ISSN 1751-8644 Tracking control of pneumatic artiﬁcial ......model and applied the H1 theory to design the controller for PMAs [24]. In this paper, we propose a new PMA model used

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Published in IET Control Theory and ApplicationsReceived on 27th October 2009Revised on 23rd June 2010doi: 10.1049/iet-cta.2009.0555

ISSN 1751-8644

Tracking control of pneumatic artificial muscleactuators based on sliding mode andnon-linear disturbance observerK. Xing J. Huang Y. Wang J. Wu Q. Xu J. HeKey Laboratory of Image Processing and Intelligent Control, Department of Control Science and Technology, HuazhongUniversity of Science and Technology, Wuhan, People’s Republic of ChinaE-mail: huang_ [email protected]

Abstract: The dynamic properties and non-linear control of the pneumatic muscle actuator (PMA) wereinvestigated in this study for use in a specially designed hand rehabilitation device. The phenomenologicalmodel of PMA was established in the lower pressure range applicable for hand rehabilitation. Theexperimental results show that PMA’s characteristics can be approximated by piecewise functions. In order toimprove the performance and robustness of control for accurate trajectory tracking, a sliding mode controlbased on non-linear disturbance observer (SMCBNDO) was designed. The simulation and experimental resultsdemonstrated that the model and the sliding mode control achieved the desired performance in tracking adesired trajectory within guaranteed accuracy. The work indicates that the model and the non-linear controlproposed in this study can be applied in PMA-driven hand function rehabilitation devices requiring lowerpressures.

1 IntroductionThe challenge of rehabilitation robot design is to develop arobot suitable for task-oriented rehabilitation therapy, thatis, the robots should be compliant to spastic reactions oftenseen from neurologically impaired patients, therefore safeand not inducing further injury or pain during motorfunction training. As a naturally compliant actuator withlower cost, the pneumatic muscle actuator (PMA) is widelyused in rehabilitation robots [1–4].

The pneumatic muscle (PM) can be made similar tohuman skeletal muscles in size, weight and power output.In contrast to traditional motor actuators, it possesses manyadvantages, such as lower cost, light weight, compliance,and very high power/weight and power/volume ratios [5].However, the application of this kind of compliantactuators has its challenges. Compared to electric motors,the PMA has slower response in force-generation, non-linear parameters depending on the load, position andspeed. The complex non-linear dynamics of the PM makes

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devices driven by PMAs difficult to control. This hasrestricted widespread use of PMA-driven devices in the past.

In recent years, the research on the modelling of PMA andits properties has been undertaken by several researchers.Mathematical models of a PMA can mainly be categorisedinto two classes: the theoretical models and thephenomenological models. Some theoretical models ofPMA can be found in [6–12], which are derived from thelaw of energy conservation. In [6], the relationship betweenthe pressure and the force of a PMA was analysed bySchulte. Chou and Hannaford developed a complex modelrelating the geometric structure and the contractile forceoutput of PMA [7, 8]. The generated equations werefunctions of the input pressure, initial length, diameter ofthe PMA, braid thread angle, thread length and thenumber of thread turns. Later, they proposed a static modelbased on static length–tension experiment [8, 9]. Tonduand Lopez [10] improved the theoretical model byincorporating a muscle contraction ratio. They comparedthe theoretical model to human skeletal muscle and

IET Control Theory Appl., 2010, Vol. 4, Iss. 10, pp. 2058–2070doi: 10.1049/iet-cta.2009.0555

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analysed their similarities in the force–length relationship anddifferences in the force–speed relationship. Klute andHannaford presented another model that includes a non-linear, Mooney–Rivlin mathematical description of theactuator’s internal bladder [11]. However, these theoreticalmodels describe PMA behaviour based on quasi-static stateswithout incorporating explicitly temporal information. Thisapproach limits its application for real-time control because itis too complex in structure and requires too many parametersthat are difficult to obtain during experimentation. Thus,some researchers adopted the phenomenological model todescribe the PMA’s dynamics. Colbrunn and colleaguesproposed a PMA’s phenomenological model similar to Hill’smodel of skeletal muscles [12]. In order to facilitate thedesign of controller, Repperger et al. modelled the PMA as astiffness-visco model consisting of a spring element anddamping element arranged in parallel [13]. Later, Reynoldsimproved the model proposed by Repperger to construct anew phenomenological model consisting of a contractileelement, a spring element and a damping element [14].These models allowed for control design and practicalapplication of PMA.

