NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

Trajectory Tracking Control of a PneumaticMuscle System Using Fuzzy Logic

Kishore Balasubramanian and Kuldip S. RattanDepartment of Electrical Engineering

Wright State University, Dayton, OH 45435krattan(6Dcs.wriuht.edu

Abstract - Pneumatic Muscle (PM) system was first developedby McKibben in the 1950's as soft actuators in artificial limbsand became commercially available in the 1980's. They havesince been used as actuators in high-tech robotic applicationsand in physical therapy for functional recovery since they areextremely safe in human presence compared to electric andhydraulic actuators. A high power to weight and power tovolume ratios make the PM a very light yet powerful actuator.However, PM is highly nonlinear in nature due to itsconstruction and mechanical properties and hence, they aredifficult to control using a linear controller. Fuzzy logic is agood nonlinear modeling approach, since it uses fuzzy rules tohandle nonlinearities. A fuzzy logic based feedforwardcontroller (inverse dynamics) and feedback hnearizing controlschemes are proposed in this paper. Controllers are designedusing data obtained from the PM system, and the design doesnot require a mathematical model. The PM parameters canchange over time and varying operating conditions. Hence, anadaptive fuzzy algorithm is used to tune the fuzzy models tocapture the parameter changes. The proposed control schemesare tested for its trajectory tracking capabilities and are foundto yield excellent results.

I. INTRODUCTIONModem actuator technology stresses safety as a primary

concern, especially in human presence. Actuators used inhuman assistive devices such as in prosthesis, robots etc.requires 'soft' actuators, which would be safe to use inhuman presence. Traditional electrical and hydraulicactuators could prove dangerous during failure, expellingfluids or metal fragments. This gave rise to pneumaticmuscle actuators [1]. A Pneumatic Muscle (PM) system wasfirst developed by McKibben in the 1950's as soft actuatorsin artificial limbs and became commercially available in the1980's [2]. They have since been used as actuators in high-tech robotic applications, in physical therapy for functionalrecovery and for strength augmentation devices involvinghumans, where the devices have to be self-contained andcarried long distances. A high power/weight andpower/volume ratio makes the PM an ideal candidate overtraditional mechanical and hydraulic actuators. This ratio isabout five times higher when compared to conventionalactuators. Control issues, especially dealing withnonlinearities and position changes due to load variation,have caused researchers problems since the invention of thePM [2]. High compliance resulting from the compressibilityof the gas, high nonlinear elasticity of the muscle bladderand uncertain mechanical properties of the PM constructionmakes the PM difficult to control using a linear control

scheme. The objective is to design an efficient and robustcontrol scheme which can cope with the nonlinear PMsystem to produce superior tracking capability. In thispaper, a fuzzy logic based inverse model feedforward andlinearzing control schemes are proposed. An adaptive fuzzycontrol scheme is also proposed.

II. PNEUMATIC MUSCLE SYSTEMThe PM consists of three main components: a rubber tube, asheath, and a control valve as shown in Figure 1. The tube isenclosed and clamped within a sheath (1.25 inch plasticcoaxial cable material attached with hose-clamps). Thesheath is the component that imparts a cylindrical shape tothe PM and is a vital component to cause contraction andrelaxation of the PM, which in turn causes the change inlength of the muscle. The valve is usually a two wayproportional valve controlled by an electric signal, and isused to control the air pressure inside the muscle [2]. Therelaxed length of the muscle model is approximately 140mm between adapters. When the volume of gas or pressureinside the PM increases, it creates an expansion of therubber tube along the radius. This radial expansion creates acontraction in the linear direction. Thus, by controlling thevolume of gas inside the PM, the movement of the musclecan be controlled. The minimum and maximum length ofthe PM depends on the sheath characteristics. Thismovement is what makes the muscle useful as actuators inseveral applications. The PM's behavior depends on theparticular sheath characteristics, the elastic element'sproperties, instantaneous pressure and length, and also onwhether the muscle is contracting or relaxing. A change intemperature can cause change in gas volume and alsostretch the rubber and this in turn causes change in length.Even a change in load can cause a change in length of thePM [1].

Figure 1. PM Construcion. Elastic Tube (top), Elastic tube and sheath(middle, sheath unraveled to show detail), completed PM (bottom) [2].

