisms. Yet, applications such as teleoperation master arms with gravitational compen-sation, require long duration, static high force output. In these cases, direct drive electricalactuators necessitate special cooling systems to dissipate the excessive heat.

We believe that pneumatic cylinders can offer a better alternative to electrical or hydraulicactuators for certain types of applications. Pneumatic actuators provide the previouslyenumerated qualities at low cost. They are also suitable for clean environments and saferand easier to work with. However, position and force control of these actuators in applicationsthat require high bandwidth is somehow difficult, because of compressibility of air and highlynonlinear flow through pneumatic system components. In addition, due to design and spaceconsiderations, in many applications the command valve is positioned at relatively largedistance from the pneumatic cylinder. Thus, the effects of time delay and attenuation dueto the connecting tubes becomes significant. Due to these difficulties, early use of pneumaticactuators were limited to simple applications that required only positioning at the two endsof the stroke. Subsequently, more complete mathematical models for the thermodynamicand flow equations in the charging-discharging processes were developed (Shearer, 1956).As a result more complex position controllers, based on the linearization around the midstroke position, were developed (Burrows, 1966; Liu and Bobrow, 1988). These simplifiedmodels provided only modest performance improvements. During the last decade, nonlinearcontrol techniques were implemented using digital computers. Bobrow and Jabbari, 1991,and McDonell and Bobrow, 1993 used adaptive control for force actuation and trajectorytracking, applied to an air powered robot. Improved results were presented for low frequencies(approx. 1 Hz). Sliding mode position controllers were also tested (Arun et al., 1994; TangandWalker, 1995), again with improved results at low frequencies. As a general characteristicthe mathematical models used in these controllers assumed no piston seals friction, linearflow through the valve, and neglected the valve dynamics. Ben-Dov and Salcudean, 1995developed a forced-controlled pneumatic actuator that provided a force with an amplitudeof 2 N at 16 Hz. Their model included the valve dynamics and the nonlinear characteristicsof the compressible flow through the valve. A comparison between linear and nonlinearcontrollers applied to a rotary pneumatic actuator is presented by Richard and Scavarda,1996. The mathematical model accounted for the leakage between actuator’s chambers andthe nonlinear variation of the valve effective area with the applied voltage. Their modeldepends heavily on curve fitting using experimental values, making difficult the applicationof the model to even slightly different systems.

The goal of this article is to provide an accurate model of a pneumatic actuator sys-tem controlled using a proportional spool valve. The model takes into consideration thefriction in piston seals, the difference in active areas of the piston due to the rod, inactivevolume at the ends of the piston stroke, leakage between chambers, valve dynamics and flownonlinearities through the valve orifice, and time delay and flow amplitude attenuation invalve-cylinder connecting tubes. Since the ultimate purpose of the modeling effort is forcontrol applications, the proposed equations should be suitable for on-line implementation.We designed special experiments in order to identify the unknown characteristics of the pneu-matic system, such as: valve discharge coefficient, valve spool viscous friction coefficient, and

2

piston friction forces. The model was finally tested using two experiments that allowed themeasurement of the actuator force output and piston displacement. The experimental resultswere compared with the results obtained by numerical simulation.

2 System Dynamics

A typical pneumatic system includes a force element (the pneumatic cylinder), a commanddevice (valve), connecting tubes, and position, pressure and force sensors. The external loadconsists of the mass of external mechanical elements connected to the piston and perhaps aforce produced by environmental interaction. A schematic representation of the pneumaticactuator system is shown in Fig. (1), with variables of interest specified for each component.

���������������������

������������������������������������������������������������������

x +-

P1,V1, A1 P2,V2, A2ML

FL

Supply

Exhaust

Valve

At, Lt

Cylinder

Exhaust

ic

Mp, ArLoad

P1 P2

Position sensor

Pressure sensors

PaPa

Ps

xs +-

����������������������

������

������������

Fa

Figure 1: Schematic representation of the pneumatic cylinder-valve system.

3

2.1 Piston-Load Dynamics

The equation of motion for the piston-rod-load assembly can be expressed as,

(ML +Mp)x+ βx+ Ff + FL = P1A1 − P2A2 − PaAr (1)

where ML is the external load mass, Mp is the piston and rod assembly mass, x is thepiston position, β is the viscous friction coefficient, Ff is the Coulomb friction force, FL isthe external force, P1 and P2 are the absolute pressures in actuator’s chambers, Pa is theabsolute ambient pressure, A1 and A2 are the piston effective areas, and Ar is the rod crosssectional area. The right hand side of Eq. (1) represents the actuator active force, producedby the different pressures acting on the opposite sides of the piston. In order to control theactuator force output, one has to finely tune the pressure levels in the cylinder chambersusing the command element (the pneumatic valve). This requires detailed models for thedynamics of pressure in both chambers of the actuator, valve dynamics, and connectingtubes.