Based on a linearised model of PMA, proportional-integral-differential (controller) (PID) control was used toimplement the control of a seven-degree of freedom(DOF) robotic arm [15]. However, the controller is onlyeffective for a particular operational region for a linearmodel. To deal with the complex non-linear dynamics ofthe PMA, researchers have attempted non-linear controlmethods such as adaptive pole-placement control, H1

control, fuzzy backstepping, variable structure control (VSC),neural network control etc. [16–25]. Caldwell studied thecontroller design and application of the PM control systems.In his study, the control of PMs was explored via adaptivepole-placement technique [5, 16, 17]. Repperger et al.designed a gain scheduling method based on the PM modelapproximated from experiment data [13]. They alsoinvestigated gain-scheduling H1 control [18], adaptivebackstepping [19], fuzzy backstepping [20] and VSC [21].Among these control methods, satisfactory performance canbe achieved by using adaptive backstepping or VSCmethods. Unfortunately, the conventional VSC control isnot suitable for practical application because of thechattering problem. Artificial neural network (ANN)-basedapproach was applied to the control of a five-joint robot armdriven by PMAs [22]. By using the ANN learningalgorithm, a non-linear multidimensional function can beapproximated well and the position could be controlled to asatisfactory level. Cai and Yamaura applied the sliding modecontrol (SMC) technique to design a robust tracking controlfor a manipulator system driven by a pair of agonist/antagonist PMA [23]. Osuka et al. linearised the PMAmodel and applied the H1 theory to design the controllerfor PMAs [24].

In this paper, we propose a new PMA model used in ahand rehabilitation device and controlled by a sliding mode

T Control Theory Appl., 2010, Vol. 4, Iss. 10, pp. 2058–2070oi: 10.1049/iet-cta.2009.0555

algorithm. The dynamics of PMA was described by aphenomenological model consisting of a contractileelement, a spring element, and a damping element inparallel. The parameters of the model were fitted by theexperimental data generated by piecewise characteristics inthe different ranges of pressure, especially the lowerpressure region applicable for the hand functionrehabilitation. Considering the bounded modelling errors,the sliding mode approach was adopted. In order toimprove the effectiveness and robustness of the control, wedesigned the SMC based on a non-linear disturbanceobserver (SMCBNDO) that provided robust performancein tracking a desired trajectory with a guaranteed accuracyregardless of the modelling uncertainties and perturbation.

The rest of this paper is organised as follows. In Section 2we describe the background work leading to this research. InSection 3 we derive the model for the PMA used in the handrehabilitation device. In Section 4, a general SMC isdesigned based on the PMA’s model. Section 5 shows theimprovement of the controller by inserting perturbationobserver in sliding mode method. Section 6 presents thesimulation and the experimental results of the closed-looptracking performance of the controller. Finally, conclusionsare given in the last section.

2 BackgroundIn previous studies, a rehabilitation robotic hand driven bytwo PMAs, shown in Fig. 1a, was designed to provide

Figure 1 Hand rehabilitation robot and its actuators

a Prototype of a wearable device for hand rehabilitationb Mckibben PMA made manually

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active assistance for repetitive physical therapy to improve theintrinsic motor functions of the affected hand [4]. The robothas three distinct degrees of freedom at the thumb and thefingers. The embedded sensors are employed to obtainposition and force information for feedback control andquantitative evaluation of task performance. The robot isdriven by two PMAs anchored on the forearm attachment.One of the PMAs is directly connected to the lever of thethumb. In order to magnify the limited travel of thePMAs, a travel transform pulley is designed and locatedabove the metacarpophalangeal joints. Another PMA isconnected with a cable passing through the travel transformpulley and connected with the front lever. PM is madefrom inexpensive materials, such as natural latex rubber forthe inner bladder covered by a polymer-based braid(Fig. 1b). The outer and inner diameters of the used latexrubber tube are 12 and 10 mm, respectively. The maximumcontraction ratios of both PM are about 25%.

To realise the trajectory tracking control, the conventionalPID and fuzzy PID controller were implemented andevaluated in previous studies [25, 26]. To realise themodel-based control method in further study, amathematical model of the PMA and the robot must beestablished. The objective of this study is to develop andverify the model and then to design a robust controller forPMAs under the modelling uncertainties and theperturbation.

