整合連體力學與熱動力學之變分原理(2/3)
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(8) . . Keywords: Continuum Mechanics, Thermodynamics, Principle of Virtual Power, Privileged Coordinates, Tracking Control In dealing with the mechanics of material bodies modeled by particles, rigid body, or deformable body subject to various types of constraint, variational principles can usually provide a suitable path to obtain the solutions. The main objective of this project is to seek additional applications and possible extensions on the basis of the Principle of Virtual Power, and to explore whether the proposed principle can be unified with the law of thermodynamics to form a unified variational principle. After nearly two-year term of endeavor, we are able to obtain the desired Principle by choosing the velocity and the velocity gradient as fundamental variables in mechanics, and the entropy and the hear flux as the fundamental quantities in theomodynamics. The Principle indeed unify the theory of thermodynamics and continuum mechanics, from which a variational equation can be established. The Cauchy’s laws of linear and angular momentum and the newly developed variational constitutive law are then deduced. By selecting appropriate internal energy function and dissipating function, we have derived the constitutive equations for elastic body or viscous fluid in mechanics, and Fourier law and Maxwell-Catteneo Equation in the first year. In the second year, we established the constitutive laws for the thermo-viscous fluid and micro-polar material. See Part I of this report for more details. On the other hand, in the aspect of tracking control, we have used the Reduced Appell Equation to construct equations of motion of the privileged coordinates, and found that the equations are decoupled from the non-privileged coordinates in the first year. The fuzzy controller and sliding mode controller were then used to design the tracking control loop. This year, we adopt the backstepping method applied to the chained form of the kinematic equation to compute the compensated values for the privileged variables. Simulation results on the tracking control of wheeled vehicle show that the performance is excellent. The results are shown in Part II of this report.. 2.
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(15) ut'+ q H r # C J2 s fy%g$!ø c0r1 +!# * vC #ñ/X¹ W $ß v c{ 1|,+!} È # G$ w x * y2zAmíijÓ)G$w$* ~x r[* y2 z C ;4|<{)9º( GG H ¥ n ¥ y¤¥ ~ $R P G H ¥ á y³ w%q J!Ó G H ¥ á$;ì4 1*\ C }1È ,ó'· R$SRT5 R mST )54 !j C%,4)rQ È ú 8R Sk T +5 , pj q )4{ rÕ 3 «k B k ¥ +C), l!j o'G pH q'¥ ,y á ; 8 ` , µ  ; 4#${ 8à $ n C 4 m k +,prqm k + , R S'T 5 C ;4< 5 }1·8 (viscous fluid with heat-conduction). (thermo-mechanical). --. Truesdell. --. Toupin (1960). Toupin (1963) Mindlin Tiersten (1963). Toupin simple force multipoles of the first kind multipoles of the various kinds. [12]. Green & Rivlin (1964). compound force. Erigen (1966). [13]. k*8¦ R+$,'ßl 8æ4; ¡ © prÓq È)=! 7èY énn* ;8¢ k è+)9 ¨%:n©,* £!¤ y)x$Ü ¥ tR9 ;î8PR¢'Q ¨%w$©,O,£Xª ¤ !0 2 k ú +,'l ; prq'ãä! ë 8 Õ § ¨ Sj k C ¯® ¬ = + « 9¹ ° ±«Å 7 ë 8 V!ïy1 1T ² Å ç S ¿ æ +Aè * S'T%j '8 1T y³â´J2µ ( 7/è¶ ~ O#8* ¹# 4 PRQ ú · * k 1T C $¸ 1¦ îå). 2.1 (1). –. (1). » º = ρ Uº dV , (2a) ∫V ½ = r ⋅ dF − 1 ρr¼ ⋅ r¼ dV , (2b) ∫B 2 ∫V ¾ = ρq b dV − q ⋅ n dA, (2c) ∫V ∫A ¿À U Á2ÂÃ$Ä4ÅRÆÇÉÈËÊ8Ì Í>ÎÏ Ð ÂÃÒÑ dF ÁÓ!Ô Õ Ö$× ÌÍ>Î Ï Ø4ÙRÚ Û Ü4ÝÉÑ q b Þ4Ö$×$Ì Í>ÎÏ Ù*ß!à$Ð4áâ (heating) Ñ´ã q ä$Á*å$æRç*èRéÏ n Ðê!ë ì dA Ü á*åRÏ (heat flux) í (2). áÛ*îRï8ðRñò. –. á Û*îïðRñòó ô,õ Ö*ö,Ï -- ø ÷ ÈTù$úXû Õüñ4ý$Å*þÿÈ ÷RÐ2âÙ*ß!à4áÐ 3.
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(17) θ ×ÿ≥È ÷. R áâ ÔXÐ8Ç êÒÈ%ä*Ø ! ρq q ∫. ρ S dV ≥ ∫. ∫A θ θ ρq b q =∫ ( − ∇ ⋅ )dV . V θ θ. S. Þ4Ö¨Î8Ì ÍR×ìXÐ ÷RÄ4Å í#"ð$&% Ü áÛ*î ñò È Õ'*öRáÛ*îÜ. V. ¿À 2.2. V. ∫. V. ä N. dV −. ⋅ n dA. (4). (*) ö,+ È.-0/12*34 í. 567*89:,;<=?> Õ ' öáÛî4Ü ( ) ö +FÈA@?B!óôRõ à CD!ÆÇ E ÈGFH¨ïðñ ò,Ü& K ρq q R ρSdV = ( − ∇ ⋅ ) dV + dV , ∫. L. b. (3). θ. V. M. ≡. ∫. ∫. b. θ. V. (4). I& Þ? J (5). θ. RdV ,. (6). ≥ 0 ?O2Þ*áÛ*îRïðRñò¨Ð&P*õ*Q&J Aí R S T S T À [ + Z =(5)Y \Â+Ð X ï, õ X U ≥ì0( V Ð ( ρq − ∇ ⋅ q )W ô (1) Ð8ì( È%ä&*H (7) ¿À ]_^ Þ&* ` áâ (net applied heating) Èaê c = ρθSb dV − ∇θ ⋅ qdV . ∫ ^ ∫ θ S T (7) @?B Ü ÞáÛî$ñòRÜ?34 í Õ" 34 d*eþÒÈTáÛîÜõ*$&% )$öÿÈgf&hâ )$öÿÈi*jk þ T Õ,l?m,Ð á?næ À Èpo ?
(18) &qr tsvuwx yzo{Üf 8Ç&|,â( È.}*~*Û8Ù*ÓÜf&hRâ &f` áâ4Ü&4 Ý $ Ö ×X Rh â4ÜS Tf &vo?ÿ í " fR h â ) ö, Ô ( ê$ Þ ∫ δ v ⋅ (dF − rdm) 1 + ∫ ( ρθδ S − ∇θ ⋅ δ q)dV (8) θ , =δ +δ SvT SvT ¿À Á4Ö ×RÐ ÂCX À ¿ D Æ8Ç È T À (7) Ð È Õ& H* Ü? È á Û*îï4õ ñò&t> í Õ,á*nsvu (æ È$ | Å,Ð f (R Ô δ v ê È@? B ù & δ r = δ t = 0 íaP~ È δ Ãv Ó$Õ$Ý, | âÚ,Ü 8ÇÉÈg v, ∇v, S , q. 2.3 ¡¢&£,¤¥¦?§ ×*Ø*Ü4Ó!Ô $~ ÛÝ dF &(?R¨ Þ©U(ª õ*Þ Ö$×XÂ8Ì Í>ÎÏÙ2à*Ð*~ Û f (body force) ÈpP õÞ4Ö$× ê!ëÐß! « Û t (surface traction) í Ê dF = f ρdV + t dA. (9) S T ¿ À ² 㬮¯2Û ) ö°±@ B t = n ⋅ σ È Þ?¯2Û&³$ÏAíËä ( (8) I ¸¸ ∫¶ ρ (f − r ) ⋅ δ vdV + ∫· t ⋅ δ vdA ¸ 1 (10) + ∫¶ ( ρθδ S − ∇θ ⋅ δ q)dV θ ¸ µ ´ , =δ +δ ¹ Á V. b. ′. ′. ′. V. V. 1. B. 1. V. 1. W. 1. 1. W. 1. 1. B. (n ). (n ). B. (n ). B. (n ). 1. 1. 1. 1. W. 1. 1. 4. 1. 1.