2.2 Cylinder Chambers Model

In this section we seek to develop a mathematical relationship linking the pressure changewith the mass flow rate and piston translational speed is necessary for each pneumaticcylinder chamber. In previous works (see Liu and Bobrow, 1988; Ben-Dov and Salcudean,1995; Richard and Scavarda, 1996) the authors derived this equation using the assumptionthat the charging and discharging processes are both adiabatic. Al-Ibrahim and Otis, 1992,found experimentally that the temperature inside the chambers lays between the theoreticaladiabatic and isothermal curves. The experimental values of the temperature were closeto the adiabatic curve only for the charging process. For the discharging of the chamberthe isothermal assumption was closer to the measured values. In this article we derive thepressure dynamics equation in a way that accounts for the different thermal characteristicsof the charging and discharging processes of the cylinder chambers.

The most general model for a volume of gas consists of three equations (see Hullenderand Woods, 1985): an equation of state (ideal gas law), the conservation of mass (continuity)equation, and the energy equation. Assuming that: (i) the gas is perfect, (ii) the pressuresand temperature within the chamber are homogeneous, and (iii) kinetic and potential energyterms are negligible, these equations can be written for each chamber. Considering thecontrol volume V , with density ρ, mass m, pressure P , and temperature T , the ideal gas lawcan be written as,

P = ρRT (2)

where, R is the ideal gas constant. Applying the continuity equation the mass flow rate canbe expressed as,

m =d

dt(ρV ) (3)

4

which can be also expressed as,

min − mout = ρV + ρV (4)

where, min and mout are the mass flows entering and leaving the chamber.The energy equation can be written as follows:

qin − qout + kCv(minTin − moutT )− W = U (5)

where qin and qout are the heat transfer terms, k is the specific heat ratio, Cv is the specificheat at constant volume, Tin is the temperature of the incoming gas flow, W is the rate ofchange in the work, and U is the change of internal energy. The total change in internalenergy is,

U =d

dt(CvmT ) =

1

k − 1d

dt(PV ) =

1

k − 1(V P + PV ) (6)

in which we used the ideal gas relation, Cv = R/(k − 1). Now, substituting W = PV andEq. (6), into Eq. (5),

qin − qout +k

k − 1P

ρT(minTin − moutT )− k

k − 1PV =1

k − 1V P (7)

Assuming that the incoming flow is already at the temperature of the gas in the chamberconsidered for analysis, the energy equation becomes,

k − 1k

(qin − qout) +1

ρ(min − mout)− V =

V

kPP (8)

Further simplification can be made by analyzing the heat transfer terms in Eq. (8). Ifthe process is considered to be adiabatic (qin − qout = 0), the time derivative of the chamberpressure is,

P = kP

ρV(min − mout)− k

P

VV (9)

or, substituting ρ from Eq. (2),

P = kRT

V(min − mout)− k

P

VV (10)

If the process is considered to be isothermal (T = constant), then the change in internalenergy is,

U = CvmT (11)

and Eq. (8) can be written as,

qin − qout = PV − P

ρ(min − mout) (12)

5

Then, the rate of change in pressure will be,

P =RT

V(min − mout)− P

VV (13)

A comparison of Eqs. (10) and (13) shows that the only difference is the specific heatratio term k. Thus, both equations can be written as,

P =RT

V(αinmin − αoutmout)− α

P

VV (14)

with α, αin, and αout taking values between 1 and k, depending on the actual heat transferduring the process. In equation (14) one does not have to know the exact heat transfercharacteristics, but merely estimate the coefficients α, αin, and αout. The fact that theuncertainty of the estimation is bounded by k − 1 is also very important from the controldesign perspective. For the charging process, a value of αin close to k is recommended, whilefor the discharging of the chamber αout should be choose close to 1. The thermal characteristicof compression/expansion process, due to the piston movement is better described usingα = 1.2 (see Al-Ibrahim and Otis, 1992).

Choosing the origin of piston displacement at the middle of the stroke, the volume ofeach chamber can be expressed as,

Vi = V0i + Ai(1

2L± x) (15)

where i = 1, 2 is the cylinder chambers index, V0i is the inactive volume at the end of strokeand admission ports, Ai is the piston effective area, L is the piston stroke, and x is the pistonposition. The difference between the piston effective areas for each chamber A1 and A2 isdue to the piston rod. Substituting Eq. (15) into (14), the time derivative for the pressurein the pneumatic cylinder chambers becomes:

Pi =RT

V0i + Ai(12L± x)

(αinmin − αoutmout)− αPAi

V0i + Ai(12L± x)

x (16)

In this new form the pressure dynamics equation accounts for the different heat transfercharacteristics of the charging and discharging processes, air compression or expansion dueto piston movement, the difference in effective area on the opposite sides of the piston, andthe inactive volume at the end of stroke and the admission ports. The first term in theequation represents the effect on pressure of the air flow in or out of the chamber, andthe second term accounts for the effect of piston motion. There are two sources for theflow entering a cylinder chamber: a) the pressure tank, through the pneumatic valve andconnecting tube, b) the neighboring chamber if it has a higher pressure and piston seals areleaking. The air can flow out to the atmosphere through the valve or piston rod seals, orto the second chamber if it has a smaller pressure. The leakage between chambers can beneglected for regular pneumatic cylinders with rubber type seals, but can be significant forlow friction cylinders that have graphite or Teflon seals. The expressions for the input andoutput flows will be derived in the following section.

