cascade position control of a single pneumatic artificial muscle

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Cascade position control of a single pneumatic artificial muscle–mass systemwith hysteresis compensation

Tri Vo Minh a,*, Tegoeh Tjahjowidodo b, Herman Ramon c, Hendrik Van Brussel a

a Mechanical Engineering Department, Division PMA, Celestijnenlaan 300B, B3001 Heverlee, Belgiumb School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singaporec Department of Biosystems, Kasteelpark Arenberg 30, B3001 Heverlee, Belgium

a r t i c l e i n f o

Article history:Received 1 May 2009Accepted 3 March 2010

Keywords:Hysteresis modelingPneumatic artificial muscleTracking position control

a b s t r a c t

The inherent hysteresis in a pneumatic artificial muscle (PAM) makes it difficult to control accurately theposition of the PAM’s end effector. This hysteresis causes energy loss and the area of the hysteresis loop isdependent on the amplitude of the motion and on the underlying causes of the hysteresis phenomenon.This means that if the hysteresis energy loss is properly compensated, a more accurate positioning wouldbe achieved. In this paper, the pressure/length hysteresis of a single PAM is modeled by using a Maxwell-slip model. The obtained model is used in the feedforward path of a cascade position control scheme, inwhich the inner loop is designed to cope with the nonlinearity of the pressure buildup inside the PAM,whereas the outer loop is designed to cope with the nonlinearity of the PAM dynamics itself. The exper-imental results show that position control of a single PAM–mass system with hysteresis compensation(HC) is effectively improved compared to a control without HC, and the control system shows highrobustness to load changes.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, pneumatic artificial muscles (PAMs) have many desir-able characteristics, such as high power-to-weight ratios, highpower-to-volume ratios, and inherent compliance and have there-fore been used in a wide variety of applications in humanoid ro-bots, prostheses and orthoses. Most of the PAMs used today arebased on the McKibben artificial muscle [4,7], which mainly con-sists of a rubber tube, a so-called bladder, and a surroundingsheath made of two helix inextensible fibers, a so-called braidedshell. This cylindrically-shaped tube is closed at both ends withcaps. When pressurized, the bladder tends to increase in volume(like a balloon), however, thanks to the non-extendable outsidesheath, the bladder is restricted to expand in radius but shortensaxially instead. If one end of the muscle (simply called a PAM inour paper) is fixed to a stationary support, the other end will exerta contracting force to the coupled load. The pneumatic energy inthe compressed air is thus transformed into mechanical work.When depressurized, the muscle returns passively. Hence, similarto a human muscle, a PAM is a unidirectional actuator (Fig. 1).The contracting force of a muscle with a certain diameter is a non-linear function of the input pressure and the length of the muscle.During a contraction/extension cycle hysteresis inherently occurs.

The high nonlinearity of the contracting force model and the com-plex hysteresis complicate the control problem. Two approaches todeal with this problem can be observed in literature; one usingsimplified models, the other using advanced nonlinear controlalgorithms.

1.1. Model accuracy

The contracting force model of the PAM is difficult to obtainprecisely. Most of the developed models are only approximationsof its real behavior. There are three models that have received rec-ognition in the PAM research community. The first model, devel-oped by Inoue, is based on an empirical approach [1]. The secondone was developed by Caldwell et al., and is based on the theoremof virtual work [2]. The last one was developed by Chou and Han-naford, and is based on geometric analysis [3]. With some restric-tions, the three models are quite related to each other. In order toreduce the discrepancy between the output contracting force pre-diction and the measurement, many investigators have been work-ing on the extensive assessment of the model accuracy. In papers[4–7], the authors took into account the effect of the nonzero wallthickness of the PAM, the non-cylindrical shape of the end parts ofthe PAM, and the inherent hysteresis. The model improvements arevaluable to get an insight to PAM designers, but they increase thecomplexity in terms of control, since the modifications involvemany unmeasurable parameters. Measuring and implementing

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* Corresponding author. Tel.: +32 (016) 32 14 44.E-mail address: [email protected] (T.V. Minh).

Mechatronics 20 (2010) 402–414

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parameters such as the contacting surface of braiding fibers ortheir friction coefficient to the control scheme in [4] is an example.Consequentially, simple models, as presented in [1–3], are morefrequently used in PAM control systems.