(19) ∫ ( ρf + ∇ ⋅ σ − ρr) ⋅ δ v dV + ∫ σ : ∇ (δ v ) dV B. 1. 1. 1 ∫ ( ρθδ S − θ ∇θ ⋅ δ q)dV = ∫ ( ρδ U + δ R )dV . +. 1. (11). 1. W. S T " È Ø $ Ð ï8ð?U ì(RÂÐ δ Ý ∇ YÁ? Ó È ?o
(20) ∇(δ v) = δ (∇v) í Ò 1. Þ. Vê . 1. 1. 1. 1. σ : ∇(δ1 v ) = σ : δ1 (∇v ) = σ : δ1D + σ : δ1Ω,. ¿À. (12). 1 (∇v + v∇), 2 1 Ω = (∇v − v∇) = −1 × ω. 2 Ð È Þ D Þ ∇v Ü D=. (13a) (13b). ³ Ï ^ ( ^ ³ Ï (stretching tensor) È ãÞ ∇v Ü ¿^ ( È ^ Þ,³ Ï $ ³ Ï8ÝÅ2éÏ (vorticity vector) ω = (1 / 2)∇ × v Ú,AÈ ,Ô W tensor) í S T S T σ : δ Ω = σ : (−1 × δ ω ) = −(σ × 1) ⋅ δ ω (14) ¿À Õt ©U,$³ Ï 4Ð! ×" ñ#Þ ab ×$ cd ≡ (a ⋅ c)(b × d) í W&% (11) Èa*H ×$Ü ( 1. 1. 1. ∫( ( ρf + ∇ ⋅ σ − ρr) ⋅ δ v dV ' + ∫( σ : δ D - (σ × 1) ⋅ δ ω ) dV. ''. B. 1. '. ∫( ' θ = ∫( ( ρδ U + δ R +. ( ρθδ 1 S −. 1. 1. 1. 1. 1. (15). ∇θ ⋅ δ 1q )dV W. )dV .. )+*, - .0/1234. ÕRØkd eRþ Èa@?B¨Ê 56 7kf&hRâv) öÕ,á89×*Ø$Ü ¯!Ô S í T } Ô@?v}à (free energy) F = U − θS A&W¨Â à U È FB = UB − θSB − SθB È ( E CC B ∫D ( ρf + ∇ ⋅ − ρr) ⋅ δ1 vdV C + ∫D ( E ⋅ δ 1D − ( E × 1) ⋅ δ 1ω )dV. 2.4. C. (spin.
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(22) Ï=ÒÈi@ B0> (15) I . ∫D C θ = ∫D ( ρδ F + δ R )dV . F 5 6 v s Æ, u Ç + Èp@ BG0HI$Ö ×*Üd*e ÀKJ )$ö (Principle of Frame Indifference) È%ÊL ×0MRàõ N × è! ¹ &+¨È ¿  à ÆÇ2Ä Å U Ý>Â0CD!ÆÇ2Ä Å R ¯ O0P !ÈRQ Ê U Ý R δ v δ ω S  C&D!Æ8ÇRÁ ³$ÏWVvý$ÅRá*åRÏ Ð$Æ8Ç í "@ BT?v}Ã* Á U*{ ν (= 1 / ρ ) ý$Å,Ð$Æ8ÇÉÈ´ (− ρSδ 1θ −. +. 1. (16). ∇θ ⋅ δ 1q)dV. W. 1. 1. W. W. R. 1. F = Fˆ (ν ,θ ),. X. F. W. R. Z. (!Y/. R W = Rˆ W (D,θ , q).. δ1. Z. (17). (vÓ*H Z. ∂Fˆ ∂Fˆ δ 1ν + δ 1θ ∂ν ∂θ ∂Fˆ 1 ∂Fˆ = (1 : δ 1 D) + δ 1θ , ∂θ ρ ∂ν. δ1 F =. Z. (18a). 5. 1.
(23) δ 1 RW =. ∂Rˆ W ∂Rˆ W : δ 1D + ⋅ δ1q. ∂D ∂q. (18b) S T }
(24) Ø À 5 Uf 8Ç δ v V δ ω V δ D V δ θ δ q YÁ?o J Ð È%ä&@ B*H U ρf + ∇ ⋅ − ρv = 0 (19a) 1. 1. 1. ×1 = 0 ∂Fˆ ∂Rˆ W = 1+ , ∂ν ∂D ∂Fˆ S =− ∂θ ∂Rˆ W 1 ∇θ = − θ ∂q. 1. 1. (19b). (19c) (19d). Ø ï8ð t$ï©õ * X0H2ïðá Þ* ¬tS Tïí õ !. L¨Â0CD!ÆÇ$õ( 8 . VøÈTê?þ. RV. SvT. áÏ. È´Ê,Ï , Ï?? RT. SvT. (19e). Èvï
(25) Þ?¯,Û4Ü ò È.}. ©( J Æ ÇDÈ ù 8 (. RV. R W = Rˆ V (D,θ ) + Rˆ T (θ , q) 1 = D : C′ (θ ) : D + Rˆ T (θ , q) 2. Á0 ³*Ï. D. Ðð. (20). ä ¯2Û&³$Ïê . =. ∂Fˆ 1 + C′ : D. ∂ν. (21). õ05 68 9×+ È ¿ 2 ¯ Û&³$Ï($Þ©v(. vS T σ = − p1 + σ 7Á*éÐ RÛ (hydrostatic pressure) Èä*Á8 ,ÐCD( í [1]. d. ∂Fˆ (ν , θ ), ∂ν σ d = C′ (θ ) : D. p=−. L"8 9$Þ*é ¿À. (23b) (isotropic). È%ä ¯2Û4Ü C&D( õê . SvT (24) !8 Ç (bulk and shear viscosity) í (24) 0,Ï?* (19a) *t ′. !( Y S T 9 8 9Ü&. . ∂v + v ⋅ ∇v ) ∂t = ρf B − ∇p + (λ ′ + µ ′ )∇∇ ⋅ v + µ ′∇ 2 v. ρ(. (25). S T "Ø 3λ ′ + 2µ = 0 È%ä&*H #$2Ð Navier-Stokes í [1] 2.5 %!& '( ?v 8} ÃJ ÆÇ&(*Þ*) U*{ ¹ ý*Å o®4Ð J ÆÇøÈ,+?RÞý*Å,Ðð ÆÇøÑ P!ÂCD!ÆÇ2Ð4á Ï*( Æ8Ç,Þ*á*åRÏ4Ü$ð Æ8Ç ". S T. o@U : Èä& H (23a). σ = λ ′(1 : D)1 + 2 µ D, λ ′ µ ′ Þ ×*Ü*× d. (22) (21). ′. 1 F = Fˆ1 (ν ) + Fˆ2 (θ ) = Fˆ1 (ν ) − cθ 2 , 2. (26a). 1 T RW = Rˆ V ( D) + Rˆ T (θ , q) = Rˆ V ( D) + q Γq, 2 θ ¿À c Þ á È Γ Þ*á *Ç íËä ∂Fˆ ∂Fˆ S =− = − 2 (θ ) = cθ , ∂θ ∂θ. *{U. -* .0/ ,} ( ò. (19d). . (19e). *H. (26b). (27a). 6.
