Research ArticleTracking Control of a Leg Rehabilitation MachineDriven by Pneumatic Artificial Muscles Using CompositeFuzzy Theory

Ming-Kun Chang

Department of Mechanical and Computer-Aided Engineering St Johnrsquos University No 499 Section 4 Tam King RoadTamsui District New Taipei City 25135 Taiwan

Correspondence should be addressed to Ming-Kun Chang mkchangmailsjuedutw

Received 14 November 2013 Accepted 22 December 2013 Published 18 March 2014

Academic Editors N-I Kim and K I Ramachandran

Copyright copy 2014 Ming-Kun Chang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is difficult to achieve excellent tracking performance for a two-joint leg rehabilitation machine driven by pneumatic artificialmuscles (PAMs) because the system has a coupling effect highly nonlinear and time-varying behavior associated with gascompression and the nonlinear elasticity of bladder containers This paper therefore proposes a T-S fuzzy theory with supervisorycontrol in order to overcome the above problems The T-S fuzzy theory decomposes the model of a nonlinear system into a set oflinear subsystems In this manner the controller in the T-S fuzzy model is able to use simple linear control techniques to providea systematic framework for the design of a state feedback controller Then the LMI Toolbox of MATLAB can be employed tosolve linear matrix inequalities (LMIs) in order to determine controller gains based on the Lyapunov direct method Moreover thesupervisory control can overcome the coupling effect for a leg rehabilitation machine Experimental results show that the proposedcontroller can achieve excellent tracking performance and guarantee robustness to system parameter uncertainties

1 Introduction

In cases of traumatic brain injury bone injury amputationor spinal cord injury caused by misfortunes such as trafficaccidents and cerebral apoplexy lower limb rehabilitationmachine can help patients recover extremity functions bymeans of continuous passive motion (CPM) Traditionallyphysical therapy for achieving functional rehabilitation iscarried out bymedical therapists on a person-to-person basisHowever recently many automatic rehabilitation deviceshave been gradually applied in physical therapy programsRehabilitationmachines are usually driven by electricmotorswhich are typically rigid in nature Because of this actuatorscan generate discomfort or pain when interfacing withhumans For this reason current electromechanical actuationsystems should be replaced to ensure adaptability conformityand safety An adequate actuator for a rehabilitation devicemust provide physically adjustable compliance and safety andensure soft contact with the patient similar to the behaviorof human muscles It has been suggested that pneumaticartificial muscles (PAMs) can contribute towards achieving

more comfortable devices for interfacing with human limbsegments

PAMs behave in amanner very similar to themuscles thatmove the skeletons of animals and have many advantagessuch as high power to weight ratio [1] high power to volumeratio [2] low maintenance negligible mechanical wear lowcost cleanliness high reliability flexibility and compliancefor use with humans For these reasons PAMs are commonlyemployed in rehabilitation engineering nursing and human-friendly therapeutic machine

However PAMs exhibit highly nonlinear and time-vary-ing behavior due to the compression of air and the nonlinearelasticity of bladder containers This makes it difficult forclassical controllers to achieve excellent control performanceIn recent years researchers have developed a wide varietyof approaches to overcome these problems Noritsugu andTanaka [3] developed four modes of linear motion withimpedance control to control force during movement andused an adaptive identificationmethod to estimate the systemmodel Lilly and Yang [4] applied a slidingmode controller toa planar arm actuated by two PMA groups simulation results

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 464276 12 pageshttpdxdoiorg1011552014464276

2 The Scientific World Journal

were consistent with theoretical findings for two differentmasses Ahn and Anh [5] adopted an ARNN controller ina PAM manipulator for reducing tracking errors Shen [6]developed a full nonlinear model that encompassed allthe major existing nonlinearities Based on this model thestandard slidingmode control approachwas applied to obtainrobust control even in the event of model uncertainties anddisturbances

Since the inception of fuzzy set theory by Zadeh [7] in1965 a great deal of research has been focused on fuzzycontrol systems Takagi and Sugeno [8] proposed the T-S fuzzy model-based controller in 1985 and the T-S fuzzymodel-based system subsequently emerged as one of themostactive and fruitful areas of fuzzy control Using a T-S fuzzymodel-based controller a complex dynamic model can bedecomposed into a set of local linear subsystems via fuzzyinference Stability analysis is carried out using the Lyapunovdirect method where the control problem is formulated intolinear matrix inequalities (LMIs) Based on this approachAhn and Anh [9] also developed an inverse double nonlinearautoregressive model with exogenous control based on theT-S fuzzy model applied in a PAM robot A novel 119867

infin

control structure based on a Takagi-Sugeno model [10] wasproposed to track the desired trajectories and simulationresults illustrated the efficiency of the proposed approach forthe new rehabilitation device

The leg rehabilitation machine driven by PAMs is a two-input two-output system This paper proposes compositefuzzy theory which includes T-S fuzzy tracking control andsupervisor control in order to improve tracking performanceThe proposed approach decomposes the model of a nonlin-ear system into a set of linear subsystems with associatednonlinear weighting functions enabling the use of simplelinear control techniques without the need for complicatednonlinear control strategies and also provides a systematicframework for the design of a state feedback controller [11]It has been shown that a composite fuzzy control systemcan be guaranteed to be asymptotically stable if a commonpositive definite solution exists for a set of Lyapunov inequal-ities In addition the supervisory control can overcome thecoupling effect due to two-joint motion In view of the aboveadvantages the proposed controller was applied to the outputtracking control of this system and experimental resultsverified that the proposed controller is capable of achievingexcellent tracking performance

The remainder of the paper is organized as followsSection 2 describes the control strategies Section 3 describesthe system In Section 4 the dynamics of the model arederived Experimental results for output tracking are shownin Section 5 Finally conclusions are presented in Section 6

2 Control Strategies

21 Takagi-Sugeno Fuzzy Tracking Controller Consider ageneral nonlinear dynamic equation

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905)

119910 (119905) = 119902 (119909 (119905)) (1)

where 119909 isin 119877119899 is the state vector 119910 isin 119877

119898 is the controlledoutput 119906 isin 119877

119898 is the control input vector and119891(119909)119892(119909) and119902(119909) are nonlinear functions with appropriate dimensionsThe nonlinear system (1) can then be expressed by the fuzzysystem

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) 119894 = 1 2 119903

(2)

where 119911(119905)1

sim 119911119892(119905) are the premise variables including

system states 119865119894119895denotes the fuzzy sets 119903 is the number

of fuzzy rules and 119860119894and 119861

119894are system matrices with

appropriate dimensions For simplicity this study assumedthat the membership functions had been normalized that issum119903

119894=1Π119892

119895=1119865119895119894(119911119895) = 1 As in (1) using the singleton fuzzier

product inferred and weighted defuzzier the fuzzy system isinferred as

(119905) =

119903

sum

119894=1

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905)] (3)

where ℎ119894(119911(119905)) = Π

119892

119895=1119865119895119894(119911119895(119905)) Note that sum119903

119894=1ℎ119894(119911(119905)) = 1

for all 119905 wheresum119903119894=1

ℎ119894(119911(119905)) ge 0 for 119894 = 1 2 119903 are regarded

as grade functionsFor output tracking control the control objective is

required to satisfy

119910 (119905) minus 119903 (119905) 997888rarr 0 as 119905 997888rarr infin (4)

where 119903(119905) denotes the desired trajectory or reference signalTo convert the output tracking problem into a stabilizationproblem a set of virtual desired variables 119909

119889(119905) was intro-

duced to be tracked by the state variable 119909 Let 119909(119905) = 119909(119905) minus

119909119889(119905) denote the tracking error for the state variables The

time derivative of 119909(119905) yields

119909 (119905) = minus 119889=

119903

sum

119894

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905)] minus

119889(119905) (5)

If the control input 119906(119905) is assumed to satisfy the followingequation

119903

sum

119894=1

ℎ119894119861119894120591 (119905) =

119903

sum

119894=1

ℎ119894119861119894119906 (119905) +

119903

sum

119894=1

ℎ119894119860119894119909119889(119905) minus

119889(119905) (6)

where 120591(119905) is a new control to be designed then the trackingerror system (5) results in the following form

119909 (119905) =

119903

sum

119894=1

ℎ119894119860119894119909 (119905) +

119903

sum

119894=1

ℎ119894119861119894120591 (119905) (7)

The design of the new control 120591(119905) is similar to solvinga stabilization problem The purpose is to steer 119909(119905) to zerowhich means that state 119909(119905) tracks 119909

119889(119905) The new fuzzy

The Scientific World Journal 3

controller 120591(119905) is designed on the basis of parallel distributedcompensation (PDC) and is represented as follows

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN 120591 (119905) = minus119870119894119909 (119905)

(8)

where119870119894represents feedback gainThe inferred output of the

PDC controller is expressed in the following form

120591 (119905) = minus

119903

sum

119894=1

ℎ119894119870119894119909 (119905) (9)

Substituting (9) into (7) yields

119909 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894minus 119861119894119870119895) 119909 (119905) (10)

The stability analysis of this tracking system (10) is carried outusing the Lyapunov direct method and the Lyapunov func-tion is defined as

119881 (119909 (119905)) = 119909119879

(119905) 119875119909 (119905) gt 0 (11)

where 119875 is a positive symmetric matrix Taking the derivativeof 119881 with respect to time yields

(119909 (119905)) = 119909119879

(119905) 119875119909 + 119909119879

(119905) 119875 119909 (119905)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119909119879

(119860119879

119894minus 119870119879

119895119861119879

119894) 119875119909)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

119875 (119860119894minus 119861119894119870119895) 119909

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

(12)

The controller is stable if lt 0 Hence the LMI form isexpressed as follows

(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894) lt 0 for 119894 = 1 2 119903

119866119894119895119875 + 119875119866

119894119895lt 0 1 le 119894 lt 119895 le 119903

(13)

where 119866119894119895= (119860119894minus 119861119894119870119895+ 119860119895minus 119861119895119870119894)2 and 119866

119894119894= 119860119894minus 119861119894119870119894

The controller gain 119870119894is obtained using the LMI toolbox

ofMATLAB If there exists a commonpositive definitematrix119875 that satisfies inequalities (13) it can be guaranteed that thetracking error will approach zero

22 Composite Fuzzy Tracking Controller Because the legrehabilitation machine has a coupling effect due to mecha-nism interaction many fuzzymodel controllers in the relatedliterature exhibit restrictive tracking control in application

