Research ArticleWheeled Mobile Robot RBFNN Dynamic Surface ControlBased on Disturbance Observer

Shaohua Luo12 Songli Wu2 Zhaoqin Liu2 and Hao Guan1

1 State Key Laboratory of Mechanical Transmission Chongqing University Chongqing 400044 China2Department of Mechanical Engineering Chongqing Aerospace Polytechnic College Chongqing 400021 China

Correspondence should be addressed to Shaohua Luo hua66com163com

Received 13 November 2013 Accepted 9 December 2013 Published 11 February 2014

Academic Editors X-G Yan and X-S Yang

Copyright copy 2014 Shaohua Luo et al 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

This paper focuses on the problem of an adaptive neural network dynamic surface control (DSC) based on disturbance observerfor the wheeled mobile robot with uncertain parameters and unknown disturbancesThe nonlinear observer is used to compensatefor the external disturbance and the neural network is employed to approximate the uncertain and nonlinear items of systemThen the Lyapunov theory is introduced to demonstrate the stabilization of the proposed control algorithm Finally the simulationresults illustrate that the proposed algorithm not only is superior to conventional DSC in trajectory tracking and external frictiondisturbance compensation but also has better response adaptive ability and robustness

1 Introduction

Wheeled mobile robot is an intelligent object It can col-lect the surrounding environmental information from theconstant feedback of sensors and allodial makes decisionsThen it outputsmotion instructions and guides itself tomoveto the destination quickly with high precision of trajectorytracking [1 2]However there still exists a challenging issue tocontrol robot to obtain perfect dynamic performance becauseits mathematical model is usually multivariable coupled andnonlinear Dong andKuhnert [3] presented a tracking controlapproach for mobile robots with both parameter and non-parameter uncertainties In [4] an wavelet-network-basedcontroller is developed for mobile robots with unstructureddynamics and disturbances Chwa [5] designed a positionand heading direction controller using the SMC method fornonholonomic mobile robots Gu and Hu [6] studied thereceding horizon tracking control on the wheeled mobilerobots which used the optimized method to accelerate theconvergence speed of errors

Nowadays various intelligence control algorithms formobile robot have been represented in the literature suchas genetic algorithm [7] iterative learning control [8] neu-ral networks [9] fuzzy logic [10] and backstepping In

the above-mentioned methods the backstepping method ispreferred In [11] an adaptive tracking controller using abackstepping method is presented for the dynamic modelof mobile robots with unknown parameters Unfortunatelythe backstepping suffers from the ldquoexplosion of complexityrdquocaused by the repeated differentiation of virtual controlfunctions [12] Dynamic surface control (DSC) [13 14] is anew control technique by introducing a first-order filter ateach recursive step of the backstepping design proceduresuch that the differentiation item on the virtual function canbe avoided Zhang and Ge further studied the control designfor some special nonlinear systems with time delay or deadzone using the DSC technique [15] However the adaptiveparameters involved in the aforementioned DSC still imposethat a large number of parameters need to be tuned online inthe NN approximation In addition a robust adaptive DSCfor a class of uncertain perturbed strict-feedback nonlinearsystems preceded by unknown backlash-like hysteresis isproposed [16]

On the other hand the output of the disturbance observercan be extensively used in feed-forward compensation ofexternal disturbances The disturbance observers can givefast excellent tracking performance and smooth controlactions without the use of large feedback gains [17] Zhongyi

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014 Article ID 634936 9 pageshttpdxdoiorg1011552014634936

2 ISRN Applied Mathematics

et al [18] presented a disturbance observer-based controlscheme for free-floating space manipulator with nonlineardynamics derived using the virtual manipulator approachYan presented a nonlinear dynamic output feedback decen-tralized controller and further studied state and parameterestimation for nonlinear delay systems using sliding modeobserver [19 20] The friction compensation schemes basedon disturbance observer is in that they are not based on anyparticular friction models [21]

To the best of our knowledge the DSC method anddisturbance observer have been seldom used in the controlof wheeled mobile robot so far The first main contribu-tion of this paper is the compensation of external frictionby using the nonlinear disturbance observer The secondcontribution is the design of the dynamic surface controlmethod combined with neural network During the designprocess neural networks are employed to approximate thenonlinearities and adaptive method and DSC are used toconstruct neural network controller It means that uncertainparameters are taken into account and explosion of com-plexity is solved Finally to testify to the superiority of theproposed control algorithm a comparison between DSCneural network dynamic surface controller (NNDSC) andNNDSC with nonlinear observer is studied The simulationresults are provided to demonstrate the effectiveness androbustness objectively against the parameter uncertaintiesand external disturbances

2 Kinetic Models

21 Mathematical Model of Wheeled Mobile Robot The sche-matic diagram of wheeled mobile robot is shown in Figure 1It has two independent actuated rear-wheels with servomotor driven To change the relative input voltage to realizespeed difference of two rear-wheels it is achieved to adjustthe position of both car body and tracking trajectory Frontwheels only support the car body as supporting roller

The dynamics of wheeled mobile robot with uncertainparameters and nonlinearity in Figure 1 is generally describedby [22]

V = minus

2119888

119872119903

2+ 2119868

119908

V +119896119903

119872119903

2+ 2119868

119908

(119906

119903+ 119906

119897)

120601 = minus

2119888119871

2

119868V1199032+ 2119868

119908119871

2

120601 +

119896119903119871

119868V1199032+ 2119868

119908119871

2(119906

119903minus 119906

119897)

119896 =

119894119896

119898

119877

119886

(1)

where 119868V stands for the moment of inertia around the centerof gravity of robot 119868

119908denotes themoment of inertia of wheel

119872 represents the mass of robot 119896 stands for the driving gainfactor 119903 denotes the radius of wheel 119888 represents the viscousfriction factor of both wheel and ground 119906

119903and 119906

119897stand for

the right and left driving input of rear axle respectively 119877119886

denotes the armature resistance of motor 119896119898represents the

electromagnetism torque constant of motor 119894 stands for thetransmission ratio of reducer V denotes the velocity of robotand 120593 represents the azimuth of robot

y

x0

2L 2r

120601

l

M

Figure 1 The wheeled mobile robot

For simplicity the following notations are introduced

119909

1= V 119909

2= 120601 119909

3=

2

119886

1= minus

2

119872119903

2+ 2119868

119908

119887

1=

119903

119872119903

2+ 2119868

119908

119886

2= minus

2119871

2

119868V1199032+ 2119868

119908119871

2 119887

2=

119903119871

119868V1199032+ 2119868

119908119871

2

(2)

Outdoor wheeled mobile robot suffers more unknownand uncertain disturbance than indoor one In the real con-trol system it is unavoidable for some implicit prior unknownmodeling and external disturbance to exist Suppose the sumof all external uncertainty items to describe with function 119889Uncertainties consisted of gear clearance existence of reducerin the transmission system friction coefficient variation ofroadbed and motor parameter change because of ambienttemperature surrounded material wear and so on influencestatic and dynamic performances of robot to some extentTherefore the drive gain 119896 and friction coefficient 119888 areuncertain

By using these notations the dynamic model of wheeledmobile robot can be described by the following differentialequations

1= 119886

1119888119909

1+ 119887

1119896 (119906

119903+ 119906

119897)

2= 119909

3

3= 119886

2119888119909

3+ 119887

2119896 (119906

119903minus 119906

119897) + 119889

(3)

Remark 1 119889 is unknown but its upper bound is |119889(

120601 120601 119905)| le

120575 and 120575 ge 0

22 System Decoupling System dynamic equations ofwheeled mobile robot are a coupled system Firstly it isto decouple the system mentioned and then to convert toparametric strict-feedback forms It is defined as follows

[

119906

119903

119906

119897

] = [

1 minus1

0 1

] [

119906

1

119906

2

] (4)

ISRN Applied Mathematics 3

Then 119906

119903= 119906

1minus 119906

2 (3) can successfully decouple two inde-

pendent subsystems below

1= 119886

1119888119909

1+ 119887

1119896119906

1 (5)

2= 119909

3

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906

2+ 119889

(6)

Assumption 2 The reference signals119910119903and119910119889are continuous

about 119905 and satisfy (119910

119894 119910

119894 119910

119894) isin Ξ

119894 119894 = 119903 119889 where Ξ

119894=

(119910

119894 119910

119894 119910

119894) 119910

2

119894+ 119910

2

119894+ 119910

2

119894le 119861

119894 and 119861

119894is the known positive

constant

3 RBFNN Dynamic Surface ControllerBased on Nonlinear Disturbance Observer

31 RBF Neural Network Figure 2 shows the RBF neuralnetwork expression which is given as follows

119891

119898(120579 119911) = 120579

119879120585 (119911)

(7)

where 119911 isin Ω sub 119877

119899 is the neural network input vector Ωdenotes some compact set in 119877

119899 120579 isin 119877

119897 is the neural networkweight vector 119897 is the node number of neuron and 120585(119911) =

[120585

1(119911) 120585

2(119911) 120585

119899(119911)]

119879isin 119877

119879 where 120585119894(119911) is basic function

Then Gaussian function can be formulated as

120585

119894(119911) = exp[minus

1003816

1003816

1003816

1003816

119911 minus 120583

119894

1003816

1003816

1003816

1003816

2

120590

2

119894

]

= exp[minus(119911 minus 120583

119894)

119879

(119911 minus 120583

119894)

120590

2

119894

] 119894 = 1 2 119897

(8)

where 120583119894= [120583

1198941 120583

1198942 120583

119894119899]

119879 is the center of basic function120585

119894(119911) and 120590

119894is the width of 120585

119894(119911)

Remark 3 For the Gaussian neural network Ω119911sub 119877

119899 and119891(sdot) stands for the real value continuous function in the fieldof Ω119911 Therefore for any given 120576 gt 0 there always exist

positive integer 119897 and constant vector 120579 such that

max119911isinΩ119911

1003816

1003816

1003816

1003816

119891 (119911) minus 119891

119898(120579 119911)

1003816

1003816

1003816

1003816

lt 120576 (9)

32 Nonlinear Disturbance Observer Define state variable119883 = [119909

2 119909

3]

119879 Using (6) it is constructed as

119883 = 119891 (119883) + 119861 (119883) 119906

2+ 119862 (119883) 119889 (10)

where 119891 = [

1199093

11988621198881199093+11988721198961199061

] 119861 = [

0

minus21198872119896 ] and 119862 = [

0

1]

By equation transformation (10) can be rewritten as

119862119889 =

119883 minus 119891 minus 119861119906

2 (11)

On the basis of the difference between real output andevaluated output of system the evaluator can be adjusted todesign disturbance observer

Input layer Hidden layer

Output layer

z1

z2

zn

Ω

Ω

Ω

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

Figure 2 A general architecture of RBF neural network

Suppose that nonlinear disturbance observer owns thefollowing form

_119889= 119871 (

119883 minus 119891 minus 119861119906

2minus 119862

_119889)

(12)

where_119889is estimated value of 119889

Define the auxiliary variable

119885 =

_119889minus119906

2

(13)

The gain 119871 of nonlinear disturbance observer can be de-scribed by the following differential equation

119871 =

120597119906

2(119883)

120597119883

(14)

Differentiating 119885 gives

119885 =

_119889minus

2= 119871 (minus119891 minus 119861119906

2minus 119862

_119889)

= 119871 (minus119891 minus 119861119906

2minus 119862 (119885 + 119906

2))

(15)

Define the error as 119890119889= 119889minus

_119889 and the time derivative of 119890

119889

is computed by

119890

119889+ 119871119862119890

119889= minus

_119889+ 119871119862 (119889minus

_119889)

(16)

Substituting (12) into (16) gives

119890

119889+ 119871119862119890

119889= minus119871 (

119883 minus 119891 minus 119861119906

2minus 119862119889) = 0 (17)

Introduce gain constant vector 119871 = [119888

1 119888

2] then

119906

2= 119888

1119909

2+ 119888

2119909

3 (18)

At this present stage nonlinear disturbance observer is de-signed as

_119889= 119885 + 119888

1119909

2+ 119888

2119909

3

119885 = minus 119888

1119909

3+ 119888

2[minus119886

2119888119909

3minus 119887

2119896119906

1+ 2119887

2119896119906

2

minus119885 minus 119888

1119909

2minus 119888

2119909

3]