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(29) 98 κ = Γ :.4;=< &.1> ?2 Fourier law q = −κ∇θ (28) @A 4"B3CD.E —F.GH.?.I$4B.C (extended irreversible thermodynamics, EIT)[9] JK MLON$P $Q #/1> RTSU5VW3X!$#$/15YZ \[]$^_SU4.67.:.`a1> cb ˆ (29) d ?2 F = F (ν ,θ , q), e ∂Fˆ e ∂Fˆ e ∂Fˆ e δ F =δ ( ν + θ+ (30) q). −1. 1. 1. ∂ν. ∂θ. fgh3ijkl3m. ∂q. no. F. Fˆ = Fˆ1 (ν ) + Fˆ2 (θ ) + Fˆ3 (q),. d. ~. `*J 8. ?pqrstvuxwyz. q. {O|"/1>} (31). ~ ~. ~. δ1 F =. ∂Fˆ ∂Fˆ δ1ν + δ1θ + q ⋅ δ1G q , ∂ν ∂θ. (32). Gq ≡. ∂Fˆ = G q (q). ∂q. (33). :. .5/1>}. Gq q G q = Λq q,. ) +$-0/1$ 5a3p" O δ q a1> 5 5 (27b)O 5 (32) ?2 τq + q = −κ∇θ , (34) `*J τ = ρκΛ θ (:" " (relaxation time) "b: Maxwell-Cattaneo \ LON$!$#3X4.6"7 %& ¢ © £ª$« ª«¬ 2.6 $$Ol £¤ $¥¦ .§"¡.¨$ 93®¯°.±5Bp²O B .0³´µ\st £¤ $l ¥¦ ?¶ · ¸.¹OB5ºU»_S,¼½¾pZ}¿ÀP Á7$3Y$7Z ¦ Ç d ºÈ»ÉS ʽ$¾pË}Ì3 Ç ∫ 1 ρv ⋅ vdVB F w« ¿Ç.ÀPÇ Á 7$∫ 31 ρ B I. ⋅ ωl⋅wωÃ3dV±5B Ì`Pt .J °$I ±ª$B «"c ¬ zÄ 7Í9st5Šƥ$ Î3Ï 2 2 ij© Á Ñ = ρF ⋅ δ vdV + ρl ⋅ δ ωdV δ 1. q. ∫ ∫ 1 1 + ∫ t ⋅ δ 1 vdS + ∫ c ⋅ δ 1ωdS − ∫ ρr ⋅ δ 1 vdV − ∫ (I ⋅ ω + ω × I ⋅ ω) ⋅ δ 1ωdV ,. 1. 8. t= n ⋅. Ô . δ1. ÐÐ. Ó. W. Ð. = ∫ ρθδ 1sdV − ∫. . Ò. 1. θ. ∇θ ⋅ δ 1qdV .. Õ:3."Ö.×.Ø"?Ù Ú>. `*J. = δ 1U& + δ 1R .. (12). ij. (36). ?2PÛÜ5a3pÝ5. δ1 ∇ ρF ⋅ δ 1 v + ρl ⋅ δ 1ω + (∇ ⋅ σ ) ⋅ δ 1 v + σ : δ 1 (∇v) + (∇ ⋅ C) ⋅ δ 1ω + C : δ 1 (∇ω) − ρ&r& ⋅ δ 1 v. c= n ⋅ C .. & + ω × I ⋅ ω) ⋅ δ 1ω + ( ρθδ 1 s& − − ρ (I ⋅ ω. ÞOß. (35). à$áâ$7ã"Öä. ij. 1. θ. ∇θ ⋅ δ 1q ). W. ?2. σ : δ 1 Ω = σ : ( −1 ×& δ 1 ω) = −(σ ×& 1) ⋅ δ 1 ω = −2t A ⋅ δ 1 ω. tA. : å æç=è37. (axial vector), 2t A = −ε ijk σ jk e i. µ. 7. (37). ?3é:. (37).
(30) ρF ⋅ δ 1 v + [(∇ ⋅ σ ) − ρ&r&] ⋅ δ 1 v + σ : δ 1 D + ρl ⋅ δ 1ω & + ω × I ⋅ ω)] ⋅ δ 1ω + [−2t A + (∇ ⋅ C) − ρ (I ⋅ ω + C : δ 1 (∇ω) + ( ρθδ 1 s& −. :. = δ 1U& + δ 1R W .. δ 1R. ± X . δ1 v W. 1. θ. (38). ∇θ ⋅ δ 1q ). é.^t0¿.ÀO¦"æ"ã Ç « Ç ). "+-*/1 δ1 v w δ1ω 5% za3p" (38) b?$E.2 :. δ1ω. . . ρF + (∇ ⋅ ) − ρr = 0,. RW. . θ. = δ 1U& + δ 1R . (41).
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(32) ?2 £¤ $¥¦$æ"3 O:. σ=. ∂Fˆ ∂Fˆ ∂Fˆ 1, C = ρ , s=− , ∂ν ∂K ∂θ. −. 1. θ. ∇θ =. ∂Rˆ W . ∂q. "!# @ l%/ á%$& '( )*¦3X £$¤ $¥$¦ 4¸ · +3 TJUæ- , :Z 4B3C. Ç (1) 1032Ç $7 4 5 b ϦO)"]^76 tOY-8$s"Ϧ/ 9.¶0 d X%9%:<0 ; 4.æä\ 7-4 5= 5 (25) :.`3ã > ÆX 4"B.@ C$3 (34) >=3i^"j 0Á·%73I?AJ @ · B-CED" F GAH @ %F(3EÉJ AKD.`"a3p .± ß 4AÇ ) *3¦3X £$¤ Ç.¥¦. cAL½.{5| 5a1> δ v δ ω δ D δ θM δ q ?P< NL ½0 }9 7-4 50wPO Ç 7-4 5=w ±5dBX_ $+.-0/ 1O"% &Zw 3X_%&3"!.#3X0),+.-0/ 1O"%& Ê?2"ã
(33) Ì' Q3 ? <`3· ß © lRV l RK$D$±ijtZ ¥¦.ä TS-. U J 9 a1
(34) } ºU»B3CX4Cæa 1 C- W TB ÇH-W B3CX4 C3æB-C< UE[%\ `"a3p
(35) ^3] <W 3_ æ5)/1$X0),+$-O/ 1> Âb?=< Èã .3 h XEYµ ij T` · J .L$a 4A)ßEb gi h 3* ¦ (viscoelastic fluids with heat conduction) c £¤ * ¦ (micropolar fluids) edF3 f RK D"æ3± j%kl-m 1. 1. 1. 1. 1. [1] Batchelor, G. (1967) An Introduction to Fluid Dynamics, Cambridge University Press, New York. 8.