1.2. Hysteresis in PAMs

It is indicated above that hysteresis is inherently present in aPAM and that it is complicated to model. The cause leading to hys-teresis is assumed to be the friction, which exists between thebraided shell and the bladder, between the cords of the braidedshell, and the inherent hysteresis of the bladder. A few studies inliterature have dealt with PAM hysteresis modeling. Tondu and Lo-pez [4] added the hysteresis to the contracting force model as Cou-lomb dry friction in order to capture not only the static but also thedynamics of the output force prediction. Due to the increased com-plexity of the contracting force model by accounting for hysteresis,the implementation of this model is of limited use for control pur-poses. Davis and Caldwell [7], tried to interpret in detail the con-tacting surface of braided cords which characterizes thehysteresis. Their assessment showed a good agreement betweenthe measurement and the prediction of the static force, but also in-creased the complexity of the model. This work is valuable to pro-vide an insight to PAM designers. Although Chou and Hannaford[3] did not model the hysteresis, they generalized and tested someinteresting characteristics of the hysteresis such as quasi rate-inde-pendency and dependence on history (memory).

1.3. Control of PAMs

The model error of the contracting force, due to the complexhysteresis and the effect of the rubber relaxation [22], introduces

uncertainties in the PAM control system. To increase controlrobustness, many different control methods have been tried: fromconventional to nonlinear PID controllers, from learning controllersto robust control algorithms [10–28]. To get an insight, these refer-ence papers can be divided into two groups: one group is related tothe control of a single muscle [10–16], while the second group isdirectly related to the control of an antagonistic system, in whichtwo muscles are coupled to make up a rotary joint [17–28]. Care-fully considering the control objectives of these two groups, onecan see that there are some similarities between them. The pres-sure/length relationship in the tracking position control of the sin-gle muscle is somehow similar to the pressure difference/angulardisplacement control of the antagonistic system. The torque/angu-lar displacement control of the antagonistic system is comparableto the load disturbance control of the single muscle system. Thealgorithms developed for controlling the single muscle are applica-ble to controlling the antagonistic system. In fact, the control algo-rithms for a single PAM developed in [10,16] were applied in[17,27] respectively, dealing with antagonistic systems.

Obviously, each of the two muscles contributes hysteresis to theantagonistic system. In literature, this hysteresis was almost al-ways un-modeled and considered as part of the uncertainties ofthe contracting force model. This shortcoming was proposed tobe overcome by using pole placement methods [10,11,17], neuralnetworks control [22], variable structure control or sliding modecontrol [23–25], or using a combination of more advanced tech-niques such as nonlinear PID control [14], adaptive fuzzy model-based control [15], adaptive fuzzy logic siding mode control [26].

Fig. 2. The single muscle–mass system (SMu).

Fig. 1. (a) Structural layout of a PAM. (b) Description of the PAM working principle.

T.V. Minh et al. / Mechatronics 20 (2010) 402–414 403


Most of these studies are applied to tracking position problems. Afew authors [12,16,20] consider loading effects. Hildebrandt et al.introduced the hysteresis model developed in [4] in an inversemodel-based control algorithm [16], while Balasubramanian andRattan suggested using an offset for compensating the hysteresis[15].

In this paper we introduce a new deterministic approach: thehysteresis is characterized and modeled carefully, the associatedcontrol is therefore implemented as a conventional PID controllerwith feedforward hysteresis compensation. This study focuses onlyon the single muscle (SMu) system, in which the muscle drives avertically suspended mass, as shown in Fig. 2.

This paper is organized as follows. Section 2 derives the novelcontracting force model. Section 3 addresses the pressure/lengthhysteresis modeling. Section 4 presents the design of the cascadetracking position control. Section 5 gives experimental resultsand discussions. Section 6 brings the conclusion and future work.

2. The contracting force modeling

In order to derive the contracting force model in a straightfor-ward way, we need to establish two experimental setups. The firstsetup is the isometric setup as shown in Fig. 3a. The modeled mus-cle is put in the test-bed where the two ends of the muscle are fas-tened in two stationary supports. From this experiment, thecontracting force/pressure activation relationship at different

lengths of the PAM is found and shown in Fig. 3b. The force–pres-sure–length relationship resulting from this experiment is calledthe ‘‘isometric model”. The second setup is the ‘‘isobaric setup”as shown in Fig. 4a. The modeled muscle is stretched by anotherstronger muscle. During testing, the pressure in the test muscleis regulated at a constant value and different constant pressures

Fig. 3. (a) Photo of the isometric setup. (b) The contracting force is linearlyproportional to the pressure activation at different contraction ratios.

Fig. 4. (a) Photo of the isobaric setup. (b) The contracting force is nonlinear to thecontraction ratio at different pressures inside the muscle and appears to havedifferent values during contraction or extension (hysteresis).

404 T.V. Minh et al. / Mechatronics 20 (2010) 402–414


are selected. The proper pressure in the stretching muscle is pre-selected such that the test muscle with a given inside pressure ispulled and relieved in its permissible length or contraction ratio;at the smallest length the contracting force does not become neg-ative and at the largest length, the highest force does not rupturethe muscle. The experimental results of the isobaric setup areshown in Fig. 4b.