6

2.3 Valve-Cylinder Connecting Tube Model

The tubes that connect the valve with the actuator have two effects on the system response.First, the pressure drop along the tube will induce a decrease in the steady state air flowthrough the valve. Second, the flow profile at the outlet will be delayed with respect to theone at the inlet by the time increment necessary for the acoustic wave to travel the entirelength of the tube. This will affect the transient response of the flow in the cylinder chambers.The problem of the pressure losses and time delay in long pneumatic lines was analyzed bymany authors: Schuder and Binder, 1959; Hougen et al., 1963; Andersen, 1967; Whitmoreet al., 1990, Elmadbouly and Abdulsadek, 1994. Most of these investigators assumed fullydeveloped laminar flow through the tube. They provide infinite series solutions for thepressure dynamics, or approximate the response to harmonic pressure inputs using a secondorder linear system.

Because we model for online control applications, in our derivation we seek a simplerexpression for the mass flow through the tube. The expression should not require intensivecomputation like the series solution and should not increase the order of the system asthe second order linear approximation does. We also extend the analysis to include whollyturbulent flow.

At, Lt

s

mt (0, t)= h(t) mt (Lt, t)

Figure 2: Pneumatic tube notations.

In Schuder and Binder 1959, and Andersen, 1967, the two basic equations governing theflow in a circular pneumatic line were derived as,

∂P

∂s= −Rtu− ρ

∂u

∂t(17)

∂u

∂s= − 1

ρc2∂P

∂t(18)

where P is the pressure along the tube, u is the velocity, ρ is the air density, c is the soundspeed, s is the tube axis coordinate, and Rt is the tube resistance. Introducing the massflow through the tube as mt = ρAtu, where At is the tube cross sectional area,

∂P

∂s= − 1

At

∂mt

∂t− Rt

ρAt

mt (19)

∂mt

∂s= −At

c2∂P

∂t(20)

These equations are similar to those in Elmadbouly and Abdulsadek, 1994, with the secondterm in Eq. (19) that accounts for resistance of the tube. Differentiating Eq. (19) with

7

respect with t and Eq. (20) with respect to s, we obtain the equation for the mass flowthrough the tube as,

∂2mt

∂t2− c2

∂2mt

∂s2+Rt

ρ

∂mt

∂t= 0 (21)

This equation represents a generalization of a wave equation, with an additional dissipativeterm. The proposed mass flow equation can be solved by using the following form (seeChester, 1971):

mt(s, t) = φ(t)v(s, t) (22)

where v(s, t) and φ(t) are new unknown functions. Substituting Eq. (22) into (21) yields,

φ∂2v

∂t2− c2φ

∂2v

∂s2+

(φRt

ρ+ 2φ′

)∂v

∂t+

(φ′Rt

ρ+ φ′′

)v = 0 (23)

In order to simplify the equation for v, we determine φ(t) such that, after substitution inEq. (23), the remaining equation in v contains no first derivative term (Chester, 1971),

2φ′ + φRt

ρ= 0 (24)

which is equivalent to,

φ(t) = e−Rt2ρ

t (25)

The resulting equation for v will be,

∂2v

∂t2− c2φ

∂2v

∂s2+

R2t

4ρ2v = 0 (26)

which is a dispersive hyperbolic equation. Chester, 1971, showed that Eq. (26) had a solutionin the form of a progressive wave that propagates along the tube with any constant velocitydifferent than c. The fact that the solution waves do not propagate with the same velocityis called dispersion, and it is caused by the term (Rt/2ρ)

2v. The tubes under considerationare not very long, so we assumed that the dispersion is small, and can be neglected. Thisassumption yields,

∂2v

∂t2− c2φ

∂2v

∂s2= 0 (27)

This is the classic one-dimensional wave equation, and can be solved for specific boundaryand initial conditions. We assume that there is no flow through the tube at t = 0, the flowat the inlet (s = 0) is an arbitrary time dependent function h(t), and there is no reflectionfrom the end connected at the pneumatic cylinder. The corresponding initial and boundaryconditions become,

v(s, 0) = 0∂v

∂t(s, 0) = 0

v(0, t) = h(t)

(28)