The proposed approach introduces supervisory control inorder to overcome the coupling effect The 119894th rule of theproposed controller is defined as follows

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN 119906 (119905) = 119870119894119909 (119905) + 119906

119904119894 = 1 2 119903

(14)

where 119906119904(119905) isin 119877

119898 The proposed controller consists of a localstate feedback 119870

119894119909 and a supervisory control 119906

119904 Therefore

the output of the proposed controller is

119906 (119905) =

119903

sum

119894

ℎ119894119870119894119909 (119905) + 119906

119904 (15)

The closed-loop system is given by

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119860119894minus 119861119894119870119895] 119909 (119905) +

119903

sum

119894=1

ℎ119894119861119894119906119904(119905)

=

119903

sum

119894=1

ℎ2

119894119866119894119894119909 (119905) + 2

119903

sum

119894lt119895

ℎ119894ℎ119895119866119894119895119909 (119905) + 119861119906

119904(119905)

(16)

Suppose that there exist a symmetric and positive definitematrix 119875 and some matrices119870

119894so that the following reduced

stability condition holds

(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894) le minus119876

119894 119894 = 1 119903 (17)

where 119876119894is a positive definite matrix Based on this assump-

tion each subsystem is locally controllable and a stablefeedback gain is obtainable Intuitively a common matrix 119875

that satisfies (17) can be obtained more easily than can onethat fulfills the basic stabilization conditions When the LMImethod is applied conditions (17) can be efficiently verified Ifa feasible solution is obtained the design proceeds to exploitthe supervisory control in order to deal with the couplingeffects

Choose the Lyapunov function candidate 1198811(119909) = 119909

119879

119875119909The time derivative of 119881

1(119909) is as follows

1(119909) =

119903

sum

119894=1

ℎ2

119894119909119879

(119866119879

119894119894119875 + 119875119866

119894119894) 119909

+ 2

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 + 2119909

119879

119875119861119906119904

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909

+ 2

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 + 2119909

119879

119875119861119906119904

(18)

Given the matrix property clearly

120582min (119866119879

119894119895119875 + 119875119866

119894119895) 1199092

le 119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

le 120582min (119866119879

119894119895119875 + 119875119866

119894119895) 1199092

(19)

4 The Scientific World Journal

where 120582min(max) denotes the smallest (largest) eigenvalue ofthe matrix Define

120572 = max119894119895

120582max (119866119879

119894119895119875 + 119875119866

119894119895) for 1 le 119894 lt 119895 le 119903 (20)

A relaxed condition concerning the coupling effect isexpressed as

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 le 119896

11199092

1198961=

119903 (119903 minus 1)

2120572

(21)

Finding the maximum value of sum119903119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895)119909

is equivalent to determining the maximum value ofsum119903

119894lt119895ℎ119894ℎ119895120582max(119866

119879

119894119895119875 + 119875119866

119894119895) This can be presented as

a nonlinear programming The optimal algorithms areemployed to seek the best solution Moreover the MATLABOptimization Toolbox consists of functions that minimize ormaximize general nonlinear functions By using the toolboxthe nonlinear programming is expressed in the followingform

max119894119895

119903

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) 1 le 119894 lt 119895 le 119903

Subject to119903

sum

119894=1

120583119894= 1 120583

119894ge 0

119903

sum

119895=1

120583119895= 1 120583

119895ge 0

(22)

The largest eigenvalue of (119866119879119894119895119875 + 119875119866

119894119895) can be obtained in

advance so the maximum value is determined to be

1198962= max119894119895

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) (23)

The following supervisory control is chosen

119906119904=

minus119861119879

119875119909

1003817100381710038171003817119909119879119875119861

1003817100381710038171003817

21198961199092

if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817

= 0

0 if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817= 0

(24)

where 119896 gt 119896119895 119895 = 1 or 2 If 119909119879119875119861 = 0 then substituting (24)

into (18) gives

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 + 2119896

1198951199092

minus 21198961199092

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 = minus119881

2(119909)

(25)

where1198812(119909) is a positive definite functionWhen 119909

119879

119875119861 = 0

can give the following form

119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

= 119909119879

(119860119879

119894119875 + 119875119860

119894) 119909 le minus119909

119879

119876119894119909 119894 = 1 119903

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

=

119903

sum

119894lt119895

ℎ119894ℎ119895[119909119879

(119860119894119875 minus 119875119860

119894) 119909 + 119909

119879

(119860119879

119895119875 minus 119875119860

119895) 119909]

le minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 1 le 119894 lt 119895 le 119903

(26)

the time derivative of 1198811(119909) becomes

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 = minus119881

3(119909)

(27)

where 1198813(119909) is a positive definite function Thus the closed-

loop fuzzy system is asymptotically stable

3 System Descriptions

Figure 1 shows the experimental setup including four PAMstwo rotary potentiometers four pressure proportional valvesand four pressure transducers The hardware includes anIBM-compatible personal computer to calculate the con-trol signal which controls the pressure proportional valvethrough a DA card The angles of the joints are detectedusing rotary potentiometers the air pressure of each PAM ismeasured using pressure transducers and the measurementsare then fed back to the computer through anAD cardThesespecifications are listed in Table 1

Figure 2 presents the operation principle of the leg reha-bilitation machine depicting a two-joint leg The behavior ofthe leg manipulated by the rehabilitation machine is similarto that of a human leg Output angles 120579

1and 1205792simulate the

knee and ankle joints and the ranges of the rotary angles 1205791

and 1205792are fromminus45

∘ to 45∘ and fromminus50∘ to 50∘ respectively

The link mass 1198981= 27 kg119898

2= 081 kg and the link length

1198971= 05m 119897

2= 026m The rotating torque 120591 is generated

by the difference in pressure Δ119901 between the two opposingPAMs That is when 119901

119886gt 119901119887 as in Figure 2 the torque

exerted on the joint is counterclockwise and the rotation ofthe joint is also counterclockwise

So a pair of such PAMs is tied together around a pulleywith a radius 119903

119894 as in Figure 2 Then the torque values

imparted to the pulley by the PAM pair are [12]

1205911= (1206011119886

minus 1206011119887) 1199031

1205912= (1206012119886

minus 1206012119887) 1199032

(28)

The Scientific World Journal 5

1

2

3

4

5

6

78910

11

12

1314

15

16

ADDA

Figure 1 The experimental setup

1b

1205911

2b

1205912 m2

r1

1205792

2a

1a

m1

r1

1205791

l1

l2

p1b = p10 minus Δp1

p1a = p10 + Δp1

p2a=p 20

+ Δp2

p 2b=p 20

minus Δp2

Figure 2 Operation principle of the leg rehabilitation machine driven by PAMs

6 The Scientific World Journal

Table 1 Component specifications

Number Component Specifications1 2 3 4 PAM Festo MAS-20-150N7 8 9 10 Pressure proportional valve Mac PPC5C5 6 Rotary potentiometer Keen Engineering KRT205011 12 13 14 Pressure transducer Jihsense SN 911166 0ndash10 kgfcm2

16 IBM-compatible PC Pentium 18GHzDAC ADC Automation AIO3321

where

1206011119886

= 1198651119886(1199011119886) minus 1198701119886(1199011119886) 11990311205791minus 1198611119886(1199011119886) 1199031

1205791

(29a)

1206011119887

= 1198651119887(1199011119887) minus 1198701119887(1199011119887) 11990311205791minus 1198611119887(1199011119887) 1199031

1205791

(29b)

1206012119886

= 1198652119886(1199012119886) minus 1198702119886(1199012119886) 11990321205792minus 1198612119886(1199012119886) 1199032

1205792

(29c)

1206012119887

= 1198652119887(1199012119887) minus 1198702119887(1199012119887) 11990321205792minus 1198612119887(1199012119887) 1199032

1205792 (29d)

where the spring coefficient119870(119901) and the damping coefficient119861(119901) are given by Reynolds et al [13]

The desired input pressures P119886= [1199011119886

1199012119886]119879 and P

119887=

[1199011119887

1199012119887]119879 for each PAM are generated by the following

equation

P119886(119905) = P

0+ ΔP (119905) P

119887(119905) = P

0minus ΔP (119905) (30)

where P0

= [11990110

11990120]119879 is a nominal constant input PAM

pressure and ΔP(119905) = [Δ1199011

Δ1199012]119879 is the control pressure

input with an arbitrary function of time Because the pressureinputΔP(119905) = [Δ119901

1Δ1199012]119879 and output 120579 = [120579

11205792]119879 the sys-

tem can bewritten as a two-input two-output (TITO) controlsystem The control signal u = [119906

11199062]119879 is proportional to

ΔP based on the pressure proportional valversquos characteristicsThat is ΔP can be used instead of u as a control input

4 Dynamic Model of a Two-Joint LegRehabilitation Machine Driven by PAMs

Figure 2 shows a two-joint leg rehabilitation machine drivenby PAMs and the dynamic equation is given as follows [14]

119872(120579) 120579 + 119862 (120579 120579) 120579 + 119866 (120579) = 120591 (31)

where

119872(120579) = [(1198981+ 1198982) 1198972

119898211989711198972(11990411199042+ 11988811198882)

119898211989711198972(11990411199042+ 11988811198882) 119898

21198972

2

]

119862 (120579 120579) = [0 minus119898

211989711198972(11988811199042minus 11990411198882) 1205792

minus119898211989711198972(11988811199042minus 11990411198882) 1205791

0]

119866 (120579) = [(1198981+ 1198982) 11989711198921199041

minus119898211989721198921199042

]

(32)

and119872(120579) is the moment of inertia 119862(120579 120579) includes Coriolisand centripetal force and 119866(120579) is the gravitational force

Notation 1199041= sin(120579

1) 1199042= sin(120579

2) 1198881= cos(120579

1) and 119888

2=

cos(1205792) Let 119909

1= 1205791 1199092= 1205791 1199093= 1205792 and 119909

4= 1205792 then (31)

can be written as the following state-space form [14]

1= 1199092

2= 1198911(119909) + 119892

11(119909) 1205911+ 119892121205912

3= 1199094

4= 1198912(119909) + 119892

21(119909) + 119892

221205912

(33)

where1198911(119909)

=

(11990411198882minus 11988811199042) [119898211989711198972(11990411199042+ 11988811198882) 1199092