0-7803-9187-X/05/$20.00 02005 IEEE. 472

For modeling the PM, it is assumed that the muscle ishanging from a fixed point, and has a fixed mass M hangingfrom it. The mass is assumed to be constant with timewhereas pressure, position and velocity vary with time. Themodeling of the system is carried out by considering thesystem as a spring, damper and a contractile force elementin parallel. Figure 2 shows the free body diagram of themodel, where x is the linear displacement in mm, K is thespring coefficient, B is the damping coefficient and consistsof two phases Bc and Br, for contraction and relaxationphases respectively, P is the applied pressure in Kilopascals(kPa), Fc is the contractile force, Mass M is in Kg and g isin mm/s2. The model is based on data for the pressure range0-620 kPa (0-90 psi) which causes an elongation rangingfrom 0-140mm. It should be noted that for simulationpurposes, the value ofB is assumed to be the same for bothphases and was fixed as Bc. This assumption simplifies themodel and also mimics the behavior of an actual PM systemclosely [2].

Kx [ + Dx Mass t

MxxF Kx BxMg

trNgM x

Figure 2. Model showing the free body diagram

The model equation is derived from the free bodydiagram using the dynamic equation, E F= M*a,and are given by [2]:

Fe-Mg-Ma-B X-K x=O,

where Bc = 1.01+6.91*10-3* p

Br= 0.6-8.03*104*PK = 5.71+0.0307*PFc = 2.43P-1.29*10-3*p2Weight W = M*g

[N.s/mm][N.s/mm][N/mm][N][N]

The model does not account for line ripple (the gas lineexpansion due to pressurization) and static friction. Themodel developed is used (solely for simulation purposes) toobtain input/output data, since access to a real PM systemset-up was not available.

III. FUZZY LOGIC MODELING AND CONTROLFuzzy logic incoxporates the idea of imprecision in the realworld based on fuvzzy rules or if-then statements. A fuzzylogic model/controller can be designed by simply obtaininginput/output data from the system and does not requiremathematical parameter estimation. In this paper, the

proposed controller models are designed using system data.A fizzy logic model has the structure as shown in Figure 3.

Figure 3. Structure of a Fuzzy Logic Model

Fuzzy models can be derived by a number of methods basedon data from the system. A Least Squared Error (LSE)approach is used in this paper and is explained with anexample to obtain an inverse dynamics model for the PMsystem is determined.

The PM system has a single input pressure and two outputs,position and velocity. The inputs to the inverse model areposition and velocity and the output is the pressurecommand. The first step is to obtain training data from thePM system in the form of input-output data sets. The LSEmethod can be demonstrated as shown in equation 1.

I W2 ~ 1 L 14~ina)l4 2a

=1 wt1 $ st F V I-F Rs1 VccnCsniau Vcc2t Me-be5h.P V iator R. :

where n is the number of input fuzzy sets, and m is thenumber of training data sets obtained from the PM system.One set of training data (i.e. pressure command, positionand velocity of the system) is taken and the position andvelocity data are fuzzified using equally spaced fuzzy setsfor both inputs). Thus, for a fuzzy model with 11 fuzzy setsfor both position and velocity, there are two membershipvectors with 11 membership values each. Each row of thematrix contains all possible combinations of themembership values of position and velocity, i.e. the firstelement of the membership vector for velocity is multipliedwith the entire membership vector for position, followed bythe second element of the membership vector for velocitybeing multiplied with the entire membership vector forposition and so on. Thus, for a system with 11 fuzzy setseach for position and velocity, there are 121 combinationsof the membership values for each data set. This is followedby building a matrix called the membership matrix bytaking all m data points obtained from the PM system. TheLSE error scheme can now be represented as: [Command] =

473

(1)

lot

[mem matrix] * [rule vector] as shown in equation (1),where command is the input to the actual system (i.e.pressure for a PM) and is a column vector, mem_matrix isthe membership matrix built using combinations of themembership vectors, and rule_vector is a column vector andis unknown. This rule_vector when determined will providethe rules describing the inverse dynamics of the PM system.The rules can now be obtained as: [rule_vector] = pinv(mem-matrix)*[Command]; where pinv stands for pseudoinverse. The membership matrix is usually a non-squarematrix and hence the need to find the pseudo inverse. Thus,for an inverse model with 11 fuzzy sets for position andvelocity, the rule vector obtained is a 121 by 1 vector. Therule vector has to be converted into a matrix form toimplement the inverse model. The rules of the inversemodel can now be obtained by considering the first 11elements of the rule vector as the first column, the next 11elements as the second column and so on and so forth. Eachrule or weight obtained represents the center point of theoutput fuzzy set, and hence in this example there are 121output fuzzy sets. This method has the drawback of beingcomputationally intensive. For a large training data set, thescheme requires building a very large membership matrix,and then determining the matrix inverse which createsmemory problems. An alternate way to determine the rulevector is to use a recursive procedure, which uses asequential method instead of batch training to compute therules. Let the i-th row of the membership matrix be definedas aiT and the i-th element of the input vector be bIT, then therule vector Wp can be calculated using the sequentialformula [9]:

WI1 +Si+ 1 a i+l ( b6 j+}I &lk lwx

Sii -=Si ai. I. at+ Si

si-1+ ai+lTSiai+1

i =0,1, .... p-l.where p is the number of data points, Si is called thecovariance matrix and the final rule vector is Wp. The finalrule vector is obtained after all the p data points are covered.To bootstrap the algorithm, the following initial conditionscan be used: Wo = 0 and So = yI, where y is a positive largenumber (e.g. 100000) and I is the identity matrix of size n,where n is the product of the number of fuzzy sets in eachinput. Thus for a two input system with 11 rules each, S willbe of size 11*11=121. This method is recursive in nature,and hence does not have large memory requirements [9]. Inthis paper, the recursive least squares algorithm is usedinstead ofbatch LSE estimation as a training algorithm.

errors and achieve better overall stability as shown in Figure4. Designing the inverse model requires capturing thedynamics of the system as closely as possible to achievesuperior tracking.

Figure 4. Inverse dynamics feedforward control scheme.

Another method to control a nonlinear system islinearizing control or control law partitioning scheme thathas been widely used in controlling robotic systems [3].Since the PM is a highly nonlinear system, the concept oflinearizing control can be extended to control the PMsystem as shown in Figure 5. The scheme consists of twoparts: a model-based portion and a servo-based portion. Themodel-based portion cancels all the system dynamics byfeeding back the dynamic terms of the system. In the case ofthe PM system, this would be a feedback consisting of theforce term (B x+K x), friction and other unmodeleddynamics. Since a fuzzy model is data derived, the data willinclude all these forces, unlike a mathematical model whereonly the dynamic parameters included in the model equationare included. After canceling the weight of the hangingmass, the system now appears as a unit mass to the servo-portion and can track a trajectory by providing the necessaryacceleration.

Figure 5. Linearizing Control Scheme.

IV. PROPOSED CONTROL SCHEMESInverse dynamics model as a feedforward controller is oneof the commonly used schemes to control dynamic systems.The inverse model acts as a dynamics inverse of the system,thus generating the control commands necessary to track agiven trajectory. The feedforward controller is generallyaccompanied by a feedback controller to eliminate tracking

For the PM, the coefficients B and K are nonlinearfunctions of the applied pressure, and hence the dynamicfeedback term is nonlinear especially at higher pressureranges [2]. This makes the precise estimation of thelinearizing portion using a mathematical ifunction evenharder. Hence, a fuzzy model is used to implement the

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model-based portion of the controller. The command that isgenerated as a result of this scheme is a force command.This force command cannot be used directly in controllingthe PM, as the input to the muscle must be a pressurecommand. Therefore, it is necessary to convert thegenerated force command into the corresponding pressurecommand. This relation is hard to determine mathematicallysince the contractile force, Fc, is a second-order function ofthe pressure P. Hence, a fuzzy inverse model is designed toprovide the corresponding pressure command for a givenforce command. Since the position is a function of theapplied pressure, the force K x can be approximated asfunction of the position of the PM as opposed to thepressure. Similarly, the velocity is a direct indication of theforce B x. Therefore, the model for the linearizing feedbackfunction (B x+K x), would be a function of the position andvelocity of the system.

Once the system dynamics are cancelled by the model-based portion and the weight of the mass is cancelled, thesystem behaves as a unit mass and should track any desiredtrajectory of an acceleration command. However, in mostcases, perfect cancellation of the dynamics is not possibleand this results in tracking errors. The linearizing feedbackmodel usually contains certain modeling uncertainties dueto inaccuracy in parameter estimation and unmodeleddynamics. Hence, partial cancellation should be acceptableif the linearizing feedback model manages to reduce theextent of nonlinearity to an acceptable level. Theacceleration profile which assumes the system to be unitmass may drive the system in the wrong direction if thenonlinearities are not cancelled. The position of the systemis a double integral of acceleration, and an erroneousacceleration command when integrated twice wouldproduce greater tracking error. Hence, a position andvelocity feedback is included in the servo-part, and a properchoice of the gains Kp and Kv as shown in Figure 5stabilizes the system and also produces the desired response.This feedback loop also facilitates better functioning of thefuzzy linearizing model by continuously eliminating smallerrors before they eventually grow larger over time. Hence,in all further discussions and simulation results, it isassumed that a feedback loop consisting of position andvelocity feedback is always present.