8

The solution for this boundary-initial-value problem is (see Chester, 1971),

v(s, t) =

0 if t < s/c

h(t− s

c

)if t > s/c

(29)

The input wave will reach the end of the tube in a time period τ = Lt/c. Replacing t byLt/c in Eq. (25), and substituting ρ from the equation of state, the attenuation componentin Eq. (22) becomes,

φ = e−RtRT

2PLtc (30)

where P is the end pressure. The mass flow at the outlet of the tube (s = Lt) is,

mt(Lt, t) =

0 if t < Lt/c

e−RtRT

2PLtc h

(t− Lt

c

)if t > Lt/c

(31)

Equation (31) describes in a simple form the mass flow at the tube outlet, for any inletflow. It shows that the flow at the outlet of the tube is attenuated in amplitude and delayedby Lt/c, which represents the time required by the input wave to travel through the entirelength of the tube. This solution can not account for the dependence of the amplitudeattenuation on the frequency of the input flow. Its application have to be restricted torelatively small frequencies. The experimental results presented by Hougen et al., 1963,showed very small amplitude dependence on frequencies up to 50 Hz, for tubes up to 15 m inlength. Considering the fact that Eq. (31) is applied to the valve-cylinder connecting tubes,this limitations are considered more than satisfactory for usual applications.

The tube resistance Rt can be obtained from the expression for the pressure drop alongthe tube (see Munson et al, 1990),

∆p = fLt

D

ρu2

2= RtuLt (32)

where f is the friction factor, and D the inner diameter of the tube. For fully developedlaminar flow f = 64/Re, where Re is the Reynolds number. The tube resistance becomes,

Rt =32µ

D2(33)

where µ is the dynamic viscosity of air. This result is also derived in Schuder and Binder,1959. Extending the analysis for wholly turbulent flow in smooth tubes (the plastic tubesused in this work can be considered smooth), the friction factor can be computed using theBlasius formula (Munson et al, 1990),

f =0.316

Re1/4(34)

9

-6

-4

-2

0

2

4

6

mt

[x10

-3 Kg/

s]

100806040200

t [x10-3

sec ]

Input flow Solution from Eq. (31) Solution from Eq. (21)

Figure 3: Tube outlet flow for a sinusoidal flow input.

The tube resistance for wholly turbulent flow becomes,

Rt = 0.158µ

D2Re3/4 (35)

In order to use Eq. (31) one has to evaluate the Reynolds number from the input flow values,and compute Rt using Eq. (33) or (35) accordingly.

We compared the results given by Eq. (31) with those obtained by numerical integrationof Eq. (21), for a plastic tube with 3.2 mm internal diameter and 3 m length. The inputflow was considered to be sinusoidal, with 6 × 10−3 Kg/s amplitude and 30 Hz frequency(see Fig. (3)). One can observe from the figure that the simplified equation can effectivelypredict time delay and the amplitude attenuation.

2.4 Valve Model

The pneumatic valve is a critical component of the actuator system. It is the commandelement, and should be able to provide fast and precisely controlled air flows in and out ofthe actuator chambers. There are many available designs for pneumatic valves, which differby geometry of the active orifice, type of flow regulating element, number of paths and ports,type of actuating, etc. We restricted our study to proportional spool valves, actuated byvoice coil. This design presents several advantages: quasi-linear flow characteristic, smalltime constant, small internal leakage, ability to adjust both chambers pressure using one

10

control signal, very small hysteresis, and small internal friction. In addition, there are manycommercially available quality valves. We used a PositioneX, four-way, proportional valveproduced by Numatics Inc. The valve features a lapped spool-sleeve assembly, with very lowfriction. The spool is balanced with respect to pressure and positioned at the equilibrium(closed) position using two coil springs. This design permits fast and precise adjustmentsof the valve orifice area, providing accurate flow control. If the spool is displaced in thepositive direction, one chamber will be connected to the pressure tank through the supplypath, and the compressed air will flow inward (min > 0, mout ≈ 0). The other chamber willbe connected to the atmosphere through the exhaust path, and the air will flow outward(min ≈ 0, mout > 0).

Now we present a model that is developed for the PositioneX valve. This analysis,however, can be easily tailored to model any commercially available pneumatic proportionalspool valve. Modeling of the valve involves two aspects: the dynamics of the valve spool,

xs +-

Fc

Ff

cs

ksks

Ms

Figure 4: Valve spool dynamic equilibrium.

and the mass flow through the valve’s variable orifice. Analyzing Fig. (4), the equation ofmotion for the valve spool can be written as,

Msxs = −csxs − Ff + ks(xso − xs)− ks(xso + xs) + Fc (36)

where xs is the spool displacement, xso is the spring compression at the equilibrium position,Ms is the spool and coil assembly mass, cs is the viscous friction coefficient, Ff is the Coulombfriction force, ks is the spool springs constant, and Fc is the force produced by the coil.Simplifying the spring force expressions yields:

Msxs + csxs + Ff + 2ksxs = Fc (37)

The friction force Ff , can be neglected because it is customary in control applications toapply dither signal to the coil, with small magnitude and frequency close to the bandwidth ofthe valve. The spool will slightly vibrate around the equilibrium position, and the Coulombfriction force will be greatly reduced. Using the force-current expression for the coil andneglecting Ff , Eq. (37) becomes,

Msxs + csxs + 2ksxs = Kfcic (38)

11

where Kfc is the coil force coefficient, and ic is the coil current.The pressure drop across the valve orifice is usually large, and the flow has to be treated

as compressible and turbulent. If the upstream to downstream pressure ratio is larger thana critical value Pcr, the flow will attain sonic velocity (choked flow) and will depend linearlyon the upstream pressure. If the pressure ratio is smaller than Pcr the mass flow dependsnonlinearly on both pressures. The standard equation for the mass flow through an orificeof area Av is (see Ben-Dov and Salcudean, 1995),

mv =

CfAvC1

Pu√T

ifPd

Pu

≤ Pcr

CfAvC2Pu√T

(Pd

Pu

)1/k

√√√√1− (Pd

Pu

)(k−1)/k

ifPd

Pu

> Pcr

(39)

where mv is the mass flow through valve orifice, Cf is a nondimensional, discharge coefficient,Pu is the upstream pressure, Pd is the downstream pressure and,

C1 =

√√√√ k

R

(2

k + 1

) k+1k−1

; C2 =

√2k

R(k − 1); Pcr =(

2

k + 1

) kk−1

(40)

are constants for a given fluid. For air (k = 1.4) we have C1 = 0.040418, C2 = 0.156174,and Pcr = 0.528. The meaning of the upstream and downstream pressure in Eq. (39) isdifferent for the charging and discharging process of the cylinder chambers. For charging,the pressure in the supply tank should be considered the upstream pressure and the pressurein the cylinder chamber is the downstream one. For discharging process, the pressure inthe chamber is the upstream, and the ambient pressure is the downstream pressure. Sameexpression can be used for the leakage flow between the chambers, with the valve areareplaced by an experimentally determined leakage area, and the upstream and downstreampressures as the ones from the chambers.

The area of the valve is given by the spool position relative to the radial holes in thevalve sleeve, as it is shown in Fig (5). The area of the segment of the circle delimited by theedge of the spool can be expressed as,

Ae = 2∫ xe

0

√R2

h − (ξ −Rh)2 dξ = 2∫ xe

0

√ξ (2Rh − ξ) dξ (41)

where Ae is the effective area for one radial sleeve hole, xe is the effective displacement ofthe valve spool, and Rh is the hole radius. Integrating Eq. (41) and considering all activeholes for an air path in the sleeve (nh), the effective area of one path in the valve is,

Av = nh

[2R2

h arctan

(√xe

2Rh − xe

)− (Rh − xe)

√xe (2Rh − xe)

](42)

The spool width (2 pw) is slightly larger than the radius of the hole, to ensure that theair paths are closed even in the presence of small valve misalignments. Thus, the effectivedisplacement of the valve spool xe will be different from its absolute displacement xs,

xe = xs − (pw −Rh) (43)

12

xe

Rha

2pw

xs

Figure 5: Orifice area versus spool position.

Substituting Eq. (43) in (42) the valve effective areas for input and exhaust paths becomes,

Avin=

0 if xs ≤ pw −Rh

nh

[2R2

h arctan

(√Rh − pw + xs

Rh + pw − xs

)− (pw − xs)

√R2

h − (pw − xs)2

]

if pw −Rh < xs < pw +Rh

πnhR2h if xs ≥ pw +Rh

(44)

and,

Avex =

πnhR2h if xs ≤ −pw −Rh

nh

2R2

h arctan

√√√√Rh − pw + |xs|Rh + pw − |xs|

− (pw − |xs|)

√R2

h − (pw − |xs|)2

if − pw −Rh < xs < Rh − pw

0 if xs ≥ Rh − pw

(45)

Valve areas for input and exhaust paths versus the spool displacement are presented inFig. (6). The direction of flows in cylinder chambers have opposite signs, when one chamberis charged the other one is discharged, thus the role of the curves should be switched for thesecond chamber.

Substituting Eqs. (44) and (45) in Eq. (39) we obtain the expressions for the input andoutput valve flows.

13

30

25

20

15

10

5

0

Av

[mm

2 ]

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

xs [mm ]

Input path area Exhaust path area

Figure 6: Input and exhaust valve areas.

