POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER
Dis u bances Elimina ion wi h Fuzzy Sliding Mode
Con ol o Mobile Robo T ajec o y T acking
Walid BENAZIZA1, Nou eddine SLIMANE2, Ali MALLEM1
1Elec onics Depa men , Facul y o Technology, Uni e si y o Ba na 2,
Rou e de Cons an ine 53, Fesdis, Ba na 05078, Alge ia
2Ad anced Elec onics Labo a o y, Facul y o Technology, Uni e si y o Ba na 2,
Rou e de Cons an ine 53, Fesdis, Ba na 05078, Alge ia
walid_b[email p o ec ed], Slimane_doudi@yahoo. , [email p o ec ed]
DOI: 10.15598/aeee. 16i3.2767
Abs ac . The dis u bances a e he signi ican issue
o he ajec o y acking o mobile obo s. The e-
o e, an adequa e con ol law is p esen ed in his pape
and his one is based on Global Te minal Sliding Mode
(GTSM) wi h uzzy con ol. This con ol law aims o
gua an ee he a oidance o he kinema ic dis u bances
which a e injec ed in he angula and linea eloci ies,
espec i ely. Mo eo e , he dynamic model based on
exponen ial eaching law is p esen ed o a oid he un-
ce ain ies. The con ol law p o ides he asymp o ic
s abili y by aking in o accoun he uzzy ules and Lya-
puno heo y. Thus, he cha e ing phenomenon should
be a oided. The simula ion wo ks p o e he obus ness
o he p oposed con ol law by conside ing he dis u -
bances unc ion and he obo can ollow he desi ed
ajec o ies.
Keywo ds
Fuzzy ule, global e minal sliding mode, Lya-
puno heo y, mobile obo .
1. In oduc ion
The domain o obo ic is usually known wi h i s ex-
e nal dis u bances and pe u ba ions. So, he ecen
wo ks a e o ien ed on he con ol o his kind o sys-
ems, especially he nonholonomic sys ems. In his do-
main, i is in e es ing o ob ain a s able mo emen o
ajec o y acking [1] and [2]. Sliding Mode Con ol
(SMC) is known by i s solu ion o design he con ol
law and he s abili y o dis u bed sys ems [3] and [4].
The sliding mode depends on he sliding su ace, which
is exponen ially s able by aking in o accoun he Lya-
puno me hod o gua an ee he asymp o ically s abil-
i y o he sys em. A me hod known as con en ional
sliding mode sugges s a discon inuous unc ion and his
one p oduces high equencies known as cha e ing phe-
nomenon. In his ac , many wo ks use a highe -o de
sliding mode as a solu ion o his p oblem o educe
he cha e ing e ec [5]. Fu he mo e, many au ho s
ha e sugges ed me hods o minimize his phenomenon
by using adi ional sliding mode con ol [6], [7] and
[8]. A s anda d sliding mode has been exposed o be
e icien con ol app oach in he s abiliza ion o nonlin-
ea sys ems [9] and [10]. Ano he ype o sliding mode
con ol wi h obse e is p oposed in [11] in o de o
imp o e he e iciency o induc ion mo o d i e. The
eason o using sliding mode con ol is in i s good e-
sul s and he simplici y o he con ol law [5] and [12].
A obus sliding mode con olle o ajec o y ack-
ing o nonholonomic obo is p oposed by [13], which
gi es a good simula ion esul s agains he unce ain y
p esen ed in he model. Ano he wo k p esen ed in
[14] p oposes a new con olle using sliding mode con-
ol wi h kalman il e o he ajec o y acking.
A uzzy con olle p oposed in [15] is used o adjus
he sliding su ace pa ame e s and o accele a e he
sys em o a ain he eaching phase.
The uzzy logic is a p obable solu ion o educe he
cha e ing p oblem as p esen ed in [16] and [17]. An-
o he wo k [18] applies a uzzy sliding mode obse e
o synch onous mo o , using sigmoid unc ion, in o de
o minimize he e ec o cha e ing. Many esea che s
sugges an adap i e uzzy e minal sliding mode con-
ol o nonlinea sys ems wi h non-singula i y in o -
de o each a as con e gence in p esence o ex e -
nal dis u bances [19] and [20]. In his a ea, o esol e
he con e gence s a es e o p oblem in a sho ime,
o mi iga e he ha m ul e ec s o he ex e nal dis u -
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bances and o imp o e he obus ness o mobile obo
ajec o y acking, a new me hod is p esen ed in his
pape .
The p oposed con ol me hod o ajec o y ack-
ing is di ided in wo subsys ems, a kinema ic con ol
and a dynamic con ol. The kinema ic con ol uses
a Global Fas Te minal Sliding Mode con ol (GFTSM)
in o de o a oid he dis u bances o he angula e-
loci y. The main objec i e is o s abilize he o ien-
a ion e o o ajec o y acking o ze o in ini e
ime wi h asymp o ic s abili y agains he unce ain-
ies and ex e nal dis u bances p esen ed in a kinema ic
model. The e minal a ac o is implemen ed in he
sliding su ace and b ings he o ien a ion e o o ze o
apidly and he con e gence a e o he linea sliding
su ace is assu ed. The used obo model, ini ially
pe u bed, can con e ge o equilib ium s able poin
[21] by using Global Fas Te minal Sliding Mode Con-
ol (GFTSMC). The ea e , he wo k aims o p o ide
a kinema ic con olle using he uzzy logic o ackle
he e ec o he dis u bances p esen ed in a kinema ic
model and a enua e he cha e ing phenomenon o he
linea dis u bed eloci y. The e o e, he pa ame e s se-
lec ion by uzzy logic can elimina e he e ec o he dis-
u bances and ends he obo posi ion e o s o ze o
in sho ime. I has been no iced ha he con e -
gence e o pos u e o he obo could be as e wi h
asymp o ic s abili y using his con ol law.
The dynamic con ol using he exponen ial sliding
mode p o ides an e ec i e me hod o ackle he un-
ce ain ies and dis u bances p esen ed in a dynamic
model. The main ad an ages o his con ol law a e he
s abili y o he eloci y e o o ze o o any bounded
dis u bances p esen ed in he model and he gua an ee
o he sys em asymp o ic s abili y. The pe o mance
compa ison among he achie ed con olle in [22] and
he con ol law, which is p esen ed in his pape shows
ha GFTSMC has a pe ec pe o mance and can deal
wi h he e ec o dis u bances by using he uzzy logic.
This wo k is o ganized as ollows: Kinema ic and Dy-
namic models a e p esen ed in Sec. 2. A kinema ic
con ol based on a global as e minal sliding mode
and uzzy heo y is p oposed in Sec. 3. An expo-
nen ial eaching law con ol is p oposed in Sec. 4.
Finally, simula ion esul s a e p esen ed in Sec. 5.
2. Kinema ic and Dynamic
Models
The mobile obo used in his wo k is gi en in Fig. 1.
Fo he obo mo ion, he ollowing equa ions desc ibe
he mo emen o he obo :
=Ra˙ϕ + ˙ϕl
2,(1)
Fig. 1: Diag am o mobile obo .
ω=Ra
2L( ˙ϕ −˙ϕl).(2)
˙ϕ and ˙ϕla e he linea eloci ies o he igh and le
wheels, espec i ely. θis he o ien a ion angle o he
mobile obo , ep esen s he linea eloci y and Rais
he wheel adius, ωindica es he angula eloci y and
2Lis he dis ance sepa a ing he wo wheels.
The pos u e o he obo is in oduced wi h he eal
ec o ρ= (xyθ)Tand he con ol ec o γ= ( ω)T.
The dis u bed kinema ic model [23], [24], [25] and
[26] is gi en by:
˙ρ=