(19)

4 ISRN Applied Mathematics

33 Neural Network Dynamic Surface Controller with Distur-bance Compensation By obtaining the output of neural net-work dynamic surface controller the disturbance of wheeledmobile robot existing is compensated well The control law isdesigned as

119906

2= 119906ND minus 119906

119863 (20)

where 119906ND is output of neural network dynamic surfacecontroller and119906

119863is output of nonlinear disturbance observer

Furthermore (10) yields

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896 (119906ND minus 119906

119863) + 119889 (21)

Define 119906119863= minus12119887

2119896sdot

_119889 and then substitute it into (21)

that

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889(22)

To aim at the wheeledmobile robot represented in (5) and(22) the designing steps of neural network dynamic surfacecontroller are described as follows

Step 1 For the reference signal119910119889 define the dynamic surface

as 1198781= 119909

1minus 119910

119889 Obviously the time derivative of 119878

1is given

by

119878

1= 119886

1119888119909

1+ 119887

1119896119906

1minus 119910

119889= 119891

1+ 119887

1119896119906

1 (23)

where 119891

1(119878

1) = 119886

1119888119909

1minus 119910

119889 Notice that 119891

1contains the

derivative of 119910119889and unknown friction coefficient 119888 This

will make the classical dynamic surface control become verycomplex and troublesome Therefore we will employ theneural network to approximate the function119891

1 According to

the description above there exists a neural network systemsuch that

119891

1(119878

1) = 120579

119879

1120585

1(119878

1) + 120575

1(119878

1) (24)

where 1205751is the approximation error and satisfies |120575

1(119878

1)| lt 120576

1

According to (23) and (24) the corresponding control lawis chosen as follows

119906

1=

_119896

(

_119896

2

+ 120576) 119887

1

(minus

_120579

119879

1120585

1minus 119896

1119878

1) (25)

where_119896and

_120579 1

are the evaluated value of 119896 and 1205791at the time

of 119905 respectively and 119896

1gt 0 is a design constant

The adaptive laws are chosen as_120579 1

= 120578

1(120585

1119878

1minus 119898

1

_120579 1

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(26)

where 1205781 12057831198981 and119898

3are the positive design parameters

Step 2 For the reference signal119910119903 define the dynamic surface

as 1198782= 119909

2minus 119910

119903 Obviously the time derivative of 119878

2is given

by

119878

2= 119909

3minus 119910

119903 (27)

Construct the virtual control law 120572 as

120572 = minus119896

2119878

2+ 119910

119903 (28)

with 119896

2gt 0 being a design parameter

According to the known 120572120572119891is calculated by the first-

order filter

120591

119891+ 120572

119891= 120572 120572

119891(0) = 120572 (0) (29)

where 120591 is a positive filter time constant

Step 3 Define the dynamic surface as 1198783= 119909

3minus 120572

119891 the time

derivative of 1198783is given by

119878

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889 minus

119891

= 119891

2+ 119887

2119896119906

1minus 2119887

2119896119906ND minus

119891

(30)

where 1198912(119878

3) = 119886

2119888119909

3+

119889 Notice that 1198912contains unknown

119889 and uncertainty friction coefficient 119888 To make full use ofuniversal approximation theorem of neural network for any

given 120576

2gt 0 there is a neural network

_120579

119879

2120585

2(119878

3) such that

119891

2(119878

3) = 120579

119879

2120585

2(119878

3) + 120575

2(119878

3) (31)

where 1205752is the approximation error and satisfies |120575

2| le 120576

2

Combining (30) and (31) design control law as

119906ND =

_119896

(

_119896

2

+ 120576) 2119887

2

(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

(32)

where_120579 2

is the estimation of 1205792at the time of 119905 and 119896

3gt 0 is

a design constantThe adaptive laws are chosen as

_120579 2

= 120578

2(120585

2119878

3minus 119898

2

_120579 2

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(33)

From the analysis mentioned above the controlschematic of wheeled mobile robot neural network dynamicsurface control based on nonlinear disturbance observer isshown clearly in Figure 3

34 A Comparison with Traditional Dynamic Surface ControlTo verify the proposed controller we make a comparisonbetween neural network dynamic surface control and tradi-tional dynamic surface control

Step 1 From (23) the control law is chosen as

119906

1=

1

119887

1119896

(minus119896

1119878

1minus 119886

1119888119909

1+ 119910

119889) (34)

Step 2 According to the definition the virtual control func-tion is equal to (28)

ISRN Applied Mathematics 5

Adaptive law (26)

Control law (25)

(28)Virtual control

Filter (29)

(33)Adaptive law

+

+

+

Vd

Z

Nonlinear disturbance observer

m1 m3 1205781 1205783

Vd S1

S2

S3

+

minus

minus

minus

minus

k1

k2

k3

b1

b2 120576

120576

f1

f2

u1

u1

u2

ddt

ddt

ddt

Φd

Φ 120572120572f120591

Auxiliary variable(19)

u2(18)

dZint

d

Control law (32)

uND

minus(2b2k)minus1

m2 m3 1205782 1205783

Wheeled mobile robot(5) and (6)

Φ

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

V

Figure 3 Control schematic of wheeled mobile robot

Step 3 From (30) the corresponding control law is gotten as

119906ND =

1

2119887

2119896

(119896

3119878

3+ 119886

2119888119909

3+ 119887

2119896119906

1+

119889 minus

119891) (35)

So far in contrast to (25) and (32) which belonged to neuralnetwork dynamics surface controller it is easy to see that(34) and (35) which belonged to traditional dynamics surfacecontroller not only need more precise mathematical modelbut also require more items of expression and coupling items

4 Stability Analysis

Define the filter error 119910 = 120572

119891minus 120572 Differentiating it results in

the following differential equation

119891= minus

119910

120591

(36)

Introduce boundary layer differential equation as

119910 = minus

119910

120591

+ 119861 (119878

2 119878

3 119910

_119896

_120579 2

119910

119903 119910

119903 119910

119903) (37)

with 119861 = 119896

2

119878

2minus 119910

119903and

119910 119910 le minus

119910

2

120591

+ 119910

2+

1

4

119861

2

(38)

With (24) and (25) (23) can be expressed as

119878

1= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ 119887

1

_119896119906

1

= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ (1 minus

120576

_119896

2

+ 120576

)(minus

_120579

119879

1120585

1minus 119896

1119878

1)

6 ISRN Applied Mathematics

= minus119896

1119878

1minus

120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+

120576

_119896

2

+ 120576

(

_120579

119879

1120585

1+ 119896

1119878

1)

le minus119896

1119878

1minus

120579

119879

1120585

1minus 119887

1

119896119906

1+ 120574

1

(39)

where 1205741(119878

1

_119896

_120579 1

119910

119889 119910

119889) is continuous function and satis-

fies 1205741ge |120575

1+ (120576(

_119896

2

+ 120576))(

_120579

119879

1120585

1+ 119896

1119878

1)|

In the same way the following can be deduced

119878

2= 119878

3+ 119910 minus 119896

2119878

2

119878

3= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891minus 2119887

2

_119896119906ND

= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891

minus (1 minus

120576

_119896

2

+ 120576

)(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

le minus

120579

119879

2120585

2minus 119887

2

119896119906

1+ 2119887

2

119896119906ND minus 119896

3119878

3+ 120574

2

(40)

where 120574

2(119878

2 119878

3

_119896

_120579 2

119910

119903 119910

119903) is continuous function and

satisfies 1205742ge |120575

2+ (120576(

_119896

2

+ 120576))(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)|

According to Youngrsquos inequality the calculation producesthe following equality

119878

1

119878

1le (minus119896

1+ 1) 119878

2

1minus

120579

119879

1120585

1119878

1minus 119887

1

119896119906

1119878

1+

1

4

120574

2

1 (41)

In the same way it can be deduced that

119878

2

119878

2=

1

4

119878

2

3+

1

4

119910

2minus (119896

2minus 2) 119878

2

2

119878

3

119878

3le minus

120579

119879

2120585

2119878

3minus 119887

2

119896119906

1119878

3+ 2119887

2

119896119906ND1198783

minus (119896

3minus 1) 119878

2

3+

1

4

120574

2

2

(42)

For any given 119901 gt 0 the sets can be defined

Ω

1= (119878

1

_119896

_120579 1

) 119878

2

1+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

2= (119878

1 119878

2

_119896

_120579 1

)

2

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

3= (119878

1 119878

2 119878

3

_119896

_120579 1

_120579 2

119910)

3

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2

+ 120578

minus1

1

120579

119879

1

120579

1+ 120578

minus1

2

120579

119879

2

120579

2+ 119910

2le 2119901

(43)

Theorem 4 To aim at the wheeled mobile robot expressed in(3) satisfying Assumption 2 and giving a positive number 119901for all satisfying initial condition 119881(0) le 119901 the controllers

(25) and (32) and the adaptive laws (26) and (33) guaranteesemiglobal uniform ultimate boundedness of closed loop systemwith tracking error converging to zero

Proof Choose the Lyapunov function candidate as

119881 =

1

2

(

3

sum

119894=1

119878

2

119894+ 119910

2+

2

sum

119894=1

120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) (44)

Combining (26) and (33) and differentiating (44) give

119881 le (minus119896

1+ 1) 119878

2

1minus (119896

2minus 2) 119878

2

2minus (119896

3minus

5

4

) 119878

2

3

+

2

sum

119894=1

1

4

120574

2

119894+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2minus 119887

1

119896119906

1119878

1minus 119887

2

119896119906

1119878

3

+ 2119887

2

119896119906ND1198783 +

119896119878

1119906

1minus 119898

3

119896

_119896minus

2

sum

119894=1

119898

119894

120579

119879

119894

_120579 119894

(45)

Remark 5 Consider that minus119898119894

120579

119879

119894

_120579 119894

= minus119898

119894

120579

119879

119894(

120579

119894+ 120579

119894) le

minus2

minus1119898

119894

120579

119894

2

+2

minus1119898

119894120579

119894

2 119894 = 1 2 and minus1198983

119896

_119896le minus2

minus1119898

3

119896

2+

2

minus1119898

3119896

2Then it can be verified easily that

119881 le (minus119896

1+ 1) 119878

2

1minus

119896119906

1(119887

2119878

3+ 119887

1119878

1) +

2

sum

119894=1

1

4

120574

2

119894minus (119896

2minus 2) 119878

2

2

+ 2119887

2

119896119906ND1198783 minus (119896

3minus

5

4

) 119878

2

3+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2

+

119896119878

1119906

1minus

1

2

(119898

1

1003817

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

1003817

2

+ 119898

3

119896

2)

+

1

2

(119898

1

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

2

+ 119898

3119896

2)

(46)

If 119881(119905) = (12)(sum

3

119894=1119878

2

119894+ 119910

2+ sum

2

119894=1120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) = 119901

then 1205742119894le Δ

2

119894and1198612 le Γ

2 Define120583 = 2

minus1(119898

1120579

1

2+119898

2120579

2

2+

119898

3119896

2) + sum

2

119894=1(14)Δ

2

119894+ (14)Γ

2Taking those into account there exists

119881 (119905) le minus2119886

0119881 (119905) + 120583 (47)

In the condition of 119881(119905) = 119901 just choosing the appro-priate design constant can guarantee 119886

0ge 1205832119901 and then

119881(119905) le 0 119881(119905) le 119901 belongs to invariant set From (47) wehave

0 le 119881 (119905) le

120583

2119886

0

+ (119881 (0) minus

120583

2119886

0

) 119890

minus21198860119905 (48)

5 Simulation Analysis

In this section in order to illustrate the effectiveness ofwheeled mobile robot neural network dynamic surface con-trol method which was based on nonlinear disturbanceobserver the simulation will be conducted under the initialcondition of 119909

