Robust fuzzy control for robot manipulators

Transcription

1 Roust fuzzy control for root mnipultors C.Hm, Z.Qu nd R.Johnson Astrct: A roust fuzzy control is developed for root mnipultors to gurntee oth glol stility nd Performnce. Root dynmics under considertion my include lrge nonliner uncertinties, such s nonliner lod vritions nd unmodelled dynmics. Fuzzy sets re chosen sed on performnce requirements nd stility regions of the control system. For ech fuzzy set, su-control is designed, sed on nonliner roust control design using Lypunov's direct method; this is lended with others into finl fuzzy control. The resulting control provides not only roust nd glol stility, ut lso more ccurte control performnce thn fuzzy controls otined from constnt su-controls. The proposed design is pplied to root trjectory control prolem nd compred with stndrd nonliner roust controller. The simultion results show tht the proposed control is effective nd yields superior trcking performnce. ntroduction Dynmics of root mnipultors re highly nonliner nd my contin uncertin elements such s friction. Mny efforts hve een mde in developing control schemes to chieve the precise trcking control of root mnipultors [-3. Among ville options, fuzzy control hs gret potentil since it is le to compenste for the uncertin nonliner dynmics using the progrmming cpility of humn control ehviour. Mny results hve een pulished in the re of design nd stility of fuzzy control systems [4-9. However, one of the criticl issues in fuzzy control design is, lthough designed in heuristic mnner, how to ensure glol nd roust stility of the system under control. A roust fuzzy control design hs een developed [5] for clss of nonliner systems, nd the fuzzy control is roustly nd glolly stilising. The design ssumes generl structure nd needs no supervisory control. n this pproch, roust su-control is designed first nd fuzzified for ech rule to gurntee closed-loop stility in ech fuzzy set. ndividul roust controls re then lended into the overll fuzzy controller. n this pper, the ide of roust fuzzy control design nd its ssocited Lypunov technique [lo] re pplied to develop roust fuzzy control for rootic mnipultors. The resulting control is shown to gurntee glol stility, nd to yield etter performnce thn fixed roust control s the fuzzy controller is configured to e refined roust control in which non-conservtive ounding functions of uncertin dynmics my e ville. Consequently, the 0 EE, 000 EE Proceedings online no DO: 0.049/ip-ct:00005 Pper first received 7th My nd in revised form 9th Novemer 999 C. Hm is with the Florid Spce nstitute, University of Centrl Florid, 44 Reserch Prkwy, Suite 400, Orlndo, FL 386, USA E-mil: chm@pegsus.cc.ucf.edu. Z. Qu is with the Deprtment of Electricl nd Computer Engineering, University of Centrl Florid, Orlndo, FL 386, USA R. Johnson is with the Deprtment of Mechnicl, Mteril nd Aerospce Engineering, University of Centrl Florid, Orlndo, FL 38 6, USA proposed fuzzy controller cn e configured to e refined roust control so tht etter performnce cn e chieved. Prolem formultion Consider the dynmic eqution of n n rigid-link root mnipultor: M(q)q + N(q, 4) = T () N(q, i) = Vnl(q, 44 + G(q) + F(i) + T,(q, i) q E gin is vector of joint ngle vriles; M(q) is n [n x n] inerti mtrix, which is symmetric nd positive definite; Vm(q, q), G(q), nd F(q) re [n x vectors representing the centripetl nd Coriolis terms, grvity terms, nd sttic nd dynmic friction terms, respectively; T,(q, q) represents n dditive ounded disturnce due to lod vrition ndor modelling error; nd TE!lY is control vector of torque