˙x
˙y
˙
θ
=

cos θ0
sin θ0
0 1
(γ+D).(3)
Dis he unknown dis u bance, which is bounded [27]
and exp essed by Eq. (4).
D= [d dω]T,(4)
whe e |d |< ζ ,|dω |< ζω.d and dω a e he dis u -
bances o he linea and angula eloci ies, espec i ely.
ζ and ζωa e posi i e limi ed cons an s.
The dynamic model o he mobile obo [16] and [28]
is desc ibed by Eq. (5):
M(q)˙
V+V(q, ˙q)V+F( ˙q) + G(q) + τd=β(q)τ+R( ),
(5)
whe e V= ( ω)Tis a ec o , which has as compo-
nen s and ωand τ= (τ τl) ep esen s he o ques
o he igh and le wheels.
M(q) = m0
0Iand β(q) = 1
Ra1 1
L−L,(6)
whe e mis he obo mass and I he ine ia momen .
R( ) ep esen s he dis u bance ec o 2×1. V(q, ˙q)
is he cen ipe al and Co iolis o ces. F( ˙q)is he ic-
ion ma ix.
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G(q) ep esen s he g a i a ional ec o and τdis an
unknown dis u bance.
Equa ion (3) and Eq. (5) o he obo model a e used
in o de o elabo a e he con ol law based on uzzy
global as e minal sliding mode and exponen ial slid-
ing mode con ol. This con ol algo i hm is applied
o sa is y he asymp o ic con e gence and a he same
ime elimina e he e ec o dis u bances, which a e oc-
cu ing in linea and angula eloci ies.
lim
→∝ ρe= lim
→∝ kρ ( )−ρ( )k= 0,(7)
whe e: ρ = (x y θ )Tis he e e ence pos u e
o mobile obo and ρe= (xeyeθe)Tis an e o
pos u e be ween he eal ec o ρand he e e ence ρ .
3. Kinema ic Con ol
Conce ning he ajec o y acking, he e e ence pos-
u e ρ o he mobile obo and a desi ed eloci y
γ = ( ω )Ta e used.
The e o pos u e ρeis ep esen ed by he ollowing
sys em [29]:
ρe=

xe
ye
θe
=

cos θsin θ0
−sin θcos θ0
0 0 1


x −x
y −y
θ −θ
.(8)
By in oducing he nonholonomic cons ain s Eq. (9)
in o he sys em Eq. (8), he eloci y e o wi hou dis-
u bances is de ined as in [30] by Eq. (10).
˙xsin θ+ ˙ycos θ= 0,(9)
˙ρe=

˙xe
˙ye
˙
θe
=

yeω+ cos θe−
−xeω+ sin θe
ω −ω
.(10)
Conside ing he dis u bances on he eloci ies and ω,
Eq. (10) becomes as ollows:
˙ρe=