In these two setups, the test muscle is a FESTO Fluidic muscle(type MAS-20-200N) with an internal diameter of 20 mm and alength of 200 mm. The stretching muscle is a MAS-40-300N typewith an internal diameter of 40 mm and a length of 300 mm. Thedatasheet shows that these muscles have a maximum contractionratio of 25%. The PTE5151D1A pressure sensor from SENSORTECH-NICS was employed to measure the pressure. The air was suppliedvia a pneumatic 5/3 directional proportional valve (FESTO type ofMYPE-5-M5-010B). The length of the muscle was measured by alaser displacement sensor type PD6506002 from BAUMER ELEC-TRIC, while the contracting force was measured by using a load celltype DBBP-200 from BONGSHIN. All I/O information from/to theplant setup was processed by a 16-bit data acquisition cardDAQmx NI-6229 from National Instrument, which was embeddedin a real-time desktop PC. The control and measurement algo-rithms were developed based on LabView Professional Develop-ment System for Windows with the add-on LabView Real-TimeModule.

From the isometric and isobaric test results as shown in Figs. 3and 4b respectively, one can observe that the contracting forces inthe isometric test curves are linearly proportional to the pressureactivation. The slopes and intercepts of these straight lines arechanging for different contraction ratios. The contracting forcecurves are nonlinear function of the contraction ratio at differentgiven pressures in the isobaric test, and the hysteretic loops areclearly visible. This brings us to the following model of the con-tracting force (similar work is found in [8,9]):

Fisob ¼ Fisom þ Fhys ð1Þ

where Fisob is the measured static contracting force from the iso-baric experiment as shown in Fig. 4b, Fisom is the static force fromthe isometric model, and Fhys is the extracted force/length hystere-sis, resulting from the subtraction of Fisob from Fisom.

The isometric model of the contracting force is governed by thefollowing equation:

Fisom ¼ aðeÞP þ bðeÞ ð2Þ

Experimental results show that a(e) and b(e) take the form:

aðeÞ ¼X2

i¼0

Ciei and bðeÞ ¼X3

j¼0

Cjej ð3Þ

where a(e) and b(e) are the slopes and the intercepts respectively ofthe straight lines in Fig. 3b, Ci, Cj are the coefficients of the polyno-mial function, P the internal pressure of the muscle, and e is the con-traction ratio defined as the ratio of the difference between themaximum length lmax and the actual length l to the maximumlength of the muscle: e ¼ lmax�l

lmax

Substituting (3) into (2), the isometric model of the contractingforce can be rewritten as follows:

Fisom ¼ PX2

i¼0

Ciei þX3

j¼0

Cjej ð4Þ

The isometric model described in Eq. (4) is quite similar to theone found in [27], but for us it is just an isometric part in our novelmodel as shown in Eq. (1). The remaining part in this model is theextracted hysteretic force, which can be mathematically describedas we have shown in [9]. This study indicated that the hysteresis in

a PAM behaves like the friction in a two-object surface contact inthe presliding regime. This hysteresis is well described by a hyster-esis function with non-local memory. This type of hysteresis hasthe properties of quasi rate-independency and history dependency.The general form of this hysteresis, which is a nonlinear functionF(�) of the virgin curve f(e) and the hysteretic force Fhys(en) at thelast reversal extremum en, can be written as follows:

Fhys ¼ Fðf ðeÞ; FhysðenÞÞ ð5Þ

Substituting (4) and (5) into (1), one obtains the new form of thecontracting force model (6):

Fisob ¼ PX2

i¼0

Ciei þX3

j¼0

Cjej þ Fðf ðeÞ; FhysðenÞÞ ð6Þ

In the SMu system, we try to control the position with respect tothe pressure activation, while the force is dynamically changingdue to the acceleration effect on the load mass. In other words,the control input is the input pressure to the PAM which is inferredfrom the desired position, and the control output is the musclelength. The complete model derived in (6) contains the quasi-dy-namic term Fhys, which can be used to compensate for the hyster-esis in the position tracking control. However, written in suchform, the pressure is vanishing in Fhys. In fact, zero pressure activa-tion will lead to zero motion, and as a result there is no hysteresis.In order to show the pressure explicitly in Fhys, model (6) can berewritten in the following form:

Pisot ¼ Pisom þ Phys ð7Þ

where Pisot is the measured pressure in the isotonic test (see Section

3), Pisom ¼ðFisom�

P3

j¼0CjejÞP2

i¼0Ciei

the internal pressure P taken from the iso-

metric model (4), and Phys = F�1(f(e), Fhys(en)) is the extracted pres-sure/length hysteresis, experimentally resulting from thedifference between Pisot and Pisom.

