C e f paamete adaptation f (' x) ' ' d _ d ; ; e _e K p K v u ^M() RBF NN ^h( ) _ obot s _ s n W ' f x x xm xm f x xm d Figue : Block diagam of comput

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1 A Neual-Netwok Compensato with Fuzzy Robustication Tems fo Impoved Design of Adaptive Contol of Robot Manipulatos Y.H. FUNG and S.K. TSO Cente fo Intelligent Design, Automation and Manufactuing City Univesity of Hong Kong, Hong Kong, CHINA FAX: (8) 888 Abstact: A sum adial-basis-function neual-netwok (NN) compensato with computed-toque contol and novel weight-tuning algoithms is poposed to impove tacking pefomance and to account fo stuctued/unstuctued uncetainties of obot manipulatos. The poposed weight-tuning algoithms do not equie the initial NN weights to be small. The bounds of NN weights ae guaanteed to be convegent in the sense of Lyapunov. The eectiveness of the poposed algoithm is demonstated using a laboatoy obot manipulato. Key-Wods: neual netowk, fuzzy, adaptive contol, obot manipulato Intoduction In ode to account fo the stuctued/unstuctued uncetainties of obot manipulatos, adaptive contol and vaiable-stuctue contol may be employed [, ]. In sliding-mode (a special case of vaiable-stuctue) contol, chatteing of the contol output often occus. In model-based adaptive contol (MRAC), a peliminay model based on egession matices is usually equied, and the paametes of the basic model may be pogessively updated. Howeve, the model stuctue may not be as complex as the actual obot manipulato. A neual netwok (NN) may instead be employed to epesent the nonlinea model and impovetacking pefomance []. Howeve, no expeimental esults ae pesented in [] to discuss the eects of dieent initial NN weights on the initial pefomance and oveall eo convegence. In the system discussed in this pape, the appoach is dieent. The NN is not used to epesent the entie manipulato model. It is only used to epesent the uncetainties in the model used in the computedtoque-contol (CTC) scheme []. Based on an ealie development, a sum adial-basis-function (RBF) NN compensato with new adaptation laws using fuzzy logics is poposed. System stability is guaanteed in the sense of Lyapunov. The application of fuzzy logic enhances the design exibility of the compensato not available to ealie designs epoted elsewhee. Mathematical backgound The following mathematical basis is essential fo the subsequent study []. Let A R n = [a a ::: a n ] T B R n = [b b ::: b n ] T M R mn N R nn x R n = [x x ::: x n ] T and max () ( min ()) denote the maximum (minimum) eigenvalue. The tace function of M is dened as P the sum of n the diagonal elements given by t(m) = i j= m ij, whee m ij denote the ith-owandjth-column element of M. The Euclidean nom denoted as jjjj and Fobenius nom denoted as jj jj F jjmjj = of M ae given by q max (M T M) jjmjj F = q t(m T M) Fo simplicity, the subscipt in jj jj is omitted, i.e. jj jj = jj jj. Fo a scala poduct A T B, we have B T A = A T B =t(a T B) = t(ba P T n )= i= a ib i. In geneal, the Fobenius nom of M is geate than o equal to the Euclidean nom of M such that jjmjj jjmjj F. If M is a vecto, then both noms (Euclidean and Fobenius) ae equal. Conside the following nom jjmajj jjmjj jjajj jjmjj F jjajj () An abitay squae matix N with given max and min is govened by the following elation min (N)jjxjj jjx T Nxjj max (N)jjxjj ()