˙xe
˙ye
˙
θe
=

cos θe
sin θe
ω 
+

−1ye
0−xe
0−1
 +d
ω+dω=
=

cos θe−( +d ) + ye(ω+dω)
sin θe−xe(ω+dω)
ω −(ω+dω)
.(11)
Assump ion 1. The ajec o y acking e o can
be bounded and can asymp o ically con e ge o ze o
when he ac o →ω, depending on he inpu ec-
o γ= ( ω)Tby aking in o accoun he ollowing
cons ain s: | |≤ max and |ω|≤ ωmax.
The pu pose o his con ol is o design a con olle
such as he mobile obo con e ges asymp o ically o
he desi ed ajec o y.
3.1. Design o he Angula Veloci y
Con ol
In o de o make θecon e ge o ze o, he linea and
he e minal sliding su ace a e chosen as in [31].
s= ˙x+αx +βxq/p = 0.(12)
The Eq. (13) is w i en as:
˙x=−αx −βxq/p = 0,(13)
wi h αand β > 0and p, q(p>q)a e posi i e pa ame-
e s.
The bene i o he non-linea e m in Eq. (13) is
he inc ease o he con e gence a e when he s a e
is a away om he o igin. The e o e, he in eg al o
Eq. (13) gi es he eaching ime s.
This eaching ime is:
s=p
α(p−q)ln αx(0)p−q
p+β
β.(14)
Assump ion 2. Equa ion (12) is used o design he
i s sliding su ace, which is selec ed as:
s1=˙
θe+αθe+βθq/p
e= 0.(15)
Equa ion (15) becomes:
˙
θe=−αθe−βθq/p
e.(16)
Acco ding o Eq. (10) and Eq. (16), he ollowing esul
is ob ained:
ω −ω=−βθq/p
e−αθe.(17)
The con ol law is ob ained:
ωc=ω +βθq/p
e+αθe.(18)
The con ol law ωccan ake θe o ze o in ini e ime e
and he sys em Eq. (11) eaches he i s sliding su ace
s1= 0. Then, he eaching ime is:
e=p
α(p−q)ln αθe(0)p−q
p+β
β.(19)
Rema k 1. The con ol law ωccon e ge he dis-
u bed angula eloci y o he e e ence one. Then,
he sys em Eq. (11) eaches ω ≈ω+dω in he ime e.
P oo 1. In o de o ensu e he s abili y o he sys em,
he selec Lyapuno unc ion is gi en as:
Vθe=1
2θ2
e.(20)
The de i a i e is gi en by:
˙
Vθe=θe˙
θe,(21)
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˙
Vθe=−βθ(q/p)+1
e−αθ2
e≤0.(22)
Rema k 2. The de i a i e o Vθeis less han o equal
o ze o by aking in o accoun ha he pa ame e s
α≥βand he pa ame e s pand qsa is y he condi ion
q
p≺ln α−ln β
ln θe+ 1. Then, he sys em Eq. (11) con e ges
asymp o ically o he i s sliding su ace s1= 0.
3.2. Linea Veloci y Con olle
Design
In he eaching ime e, he sys em eaches he i s
sliding su ace and he s a e θe ends o ze o. Then,
he sys em Eq. (11) becomes:
˙xe=ω ye+ − −d , (23)
˙ye=−ω xe.(24)
Assump ion 3. F om Eq. (23) and Eq. (24), he se-
lec ed swi ching unc ion is gi en as:
s2=xe−ye.(25)
By designing he sliding mode con ol law, which leads
he sliding su ace s2 o a ain ze o, i is in e es ing o
conside he con e gence o he s a e xe o he s a e
yeand he wo s a es con e ging o ze o. In ac , he
exponen ial eaching law is de ined by he ollowing
equa ion:
˙s2=−G( )Sign(s2)−ks2.(26)
To elimina e he cha e ing, a con inuous unc ion e-
places he sign unc ion:
˙s2=−G( )˙s2
|˙s2|+δ−k˙s2,(27)
whe e k,δand G( )a e posi i e pa ame e s.
Equa ion (23), Eq. (24) and Eq. (25) a e used and
he esul is ob ained as:
˙s2= ˙xe−˙ye=ω ye+ω xe+ − c.(28)
Using Eq. (27) and Eq. (28), he con ol law cis de-
e mined as:
c= +ω xe+ω ye+G( )s2
|s2|+δ+ks2,(29)
whe e
G( ) = max |d ( )|.(30)
In Eq. (29), G( )is used in o de o compensa e he
e ec o dis u bance d ( )and ensu es he exis ing
condi ion o he sliding mode and pe mi s o a oid he