2minus 11989821198972

21199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[(1198981minus 1198982) 11989721198921199041minus 119898211989721198921199042(11990411199042+ 11988811198882)]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

1198912(119909)

=

(11990411198882minus 11988811199042) [minus (119898

1+ 1198982) 1198972

11199092

2+ 119898211989711198972(11990411199042+ 11988811198882) 1199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[minus (119898

1+ 1198982) 11989711198921199041(11990411199042+ 11988811198882) + (119898

1+ 1198982) 11989711198921199042]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989211(119909) =

11989821198972

2

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989212(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989221(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989222(119909) =

(1198981+ 1198982) 1198972

1

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

(34)

5 Experimental Studies

Figure 3 shows the leg rehabilitation machine using an actualhuman loading with a 65 kg weight The automatic device

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

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DistributedSensor Networks

International Journal of

2 The Scientific World Journal

were consistent with theoretical findings for two differentmasses Ahn and Anh [5] adopted an ARNN controller ina PAM manipulator for reducing tracking errors Shen [6]developed a full nonlinear model that encompassed allthe major existing nonlinearities Based on this model thestandard slidingmode control approachwas applied to obtainrobust control even in the event of model uncertainties anddisturbances

Since the inception of fuzzy set theory by Zadeh [7] in1965 a great deal of research has been focused on fuzzycontrol systems Takagi and Sugeno [8] proposed the T-S fuzzy model-based controller in 1985 and the T-S fuzzymodel-based system subsequently emerged as one of themostactive and fruitful areas of fuzzy control Using a T-S fuzzymodel-based controller a complex dynamic model can bedecomposed into a set of local linear subsystems via fuzzyinference Stability analysis is carried out using the Lyapunovdirect method where the control problem is formulated intolinear matrix inequalities (LMIs) Based on this approachAhn and Anh [9] also developed an inverse double nonlinearautoregressive model with exogenous control based on theT-S fuzzy model applied in a PAM robot A novel 119867

infin

control structure based on a Takagi-Sugeno model [10] wasproposed to track the desired trajectories and simulationresults illustrated the efficiency of the proposed approach forthe new rehabilitation device

The leg rehabilitation machine driven by PAMs is a two-input two-output system This paper proposes compositefuzzy theory which includes T-S fuzzy tracking control andsupervisor control in order to improve tracking performanceThe proposed approach decomposes the model of a nonlin-ear system into a set of linear subsystems with associatednonlinear weighting functions enabling the use of simplelinear control techniques without the need for complicatednonlinear control strategies and also provides a systematicframework for the design of a state feedback controller [11]It has been shown that a composite fuzzy control systemcan be guaranteed to be asymptotically stable if a commonpositive definite solution exists for a set of Lyapunov inequal-ities In addition the supervisory control can overcome thecoupling effect due to two-joint motion In view of the aboveadvantages the proposed controller was applied to the outputtracking control of this system and experimental resultsverified that the proposed controller is capable of achievingexcellent tracking performance

The remainder of the paper is organized as followsSection 2 describes the control strategies Section 3 describesthe system In Section 4 the dynamics of the model arederived Experimental results for output tracking are shownin Section 5 Finally conclusions are presented in Section 6

2 Control Strategies

21 Takagi-Sugeno Fuzzy Tracking Controller Consider ageneral nonlinear dynamic equation

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905)

119910 (119905) = 119902 (119909 (119905)) (1)

where 119909 isin 119877119899 is the state vector 119910 isin 119877

119898 is the controlledoutput 119906 isin 119877

119898 is the control input vector and119891(119909)119892(119909) and119902(119909) are nonlinear functions with appropriate dimensionsThe nonlinear system (1) can then be expressed by the fuzzysystem

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) 119894 = 1 2 119903

(2)

where 119911(119905)1

sim 119911119892(119905) are the premise variables including

system states 119865119894119895denotes the fuzzy sets 119903 is the number

of fuzzy rules and 119860119894and 119861

119894are system matrices with

appropriate dimensions For simplicity this study assumedthat the membership functions had been normalized that issum119903

119894=1Π119892

119895=1119865119895119894(119911119895) = 1 As in (1) using the singleton fuzzier

product inferred and weighted defuzzier the fuzzy system isinferred as

(119905) =

119903

sum

119894=1

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905)] (3)

where ℎ119894(119911(119905)) = Π

119892

119895=1119865119895119894(119911119895(119905)) Note that sum119903

119894=1ℎ119894(119911(119905)) = 1

for all 119905 wheresum119903119894=1

ℎ119894(119911(119905)) ge 0 for 119894 = 1 2 119903 are regarded

as grade functionsFor output tracking control the control objective is

required to satisfy

119910 (119905) minus 119903 (119905) 997888rarr 0 as 119905 997888rarr infin (4)

where 119903(119905) denotes the desired trajectory or reference signalTo convert the output tracking problem into a stabilizationproblem a set of virtual desired variables 119909

119889(119905) was intro-

duced to be tracked by the state variable 119909 Let 119909(119905) = 119909(119905) minus

119909119889(119905) denote the tracking error for the state variables The

time derivative of 119909(119905) yields

119909 (119905) = minus 119889=

119903

sum

119894

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905)] minus

119889(119905) (5)

If the control input 119906(119905) is assumed to satisfy the followingequation

119903

sum

119894=1

ℎ119894119861119894120591 (119905) =

119903

sum

119894=1

ℎ119894119861119894119906 (119905) +

119903

sum

119894=1

ℎ119894119860119894119909119889(119905) minus

119889(119905) (6)

where 120591(119905) is a new control to be designed then the trackingerror system (5) results in the following form

119909 (119905) =

119903

sum

119894=1

ℎ119894119860119894119909 (119905) +

119903

sum

119894=1

ℎ119894119861119894120591 (119905) (7)

The design of the new control 120591(119905) is similar to solvinga stabilization problem The purpose is to steer 119909(119905) to zerowhich means that state 119909(119905) tracks 119909

119889(119905) The new fuzzy

The Scientific World Journal 3

controller 120591(119905) is designed on the basis of parallel distributedcompensation (PDC) and is represented as follows

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN 120591 (119905) = minus119870119894119909 (119905)

(8)

where119870119894represents feedback gainThe inferred output of the

PDC controller is expressed in the following form

120591 (119905) = minus

119903

sum

119894=1

ℎ119894119870119894119909 (119905) (9)

Substituting (9) into (7) yields

119909 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894minus 119861119894119870119895) 119909 (119905) (10)

The stability analysis of this tracking system (10) is carried outusing the Lyapunov direct method and the Lyapunov func-tion is defined as

119881 (119909 (119905)) = 119909119879

(119905) 119875119909 (119905) gt 0 (11)

where 119875 is a positive symmetric matrix Taking the derivativeof 119881 with respect to time yields

(119909 (119905)) = 119909119879

(119905) 119875119909 + 119909119879

(119905) 119875 119909 (119905)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119909119879

(119860119879

119894minus 119870119879

119895119861119879

119894) 119875119909)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

119875 (119860119894minus 119861119894119870119895) 119909

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

(12)

The controller is stable if lt 0 Hence the LMI form isexpressed as follows

(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894) lt 0 for 119894 = 1 2 119903

119866119894119895119875 + 119875119866

119894119895lt 0 1 le 119894 lt 119895 le 119903

(13)

where 119866119894119895= (119860119894minus 119861119894119870119895+ 119860119895minus 119861119895119870119894)2 and 119866

119894119894= 119860119894minus 119861119894119870119894

The controller gain 119870119894is obtained using the LMI toolbox

ofMATLAB If there exists a commonpositive definitematrix119875 that satisfies inequalities (13) it can be guaranteed that thetracking error will approach zero

22 Composite Fuzzy Tracking Controller Because the legrehabilitation machine has a coupling effect due to mecha-nism interaction many fuzzymodel controllers in the relatedliterature exhibit restrictive tracking control in application

The proposed approach introduces supervisory control inorder to overcome the coupling effect The 119894th rule of theproposed controller is defined as follows

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN 119906 (119905) = 119870119894119909 (119905) + 119906

119904119894 = 1 2 119903

(14)

where 119906119904(119905) isin 119877

119898 The proposed controller consists of a localstate feedback 119870

119894119909 and a supervisory control 119906

119904 Therefore

the output of the proposed controller is

119906 (119905) =

119903

sum

119894

ℎ119894119870119894119909 (119905) + 119906

119904 (15)

The closed-loop system is given by

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119860119894minus 119861119894119870119895] 119909 (119905) +

119903

sum

119894=1

ℎ119894119861119894119906119904(119905)

=

119903

sum

119894=1

ℎ2

119894119866119894119894119909 (119905) + 2

119903

sum

119894lt119895

ℎ119894ℎ119895119866119894119895119909 (119905) + 119861119906

119904(119905)

(16)

Suppose that there exist a symmetric and positive definitematrix 119875 and some matrices119870

119894so that the following reduced

stability condition holds

(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894) le minus119876

119894 119894 = 1 119903 (17)

where 119876119894is a positive definite matrix Based on this assump-

tion each subsystem is locally controllable and a stablefeedback gain is obtainable Intuitively a common matrix 119875

that satisfies (17) can be obtained more easily than can onethat fulfills the basic stabilization conditions When the LMImethod is applied conditions (17) can be efficiently verified Ifa feasible solution is obtained the design proceeds to exploitthe supervisory control in order to deal with the couplingeffects

Choose the Lyapunov function candidate 1198811(119909) = 119909

119879

119875119909The time derivative of 119881

1(119909) is as follows

1(119909) =

119903

sum

119894=1

ℎ2

119894119909119879

(119866119879

119894119894119875 + 119875119866

119894119894) 119909

+ 2

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 + 2119909

119879

119875119861119906119904

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909

+ 2

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 + 2119909

119879

119875119861119906119904

(18)

Given the matrix property clearly

120582min (119866119879

119894119895119875 + 119875119866

119894119895) 1199092

le 119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

le 120582min (119866119879

119894119895119875 + 119875119866

119894119895) 1199092

(19)

4 The Scientific World Journal

where 120582min(max) denotes the smallest (largest) eigenvalue ofthe matrix Define

120572 = max119894119895

120582max (119866119879

119894119895119875 + 119875119866

119894119895) for 1 le 119894 lt 119895 le 119903 (20)

A relaxed condition concerning the coupling effect isexpressed as

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 le 119896

11199092

1198961=

119903 (119903 minus 1)