1= 119909

2= 119909

3= 02 At the same time in order to

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Applied Mathematics

et al [18] presented a disturbance observer-based controlscheme for free-floating space manipulator with nonlineardynamics derived using the virtual manipulator approachYan presented a nonlinear dynamic output feedback decen-tralized controller and further studied state and parameterestimation for nonlinear delay systems using sliding modeobserver [19 20] The friction compensation schemes basedon disturbance observer is in that they are not based on anyparticular friction models [21]

To the best of our knowledge the DSC method anddisturbance observer have been seldom used in the controlof wheeled mobile robot so far The first main contribu-tion of this paper is the compensation of external frictionby using the nonlinear disturbance observer The secondcontribution is the design of the dynamic surface controlmethod combined with neural network During the designprocess neural networks are employed to approximate thenonlinearities and adaptive method and DSC are used toconstruct neural network controller It means that uncertainparameters are taken into account and explosion of com-plexity is solved Finally to testify to the superiority of theproposed control algorithm a comparison between DSCneural network dynamic surface controller (NNDSC) andNNDSC with nonlinear observer is studied The simulationresults are provided to demonstrate the effectiveness androbustness objectively against the parameter uncertaintiesand external disturbances

2 Kinetic Models

21 Mathematical Model of Wheeled Mobile Robot The sche-matic diagram of wheeled mobile robot is shown in Figure 1It has two independent actuated rear-wheels with servomotor driven To change the relative input voltage to realizespeed difference of two rear-wheels it is achieved to adjustthe position of both car body and tracking trajectory Frontwheels only support the car body as supporting roller

The dynamics of wheeled mobile robot with uncertainparameters and nonlinearity in Figure 1 is generally describedby [22]

V = minus

2119888

119872119903

2+ 2119868

119908

V +119896119903

119872119903

2+ 2119868

119908

(119906

119903+ 119906

119897)

120601 = minus

2119888119871

2

119868V1199032+ 2119868

119908119871

2

120601 +

119896119903119871

119868V1199032+ 2119868

119908119871

2(119906

119903minus 119906

119897)

119896 =

119894119896

119898

119877

119886

(1)

where 119868V stands for the moment of inertia around the centerof gravity of robot 119868

119908denotes themoment of inertia of wheel

119872 represents the mass of robot 119896 stands for the driving gainfactor 119903 denotes the radius of wheel 119888 represents the viscousfriction factor of both wheel and ground 119906

119903and 119906

119897stand for

the right and left driving input of rear axle respectively 119877119886

denotes the armature resistance of motor 119896119898represents the

electromagnetism torque constant of motor 119894 stands for thetransmission ratio of reducer V denotes the velocity of robotand 120593 represents the azimuth of robot

y

x0

2L 2r

120601

l

M

Figure 1 The wheeled mobile robot

For simplicity the following notations are introduced

119909

1= V 119909

2= 120601 119909

3=

2

119886

1= minus

2

119872119903

2+ 2119868

119908

119887

1=

119903

119872119903

2+ 2119868

119908

119886

2= minus

2119871

2

119868V1199032+ 2119868

119908119871

2 119887

2=

119903119871

119868V1199032+ 2119868

119908119871

2

(2)

Outdoor wheeled mobile robot suffers more unknownand uncertain disturbance than indoor one In the real con-trol system it is unavoidable for some implicit prior unknownmodeling and external disturbance to exist Suppose the sumof all external uncertainty items to describe with function 119889Uncertainties consisted of gear clearance existence of reducerin the transmission system friction coefficient variation ofroadbed and motor parameter change because of ambienttemperature surrounded material wear and so on influencestatic and dynamic performances of robot to some extentTherefore the drive gain 119896 and friction coefficient 119888 areuncertain

By using these notations the dynamic model of wheeledmobile robot can be described by the following differentialequations

1= 119886

1119888119909

1+ 119887

1119896 (119906

119903+ 119906

119897)

2= 119909

3

3= 119886

2119888119909

3+ 119887

2119896 (119906

119903minus 119906

119897) + 119889

(3)

Remark 1 119889 is unknown but its upper bound is |119889(

120601 120601 119905)| le

120575 and 120575 ge 0

22 System Decoupling System dynamic equations ofwheeled mobile robot are a coupled system Firstly it isto decouple the system mentioned and then to convert toparametric strict-feedback forms It is defined as follows

[

119906

119903

119906

119897

] = [

1 minus1

0 1

] [

119906

1

119906

2

] (4)

ISRN Applied Mathematics 3

Then 119906

119903= 119906

1minus 119906

2 (3) can successfully decouple two inde-

pendent subsystems below

1= 119886

1119888119909

1+ 119887

1119896119906

1 (5)

2= 119909

3

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906

2+ 119889

(6)

Assumption 2 The reference signals119910119903and119910119889are continuous

about 119905 and satisfy (119910

119894 119910

119894 119910

119894) isin Ξ

119894 119894 = 119903 119889 where Ξ

119894=

(119910

119894 119910

119894 119910

119894) 119910

2

119894+ 119910

2

119894+ 119910

2

119894le 119861

119894 and 119861

119894is the known positive

constant

3 RBFNN Dynamic Surface ControllerBased on Nonlinear Disturbance Observer

31 RBF Neural Network Figure 2 shows the RBF neuralnetwork expression which is given as follows

119891

119898(120579 119911) = 120579

119879120585 (119911)

(7)

where 119911 isin Ω sub 119877

119899 is the neural network input vector Ωdenotes some compact set in 119877

119899 120579 isin 119877

119897 is the neural networkweight vector 119897 is the node number of neuron and 120585(119911) =

[120585

1(119911) 120585

2(119911) 120585

119899(119911)]

119879isin 119877

119879 where 120585119894(119911) is basic function

Then Gaussian function can be formulated as

120585

119894(119911) = exp[minus

1003816

1003816

1003816

1003816

119911 minus 120583

119894

1003816

1003816

1003816

1003816

2

120590

2

119894

]

= exp[minus(119911 minus 120583

119894)

119879

(119911 minus 120583

119894)

120590

2

119894

] 119894 = 1 2 119897

(8)

where 120583119894= [120583

1198941 120583

1198942 120583

119894119899]

119879 is the center of basic function120585

119894(119911) and 120590

119894is the width of 120585

119894(119911)

Remark 3 For the Gaussian neural network Ω119911sub 119877

119899 and119891(sdot) stands for the real value continuous function in the fieldof Ω119911 Therefore for any given 120576 gt 0 there always exist

positive integer 119897 and constant vector 120579 such that

max119911isinΩ119911

1003816

1003816

1003816

1003816

119891 (119911) minus 119891

119898(120579 119911)

1003816

1003816

1003816

1003816

lt 120576 (9)

32 Nonlinear Disturbance Observer Define state variable119883 = [119909

2 119909

3]

119879 Using (6) it is constructed as

119883 = 119891 (119883) + 119861 (119883) 119906

2+ 119862 (119883) 119889 (10)

where 119891 = [

1199093

11988621198881199093+11988721198961199061

] 119861 = [

0

minus21198872119896 ] and 119862 = [

0

1]

By equation transformation (10) can be rewritten as

119862119889 =

119883 minus 119891 minus 119861119906

2 (11)

On the basis of the difference between real output andevaluated output of system the evaluator can be adjusted todesign disturbance observer

Input layer Hidden layer

Output layer

z1

z2

zn

Ω

Ω

Ω

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

Figure 2 A general architecture of RBF neural network

Suppose that nonlinear disturbance observer owns thefollowing form

_119889= 119871 (

119883 minus 119891 minus 119861119906

2minus 119862

_119889)

(12)

where_119889is estimated value of 119889

Define the auxiliary variable

119885 =

_119889minus119906

2

(13)

The gain 119871 of nonlinear disturbance observer can be de-scribed by the following differential equation

119871 =

120597119906

2(119883)

120597119883

(14)

Differentiating 119885 gives

119885 =

_119889minus

2= 119871 (minus119891 minus 119861119906

2minus 119862

_119889)

= 119871 (minus119891 minus 119861119906

2minus 119862 (119885 + 119906

2))

(15)

Define the error as 119890119889= 119889minus

_119889 and the time derivative of 119890

119889

is computed by

119890

119889+ 119871119862119890

119889= minus

_119889+ 119871119862 (119889minus

_119889)

(16)

Substituting (12) into (16) gives

119890

119889+ 119871119862119890

119889= minus119871 (

119883 minus 119891 minus 119861119906

2minus 119862119889) = 0 (17)

Introduce gain constant vector 119871 = [119888

1 119888

2] then

119906

2= 119888

1119909

2+ 119888

2119909

3 (18)

At this present stage nonlinear disturbance observer is de-signed as

_119889= 119885 + 119888

1119909

2+ 119888

2119909

3

119885 = minus 119888

1119909

3+ 119888

2[minus119886

2119888119909

3minus 119887

2119896119906

1+ 2119887

2119896119906

2

minus119885 minus 119888

1119909

2minus 119888

2119909

3]

(19)

4 ISRN Applied Mathematics

33 Neural Network Dynamic Surface Controller with Distur-bance Compensation By obtaining the output of neural net-work dynamic surface controller the disturbance of wheeledmobile robot existing is compensated well The control law isdesigned as

119906

2= 119906ND minus 119906

119863 (20)

where 119906ND is output of neural network dynamic surfacecontroller and119906

119863is output of nonlinear disturbance observer

Furthermore (10) yields

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896 (119906ND minus 119906

119863) + 119889 (21)

Define 119906119863= minus12119887

2119896sdot

_119889 and then substitute it into (21)

that

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889(22)

To aim at the wheeledmobile robot represented in (5) and(22) the designing steps of neural network dynamic surfacecontroller are described as follows

Step 1 For the reference signal119910119889 define the dynamic surface

as 1198781= 119909

1minus 119910

119889 Obviously the time derivative of 119878

1is given

by

119878

1= 119886

1119888119909

1+ 119887

1119896119906

1minus 119910

119889= 119891

1+ 119887

1119896119906

1 (23)

where 119891

1(119878

1) = 119886

1119888119909

1minus 119910

119889 Notice that 119891

1contains the

derivative of 119910119889and unknown friction coefficient 119888 This

will make the classical dynamic surface control become verycomplex and troublesome Therefore we will employ theneural network to approximate the function119891

1 According to

the description above there exists a neural network systemsuch that

119891

1(119878

1) = 120579

119879

1120585

1(119878

1) + 120575

1(119878

1) (24)

where 1205751is the approximation error and satisfies |120575

1(119878

1)| lt 120576

1

According to (23) and (24) the corresponding control lawis chosen as follows

119906

1=

_119896

(

_119896

2

+ 120576) 119887

1

(minus

_120579

119879

1120585

1minus 119896

1119878

1) (25)

where_119896and

_120579 1

are the evaluated value of 119896 and 1205791at the time

of 119905 respectively and 119896

1gt 0 is a design constant

The adaptive laws are chosen as_120579 1

= 120578

1(120585

1119878

1minus 119898

1

_120579 1

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(26)

where 1205781 12057831198981 and119898

3are the positive design parameters

Step 2 For the reference signal119910119903 define the dynamic surface

as 1198782= 119909

2minus 119910

119903 Obviously the time derivative of 119878

2is given

by

119878

2= 119909

3minus 119910

119903 (27)

Construct the virtual control law 120572 as

120572 = minus119896

2119878

2+ 119910

119903 (28)

with 119896

2gt 0 being a design parameter

According to the known 120572120572119891is calculated by the first-

order filter

120591

119891+ 120572

119891= 120572 120572

119891(0) = 120572 (0) (29)

where 120591 is a positive filter time constant

Step 3 Define the dynamic surface as 1198783= 119909

3minus 120572

119891 the time

derivative of 1198783is given by

119878

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889 minus

119891

= 119891

2+ 119887

2119896119906

1minus 2119887

2119896119906ND minus

119891

(30)

where 1198912(119878

3) = 119886

2119888119909

3+

119889 Notice that 1198912contains unknown

119889 and uncertainty friction coefficient 119888 To make full use ofuniversal approximation theorem of neural network for any

given 120576

2gt 0 there is a neural network

_120579

119879

2120585

2(119878

3) such that

119891

2(119878

3) = 120579

119879

2120585

2(119878

3) + 120575

2(119878

3) (31)

where 1205752is the approximation error and satisfies |120575

2| le 120576

2

Combining (30) and (31) design control law as

119906ND =

_119896

(

_119896

2

+ 120576) 2119887

2

(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

(32)

where_120579 2

is the estimation of 1205792at the time of 119905 and 119896

3gt 0 is

a design constantThe adaptive laws are chosen as

_120579 2

= 120578

2(120585

2119878

3minus 119898

2

_120579 2

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(33)