V. ADAPTIVE FUZZY CONTROLIn the case of a PM system, the system change over timecan occur due to continuous use of the PM, which canstretch and heat up the rubber material and create a changein the spring constant of the muscle. The position of the PMcan change with change in temperature, since the system ispneumatic. Load variations can be another cause that cancreate position changes. All these factors can contribute to afixed structure fuzzy model to be imprecise and henceineffective in controlling the PM system. A change in thesystem's dynamics can hamper effective control, since thefeedforward and linearizing controllers cannot completely

cancel the system dynamics. This calls for a change in thestructure or weights (which are the center points of theoutput fuzzy sets) of the fizzy model with change in systemparameters. Thus, an adaptive scheme is introduced, whichwould adjust or tune the fuzzy models over time in case achange in system parameters should occur. The adaptivecontrol scheme can be implemented by extending therecursive LSE algorithm to learn the new system parametersonline by introducing a factor X. This factor X is known asdecay factor since it is used to forget the old data andupdate the fuizzy model based on new data derived from thesystem. The recursive LSE algorithm introduced earlier inequation (2) is modified in equation (3) by including thedecay factor [9].

Wi4-1| = W (birL-ai,irWf )

l Si+ai~a-iiari J (3)1+tI ss ..

X + a,+ JS , 21+1

The decay factor X decays old values by giving morepreference to recent data, and hence it forgets the past. Thevalue of X is chosen between 0 and 1. Usually the value isset very close to 1 (0.996 for example). The smaller thevalue of X, the faster the decay rate. Very small values of Xshould be avoided since it would cause numericalinstability.

VI. IMPLEMENTATION RESULTS AND DISCUSSION

A. Inverse Dynamics Model ResultsA pressure profile with various position and velocity rangesis introduced to the PM to obtain input/output data sets fromthe PM system, and is trained to obtain an inverse dynamicsmodel. The response to a test trajectory for the LSE trainedfuzzy inverse dynamics model is shown in Figure 6.

L .LcE 4C L

n tit O /0 5 I10

D ~~~~~~~~~ER R O R

0 5 1 ~~~015 20 25 30 Is

Figure 6. Trajectory tracking and error profile using LSE fuzzy model with11 fuzzy sets.

The tracking performance can be improved by using greaternumber of input fuzzy sets, which will obviously yieldgreater resolution in control. However, this will in turnincrease the overall computational time since there are agreater number of fuzzy sets and, as a result, greater numberof rules. This factor will be critical when we adapt the

475

J

, 1,1,

3 0 1 5\11 I/

E 3 . .

controller on-line, since a larger number of fuzzy sets wouldrequire tweaking a lager number of output weights.There is usually a limit to the maximum number of fuwzzysets that can be utilized to improve resolution. The choice ofa very large number of fuzzy sets, without sufficienttraining data to cover the entire range of the sets woulddeteriorate the controller performance.

B. Linearizing Feedback Controller ResultsThe second scheme proposed is the control-law-partitioningor the linearizing feedback control. The two fuzzy modelsused in this control scheme are the force term (B i +K x)with inputs as position and velocity, and the model for forceto pressure conversion. Both models are trained using LSEalgorithm with 11 equally spaced input fuzzy sets. The LSEscheme can be recursively implemented and hence it iscomputationally less demanding especially with largenumbers of training data. To design the model-based portionor the linearizing part of the controller, training data whichis position and velocity versus the force term (B i +K x) isobtained. Similarly, data is derived from the PM system tomodel the force-pressure term in the control scheme.

The input to the model-based portion is an accelerationprofile of the desired trajectory, since the system is seen as aunit mass from the servo-portion in an ideal condition. Aposition and velocity feedback are also included toeliminate tracking errors and imprecision in the fuzzymodel. The feedback gains are determined as follows. PMsystem generally tracks signals up to a frequency ofapproximately lHz [1]. The natural frequency of the PMsystem, (in ( 2*pi* f ) is thus found to be approximately 6rad/sec. The position feedback gain Kp is taken as on2 andhence equals 36. Velocity error constant Kv is calculated tobe 2*sqrt (Kp) = 12. Figure 7 shows the trajectory trackingcapability of the linearizing control scheme. The responseindicates that the linearizing feedback control is effective incanceling the dynamics of the system, and helps in trackinga desired trajectory. It can be seen from Figure 7 that theerror stays within satisfactory limits. The fuzzy modelapproximated the non-linear system dynamics, and thusaided in obtaining the desired tracking capability. Thiswould have been a cumbersome task with a non-linearmathematical model. The fuzzy controller is designedexclusively using data from the system, and this takes careof any other nonlinearity such as friction, stiction, deadband etc. This approach saves the designer a lot of time andeffort when compared to approximating the systemdynamics using a set of mathematical equations.