y the joint ctutors. The following properties nd ssumptions [ ] re introduced for the proposed design.. Root dynmics nerti mtrix M(q) is symmetric nd positive definite, nd it is ounded from ove nd elow s nz 5 W q) 5 %(q)z () for some positive constnt m nd function Z(q). The centripetl/coriolis term Vm(q, q) is ounded s t Vm(q, ill 5 llill The friction nd grvity terms re ounded s G(q) + F(4)l llill, re known constnts.. Disturnce TJq, q, t) is ounded y known function q(q, q) s (3) (4) 0 T,(q, 4, t>ll 5 Y(4, i) (5) U EE Proc-Control Theory Appl., Vol. 47, No., Mrch 000

2 .3 Trjectory The desired trjectory qd E Ytn nd its derivtives qd nd qd re ounded y constnts s 3 Roust Fuzzy Control The proposed control is in the form of for constnts cl nd c,. Since the control ojective is trjectory trcking, the trcking errors re defined s e=qd-qq, e=q..d -4 mesurements of q nd q re required in the susequent control design. The stte of the trcking error system is then chosen to e x = [et et]. To design trjectory trcking control, rewrite eqn. in terms of the trcking error given y eqn. 7 nd formulte the stte-spce eqution (7) X =&+BM(q)-'(AA - T) (8) 0 n A=[o o]> = A - B R-B~P,.=[:]> AA = M(q)(R-'BTPx + qd) + N(q, 4) (9) n is the identity mtrix, nd mtrix P is the positive definite solution of the Riccti eqution: for ny given pir of mtrices Q, R > 0 ATP +PA - PBR-'BTP + Q = 0 (0) Equivlently, if we hve positive definite mtrix in the form of PU re su-mtrix locks in P, nd P =PZp f mtrices Q nd R re set to e ', R=[';. () 0 n rn For positive constnts r nd r,, the positive definite solution for mtrix P is given y re s follows: t is ovious tht the ounding function for uncertinty AA cn e otined s )).\ denotes the Eucliden norm. EE ProcContro Theory Appl., Vol. 47, No., Mrch 000 T = - =' f- = PM, () (6) is the numer of su-fuzzy sets chosen; U, is the individul control in the ith su-fuzzy set; M is the memership function to e chosen lter; pm,(x) is the degree of memership function M; nd z is the uxiliry stte defined y z = BTPx = [P, P~,]x = Pe + Pi (7) From eqn. 3. PG'P~~ is positive definite mtrix. The ide ehind choosing uxiliry stte z is tht, since P;'P,, is positive definite, then (8) Our ojective is to design fuzzy control of the form of eqn. 6, which gurntees stility nd performnce for the system in eqn. 3. The proposed scheme is sed on the stndrd nonliner roust control [, 3 nd stndrd fuzzy control design. 3. Roust fuzzy control design The procedure of designing fuzzy control consists of four steps. Step : selections of fuzzy sets nd memership functions As one of mny possile choices, susets F, in the stte spce cn e chosen s follows. For i=,..., -, A F, = {x:x E M", nd x is either on, inside, or close to the hyper-ll defined y llzll =d,}. d = 0, nd dj > dj for i >j is finite sequence of positive incresing numers chosen y the designer to reflect which stility regions nd performnce re desired or cn e chieved. F, A {x:x E Mn, nd x is on the outside nd not close to the hyper-lls defined y llzll =d-l}. t is then ovious tht U Fj = s long s close to nd not close to re complementry sttements. From mny possile choices [9], select memership function M,(x) to mke sets F, fuzzy. The only requirement on memership function is tht the degree of memership function pmi(z) is etween zero nd one. Step : selections of Lypunov function nd ounding function The Lypunov function is chosen to e V(X) = -XTPX (9) mtrix P is given y eqn. 3, nd ounding function p,(x) is given y eqn. 5. Step 3: selection of individul fuzzy control For i =,...,, design individul control U&) to the fuzzy rule. Rule i if x E F,, then control is given y U = u,(x). ccording Choice of individul control ui(x) is not unique, ut it must stisfy the following three conditions. 3