(36) [2] Biot, M.A. (1984) New variational-Lagrangian irreversible thermodynamics with application to viscous flow, [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]. reaction-diffusion and solid mechanics, in:Advances in applied mechanics, Academic Press, New York, 24, 1--91. Chen, K.C., Wang, L.S, & Pao, Y.H. (2003) A Variational Principle for the Dynamics of Rigid Bodies, Proceeding of the Cross-Strait Workshop on Dynamics, Control, and Variational Principles in Mechanics. Germain, P. (1973) The method of virtual power in continuum mechanics-II-Microstructure, SIAM J. Appl. Math., 25, 556--575. Green, A.E. & Naghdi, P.M. (1995a) A unified procedure for construction of theories of deformable media. I. Classical continuum Physics, Proc. R. Soc. Lond. A, 448, 335--356. Green, A.E. & Naghdi, P.M. (1995b) A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. R. Soc. Lond. A, 448, 357--377. Green, A.E. & Naghdi, P.M. (1995c) A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. R. Soc. Lond. A, 448, 379--388. Green, A.E. & Rivlin, R.S. (1964) Simple Forces and Stress Multipoles, Arch. Rational Mech. Anal., 16, 325--353. Jou, D., Casas-Vázquez, J. & Lebon, G. (1996) Extended Irreversible Thermodynamics, Springer-Verlag, Berlin. Landau, L.D.& Lifshitz, E.M. (1987) Fluid Mechanics, 2nd ed., Pergamon Press, Oxford. Maugin, G.A. (1980) The method of virtual power in continuum mechanics: application to coupled fields, Acta Mech., 35, 1--70. Pao, Y.H. (2003) A unified Variational Principle of Thermo-mechanics, Proceeding of the Cross-Strait Workshop on Dynamics, Control, and Variational Principles in Mechanics. Truesdell, C. (1969) The Rational Thermodynamics, McGraw-Hill. Wang, L.S. & Pao, Y.H. (2003) Jourdain's variational equation and Appell's equation of motion for nonholonomic dynamical systems, Am. J. Phys., 73, 72—82.. [15] Erigen, A.C. & Suhubi, E.S. (1964) Nonlinear Theroy of Simple Micro-Elastic Solids -- I, Int. J. Eng. Sci. 2, 189-203. [16] Erigen, A.C. (1966) Linear Theory of Micropolar Elasticity, J. Math. Mech., 15, 6, 909-923. [17] Mindlin, R.D. & Tiersten, H.F. (1962) Effects of Couple-Stresses in Linear Elasticity, Arch. Rat. Mech. Anal., 11,415-448. [18] Toupin, R.A. (1962) Elastic Materials with Couple-stresses, Arch. Rat. Mech. Anal., 11, 385-414.. 9.
(37) Part II Global Backstepping Tracking Control for Car-Like Mobile Robots 1. Introduction Rolling wheels are frequently installed to enhance the mobility of a robot such that the working space can be enlarged significantly. However, due to the appearance of the nonholonomic constraints, the motion planning and the tracking control of wheeled mobile robots are difficult to be managed. In the phase of motion planning [1, 2], a suitable trajectory is designed to connect the initial posture (i.e. the position and the orientation of the robot) and the final one such that no collisions with obstacles would occur and the kinematic constraints are satisfied. Once the optimal path is obtained, the navigation and control process enters the tracking phase in which the kinematics as well as the dynamical equations must be considered. To appropriately integrate the kinematics and the dynamics such that every path can be followed efficiently and globally is the main subject of this report. Among various schemes in performing nonholonomic motion planning and stabilization, the transformation of the kinematic equations to chained form or skew-symmetric chained form attracts the interests of many researchers [3, 4] due to their simple structures. Based on the chained form, time-varying feedback [5], discontinuous feedback [6], and hybrid strategy [7] can be applied to circumvent the under-actuated problem. In particular, the idea of backstepping [8, 9] can be adopted to systematically design recursive algorithms for the stabilization of a multi-input chained system such as a fire truck [10], the tracking of a two-input chained system such as an articulated vehicle [5], or the tracking with saturation constraint for a class of unicycle mobile robot [11]. However, in transforming the kinematic equations to the chained form according to the algorithm given in [3], singularity problems may occur. In certain postures, the transformation becomes singular and thus the corresponding controller is not global [12]. This problem needs to be solved before the backstepping controller can be applied to all possible trajectories. Most of the methods in nonholonomic motion planning only deal with the kinematics of the wheeled mobile robot. To perform tracking control, the mass and the moment of inertia of the vehicle cannot be ignored. Therefore, for the nonholonomically constrained mechanical systems, it is desired to develop a controller which takes the kinematic model and the dynamic model into account simultaneously. In [13, 14], a kinematic controller and a neural network computed torque controller are integrated to stabilizing a nonholonomic mobile robot in which uncertainty exists. The point stabilization problems were solved in [15, 16] by first transforming the kinematical equation into a skew-symmetric chained form, and then designing the adaptive controller for the combined system. In [17], a robust adaptive control scheme is proposed for the tracking control of wheeled mobile robot. However, it shall be seen in this paper that by suitably choosing state variables, the dynamical equations can be decoupled from the kinematics. This feature has not been observed in these works so that the previously designed controllers are more complex. In this report, based on the decoupling feature of the underlying system, a global hierarchical tracking controller using backstepping idea is proposed. By selecting a proper set of privileged variables, it is seen that the reduced Appell equations discussed in [18] is decoupled from the kinematic equations. The tracking of a desired trajectory can thus be fulfilled by a kinematic compensator, which generates updated desired values for the privileged variables, and a dynamic controller, which issues control commands such that the new set of privileged variables can be followed. To resolve the above-mentioned singularity problem, it is found that two chained systems can be used to encompass all postures of the car-like mobile robot. A recursive backstepping controller can be designed to each chained system in its region of applicability. However, the compensations of the privileged variables may be severely discontinuous when the active chained system is changed. Therefore, a switching algorithm with associated continuation method which adjusts the control parameters is proposed in this paper to yield smooth signals while maintaining the desired performance. Simulation results performed in the paper show that the proposed scheme can be used effectively to track arbitrary paths.. 2. Problem Description The practical problem to be attacked in this paper is the tracking of a desired trajectory for a four-wheeled mobile robot, as shown in Fig. 1, moving on a horizontal plane. The system may be modeled by a platform with mass mc, width w, length l, and height hc, attached by four rolling-without-sliding wheels with equal masses mw and radius a. To simplify the analysis, we assume that the two wheels on each axle (front or rear) can be treated as a single wheel centered at the midpoint of the axle (Qf or Qr , respectively), cf. Fig. 1. The approximated model is the so-called car-like mobile robot, which consists of a platform (body c), a front wheel (body f) and a rear wheel (body r). The masses of the rim of each wheel and platform are denoted by mw′ and mc′ , respectively. Let ρ f , ρ r be the distance between the point Qf , Qr and the mass center of platform C, respectively. The contacts between the wheels and the ground are assumed to be pure rolling without slipping.. 10.