Eq. (7) is extensively rewritten as:

Pisot ¼Fisom �

P3j¼0Cjej

� �P2

i¼0Cieiþ F�1ðf ðeÞ; FhysðenÞÞ ð8Þ

Comparing (6) with (8), one can infer that the force/length hys-teresis would be transformed to the pressure/length hysteresis.Some questions may arise here, such as: Does the pressure/lengthhysteresis have a similar behavior as the force/length hysteresis? Isit possible to do the conversion mathematically between them? Ifthey cannot be converted directly, can they be modeled or not? Theanswers are given in the following section.

3. Pressure/length hysteresis modeling

3.1. From the force/length hysteresis to the pressure/length hysteresis

As discussed earlier, the pressure activation is vanishing in thehysteresis term in Eq. (6), therefore the inverse hysteresis term inEq. (8) cannot be directly derived from (6). The pressure/lengthhysteresis term in (7) and (8) obviously occurs in an isotonic test.The isotonic test setup uses the same configuration as the SMu sys-tem depicted in Fig. 2. The muscle carries a certain mass and themanipulating pressure will activate the mass up and down. Differ-ent masses are tested and the set of test results for different loadsis shown in Fig. 5b, which are similar to the test results in [3]. Thisfigure shows that during moving up and down the pressure/lengthhysteresis inherently appears. The pressure activation in this setup,denoted as the isotonic pressure Pisot (8), not only contains the di-rect power conversion term Pisom but also the extra amount of pres-sure needed to overcome the friction during movement. This



additional pressure is the compensation term Phys and exists in theform of extracted pressure/length hysteresis. The pressure/lengthhysteresis therefore originates from the force/length hysteresis.This resulting hysteresis has also been explained in [27]. However,the pressure/length hysteresis can be visually explained based onthe isobaric test results shown in Fig. 5a. If we draw a horizontalline crossing the force/length hysteresis loops, the intersectingpoints can be used to rebuild the pressure/length hysteresis loopsin the isotonic test (Fig. 5b) while noting that the horizontal cross-ing line in the isobaric test implies a certain mass in the isotonictest .

The force/length behavior has been characterized in [8,9]. Pre-liminary tests showing the non-local memory and the quasi rate-independency characteristics of the pressure/length hysteresis aretreated in this paper. A clue for the non-local memory behavior ofthe pressure/length hysteresis is shown in Fig. 6. This figureshows the hysteresis loop 1-2-3-4-1 as the muscle carries 6 kgmass (equivalent to a force of 60N approximately). In order to ob-tain this loop, the activation pressure is selected such that duringmoving up and down the muscle length interval is in the permis-sible range, to avoid the muscle being directly stretched, and that

it is gradually changing so that the measurement is quasi static,to avoid the effect of mass acceleration. Following the upper halfloop 1-2-3 corresponds to moving up the mass, if the trackingpressure is reversed at point 2, the system will memorize point2 and when it goes to point 20, meaning that the mass is movingdown, the trace goes from 2 to 20. If the tracking pressure is againreversed at point 20, the system will memorize points 2 and 20

and when it goes back to point 2, meaning that the mass is mov-ing up again, the trace goes from 20 back to 2. Reaching point 2, ifthe tracking pressure is rising, the system will follow the upperhalf loop 1-2-3 instead of following the dotted curve 20-2-200. Itmeans that when the tracking pressure finishes 2-20-2 loop, thesystem will forget memory point 2. This behavior is called‘‘non-local memory”. The same behavior is found when followingthe lower half loop 3-4-1 corresponding to moving down themass. The rate independent characteristic of the hysteresis loopis shown in Fig. 7, in which the 0.2 Hz loop is compared withthe 0.05 Hz loop. Higher loop rates are difficult to test due tothe acceleration effect on the carrying mass. The two main char-acteristics of the pressure/length hysteresis resulting from the

Fig. 5. The force/length hysteresis (in an isobaric test) leads to the pressure/length hysteresis (in an isotonic test).

Fig. 6. The pressure/length hysteresis exhibits a non-local memory characteristic. Fig. 7. The pressure/length hysteresis exhibits a quasi rate-independentcharacteristic.



above analysis bring us to the conclusion that this kind of hyster-esis can be completely described by the hysteresis function withnon-local memory, which is the same solution as explained in[9,30,32]. Consequently, the force/length hysteresis term in Eq.(6) is hard to convert to the pressure/length hysteresis term inEq. (8), but it can be rewritten in another form such that it canbe mathematically modeled as:

Phys ¼ PðpðeÞ; PhysðenÞÞ ð9Þ

where P(�) is the nonlinear function of pressure/length hysteresisprediction, p(e) is pressure hysteretic function, namely the virgincurve, and Phys(en) is the hysteretic pressure at the last reversalextremum en.