3 System Identification

The mathematical model of the pneumatic system derived in the previous sections includesa number of geometric and functional characteristics or parameters. Accurate values ofthis parameters are required in order to use the model in practical applications. Someparameters, such as cylinder bore diameter, piston stroke, or the length of the connectingtubes are provided by the manufacturer or they can be easily measured. Other parameters,such as the critical pressure ratio for chocked flow, can be calculated using generally acceptedformulas and values for the physical constants involved. The parameters which can not bedirectly measured or calculated, have to be estimated (identified) using specially designedexperiments. The valve discharge coefficient, spool viscous friction coefficient, static anddynamic friction force between the piston and the cylinder bore, and the piston viscousfriction coefficient are parameters of the model that have to be identified experimentally.

An important factor in the pressure dynamics equation is the mass flow, which can becontrolled using the pneumatic valve. Thus, we initiate the system identification processby considering the command element (valve). The value for the coil force coefficient isprovided in the valve user manual (Kfc = 2.78N/A), and Ms and ks can be easily measuredby dismantling the spool. With these values known, the steady state value for the spooldisplacement can be computed for any constant coil current as,

xs =Kfc

2 ks

ic (46)

Measuring the flow for several coil currents and fitting the theoretical mass flow expression,Eq. (39), on the experimental values as it is shown in Fig. (7), we determined the discharge

14

coefficient Cf = 0.25. The last unknown characteristic of the valve is the viscous friction co-

3.0

2.5

2.0

1.5

1.0

0.5

0.0

mf

[x10

-3 K

g/s]

0.50.40.30.20.10.0

ic [A]

Experimental values Theoretical flow

Figure 7: Valve flow versus coil current.

efficient. We determined its value indirectly by analyzing the step response of the input flowin the actuator chamber charging process while keeping the piston in a fixed position. Thetheoretical flow curve obtained using Eqs. (38), (39) and (14) matches closely the experimen-tal values, for cs = 7.5 as it is shown in Fig. (8). Using the identified values, we performedan additional test for the valve flow. Figure (9) presents the dependence of the flow onthe downstream pressure, with upstream pressure held constant, for both the experimentalvalues and theoretical equation.

The unknown characteristics of the pneumatic cylinder model include Coulomb and vis-cous friction forces, inactive volumes at stroke ends, and leakage between chambers. If thedetailed geometry of the actuator is available, the inactive volume at the stroke ends canbe easily computed. Otherwise they can be measured by positioning the piston at eachend of stroke, filling the ports cavities with a liquid (we recommend lubrication oil), andthen measuring the volume of the used liquid. The friction forces and the leakage betweenchambers can only be determined experimentally for each type of cylinder used in a specificapplication. We considered two types of cylinders, with similar geometric characteristics.First, 0750D02-03A, produced by Numatics, Inc. is a double acting, .75” bore diameter,3” stroke pneumatic actuator, with regular rubber piston seals. The second, E16 D 3.0 NAirpel, is produced by Airpot Corp., with .627” bore size and 3” stroke, and has glass linercylinder and graphite piston for ultra-low friction.

15

1.0

0.8

0.6

0.4

0.2

0.0

mf

[x10

-3 K

g/s]

0.140.120.100.080.060.040.020.00

t [sec]

Experimantal values Theoretical flow

Figure 8: Valve viscous friction coefficient identification.

The Coulomb friction force can be expressed as,

Ff =

{Fsf if x = 0Fdf sign(x) if x = 0 (47)

where Fsf and Fdf are the static and dynamic friction forces, and

sign(x) =

−1 if x ≤ −10 if x = 01 if x ≥ 1

(48)

In order to determine the friction force we performed a simple experiment. With thepiston at rest and air ports disconnected, we applied an increasing force at the rod end. Thepiston remained fixed until the external force reaches the value of the static friction, andthen started moving, with a smaller friction force, which included the dynamic Coulombforce and viscous friction, and piston inertia force. Figure (10) shows the total resistanceforce for the Numatics actuator, and the corresponding position and velocity of the piston.The static friction is identified from the curve peak as Fsf = 3.8N . If we neglect the inertiaof the piston,in the time interval 0.15 - 0.40 sec, the resistant force can be expressed as,

Frf = Fdf sign(x) + β x (49)

Plotting the resistance force versus piston velocity, and fitting Eq. (49) to the experimentalvalues (see Fig. (11), we separated the dynamic friction force Fdf from the viscous friction.

16

1.0

0.8

0.6

0.4

0.2

0.0

mf

[x10

-3 K

g/s]

550500450400350300250200150

P [x103 Pa]

Experimantal values Theoretical flow

Figure 9: Valve flow versus downstream pressure.

We obtained Fdf = 0.486 N , and β = 4.47. Figure. (12) shows the results of the sameexperiment performed with the Airpel actuator, and Fig. (13) shows the total resistance forceversus velocity. The experiments show no static friction peak, and relative independence ofthe friction force on the velocity after compensation for piston inertia. The dynamic frictionforce was identified as Fd = 0.8N .For the Airpel actuator, the leakage between chambers was also measured, and no significantvalues detected.