The model formulation followed the work of Repperger et al.[14] with two major differences: the material for the PMs andthe working pressure ranges. The parameters in the formulationwere identified by experimental measurement. The range ofload and pressure in Reynolds’ work were different from thatrequired for the hand device. The PM will have differentbehaviour in different pressure ranges.

The static model of the PM in our study is similar to thatproposed by Klute and Hannaford [11] (we did note that asign change was warranted in the second term of Klute andHannaford’s force equation) (see equation at the bottom ofthe page)

where F is the predicted force, P is the pressure, R0 is theresting state radius, L0 is the resting state PM length, l1 isthe rate of the actuator’s instantaneous length and restinglength L0, C10 and C01 are two Mooney–Rivlin constants,B is the length of helical thread, Vb is the bladder volumeand N is the number of turns a single helical thread makesabout the diameter of the actuator over its length. B, Vb


and N can be calculated by the following equations

Vb = pL0tk(2R0 − tk)

N = b sin u0

2pR0

B = L0

cos u0

In this model the hysteresis loop caused by the friction is notconsidered. The equation is too complex to be used in thereal-time control. Whereas the intuitive model can bederived from the PM’s geometry and the material, althoughthe accurate value of each parameter is difficult to obtainand some may change after a period of the PM usage.

Since the dynamics of PMA is pressure and lengthdependent, we simulated the model to obtain therelationships between the predicted force F and the rate l1

with different pressures. The simulation parameters wereselected based on the actual PM used in the device:L0 ¼ 0.2 m, C10 ¼ 100 kPa, C01 ¼ 150 kPa, R0 ¼ 6 mm,bladder thickness t0 ¼ 1.5 mm and the resting state braidangle is 228. The simulation results for the specific PMunder different pressures are given in Fig. 2.

Fig. 2 shows the force–length relationship of the PMpredicted from the model simulation. The result suggests anear linear relationship between the generated force and thelength of PM under a constant pressure. However, the

Figure 2 Relationships between predicted force F and thel1 in different pressures

F = P3(l1 L0)2 − B2

4N 2p

{ }+ Vb

1

2L30l

31

{4(C10 + C01)L2

0(−1 + l41) + 4L6

0(−1 + l1)l21(1 + l1)(C10 + C01l

21)

[−4N 2p2R20 + L0(−1 + l2

1)]2

− 4L40(C10 + C01l

41)

−4N 2p2R20 + L2

0(−1 + l21)− L4

0l41[C10 + C01(−1 + 2l2

1)]

N 2p2R20

}⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭


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slope changes as the pressure changes, but not in a linear orproportional fashion. As the pressure increases, the initialslope decreases gradually first, then increases gradually.Within the range of parameters for specific PM used in ourhand rehabilitation robot there is obviously a difference inthe stiffness characteristic from that proposed by Reppergeret al. [14].

3 Dynamic modelThe dynamic behaviour of a PM pulling a mass againstgravity has been modelled as a combination of effects fromnon-linear friction, spring and contraction components.The coefficients corresponding to these three elementsdepend on the input pressure of the PM. The equationsdescribing the dynamics of a PM are approximated by

Mx + B(P)x + K (P)x = F (P) − Mg (1)

K (P) = K0 + K1P (2)

B(P) = B0i + B1iP (inflation) (3a)

B(P) = B0d + B1d P (deflation) (3b)

F (P) = F0 + F1P (4)

where M is the mass, g is the acceleration of gravity, x ¼ 0corresponds to the fully deflated position and P is the inputpressure. Notations K(P) and B(P) indicate the pressure-dependent coefficients for the spring and the damping,respectively. F(P) is the effective force provided by thecontractile element. The damping coefficient depends onwhether the PM is inflated and deflated.

The experimental device is shown in Fig. 3 forexperimentally testing the model. The length change of thePM was measured by a low friction linear variabledifferential transformer (LVDT) attached to the lower endof the PM. In addition, there was also a pressure sensorintegrated in the electric proportional valve (EPV) toprovide the pressure feedback. Data were collected by amultichannel real-time data acquisition card (AdvantachUSB-4716).

