Hybrid adaptive control for tracking of rigid-link flexible-joint robots

Transcription

1 Hybid adaptive contol fo tacking of igid-link flexible-joint obots D.M. Dawson 2. Qu M.M. Bidges Indexing tems: Contol theoy, Robotics Abstact: A hybid adaptive tacking contolle fo igid-link flexible-joint (RLFJ) obot manipulatos is designed. The contolle is hybid in the sense that an adaptive indiect contolle is used to compensate fo paametic uncetainties in the igid-link equation as an adaptive obust contolle coects fo the dynamics (i.e. mechanical flexibilities) and paametic uncetainty associated with the actuatos. That is, in spite of these detimental uncetainties and effects, we ae still able to ensue a global asymptotic stability (GAS) esult fo the link position tacking eo. 1 Intoduction Ove the last decade thee has been much inteest in the adaptive contol of obot manipulatos. An excellent eview pape on the subject is given in Refeence 1. The basic stategy with egad to the adaptive contol of obot manipulatos has been to use a contolle that adjusts fo an unknown set of paametes online while maintaining good link tacking eo pefomance in spite of this paametic uncetainty. The typical stability esult fo the adaptive contolles has been a GAS esult fo the link tacking eo. It should be noted that almost all of the above stability esults geneated fo igid link obot manipulatos assume that the flexibilities of the joint can be neglected. Ealy wok egading the compensation fo joint flexibilities can be found in Refeences Recently in Refeence 2, Readman illustated that fo RLFJ obots thee exists a decentalised velocity contol law which asymptotically stabilises the joint flexibility dynamics if the actuato inetia matix is suficently small. In Refeence 3, Mad designed an indiect adaptive contolle fo RLFJ obots; howeve, this contolle is a hybid adaptive/vaiable-stuctue contol law. Because the contol law contains vaiable stuctue-like tems, the contol input will unfotunately exhibit the chatteing phenomenon. The most compehensive study of adaptive contol of RLFJ obots is given in Refeences 4 and 15. In Refeence, Ghobel and Spong illustate that by modifying the adaptation law and assuming the link desied tajectoy appoaches zeo as time appoaches 0 IEE, 1993 Pape 8628D (C9). eceived 29th May 1991 D. Dawson and M. Bidges ae with the School of Electical and Compute Engineeing, Clemson Univesity, Clemson, SC , USA Z. Qu is with the Depatment of Electical Engineeing, Univesity of Cental Floida, Olando, FL , USA infinity then we can achieve asymptotic link tacking. In Refeence 15, Benallegue and MSidi designed a GAS indiect adaptive contolle based on the passivity appoach; howeve, this appoach equies measuements of the link acceleation and link jek. In Refeence 16, Chen and Fu used a simila appoach as the one given in this pape; howeve, the stability esult is local, and the contolle equies measuement of joint acceleation. In this pape, we design a hybid adaptive tacking contolle fo RLFJ obot manipulatos. The teminology hybid is used to emphasise that we use an indiect adaptive contolle that compensates fo paametic uncetainties in the igid-link obot equation and an adaptive obust contolle compensates fo the uncetainties associated with the actuato dynamics (i.e. mechanical flexibilities) and paametic uncetainty associated with the actuatos. That is, in spite of these detimental uncetainties and effects, we ae still able to ensue that the obot links follow a desied tajectoy in the GAS sense. In othe wods, we achieve a GAS esult fo the link position tacking eo. This stability esult only equies that the desied link tajectoy and up to the fouth deivative be bounded. We also assume that the link position, link velocity, actuato position, and actuato velocity ae all measuable signals. It should be noted that this contolle does not depend on the assumption that the joint flexibility should be suffciently igid. That is, we specifically show how the contolle gains should be adjusted with egad to the joint flexibility constants. 