2120572

(21)

Finding the maximum value of sum119903119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895)119909

is equivalent to determining the maximum value ofsum119903

119894lt119895ℎ119894ℎ119895120582max(119866

119879

119894119895119875 + 119875119866

119894119895) This can be presented as

a nonlinear programming The optimal algorithms areemployed to seek the best solution Moreover the MATLABOptimization Toolbox consists of functions that minimize ormaximize general nonlinear functions By using the toolboxthe nonlinear programming is expressed in the followingform

max119894119895

119903

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) 1 le 119894 lt 119895 le 119903

Subject to119903

sum

119894=1

120583119894= 1 120583

119894ge 0

119903

sum

119895=1

120583119895= 1 120583

119895ge 0

(22)

The largest eigenvalue of (119866119879119894119895119875 + 119875119866

119894119895) can be obtained in

advance so the maximum value is determined to be

1198962= max119894119895

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) (23)

The following supervisory control is chosen

119906119904=

minus119861119879

119875119909

1003817100381710038171003817119909119879119875119861

1003817100381710038171003817

21198961199092

if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817

= 0

0 if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817= 0

(24)

where 119896 gt 119896119895 119895 = 1 or 2 If 119909119879119875119861 = 0 then substituting (24)

into (18) gives

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 + 2119896

1198951199092

minus 21198961199092

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 = minus119881

2(119909)

(25)

where1198812(119909) is a positive definite functionWhen 119909

119879

119875119861 = 0

can give the following form

119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

= 119909119879

(119860119879

119894119875 + 119875119860

119894) 119909 le minus119909

119879

119876119894119909 119894 = 1 119903

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

=

119903

sum

119894lt119895

ℎ119894ℎ119895[119909119879

(119860119894119875 minus 119875119860

119894) 119909 + 119909

119879

(119860119879

119895119875 minus 119875119860

119895) 119909]

le minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 1 le 119894 lt 119895 le 119903

(26)

the time derivative of 1198811(119909) becomes

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 = minus119881

3(119909)

(27)

where 1198813(119909) is a positive definite function Thus the closed-

loop fuzzy system is asymptotically stable

3 System Descriptions

Figure 1 shows the experimental setup including four PAMstwo rotary potentiometers four pressure proportional valvesand four pressure transducers The hardware includes anIBM-compatible personal computer to calculate the con-trol signal which controls the pressure proportional valvethrough a DA card The angles of the joints are detectedusing rotary potentiometers the air pressure of each PAM ismeasured using pressure transducers and the measurementsare then fed back to the computer through anAD cardThesespecifications are listed in Table 1

Figure 2 presents the operation principle of the leg reha-bilitation machine depicting a two-joint leg The behavior ofthe leg manipulated by the rehabilitation machine is similarto that of a human leg Output angles 120579

1and 1205792simulate the

knee and ankle joints and the ranges of the rotary angles 1205791

and 1205792are fromminus45

∘ to 45∘ and fromminus50∘ to 50∘ respectively

The link mass 1198981= 27 kg119898

2= 081 kg and the link length

1198971= 05m 119897

2= 026m The rotating torque 120591 is generated

by the difference in pressure Δ119901 between the two opposingPAMs That is when 119901

119886gt 119901119887 as in Figure 2 the torque

exerted on the joint is counterclockwise and the rotation ofthe joint is also counterclockwise

So a pair of such PAMs is tied together around a pulleywith a radius 119903

119894 as in Figure 2 Then the torque values

imparted to the pulley by the PAM pair are [12]

1205911= (1206011119886

minus 1206011119887) 1199031

1205912= (1206012119886

minus 1206012119887) 1199032

(28)

The Scientific World Journal 5

1

2

3

4

5

6

78910

11

12

1314

15

16

ADDA

Figure 1 The experimental setup

1b

1205911

2b

1205912 m2

r1

1205792

2a

1a

m1

r1

1205791

l1

l2

p1b = p10 minus Δp1

p1a = p10 + Δp1

p2a=p 20

+ Δp2

p 2b=p 20

minus Δp2

Figure 2 Operation principle of the leg rehabilitation machine driven by PAMs

6 The Scientific World Journal

Table 1 Component specifications

Number Component Specifications1 2 3 4 PAM Festo MAS-20-150N7 8 9 10 Pressure proportional valve Mac PPC5C5 6 Rotary potentiometer Keen Engineering KRT205011 12 13 14 Pressure transducer Jihsense SN 911166 0ndash10 kgfcm2

16 IBM-compatible PC Pentium 18GHzDAC ADC Automation AIO3321

where

1206011119886

= 1198651119886(1199011119886) minus 1198701119886(1199011119886) 11990311205791minus 1198611119886(1199011119886) 1199031

1205791

(29a)

1206011119887

= 1198651119887(1199011119887) minus 1198701119887(1199011119887) 11990311205791minus 1198611119887(1199011119887) 1199031

1205791

(29b)

1206012119886

= 1198652119886(1199012119886) minus 1198702119886(1199012119886) 11990321205792minus 1198612119886(1199012119886) 1199032

1205792

(29c)

1206012119887

= 1198652119887(1199012119887) minus 1198702119887(1199012119887) 11990321205792minus 1198612119887(1199012119887) 1199032

1205792 (29d)

where the spring coefficient119870(119901) and the damping coefficient119861(119901) are given by Reynolds et al [13]

The desired input pressures P119886= [1199011119886

1199012119886]119879 and P

119887=

[1199011119887

1199012119887]119879 for each PAM are generated by the following

equation

P119886(119905) = P

0+ ΔP (119905) P

119887(119905) = P

0minus ΔP (119905) (30)

where P0

= [11990110

11990120]119879 is a nominal constant input PAM

pressure and ΔP(119905) = [Δ1199011

Δ1199012]119879 is the control pressure

input with an arbitrary function of time Because the pressureinputΔP(119905) = [Δ119901

1Δ1199012]119879 and output 120579 = [120579

11205792]119879 the sys-

tem can bewritten as a two-input two-output (TITO) controlsystem The control signal u = [119906

11199062]119879 is proportional to

ΔP based on the pressure proportional valversquos characteristicsThat is ΔP can be used instead of u as a control input

4 Dynamic Model of a Two-Joint LegRehabilitation Machine Driven by PAMs

Figure 2 shows a two-joint leg rehabilitation machine drivenby PAMs and the dynamic equation is given as follows [14]

119872(120579) 120579 + 119862 (120579 120579) 120579 + 119866 (120579) = 120591 (31)

where

119872(120579) = [(1198981+ 1198982) 1198972

119898211989711198972(11990411199042+ 11988811198882)

119898211989711198972(11990411199042+ 11988811198882) 119898

21198972

2

]

119862 (120579 120579) = [0 minus119898

211989711198972(11988811199042minus 11990411198882) 1205792

minus119898211989711198972(11988811199042minus 11990411198882) 1205791

0]

119866 (120579) = [(1198981+ 1198982) 11989711198921199041

minus119898211989721198921199042

]

(32)

and119872(120579) is the moment of inertia 119862(120579 120579) includes Coriolisand centripetal force and 119866(120579) is the gravitational force

Notation 1199041= sin(120579

1) 1199042= sin(120579

2) 1198881= cos(120579

1) and 119888

2=

cos(1205792) Let 119909

1= 1205791 1199092= 1205791 1199093= 1205792 and 119909

4= 1205792 then (31)

can be written as the following state-space form [14]

1= 1199092

2= 1198911(119909) + 119892

11(119909) 1205911+ 119892121205912

3= 1199094

4= 1198912(119909) + 119892

21(119909) + 119892

221205912

(33)

where1198911(119909)

=

(11990411198882minus 11988811199042) [119898211989711198972(11990411199042+ 11988811198882) 1199092

2minus 11989821198972

21199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[(1198981minus 1198982) 11989721198921199041minus 119898211989721198921199042(11990411199042+ 11988811198882)]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

1198912(119909)

=

(11990411198882minus 11988811199042) [minus (119898

1+ 1198982) 1198972

11199092

2+ 119898211989711198972(11990411199042+ 11988811198882) 1199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[minus (119898

1+ 1198982) 11989711198921199041(11990411199042+ 11988811198882) + (119898

1+ 1198982) 11989711198921199042]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989211(119909) =

11989821198972

2

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989212(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989221(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989222(119909) =

(1198981+ 1198982) 1198972

1

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

(34)

5 Experimental Studies

Figure 3 shows the leg rehabilitation machine using an actualhuman loading with a 65 kg weight The automatic device

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

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The Scientific World Journal 3

controller 120591(119905) is designed on the basis of parallel distributedcompensation (PDC) and is represented as follows

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN 120591 (119905) = minus119870119894119909 (119905)

(8)

where119870119894represents feedback gainThe inferred output of the

PDC controller is expressed in the following form

120591 (119905) = minus

119903

sum

119894=1

ℎ119894119870119894119909 (119905) (9)

Substituting (9) into (7) yields

119909 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894minus 119861119894119870119895) 119909 (119905) (10)

The stability analysis of this tracking system (10) is carried outusing the Lyapunov direct method and the Lyapunov func-tion is defined as

119881 (119909 (119905)) = 119909119879

(119905) 119875119909 (119905) gt 0 (11)

where 119875 is a positive symmetric matrix Taking the derivativeof 119881 with respect to time yields

(119909 (119905)) = 119909119879

(119905) 119875119909 + 119909119879

(119905) 119875 119909 (119905)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119909119879

(119860119879

119894minus 119870119879

119895119861119879

119894) 119875119909)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

119875 (119860119894minus 119861119894119870119895) 119909

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

(12)

The controller is stable if lt 0 Hence the LMI form isexpressed as follows

(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894) lt 0 for 119894 = 1 2 119903

119866119894119895119875 + 119875119866

119894119895lt 0 1 le 119894 lt 119895 le 119903

(13)

where 119866119894119895= (119860119894minus 119861119894119870119895+ 119860119895minus 119861119895119870119894)2 and 119866

119894119894= 119860119894minus 119861119894119870119894

The controller gain 119870119894is obtained using the LMI toolbox

ofMATLAB If there exists a commonpositive definitematrix119875 that satisfies inequalities (13) it can be guaranteed that thetracking error will approach zero