Eq. (8) now takes the new form:

Pisot ¼Fisom �

P3j¼0Cjej

� �P2

i¼0Cieiþ PðpðeÞ; PhysðenÞÞ ð10Þ

Comparing (8) and (10), one obtains:

Phys ¼ PðpðeÞ; PhysðenÞÞ ¼ F�1ðf ðeÞ; FhysðenÞÞ

Hence there are two forms of the pressure/length hysteresis,and expressed in the form P(p(e), Phys(en)) this hysteresis can bemodeled. However, the hysteresis force is in turn vanishing in thisnew form.

3.2. Modeling the pressure/length hysteresis

The modeling work of the pressure/length hysteresis in this pa-per resembles the work of [9,31]. It is briefly described in the fol-lowing steps:

– Extracting the pressure/length hysteresis loop experimentally.– Shrinking the upper (or lower) half of extracted hysteresis loop

to get the virgin curve.– Picking up intuitively the segments which represent the Max-

well-slip elements, a kind of piecewise linearization of the virgincurve.

– Identifying the representing parameters for those selectedelements.

When drawing a number of horizontal lines from the bottom tothe top of Fig. 5 and aligning vertically the starting point and thefinal point when these lines are crossing through the force/lengthloops, one can obtain the displacement intervals at different loads.The heavier the loads the shorter the intervals and this agrees withthe experimental results shown in Fig. 8. In this figure, the loop-type curves are plotted from the isotonic experiment, but the sin-gle-line curves are rebuilt from the isometric model. At the sameload a loop-type curve can be subtracted from a single-line curvein order to obtain the pressure/length hysteresis loop, namely ex-tracted hysteresis. In order to reach the extreme lengths for mod-eling the pressure/length hysteresis, one can chose the isotonicloop corresponding to the lighter mass since the interval in suchselection should spread out over the working length of the muscle.However, in this way the extracted hysteresis cannot accommo-date the effect of the full-ranged loads, which are in forms of theforces that are vanishing in the model as discussed earlier. Bearingthis in mind, a number of different-load hysteresis extractions iscarried out and shown in Fig. 9.

We have just distinguished how to obtain the extracted pres-sure/length hysteresis loops which accommodate the load effect.The remaining steps in the modeling procedure described aboveare intensively exploited based on what has been seen in Fig. 9and shown in Fig. 10.

The pressure/length hysteresis loops at different carrying loadsare plotted in Fig. 10. These loops are bounded by the outer dotted-line loop. This bound is assumed to be a symmetric loop of the dou-ble-stretched virgin curve and the flipped-double-stretched virgincurve [8,9]. When the pressure–velocity is positive, the virgincurve can be shrunk from the double-stretched virgin curve, whilethe virgin curve can be shrunk from the flipped-double-stretchedvirgin curve when the pressure velocity is negative. After the virgincurve is extracted (the dashed line), the piecewise linearization isapplied by intuitively picking up the representative Maxwell ele-ments [9,29,31]. As presented in [32], four representative Maxwellelements are sufficient to capture and simulate the hysteresisbehavior. These elements are shown as asymptotic segments tothe virgin curve. Based on the coordinates of these segments, onecan calculate the representative parameters of each element bysolving Eq. (11):

Fig. 8. The isotonic test (loop-type curve) at different loads compared to theisometric model (single-line curve). Fig. 9. The extracted pressure/length hysteresis at different loads.



k1 þ k2 þ k3 þ k4 ¼ Ka ¼ a1=b1

k2 þ k3 þ k4 ¼ Kb ¼ a2=ðb2 � b1Þk3 þ k4 ¼ Kc ¼ a3=ðb3 � b2Þk4 ¼ Kd ¼ a4=ðb4 � b3Þw1=k1 ¼ b1

w2=k2 ¼ b2

w3=k3 ¼ b3

w4=k4 ¼ b4

ð11Þ

where ki, wi (i, j = 1, . . . , 4) are the stiffness and the saturation val-ues of the elements respectively and ai, bj (i, j = 1, . . . , 4) are thecoordinates directly measured on the graph.

The identified parameters are given in Table 1 and the schemeof the prediction of the output hysteretic pressure is shown inFig 11. This hysteresis model is applied in the feedforward pathof the control scheme that will be analyzed in the next section.

