Disturbances elimination with fuzzy sliding mode control for mobile robot trajectory tracking

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 - c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 297 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 298 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 299 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER ˙ 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. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 300 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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( ). c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 301 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 302 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 303 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 304 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 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. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 305