L 40 _i

C 30'

h. _ 0 20 4(

I/ 11

0 4 o so * w 12T I11E( In Sea)

Figure 7. System tracking using a linearizing control scheme to a desiredtrajectory.

C. Adaptive Control ResultsTo demonstrate the adaptive scheme, the PM systemparameters are altered, but the fuzzy model is unchanged.The constant value in the spring coefficient K of the PMsystem is changed from 5.71 to 5.0, the damping ratio B isslightly changed from 1.01 to 0.7 and the load is variedfrom 68 kg to 72 kg. This resulted in a PM system that isless stiff, less damped and carries a heavier load and thuscauses inaccuracy in control. The fixed structure fuzzycontroller when used in the feedforward control scheme isunable to track a desired trajectory. It should be noted thatthe feedforward scheme is implemented without a feedbackloop to demonstrate the effect of system changes and theinclusion of the adaptive controller. The online adaptive partis now introduced which continuously monitors the systemand updates the fuzzy model weights. The performance ofthe feedforward control scheme with an adaptive controlleris analyzed and is shown in Figure 8. The response of thesystem without the adaptive algorithm clearly demonstratesthe inefficiency of the fixed structure controller ingenerating the desired response, by creating steady-stateerror and velocity error. The effect of including an adaptivecontroller can also be seen in Figure 8 and that the steady-state position error is eliminated. The adaptive controlleralso learns new velocity data during the initial transientphase of the trajectory, and thus eliminates velocity errorduring the latter phase of the transient response. Trackingerror is eliminated by the adaptive part over time, bymaking changes to the fuzzy structure with respect tosystem changes.

120.

; wn"out ~~~~~~~AdWlr.00 -,

8" D Ft-P

0 610 16~~~~i 20 '25r;En s)

Figure 8. Performance of the fixed structure and adapted fuzzy feedforwardcontroller

476

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The fuzzy model for (B ic+K x) in the linearizing feedbackcontroller also needs to be adapted to system parameterchanges, or else cancellation of the nonlinear dynamics ofthe system is not at an acceptable level and causes trackingerror. This is shown in Figure 9, where the PM parametersare varied by the same amount as the feedforward scheme,but the linearizing model (B xc+K x) is fixed-structure. Thetracking error is easily visible from Figure 9. This is thenfollowed by an online adaptive controller response thatchanges the model weights in accordance with parameterchanges of the system.

70 -- -- - -~~~~~~~~-- .Hi

so~ --

401

0 0 10 15 20 25

Figure 9. Response of the linearizing control scheme with/without adaptingthe model based portion.

VII. EFFECTS OF DECAY FACTOR X ON ADAPTIVECONTROLLER PERFORMANCE

For the adaptive control scheme, the decay factor Xdetermines the rate at which the weights of the controllerare altered with respect to system parameter changes. Asmaller value of X causes faster decay, but there is always alimit to the minimum value of X that ensures stability.Results for the adaptive feedforward controller are shown inFigure 10, where the best tracking was obtained with k =

0.995. To investigate the effect of X on the adaptivefeedforward controller, k. is varied from 1 to 0.993. It shouldbe noted that the controller is implemented without afeedback loop for clear illustration of the effects of X.System parameter changes are 10% alteration in mass,damping (B) and spring constant (K). Figure 10 shows theeffects of the reduction in the value of X. While X = 0.995yielded the best performance, reducing the value to 0.993deteriorates the tracking and further reduction in the valueof X leads to instability. This suggests the fact that theweights of the fuzzy model cannot be altered beyond acertain rate without causing stability problems. A properchoice of X is thus an essential requirement for the adaptivecontrol scheme.

"120

A -.bda 100 '

LV --d NY H

AC So~~ ~ ~ ~ ~ ~ N

TIME (In See

Figure 10. Adaptive feedforward controller response for various values of

VIII. CONCLUSIONA feedfoward control and linearizing feedback

control schemes are proposed for a nonlinear PM system,designed using fuzzy logic models derived using data fromthe system. The control schemes are tested for its trajectorytracking capabilities and found to yield good results, but achange in PM system parameters seriously hampers thecontroller performnance. Hence, a fuzzy adaptive algorithmis introduced to make changes to the fixed model fuzzycontroller to incorporate parameter changes in the systemover time. The adaptive controller is tested for itseffectiveness by altering the PM system parameters. Resultsprove that the adaptive controller is highly effective inmaking changes to the fuzzy models and thus ensuringexcellent

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