From the analysis mentioned above the controlschematic of wheeled mobile robot neural network dynamicsurface control based on nonlinear disturbance observer isshown clearly in Figure 3

34 A Comparison with Traditional Dynamic Surface ControlTo verify the proposed controller we make a comparisonbetween neural network dynamic surface control and tradi-tional dynamic surface control

Step 1 From (23) the control law is chosen as

119906

1=

1

119887

1119896

(minus119896

1119878

1minus 119886

1119888119909

1+ 119910

119889) (34)

Step 2 According to the definition the virtual control func-tion is equal to (28)

ISRN Applied Mathematics 5

Adaptive law (26)

Control law (25)

(28)Virtual control

Filter (29)

(33)Adaptive law

+

+

+

Vd

Z

Nonlinear disturbance observer

m1 m3 1205781 1205783

Vd S1

S2

S3

+

minus

minus

minus

minus

k1

k2

k3

b1

b2 120576

120576

f1

f2

u1

u1

u2

ddt

ddt

ddt

Φd

Φ 120572120572f120591

Auxiliary variable(19)

u2(18)

dZint

d

Control law (32)

uND

minus(2b2k)minus1

m2 m3 1205782 1205783

Wheeled mobile robot(5) and (6)

Φ

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

V

Figure 3 Control schematic of wheeled mobile robot

Step 3 From (30) the corresponding control law is gotten as

119906ND =

1

2119887

2119896

(119896

3119878

3+ 119886

2119888119909

3+ 119887

2119896119906

1+

119889 minus

119891) (35)

So far in contrast to (25) and (32) which belonged to neuralnetwork dynamics surface controller it is easy to see that(34) and (35) which belonged to traditional dynamics surfacecontroller not only need more precise mathematical modelbut also require more items of expression and coupling items

4 Stability Analysis

Define the filter error 119910 = 120572

119891minus 120572 Differentiating it results in

the following differential equation

119891= minus

119910

120591

(36)

Introduce boundary layer differential equation as

119910 = minus

119910

120591

+ 119861 (119878

2 119878

3 119910

_119896

_120579 2

119910

119903 119910

119903 119910

119903) (37)

with 119861 = 119896

2

119878

2minus 119910

119903and

119910 119910 le minus

119910

2

120591

+ 119910

2+

1

4

119861

2

(38)

With (24) and (25) (23) can be expressed as

119878

1= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ 119887

1

_119896119906

1

= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ (1 minus

120576

_119896

2

+ 120576

)(minus

_120579

119879

1120585

1minus 119896

1119878

1)

6 ISRN Applied Mathematics

= minus119896

1119878

1minus

120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+

120576

_119896

2

+ 120576

(

_120579

119879

1120585

1+ 119896

1119878

1)

le minus119896

1119878

1minus

120579

119879

1120585

1minus 119887

1

119896119906

1+ 120574

1

(39)

where 1205741(119878

1

_119896

_120579 1

119910

119889 119910

119889) is continuous function and satis-

fies 1205741ge |120575

1+ (120576(

_119896

2

+ 120576))(

_120579

119879

1120585

1+ 119896

1119878

1)|

In the same way the following can be deduced

119878

2= 119878

3+ 119910 minus 119896

2119878

2

119878

3= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891minus 2119887

2

_119896119906ND

= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891

minus (1 minus

120576

_119896

2

+ 120576

)(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

le minus

120579

119879

2120585

2minus 119887

2

119896119906

1+ 2119887

2

119896119906ND minus 119896

3119878

3+ 120574

2

(40)

where 120574

2(119878

2 119878

3

_119896

_120579 2

119910

119903 119910

119903) is continuous function and

satisfies 1205742ge |120575

2+ (120576(

_119896

2

+ 120576))(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)|

According to Youngrsquos inequality the calculation producesthe following equality

119878

1

119878

1le (minus119896

1+ 1) 119878

2

1minus

120579

119879

1120585

1119878

1minus 119887

1

119896119906

1119878

1+

1

4

120574

2

1 (41)

In the same way it can be deduced that

119878

2

119878

2=

1

4

119878

2

3+

1

4

119910

2minus (119896

2minus 2) 119878

2

2

119878

3

119878

3le minus

120579

119879

2120585

2119878

3minus 119887

2

119896119906

1119878

3+ 2119887

2

119896119906ND1198783

minus (119896

3minus 1) 119878

2

3+

1

4

120574

2

2

(42)

For any given 119901 gt 0 the sets can be defined

Ω

1= (119878

1

_119896

_120579 1

) 119878

2

1+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

2= (119878

1 119878

2

_119896

_120579 1

)

2

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

3= (119878

1 119878

2 119878

3

_119896

_120579 1

_120579 2

119910)

3

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2

+ 120578

minus1

1

120579

119879

1

120579

1+ 120578

minus1

2

120579

119879

2

120579

2+ 119910

2le 2119901

(43)

Theorem 4 To aim at the wheeled mobile robot expressed in(3) satisfying Assumption 2 and giving a positive number 119901for all satisfying initial condition 119881(0) le 119901 the controllers

(25) and (32) and the adaptive laws (26) and (33) guaranteesemiglobal uniform ultimate boundedness of closed loop systemwith tracking error converging to zero

Proof Choose the Lyapunov function candidate as

119881 =

1

2

(

3

sum

119894=1

119878

2

119894+ 119910

2+

2

sum

119894=1

120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) (44)

Combining (26) and (33) and differentiating (44) give

119881 le (minus119896

1+ 1) 119878

2

1minus (119896

2minus 2) 119878

2

2minus (119896

3minus

5

4

) 119878

2

3

+

2

sum

119894=1

1

4

120574

2

119894+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2minus 119887

1

119896119906

1119878

1minus 119887

2

119896119906

1119878

3

+ 2119887

2

119896119906ND1198783 +

119896119878

1119906

1minus 119898

3

119896

_119896minus

2

sum

119894=1

119898

119894

120579

119879

119894

_120579 119894

(45)

Remark 5 Consider that minus119898119894

120579

119879

119894

_120579 119894

= minus119898

119894

120579

119879

119894(

120579

119894+ 120579

119894) le

minus2

minus1119898

119894

120579

119894

2

+2

minus1119898

119894120579

119894

2 119894 = 1 2 and minus1198983

119896

_119896le minus2

minus1119898

3

119896

2+

2

minus1119898

3119896

2Then it can be verified easily that

119881 le (minus119896

1+ 1) 119878

2

1minus

119896119906

1(119887

2119878

3+ 119887

1119878

1) +

2

sum

119894=1

1

4

120574

2

119894minus (119896

2minus 2) 119878

2

2

+ 2119887

2

119896119906ND1198783 minus (119896

3minus

5

4

) 119878

2

3+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2

+

119896119878

1119906

1minus

1

2

(119898

1

1003817

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

1003817

2

+ 119898

3

119896

2)

+

1

2

(119898

1

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

2

+ 119898

3119896

2)

(46)

If 119881(119905) = (12)(sum

3

119894=1119878

2

119894+ 119910

2+ sum

2

119894=1120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) = 119901

then 1205742119894le Δ

2

119894and1198612 le Γ

2 Define120583 = 2

minus1(119898

1120579

1

2+119898

2120579

2

2+

119898

3119896

2) + sum

2

119894=1(14)Δ

2

119894+ (14)Γ

2Taking those into account there exists

119881 (119905) le minus2119886

0119881 (119905) + 120583 (47)

In the condition of 119881(119905) = 119901 just choosing the appro-priate design constant can guarantee 119886

0ge 1205832119901 and then

119881(119905) le 0 119881(119905) le 119901 belongs to invariant set From (47) wehave

0 le 119881 (119905) le

120583

2119886

0

+ (119881 (0) minus

120583

2119886

0

) 119890

minus21198860119905 (48)

5 Simulation Analysis

In this section in order to illustrate the effectiveness ofwheeled mobile robot neural network dynamic surface con-trol method which was based on nonlinear disturbanceobserver the simulation will be conducted under the initialcondition of 119909

1= 119909

2= 119909

3= 02 At the same time in order to

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

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ISRN Applied Mathematics 3

Then 119906

119903= 119906

1minus 119906

2 (3) can successfully decouple two inde-

pendent subsystems below

1= 119886

1119888119909

1+ 119887

1119896119906

1 (5)

2= 119909

3

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906

2+ 119889

(6)

Assumption 2 The reference signals119910119903and119910119889are continuous

about 119905 and satisfy (119910

119894 119910

119894 119910

119894) isin Ξ

119894 119894 = 119903 119889 where Ξ

119894=

(119910

119894 119910

119894 119910

119894) 119910

2

119894+ 119910

2

119894+ 119910

2

119894le 119861

119894 and 119861

119894is the known positive

constant

3 RBFNN Dynamic Surface ControllerBased on Nonlinear Disturbance Observer

31 RBF Neural Network Figure 2 shows the RBF neuralnetwork expression which is given as follows

119891

119898(120579 119911) = 120579

119879120585 (119911)

(7)

where 119911 isin Ω sub 119877

119899 is the neural network input vector Ωdenotes some compact set in 119877

119899 120579 isin 119877

119897 is the neural networkweight vector 119897 is the node number of neuron and 120585(119911) =

[120585

1(119911) 120585

2(119911) 120585

119899(119911)]

119879isin 119877

119879 where 120585119894(119911) is basic function

Then Gaussian function can be formulated as

120585

119894(119911) = exp[minus

1003816

1003816

1003816

1003816

119911 minus 120583

119894

1003816

1003816

1003816

1003816

2

120590

2

119894

]

= exp[minus(119911 minus 120583

119894)

119879

(119911 minus 120583

119894)

120590

2

119894

] 119894 = 1 2 119897

(8)

where 120583119894= [120583

1198941 120583

1198942 120583

119894119899]

119879 is the center of basic function120585

119894(119911) and 120590

119894is the width of 120585

119894(119911)

Remark 3 For the Gaussian neural network Ω119911sub 119877

119899 and119891(sdot) stands for the real value continuous function in the fieldof Ω119911 Therefore for any given 120576 gt 0 there always exist

positive integer 119897 and constant vector 120579 such that

max119911isinΩ119911

1003816

1003816

1003816

1003816

119891 (119911) minus 119891

119898(120579 119911)

1003816

1003816

1003816

1003816

lt 120576 (9)

32 Nonlinear Disturbance Observer Define state variable119883 = [119909

2 119909

3]

119879 Using (6) it is constructed as

119883 = 119891 (119883) + 119861 (119883) 119906

2+ 119862 (119883) 119889 (10)

where 119891 = [

1199093

11988621198881199093+11988721198961199061

] 119861 = [

0

minus21198872119896 ] and 119862 = [

0

1]

By equation transformation (10) can be rewritten as

119862119889 =

119883 minus 119891 minus 119861119906

2 (11)

On the basis of the difference between real output andevaluated output of system the evaluator can be adjusted todesign disturbance observer

Input layer Hidden layer

Output layer

z1

z2

zn

Ω

Ω

Ω

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

Figure 2 A general architecture of RBF neural network

Suppose that nonlinear disturbance observer owns thefollowing form

_119889= 119871 (

119883 minus 119891 minus 119861119906

2minus 119862

_119889)

(12)

where_119889is estimated value of 119889

Define the auxiliary variable

119885 =

_119889minus119906

2

(13)

The gain 119871 of nonlinear disturbance observer can be de-scribed by the following differential equation

119871 =

120597119906

2(119883)

120597119883

(14)

Differentiating 119885 gives

119885 =

_119889minus

2= 119871 (minus119891 minus 119861119906

2minus 119862

_119889)

= 119871 (minus119891 minus 119861119906

2minus 119862 (119885 + 119906

2))

(15)