2 Manipulato dynamic equation and mathematical peliminaies In this Section, we give the dynamic equations associated with RLFJ obots along with some physical popeties that ae inheent to obot manipulatos. The model [5] fo the igid link pat of the RLFJ obot is taken to be M(q)ij + v,(q. 447 G(q) + F(4) = Kz (1) z=y..-9 (2) M(q) is an n x n inetia matix, V,(q, q) is an n x n matix containing the centipetal and Coiolis tems, G(q) is an n x 1 vecto containing the gavity tems, F(q) is an n x 1 vecto containing the fiction tems, q,(t) is an n x 1 vecto used to epesent the actuato displacement This wok is suppoted in pat by the US. National Science Foundation Gants MSS and IRI

2 fo the RLFJ obot, q(t) is an n x 1 vecto epesenting the link displacement, K is a positive definite, diagonal, constant, flexibility matix, and z(t) is an n x 1 vecto used to epesent the diffeence between q,(t) and q(t). Fom Refeence 5, the associated mechanical dynamics of the actuato is taken to be Jq, + Bq, + Kz = up (3) J is an n x n constant, diagonal, positive definite matix used to epesent the actuato inetia, B is an n x n constant diagonal matix used to epesent the actuato damping, and udt) is an n x 1 vecto used to denote the toque contol input. As stated in the intoduction, the main idea of this pape is to obtain good link tacking pefomance in spite of any paametic uncetainty egading the dynamics given by eqns. 1 and 3. It is impotant to emphasise that bounds ae assumed to exist fo each of the 'uncetain' paametic quantities. Fo example, the joint flexibility matix is assumed to be bounded as kill~ll~ = L{K}ll~ll~ Q xtkx Q ~wx{k}ii~i12 = k,llxl12 (4) k,, k, ae postive scala bounding constants, and x is an n x 1 vecto. Because the components of K ae typically lage, we can use eqn. 4 to nomalise eqn. I. That is, we can divide eqn. 1 by k, to yield M(q)ii + V,(q, 4)q + G(q) + F(4) = = +(d/k2 7 Vn(qL 4) = Vn(q, 4hk2 9 Gq) = G(q)/k,, F(4) = F(q)/k,, and K = K/k,. Bounds on the new nomalised flexibility matix (i.e. K) ae now defined as L,11x112 = A,i.{R}lIXll2 Q XTRX Q A,,{R}lIxllZ (5) = ~211x112 (6) L, = k,/k, and L2 = 1 (7) In the stability analysis of the obot contolle discussed in this pape, the following manipulato popeties will be used. Fo a moe detailed explanation of these popeties as elated to igid link manipulatos, the eade is efeed to Refeence Positive definiteness The inetia matix M(q) defined in eqn. 5 is symmetic, positive definite and is unifomly bounded as a function of q, i.e. fo any n x 1 vecto x, we have m2 = - llx1i2 k2 m, and m2 ae positive scala constants that depend on the mass popeties of the specific obot (i.e. evolute obot), and (1. (1 is used to denote the Euclidean nom [7]. It should also be noted that because J defined in eqn. 3 is an n x n matix that is constant, diagonal, and positive definite, we can state that fo any n x 1 vecto x i111~11~ = L.{J}ll~ll~ < xtjx (9) j, is a positive scala constant that depends on the mass popeties of the specific actuato Skew symmetic A useful elationship exists between the time deivative of the inetia matix M(q) and the Coiolis/centifugal matix Vm(q, 4). Specifically, a cetain quadatic fom is equal to zeo. That is, we can wite xt(if(q) - 2Vn(q, 4))x = 0 fo all n x 1 vectos x. 2.3 Paamete sepaation The left-hand side of eqn. 5 can be ewitten as (10) W(q, 4, ii)4 = MMii + Uq, 44 + ad + F(4) (1 1) W is an n x matix of known obot functions, and 4 is an x 1 vecto of unknown constant paametes. 3 Indiect igid link adaptive contolle In this pape, ou contol objective is to obtain good link tacking in spite of paametic uncetainty and joint flexibilities. Theefoe, we will define the link tacking eo to be = q d - 4 (12) qd is an n x 1 vecto used to epesent the desied link tajectoy. We will assume that qd and its fist, second, thid, and fouth deivatives ae all bounded as functions of time. We now fomulate the eo system that will be used in the stability analysis. It should be noted that this step is not a tivial matte because the stability analysis is diectly elated to the contuction of the eo system. Fist, we