Figure 3 Experimental apparatus to measure the force –length relationship under different pressures

T Control Theory Appl., 2010, Vol. 4, Iss. 10, pp. 2058–2070i: 10.1049/iet-cta.2009.0555

3.1 Experimental process

In order to meet the requirement of hand rehabilitationdevice reported previously [4], pressures in the range of150–300 kPa and loads in the range of 0–40.0 N wereapplied. The frequency of data acquisition was 1250 Hz.The experimental process was arranged as follows:

Step 1: Calibration of LVDT and pressure sensor. Beforestarting the experiment, the LVDT was powered on andpreheated for 15 min. The initial end position of the PMwas selected as the reference point. The end position of thePM was changed by regulating the pressure of the EPV toa specific value of 0.10, 0.15, 0.20, 0.25, 0.30 or 0.35 MPa.When the pressure reached the steady state, the output ofLVDT and the pressure sensor were recorded by the dataacquisition card. The real end position of the PM wasrecorded manually according to the side scale. Thefunctional relationships between voltage signal of the sensorand real value can be obtained by using least-squarestechniques to fit the recorded data to linear models.

Step 2: Estimation of the effective force F(P) provided by thecontractile element and the spring coefficient K(P). From (1)the static model of the PM can be expressed as follows

F (P) K (P)[ ] 1

−x

[ ]= Mg (5)

To provide independent estimations of K and F as functionsof pressure, the pressure was set by EPV to different desiredconstant levels with the same load and the displacement anddata were recorded when the pressure reached steady state.The procedure was repeated for pressure range from 150 to300 kPa in steps of 10 kPa. The same experiment wasrepeated for different loads of 0, 4.90, 11.76, 19.60, 24.50,29.40, 34.30, 38.50 and 40.0 N. According to (5), theposition and load data in the same pressure were used toestimate F and K by least-squares techniques. Differentpressure P generated different F and K. According to thechanging trend of F and K with P, the functionalrelationship between K, F and P was established.

Step 3: Estimation of the damping coefficient B(P). From (1)the dynamic model of the PM can be expressed as follows

B(P) K (P)[ ] x

x

[ ]= F (P) − Mg − Mx (6)

To provide estimations of B as a function of pressure in thecase of inflation, we reduced the weight from a large weightstack suddenly after the pressure had reached a steady state,and then the PM would continue contracting for anadditional distance. Following the procedure reported in[14], the load was reduced by 1/3 of value in the steadystate. This is a typical step response process. However, thesame procedure cannot be performed in the case ofdeflation. Owing to the uncertainty of initial end position,

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we can hardly obtain an ideal lengthening (deflation) processof the PM by adding a load. So we reduced the PM’s pressurefrom a high pressure suddenly after the PM had reached asteady contracted state, and then the PM would extend anadditional distance. Based on the F obtained in step 2, thedamping coefficient B(P) during PM inflation anddeflation can be fitted according to (6). The perturbationand release experiments were implemented in differentpressures of 150, 200, 250 and 300 kPa and different loadsof 4.90, 14.70, 29.40 and 40.0 N.

Step 3 of the experiment demonstrated that the EPV in thecontrol loop had an effect on the system dynamics. Fig. 4 showsthe different step responses in two different cases when thepressure of the steady state was 150 kPa and the load wasreduced from 14.7 to 9.8 N suddenly. The dashed linerepresents the dynamic response when the EPV wascontained in the control loop, whereas the solid linerepresents the dynamic response after removing EPV.Obviously, the overshooting was higher and the time to reachsteady state longer when the EPV was in the control loop.Therefore the response without EPV was used to estimatethe damping coefficient B(P) of the PMA by (6). Theacceleration x was obtained by taking the second derivative ofthe position x. The noise contained in signal x had a greatinfluence on the calculation of acceleration. A finite impulseresponse low-pass digital filter was designed to remove thenoise from x, and subsequently the phase compensation hadto be considered. The filtered signal x is shown in Fig. 5.The same filtering process is applied to the data during thePM deflation.

3.2 Experimental result

The experimental result shows that there is an approximatelylinear relationship between the effective force F(P) withrespect to the pressure P (see Fig. 6). The F–P (∗) denotesthe value of F in a special pressure, which is estimated byleast-squares techniques applying to the position and loaddata in same pressure. The line in Fig. 6 denotes theestimated functional relationship between F and P.