2 C e f paamete adaptation f (' x) ' ' d _ d ; ; e _e K p K v u ^M() RBF NN ^h( ) _ obot s _ s n W ' f x x xm xm f x xm d Figue : Block diagam of computed-toque contol with NN compensation Basic compensato stuctue The dynamics of an n-link igid obot manipulato can be witten in the fom [, ]: M() h( _ )= () whee _ R n denote the vecto of joint position, joint angula velocity, and joint angula acceleation, espectively M() R nn the inetia matix h( ) _ R n the vecto associated with centifugal, Coiolis, gavitational and fictional toque components and R n the input toque vecto. Accoding to the well-known CTC method [], we have = ^M()u ^h( _ ) () whee ^M() and^h( ) _ ae the estimates of the actual M() and h( ), _ espectively, and u R n is the contol input vecto intoduced. Let e R n = d ; denote the vecto of the joint position eos, whee d is the desied joint position vecto, and R n denote the RBF NN signal vecto that is used to counteact the manipulato uncetainties. The vecto u in () is designed to be u = d K v _e K p e () whee K v, K p R nn ae the diagonal gain matices with positive constants k p ::: k pn,andk v ::: k vn, expessed as K p = diag(k p k p ::: k pn ) () K v = diag(k v k v ::: k vn ) () Figue shows a block diagam of obotmanipulato contol system evealing the basic CTC stuctue but with the RBF NN compensation added. Figue : Block diagam of RBF NNs Substituting () into () yields M() h( _ )= ^M()u ^h( _ ) (8) Substituting u fom () into (8) esults in the eo dynamics equation of the closed-loop obot contol system: e K v _e K p e = (9) whee R n denotes the discepancy in uncetainty compensation fo the obot manipulato, given by = h i ^M ; ()M() ; I (0) i ^M ()h ; h( ) _ ; ^h( ) _ ; = ( _ ) ; () Obviously, the vecto is epesented as a complex function of the joint vaiables, _,and. As a esult of intoducing the model eo compensation by the NN, ^M and ^h ae xed by design and no longe changed in the couse of system adaptation. This is in contast to the MRAC appoach[]. RBF neual-netwok compensato Conside an RBF NN compensato of the following fom (Figue ) = Wf(' x) ()

3 whee W R n and f R ae given by W = f( ' x) = w w ::: w w w ::: w w n ::: ::: w n f (' x) f (' x). f (' x) () () with ' R m = [' ' ::: ' n ], ' i R m = [' i ' i ::: ' im ], R m = [ ::: n ], i R m = [ i i ::: im ], and x R m = [x x ::: x m ]. In (), f i is given by f i (' i i x) = mx j= exp (x j ; ' ij ) ; i = ::: ij () In (), as fast eo convegence ate esults, f i is delibeately chosen to be the sum of RBFs athe than the moe usual poduct of RBFs [, ]. Let x = [ T _ T T ]. Hence m = n. Assuming that thee exists a sum-rbf NN compensato with suitable paamete settings, W, ' and capable of appoximating the complex function in (), we have (x) =(W ' x) () whee is the appoximation eo vecto. It is assumed that jj jj max () If the bounds of the uncetainties ae known, then the maximum limits of the Fobenius noms of W, ' and ae given by jjw jj F Wmax jj' jj F ' max jj jj F max (8) Substituting () into () yields = (W ' x) ; (W ' x) (9) In the following paagaphs, we attempt to nd out a leaning algoithm fo tuning the NN paamete settings W, ' and close to the desied paamete settings W, ' and. Dieentiating () w..t. W, ' and @' (@)T Wf 0 '(@') T Wf 0 (@) T (0) = W ; = ' ; = ;, denotes the eo vecto due to the neglected highode tems, with a maximum bound given by and f 0 ' f 0 Rm ae given by and f 0 ' = f 0 jj jj max 0 m ::: 0 m ::: 0 m m 0 0 m ::: 0 0 ::: 0 m m 0 () () The diagonal elements in () and () ae espectively given = ::: i= im () ij = exp ; (x j ; ' ij ) i ij = exp ; (x j ; ' ij ) ij Let (x j ; ' ij ) ij! i= im (x j ; ' ij ) = (W ' x) ; (W ' x) Substituting (8) into (9), we have Substituting (0) into (9) yields! () () () (8) (9) Wf 0 '(@') T Wf 0 (@) T (0) = (0) becomes ~ = ~' = ~. Equation = ~Wf Wf 0 ' ~'T Wf 0 ~T () If the paamete settings ae getting close to the desied paamete settings in the iteation pocess, the value of should become smalle.