22 Composite Fuzzy Tracking Controller Because the legrehabilitation machine has a coupling effect due to mecha-nism interaction many fuzzymodel controllers in the relatedliterature exhibit restrictive tracking control in application

The proposed approach introduces supervisory control inorder to overcome the coupling effect The 119894th rule of theproposed controller is defined as follows

Rule 119894

IF 1199111(119905) is 119865

1119894and sdot sdot sdot and 119911

119892(119905) is 119865

119892119894

THEN 119906 (119905) = 119870119894119909 (119905) + 119906

119904119894 = 1 2 119903

(14)

where 119906119904(119905) isin 119877

119898 The proposed controller consists of a localstate feedback 119870

119894119909 and a supervisory control 119906

119904 Therefore

the output of the proposed controller is

119906 (119905) =

119903

sum

119894

ℎ119894119870119894119909 (119905) + 119906

119904 (15)

The closed-loop system is given by

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119860119894minus 119861119894119870119895] 119909 (119905) +

119903

sum

119894=1

ℎ119894119861119894119906119904(119905)

=

119903

sum

119894=1

ℎ2

119894119866119894119894119909 (119905) + 2

119903

sum

119894lt119895

ℎ119894ℎ119895119866119894119895119909 (119905) + 119861119906

119904(119905)

(16)

Suppose that there exist a symmetric and positive definitematrix 119875 and some matrices119870

119894so that the following reduced

stability condition holds

(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894) le minus119876

119894 119894 = 1 119903 (17)

where 119876119894is a positive definite matrix Based on this assump-

tion each subsystem is locally controllable and a stablefeedback gain is obtainable Intuitively a common matrix 119875

that satisfies (17) can be obtained more easily than can onethat fulfills the basic stabilization conditions When the LMImethod is applied conditions (17) can be efficiently verified Ifa feasible solution is obtained the design proceeds to exploitthe supervisory control in order to deal with the couplingeffects

Choose the Lyapunov function candidate 1198811(119909) = 119909

119879

119875119909The time derivative of 119881

1(119909) is as follows

1(119909) =

119903

sum

119894=1

ℎ2

119894119909119879

(119866119879

119894119894119875 + 119875119866

119894119894) 119909

+ 2

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 + 2119909

119879

119875119861119906119904

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909

+ 2

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 + 2119909

119879

119875119861119906119904

(18)

Given the matrix property clearly

120582min (119866119879

119894119895119875 + 119875119866

119894119895) 1199092

le 119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

le 120582min (119866119879

119894119895119875 + 119875119866

119894119895) 1199092

(19)

4 The Scientific World Journal

where 120582min(max) denotes the smallest (largest) eigenvalue ofthe matrix Define

120572 = max119894119895

120582max (119866119879

119894119895119875 + 119875119866

119894119895) for 1 le 119894 lt 119895 le 119903 (20)

A relaxed condition concerning the coupling effect isexpressed as

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 le 119896

11199092

1198961=

119903 (119903 minus 1)

2120572

(21)

Finding the maximum value of sum119903119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895)119909

is equivalent to determining the maximum value ofsum119903

119894lt119895ℎ119894ℎ119895120582max(119866

119879

119894119895119875 + 119875119866

119894119895) This can be presented as

a nonlinear programming The optimal algorithms areemployed to seek the best solution Moreover the MATLABOptimization Toolbox consists of functions that minimize ormaximize general nonlinear functions By using the toolboxthe nonlinear programming is expressed in the followingform

max119894119895

119903

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) 1 le 119894 lt 119895 le 119903

Subject to119903

sum

119894=1

120583119894= 1 120583

119894ge 0

119903

sum

119895=1

120583119895= 1 120583

119895ge 0

(22)

The largest eigenvalue of (119866119879119894119895119875 + 119875119866

119894119895) can be obtained in

advance so the maximum value is determined to be

1198962= max119894119895

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) (23)

The following supervisory control is chosen

119906119904=

minus119861119879

119875119909

1003817100381710038171003817119909119879119875119861

1003817100381710038171003817

21198961199092

if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817

= 0

0 if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817= 0

(24)

where 119896 gt 119896119895 119895 = 1 or 2 If 119909119879119875119861 = 0 then substituting (24)

into (18) gives

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 + 2119896

1198951199092

minus 21198961199092

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 = minus119881

2(119909)

(25)

where1198812(119909) is a positive definite functionWhen 119909

119879

119875119861 = 0

can give the following form

119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

= 119909119879

(119860119879

119894119875 + 119875119860

119894) 119909 le minus119909

119879

119876119894119909 119894 = 1 119903

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

=

119903

sum

119894lt119895

ℎ119894ℎ119895[119909119879

(119860119894119875 minus 119875119860

119894) 119909 + 119909

119879

(119860119879

119895119875 minus 119875119860

119895) 119909]

le minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 1 le 119894 lt 119895 le 119903

(26)

the time derivative of 1198811(119909) becomes

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 = minus119881

3(119909)

(27)

where 1198813(119909) is a positive definite function Thus the closed-

loop fuzzy system is asymptotically stable

3 System Descriptions

Figure 1 shows the experimental setup including four PAMstwo rotary potentiometers four pressure proportional valvesand four pressure transducers The hardware includes anIBM-compatible personal computer to calculate the con-trol signal which controls the pressure proportional valvethrough a DA card The angles of the joints are detectedusing rotary potentiometers the air pressure of each PAM ismeasured using pressure transducers and the measurementsare then fed back to the computer through anAD cardThesespecifications are listed in Table 1

Figure 2 presents the operation principle of the leg reha-bilitation machine depicting a two-joint leg The behavior ofthe leg manipulated by the rehabilitation machine is similarto that of a human leg Output angles 120579

1and 1205792simulate the

knee and ankle joints and the ranges of the rotary angles 1205791

and 1205792are fromminus45

∘ to 45∘ and fromminus50∘ to 50∘ respectively

The link mass 1198981= 27 kg119898

2= 081 kg and the link length

1198971= 05m 119897

2= 026m The rotating torque 120591 is generated

by the difference in pressure Δ119901 between the two opposingPAMs That is when 119901

119886gt 119901119887 as in Figure 2 the torque

exerted on the joint is counterclockwise and the rotation ofthe joint is also counterclockwise

So a pair of such PAMs is tied together around a pulleywith a radius 119903

119894 as in Figure 2 Then the torque values

imparted to the pulley by the PAM pair are [12]

1205911= (1206011119886

minus 1206011119887) 1199031

1205912= (1206012119886

minus 1206012119887) 1199032

(28)

The Scientific World Journal 5

1

2

3

4

5

6

78910

11

12

1314

15

16

ADDA

Figure 1 The experimental setup

1b

1205911

2b

1205912 m2

r1

1205792

2a

1a

m1

r1

1205791

l1

l2

p1b = p10 minus Δp1

p1a = p10 + Δp1

p2a=p 20

+ Δp2

p 2b=p 20

minus Δp2

Figure 2 Operation principle of the leg rehabilitation machine driven by PAMs

6 The Scientific World Journal

Table 1 Component specifications

Number Component Specifications1 2 3 4 PAM Festo MAS-20-150N7 8 9 10 Pressure proportional valve Mac PPC5C5 6 Rotary potentiometer Keen Engineering KRT205011 12 13 14 Pressure transducer Jihsense SN 911166 0ndash10 kgfcm2

16 IBM-compatible PC Pentium 18GHzDAC ADC Automation AIO3321

where

1206011119886

= 1198651119886(1199011119886) minus 1198701119886(1199011119886) 11990311205791minus 1198611119886(1199011119886) 1199031

1205791

(29a)

1206011119887

= 1198651119887(1199011119887) minus 1198701119887(1199011119887) 11990311205791minus 1198611119887(1199011119887) 1199031

1205791

(29b)

1206012119886

= 1198652119886(1199012119886) minus 1198702119886(1199012119886) 11990321205792minus 1198612119886(1199012119886) 1199032

1205792

(29c)

1206012119887

= 1198652119887(1199012119887) minus 1198702119887(1199012119887) 11990321205792minus 1198612119887(1199012119887) 1199032

1205792 (29d)

where the spring coefficient119870(119901) and the damping coefficient119861(119901) are given by Reynolds et al [13]

The desired input pressures P119886= [1199011119886

1199012119886]119879 and P

119887=

[1199011119887

1199012119887]119879 for each PAM are generated by the following

equation

P119886(119905) = P

0+ ΔP (119905) P

119887(119905) = P

0minus ΔP (119905) (30)

where P0

= [11990110

11990120]119879 is a nominal constant input PAM

pressure and ΔP(119905) = [Δ1199011

Δ1199012]119879 is the control pressure

input with an arbitrary function of time Because the pressureinputΔP(119905) = [Δ119901

1Δ1199012]119879 and output 120579 = [120579

11205792]119879 the sys-

tem can bewritten as a two-input two-output (TITO) controlsystem The control signal u = [119906

11199062]119879 is proportional to

ΔP based on the pressure proportional valversquos characteristicsThat is ΔP can be used instead of u as a control input

4 Dynamic Model of a Two-Joint LegRehabilitation Machine Driven by PAMs

Figure 2 shows a two-joint leg rehabilitation machine drivenby PAMs and the dynamic equation is given as follows [14]

119872(120579) 120579 + 119862 (120579 120579) 120579 + 119866 (120579) = 120591 (31)

where

119872(120579) = [(1198981+ 1198982) 1198972

119898211989711198972(11990411199042+ 11988811198882)

119898211989711198972(11990411199042+ 11988811198882) 119898

21198972

2

]

119862 (120579 120579) = [0 minus119898

211989711198972(11988811199042minus 11990411198882) 1205792

minus119898211989711198972(11988811199042minus 11990411198882) 1205791

0]

119866 (120579) = [(1198981+ 1198982) 11989711198921199041

minus119898211989721198921199042

]

(32)

and119872(120579) is the moment of inertia 119862(120579 120579) includes Coriolisand centripetal force and 119866(120579) is the gravitational force

Notation 1199041= sin(120579

1) 1199042= sin(120579

2) 1198881= cos(120579

1) and 119888

2=

cos(1205792) Let 119909

1= 1205791 1199092= 1205791 1199093= 1205792 and 119909

4= 1205792 then (31)

can be written as the following state-space form [14]

1= 1199092

2= 1198911(119909) + 119892

11(119909) 1205911+ 119892121205912

3= 1199094

4= 1198912(119909) + 119892

21(119909) + 119892

221205912

(33)

where1198911(119909)