Define the error as 119890119889= 119889minus

_119889 and the time derivative of 119890

119889

is computed by

119890

119889+ 119871119862119890

119889= minus

_119889+ 119871119862 (119889minus

_119889)

(16)

Substituting (12) into (16) gives

119890

119889+ 119871119862119890

119889= minus119871 (

119883 minus 119891 minus 119861119906

2minus 119862119889) = 0 (17)

Introduce gain constant vector 119871 = [119888

1 119888

2] then

119906

2= 119888

1119909

2+ 119888

2119909

3 (18)

At this present stage nonlinear disturbance observer is de-signed as

_119889= 119885 + 119888

1119909

2+ 119888

2119909

3

119885 = minus 119888

1119909

3+ 119888

2[minus119886

2119888119909

3minus 119887

2119896119906

1+ 2119887

2119896119906

2

minus119885 minus 119888

1119909

2minus 119888

2119909

3]

(19)

4 ISRN Applied Mathematics

33 Neural Network Dynamic Surface Controller with Distur-bance Compensation By obtaining the output of neural net-work dynamic surface controller the disturbance of wheeledmobile robot existing is compensated well The control law isdesigned as

119906

2= 119906ND minus 119906

119863 (20)

where 119906ND is output of neural network dynamic surfacecontroller and119906

119863is output of nonlinear disturbance observer

Furthermore (10) yields

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896 (119906ND minus 119906

119863) + 119889 (21)

Define 119906119863= minus12119887

2119896sdot

_119889 and then substitute it into (21)

that

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889(22)

To aim at the wheeledmobile robot represented in (5) and(22) the designing steps of neural network dynamic surfacecontroller are described as follows

Step 1 For the reference signal119910119889 define the dynamic surface

as 1198781= 119909

1minus 119910

119889 Obviously the time derivative of 119878

1is given

by

119878

1= 119886

1119888119909

1+ 119887

1119896119906

1minus 119910

119889= 119891

1+ 119887

1119896119906

1 (23)

where 119891

1(119878

1) = 119886

1119888119909

1minus 119910

119889 Notice that 119891

1contains the

derivative of 119910119889and unknown friction coefficient 119888 This

will make the classical dynamic surface control become verycomplex and troublesome Therefore we will employ theneural network to approximate the function119891

1 According to

the description above there exists a neural network systemsuch that

119891

1(119878

1) = 120579

119879

1120585

1(119878

1) + 120575

1(119878

1) (24)

where 1205751is the approximation error and satisfies |120575

1(119878

1)| lt 120576

1

According to (23) and (24) the corresponding control lawis chosen as follows

119906

1=

_119896

(

_119896

2

+ 120576) 119887

1

(minus

_120579

119879

1120585

1minus 119896

1119878

1) (25)

where_119896and

_120579 1

are the evaluated value of 119896 and 1205791at the time

of 119905 respectively and 119896

1gt 0 is a design constant

The adaptive laws are chosen as_120579 1

= 120578

1(120585

1119878

1minus 119898

1

_120579 1

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(26)

where 1205781 12057831198981 and119898

3are the positive design parameters

Step 2 For the reference signal119910119903 define the dynamic surface

as 1198782= 119909

2minus 119910

119903 Obviously the time derivative of 119878

2is given

by

119878

2= 119909

3minus 119910

119903 (27)

Construct the virtual control law 120572 as

120572 = minus119896

2119878

2+ 119910

119903 (28)

with 119896

2gt 0 being a design parameter

According to the known 120572120572119891is calculated by the first-

order filter

120591

119891+ 120572

119891= 120572 120572

119891(0) = 120572 (0) (29)

where 120591 is a positive filter time constant

Step 3 Define the dynamic surface as 1198783= 119909

3minus 120572

119891 the time

derivative of 1198783is given by

119878

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889 minus

119891

= 119891

2+ 119887

2119896119906

1minus 2119887

2119896119906ND minus

119891

(30)

where 1198912(119878

3) = 119886

2119888119909

3+

119889 Notice that 1198912contains unknown

119889 and uncertainty friction coefficient 119888 To make full use ofuniversal approximation theorem of neural network for any

given 120576

2gt 0 there is a neural network

_120579

119879

2120585

2(119878

3) such that

119891

2(119878

3) = 120579

119879

2120585

2(119878

3) + 120575

2(119878

3) (31)

where 1205752is the approximation error and satisfies |120575

2| le 120576

2

Combining (30) and (31) design control law as

119906ND =

_119896

(

_119896

2

+ 120576) 2119887

2

(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

(32)

where_120579 2

is the estimation of 1205792at the time of 119905 and 119896

3gt 0 is

a design constantThe adaptive laws are chosen as

_120579 2

= 120578

2(120585

2119878

3minus 119898

2

_120579 2

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(33)

From the analysis mentioned above the controlschematic of wheeled mobile robot neural network dynamicsurface control based on nonlinear disturbance observer isshown clearly in Figure 3

34 A Comparison with Traditional Dynamic Surface ControlTo verify the proposed controller we make a comparisonbetween neural network dynamic surface control and tradi-tional dynamic surface control

Step 1 From (23) the control law is chosen as

119906

1=

1

119887

1119896

(minus119896

1119878

1minus 119886

1119888119909

1+ 119910

119889) (34)

Step 2 According to the definition the virtual control func-tion is equal to (28)

ISRN Applied Mathematics 5

Adaptive law (26)

Control law (25)

(28)Virtual control

Filter (29)

(33)Adaptive law

+

+

+

Vd

Z

Nonlinear disturbance observer

m1 m3 1205781 1205783

Vd S1

S2

S3

+

minus

minus

minus

minus

k1

k2

k3

b1

b2 120576

120576

f1

f2

u1

u1

u2

ddt

ddt

ddt

Φd

Φ 120572120572f120591

Auxiliary variable(19)

u2(18)

dZint

d

Control law (32)

uND

minus(2b2k)minus1

m2 m3 1205782 1205783

Wheeled mobile robot(5) and (6)

Φ

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

V

Figure 3 Control schematic of wheeled mobile robot

Step 3 From (30) the corresponding control law is gotten as

119906ND =

1

2119887

2119896

(119896

3119878

3+ 119886

2119888119909

3+ 119887

2119896119906

1+

119889 minus

119891) (35)

So far in contrast to (25) and (32) which belonged to neuralnetwork dynamics surface controller it is easy to see that(34) and (35) which belonged to traditional dynamics surfacecontroller not only need more precise mathematical modelbut also require more items of expression and coupling items

4 Stability Analysis

Define the filter error 119910 = 120572

119891minus 120572 Differentiating it results in

the following differential equation

119891= minus

119910

120591

(36)

Introduce boundary layer differential equation as

119910 = minus

119910

120591

+ 119861 (119878

2 119878

3 119910

_119896

_120579 2

119910

119903 119910

119903 119910

119903) (37)

with 119861 = 119896

2

119878

2minus 119910

119903and

119910 119910 le minus

119910

2

120591

+ 119910

2+

1

4

119861

2

(38)

With (24) and (25) (23) can be expressed as

119878

1= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ 119887

1

_119896119906

1

= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ (1 minus

120576

_119896

2

+ 120576

)(minus

_120579

119879

1120585

1minus 119896

1119878

1)

6 ISRN Applied Mathematics

= minus119896

1119878

1minus

120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+

120576

_119896

2

+ 120576

(

_120579

119879

1120585

1+ 119896

1119878

1)

le minus119896

1119878

1minus

120579

119879

1120585

1minus 119887

1

119896119906

1+ 120574

1

(39)

where 1205741(119878

1

_119896

_120579 1

119910

119889 119910

119889) is continuous function and satis-

fies 1205741ge |120575

1+ (120576(

_119896

2

+ 120576))(

_120579

119879

1120585

1+ 119896

1119878

1)|

In the same way the following can be deduced

119878

2= 119878

3+ 119910 minus 119896

2119878

2

119878

3= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891minus 2119887

2

_119896119906ND

= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891

minus (1 minus

120576

_119896

2

+ 120576

)(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

le minus

120579

119879

2120585

2minus 119887

2

119896119906

1+ 2119887

2

119896119906ND minus 119896

3119878

3+ 120574

2

(40)

where 120574

2(119878

2 119878

3

_119896

_120579 2

119910

119903 119910

119903) is continuous function and

satisfies 1205742ge |120575

2+ (120576(

_119896

2

+ 120576))(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)|

According to Youngrsquos inequality the calculation producesthe following equality

119878

1

119878

1le (minus119896

1+ 1) 119878

2

1minus

120579

119879

1120585

1119878

1minus 119887

1

119896119906

1119878

1+

1

4

120574

2

1 (41)

In the same way it can be deduced that

119878

2

119878

2=

1

4

119878

2

3+

1

4

119910

2minus (119896

2minus 2) 119878

2

2

119878

3

119878

3le minus

120579

119879

2120585

2119878

3minus 119887

2

119896119906

1119878

3+ 2119887

2

119896119906ND1198783

minus (119896

3minus 1) 119878

2

3+

1

4

120574

2

2

(42)

For any given 119901 gt 0 the sets can be defined

Ω

1= (119878

1

_119896

_120579 1

) 119878

2

1+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

2= (119878

1 119878

2

_119896

_120579 1

)

2

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

3= (119878

1 119878

2 119878

3

_119896

_120579 1

_120579 2

119910)

3

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2

+ 120578

minus1

1

120579

119879

1

120579

1+ 120578

minus1

2

120579

119879

2

120579

2+ 119910

2le 2119901

(43)

Theorem 4 To aim at the wheeled mobile robot expressed in(3) satisfying Assumption 2 and giving a positive number 119901for all satisfying initial condition 119881(0) le 119901 the controllers

(25) and (32) and the adaptive laws (26) and (33) guaranteesemiglobal uniform ultimate boundedness of closed loop systemwith tracking error converging to zero

Proof Choose the Lyapunov function candidate as

119881 =

1

2

(

3

sum

119894=1

119878

2

119894+ 119910

2+

2

sum

119894=1

120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) (44)

Combining (26) and (33) and differentiating (44) give

119881 le (minus119896

1+ 1) 119878

2

1minus (119896

2minus 2) 119878

2

2minus (119896

3minus

5

4

) 119878

2

3

+

2

sum

119894=1

1

4

120574

2

119894+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2minus 119887

1

119896119906

1119878

1minus 119887

2

119896119906

1119878

3

+ 2119887

2

119896119906ND1198783 +

119896119878

1119906

1minus 119898

3

119896

_119896minus

2

sum

119894=1

119898

119894

120579

119879

119894

_120579 119894

(45)

Remark 5 Consider that minus119898119894

120579

119879

119894

_120579 119894

= minus119898

119894

120579

119879

119894(

120579

119894+ 120579

119894) le

minus2

minus1119898

119894

120579

119894

2

+2

minus1119898

119894120579

119894

2 119894 = 1 2 and minus1198983

119896

_119896le minus2

minus1119898

3

119896

2+

2

minus1119898

3119896

2Then it can be verified easily that

119881 le (minus119896

1+ 1) 119878

2

1minus

119896119906

1(119887

2119878

3+ 119887

1119878

1) +

2

sum

119894=1

1

4

120574

2

119894minus (119896

2minus 2) 119878

2

2

+ 2119887

2

119896119906ND1198783 minus (119896

3minus

5

4

) 119878

2

3+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2

+

119896119878

1119906

1minus

1

2

(119898

1

1003817

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

1003817

2

+ 119898

3

119896

2)

+

1

2

(119898

1

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

2

+ 119898

3119896

2)

(46)

If 119881(119905) = (12)(sum

3

119894=1119878

2

119894+ 119910

2+ sum

2

119894=1120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) = 119901

then 1205742119894le Δ

2

119894and1198612 le Γ

2 Define120583 = 2

minus1(119898

1120579

1

2+119898

2120579

2

2+

119898

3119896

2) + sum

2

119894=1(14)Δ

2

119894+ (14)Γ

2Taking those into account there exists

119881 (119905) le minus2119886

0119881 (119905) + 120583 (47)

In the condition of 119881(119905) = 119901 just choosing the appro-priate design constant can guarantee 119886