4 Experimental Validation of the Model

The complete mathematical model for the pneumatic actuator system consists of the valvedynamics equation, two equations for the chamber pressure time derivatives, and the piston-load equation of motion. The chamber pressure rate of change described by Eq. (16), has tobe customized for the two cylinder chambers using the corresponding input and exhaust flows,and the corresponding upstream and downstream pressures for the charging or dischargingprocess. The valve flows through the input and output paths can be written as,

mvin= CfAvin

Ps√Tmr(Ps, P ) (50)

mvex = CfAvex

Pi√Tmr(Pi, Pa) (51)

17

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Ff [

N]

0.500.450.400.350.300.250.200.150.100.050.00

t [sec]

-40

-30

-20

-10

0

10

X [

x10-3

m]

0.25

0.20

0.15

0.10

0.05

0.00

v [

m/s

]

Displacement Velocity

Figure 10: Numatics 0750D02-03A cylinder friction force.

18

2.0

1.5

1.0

0.5

0.0

Ff [

N]

0.200.150.100.050.00

v [m/s]

Experimental values Theoretical friction force

Figure 11: Dynamic and viscous friction force versus piston velocity, for Numatics 0750D02-03A.

19

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Ff [

N]

0.500.450.400.350.300.250.200.150.100.050.00

t [sec]

-30

-20

-10

0

10

20

X [

x10-3

m]

0.5

0.4

0.3

0.2

0.1

0.0

v [

m/s

]

Displacement Velocity

Figure 12: Airpel E16 D 3.0 N cylinder resistance force.

20

2.0

1.5

1.0

0.5

0.0

Ff [

N]

0.40.30.20.10.0

v [m/s]

Experimental values Theoretical friction force

Figure 13: Cylinder friction force versus piston velocity, for Airpel E16 D 3.0 N.

where Pi, i = 1, 2 is the absolute pressure in the corresponding cylinder chamber, and,

mr =

C1 ifPd

Pu

≤ Pcr

C2

(Pd

Pu

)1/k

√√√√1− (Pd

Pu

)(k−1)/k

ifPd

Pu

> Pcr

(52)

is the reduced flow function. Using Eq. (31) the flows through the connecting tubes become,

mtin = φin mvin(t− τ) (53)

mtex = φex mvex(t− τ) (54)

where φin and φex are the flow attenuations from Eq. (30), and τ is the tube time delay.Substituting these equations in Eq. (16) we obtain the final equations for chambers pressureas,

P1 =CfR

√T

V01 + A1(12L+ x)

[αinφinAv1in

Psmr(Ps, P1)− αexφexAv1exP1mr(P1, Pa)]

−α P1A1

V01 + A1(12L+ x)

x (55)

P2 =CfR

√T

V02 + A2(12L− x)

[αinφinAv2in

Psmr(Ps, P2)− αexφexAv2exP2mr(P2, Pa)]

21

−α P2A2

V02 + A2(12L− x)

x (56)

where the variables with overbars represent values delayed by time τ . These equations,together with Eqs. (1) and (38), completely describe the pneumatic system. They includethe effect of the external load, piston friction, the difference in the effective areas on the twosides of the piston on the actuator force output, the different heat transfer characteristics atcharging or discharging process and compression-expansion of air due to piston motion, theflow attenuation and delay due to the tubes, nonlinear compressible flow through the valve,and the valve dynamics. They can be numerically solved for any valve command ic, and canbe used as a mathematical model for control design.

Figure 14: The experimental set-up.

Two sets of experiments were conducted in order to verify the proposed mathematicalmodel. In the first experiment we measured the pressures in both cylinder chambers, andthe force provided by the actuator using absolute pressure piezoelectric transducers, and astrain gage force cell. The piston was fixed at the middle of the stroke, and a step input of0.5 A has been applied to the valve coil (see Fig. (14)). Two length of connecting tubes wereused: 0.5 m and 2 m. Figures (15) and (16) show both numerical and experimental resultsfor actuator chamber pressures and output force, corresponding to the valve step input of0.5 A, and for both tube lengths. There is a close correspondence between the theoreticaland experimental curves, with very good amplitude match and less than 0.001 secondsmisalignment in the time profiles. Both figures show a significant time delay between thevalve command and corresponding pressure or force output, due to the connecting tubes and

22

600

500

400

300

200

100

P1,

P2

[Pa

x103 ]

806040200

t [x10-3

sec]

Experimental P1

Experimental P2

Theoretical P1, P2

(a)

150

100

50

0

-50

-100

F [N

]

806040200

t [x10-3

sec]

Experimental Fa Theoretical Fa

(b)

Figure 15: Numerical and experimental actuator pressures (a) and force (b) for 0.5 A stepcoil current and 0.5 m tube length.