4 Stability analysis Rewiting (9) in the state-space fom, we have _E = AE B () whee E R n, A R nn, and B R nn ae given by E = e 0nn I _e A = n ;K p ;K B = v 0nn I n Let e f denote a lteed signal vecto given by () e f = CE () whee C R nn is given by C = [ ], whee and R nn ae constant diagonal matices given by = diag( ::: n ) and = diag( ::: n ). Fo each joint j of the obot manipulato, if k vj > 0 j k vj > j () by design, then thee always exist two positive-denite symmetic matices P and Q satisfying A T P PA = ;Q () PB = C T () The poof is given in [8]. As a esult of (), thee is no need to seek fo suitable matices P and Q any longe. Hence if e f is zeo, then E will each zeo. Conside the Lyapunov function candidate V (E W ~ ~' ~) =E T PE t( W ~ T ; ; ~W) t(~'; ; ~'T ) t(~; ; ~T ) (8) whee ; R nn, ; ; R mm ae positivedenite diagonal matices. Dieentiating (8) w..t. time yields _V (E ~W ~' ~) = _E T PE E T P _E t( ~W _ T ; ; ~W) t( _~'; ; ~'T )t(_~; ; ~T ) (9) Substituting _E fom () into (9) yields _V (E ~ W ~' ~) =E T (A T P PA)E E T PB t( _ ~W T ; ; ~W) t( _~'; ; ~'T )t( _~; ; ~T ) (0) Let the adaptation laws be given by _~W = ;; e f f T k W ; jjejjw () _~' = ;e T f Wf 0 '; k ' 'jjejj; () _~ = ;e T f Wf 0 ; k jjejj; () whee k W, k ', and k ae positive scala paametes. Without the use of k W, k ',andk, the system might be unstable if the dieence between the desied NN weights and the initial NN weights is lage. Substituting () though () into (0) and with suitable simplication using the matix popeties descibed in Section, we have _V ;jjejj h min (Q)jjEjj ; k C max max k W ;jjwjj F ; W max= ; kw (W max) = k ' ;jj'jj F ; ' = max ; k' (' max) = i k ;jjjj F ; max= ; k ( max) = () whee k C = p max (C T C)and max = max max. Let =k C max max k W (W max) = k ' (' max) =k ( max) = The scala _ V will be negative if eithe of () though (8) holds jjejj > () min (Q) jjwjj F > W max () k W jj'jj F > ' max () k ' jjjj F > max (8) k Fom (), in ode to make jjejj small, the values of k W, k ', and k should be chosen to be as small as possible. Howeve, if these values ae too small, fom (){(8), the guaanteed Fobenius bounds on W, ', and also incease. In ode to minimise jjejj, jjwjj F, jj'jj F, and jjjj F simultaneously, the values of k W, k ',andk must be suitably chosen. In ode to have smooth contol signals, an index MSE (see (0) ) is employed such that IF jjejj is small and MSE is small THEN the values of (k W, k ',k ) ae small, o IF jjejj is lage and MSE is lage THEN the values of (k W, k ',k ) ae lage. As a futhe development of the ealie wok of the authos [8], a fuzzy logic contolle, denoted as FLC k, is employed in this pape such that the gains k W, k ' and k will change accoding to the paametes jjejj and MSE. Figues and show the membeship distibutions of the antecedents jjejj and MSE, and the consequent u k, whee L jjejj, L MSE,andL uk denote the espective univeses of discouse. Evenly distibuted membeship functions ae

5 S M L S M L jjejj MSE L jjejj L MSE Figue : Membeship function distibutions of antecedents jjejj and MSE Z S M L H uk Lu k Figue : Membeship function distibutions of consequent u k Table : Fuzzy ules fo FLC k S M L jjejj S Z S M M S M L L M L H (MSE) (k 0 ) employed. Table lists the fuzzy ules employed in the FLC k. The fuzzy easoning method of poductsum-gavity is employed. Fo simplicity, the values of k W k ', and k ae equal to the output u k in the cuent study. Expeimental esults A laboatoy two-link obot shown in Fig. is employed. The estimated matix ^M and vecto ^h ae given by M M ^M = h ^h = (9) M h whee M M =(^m ^m )^l ^m ^l ^m ^l^l cos( ) M = M = ^m ^l ^m ^l^l cos( ) M = ^m ^l h = ;^m ^l^l sin( ) ; ^m ^l^l sin( ) _ ^f k sgn( _ ) ^f v _ h = ^m ^l^l sin( ) _ ^f k sgn( _ ) ^f v _ Figue : A laboatoy two-link obot manipulato with ^m = kg ^m = kg ^l = 0: m ^l = 0: m ^fk = : Nm ^fk = : Nm ^fv = 0: Nm ^f v = 0: Nm. The sampling time inteval fo contol is ms. The tajectoy cycle time is seconds. Hence the