Figure 4 Comparison of different step responses in twocases


Relationship between the spring coefficient K(P) and thepressure P shown in Fig. 7 presents a piecewise linearfunction. The K–P (∗) denotes the value of K in special

Figure 5 Result of filtered signal with a phasecompensation

Figure 6 Relationship of F and P

Figure 7 Relationship of K and P


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pressure, which is estimated by least-squares techniquesapplying to the position and load data in same pressure.The line in Fig. 7 denotes the estimated functionalrelationship between K and P.

Finally the value of B(P) for contraction and release indifferent pressures and different loads are shown in Figs. 8aand b, respectively. Experimental results show that it isdifficult to use a single function to describe the relationshipbetween the damping coefficient B and the pressure P.Fortunately the values of B are very small and has littleinfluence on the whole dynamic model, a limited band isused to describe B(P) and the centre of the band is a lineestimated by using least-squares techniques.

Figure 8 Relationship of B and P

a B(P) during PM contraction/inflationb B(P) during PM release/deflation


In summary, the PM model is given by

Mx + B(P)x + K (P)x = F (P) − Mg (7)

where

F (P) = F1P + F0 = 0.00071974544P − 103.35633 (8)

(see (9))

B(P) = B1iP + B0i = −0.0001245P + 52.08 + 14

(inflation) (10)

B(P) = B1d P + B0d = 0.00009022P − 3.19 + 26

(deflation) (11)

4 Traditional SMCAmong several robust non-linear control techniques, SMCstrategy has attracted quite a number of researchers [27–31].Typically, SMC is used to compensate against themodelling uncertainties and the unknown disturbances.SMC might be an ideal method for the PM manipulationbecause the PM model is usually non-linear and time-varying, which necessitates some robust control strategy.For the class of non-linear systems, sliding mode controllerdesign provides a systematic approach to the problem ofmaintaining stability and consistent performance in the faceof modelling uncertainty.

From the dynamic equation (7)–(11), we can obtain

x = f (x, x) + b(x, x)P (12)

where

f (x, x) = 1

M(F0 − Mg − B0x − K0x) (13)

b(x, x) = 1

M(F1 − B1x − K1x) (14)

where

K0 = Km0, K1 = Km1 0 , P ≤ 0.2483 MPaK0 = Kn0, K1 = Kn1 P . 0.2483 MPa

{

B0 = B0i , B1 = B1i if x . 0 (inflation)B0 = B0d , B1 = B1i if x , 0 (deflation)B0, B1hold if x = 0

⎧⎨⎩

Because of the imperfect knowledge of coefficients F, K, B, itis obvious that item f (x, x) and b(x, x) in (12) are usuallyuncertain. Whereas, the uncertainties in both models can

K (P) = Km1P + Km0 = −0.0570256932P + 15010.710 0 , P ≤ 0.2483 MPaKn1P + Kn0 = 0.008583153863P − 259.787 P . 0.2483 MPa

{(9)

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be bounded by known continuous functions with respect to xand x. The aim of control is to force the PMA’s position x(t)to track a desired trajectory xd (t) in the presence of the modeluncertainties in f and b. Assuming that we have an estimate off (x, x) for f and an estimate of b(x, x) for b, the followingequation and inequality are satisfied

| f − f | ≤ F ∗ and b = (1 + D)b (15)

where D satisfies the following inequality

|D| , B∗ (16)

Consider the sliding surface as

s = ˙x + lx (17)

where x = x(t) − xd (t) is the tracking error and l is a scalardesign parameter. Then the SMC law is given as [27]

P = b−1

(− f + xd − l ˙x − k sgn(s)) (18)

where

k ≥ (F ∗ + B∗|− f + xd − l ˙x| + h)/(1 − B∗) (19)

h is a positive constant. In order to decrease the chattering,the term sgn(s) is replaced by sat(s/G), G is the thicknessof a ‘boundary layer’, which is a neighbourhood of slidingsurface. So the sliding control law in this paper is given by

P = b−1

(− f + xd − l ˙x − k sat(s/G)) (20)

For the ‘inflation’ and ‘deflation’ cases, we designed twoSMC controllers with the same structure but differentfunctions f and b generated by the changed coefficients.Therefore the whole system becomes a typical switchingcontrol system. Each subsystem of the switching system isstable because the standard SMC controller designprocedure is followed. As pointed out by Hespanha [32], aswitching system is still stable if all its subsystems are stableand the dwell-time of each subsystem is sufficiently large.In our study, only low-frequency switching cases areconsidered, for example, the step response and the slowsinusoidal trajectory tracking problem. Therefore the wholeswitching system is supposed to be stable althoughsubstantial proof is not given.