0ge 1205832119901 and then

119881(119905) le 0 119881(119905) le 119901 belongs to invariant set From (47) wehave

0 le 119881 (119905) le

120583

2119886

0

+ (119881 (0) minus

120583

2119886

0

) 119890

minus21198860119905 (48)

5 Simulation Analysis

In this section in order to illustrate the effectiveness ofwheeled mobile robot neural network dynamic surface con-trol method which was based on nonlinear disturbanceobserver the simulation will be conducted under the initialcondition of 119909

1= 119909

2= 119909

3= 02 At the same time in order to

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRN Applied Mathematics

33 Neural Network Dynamic Surface Controller with Distur-bance Compensation By obtaining the output of neural net-work dynamic surface controller the disturbance of wheeledmobile robot existing is compensated well The control law isdesigned as

119906

2= 119906ND minus 119906

119863 (20)

where 119906ND is output of neural network dynamic surfacecontroller and119906

119863is output of nonlinear disturbance observer

Furthermore (10) yields

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896 (119906ND minus 119906

119863) + 119889 (21)

Define 119906119863= minus12119887

2119896sdot

_119889 and then substitute it into (21)

that

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889(22)

To aim at the wheeledmobile robot represented in (5) and(22) the designing steps of neural network dynamic surfacecontroller are described as follows

Step 1 For the reference signal119910119889 define the dynamic surface

as 1198781= 119909

1minus 119910

119889 Obviously the time derivative of 119878

1is given

by

119878

1= 119886

1119888119909

1+ 119887

1119896119906

1minus 119910

119889= 119891

1+ 119887

1119896119906

1 (23)

where 119891

1(119878

1) = 119886

1119888119909

1minus 119910

119889 Notice that 119891

1contains the

derivative of 119910119889and unknown friction coefficient 119888 This

will make the classical dynamic surface control become verycomplex and troublesome Therefore we will employ theneural network to approximate the function119891

1 According to

the description above there exists a neural network systemsuch that

119891

1(119878

1) = 120579

119879

1120585

1(119878

1) + 120575

1(119878

1) (24)

where 1205751is the approximation error and satisfies |120575

1(119878

1)| lt 120576

1

According to (23) and (24) the corresponding control lawis chosen as follows

119906

1=

_119896

(

_119896

2

+ 120576) 119887

1

(minus

_120579

119879

1120585

1minus 119896

1119878

1) (25)

where_119896and

_120579 1

are the evaluated value of 119896 and 1205791at the time

of 119905 respectively and 119896

1gt 0 is a design constant

The adaptive laws are chosen as_120579 1

= 120578

1(120585

1119878

1minus 119898

1

_120579 1

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(26)

where 1205781 12057831198981 and119898

3are the positive design parameters

Step 2 For the reference signal119910119903 define the dynamic surface

as 1198782= 119909

2minus 119910

119903 Obviously the time derivative of 119878

2is given

by

119878

2= 119909

3minus 119910

119903 (27)

Construct the virtual control law 120572 as

120572 = minus119896

2119878

2+ 119910

119903 (28)

with 119896

2gt 0 being a design parameter

According to the known 120572120572119891is calculated by the first-

order filter

120591

119891+ 120572

119891= 120572 120572

119891(0) = 120572 (0) (29)

where 120591 is a positive filter time constant

Step 3 Define the dynamic surface as 1198783= 119909

3minus 120572

119891 the time

derivative of 1198783is given by

119878

3= 119886

2119888119909

3+ 119887

2119896119906

1minus 2119887

2119896119906ND +

119889 minus

119891

= 119891

2+ 119887

2119896119906

1minus 2119887

2119896119906ND minus

119891

(30)

where 1198912(119878

3) = 119886

2119888119909

3+

119889 Notice that 1198912contains unknown

119889 and uncertainty friction coefficient 119888 To make full use ofuniversal approximation theorem of neural network for any

given 120576

2gt 0 there is a neural network

_120579

119879

2120585

2(119878

3) such that

119891

2(119878

3) = 120579

119879

2120585

2(119878

3) + 120575

2(119878

3) (31)

where 1205752is the approximation error and satisfies |120575

2| le 120576

2

Combining (30) and (31) design control law as

119906ND =

_119896

(

_119896

2

+ 120576) 2119887

2

(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

(32)

where_120579 2

is the estimation of 1205792at the time of 119905 and 119896

3gt 0 is

a design constantThe adaptive laws are chosen as

_120579 2

= 120578

2(120585

2119878

3minus 119898

2

_120579 2

)

_119896= 120578

3(119878

1119906

1minus 119898

3

_119896)

(33)

From the analysis mentioned above the controlschematic of wheeled mobile robot neural network dynamicsurface control based on nonlinear disturbance observer isshown clearly in Figure 3

34 A Comparison with Traditional Dynamic Surface ControlTo verify the proposed controller we make a comparisonbetween neural network dynamic surface control and tradi-tional dynamic surface control

Step 1 From (23) the control law is chosen as

119906

1=

1

119887

1119896

(minus119896

1119878

1minus 119886

1119888119909

1+ 119910

119889) (34)

Step 2 According to the definition the virtual control func-tion is equal to (28)

ISRN Applied Mathematics 5

Adaptive law (26)

Control law (25)

(28)Virtual control

Filter (29)

(33)Adaptive law

+

+

+

Vd

Z

Nonlinear disturbance observer

m1 m3 1205781 1205783

Vd S1

S2

S3

+

minus

minus

minus

minus

k1

k2

k3

b1

b2 120576

120576

f1

f2

u1

u1

u2

ddt

ddt

ddt

Φd

Φ 120572120572f120591

Auxiliary variable(19)

u2(18)

dZint

d

Control law (32)

uND

minus(2b2k)minus1

m2 m3 1205782 1205783

Wheeled mobile robot(5) and (6)

Φ

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

V

Figure 3 Control schematic of wheeled mobile robot

Step 3 From (30) the corresponding control law is gotten as

119906ND =

1

2119887

2119896

(119896

3119878

3+ 119886

2119888119909

3+ 119887

2119896119906

1+

119889 minus

119891) (35)

So far in contrast to (25) and (32) which belonged to neuralnetwork dynamics surface controller it is easy to see that(34) and (35) which belonged to traditional dynamics surfacecontroller not only need more precise mathematical modelbut also require more items of expression and coupling items

4 Stability Analysis

Define the filter error 119910 = 120572

119891minus 120572 Differentiating it results in

the following differential equation

119891= minus

119910

120591

(36)

Introduce boundary layer differential equation as

119910 = minus

119910

120591

+ 119861 (119878

2 119878

3 119910

_119896

_120579 2

119910

119903 119910

119903 119910

119903) (37)

with 119861 = 119896

2

119878

2minus 119910

119903and

119910 119910 le minus

119910

2

120591

+ 119910

2+

1

4

119861

2

(38)

With (24) and (25) (23) can be expressed as

119878

1= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ 119887

1

_119896119906

1

= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ (1 minus

120576

_119896

2

+ 120576

)(minus

_120579

119879

1120585

1minus 119896

1119878

1)

6 ISRN Applied Mathematics

= minus119896

1119878

1minus

120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+

120576

_119896

2

+ 120576

(

_120579

119879

1120585

1+ 119896

1119878

1)

le minus119896

1119878

1minus

120579

119879

1120585

1minus 119887

1

119896119906

1+ 120574

1

(39)

where 1205741(119878

1

_119896

_120579 1

119910

119889 119910

119889) is continuous function and satis-

fies 1205741ge |120575

1+ (120576(

_119896

2

+ 120576))(

_120579

119879

1120585

1+ 119896

1119878

1)|

In the same way the following can be deduced

119878

2= 119878

3+ 119910 minus 119896

2119878

2

119878

3= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891minus 2119887

2

_119896119906ND

= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891

minus (1 minus

120576

_119896

2

+ 120576

)(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

le minus

120579

119879

2120585

2minus 119887

2

119896119906

1+ 2119887

2

119896119906ND minus 119896

3119878

3+ 120574

2

(40)

where 120574

2(119878

2 119878

3

_119896

_120579 2

119910

119903 119910

119903) is continuous function and

satisfies 1205742ge |120575

2+ (120576(

_119896

2

+ 120576))(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)|

According to Youngrsquos inequality the calculation producesthe following equality

119878

1

119878

1le (minus119896

1+ 1) 119878

2

1minus

120579

119879

1120585

1119878

1minus 119887

1

119896119906

1119878

1+

1

4

120574

2

1 (41)

In the same way it can be deduced that

119878

2

119878

2=

1

4

119878

2

3+

1

4

119910

2minus (119896

2minus 2) 119878

2

2

119878

3

119878

3le minus

120579

119879

2120585

2119878

3minus 119887

2

119896119906

1119878

3+ 2119887

2

119896119906ND1198783

minus (119896

3minus 1) 119878

2

3+

1

4

120574

2

2

(42)

For any given 119901 gt 0 the sets can be defined

Ω

1= (119878

1

_119896

_120579 1

) 119878

2

1+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

2= (119878

1 119878

2

_119896

_120579 1

)

2

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

3= (119878

1 119878

2 119878

3

_119896

_120579 1

_120579 2

119910)

3

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2

+ 120578

minus1

1

120579

119879

1

120579

1+ 120578

minus1

2

120579

119879

2

120579

2+ 119910

2le 2119901

(43)

Theorem 4 To aim at the wheeled mobile robot expressed in(3) satisfying Assumption 2 and giving a positive number 119901for all satisfying initial condition 119881(0) le 119901 the controllers

(25) and (32) and the adaptive laws (26) and (33) guaranteesemiglobal uniform ultimate boundedness of closed loop systemwith tracking error converging to zero

Proof Choose the Lyapunov function candidate as

119881 =

1

2

(

3

sum

119894=1

119878

2

119894+ 119910

2+

2

sum

119894=1

120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) (44)

Combining (26) and (33) and differentiating (44) give

119881 le (minus119896

1+ 1) 119878

2

1minus (119896

2minus 2) 119878

2

2minus (119896

3minus

5

4

) 119878

2

3

+

2

sum

119894=1

1

4

120574

2

119894+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2minus 119887

1

119896119906

1119878

1minus 119887

2

119896119906

1119878

3

+ 2119887

2

119896119906ND1198783 +

119896119878

1119906

1minus 119898

3

119896

_119896minus

2

sum

119894=1

119898

119894

120579

119879

119894

_120579 119894

(45)

Remark 5 Consider that minus119898119894

120579

119879

119894

_120579 119894

= minus119898

119894

120579

119879

119894(

120579

119894+ 120579

119894) le

minus2

minus1119898

119894

120579

119894

2

+2

minus1119898

119894120579

119894

2 119894 = 1 2 and minus1198983

119896

_119896le minus2

minus1119898

3

119896

2+

2

minus1119898

3119896

2Then it can be verified easily that

119881 le (minus119896

1+ 1) 119878

2

1minus

119896119906

1(119887

2119878

3+ 119887

1119878

1) +

2

sum

119894=1

1

4

120574

2

119894minus (119896

2minus 2) 119878

2

2

+ 2119887

2

119896119906ND1198783 minus (119896

3minus

5

4

) 119878

2

3+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2

+

119896119878

1119906

1minus

1

2

(119898

1

1003817

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

1003817

2

+ 119898

3

119896

2)

+

1

2

(119898

1

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

2

+ 119898

3119896

2)

(46)

If 119881(119905) = (12)(sum

3

119894=1119878

2

119894+ 119910

2+ sum

2

119894=1120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) = 119901

then 1205742119894le Δ

2

119894and1198612 le Γ

2 Define120583 = 2

minus1(119898

1120579

1

2+119898

2120579

2

2+

119898

3119896

2) + sum

2

119894=1(14)Δ

2

119894+ (14)Γ

2Taking those into account there exists

119881 (119905) le minus2119886

0119881 (119905) + 120583 (47)

In the condition of 119881(119905) = 119901 just choosing the appro-priate design constant can guarantee 119886

0ge 1205832119901 and then

119881(119905) le 0 119881(119905) le 119901 belongs to invariant set From (47) wehave