(38) Ey θ. ecz. ϕ ecy. ψ ecx C. Ex. O. Fig. 1. A four-wheeled mobile robot. EE yy. eexxf. φ ey. l. yf. yc. Wf. dρf f. e cy. Qf. PC. dρrr Qr. yr. Wr. w 2ra. θ xr. Fig. 2.. xc. xf. x. E E x. The configuration of the car-like mobile robot.. The translational motion of body i (i = c, f, r) may be described by the position of its mass center, which is expressed x y z with respect to the inertial frame {E , E , E } as. riC = xi E x + yi E y + zi E z .. (1). To describe the rotational motion, the type (3-1-2) Eulerian angles are used to rotate the inertial frame {E x , E y , E z } to {i i' , ji' , k i' = E z } by the heading angle θi , and then to {i "i = i i' , j"i , k "i } by the camber x y " z angle ψ i , and finally to the pre-specified body frame {ei , ei = ji , ei } of body i by the spin angle ϕi . If the rotation is time-varying, the rates of change of the Eulerian angles are related to the angular velocity i by:. ω i = θ&i E z + ψ& i i i' + ϕ&i j"i . (2) The six variables ( xi , yi , zi ,ψ i , ϕi , θ i ) are adopted here to describe the configuration of body i, and therefore. there are eighteen variables to be specified for the system. However, due to physical constraints, the number required may be reduced. By the assumption that the motion is horizontal, we have (i) zr = a , (ii) z f = a , (iii) zc = hc . The platform is assumed to be kept horizontal as well so that (iv) ϕ c = 0 and (v) ψ c = 0 , and hence the triad {ecx , ecy , ecz } coincides with {i 'c , j'c , k 'c } and {i "c , j"c , k "c } for the platform. If the wheels are properly " " " aligned, the camber angles for the wheels vanish, (vi) ψ r = 0 , (vii) ψ f = 0 , so that {i i , ji , k i } coincides ' ' ' with {i i , ji , k i }, for the wheels (i = r, f). Moreover, if the vehicle is FWD (front-wheel-driven), we have (viii) θ r = θ c ≡ θ . From the geometry of the interconnected bodies (Fig. 1), we have the last four geometric constraints: (ix) xc = xr + ρr cos θ , (x) yc = yr + ρr sin θ , (xi) x f = xr + ρ cos θ , (xii) y f = yr + ρ sin θ , where ρ = ρ f + ρr . Finally, the condition that the wheels roll without slipping is realized by the following velocity constraints:. (. ). x&r = aϕ&r cos θ , y& r = aϕ&r sin θ , x& f = aϕ& f cos(θ + φ ), y& f = aϕ& f sin(θ + φ ), where φ is the steering angle of front wheel ( φ = θ f − θ ). Applying the previous geometric constraints, the four kinematic constraints can be converted to the following independent ones:. (xiii) x&r sin θ − y& r cos θ = 0 , (xiv) xr cos θ + yr sin θ = rϕr , (xv) x&r sin(θ + φ ) − y& r cos(θ + φ ) − dθ& cos φ = 0 , (xvi) xr cos(θ + φ ) + yr sin(θ + φ ) + dθ sin φ = rϕ f . . . . . . . . 11.
(39) . . . Based on these constraints, the angular velocities of the bodies in (2) can be simplified as = θE ,. c. z. = θ E + ϕ r jr , z. r. = (θ + φ )E + ϕ f j f .. ". z. f. ". (3). It is desired to control the steering angle of the front wheel and the spin of the rear wheel by exerting torques. τ i , (i = 1, 2) , respectively, such that the system is able to track a reference trajectory which satisfies the constraints.. ( . Due to the twelve geometric ones, the dimension of the system becomes six, and we may choose xr , yr ,θ , ϕ r , ϕ f , φ as the generalized coordinates. The desired trajectory may be then specified by (xrd(t), yrd(t), d(t), rd(t), fd(t), φd(t)), t∈(0, tf). Due to the four nonholonomic constraints, the degree of freedom of the system is further dropped to two.. ). 3. Reduced Appell’s Equation of Motion. . The framework described in [8] for reduced Appell’s equation is applied to derive the equations of motion. By choosing r and φ as the privileged coordinates, the dynamic equation can be expressed in matrix form as. . . M (y )y + C(y , y )y = B (y ). . (4). where. M (y ) =. I η . 1. 2 a. . η a I m tan φ . tan φ + I 2 + I w sec φ 2. 2. η a I m tan φ. . . Im. ,. ( I1ηa2 + I w ) φ sec 2 φ tan φ 0 1 ηa tan φ C(y , y ) = , , B( y ) = 0 2 1 η a I mφ sec φ 0 . . I1 = ( I c + mc ρ r + mw ρ + I w ), I 2 = ( mc a + 2mw a + I w ), 2. I m = mw′ a. 2. 2. 2. 2. 2 , I w = mw′ a , I c = mc′ ( w + l ) 3. 2. 2. 2. Various control schemes can be used to fulfill the objective of steering the privileged variables with (17). To deal with uncertainties on the parameters of the system, one may apply the idea of adaptive control for which some intrinsic properties of the reduced Appell equations is essential. For the dynamic systems, the following lemmas can be established. Lemma 1: M(y) is an m×m positive-definite symmetric matrix.. & (y ) − 2C( y, y& ) is skew-symmetric. M. Lemma 2.. If all the system parameters such as masses, moments of inertia, and physical specifications of the mobile robot, etc, are known, we may use traditional methods to steer the privileged coordinates. However, in many cases, some p parameters are unknown or uncertain, which may form an uncertain parameter vector ∈ R . By appropriate re-arrangement, we may express the left-hand side of (4) in the following linear parametric form.
(40) . . M ( y ) y + C( y , y ) y = Y ( y , y , y ) , m× p. (5). where Y (⋅) ∈ R is termed the regressor matrix [20], whose elements consist of known functions of y , y& , and &y& . This form shall be used later to design the adaptive control law in the dynamic level. In particular, by choosing the unknown vector of parameters as. . = η a I1 2. I2. ηa I m. the corresponding regression matrix. . . . I w , T. Y in (5) is given by. ϕ r tan φ + ϕ rφ sec φ tan φ 2. 0. 2. (6). . ϕr 0. . . . . . ϕ r sec φ + ϕ rφ sec φ tan φ . φ tan φ. ϕ r tan φ + ϕ rφ sec φ 2. 2. . φ 2. .. 2. . It is noted that z does not appear in (4), and hence the system of mobile robot is reducible so that the idea of the hierarchical control proposed in this paper is applicable.. 12.
(41) 4. Hierarchical Tracking Controller Design As described in the previous section, the dynamics of a reducible mechanical system may be separated into two parts: the reduced Appell equation (4) and the corresponding kinematic equation relating the privileged velocities and the non-privileged ones. Since the reduced equations are decoupled from the kinematic equations, a controller may be designed to steer y to the desired yd(t) independently. However, if the initial condition is not perfect or there are some disturbances during the motion, the desired non-privileged variables zd(t) cannot be tracked. It is then necessary to invoke the kinematic relation to fulfill the control objective. To accommodate the constraints and take the advantage of the decoupling property of the mobile robot system, the idea of hierarchical tracking controller with three levels will be proposed. The specific structure with the backstepping controller in the kinematic level for the car-like robot is described below, with details given in the next section. A.. Transformation to Chained Forms. It is noted first that, in the tracking of the vehicle’s motion, the front wheel is not the driven wheel and its rotation angle ϕ f is not concerned. In fact, for the planar motion of the vehicle, it is desired to track the non-privileged posture variables ( xr , yr , θ ) , which is subject to the kinematic constraints. x&r = aϕ&r cos θ , y& r = aϕ&r sin θ , θ& = ηaϕ&r tan φ .. (7). The goal of the kinematic compensator is to find a set of new reference privileged velocities u c ( = y& c ) such that desired posture variables can be followed. To construct a suitable Lyapunov function so that tracking can be assured, the idea of backstepping controller discussed in [8, 9] may be adopt, since the kinematic relation may be transformed into a chained form. To perform the transformation, the steering angle φ is added to form the state variable x = ( x1 , x2 , x3 , x4 ) = ( xr , yr ,θ , φ ) due to its presence in (7), and the state equation becomes. a cos x3. 0. a sin x 3. 0. x=. u.. η a tan x4 . 0. (8). . 1. 0. The technique of input-state linearization [19] is next used to conduct the transformation. By using the set of state transformation ξ(x) = Ξ(x) and input transformation u = Ψ(x) v , where. . . 1 sec x 3 a (x) = n −1 − Lg ξ1 L Ln − 2ξ g g 1 1. 2. 1. x2 ξ1 tan x3 ξ 2 ( x) = = 1 , 3 ξ3 tan x4 sec x3 ρ ξ4 x1 . 1 sec x3 a = 1 3 2 − sin x4 tan x3 sec x3 n−2 Lg Lg ξ1 ρ 0. 2. . 0. ,. ρ cos x4 cos x3 2. 3. . (9). 1. system (8) can be transformed into the following chained form. ξ1 = ξ 2 v1 ξ = ξ v 2 31 . ξ = v 3 2 ξ 4 = v1 . . (10). . . The tracking problem now becomes to track the desired ξ d by designing suitable input v through (10). However, singularity may occur around θ d = ± π 2 , where the tracking of the chain form (10) with the transformation (9) is not feasible. To overcome the above problem, we construct another set of coordinate transformation ξ = Ξ ( x) and input transformation u = Ψ( x) v by using the same procedures described before, as. 13.
