research article wheeled mobile robot rbfnn dynamic surface control … · 2019. 7. 31. ·...
Post on 09-Apr-2021
3 Views
Preview:
TRANSCRIPT
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
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
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
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 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
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
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
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
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
top related