cha e ing phenomenon.
P oo 2. Fo he s abili y analysis, he ollowing Lya-
puno unc ion is conside ed:
Vl=1
2s2
2.(31)
The de i a i e o Vlis gi en by:
˙
Vl=s2˙s2=s2(ω xe+ω ye+ − c−d ).(32)
Using Eq. (29, he ollowing equa ion is ob ained:
˙
Vl=s2(−d −G( )s2
|s2|+δ−ks2) =
=−s2d −G( )s2
2
|s2|+δ−ks2
2≤0.
(33)
To p o ide he s abili y, he uzzy sys em is applied
and he Lyapuno unc ion Vlshould be nega i e.
Indeed, d ( )is ime a ian and in o de o b ing ou
he e ec o unce ain ies, G( )is ime a ian .
Rema k 3. While he second su ace s2= 0, he
s a es xeand yea e equal and he s a es xand ya e
equal o he e e ence x and y espec i ely. Using
his esul and he squa e o he sys em Eq. (8), one
deduces ha xe=ye= 0. So, he asymp o ic acking
s abili y is gua an eed.
3.3. Design o Fuzzy Con ol
The uzzy sys em ules a e used o es ima e G( )and
he con ol law o he linea eloci y is designed [32]
and [33]. A uzzy sys em is used o achie e he pa am-
e e ˆ
G( )o he exponen ial eaching law cand he
block diag am is shown in Fig. 2.
So, o design he uzzy con ol, i is in e es ing o as-
su e he condi ion o sliding mode, which is gi en by
he ollowing equa ion:
s2˙s2<0.(34)
I Eq. (34) is sa is ied, hen he sys em s a es will be
on he second sliding su ace.
Assump ion 4. In his sys em, s2and ˙s2a e he
inpu s and G( )is he ou pu . The uzzy se s o he
inpu s and ou pu a e he same and de ined as ollows:
(NL, NM, Z, PM, PL).
Whe e, NL, NM, Z, PM, PL a e linguis ic wo ds and
p esen ed as nega i e la ge, nega i e medium, ze o,
posi i e medium and posi i e la ge. To ensu e he
p esence condi ion o sliding mode, he uzzy ules a e
applied as:
•I S2is NL and i ˙
S2is NL hen G( )is NL.
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Fig. 2: Diag am o he con ol sys em.
•I S2is NM and i ˙
S2is NM hen G( )is NM.
•I S2is Z and i ˙
S2is Z hen G( )is Z.
•I S2is PM and i ˙
S2is PM hen G( )is PM.
•I S2is PL and i ˙
S2is PL hen G( )is PL.
The inpu s membe ship unc ion is shown in Fig. 3
and Fig. 4 espec i ely. Thus, he ou pu membe ship
unc ion is gi en in Fig. 5.
The pa ame e G( )is es ima ed by:
ˆ
G( ) = k1Z
0
G( )d , (35)
whe e k1 ep esen s a gain.
-20 -15 -10 -5 0 5 10 15 20
uzzy se s o he inpu
0
0.2
0.4
0.6
0.8
1
Membe ship deg ee
NL NM ZPM PL
Fig. 3: Fuzzy se s o inpu unc ion s2.
So, he con ol law in Eq. (29) becomes a new uzzy
con ol law, which is indica ed in his o m:
c= +ω xe+ω ye+ks2+ˆ
G( )s2
|s2|+δ.(36)
-20 -15 -10 -5 0 5 10 15 20
uzzy se s o he inpu
0
0.2
0.4
0.6
0.8
1
Membe ship deg ee
NL NM Z PM PL
Fig. 4: Fuzzy se s o inpu unc ion ˙s2.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
uzzy se s o he ou pu
0
0.2
0.4
0.6
0.8
1
Membe ship deg ee
NL NM Z PM PL
Fig. 5: Fuzzy se s o ou pu unc ion G( ).
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P oo 3. Fo he s abili y analysis, he de i a i e o
Lyapuno unc ion Eq. (31) is conside ed.
˙
V=−s2d − | s2|ˆ
G( )−ks2
2,(37)
whe e |s2|ˆ
G( )−ks2
2>−s2d .
The e o e, he con ol co sliding su ace s2con-
e ges he e o s a es o ze o wi h asymp o ic s abil-
i y. The gene al con ol law o kinema ic model is gi en
as ollows:
Vc= c
ωc="ω +βθq/p
e+αθ
+ω xe+ω ye+ks2+ˆ
G( )s2