5 SMC based on the non-linearperturbation observer design5.1 Controller design

SMC is used to compensate for the modelling uncertaintiesand the unknown disturbances. This control strategy can bedesigned to be globally stable and robust despite thepresence of the perturbations if their upper bounds areknown in advance. However, it may be difficult or


sometimes impossible to obtain these upper bounds. Theidea of utilising an online perturbation estimator to resolvethis issue appears in some literatures [33–37].

Chattering is a major drawback of the traditional SMC.The main cause of chattering comes from the switch gaink, which is mandatory for suppression of the disturbance.One possible approach to reduce the gain k is to use afeedforward compensation for the disturbance, which canbe realised by the disturbance observer. A non-lineardisturbance observer based on control (NDOBC) approachfor non-linear systems under disturbances was proposed byChen [33]. This technique has also been applied to dealwith parameter variation and unmodelled dynamics,whereby it improves the robustness of control system.Unmodelled dynamics is generated from the modellingprocedure in which the parameters of non-linear PM areestimated by fitting a linear model. In addition, the frictionmust be contained in the experimental apparatus followingthe PM moving. Both the unmodelled dynamics and thefriction can be integrated in the non-linear disturbance. Inthis study, SMCBNDO is designed to ensure that thePMA can track a desired trajectory regardless of themodelling uncertainties and disturbance. The systemfunctional diagram with SMCBNDO is presented in Fig. 9.

We assume the non-linear disturbance is d; the dynamicmodel equation (12)–(14) can be rewritten as

x1 =x2

x2 =1

M[F0+F1P −Mg− (B0+B1P)x2− (K0+K1P)x1+d]

⎧⎨⎩

(21)

where x1 = x, x2 = x. So

x = f1(X ) + g1(X )P + g2(X )d (22)

Figure 9 SMC based non-linear disturbance observer


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where

X =x1

x2

[ ]; f1(X ) =

x2

1

M(F0 − Mg − B0x2 − K0x1)

⎡⎣

⎤⎦;

g1(X ) =0

1

M(F1 − B1x2 − K1x1)

[ ]; g2(X ) =

01

M

[ ]

(23)

The basic idea in the design of the observers is to modify theestimation by the difference between the estimated outputand the actual output. Since (22) can be written as

g2(X )d = X − f1(X ) − g1(X )P (24)

A disturbance observer is proposed as

˙d = L(X − f1(X ) − g1(X )P − g2(X )d ) (25)

Define an auxiliary vector z = d − p(X ), where z [ R2. Thenon-linear observer gain L is defined as L ¼ (∂p(X )/∂X ). Thus

z = ˙d − p

= L( − f1(X ) − g1(X )P − g2(X )(z + p(X )) (26)

Let d = d − d . In general, there is no a prior information aboutthe derivative of the disturbance d. When the disturbance variesslowly relative to the observer dynamics, it is reasonable tosuppose that d = 0, we have

˙d + Lg2(X )d = 0 (27)

Choose the gain L as a constant matrix, that is

L = [ c1 c2 ] (28)

where c1 . 0, c2 . 0. In this case, p(X ) is given by

p(X ) = c1x1 + c2x2 = c1x + c2x (29)

Substituting (28) and (29) into (26), the disturbance observercan be designed as

d = z + p(X )

z = − c1x + c2( f + bP) + c2

M(z + c1x + c2x)

( )f (x, x) = 1

M[F0 − Mg − B0x − K0x]

b(x, x) = 1

M(F1 − B1x − K1x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

(30)

Considering (20) and (21), the SMC law-based disturbanceobserver can be deduced as

P = b−1 −f + xd − l ˙x − d

M− k sgn(s)

( )(31)


where

k ≥ F ∗ + B∗ −f + xd − l ˙x − d

M

∣∣∣∣∣∣∣∣∣∣+ Dmax

M+ h

( )/(1 − B∗)

5.2 Stability analysis

Although the disturbance observer is adopted to estimate thenon-linear disturbance, there is still the disturbanceestimation error d caused by the uncertainties of functionsf, b, the measurement errors of the displacement x andspeed x in (30). The uncertainties and the measurementerrors of these parameters can be thought as the boundedfunctions. Thus, we assumed that the following inequalityis satisfied

|d | ≤ Dmax (32)