=

(11990411198882minus 11988811199042) [119898211989711198972(11990411199042+ 11988811198882) 1199092

2minus 11989821198972

21199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[(1198981minus 1198982) 11989721198921199041minus 119898211989721198921199042(11990411199042+ 11988811198882)]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

1198912(119909)

=

(11990411198882minus 11988811199042) [minus (119898

1+ 1198982) 1198972

11199092

2+ 119898211989711198972(11990411199042+ 11988811198882) 1199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[minus (119898

1+ 1198982) 11989711198921199041(11990411199042+ 11988811198882) + (119898

1+ 1198982) 11989711198921199042]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989211(119909) =

11989821198972

2

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989212(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989221(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989222(119909) =

(1198981+ 1198982) 1198972

1

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

(34)

5 Experimental Studies

Figure 3 shows the leg rehabilitation machine using an actualhuman loading with a 65 kg weight The automatic device

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

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RotatingMachinery

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Shock and Vibration

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Electrical and Computer Engineering

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Advances inOptoElectronics

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DistributedSensor Networks

International Journal of

4 The Scientific World Journal

where 120582min(max) denotes the smallest (largest) eigenvalue ofthe matrix Define

120572 = max119894119895

120582max (119866119879

119894119895119875 + 119875119866

119894119895) for 1 le 119894 lt 119895 le 119903 (20)

A relaxed condition concerning the coupling effect isexpressed as

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909 le 119896

11199092

1198961=

119903 (119903 minus 1)

2120572

(21)

Finding the maximum value of sum119903119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895)119909

is equivalent to determining the maximum value ofsum119903

119894lt119895ℎ119894ℎ119895120582max(119866

119879

119894119895119875 + 119875119866

119894119895) This can be presented as

a nonlinear programming The optimal algorithms areemployed to seek the best solution Moreover the MATLABOptimization Toolbox consists of functions that minimize ormaximize general nonlinear functions By using the toolboxthe nonlinear programming is expressed in the followingform

max119894119895

119903

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) 1 le 119894 lt 119895 le 119903

Subject to119903

sum

119894=1

120583119894= 1 120583

119894ge 0

119903

sum

119895=1

120583119895= 1 120583

119895ge 0

(22)

The largest eigenvalue of (119866119879119894119895119875 + 119875119866

119894119895) can be obtained in

advance so the maximum value is determined to be

1198962= max119894119895

sum

119894lt119895

ℎ119894ℎ119895120582max (119866

119879

119894119895119875 + 119875119866

119894119895) (23)

The following supervisory control is chosen

119906119904=

minus119861119879

119875119909

1003817100381710038171003817119909119879119875119861

1003817100381710038171003817

21198961199092

if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817

= 0

0 if 10038171003817100381710038171003817119909119879

11987511986110038171003817100381710038171003817= 0

(24)

where 119896 gt 119896119895 119895 = 1 or 2 If 119909119879119875119861 = 0 then substituting (24)

into (18) gives

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 + 2119896

1198951199092

minus 21198961199092

le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 = minus119881

2(119909)

(25)

where1198812(119909) is a positive definite functionWhen 119909

119879

119875119861 = 0

can give the following form

119909119879

[(119860119894minus 119861119894119870119894)119879

119875 + 119875 (119860119894minus 119861119894119870119894)] 119909

= 119909119879

(119860119879

119894119875 + 119875119860

119894) 119909 le minus119909

119879

119876119894119909 119894 = 1 119903

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879

(119866119879

119894119895119875 + 119875119866

119894119895) 119909

=

119903

sum

119894lt119895

ℎ119894ℎ119895[119909119879

(119860119894119875 minus 119875119860

119894) 119909 + 119909

119879

(119860119879

119895119875 minus 119875119860

119895) 119909]

le minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 1 le 119894 lt 119895 le 119903

(26)

the time derivative of 1198811(119909) becomes

1(119909) le minus

119903

sum

119894=1

ℎ2

119894119909119879

119876119894119909 minussum

119894lt119895

ℎ119894ℎ119895119909119879

(119876119894+ 119876119891) 119909 = minus119881

3(119909)

(27)

where 1198813(119909) is a positive definite function Thus the closed-

loop fuzzy system is asymptotically stable

3 System Descriptions

Figure 1 shows the experimental setup including four PAMstwo rotary potentiometers four pressure proportional valvesand four pressure transducers The hardware includes anIBM-compatible personal computer to calculate the con-trol signal which controls the pressure proportional valvethrough a DA card The angles of the joints are detectedusing rotary potentiometers the air pressure of each PAM ismeasured using pressure transducers and the measurementsare then fed back to the computer through anAD cardThesespecifications are listed in Table 1

Figure 2 presents the operation principle of the leg reha-bilitation machine depicting a two-joint leg The behavior ofthe leg manipulated by the rehabilitation machine is similarto that of a human leg Output angles 120579

1and 1205792simulate the

knee and ankle joints and the ranges of the rotary angles 1205791

and 1205792are fromminus45

∘ to 45∘ and fromminus50∘ to 50∘ respectively

The link mass 1198981= 27 kg119898

2= 081 kg and the link length

1198971= 05m 119897

2= 026m The rotating torque 120591 is generated

by the difference in pressure Δ119901 between the two opposingPAMs That is when 119901

119886gt 119901119887 as in Figure 2 the torque

exerted on the joint is counterclockwise and the rotation ofthe joint is also counterclockwise

So a pair of such PAMs is tied together around a pulleywith a radius 119903

119894 as in Figure 2 Then the torque values

imparted to the pulley by the PAM pair are [12]

1205911= (1206011119886

minus 1206011119887) 1199031

1205912= (1206012119886

minus 1206012119887) 1199032

(28)

The Scientific World Journal 5

1

2

3

4

5

6

78910

11

12

1314

15

16

ADDA

Figure 1 The experimental setup

1b

1205911

2b

1205912 m2

r1

1205792

2a

1a

m1

r1

1205791

l1

l2

p1b = p10 minus Δp1

p1a = p10 + Δp1

p2a=p 20

+ Δp2

p 2b=p 20

minus Δp2

Figure 2 Operation principle of the leg rehabilitation machine driven by PAMs

6 The Scientific World Journal

Table 1 Component specifications

Number Component Specifications1 2 3 4 PAM Festo MAS-20-150N7 8 9 10 Pressure proportional valve Mac PPC5C5 6 Rotary potentiometer Keen Engineering KRT205011 12 13 14 Pressure transducer Jihsense SN 911166 0ndash10 kgfcm2

16 IBM-compatible PC Pentium 18GHzDAC ADC Automation AIO3321

where

1206011119886

= 1198651119886(1199011119886) minus 1198701119886(1199011119886) 11990311205791minus 1198611119886(1199011119886) 1199031

1205791

(29a)

1206011119887

= 1198651119887(1199011119887) minus 1198701119887(1199011119887) 11990311205791minus 1198611119887(1199011119887) 1199031

1205791

(29b)

1206012119886

= 1198652119886(1199012119886) minus 1198702119886(1199012119886) 11990321205792minus 1198612119886(1199012119886) 1199032

1205792

(29c)

1206012119887

= 1198652119887(1199012119887) minus 1198702119887(1199012119887) 11990321205792minus 1198612119887(1199012119887) 1199032

1205792 (29d)

where the spring coefficient119870(119901) and the damping coefficient119861(119901) are given by Reynolds et al [13]

The desired input pressures P119886= [1199011119886

1199012119886]119879 and P

119887=

[1199011119887

1199012119887]119879 for each PAM are generated by the following

equation

P119886(119905) = P

0+ ΔP (119905) P

119887(119905) = P

0minus ΔP (119905) (30)

where P0

= [11990110

11990120]119879 is a nominal constant input PAM

pressure and ΔP(119905) = [Δ1199011

Δ1199012]119879 is the control pressure

input with an arbitrary function of time Because the pressureinputΔP(119905) = [Δ119901

1Δ1199012]119879 and output 120579 = [120579

11205792]119879 the sys-

tem can bewritten as a two-input two-output (TITO) controlsystem The control signal u = [119906

11199062]119879 is proportional to

ΔP based on the pressure proportional valversquos characteristicsThat is ΔP can be used instead of u as a control input

4 Dynamic Model of a Two-Joint LegRehabilitation Machine Driven by PAMs

Figure 2 shows a two-joint leg rehabilitation machine drivenby PAMs and the dynamic equation is given as follows [14]

119872(120579) 120579 + 119862 (120579 120579) 120579 + 119866 (120579) = 120591 (31)

where

119872(120579) = [(1198981+ 1198982) 1198972

119898211989711198972(11990411199042+ 11988811198882)

119898211989711198972(11990411199042+ 11988811198882) 119898

21198972

2

]

119862 (120579 120579) = [0 minus119898

211989711198972(11988811199042minus 11990411198882) 1205792

minus119898211989711198972(11988811199042minus 11990411198882) 1205791

0]

119866 (120579) = [(1198981+ 1198982) 11989711198921199041

minus119898211989721198921199042

]

(32)

and119872(120579) is the moment of inertia 119862(120579 120579) includes Coriolisand centripetal force and 119866(120579) is the gravitational force

Notation 1199041= sin(120579

1) 1199042= sin(120579

2) 1198881= cos(120579

1) and 119888

2=

cos(1205792) Let 119909

1= 1205791 1199092= 1205791 1199093= 1205792 and 119909

4= 1205792 then (31)

can be written as the following state-space form [14]

1= 1199092

2= 1198911(119909) + 119892

11(119909) 1205911+ 119892121205912

3= 1199094

4= 1198912(119909) + 119892

21(119909) + 119892

221205912

(33)

where1198911(119909)

=

(11990411198882minus 11988811199042) [119898211989711198972(11990411199042+ 11988811198882) 1199092

2minus 11989821198972

21199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[(1198981minus 1198982) 11989721198921199041minus 119898211989721198921199042(11990411199042+ 11988811198882)]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

1198912(119909)

=

(11990411198882minus 11988811199042) [minus (119898

1+ 1198982) 1198972

11199092

2+ 119898211989711198972(11990411199042+ 11988811198882) 1199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[minus (119898

1+ 1198982) 11989711198921199041(11990411199042+ 11988811198882) + (119898

1+ 1198982) 11989711198921199042]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989211(119909) =

11989821198972

2

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989212(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989221(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989222(119909) =