|s2|+δ#.
(38)
In ac i = cand ω=ωc, hen he sys em closed
loop is asymp o ically s able.
4. Dynamic Con ol
In his sec ion, an exponen ial sliding mode con ol is
used o he o que con ol τin o de o gua an ee he
con e gence e o Veo he eloci ies.
lim
→∝ Ve= lim
→∝ kVc( )−V( )k= 0.(39)
The con ol τo dynamic model is designed in o de
o ake he ac ual eloci ies o he obo o he e e ence
ob ained wi h he kinema ic con olle .
4.1. Dynamic Model
Conside he dynamic model Eq. (5) and aking in o
accoun ha he obo mo es in ho izon al plan, he e-
o e g a i a ional ec o , cen ipe al ec o , Co iolis
ma ix, ic ion ma ix and unknown dis u bances be-
come ze o. The dynamic model Eq. (5) becomes as
ollows:
M(q)˙
V=β(q)τ+R( ).(40)
4.2. Dynamic Con ol Based on
Exponen ial Sliding Mode
In his sec ion, he exponen ial sliding mode con ol
is used o o ce he dynamic model o he obo o
gua an ee he asymp o ic s abili y. Thus, he eloci y
e o is chosen as:
Ve=e eωT= c−
ωc−ω.(41)
The de i a i e eloci y e o is gi en as:
˙
Ve=˙ c−˙
˙ωc−˙ω.(42)
The selec ed sliding su aces a e:
S=s3
s4= c−
ωc−ω.(43)
The de i a i e sliding su aces a e:
˙
S=˙s3
˙s4=˙ c−˙
˙ωc−˙ω.(44)
Assump ion 5. Using he exponen ial sliding mode
con ol:
˙
S=−ε1sgn(s3)−µ1s3
−ε2sgn(s4)−µ2s4,(45)
whe e ε=ε10
0ε2and µ=µ10
0µ2a e posi i e
cons an s.
Thus, combining Eq. (41) and Eq. (46), he dynamic
con ol is gi en as:
τ=β(q)−1M(q)( ˙
Vc+εsgn(S) + µS).(46)
Conside he dynamic model Eq. (40) wi h he dis u -
bance and unce ain ies R( ), he esul is gi en as [34]:
˙
V=M(q)−1β(q)τ+M(q)−1R( ).(47)
By choosing:
M(q)−1β(q) = Q= ( ˆ
Q+ ∆Q).(48)
∆Q ep esen s he unce ain ies and ˆ
Qis he nominal
e m o he ma ix. The e o e, ϕ( )is de ined as he
uppe bound unce ain y, which is gi en as ollows:
∆Qτ +M(q)−1R( ) = ϕ( ),(49)
whe e |ϕ( )|≤ φand φis cons an posi i e pa ame e .
The con ol law is de e mined as:
τ=ˆ
Q−1(˙
Vc+εsgn(S) + µS),(50)
τ1
τ2=ˆ
Q−1˙ c
˙ωc+ε10
0ε2sgn(s3)
sgn(s4)
+µ10
0µ2s3
s4.
(51)
In o de o a oid he cha e ing phenomenon c ea ed
by he sign unc ion, a quasi-sliding mode unc ion is
applied:
˙
S="−ε1s3
abs(s3)+δ1−µ1s3
−ε2s4
abs(s4)+δ2−µ1s4#.(52)
The con ol law o he dynamic sys em is:
τ1
τ2=ˆ
Q−1 ˙ c
˙ωc+ε10
0ε2+"s3
abs(s3)+δ1
s4
abs(s4)+δ2#
µ10
0µ2s3
s4.
(53)
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P oo 4. The selec ion o Lyapuno unc ion is as
ollows [35]:
Vd=1
2STS. (54)
The de i a i e o he Lyapuno unc ion Vdis ob ained.
˙
Vd=ST˙
S=s3˙s3+s4˙s4.(55)
Equa ion (55) becomes as ollows:
˙
Vd=ST˙
S=−ε1
s2
3
abs(s3) + δ1
−µ1s2
3
−ε2
s2
4
abs(s4) + δ2
−µ2s2
4≤0,
(56)
whe e diag (ε1,ε2)>0, diag (µ1,µ2)>0and δ1,
δ2>0.
Rema k 4. The pa ame e diag (µ1,µ2) is aken in
o de o compensa e he e ec o dis u bances and un-
ce ain ies o he sys em.
5. Simula ion Resul s
In his sec ion, o show he e icacy o he con ol law,
simula ion wo ks unde Ma lab en i onmen a e e-
alized, and h ee di e en ajec o ies (ci cula , sinu-
soidal and speci ic) a e conside ed. The mobile obo
pa ame e s used o simula ion a e gi en as ollows:
m=4 kg, I=3 kg·m2,Ra=0.03 m and L=0.15 m.
The desi ed pa ame e s o con ol a e selec ed as be-
low =2 m·s−1and ω =2 ad·s−1. The desi ed a-
jec o y o he obo is:





x cos(ω ) = cos(2 )
y sin(ω ) = sin(2 )
θ =ω = 2 .
(57)
The con olle pa ame e s a e chosen a bi a ily.
p= 10,q= 9,k= 30,K1= 250,α= 50,β= 50,
δ= 0.08.
A limi ed pe iodic dis u bance e m is inse ed be ween
ime 4< <5and conside ed as ollows:
(d = sin( −π)
dω = 2 sin( −π).(58)
The pa ame e s o he dynamic con olle a e selec ed
as: ε1= 80, ε2= 80, µ1= 30, µ2= 30, δ1= 0.95and
δ2= 0.95 and he dynamic dis u bances, in oduced
be ween ou and i e seconds (4 s< <5 s) a e con-
side ed:
E( ) = [2.5 sin( −π) 1.5 sin( −π)].(59)
The ini ial posi ion and o ien a ion e o s a e gi en
as (2 m, 1 m) and (π/6 ad) espec i ely. Figu e 6
0 2 4 6 8 10 12 14 16 18 20
Time(s)
-40
-20
0
20
40
60
80
(m/s)/ ( ad/s)
Fig. 6: Con ol law o he signals cand ωc.
ep esen s he con ol signals cand ωco he kinema ic
model.
I can be seen ha he ac ual linea and angula
eloci ies o he p oposed con ol can each he desi ed
eloci ies in sho ime.
By using he kinema ic and dynamic con olle s o
Eq. (38) and Eq. (53), he simula ion esul s o a ci -
cula ajec o y acking a e shown in Fig. 7.
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
x(m)
-1.5
-1
-0.5
0
0.5
1
1.5
y(m)
Re e ence T ajec o y
Real T ajec o y
Fig. 7: Ci cula ajec o y acking o he mobile obo .
Hence, he mobile obo can achie e he ci cula a-
jec o y apidly in sho ime among = 2 s in p es-
ence o dis u bances and he asymp o ic s abili y o
he obo is assu ed. This is he main ad an age o
he p oposed con ol law in e m o e o s a es con-
e gence o ze o and he elimina ion o he dis u bance
e ec . On he o he side in [36], he obo canno con-
e ge comple ely o he e e ence and he e o s a es
canno s abilize ully o ze o due o some pe u ba ion.
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Figu e 8 ep esen s he ajec o ies acking e o s
o he s a es xeand yeo he sys em ρe.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
xe ,ye (m)
xe
ye
Fig. 8: Posi ion acking e o s o he s a es xeand ye.
The o ien a ion e o θeis illus a ed in Fig. 9.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
e( ad)
e
Fig. 9: O ien a ion acking e o θe.
By using he dynamic con olle , he o ques and
he eloci y e o s a e shown in Fig. 10 and Fig. 11
espec i ely.
Fo he sinusoidal ajec o y, he same ini ial po-
si ion and o ien a ion e o s a e conside ed, bu he
e e ence pos u e is selec ed as ollows:





x =
y = sin(2 )
θ =ω = 2
.(60)
The con ol signals cand ωca e illus a ed in Fig. 12.
The simula ion esul s o a sinusoidal ajec o y
acking a e shown in Fig. 13. I can be seen ha
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-20
-15
-10
-5
0
5
10
15
20
1/ 2 (N.m)
1
2
Fig. 10: Gene a ed o ques τ1and τ2.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-40
-30
-20
-10
0
10
20
30
40
50
60
e (m)/e ( ad)
e
e
Fig. 11: Veloci y e o s e and eω.
0 2 4 6 8 10 12 14 16 18 20
Time(s)
-40
-20
0
20
40
60
80
(m/s)/ ( ad/s)
Fig. 12: Con ol law o he signals cand ωc.
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he mobile obo con e ges o he e e ence ajec o y.
Figu e 14 ep esen s he ajec o y acking e o s o
he s a es xeand yeo he sys em ρe. The o ien a ion
θeis illus a ed in Fig. 15.
-5 0 5 10 15 20
x(m)
-1.5
-1
-0.5
0
0.5
1
1.5
y(m)
Re e ence T ajec o y Real T ajec o y
Fig. 13: Sinusoidal ajec o y acking.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
xe ,ye (m)
xe
ye
Fig. 14: Posi ion acking e o s o he s a es xeand ye.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
e( ad)
e
Fig. 15: O ien a ion acking e o θe.
The o ques and he e o eloci ies a e shown in
Fig. 16 and Fig. 17 espec i ely.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-15
-10
-5
0
5
10
15
20
1/ 2 (N.m)
1
2
Fig. 16: Gene a ed o ques τ1and τ2.
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-40
-30
-20
-10
0
10
20
30
40
50
60
e (m)/e ( ad)
e
e
Fig. 17: Veloci y e o s e and eω.
The o ques ob ained wi h he p oposed con olle in
p esence o dis u bances con e ges o ze o. Thus, he
e o eloci ies con e ge asymp o ically o ze o be o e
and a e inse ing he ex e nal dis u bances.
Fo he speci ic ajec o y, he ollowed alues (3 m,
2 m π/6 ad), o he ini ial posi ion and o ien a ion
e o a e conside ed.





x = cos
y = sin 2
θ =ω = 2
.(61)
The con ol signals cand ωca e illus a ed in Fig. 18.
The simula ion esul s o he speci ic ajec o y a e
shown in Fig. 19.
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