0 le 119881 (119905) le

120583

2119886

0

+ (119881 (0) minus

120583

2119886

0

) 119890

minus21198860119905 (48)

5 Simulation Analysis

In this section in order to illustrate the effectiveness ofwheeled mobile robot neural network dynamic surface con-trol method which was based on nonlinear disturbanceobserver the simulation will be conducted under the initialcondition of 119909

1= 119909

2= 119909

3= 02 At the same time in order to

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 5

Adaptive law (26)

Control law (25)

(28)Virtual control

Filter (29)

(33)Adaptive law

+

+

+

Vd

Z

Nonlinear disturbance observer

m1 m3 1205781 1205783

Vd S1

S2

S3

+

minus

minus

minus

minus

k1

k2

k3

b1

b2 120576

120576

f1

f2

u1

u1

u2

ddt

ddt

ddt

Φd

Φ 120572120572f120591

Auxiliary variable(19)

u2(18)

dZint

d

Control law (32)

uND

minus(2b2k)minus1

m2 m3 1205782 1205783

Wheeled mobile robot(5) and (6)

Φ

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

z1

z2

zn

1205851

1205852

120585n

1205791

1205792

120579n

sumfm

ij

V

Figure 3 Control schematic of wheeled mobile robot

Step 3 From (30) the corresponding control law is gotten as

119906ND =

1

2119887

2119896

(119896

3119878

3+ 119886

2119888119909

3+ 119887

2119896119906

1+

119889 minus

119891) (35)

So far in contrast to (25) and (32) which belonged to neuralnetwork dynamics surface controller it is easy to see that(34) and (35) which belonged to traditional dynamics surfacecontroller not only need more precise mathematical modelbut also require more items of expression and coupling items

4 Stability Analysis

Define the filter error 119910 = 120572

119891minus 120572 Differentiating it results in

the following differential equation

119891= minus

119910

120591

(36)

Introduce boundary layer differential equation as

119910 = minus

119910

120591

+ 119861 (119878

2 119878

3 119910

_119896

_120579 2

119910

119903 119910

119903 119910

119903) (37)

with 119861 = 119896

2

119878

2minus 119910

119903and

119910 119910 le minus

119910

2

120591

+ 119910

2+

1

4

119861

2

(38)

With (24) and (25) (23) can be expressed as

119878

1= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ 119887

1

_119896119906

1

= 120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+ (1 minus

120576

_119896

2

+ 120576

)(minus

_120579

119879

1120585

1minus 119896

1119878

1)

6 ISRN Applied Mathematics

= minus119896

1119878

1minus

120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+

120576

_119896

2

+ 120576

(

_120579

119879

1120585

1+ 119896

1119878

1)

le minus119896

1119878

1minus

120579

119879

1120585

1minus 119887

1

119896119906

1+ 120574

1

(39)

where 1205741(119878

1

_119896

_120579 1

119910

119889 119910

119889) is continuous function and satis-

fies 1205741ge |120575

1+ (120576(

_119896

2

+ 120576))(

_120579

119879

1120585

1+ 119896

1119878

1)|

In the same way the following can be deduced

119878

2= 119878

3+ 119910 minus 119896

2119878

2

119878

3= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891minus 2119887

2

_119896119906ND

= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891

minus (1 minus

120576

_119896

2

+ 120576

)(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

le minus

120579

119879

2120585

2minus 119887

2

119896119906

1+ 2119887

2

119896119906ND minus 119896

3119878

3+ 120574

2

(40)

where 120574

2(119878

2 119878

3

_119896

_120579 2

119910

119903 119910

119903) is continuous function and

satisfies 1205742ge |120575

2+ (120576(

_119896

2

+ 120576))(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)|

According to Youngrsquos inequality the calculation producesthe following equality

119878

1

119878

1le (minus119896

1+ 1) 119878

2

1minus

120579

119879

1120585

1119878

1minus 119887

1

119896119906

1119878

1+

1

4

120574

2

1 (41)

In the same way it can be deduced that

119878

2

119878

2=

1

4

119878

2

3+

1

4

119910

2minus (119896

2minus 2) 119878

2

2

119878

3

119878

3le minus

120579

119879

2120585

2119878

3minus 119887

2

119896119906

1119878

3+ 2119887

2

119896119906ND1198783

minus (119896

3minus 1) 119878

2

3+

1

4

120574

2

2

(42)

For any given 119901 gt 0 the sets can be defined

Ω

1= (119878

1

_119896

_120579 1

) 119878

2

1+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

2= (119878

1 119878

2

_119896

_120579 1

)

2

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

3= (119878

1 119878

2 119878

3

_119896

_120579 1

_120579 2

119910)

3

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2

+ 120578

minus1

1

120579

119879

1

120579

1+ 120578

minus1

2

120579

119879

2

120579

2+ 119910

2le 2119901

(43)

Theorem 4 To aim at the wheeled mobile robot expressed in(3) satisfying Assumption 2 and giving a positive number 119901for all satisfying initial condition 119881(0) le 119901 the controllers

(25) and (32) and the adaptive laws (26) and (33) guaranteesemiglobal uniform ultimate boundedness of closed loop systemwith tracking error converging to zero

Proof Choose the Lyapunov function candidate as

119881 =

1

2

(

3

sum

119894=1

119878

2

119894+ 119910

2+

2

sum

119894=1

120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) (44)

Combining (26) and (33) and differentiating (44) give

119881 le (minus119896

1+ 1) 119878

2

1minus (119896

2minus 2) 119878

2

2minus (119896

3minus

5

4

) 119878

2

3

+

2

sum

119894=1

1

4

120574

2

119894+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2minus 119887

1

119896119906

1119878

1minus 119887

2

119896119906

1119878

3

+ 2119887

2

119896119906ND1198783 +

119896119878

1119906

1minus 119898

3

119896

_119896minus

2

sum

119894=1

119898

119894

120579

119879

119894

_120579 119894

(45)

Remark 5 Consider that minus119898119894

120579

119879

119894

_120579 119894

= minus119898

119894

120579

119879

119894(

120579

119894+ 120579

119894) le

minus2

minus1119898

119894

120579

119894

2

+2

minus1119898

119894120579

119894

2 119894 = 1 2 and minus1198983

119896

_119896le minus2

minus1119898

3

119896

2+

2

minus1119898

3119896

2Then it can be verified easily that

119881 le (minus119896

1+ 1) 119878

2

1minus

119896119906

1(119887

2119878

3+ 119887

1119878

1) +

2

sum

119894=1

1

4

120574

2

119894minus (119896

2minus 2) 119878

2

2

+ 2119887

2

119896119906ND1198783 minus (119896

3minus

5

4

) 119878

2

3+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2

+

119896119878

1119906

1minus

1

2

(119898

1

1003817

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

1003817

2

+ 119898

3

119896

2)

+

1

2

(119898

1

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

2

+ 119898

3119896

2)

(46)

If 119881(119905) = (12)(sum

3

119894=1119878

2

119894+ 119910

2+ sum

2

119894=1120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) = 119901

then 1205742119894le Δ

2

119894and1198612 le Γ

2 Define120583 = 2

minus1(119898

1120579

1

2+119898

2120579

2

2+

119898

3119896

2) + sum

2

119894=1(14)Δ

2

119894+ (14)Γ

2Taking those into account there exists

119881 (119905) le minus2119886

0119881 (119905) + 120583 (47)

In the condition of 119881(119905) = 119901 just choosing the appro-priate design constant can guarantee 119886

0ge 1205832119901 and then

119881(119905) le 0 119881(119905) le 119901 belongs to invariant set From (47) wehave

0 le 119881 (119905) le

120583

2119886

0

+ (119881 (0) minus

120583

2119886

0

) 119890

minus21198860119905 (48)

5 Simulation Analysis

In this section in order to illustrate the effectiveness ofwheeled mobile robot neural network dynamic surface con-trol method which was based on nonlinear disturbanceobserver the simulation will be conducted under the initialcondition of 119909

1= 119909

2= 119909

3= 02 At the same time in order to

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Applied Mathematics

= minus119896

1119878

1minus

120579

119879

1120585

1+ 120575

1minus 119887

1

119896119906

1+

120576

_119896

2

+ 120576

(

_120579

119879

1120585

1+ 119896

1119878

1)

le minus119896

1119878

1minus

120579

119879

1120585

1minus 119887

1

119896119906

1+ 120574

1

(39)

where 1205741(119878

1

_119896

_120579 1

119910

119889 119910

119889) is continuous function and satis-

fies 1205741ge |120575

1+ (120576(

_119896

2

+ 120576))(

_120579

119879

1120585

1+ 119896

1119878

1)|

In the same way the following can be deduced

119878

2= 119878

3+ 119910 minus 119896

2119878

2

119878

3= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891minus 2119887

2

_119896119906ND

= 120579

119879

2120585

2+ 120575

2+ 119887

2119896119906

1+ 2119887

2

119896119906ND minus

119891

minus (1 minus

120576

_119896

2

+ 120576

)(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)

le minus

120579

119879

2120585

2minus 119887

2

119896119906

1+ 2119887

2

119896119906ND minus 119896

3119878

3+ 120574

2

(40)

where 120574

2(119878

2 119878

3

_119896

_120579 2

119910

119903 119910

119903) is continuous function and

satisfies 1205742ge |120575

2+ (120576(

_119896

2

+ 120576))(

_120579

119879

2120585

2+ 119896

3119878

3+ 119887

2

_119896119906

1minus

119891)|

According to Youngrsquos inequality the calculation producesthe following equality

119878

1

119878

1le (minus119896

1+ 1) 119878

2

1minus

120579

119879

1120585

1119878

1minus 119887

1

119896119906

1119878

1+

1

4

120574

2

1 (41)

In the same way it can be deduced that

119878

2

119878

2=

1

4

119878

2

3+

1

4

119910

2minus (119896

2minus 2) 119878

2

2

119878

3

119878

3le minus

120579

119879

2120585

2119878

3minus 119887

2

119896119906

1119878

3+ 2119887

2

119896119906ND1198783

minus (119896

3minus 1) 119878

2

3+

1

4

120574

2

2

(42)

For any given 119901 gt 0 the sets can be defined

Ω

1= (119878

1

_119896

_120579 1

) 119878

2

1+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

2= (119878

1 119878

2

_119896

_120579 1

)

2

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2+ 120578

minus1

1

120579

119879

1

120579

1le 2119901

Ω

3= (119878

1 119878

2 119878

3

_119896

_120579 1

_120579 2

119910)

3

sum

119894=1

119878

2

119894+ 120578

minus1

3

119896

2

2

+ 120578

minus1

1

120579

119879

1

120579

1+ 120578

minus1

2

120579

119879

2

120579

2+ 119910

2le 2119901

(43)

Theorem 4 To aim at the wheeled mobile robot expressed in(3) satisfying Assumption 2 and giving a positive number 119901for all satisfying initial condition 119881(0) le 119901 the controllers

(25) and (32) and the adaptive laws (26) and (33) guaranteesemiglobal uniform ultimate boundedness of closed loop systemwith tracking error converging to zero

Proof Choose the Lyapunov function candidate as

119881 =

1

2

(

3

sum

119894=1

119878

2

119894+ 119910

2+

2

sum

119894=1

120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) (44)

Combining (26) and (33) and differentiating (44) give

119881 le (minus119896

1+ 1) 119878

2

1minus (119896

2minus 2) 119878

2

2minus (119896

3minus

5

4

) 119878

2

3

+

2

sum

119894=1

1

4

120574

2

119894+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2minus 119887

1

119896119906

1119878

1minus 119887

2

119896119906

1119878

3

+ 2119887

2

119896119906ND1198783 +

119896119878

1119906

1minus 119898

3

119896

_119896minus

2

sum

119894=1

119898

119894

120579

119879

119894

_120579 119894

(45)