define the filteed tacking eo [8] L = ael + 6, (13) a is a scala positive constant. Using popety 2.3, we can ewite eqn. 5 in tems of eqn. 13 in the following fashion M(q)tL = - Vn(q, q)l + ~4 - Rz (14) Y4 = M(qXiid + ail) + VXq, 4Xdd + ae3 + a) + F(4) (15) Y is an n x matix of known time functions, and 4 is an x 1 vecto of unknown constant paametes. It is assumed that the fist and second time deivatives of the 'egession' matix Y ae bounded as functions of time. This assumption is needed because of the second ode natue of the actuato dynamics given by eqn. 3. Because thee is no contol input in eqn. 14, we will add and subtact a fictitious contol input in the fom Ru, on the ight-hand side of eqn. 14 to yield M(q)tL = - Vm(q, 4)L + (Y4 - RuL) + K(uL - z) (16) ul is an n x 1 vecto. The fictitious contol input ul is given by ul = k,l + k-ly$ (17) k, is a positive scala %onstant, 4 is an x 1 paamete estimate vecto, and K-' is an n x n estimate matix denoted by = diag {lf1} (18) Note that the notation diag {kp;'} is used_to denote a diagonal matix 2-l composed of elements k; I. IEE PROCEEDINGS-D, Vol. 140, No. 3, MAY I993

3 3.1 Remak The notation delineated by eqn. 18 simply means that invese of the ith diagonal element of the constant nomalised joint flexibility matix? denoted by R-' = diag {E;'} (19) is being estimated. As with any indiect adaptive contolle, we must specify how the paametes will!e updated. The paamete estimate update law [SI fo 4 is given by L1 is an T x T positive definite diagonal matix, and the paamete eo is given by $=4-$$ (21) The invese of the ith diagonal element of the 'nomalised' joint flexibility matix? will be updated by E; = )JLZi( Y&Li = -E; (22) ylzi is used to denote the ith diagonal component of a positive definite diagonal n x n matix L2, Li is use$ to denote the ith component of the n x 1 vecto L, (Y4)i is used tqdenote the ith component of the n x 1 vecto Y4, and k;' is given by l;' = E;,- (23) which in matix fom is given by = R-1 - E-1 = diag {;;I} (24) Now that we have defined the contol law and the appopiate estimation schemes, we can fom the filteed link tacking eo system. Specifically, afte substituting eqn. 17 into eqn. 16, we have fi(q)fl = - vm(q, 4)L - k, t, q, = U, - z + Y4 + Rk-lyd + Rq, (25) (26) 3.2 Remak If q1 in eqn. 25 was equal to zeo then we could easily show that the adaptive update laws given in eqns. 20 and 22 would indeed compensate fo the uncetainty in the dynamics given by eqn. 5. That is, we could show [SI that the tacking eo is GAS in spite of the paametic uncetainty. Of couse, we can not guaantee that q, will be equal to zeo. Howeve, if the actual contol uf, in eqn. 3, could somehow be fomulated to guaantee that q, would be 'small' then it might be possible to establish a stability esult fo the link tacking eo. In the following Section, we fomulate such a contolle. 4 Flexible joint compensation We can think of the tacking eo system given by eqn. 25 as being petubed by the quantity q, [SI; theefoe, it would be advantageous to establish the dynamic chaacteistics of q,. This can be established by finding the dynamic equation fo 4,. Specifically, fom eqn. 26, we have 4, = U, - i (27) As in Section 3, we can add and subtact an n x 1 fictitious contol input vecto u1 to the ight-hand side of eqn. 27 to yield tjl = w1 - U, + (U1 - i) (28) w1 = U, The n x 1 vecto u1 is a fictitious adaptive obust contolle [IO] designed to compensate fo the petubation q,. The fictitious contolle U, will be given explicitly, but fist we note that an examination of eqn. 28 eveals a second petubation tem (U, - i). This petubation tem is defined as q2 = u1 - z (30) If we poceed as peviously outlined, eqn. 30 is diffeentiated to obtain i2 = U, - i' (31) We can now substitute the dynamics of eqns. 1 and 3 into eqn. 31 and multiply both sides by J to yield 54, = w, - UF w2 = Ju, + B4,. + Kz + Jfi-'(q) (32) x [Kz - Vmk, G(q) - &)I (33) The n x 1 vecto uf is the actual adaptive obust contolle applied at the actuato toque input level. The contolle uf is designed to compensate fo the petubation