(42) x1 ξ1 cot x3 ξ 2 ( x) = = 1 , 3 ξ 3 − tan x4 csc x3 ρ ξ4 x2 . . 1. ( x )= . a. csc x3. 3 sin 2 x cot x csc x 4 3 3. . . 0. ,. − ρ cos x4 sin x3 . ρ. 2. 3. (11). . such that the state equation can be transformed into another set of chained form. ξ1 = ξ 2 v1 ξ 2 = ξ 3 v1 ξ 3 = v2 ξ 4 = v1 . . (12). . . It is observed that singularity occurs when θ d = 0, ±π . Thus, the set of equations (10) is complementary to that of (12), which shall be used interchangeably according to the following switching mechanism. B.. Switching Mechanism. To avoid the singularity arising in the tracking process, we adopt two sets of complementary chained form systems to design the backstepping controller. From the above discussion, If θ is closed to kπ ( k = 0, ±1) , it is desirable to adopt the ξ subsystem to design the kinematic compensator u c (t ) . On the other hand, if θ is closed to ± (π 2) , then one should change the chained form to the one with . Depending on the heading angle θ of the mobile robot, a mechanism is designed to perform the switching, which is divided into two phases. As θ increases, cf. Figure 3(a), the switch from ξ -system to -system occurs at θ = π / 3, −2π / 3 , and that from -system to ξ -system at θ = 5π / 6, −π / 6 . For example, as θ increases from π / 6 and enters the region θ ≥ π / 3 , the system which generates u c (t ) shall be changed from (10) to (12). On the other hand, as θ decreases, the switching occurs at θ = π / 6, −π / 3, −5π / 6, 2π / 3 , as shown in Figure 3(b). . . . 2π 3. π. 2π 3. . 3. coordinate. ξ. π 3. coordinate. . 5π 6. coordinate. π. 5π 6. π 6. 6. ξ coordinate. −. 5π 6. −. −. 2π 3. Fig. 3(a): The strategy as. −. θ. π. −. 6. π. 5π 6. −. −. 3. increases.. 2π 3. Fig. 3(b):. −. The strategy as. θ. π 6. π 3. decreases.. The un-symmetric patterns are imposed to prevent the chattering phenomenon. If θ enters the region θ ≥ π / 3 and moves back to the region π / 6 < θ < π / 3 , the active system remains (12) until θ goes below π / 6 . Therefore, if θ moves back and forth around a switching angle, no switching happens except the first one.. C.. Hierarchical Tracking Control Scheme 14.
(43) Based on the switching algorithm, the overall design of the hierarchical scheme is depicted in Figure 4. On the top level, the motion planner produces the desired trajectory ( xrd (t ), yrd (t ) ) according to task requirements and the conditions of constraints. The switching algorithm determines which set of desired values, either ξ id (t ) or ξ id (t ), i = 1, 2, 3, 4, are computed from either Eq. (30) or (33), respectively. For the active chained system, the backstepping technique is then used to design suitable v c (t ) or v c (t ) , from which the compensation u c (t ) is obtained by using the transformation Ψ or . A continuation method is established to make the switching smooth, with details given in Section 5. The updated desired privileged coordinates y c are then found, which is fed along with u c to the dynamical controller in the bottom level as discussed in Section 6. It will be shown that with such a hierarchical design, including privileged or state variables, all the variables can be steered to the desired value asymptotically.. Motion Planner ( . d. (. , vd ). −. . d. , vd ). ( . Turning Controller Parameters. , vd ) / ( e , v d ) . e. +. . Input Transformation. Backstepping Controller v c / v c in Kinematic Level. uc. . ∫. yc. . . Switching. −. Mechanism. +. θ. ( uc , y c ) . ( u, y ). Sliding Mode Controller in Dynamic Level. Actuator. Mechanical System. y. Sensor(1). x. Change of Coordinates. Fig. 4.. Sensor(2). The block diagram of hierarchical control design.. 5. Continuation Method in Kinematic Compensator Design After the state equation being transformed into the chained forms, the technique of backstepping can be applied to design an effective controller. While the method discussed in [5] may be adopted, a new algorithm is developed here to efficiently generate the controller.. A.. Backstepping Controller Design Consider a general 2-input chained system in the following form,. ξ i = ξ i +1v1 , ξ n −1 = v2 , ξ = v . n 1. (1 ≤ i ≤ n − 2). (13). To track the desired states ξid , i = 1, 2, L , n, and control inputs vid , i = 1, 2, for which (13) also holds, we first define ξie = ξi − ξid (i = 1, 2, L , n) , and derive the tracking error equations as. ξ&1e ξ&1 − ξ&1d ξ 2 e v1 + ξ 2 d (v1 − v1d ) & & & ξ 2 e ξ 2 − ξ 2 d ξ 3e v1 + ξ3d (v1 − v1d ) = . ξ& e = M = M M & & & (v2 − v2 d ) ξ ( n −1) e ξ n −1 − ξ ( n −1) d ξ& ξ& − ξ& − ( v v ) 1 1d n nd ne 15. (14).
(44) The goal is to find a time-varying controller,. v v c = 1c = vˆ c (ξ e , v1d , v2 d ) , v2 c such that the tracking error. (15). ξ e converges to zero asymptotically, i.e., lim ξ − ξ d = 0 . t→∞. The idea of backstepping is used to systematically construct the Lyapunov function so that the asymptotical stability can be assured. Following the standard procedure, we obtain the control law. v2 = v2 d − k2 χ n −1 + ( β n − 2 − χ n −2 ) v1 , v1 = v1d − k1∆ n ,. . where ∆ n = . . ∑χ ξ n−2. j. ∑χ γ. . n −1. ( j +1) d. j =1. −. j. ( j −1). + k 3ξ ne . By performing the Lyapunov stability analysis, we can show that the. j =1. . closed-loop system is stable. B.. Continuation Method. The control law developed above for general 2-input chained systems is now used to design the kinematic compensator for the car-like mobile robot. As discussed in Section IV, two chained systems (10) and (12) are used interchangeably depending on the heading of the vehicle. The respective control laws are (n = 4). v1 = v1d − k1 ( χ1ξ 2 d + χ 2ξ3d + χ 3ξ 2 d + k3ξ 4 e ) = v1d − k1∆ 4 , v2 = v2 d − k2 χ 3 − 2 χ 2 v1 ,. (16). and. v1 = v1d − k1 ( χ1ξ 2 d + χ 2ξ3d + χ 3ξ 2 d + k3ξ 4 e ) = v1d − k1∆ 4 , v2 = v2 d − k2 χ 3 − 2 χ 2 v1.. (17). With appropriated designed controller parameters ki , ki , i = 1, 2,3, the updated desired values u c for the privileged variables are then computed from Ψ(x) v c and Ψ( x) v c , respectively. However, large discontinuity may appear when the active chained system is switched, which may lead to the failure of the adaptive controller in the dynamic level. To solve this problem, the following continuation method is proposed. First, we note that if ξ -system is active, the computed u c is related to the control parameters k1 , k2 according to the following formula *. *. u c = Ψ(x) ( FK + G ) ,. (18). where. −∆ 4 2χ2∆4. F= On the other hand, if. uc =. 0 v1d k , G= , K = 1 . − χ3 k2 v2 d − 2 χ 2v1d . ξ -system is active, we have ( x ) ( FK + G ) ,. (19). where. −∆ 4 2χ2∆4. F=. v1d k 0 , K = 1 . , G= − χ3 v2 d − 2 χ 2v1d k2 . Recall that in the above analysis, the asymptotic stability is guaranteed if k1 > 0, k 2 > 0. Therefore, we may adjust the control parameters in the positive region to make u c continuous during switching. If F or F becomes 16.