3 () f x E Fi f l Fj for some i nd j, the signs of control vectors ui nd U] stisfy the property tht sign (U;) = sign (uj) = sign (U,) (0) sign(.) is the generlistion of sclr sign function to vector cse. (Although other forms of control cn e chosen, sign condition (eqn. 0) implies tht ll individul controllers hve the sme direction of driving the stte towrds the origin for ll vlues of z (no mtter to which Fi they elong), long ry originting from the origin in the z plne. This choice is the simplest wy to chieve stilistion.) U, is the roust control defined y 3. System stility The proposed fuzzy control is sed on the existing nonliner roust control design; therefore, stility nlysis under fuzzy control is performed in prllel to tht under the nonliner roust control design. Lemm : System in eqn. 8 is glolly nd symptoticlly stle, or uniformly ultimtely ounded under nonliner roust control (eqn. ), i.e. T=ur. Prooj to show tht system in eqn. 8 is stle under roust control (eqn. ), note tht the time derivtive of the Lypunov function (eqn. 9) is = --xtqx + [ztm(q)- AA- z~m(~)- u~] 6 = - -xtqx + xtpbm(q)- (A/l - U,) n control (eqn. l), t > 0 is design constnt, nd q(t) > 0 is uniformly continuous L, or L, time function. () Fuzzy control ui must hve the property tht T ZTUi? z U, (3) Fuzzy control ui must stisfy the inequlity tht, for ll x n this pper, fuzzy control ui(x) is selected to e U = U,, (4) ki = SUP [P,(x>~ (5) 05lZl54+, Step 4: selection of fuzzy control lw The overll fuzzy control uf is found y lending the individul controls U; ccording to the stndrd fuzzifying formul: i= Uf = - i= - -m P,(X>lZl + w(t> pmi () i= Z (6) The ove design procedure gurntees oth performnce nd roustness. Control in eqn. 4 u cn e interpreted s nonliner supervisory control which ensures glol stility. For i =,..., - controls ui provide ccurte control without over-estimting the control gin, nd the overll fuzzy control (eqn. 5) executes similrly s vrile structure controller. 4 As shown previously [3], you cn solve the ove differentil inequlity to show tht V nd xj converge exponentilly to zero or to uniform ultimte ound. 0 t hs lso een shown [3] tht the system is exponentilly convergent in the lrge nd tht trnsient excursions cn e estimted. n fct, trnsient response cn e djusted y proper choices of design prmeter 6 nd function q(t) [3]. Bsed on Lemm, the following stility result cn e esily concluded. Theorem : Under fuzzy control (eqn. 6), the system in eqn. 8 is glolly nd symptoticlly stle. Prooj The proposed control is designed to stisfy the following inequlity: T z Uf ZTU, (8) The ove inequlity is gurnteed y eqns. 0, nd 3. t follows from the proof of the Lemm tht glol stility cn e concluded. 4 Simultion A two degree-of-freedom root mnipultor is used in simultion to evlute the proposed control scheme. The dynmic equtions of the two rigid-link mnipultor cn e found else [ll], nd its prmeters re set to e ml = m =.0 kg nd, =, =.0m. n the simultion, initil conditions re given s ql(0) =q(0) = rd ( degree), ql(0) =q(0) = Ordis, nd the desired trjectory is given y qf(t) = qf(t) =.O - cos(t). The friction nd disturnce terms re ssumed to e 5 cos(5t) N-m nd F(q) = 0.5 sign (q) (9) Td = [ 5 cos(5t) sign denotes the vector sign function. For the roust fuzzy control, ounds nd ounding functions for the system re set to e m=0.5, E= 9.0 nd p,(x) = (x + ~ ~ ). t follows from the Riccti eqution tht, given Q = R =Z, Z is the identity mtrix, EE Proc.-Control Theory Appl., Vol. 47, No., Mrch 000