4. Tracking position control of the SMu system

Cascade control is a traditional control strategy to be applied tothe PAM system. Tsagarakis and Caldwell designed the inner loopto guarantee the torque while the outer loop is used to control

the position in case of controlling an antagonistic system [19]. Thiscontrol scheme is suitable for decreasing only the steady state er-ror, but not sufficient for handling the oscillation in the transientresponse, because the desired position is obtained after the con-trolled torque is attenuated to zero. In case of controlling theSMu system, Hildebrandt et al. proposed to use the cascadedscheme with the inverse model-based control, in which the innerloop is designed to control the force and the outer loop is designedto control the position [16]. This study also shows an improvementof the steady state error, but oscillations occurred during a stepchange of the load. The inverse model-based control, however, re-quires precise models of the system elements such as the controlvalve, the muscle, etc., which are difficult to obtain due to difficultyin air leakage modeling, the asymmetry in the charging/discharg-ing air flow, and particularly the hysteresis in the PAM.

A combination of a feedforward controller, which is used to lin-earize the nonlinearity of hysteresis behavior, and a PID controllerresulted in a significant improvement in tracking control of someother actuators [33,34], but it was rarely implemented in trackingcontrol of PAM actuators. In this paper we propose a novel controlscheme to deal with the tracking position control problem of theSMu system as shown in Fig. 12. In this control scheme, the innerloop is designed to cope with the building-up pressure problemand the outer loop with the feedforward path of hysteresis com-pensation is designed to cope with the highly nonlinear dynamicsof the muscle.

4.1. The inner loop

Assume that during charging and discharging of the air, theprocess is in-between the isothermal and adiabatic states. The

Fig. 10. Modeling of the extracted hysteresis based on the virgin curve of the bounded loop.

Table 1The identified parameters of the four representative Maxwell-slip elements.

Element 1 2 3 4

k 10 7.5 1.6 0.9w 0.05 0.075 0.056 0.081



relationship of the mass of air, the muscle volume and the internalpressure is governed by the polytropic gas law:

PVm

� �c¼ const: ð12Þ

Taking the total differential of Eq. (12), one can get:

_PV þ cP _V ¼ c_m

m

� �PV

For an ideal gas, we have also:

PV ¼ mRT ð13Þ

Substituting (13) into (12), the building-up pressure can be de-scribed by the following equation:

_P ¼ cVð _mRT � P _VÞ ð14Þ

where P is the pressure inside the muscle, V the volume of the mus-cle, m is the mass of air inside the muscle, c the polytropic expo-nent, R the universal gas constant, and T is the air temperatureinside the muscle.

The muscle volume varies during contraction/extension [27],thus one can obtain the volume as a function of the contraction ra-tio as follows:

V ¼X3

n¼0

Cnen ð15Þ

where Cn are the coefficients of the polynomial function.The mass flow rate through a non-ideal nozzle is governed by

the following equation:

_m ¼ PuCq0

ffiffiffiffiffiT0

T1

sif

Pd

Pu6 b ðsonic flowÞ ð16aÞ

and

_m ¼ PuCq0

ffiffiffiffiffiT0

T1

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

PdPu� b

1� b

!2vuut if

Pd

Pu� b ðsubsonic flowÞ

ð16bÞ

where Pu, Pd are the upstream and downstream pressure respec-tively, b is critical pressure ratio, _m is mass flow rate through thenozzle, C the flow conductance, q0, T0 the air density and air tem-perature respectively at standard conditions, and T1 is the air supplytemperature.

Eq. (16a) and (16b) can be rewritten in short form as follows:

m:¼ PuC

1ffiffiffiffiffiT1p w

Pd

Pu

� �ð17Þ

Fig. 11. Description of the predicted pressure output: four Maxwell-slip elements are intuitively selected, the output prediction of the extracted hysteresis pressure Phys is thesum of the individually contributing outputs Phys1,. . .,4 of these elements.

Fig. 12. The scheme of the cascade position control of the SMu system.



where w PdPu

� �is called the flow function, which is dependent only on

the pressure ratio.In order to regulate and manipulate the pressure inside the

muscle by using the proportional directional control valve, the flowfunction is frequently switching between charging and dischargingand the response of the mass flow rates are totally different be-tween these two processes. That is because during charging Pu = Ps

and Pd = P, whereas during discharging Pu = P and Pd = P0 (Ps, P0 arethe supply pressure and the atmospheric pressure respectively). Aschematic drawing of the building-up pressure is depicted inFig. 13 in order to explain graphically the relationships in Eqs.(14), (15), and (17). The building-up pressure is not only dependenton the nonlinearity and switching of the flow function, but also onthe muscle volume variation, the air temperature, the polytropicexponent and the leakage through the control valve as well. Thesedependencies are difficult to handle, the ‘PI’ controller is thereforeintroduced to close the loop. The ‘P’ control action is sufficient for

controlling such a first order system (Eq. (14)). The ‘I’ control actionis however needed to eliminate the steady state error due to theleakage. The ‘P’ and ‘I’ control gains are obtained by trial and error.