(1198981+ 1198982) 1198972

1

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

(34)

5 Experimental Studies

Figure 3 shows the leg rehabilitation machine using an actualhuman loading with a 65 kg weight The automatic device

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

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RotatingMachinery

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Navigation and Observation

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DistributedSensor Networks

International Journal of

The Scientific World Journal 5

1

2

3

4

5

6

78910

11

12

1314

15

16

ADDA

Figure 1 The experimental setup

1b

1205911

2b

1205912 m2

r1

1205792

2a

1a

m1

r1

1205791

l1

l2

p1b = p10 minus Δp1

p1a = p10 + Δp1

p2a=p 20

+ Δp2

p 2b=p 20

minus Δp2

Figure 2 Operation principle of the leg rehabilitation machine driven by PAMs

6 The Scientific World Journal

Table 1 Component specifications

Number Component Specifications1 2 3 4 PAM Festo MAS-20-150N7 8 9 10 Pressure proportional valve Mac PPC5C5 6 Rotary potentiometer Keen Engineering KRT205011 12 13 14 Pressure transducer Jihsense SN 911166 0ndash10 kgfcm2

16 IBM-compatible PC Pentium 18GHzDAC ADC Automation AIO3321

where

1206011119886

= 1198651119886(1199011119886) minus 1198701119886(1199011119886) 11990311205791minus 1198611119886(1199011119886) 1199031

1205791

(29a)

1206011119887

= 1198651119887(1199011119887) minus 1198701119887(1199011119887) 11990311205791minus 1198611119887(1199011119887) 1199031

1205791

(29b)

1206012119886

= 1198652119886(1199012119886) minus 1198702119886(1199012119886) 11990321205792minus 1198612119886(1199012119886) 1199032

1205792

(29c)

1206012119887

= 1198652119887(1199012119887) minus 1198702119887(1199012119887) 11990321205792minus 1198612119887(1199012119887) 1199032

1205792 (29d)

where the spring coefficient119870(119901) and the damping coefficient119861(119901) are given by Reynolds et al [13]

The desired input pressures P119886= [1199011119886

1199012119886]119879 and P

119887=

[1199011119887

1199012119887]119879 for each PAM are generated by the following

equation

P119886(119905) = P

0+ ΔP (119905) P

119887(119905) = P

0minus ΔP (119905) (30)

where P0

= [11990110

11990120]119879 is a nominal constant input PAM

pressure and ΔP(119905) = [Δ1199011

Δ1199012]119879 is the control pressure

input with an arbitrary function of time Because the pressureinputΔP(119905) = [Δ119901

1Δ1199012]119879 and output 120579 = [120579

11205792]119879 the sys-

tem can bewritten as a two-input two-output (TITO) controlsystem The control signal u = [119906

11199062]119879 is proportional to

ΔP based on the pressure proportional valversquos characteristicsThat is ΔP can be used instead of u as a control input

4 Dynamic Model of a Two-Joint LegRehabilitation Machine Driven by PAMs

Figure 2 shows a two-joint leg rehabilitation machine drivenby PAMs and the dynamic equation is given as follows [14]

119872(120579) 120579 + 119862 (120579 120579) 120579 + 119866 (120579) = 120591 (31)

where

119872(120579) = [(1198981+ 1198982) 1198972

119898211989711198972(11990411199042+ 11988811198882)

119898211989711198972(11990411199042+ 11988811198882) 119898

21198972

2

]

119862 (120579 120579) = [0 minus119898

211989711198972(11988811199042minus 11990411198882) 1205792

minus119898211989711198972(11988811199042minus 11990411198882) 1205791

0]

119866 (120579) = [(1198981+ 1198982) 11989711198921199041

minus119898211989721198921199042

]

(32)

and119872(120579) is the moment of inertia 119862(120579 120579) includes Coriolisand centripetal force and 119866(120579) is the gravitational force

Notation 1199041= sin(120579

1) 1199042= sin(120579

2) 1198881= cos(120579

1) and 119888

2=

cos(1205792) Let 119909

1= 1205791 1199092= 1205791 1199093= 1205792 and 119909

4= 1205792 then (31)

can be written as the following state-space form [14]

1= 1199092

2= 1198911(119909) + 119892

11(119909) 1205911+ 119892121205912

3= 1199094

4= 1198912(119909) + 119892

21(119909) + 119892

221205912

(33)

where1198911(119909)

=

(11990411198882minus 11988811199042) [119898211989711198972(11990411199042+ 11988811198882) 1199092

2minus 11989821198972

21199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[(1198981minus 1198982) 11989721198921199041minus 119898211989721198921199042(11990411199042+ 11988811198882)]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

1198912(119909)

=

(11990411198882minus 11988811199042) [minus (119898

1+ 1198982) 1198972

11199092

2+ 119898211989711198972(11990411199042+ 11988811198882) 1199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[minus (119898

1+ 1198982) 11989711198921199041(11990411199042+ 11988811198882) + (119898

1+ 1198982) 11989711198921199042]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989211(119909) =

11989821198972

2

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989212(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989221(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989222(119909) =

(1198981+ 1198982) 1198972

1

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

(34)

5 Experimental Studies

Figure 3 shows the leg rehabilitation machine using an actualhuman loading with a 65 kg weight The automatic device

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

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RotatingMachinery

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DistributedSensor Networks

International Journal of

6 The Scientific World Journal

Table 1 Component specifications

Number Component Specifications1 2 3 4 PAM Festo MAS-20-150N7 8 9 10 Pressure proportional valve Mac PPC5C5 6 Rotary potentiometer Keen Engineering KRT205011 12 13 14 Pressure transducer Jihsense SN 911166 0ndash10 kgfcm2

16 IBM-compatible PC Pentium 18GHzDAC ADC Automation AIO3321

where

1206011119886

= 1198651119886(1199011119886) minus 1198701119886(1199011119886) 11990311205791minus 1198611119886(1199011119886) 1199031

1205791

(29a)

1206011119887

= 1198651119887(1199011119887) minus 1198701119887(1199011119887) 11990311205791minus 1198611119887(1199011119887) 1199031

1205791

(29b)

1206012119886

= 1198652119886(1199012119886) minus 1198702119886(1199012119886) 11990321205792minus 1198612119886(1199012119886) 1199032

1205792

(29c)

1206012119887

= 1198652119887(1199012119887) minus 1198702119887(1199012119887) 11990321205792minus 1198612119887(1199012119887) 1199032

1205792 (29d)

where the spring coefficient119870(119901) and the damping coefficient119861(119901) are given by Reynolds et al [13]

The desired input pressures P119886= [1199011119886

1199012119886]119879 and P

119887=

[1199011119887

1199012119887]119879 for each PAM are generated by the following

equation

P119886(119905) = P

0+ ΔP (119905) P

119887(119905) = P

0minus ΔP (119905) (30)

where P0

= [11990110

11990120]119879 is a nominal constant input PAM

pressure and ΔP(119905) = [Δ1199011

Δ1199012]119879 is the control pressure

input with an arbitrary function of time Because the pressureinputΔP(119905) = [Δ119901

1Δ1199012]119879 and output 120579 = [120579

11205792]119879 the sys-

tem can bewritten as a two-input two-output (TITO) controlsystem The control signal u = [119906

11199062]119879 is proportional to

ΔP based on the pressure proportional valversquos characteristicsThat is ΔP can be used instead of u as a control input

4 Dynamic Model of a Two-Joint LegRehabilitation Machine Driven by PAMs

Figure 2 shows a two-joint leg rehabilitation machine drivenby PAMs and the dynamic equation is given as follows [14]

119872(120579) 120579 + 119862 (120579 120579) 120579 + 119866 (120579) = 120591 (31)

where

119872(120579) = [(1198981+ 1198982) 1198972

119898211989711198972(11990411199042+ 11988811198882)

119898211989711198972(11990411199042+ 11988811198882) 119898

21198972

2

]

119862 (120579 120579) = [0 minus119898

211989711198972(11988811199042minus 11990411198882) 1205792

minus119898211989711198972(11988811199042minus 11990411198882) 1205791

0]

119866 (120579) = [(1198981+ 1198982) 11989711198921199041

minus119898211989721198921199042

]

(32)

and119872(120579) is the moment of inertia 119862(120579 120579) includes Coriolisand centripetal force and 119866(120579) is the gravitational force

Notation 1199041= sin(120579

1) 1199042= sin(120579

2) 1198881= cos(120579

1) and 119888

2=

cos(1205792) Let 119909

1= 1205791 1199092= 1205791 1199093= 1205792 and 119909

4= 1205792 then (31)

can be written as the following state-space form [14]

1= 1199092

2= 1198911(119909) + 119892

11(119909) 1205911+ 119892121205912

3= 1199094

4= 1198912(119909) + 119892

21(119909) + 119892

221205912

(33)

where1198911(119909)

=

(11990411198882minus 11988811199042) [119898211989711198972(11990411199042+ 11988811198882) 1199092

2minus 11989821198972

21199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[(1198981minus 1198982) 11989721198921199041minus 119898211989721198921199042(11990411199042+ 11988811198882)]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

1198912(119909)

=

(11990411198882minus 11988811199042) [minus (119898

1+ 1198982) 1198972

11199092

2+ 119898211989711198972(11990411199042+ 11988811198882) 1199092

4]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

+[minus (119898

1+ 1198982) 11989711198921199041(11990411199042+ 11988811198882) + (119898

1+ 1198982) 11989711198921199042]

11989711198972[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989211(119909) =

11989821198972

2

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989212(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989221(119909) =

minus119898211989711198972(11990411199042+ 11988811198882)

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

11989222(119909) =

(1198981+ 1198982) 1198972

1

11989821198972

11198972

2[(1198981+ 1198982) minus 1198982(11990411199042+ 11988811198882)2

]

(34)

5 Experimental Studies

Figure 3 shows the leg rehabilitation machine using an actualhuman loading with a 65 kg weight The automatic device

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 7

Figure 3 The two-joint leg rehabilitation device with actual human loading

can help patients to recover lower limb motion function bymeans of continuous passive motion such as a sinusoidalwave command an irregular curve command and an end-effect tracking command The experiments include both theproposed approach and PDC for comparison in order toevaluate efficacy and control performance The controllerswere implemented on an Intel Pentium 18GHz PC with asampling time of 5ms and the entire control software wascoded in C++