4 , t According to eqn. 6, the overll fuzzy control is then designed s Fig. Symmetric tringle memership function Susets of the stte spce F,, =,,3, re defined s follows: F, {x:x E 9t" nd x is either t or close to the origin} F, = {x:x E Yt" is either on, inside, osclose to the hyper-ll defined y llzll = } nd F3 = {x:x E 8'" nd x is on the outside nd not close to the hyper-ll defined y llzll =0.005}. The tringle memership function given in Fig. is chosen, nd it hs the property tht, for ll z, - i= Roust control (eqn. ) nd roust fuzzy control (eqn. 34) re implemented for comprison. The simultion results re shown in Figs. nd 3. t is ovious from the results tht the proposed roust fuzzy control system is comprle to the nonliner roust control for the root trcking. Fig. 4 shows the joint errors under roust control (eqn ) minus the corresponding errors under roust fuzzy control (eqn. 34). This comprison shows tht the proposed hzzy control results in slightly etter trcking performnce thn the nonliner roust control for t L 6.8 [S. Fig. shows tht ny vlue of z does not elong to more thn two fuzzy sets of Fi. According to eqn. 4, individul controls ui(x) for i =,,3 re ,, 5,, ~ ' The design constnt nd design fhction re chosen to e t follows from eqn. 5 tht k, = 6,764 nd k = 6, Oo i i 3 k i 6 i i - 9 io Fig. 3 Position errors () e, nd () e under roust fuzzy control , ;J,,,,,, Fig Oo i i 3 i i Position errors () e, nd () e, under roust control EE Proc -Control Theory Appl.. Vol. 47, No., Mrch ' " " " " " Fig. 4 Comprison of trcking performnce. eroust - efuq 5

5 5 Conclusions The proposed roust fuzzy control gurntees not only desired performnce, ut lso glol stility nd roustness for root mnipultors. The proposed design is to synthesise individul nonliner roust controllers for ech fuzzy rule nd then to lend them into fuzzy control. The design is to comine the dvntges of oth nonliner roust control nd fuzzy control. Simultion results hve shown the effectiveness of the proposed scheme. 6 References 3 GE, S.: Advnced control techniques of rootic mnipultors.. Proc. Amer. Control Conf., 998, pp HAM, C., nd QU, Z.: A new nonliner lerning control for rootic mnipultors, Advnced Rootics, 996, 0, (S), pp. -5 QU, Z., nd DAWSON, D.: Roust trcking control of root mnipultors (EEE Press, New York, 996) 4 BRDWELL, J., nd WANG, Y.: Lypunov stility nlysis of systems using the fuzzy-pd controller. Proc. Amer. Control Conf., 994, Bltimore, Mrylnd, pp HAM, C., QU, Z., nd KALOUST, J.: Design of glolly stilizing roust fuzzy control for clss of nonliner systems, nt. ntell. Control Syst., 996,, (), pp LANGAR, G., nd TOMZUKA, M.: Stility of fuzzy linguistic control systems.. Proc. 9th EEE Conf. on Decision nd Control, 990, Honolulu, Hwii, pp MALK, H., L, H., nd CHEN, G.: New design nd stility nlysis of fuzzy proportionl-derivtive control systems, EEE Trns., 994, FS-, 8 9 pp TANAKA, K., nd SANO, M.: Concept of stility mrgin for fuzzy systems nd design of roust fuzzy controls. Proc. nd EEE Fuzzy Systems Conf., 993, pp WANG, L.: A course in fuzzy systems nd control (Prentice Hll, New Jersey, 997) 0 KHALL, H.: Nonliner systems (Mcmilln, New York, 99) CRAG, J.: Adptive control of mechnicl mnipultors (Addison- Wesley, New York, 988) CORLESS, M., nd LETMANN, G.: Continuous Stte feedck gurnteeing uniform ultimte oundedness for uncertin dynmic systems, EEE Trns., 98, AC-6, pp QU, Z.: Roust control of nonliner uncertin systems (John Wiley & Sons, New York, 998) 6 EE Proc.-Control Theory Appl., Vol. 47, No., Mrch 000