4.2. The outer loop

The dynamics of the muscle are described by Eq. (10), whichconsists of the direct conversion part (static part) and the hystere-sis part (quasi-dynamic part). Both are modeled, but the latter isdynamically changing according not only to the current positionbut also to the history of the last position reversal. If a certainamount of pressure needed to overcome the hysteresis friction isdynamically compensated, the position control of the SMu systemwill not experience the oscillation in the transient state since theposition is very sensitive to the pressure (showing in Fig. 5b thatthere are many possible pressure levels at the same position).

Fig. 13. Illustration of the high nonlinearity of the building-up pressure, which is dependent on the switching and nonlinear flow function, volume variation, un-modeled airleakage and the surrounding environment conditions as well.

Fig. 14. The muscle dynamics are online compensated by computing the hysteresis and feeding forward in the outer loop.



The pressure hysteresis compensation (HC) is therefore placed inthe feedforward path to reduce the control effort of the PI control-ler, which is used to close the position loop (Fig. 14). The desiredposition or contraction ratio ed will be converted to the requiredpressure Preq in order to serve the inner loop.

5. Results and discussions

Two main nonlinear problems in the SMu system have beenanalyzed: the nonlinearity of the building-up pressure and thenonlinearity of the muscle dynamics. For solving the former prob-lem, a PI controller is sufficient to regulate and track the requiredpressure for controlling the muscle position, because (i) the build-ing-up pressure is governed by a first-order behavior and (ii) theincreasing/decreasing pressure corresponding respectively to theshortening/elongating muscle adapts the damping of the pressurecontrol system thanks to the volume damper (Eq. (15)). By trial

and error, the parameters of the proportional gain k = 3 and theintegral time constant Ti = 0.003 s for this controller are selectedand kept constant during the tests. For solving the latter problem,the position loop is closed and a similar PI controller is used. Asusual, the ‘I’ action is used to eliminate the steady state error.The parameters of the proportional gain k = 0.03 and the integraltime constant Ti = 0.0009 s for this controller are selected and alsokept constant during the tests. However, due to the pressure/length hysteresis, at the same position or contraction ratio thereare many possible pressures to reach this position. Therefore, whenapplying a step change in the position, the SMu system easily oscil-lates around that new position. Fig. 15 shows the measured hyster-esis pressure which is captured at different positions or contractionratios by the developed model. The hysteresis pressure predictionagainst the position is fed forward to the outer loop of the SMucontrol system and gives a proper value to the total pressureneeded to reach the new position. The controller thus needs less ef-fort to correct for the hysteresis instead of using its effort to correctthe model discrepancy. The effectiveness of the hysteresis compen-sation in the SMu position control can be seen in Fig. 16. The SMusystem carries the 6 kg mass. A 0.2 Hz square wave is applied tomanipulate the muscle around the equilibrium position of185 mm length with an amplitude of 5 mm. When the hysteresiscompensation is switched off (without HC) (the top), the transientresponse becomes oscillating due to the ‘sponginess’ [2] of thepressure, as can be seen in the pressure loop response (the bot-tom). When the hysteresis compensation is on (with HC), thenew position is smoothly reached. An additional amount of extrapressure given by the online model-based computation can alsobe seen in the pressure control loop.

According to Repperger et al. [12], there exist two differentdynamics for contraction or inflation and extension or deflationin such a SMu system, which are dependent on the position. Thenext test therefore aims at investigating how the hysteresis modelis adequately adapted to the different equilibrium positions alongthe working range of the muscle. In fact, the input of the positioncontrol system is kept the same as in the previous test with the0.2 Hz square wave. Three different equilibrium positions165 mm, 175 mm, and 185 mm are chosen. At each equilibriumposition, four different excitation amplitudes 0.5 mm, 1 mm,2 mm, and 5 mm are applied. The load is changed to 24 kg. The testresults are plotted in a 4 � 3 array as shown in Fig. 17. At the bot-tom of the figure, the muscle shapes are put close to the equilib-rium position in order to illustrate the volume change. At theright side of the figure, different amplitudes are indicated. Forthe first two waves approximately in each subplot the HC isswitched off, and at rest the HC is active. With such an arrange-ment, it can be observed that:

– The HC takes effective action at any equilibrium position andwith any excitation amplitude.

– At the same equilibrium position, the higher the excitationamplitude, the less oscillating the position response (follow eachcolumn, top to bottom). This is due to the volume damping andthe flow friction.

– At the same excitation amplitude, the higher the equilibriumposition, the more oscillating the position response (follow eachrow, left to right). This is due to the volume damping against theequilibrium position. As shown at the bottom of the figure, thehigher the equilibrium, the smaller the volume of the muscle.

– At the highest equilibrium position, the excitation towardsshortening of the muscle length, the position response is moreoscillating than that of the excitation towards elongating ofthe muscle length. This is due to the asymmetry of the flowfunction, which is more visible when the volume damping israther small (the most upper right subplot).