Choose the Lyapunov function as

V = V1 + V2 = 1

2s2 + 1

2d

2(33)

where V1 = (1/2)s2, V2 = (1/2)d2. According to (13), (14),

(17) and (21), we have

s = (x − xd + l ˙x) = f + bP + d

M− xd + l ˙x

( )(34)

Substituting (31) into (34), we have

s = f + bb−1 −f + xd − l ˙x − d

M− k sgn(s)

( )

+ d

M− xd + l ˙x (35)

From (15), we have

bb−1 = (1 + D) (36)

Substituting (36) into (35), then

s = f − f + D −f + xd − l ˙x − d

M

( )

− (1 + D)k sgn(s) + d

M(37)

Therefore

V 1 = ss = s f − f + D −f + xd − l ˙x − d

M

( )(

−(1 + D)k sgn(s) + d

M

)(38)

When ‖B∗‖ ≤ 1, considering the inequality (15) and (16),

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the gain k satisfies the following inequality

k ≥ F ∗ + B∗ −f + xd − l ˙x − d

M

∣∣∣∣∣∣∣∣∣∣+ Dmax

M+ h

( )/(1 − B∗)

≥(| f − f | + |D| −f + xd − l ˙x − d

M

∣∣∣∣∣∣∣∣∣∣

+ Dmax

M+ h

)/(1 + D) . 0 (39)

When h is a strictly positive constant, according to (38) andinequality (39), it is clear that the above V 1 is negative-definite which is given by (see (40))

Substituting (23) and (28) into (25), we have

˙d = [ c1 c2 ]

([x1

x2

]−[

x2

1

M(F0 − Mg − B0x2 − K0x1)

]

−[

01

M(F1 − B1x2 − K1x1)

]P −

[01

M

]d

)

= c2 ×(

x − f − bP − d

M

)(41)

From (21), we have

x = f + bP + d

M(42)

According to (41) and (42), we have

V 2 = −d ˙d = −d c2 x − f + bP + d

M

( )( )= −c2

d2

M

(43)

Thus, when the inequality (15), (16), (32), ‖B∗‖ ≤ 1 andc2 . 0 are satisfied, we have V 1 ≤ 0, V 2 ≤ 0, V ≤ 0.Namely the system is globally exponentially stabilised andthe estimate disturbance d converges to the non-lineardisturbance d.


6 Simulation and experimentalresults6.1 Simulation

The simulation objective is to demonstrate the robustness ofthe proposed controller. In our simulation, we let the loadM ¼ 1 kg, and Table 1 gives the evaluation of PMcoefficients and three sets of coefficients (C1, C2 and C3)used for the simulations. The parameters F(P), K(P), B(P)of the PM model were randomly chosen from a uniformdistribution within +20% of the evaluation values. Thefriction was considered as the disturbance in the simulationsystem. The friction considered is Coulomb and Viscousfriction, given by

d (u) = z sign(u) + ku (44)

The parameters in the simulation are given by

z = 0.5541; k = 0.0676

The performance of proposed controller was evaluated bysetting a step and a sinusoidal trajectory tracking problem.The simulations for step experiments using models withcoefficients in C1, C2 and C3 were carried out first. Theamplitudes of step response are 15, 5 and 30 mm,respectively. The simulation interval time was selected as8 ms. Let the simulation parameters l ¼ 150, G ¼ 2,h ¼ 0.2, F ∗ = 0.3| f |, B∗ ¼ 0.3, Dmax = 0.1|d | and thecontrol gain k satisfies the inequality (39). By using a

Table 1 Coefficient sets used for the simulation

Coefficient Evaluation C1 C2 C3

F0 (×102) 21.0336 20.986 21.0812 21.188

F1(×1026) 719.75 690.96 748.54 827.712

K01(×104) 1.501 1.35 1.652 1.726

K11(×1022) 25.703 26.0 25.406 26.558

K02(×102) 2.598 2.28 2.916 2.987

K12(×1022) 0.858 1.0 0.716 0.986

B0i 52.08 65.0 39.16 59.89

B1i(×1026) 2124.5 2110 2139 2143.17

B0d 23.19 25 21.38 23.66

B1d (21026) 90.22 88.5 91.94 103.75

V 1 = s f − f + D −f + xd − l ˙x − d

M

( )+ d

M− (1 + D)k

( )≤ −hs s ≥ 0

V 1 = s f − f + D −f + xd − l ˙x − d

M

( )+ d

M+ (1 + D)k

( )≤ hs s , 0

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(40)