(45) singular, which means that χ 3 or ∆ 4 is zero, both (18) and (19) lead to the original desired value of u , and the * * value of K or K becomes immaterial. We simply set K = K or K = K . On the other hand, if the slack variables χ i are away from zero, both F and F are nonsingular. We may try to find the appropriate control parameters such that the difference between (18) and (19) is minimized. Nevertheless, the performance of the controller may be sacrificed, and a mechanism must be used after switching to drive the control parameters to their * designed value k ’s. Detailed process for switching from ξ -system to ξ -system consists of the following two steps. Step 1: Given. K. , find the control parameters. K0.. The problem is converted to the constrained optimization program:. min 1 PK − Q T PK − Q , ( ) ( ) K 2 subject to k1 > 0, k 2 > 0, where. (20). P = ΨF, Q = ΨFK + ΨG − ΨG. This is a problem of quadratic programming, and the cost function. can be further simplified as. 1 J (K ) = [K − H −1N]T H[K − H −1N] , 2. where H = PT P is a symmetric matrix and N = PT Q . The non-singularity of F implies that H is nonsingular. Let. [ k1′. k2′ ] = K ′ = H −1N = (PT P )−1 PT Q. T. The objective function is then expressed as. J (k1 , k2 ) =. (. ). 1 H11 (k1 − k1′) 2 + H 22 (k2 − k2′ )2 + 2 H12 (k1 − k1′)(k2 − k2′ ) , 2. where H ij denotes the (i,j)-component of H. It is easily seen that if k1′ and k 2′ are both positive, they are the solutions, i.e. k10 = k1′, k 20 = k 2′ . If they are both negative, the minimum occurs at (0, 0) and we set k10 = ε , k20 = ε , where ε is a very small positive number. Alternatively, if k1′ ≤ 0 and k2′ > 0 , we set k10 = ε and search for the optimal k20 > 0 such that the cost function restricted to the axis k1 = 0 is minimized. If the corresponding solution k 2′ + k1′H12 / H 22 is positive, it is set to be the value of k 20 . Otherwise, we set k 20 = ε . Similar treatment is applied to the case that k1′ > 0 and k 2′ ≤ 0 , for which the solutions are. H ′ ′ H12 , if k1′ + k2′ 12 > 0, k1 + k2 H11 H11 k20 = ε , k10 = ε , otherwise. . Step 2: Design a mechanism such that K starting from. K (0) = K 0 , * K (i + 1) = µ (K − K (i )) + K (i ),. (21). K 0 approaches K * . A simple recursive scheme (22). *. may be used to drive the control parameters to K with rate of convergence . Larger implies that the desired K * is reached faster but the performance may be worse due to the rapidly changing of u c .. Analogous process can be used to perform the switching from ξ -system to ξ -system. It shall be shown later by simulation that the proposed scheme can solve the problems with singularity and discontinuity such that the global tracking is feasible.. 6. Sliding Mode Control in Dynamical Level According to the discussions in the previous two sections, the output of the global kinematic compensator using the idea of back-stepping and the continuation method is a set of updated privileged velocities u c , which is fed 17.
(46) into the dynamic controller to complete the tracking process. Among various methods, the sliding mode controller is chosen here to deal with the uncertainties in the car-like mobile robot system. The decoupling of the reduced dynamics from the kinematic equations makes it possible to design the dynamical controller independently. However, if there is no kinematic compensator and u is driven to u d = Ψv d , the tracking of the posture of the robot cannot be fulfilled due to the absence of ( xr , yr , θ ) in the reduced dynamics although the privileged variables can be tracked. The information of the posture is used in the kinematic compensator to give the direction to which the privileged coordinates should be driven. The relation of the kinematic compensator and the dynamic controller is very similar to that of the navigator and the pilot in steering an airplane. The reduced dynamics of the privileged variables is given in (4), which may be re-written in the following form:. y& = u, M (y )u& + C(y, y& )u = B(y ) τ.. (23). To accommodate the uncertainties in the knowledge of system parameters, an adaptive sliding mode controller is % (t ) ≡ u(t ) − u c (t ) → 0 and adopted to design the control torque such that the tracking errors u y% (t ) ≡ y (t ) − y c (t ) → 0 as t . Applying the idea of sliding mode control [20], the sliding surface is given by s = 0 where the sliding variable s is chosen as. s = [ s1. . . . s2 ] = u + Λ y , T. (24). where Λ is a positive definite matrix. The sliding variable may be further written as s = u − u s , where the T auxiliary variable u s = [ϕ&rs φ&s ] = u c − Λy% . With the unknown parameter vector given in (6), it is desired to drive the system toward the sliding surface, i.e. the sliding variable s 0. By taking the difference between Eq. (68) and the second equation in (23), we obtain the evolution equation of the sliding variable as. . . M ( y )s& + C(y , y& )s = B (y ) τ − Ys (y , y& , u s , u& s )Θ .. The control torque must depend on the unknown parameter vector. % =Θ ˆ − Θ be the estimation error. Since to be available. Let Θ adaptive law can be chosen as.
(47) . , and hence an estimation for ˆ , i.e. Θˆ , needs Θ is constant, we have = . The control and (25). = − Y(y, y , u , u ) s, ˆ. . = B (y )[Ys (y , y , u s , u s ) ˆ − K s s], #. −1. (26). T. s. s. . #. where B is the left inverse of B, and K s and are positive-definite matrix. With such laws, it can be shown then that the privileged variables can be steered to their respective desired values. Details of the proof can be found in [21].. 7. Simulation Results. .. To examine the effectiveness of the proposed global backstepping tracking control methodology, computer simulations for a car-like mobile robot were performed. The system parameters of a large vehicle shown in Fig. 1 were selected as a = 0.3m, ρ r = 0.5m, ρ f = 0.75m, = 1.75m, w = 1.5m, mc = 20kg, mc′ = 6kg , mw = 1 kg , mw′ = 2kg . The desired trajectory (xrd(t), yrd(t)) is obtained by finding a cubic B-spline function (cf. [20]) passing through 12 intermediate points, {(-15,-5), (-13,4), (-10,11), (-8,13.5), (-5,15), (-2,13), (-0.5,10), (2,6), (7,8), (7,15), (2,17), (-2,13)}. Assume that the current values of the states are available so that the slack variables or χ can be obtained. By choosing the sets of control parameters as (k1* = 16, k2* = 24, k3* = 3) and (k1* = 36, k2* = 24, k3* = 3) , we find the controls v c and v c for the ξ -system and the ξ -system by the laws (16) and (17), respectively, as described in Section V.. . . (. ). Which one of v c and v c is used to generate the updated privileged velocity u c = ϕ&rc , φ&c depends on the heading angle θ according to the switching mechanism discussed in Section IV. To make u c continuous before and after switching, the continuation method described in Section V is applied with the parameter µ = 0.001 . The sliding mode controller described in Section VI is then invoked to track the privileged variables to the updated desired values adaptively and asymptotically with the parameters Ks = diag{30, 30}, = diag{4, 4}, and = diag{100, 100, 100} in (26).. . . To signify the adaptive performance of the controller, the initial values for the estimator is selected as. . ˆ (0) = [0, 0, 0]T , which is different from the true value true = [0.3, 2.1, 0.4]T. It is further assumed that initially Θ xr (0) = −10, yr (0) = −5, θ (0) = 0o , φ (0) = 0 o , which is away from the desired ones, i.e. (-15, -5, 45o , 35.6o ). For the above-described scenario, simulation results are shown in Figs. 5 to 8. In Fig. 5, the solid line 18.