Fig. 15. A measured pressure/length hysteresis is captured by the developed model.

Fig. 16. The effectiveness of the hysteresis compensation in the SMu positioncontrol.



Consequentially, the highest equilibrium position is the mostsensitive and the smaller the excitation amplitude, the moreoscillating the position response will be. This forms the basisfor performing the next test, which aims at investigating howthe hysteresis model is adequately adapted to the different loadsat the most sensitive equilibrium position and amplitude. In fact,three different loads 6 kg, 24 kg, and 50 kg are selected for thetest. The equilibrium is at 185 mm and the excitation amplitudeis 2 mm. The wave type and the wave frequency are kept un-changed. The test results are plotted and shown in Fig. 18, inwhich the first row shows the position response and the secondrow shows the response of the pressure loop. From the left tothe right, one can see that the heavier the load, the less oscillat-

ing the position response. This is due to the decrease of the nat-ural frequency of the mass–spring system. It is more convincingif we look at the pressure loop response. A heavier load needs ahigher pressure. A higher pressure will not only increase flowdamping due to flow friction but also the muscle stiffness dueto the relationship in Eq. (6). The control system with activatedHC performs an adequate action for different loads. However, theeffect of the asymmetry of the flow function is not removedafter the HC takes action. For example, in the case of the 6 kgload, the manipulating pressure interval of about 0.8 to 1.2 baris needed to move around the equilibrium position. For movingup the mass, the charging rate of air in this pressure intervalgoes very fast because the flow is in the sonic state. Sequentially,

Fig. 17. Step responses of the SMu position control under a square wave of 0.2 Hz. The test set is done with the same load, but different excitation amplitudes and differentequilibrium positions.

Fig. 18. Step responses of the SMu position control under a square wave of 0.2 Hz. The tests are performed with the different loads, but with the same excitation amplitudeand equilibrium position. The bottom plots show the pressure loop responses corresponding to each load.



the small overshoot in the position response still occurs whenthe muscle is shortened.

These overshoots occur not only at a high rate of charging butalso at a high rate of discharging. Fig. 19 shows the overshoot inboth cases when two different loads are examined. The lighter load(6 kg) is used to examine the high rate of charging. The heavierload (50 kg) is used to examine the high rate of discharging. Thehigh rate of discharging happens when the SMu system carries aheavy load and elongates from the shortest equilibrium position.This is why the equilibrium position of the heavy load is at175 mm, whereas the equilibrium position of the light load is at185 mm, which is based on the previous sensitivity analysis. Theovershoots, when shortening and elongating the muscle corre-sponding to the lighter and heavier loads, are visibly seen in Fig19, and indicated that the discharging process is more damped

than the charging process. The reason for this is related to theasymmetry of the flow function and its switching property. Forthe details, damping of the flow can be illustrated in Fig. 20; (a)the control signal is applied such that the valve is fully openedfor charging and discharging, (b) the corresponding pressures arebuilt up in the muscle, (c) the curves of the building-up pressurefor charging and discharging are put at the same coordinate tocompare, and (d) the rate of charging and discharging are showna part form each other. The charging rate runs very high in a shorttime after which it saturates, while the discharging rate goes downgradually.

In summary, we can state that for all changes of the load andequilibrium position, the SMu position control with the hysteresiscompensation has shown an efficient performance. The effects ofthe asymmetric flow cannot be removed. However, these effectson the position control system are not significant, and they comefrom the inner pressure control loop, where the controller param-eters are kept constant for all tests.

6. Conclusion

This paper addressed the difficult problem of modeling hyster-esis of PAMs and the use of the resulting models in accurate posi-tion control of single PAM systems. This hysteresis is the pressure/length hysteresis that features a rate-independent and historydependent characteristics, and is well described by using the Max-well-slip model. The developed model has been implemented intothe cascade PI–PI control architecture. The overall control systemwith the feedforward hysteresis compensation has shown a consis-tent performance regardless of the choice of equilibrium positionand load changes. By using conventional PI controllers only,together with the simple but deterministic hysteresis compensa-tion, the tracking position control of a SMu system is effectivelyimproved. These results inspire us to apply PAMs in the develop-ment of humanlike robots.

Fig. 19. Two ‘sensitive’ equilibrium positions are selected to test for the asymmet-ric effect of the building-up pressure.

Fig. 20. The asymmetry of the building-up pressure during charge and discharge.



Acknowledgment

The author gratefully acknowledges the DGDC (Directorate-General for Development Cooperation), of the Belgium Govern-ment, for awarding a scholarship to the first author and for fundingthis research.

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cascade position control of a single pneumatic artificial muscle

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