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saturation function with a boundary layer, the chattering ofSMC was significantly reduced. Whereas, there was a staticerror which results from the parameter G. If the parameterG decreases, the static error will decrease whereas thechattering becomes stronger. In the reality, small G cannotbe selected because it leads to a large switch control gain k.Three comparative step simulation results in Fig. 10 showthat there is a smaller static error by using SMCBNDO.The desired sinusoidal trajectory is given by

x = 0.015 sin(0.5p(t − 1)) + 0.015 (45)

In the sinusoidal trajectory tracking simulation, we supposethat l ¼ 150, G ¼ 10, h ¼ 0.2, F ∗ = 0.5| f |, B∗ ¼ 0.5,Dmax = 0.1|d | (chosen from trials and errors). Thesimulation results for SMC and SMCBNDO using theparameter set C1 are shown in Fig. 11. It is obvious thatthe tracking performance of the system using an SMC isnot satisfactory when there are modelling uncertainties andperturbations. As shown in Fig. 12, by increasing the F∗ to

Figure 10 Step-input control simulation result

Figure 11 Sine trajectory tracking result (F∗ ¼ 0.5| f|,B∗ ¼ 0.5 for SMC)


0.9| f | and B∗ to 0.9, the tracking error was reducedsignificantly. Although a higher control gain k can reducethe tracking error, it may cause stronger chattering, whichis unacceptable in an actual system. When using ourproposed SMCBNDO method, the non-linear disturbancecan be approximately estimated by the disturbance observer.The bounds of unknown uncertainties are then reducedsignificantly. Thus, even a small k can achieve a satisfactorytracking performance. The tracking error and the controlgain k in the simulations of SMC and SMCBNDO areshown in Figs. 13 and 14, respectively.

6.2 Experiments

According to the simulation result, the experiments usingtwo different loads (no-load and M ¼ 1 kg) were alsoimplemented on the apparatus. The results of the step andthe sinusoidal trajectory tracking experiments are shown in

Figure 12 Sine trajectory tracking result (F∗ ¼ 0.9| f|,B∗ ¼ 0.9 for SMC)

Figure 13 Sine trajectory tracking error (F∗ ¼ 0.9| f|,B∗ ¼ 0.9 for SMC)

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Figure 14 Control gain for the sine trajectory tracking(F∗ ¼ 0.9| f |, B∗ ¼ 0.9 for SMC)

Figure 15 Step trajectory tracking experiment results

a No loadb M ¼ 1 kg


Figs. 15 and 16, respectively. The experimental results alsodemonstrate that better performance can be achieved whenapplying an SMCBNDO but a conventional SMC in thetracking control problem of PMA. In the experiments, thecontrol parameters were selected as

l = 150, G = 5, h = 0.2, F ∗ = 0.2| f |, B∗ = 0.2,

Dmax = 0.1|d |

7 ConclusionWe propose a more advanced model for the PMA togetherwith the estimation approaches of its parameters. Based onthe work of Reynolds and Repperger, we studied the PMAin a different pressure range, especially in the lowerpressure. The model demonstrates good approximation ofthe non-linear characteristics of the PMs. In order tocompensate remaining uncertainties in the model as well as

Figure 16 Sine trajectory tracking experiment results

a No-loadb M ¼ 1 kg


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the friction, a non-linear disturbance observer is adapted inthe control structure. Based on the proposed model ofPMA, an SMC and SMCBNDO are, respectively,designed to realise the tracking control. Simulations andthe experiments were implemented to compare theperformance of the two methods. The results show that theSMCBNDO can estimate the non-linear disturbance ofthe system and is a more effective than a conventionalSMC for the PMA.

8 AcknowledgmentsThis work is being supported by Hi-tech Research andDevelopment Program of China under Grant2007AA04Z204 and Grant 2008AA04Z207, and in partby the Natural Science Foundation of China under Grant60674105, 60975058, 60874035 and in part by the NaturalScience Foundation of Hubei Province under Grant2007ABA027.

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ISSN 1751-8644 Tracking control of pneumatic artiﬁcial ......model and applied the H1 theory to design the controller for PMAs [24]. In this paper, we propose a new PMA model used