numbe of samples n k is 00. Conside n = = 0 m = n = k pj = k vj = 0, j ==0, j ==0, fo j =. The lte paametes j and j ae chosen in accodance with (). The diagonal elements of the leaning ates ;, ;, and ;, fo the paametes W ', and ae equal to 0.. The values of k W, k ' and k ae set equal to 0.0 fo the xed-paamete case [8]. The initial values of W, ', and fo two cases ae andomly set in the ange [;0:0 0:0] and [;0 0], espectively. The values of L jjejj, L MSE, and L uk ae 0., 0.0, an 0.0, espectively. The angula velocities and acceleations of the two joints ae detemined using the backwad-dieence method. Moving-aveage ltes ae employed to lte the noisy angula velocities and acceleations. Without the ltes, poo tacking may esult. The mean-squaed eo (MSE) is used as an index fo tacking pefomance compaison, dened by MSE = n k n X k k= e (k) _e (k)e (k) _e (k) (0) whee n k denotes the numbe of samples in each cycle. The desied tajectoy in the expeiment is specied to be d (t) = d (t) = sin(t ) () The obot manipulato is initially set at _ _ = 0. Thee schemes A, B and C, espectively denoting i) k W = k and k ' = 0, ii) k W = k = k ' = 0:0

6 MSE Figue : 0:0 MSE A C B scheme A scheme B scheme C cycle no Plot of MSE with initial weights within scheme A scheme B scheme C 0.0 B C cycle no. Figue : Plot of MSE with initial weights within 0 [8], and iii) k W = k = k ' = FLC k, ae compaed in ode to illustate the eectiveness of the poposed weight-tuning algoithm. Figues and show the expeimental esults with dieent magnitude levels of initial weights. In Figue, the initial weights ae elatively small (andomly chosen within 0:0), and both schemes A and C give elatively small MSEs and scheme B with constant paametes k W, k ', and k incopoated in the adaptation laws (){() gives a elatively lage MSE. In the case of elatively lage initial NN weights, schemes B and C outpefom scheme A. With the intoduction of the fuzzy paamete u k, scheme C pefoms bette than scheme B. Fo lage initial weights within 000, scheme C poves to emain eliable (not illustated fo bevityin this pape). The success of the scheme C depends on how well the paametes of FLC k ae chosen. A Conclusion A novel adaptation law with an RBF compensato is pesented fo obot manipulatos. The obot tacking pefomance is guaanteed in the sense of Lyapunov with dieent (even lage) levels of initial NN weights. Without adjusting the paametes k W, k ' and k in the system by means of fuzzy infeencing, poo abitay choices xed at the stat (including zeo values) may lead to less satisfactoy MSE esults. The eectiveness of the poposed weight-tuning laws is clealy demonstated using a laboatoy obot manipulato. The inceased design exibility and pefomance impovement ae achieved with negligible additional investment in computation. REFERENCES [] J. J. Caig. Adaptive contol of mechanical manipulatos. Addison-Wesley, Reading, Mass., 988. [] J. E. Slotine,, and W. Li. Adaptive manipulato contol: A case study. IEEE Tansactions on Automatic Contol, ():99{00, Novembe 988. [] F. L. Lewis, K. Liu, and A. Yesildiek. Neual net obot contolle with guaanteed tacking pefomance. IEEE Tansactions on Neual Netwoks, ():0{, May 99. [] J. Y. S. Luh, M. W. Walke, and R. P. C. Paul. On-line computational scheme fo mechanical manipulatos. Tansactions of the ASME. Jounal of Dynamic Systems, Measuement, and Contol, 0:9{, 980. [] K. Zhou and J. C. Doyle. Essentials of obust contol. Pentice Hall, New Jesey, 998. [] L. X. Wang. Stable adaptive fuzzy contol of nonlinea systems. IEEE Tansactions on Fuzzy Systems, ():{, May 99. [] T. L. Liao and C. M. Chiu. Adaptive output tacking of unknown MIMO nonlinea systems using adial basis function neual netwoks. JSME Intenational Jounal Seies C-Dynamics Contol Robotics Design & Manufactuing, 0():{9, Mach 998. [8] S. K. Tso, Y. H. Fung, and N. L. Lin. Analysis and eal-time implementation of a adial-basis-function compensato fo high-pefomance obot manipulatos. Mechatonics, 0:{8, 000.