600

500

400

300

200

100

P1,

P2

[Pa

x103 ]

806040200

t [x10-3

sec]

Experimental P1

Experimental P2

Theoretical P1, P2

(a)

150

100

50

0

-50

-100

F [N

]

806040200

t [x10-3

sec]

Experimental Fa Theoretical Fa

(b)

Figure 16: Numerical and experimental actuator pressures (a) and force (b) for 0.5 A stepcoil current and 2 m tube length.

23

the valve dynamics. The additional 0.005 sec time delay for the 2 m tube case matches thevalue computed theoretically. The attenuation of the mass flow amplitude induced by thetube is reflected by the significantly larger (more than double) amount of time required toreach the maximum value of the output force, when the long tubes are used. In the second

-20

-10

0

10

20 x

[m

m]

0.50.40.30.20.10.0

t [sec]

Experimental piston position Theoretical piston position

Figure 17: Experimental and numerical piston position for 1.5 V and 5 Hz sinusoidal valvecurrent for Numatics 0750D02-03A cylinder.

experiment a sinusoidal command current was applied to the valve coil and the piston motionwas recorded. Figures (17) and (18) show the experimental and numerically simulated resultsfor the Numatics 0750D02-03A and Airpel E16 D 3.0 N pneumatic cylinders. In both casesthe agreement between the outcome of the experiment and the theoretical model is excellent.

5 Conclusions

In this article we developed a detailed mathematical model for a dual action pneumaticactuator controlled by a proportional, coil actuated valve. The proposed model is not onlyaccurate, but also sufficiently simple such that it can be used online in control applica-tions. Specifically, one can use the model to develop high performance force controller forapplications in robotics and automation. We developed a pneumatic cylinder model thatincludes the piston seals friction, the difference in effective areas on the opposite sides ofthe piston, the inactive volumes at the ends of the stroke and the connecting ports, and thedifference in the heat transfer characteristics for the charging and discharging processes of

24

40

30

20

10

0

-10

-20

x [

mm

]

0.50.40.30.20.10.0

t [sec]

Experimental piston position Theoretic piston position

Figure 18: Experimental and numerical piston position for 1.5 V and 5 Hz sinusoidal valvecurrent for Airpel E16 D 3.0 N cylinder.

the cylinder chambers. We introduced a new equation to account for the influence of thetubes that connect the pneumatic cylinder with the valve. Explicit formulas were derivedfor the tube resistance for laminar and turbulent flow. Valve dynamics, the nonlinearity ofthe valve effective area with respect to the coil current, and the nonlinear turbulent flowthrough the valve orifice were also considered. The results obtained by numerical simulationwere compared with the experimental data, and excellent agreement was found.

References

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[2] Andersen, B., 1967, The Analysis and Design of Pneumatic Systems, New York, JohnWilley & Sons, Inc.

[3] Arun, P.K., Mishra, J.K., and Radke, M.G., 1994, Reduced Order Sliding Mode Controlfor Pneumatic Actuator, IEEE Transactions on control Systems Technology, Vol. 2, No.3, pp. 271-276.

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[4] Ben-Dov, D., and Salcudean, S.E., 1995, A Force-Controlled Pneumatic Actuator, IEEETransactions on Robotics and Automation, Vol.11, No. 6, pp. 906-911.

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[8] Elmadbouly, E.E., and Abdulsadek, N.M., 1994, Modeling, Simulation and SensitivityAnalysis of a Straight Pneumatic Pipeline, Energy Conservation and Management, Vol.35, No. 1, pp. 61-77.

[9] Hougen J.O., Martin, O. R. and Walsh, R. A., 1963, Dynamics of Pneumatic Trans-mission Lines, Energy Conservation and Management, Vol. 35, No. 1, pp. 61-77.

[10] Hullender,D.A., and Woods, R.L., 1985, Modeling of Fluid Control Components, Pro-ceedings of the First Conference on Fluid Control and Measurement, FLUCOME ’85,Tokio, London: Pergamon Press.

[11] Liu, S., and Bobrow, J.E., 1988, An Analysis of a Pneumatic Servo System and Its Ap-plication to a Computer-Controlled Robot, Journal of Dynamic Systems, Measurement,and Control, Vol. 110, pp. 228-235.

[12] McDonell, B.W., Bobrow, J.E., 1993, Adaptive Traking Control of an Air PoweredRobot Actuator, Journal of Dynamic Systems, Measurement, and Control, Vol. 115, pp.427-433.

[13] Munson, B.R., Young, D.F., and Okiishi, T.H., 1990, Fundamentals of Fluid Mechanics,John Willey & Sons, New York.

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[18] Whitmore, S.A., Lindsey, W.T., Curry, R.E., and Gilyard, G.B., 1990, ExperimentalCharacterization of the Effects of Pneumatic Tubing on Unsteady Pressure Measure-ments, pp. 1-26, NASA Technical Memorandum 4171.

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