(48) is the desired B-spline curve, and the shaded block line shows the tracking performance. The switch of active trained system between the ξ -system and the ξ -system is shown in Fig. 6. It is shown that while the initial condition is significantly away from the desired posture, the hierarchical control scheme proposed here can successfully steer the mobile robot to the desired trajectory, with the tracking errors of state variables ( xr , yr ) and (θ , φ ) being plotted in Fig. 7 and Fig. 8, respectively. Without switching mechanism, either system alone cannot be used to track the selected trajectory where the heading angle of the vehicle varies from 0 to 2 and singularity must appear at some point for either chained system. y - Axis 20. sStatus Switching Status. (m). 2 1.8. 15. 1.6. 10. 1.4 1.2. 5 . 1. System 0.8 0 0.6 0.4. -5. 0.2 . 0. -10 -20. -15. -10. -5. 0. 5. x - Axis. Fig. 5:. 10. System 0. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Time (s). B-spline trajectory tracking Performance.. Fig. 6: Switching status. Error Response. Error Response (m). 1. (m). 2. 2. (rad). ye. 1.5. 1. 1 0. φe. 0.5 0. -1. xe. -0.5. -2. -1. θe. -3. -1.5 -4. 0. 1. 2. 3. 4. 5. 6. -2. 0. T ime (s). Fig. 7:. 1. 2. 3. 4. 5. 6. T ime (s). Tracking error of state variables. Fig. 8: Tracking error of state variables. 8. Conclusions A global tracking controller was designed for a car-like mobile robot after its model being established based on the reduced Appell equation, which is decoupled from the kinematics for the tacitly selected privileged variables. The advantage of the decoupling feature was taken in the hierarchical design in which the control scheme is separated into the kinematic compensator and the dynamic controller. The updated reference values for the privileged variables were obtained from the compensator were fed into a sliding mode controller for the reduced dynamics to steer the privileged variables and all the other variables shall follow suit due to the nature of the system. This concept is quite different from that of the two-stage controllers discussed in [15, 16, 17] such that it is possible to run the controller and the compensator at different sampling rates in our design. Our design scheme may be even more beneficial in dealing with more complicated systems. A mobile robot in fact consists of many components interacting with each other and the Lagrangian formulation may not be tractable. The formulation of Appell’s equation is much more transparent and leads to a suitable structure for controller design as shown in this paper. In the development of the kinematic compensator, the state equation was first transformed into chained forms such that the structure becomes simpler and the idea of backstepping can be applied directly. To deal with the singularity problem arising from the transformations, a switching mechanism based on the posture of the robot was proposed. A continuation algorithm was then invoked to yield continuous changing of the desired privileged variables by adjusting the control parameters. The performance of the controller is maintained by the approaching method such that the control parameters reach their specified values in due course. In previous works, it is more emphasized on the techniques for the controller design of chained systems, while the singularity problem has not been addressed. According to the simulation results, the proposed methodology successfully integrates the kinematic constraints and the dynamics to generate practical control command to track all the trajectories for the car-like 19.
(49) mobile robot.. References [1] D. E. Wilson, and E. C. Luciano, “Nonholonomic Path Planning Among Obstacles Subject to Curvature Restrictions”, Robotica, vol.20, 2002, pp.49-58. [2] G. Paolo, and G. Alessandro, “A Technique to Analytically Formulate and to Solve the 2-Dimensional Constrained Trajectory Planning Problem for a Mobile Robot”, Journal of Intelligent and Robotic Systems, vol.27, 2000, pp.237-262. [3] R. M. Murray, S. S. Sastry, “Nonholonomic Motion Planning: Steering Using Sinusoids”, IEEE Transactions on Automatic Control, vol. 38, no. 5, May 1993, pp.700-716. [4] C. Samson, ”Control of chained system: Application to path following and time-varying point-stabilization of mobile robots”, IEEE Trans. on Automatic Control, vol.40, no.1, 1995, pp.64-77. [5] Z. P. Jiang, and H. Nijmeijer, “ A Recursive Technique for Tracking Control of Nonholonomic Systems in Chained Form”, IEEE Trans. on Automatic Control, vol. 44, no. 2, 1999, pp. 265-279. [6] A. Bloch, and S. Drakunov, ” Stabilization and Tracking in the Nonholonomic Integrator via Sliding Modes”, Systems & Control Letters, (29), 1996, pp. 91-99. [7] O. J. SØrdalen, and O. Egeland, “Exponential Stabilization of Nonholonomic Chained Systems”, IEEE Trans. on Automatic Control, vol. 40, no. 1, 1995, pp. 35-49. [8] P. V. Kokotovi , “The Joy of Feedback: Nonlinear and Adaptive”, IEEE Control Systems Magazine, (12), 1992, pp. 7-17. [9] I. Kanellakopoulos, P. V. Kokotovi , and A. S. Morse, “Systematic Design of Adaptive Controllers for Feedback Linearizable Systems”, IEEE Trans. on Automatic Control, vol.36, pp.1241-1253, 1991. [10] Z. P. Jiang, “Iterative design of time varying stabilizers for multi-input systems in chained form”, System & Control Letters, 28 (1996) 255-262. [11] T. C. Lee, K. T. Song, C. H. Lee, and C. C. Teng, “Tracking Control of Unicycle-Modeled Mobile Robot Using a Saturation Feedback Controller”, IEEE Trans. on Control Systems Technology, vol. 9, no. 2, pp. 305 -318, 2001. [12] A. de Luca, G. Oriolo and C. Samson, ”Feedback Control of a Nonholonomic Car-like Robot”, In Jean-Paul Laumond, Editor,”Robot Motion Planning and Control”, pp.187-188, 1998, Springer. [13] R. Fierro, and L. Lewis, ”Practical Point Stabilization of a Nonholonomic Mobile Robot Using Neural Networks”, Proc. of the 35th IEEE Conf. on Decision and Control, 1996, pp. 1722-1727. [14] C. de Sousa, E. M. Hemerly, and R. K. H. Galv a o, “Adaptive Control for Mobile Robot Using Wavelet Networks”, IEEE Trans. on Systems, Man, and Cybernetics-Part B, pp.1-12, 2002. [15] W. Dong, and W. Huo, “Adaptive Stabilization of Uncertain Dynamic Non-holonomic Systems”, Int. J. of Control, vol. 72, no. 18, 1999, pp. 1689-1700. [16] S.-S. Ge, and G.-Y. Zhou, “Adaptive Robust Stabilization of Dynamic Nonholonomic Chained Systems”, Journal of Robotic System, vol. 18, no. 3, 2001, pp. 119-133. [17] M. Oya, C. Y. Su, and R. Katoh, “Robust Adaptive Motion/Force Tracking Control of Uncertain Nonholonomic Mechanic Systems”, IEEE Trans. on Robotics and Automation, vol. 19, no.1, pp.175-181, February, 2003. [18] L. S. Wang, and Y. H. Pao, “Jourdain’s Variational Equation and Appell’s Equation of Motion for Nonholonomic Dynamical Systems,” American J. of Physics, vol. 71, no. 1, 2003, pp.72-82. [19] J. J. E. Slotine, and W. Li, Applied Nonlinear Control. Prentice Hall Publications, 1991. [20] V. I. Utkin, “Variable Structure Systems with Sliding Modes”, IEEE Trans. on Automatic Control, vol. 22, no. 2, 1977, pp.212-222. . 20.
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