This study attempts to use as few rules as possible inorder tominimize design effort and complexityTheT-S fuzzymodel of the system is thus given the following four-rulefuzzy model

1198771

IF 1199091is about (1205874) and 119909

3is about (1205874)

THEN = 11986011199091+ 1198611119906

1198772

IF 1199091is about (1205874) and 119909

3is about (minus1205874)

THEN = 11986021199091+ 1198612119906

1198773

IF 1199091is about (minus1205874) and 119909

3is about (1205874)

THEN = 11986031199091+ 1198613119906

1198774

IF 1199091is about (minus1205874) and 119909

3is about (minus1205874)

THEN = 11986041199091+ 1198614119906

(35)

where

1198601=

[[[

[

0 1 0 0

44082 00059 06742 minus00002

0 0 0 1

minus14572 00002 minus39722 00002

]]]

]

1198602=

[[[

[

0 1 0 0

46734 00049 0532 00001

0 0 0 1

minus12145 minus00001 minus3563 00001

]]]

]

1198603=

[[[

[

0 1 0 0

54782 00021 06876 minus00002

0 0 0 1

14525 00002 46723 00002

]]]

]

1198604=

[[[[

[

0 1 0 0

49821 minus00044 06543 00002

0 0 0 1

minus11034 minus00002 35631 00003

]]]]

]

1198611= 1198614=

[[[[

[

0 0

14811 minus2849

0 0

minus2849 237417

]]]]

]

1198612= 1198613=

[[[[

[

0 0

11965 10297

0 0

10297 191501

]]]]

]

119875 =

[[[[

[

162602 minus06640 10734 minus00095

minus06640 03007 minus03455 02871

10734 minus03455 04730 minus03458

minus00095 02871 minus03458 03658

]]]]

]

1198701= [

minus04509 01745 minus07182 00527

minus02807 00034 minus00471 02639]

1198702= [

minus04619 01238 minus07812 00351

minus02827 00068 minus00851 01401]

1198703= [

minus05902 02132 minus08910 00531

minus04107 00117 minus00730 02989]

1198704= [

minus04891 02109 minus06734 004145

minus03180 00085 minus00631 03893]

(36)

which guarantee the stability condition (17) MATLAB Tool-box is used to obtain parameters as 119896

1= 00251 and

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

Table 2 Peak-peak error and phase lag for Figure 5

The proposed approach PDCPeak-peak error Phase lag Peak-peak error Phase lag

1205791

1205792

1205791

1205792

1205791

1205792

1205791

1205792

07 035 49∘ 45∘ 13 065 162∘ 108∘

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(a) The proposed approach

20

10

0

minus10

minus20

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

ReferenceActual

15

0

minus15

1205792

(b) PDC

Figure 4 Sinusoidal wave response for knee and ankle joints

1198962= 001539 For comparison with the proposed controller

the PDC feedback gains are designed to be

1= [

minus03293 01023 minus02538 002317

minus01783 00021 minus002672 04732]

1198702= [

minus04278 01845 minus02451 00731

minus02185 00026 minus00752 00923]

1198703= [

minus06013 03461 minus05482 00231

minus04421 00093 minus00651 03529]

1198704= [

minus03756 03150 minus05391 00421

minus04250 00023 minus00531 03597]

(37)

51 Sinusoidal Wave Response Continuous reciprocation isrequired in order to foster the recovery of extremity functionThe sinusoidal wave responses of the proposed approach andPDC for both knee and ankle joints are shown in Figure 4It is evident that angle trajectories of the proposed approachare close to the command Figure 5 shows that the proposedapproach exhibits less tracking errors than does PDC Thepeak-peak error and phase lag are listed in Table 2 Because

of the interaction of the two joints PDC has significant angleerrors for 120579

1 which will degrade the rehabilitation effect

However supervisory control can overcome the couplingeffect of the two joints to achieve excellent rehabilitationfunction for patients

52 Irregular Curve Response In practical applications itcould be expected that the reference command will changewith different input frequencies The desired trajectories forboth knee and ankle joints are

1205791= 20 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

1205792= 15 lowast 033 (sin (2120587119891

1119905) + sin (2120587119891

2119905) + sin (2120587119891

3119905))

(38)

with 1198911= 005Hz 119891

2= 01Hz and 119891

3= 0066Hz

Figure 6 shows the tracking responses of irregular curvesobtained using both the proposed approach and PDC Track-ing errors for the knee and ankle joints are shown in Figure 7Clearly the angle error of the proposed approach is averagemaintained within 2∘ However the proposed approach iscapable of adapting to different frequencies

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

14

12

10

8

6

4

2

0

minus2

minus41009080706050403020100

Time (s)

Ang

le (d

eg)

1205791 error1205792 error

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 5 Angle tracking errors for both knee and ankle

20

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

1205791

Ang

le (d

eg)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

120579215

10

5

0

minus5

minus10

minus15

Actual

(a) The proposed approach

20

10

0

minus10

10

0

minus10

minus200 10 20 30 40 50 60 70 80 90 100

Time (s)

Ang

le (d

eg)

Ang

le (d

eg)

1205791

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Reference

1205792

Actual

(b) PDC

Figure 6 Irregular curve response for both knee and ankle joints

53 Elliptic Response The desired end-effect or trajectory isgiven by

119909119889(119905) = 0614 minus 0015 sdot cos (02120587 sdot 119905 minus 120587)

119910119889(119905) = minus01 sdot sin (02120587 sdot 119905 minus 2120587)

(39)

where 0 le 119905 le 20 seconds

The end-effect tracking responses in the 119909 119910 coordinatefor both the proposed approach and PDC are shown inFigure 8 and the end-effect position tracking errors aredisplayed in Figure 9 It is evident that tracking behavior ofthe proposed approach is better than that of the PDC Ascan be seen the tracking errors of the proposed approach are

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

(a) The proposed approach

Ang

le (d

eg)

1009080706050403020100

Time (s)

1205791 error1205792 error

15

10

5

0

minus5

minus10

(b) PDC

Figure 7 Angle tracking errors for both the proposed approach and PDC

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(a) The proposed approach

02

015

01

005

0

minus005

minus01

minus015

minus020 01 02 03 04 05 06 07 08

Y(m

)

X (m)

ReferenceActual

Initialposition

(b) PDC

Figure 8 Elliptic response of the proposed approach and PDC

within 003m On the other hand angle tracking errors ofknee and ankle joints are shown in Figure 10

Moreover it is difficult to enhance end-effector trackingperformance using the PDC algorithm because the PDCcannot overcome the nonlinearity of PAMs and the structuralinteraction However the proposed approach overcomessuccessfully the coupling effect and parameter uncertaintiesof the system As seen in the experimental results theproposed approach can attain excellent end-effector trackingperformance in rehabilitation function

6 Conclusions

In this study a novel composite fuzzy control is proposed andapplied in the two-joint leg rehabilitation device driven byPAMsThe proposed controller is not only capable of decom-posing nonlinear systems into a set of linear subsystems but isalso capable of simplifying a complex nonlinear system usinglinear control techniques with the control gains determinedusingMATLABrsquos LMI Toolbox based on the Lyapunov stabil-ity theoremMoreover the supervisory control can overcome

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 11

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(a) The proposed approach

02

018

016

014

012

01

008

006

004

002

00 2 4 6 8 10 12 14 16 18 20

Time (s)

Posit

ion

(m)

(b) PDC

Figure 9 The end-effect position tracking errors

14

12

10

8

6

4

2

0

minus2

minus40 2 4 6 8 10 12 14 16 18 20

Ang

le (d

eg)

Time (s)

1205791 error1205792 error

(a) The proposed approach

0 2 4 6 8 10 12 14 16 18 20

Time (s)

15

10

5

0

minus5

minus10

1205791 error1205792 error

Ang

le (d

eg)

(b) PDC

Figure 10 Angle tracking errors of knee and ankle joints

the coupling effect for a leg rehabilitation machine Experi-mental results show that the system response of the proposedapproach was in good agreement with that of the refer-ence input and guarantee robustness to system parameteruncertainties

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D G Caldwell G A Medrano-Cerda and M GoodwinldquoControl of pneumatic muscle actuatorsrdquo IEEE Control SystemsMagazine vol 15 no 1 pp 40ndash48 1995

[2] C P Chou and B Hannaford ldquoStatic and dynamic characteris-tics ofMcKibben pneumatic artificialmusclesrdquo inProceedings ofthe IEEE International Conference on Robotics and Automationpp 281ndash286 May 1994

[3] T Noritsugu and T Tanaka ldquoApplication of rubber artificialmuscle manipulator as a rehabilitation robotrdquo IEEEASMETransactions on Mechatronics vol 2 no 4 pp 259ndash267 1997

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 The Scientific World Journal

[4] J H Lilly and L Yang ldquoSliding mode tracking for pneumaticmuscle actuators in opposing pair configurationrdquo IEEE Trans-actions on Control Systems Technology vol 4 pp 550ndash558 2005

[5] K K Ahn and H P H Anh ldquoDesign and implementation of anadaptive recurrent neural networks (ARNN) controller of thepneumatic artificial muscle (PAM) manipulatorrdquoMechatronicsvol 19 no 6 pp 816ndash828 2009

[6] X Shen ldquoNonlinearmodel-based control of pneumatic artificialmuscle servo systemsrdquo Control Engineering Practice vol 18 no3 pp 311ndash317 2010

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[8] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[9] K K Ahn and H P H Anh ldquoInverse double NARX fuzzymodeling for system identificationrdquo IEEEASME Transactionson Mechatronics vol 15 no 1 pp 136ndash148 2010

[10] L Seddiki K Guelton and J Zaytoon ldquoConcept and Takagi-Sugeno descriptor tracking controller design of a closedmuscu-lar chain lower-limb rehabilitation devicerdquo IET Control Theoryand Applications vol 4 no 8 pp 1407ndash1420 2010

[11] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[12] M K Chang J J Liou andM L Chen ldquoT-S fuzzymodel-basedtracking control of a one-dimensional manipulator actuated bypneumatic artificial musclesrdquo Control Engineering Practice vol19 no 12 pp 1442ndash1449 2011

[13] D B Reynolds D W Repperger C A Phillips and G BandryldquoModeling the dynamic characteristics of pneumatic musclerdquoAnnals of Biomedical Engineering vol 31 no 3 pp 310ndash3172003

[14] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of