Remark 5 Consider that minus119898119894

120579

119879

119894

_120579 119894

= minus119898

119894

120579

119879

119894(

120579

119894+ 120579

119894) le

minus2

minus1119898

119894

120579

119894

2

+2

minus1119898

119894120579

119894

2 119894 = 1 2 and minus1198983

119896

_119896le minus2

minus1119898

3

119896

2+

2

minus1119898

3119896

2Then it can be verified easily that

119881 le (minus119896

1+ 1) 119878

2

1minus

119896119906

1(119887

2119878

3+ 119887

1119878

1) +

2

sum

119894=1

1

4

120574

2

119894minus (119896

2minus 2) 119878

2

2

+ 2119887

2

119896119906ND1198783 minus (119896

3minus

5

4

) 119878

2

3+ (

5

4

minus

1

120591

) 119910

2+

1

4

119861

2

+

119896119878

1119906

1minus

1

2

(119898

1

1003817

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

1003817

2

+ 119898

3

119896

2)

+

1

2

(119898

1

1003817

1003817

1003817

1003817

120579

1

1003817

1003817

1003817

1003817

2

+ 119898

2

1003817

1003817

1003817

1003817

120579

2

1003817

1003817

1003817

1003817

2

+ 119898

3119896

2)

(46)

If 119881(119905) = (12)(sum

3

119894=1119878

2

119894+ 119910

2+ sum

2

119894=1120578

minus1

119894

120579

119879

119894

120579

119894+ 120578

minus1

3

119896

2) = 119901

then 1205742119894le Δ

2

119894and1198612 le Γ

2 Define120583 = 2

minus1(119898

1120579

1

2+119898

2120579

2

2+

119898

3119896

2) + sum

2

119894=1(14)Δ

2

119894+ (14)Γ

2Taking those into account there exists

119881 (119905) le minus2119886

0119881 (119905) + 120583 (47)

In the condition of 119881(119905) = 119901 just choosing the appro-priate design constant can guarantee 119886

0ge 1205832119901 and then

119881(119905) le 0 119881(119905) le 119901 belongs to invariant set From (47) wehave

0 le 119881 (119905) le

120583

2119886

0

+ (119881 (0) minus

120583

2119886

0

) 119890

minus21198860119905 (48)

5 Simulation Analysis

In this section in order to illustrate the effectiveness ofwheeled mobile robot neural network dynamic surface con-trol method which was based on nonlinear disturbanceobserver the simulation will be conducted under the initialcondition of 119909

1= 119909

2= 119909

3= 02 At the same time in order to

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 7

0 1 2 3 405

06

07

08

09

1

Line

ar v

eloci

ty tr

acki

ng (m

s)

NNDSCDSC

t (s)

Vd

Figure 4 Linear velocity trajectory tracking

0 2 4 6 8 10 12 14 16 18 20t (s)

Dire

ctio

n an

gle t

rack

ing

(rad

)

minus1minus08minus06minus04minus02

0020406081

120593d

NNDSCDSC

Figure 5 Orientation angle trajectory tracking

give the further comparison the traditional dynamic surfacecontrol will be also used to control the real robot system

The reference signals are taken as 120593(119905) = sin 119905 and V119889(119905) =

1ms with initial condition 120593(0) = 0 rad and V119889(0) = 05ms

The proposed neural network dynamic surface con-trollers are used to control the wheeled mobile robot Thecenter of neural network 119878

119894(119911) is uniformly distributed in the

field of [minus5 5] and its width 120590

119894is 15 The control parameters

are chosen as follows

119896

1= 20 119896

2= 20 119896

3= 20 119898

1= 006

119898

2= 006 119898

3= 005 120578

1= 18

120578

2= 18 120578

3= 12 120576 = 001 120591 = 001

(49)

The control parameters of traditional dynamic surfacecontrollers are chosen as follows

119896

1= 4 119896

2= 4 119896

3= 12 120591 = 001 (50)

0 1 2 3 40

01

02

03

04

05

Velo

city

trac

king

erro

r (m

s)

NNDSCDSC

t (s)

Figure 6 Linear velocity tracking error

0 2 4 6 8 10 12 14 16 18 20

00005

0010015

0020025

0030035

Ang

le tr

acki

ng er

ror (

rad)

NNDSCDSC

minus0005

t (s)

Figure 7 Orientation angle tracking error

In comparison with traditional dynamic surface controlthe neural network dynamic surface control is run under theassumption that the system parameters and the nonlinearfunctions are unknown

51 Trajectory Tracking Analysis Figures 4ndash7 display theresult of trajectory tracking and tracking error of traditionaldynamic surface controller and neural network dynamic sur-face controller Figure 4 and Figure 6 show that the responsetime of NNDSC and DSC are 12 s and 22 s respectively itis obvious that NNDSC is superior to DSC in the way ofconvergence velocity Figures 5 and 7 display that the steadystate error of NNDSC and DSC are plusmn00003 and plusmn0003respectively it is clear to see that NNDSC is better than DSCin terms of orientation angle sine tracking

52 Robustness Analysis As explained in the previous sec-tions to obtain a precise mathematical model of robot is verydifficult and sometimes impossible because of the existenceof gear clearance friction coefficient variation of roadbed

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 ISRN Applied Mathematics

05

045

04

035

03

025

02

015

01

005

0

0 05 1 15 2 25 3

10

5

0

minus5

14 142 144 146 148 15 152 154 156Velo

city

trac

king

erro

r (m

s)

t (s)

k c200 k 200 c

300 k 200 c50 k 50 c

Figure 8 Robustness analysis

and motor parameter change coming from outside In suchcases to verify the robustness of system the drive gain 119896

and friction coefficient 119888 increase by 2 and 3 times andreduce by half respectively Figure 8 presents the trackingerror that it is 0007ms as 200 0011ms as 300 and0008ms as 50 In conclusion NNDSC can remain alwayswell tuned online and maintain the desired performanceaccording to the variation of environment and still exhibitsbetter robustness and adaptive ability

53 External Friction Disturbance To take into accountCoulomb friction and viscous friction the external frictionof system is given below

119865 (V) = 119865

119888sgn (V) + 119861V (51)

where 119865

119888stands for the coulomb friction 119861 denotes the

viscous friction coefficient and 119881 is the velocity The corre-sponding parameters are chosen as 119865

119888= 15N and 119861 = 08

The nonlinear disturbance observer parameters are takenas 1198881= 20 and 119888

2= 20

External friction is applied to determine the antidistur-bance performance of robot Figure 9 shows the comparisonbetween real friction and observed friction with nonlineardisturbance observer it is clearly known that nonlineardisturbance observer observes the change of real frictionas well Figure 10 depicts tracking error of three controlalgorithms above The errors of DSC NNDSC and NNDSCwith nonlinear observer are mean plusmn00695 rad plusmn00175 radand plusmn00023 rad respectively It can be seen that NNDSCwith nonlinear observer compensates for the external frictiondisturbance in finite time duration and makes the systemperformance better compared to the conventional DSC andNNDSC

0 5 10 15

0

1

2

3

Forc

e of f

rictio

n (N

)

minus2

minus1

Real friction forceObserved friction force

t (s)

Figure 9 Comparison between real friction and observed frictionwith nonlinear observer

0 5 10 15

0002004006008

01

Dire

ctio

n an

gle t

rack

ing

erro

r (r

ad)

NNDSC with observerNNDSCDSC

minus006

minus004

minus002

t (s)

Figure 10 Orientation angle tracking error comparison

6 Conclusion

This paper presents an adaptive neural network DSC algo-rithm based on disturbance observer for uncertain nonlinearwheeledmobile robot systemThepresented controller whichsurmounts the shortages of the conventional DSC guaranteesthe convergence of tracking error and the boundedness of allthe closed-loop signals In addition the simulation results areobtained to prove the effectiveness and robustness against theparameter uncertainties and external friction disturbancesTherefore these characteristics of NNDSC with nonlinearobserver show great advantages over the other two methodsand make it a competitive control choice for the wheeledmobile robot application

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 9

References

[1] C-Y Chen T-H S Li Y-C Yeh and C-C Chang ldquoDesignand implementation of an adaptive sliding-mode dynamiccontroller for wheeled mobile robotsrdquoMechatronics vol 19 no2 pp 156ndash166 2009

[2] M Deng A Inoue K Sekiguchi and L Jiang ldquoTwo-wheeledmobile robot motion control in dynamic environmentsrdquoRobotics and Computer-IntegratedManufacturing vol 26 no 3pp 268ndash272 2010

[3] W Dong and K-D Kuhnert ldquoRobust adaptive control ofnonholonomicmobile robot with parameter and nonparameteruncertaintiesrdquo IEEE Transactions on Robotics vol 21 no 2 pp261ndash266 2005

[4] C De Sousa Jr E M Hemerly and R K Harrop GalvaoldquoAdaptive control for mobile robot using wavelet networksrdquoIEEE Transactions on Systems Man and Cybernetics B vol 32no 4 pp 493ndash504 2002

[5] D K Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[6] D Gu and H Hu ldquoReceding horizon tracking control ofwheeled mobile robotsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 4 pp 743ndash749 2006

[7] R Martınez O Castillo and L T Aguilar ldquoOptimizationof interval type-2 fuzzy logic controllers for a perturbedautonomous wheeled mobile robot using genetic algorithmsrdquoInformation Sciences vol 179 no 13 pp 2158ndash2174 2009

[8] W B J Hakvoort R G KM Aarts J van Dijk and J B JonkerldquoLifted system iterative learning control applied to an industrialrobotrdquo Control Engineering Practice vol 16 no 4 pp 377ndash3912008

[9] J Ye ldquoAdaptive control of nonlinear PID-based analog neuralnetworks for a nonholonomic mobile robotrdquo Neurocomputingvol 71 no 7-9 pp 1561ndash1565 2008

[10] K Samsudin F A Ahmad and S Mashohor ldquoA highlyinterpretable fuzzy rule base using ordinal structure for obstacleavoidance ofmobile robotrdquoApplied SoftComputing Journal vol11 no 2 pp 1631ndash1637 2011

[11] T Fukao H Nakagawa and N Adachi ldquoAdaptive trackingcontrol of a nonholonomic mobile robotrdquo IEEE Transactions onRobotics and Automation vol 16 no 5 pp 609ndash615 2000

[12] P Patrick Yip and J Karl Hedrick ldquoAdaptive dynamic surfacecontrol a simplified algorithm for adaptive backstepping con-trol of nonlinear systemsrdquo International Journal of Control vol71 no 5 pp 959ndash979 1998

[13] J Na X Ren G Herrmann and Z Qiao ldquoAdaptive neuraldynamic surface control for servo systems with unknown dead-zonerdquoControl Engineering Practice vol 19 no 11 pp 1328ndash13432011

[14] Q Chen X Ren and J A Oliver ldquoIdentifier-based adaptiveneural dynamic surface control for uncertain DC-DC buckconverter system with input constraintrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 4 pp1871ndash1883 2012

[15] T P Zhang and S S Ge ldquoAdaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedbackformrdquo Automatica vol 44 no 7 pp 1895ndash1903 2008

[16] X-Y Zhang and Y Lin ldquoA robust adaptive dynamic surfacecontrol for nonlinear systems with hysteresis inputrdquo ActaAutomatica Sinica vol 36 no 9 pp 1264ndash1271 2010

[17] C-S Liu and H Peng ldquoDisturbance estimation based trackingcontrol for a robotic manipulatorrdquo in Proceedings of the Ameri-can Control Conference vol 1 pp 92ndash96 June 1997

[18] C Zhongyi S Fuchun and C Jing ldquoDisturbance observer-based robust control of free-floating space manipulatorsrdquo IEEESystems Journal vol 2 no 1 pp 114ndash119 2008

[19] X G Yan S Spurgeon and C Edwards ldquoState and parameterestimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[20] X G Yan J Lam H S Li and I M Chen ldquoDecentralizedcontrol of nonlinear large-scale systems using dynamic outputfeedbackrdquo Journal of OptimizationTheory and Applications vol104 no 2 pp 459ndash475 2000

[21] S I Han and K S Lee ldquoRobust friction state observer andrecurrent fuzzy neural network design for dynamic frictioncompensationwith backstepping controlrdquoMechatronics vol 20no 3 pp 384ndash401 2010

[22] A Mohammadi M Tavakoli H J Marquez and FHashemzadeh ldquoNonlinear disturbance observer designfor robotic manipulatorsrdquo Control Engineering Practice vol 21no 3 pp 253ndash267 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of