v2 ' Now that we have fomulated the petubation systems (i.e. eqns. 28 and 32), we explicitly define the adaptive obust contolles to be U, = k,q, + v, and uf = keq2 + U, (34) k,, k, ae positive scala constants, and U,, u2 ae n x 1 auxiliay contolles used to compensate fo the paametic uncetainty and the actuato dynamics. These auxiliay contolles ae defined as [lo] cl, b2 ae estimates fo the scala bounding functions pl, p2 defined as p1 3 IIwlII and p2 3 llw21/ (36) with w, and w2 being defined in eqns. 29 and 33; and E,, e2 ae scalas adjusted accoding to E, = -kel E, and k2 = -ke2e2 (37) with kel, ke2 being positive scala constants. The adaptive obust contolles [IO] 'lean' the bounding functions (defined in eqn. 36) on-line as the manipulato moves. That is, in the contol implementation, we do not equie exact knowledge of the bounding functions; athe, we only equie the existence of the bounding functions. We will assume that the actual bounding functions p1 and p2 given in eqn. 36 can be paameteised as p, =SI$, and p2 = S2B2 (38) SI, S, ae 1 x p vectos of known functions (note the dimension p could be diffeent fo SI and S2), and e, e2 ae p x 1 constant vectos of unknown 'bounding' paametes. 157

4 Based on the paameteisation of the bounding functions given in eqn. 38, we define the bounding function estimates given in eqn. 35 as 8, = SIPl and p2 = szd2 (39) 8,, J2 ae p x 1 vectos used to estimate the unknown bounding paametes. The bounding estimates 8,, 8, ae updated on-line by the elations 8, = FISTJJqlJJ = -8, and b, = F2S:IJq21/ = -8, (40) Fl F2 ae positive definite diagonal p x p matices, and 8,, 8, ae defined by 8, = 9, - 8, and 8, = e2-8, (41) 4.1 Remak In the above discussion, we have assumed that the bounding functions given in eqn. 36 exist. It should be emphasised that the bounding functions depend only on measuements of qm, 4,, q and 4. Fo the sake of bevity, we will not find geneal expessions fo p1 and p,; howeve, we will outline a pocedue fo finding these functions. By using eqn. 5, it can easily be established that all of the dynamics in eqns. 29 and 33 can be bounded by combinations of constants and functions of the measuable quantities q, q, qm and 4,. Fo example, we need to bound w1 defined in eqn. 29. As w, = U,, it seems that the coesponding bounding function p1 defined in eqn. 36 would be a function of link acceleation. Howeve, because we only need an uppe bound on acceleation, we can use eqn. 5 to obtain this uppe acceleation bound. Specifically, by ewiting eqn. 5 into the fom q = R(q)-'[Rz - Vm(q, q)q - G(q) - (42) we can use the ight-hand side of eqn. 42 to bound q by constants and functions of the measuable states. That is, we can define an uppe bound on link acceleation as llqll G 4, qm) (43) f,(q, q. q,,,) is a positive scala function. As a esult of eqn. 43, the bounding function p1 defined in eqn. 36 can be shown to be only dependent on q, 4 and qm. By using eqn. 43, the bounding function p2 defined in eqn. 36 can also be found to be only dependent on q, q, qn and 4,. 5 Stability analysis In this Section of the pape, we show how the adaptive obust contolle given in Sections 3 and 4 esults in a GAS popety fo the link position tacking eo defined in eqn. 12. That is, using Lyapunov stability analysis, we illustate how the link position tacking eo tends to zeo as time tends to infinity. We now state a theoem to illustate this concept. 5.1 Theoem The link tacking defined in eqn. 12 is GAS, that is lim IleL(t)ll = 0 (44) 1-m if the following sufficient conditions on the contolle gains hold k, > f k2/k, kb > 1 and k, > 0.5 (45) k,, k,, k,, k, and k, ae defined in eqns. 17, 34, 34, 4 and 4, espectively. 158 Poof: Select the Lyapunov-like function V = +Efi(q), + fq:ql + fqfjq2 + LI + LZ (46) L,- -T 24 L f?lal,&~ and L 2 -L-T-Ig i=, (47) k,' E, + 48zF;'82 + kg1 ~2 (48) It can be easily established that the function V given by eqn. 46 is lowe bounded by zeo. Afte taking the deivative of eqn. 46 with espect to time, we obtain V = +E G(q)L + la(q)il + q;j, + qtjj2 + i, + i, (49) Substituting eqns. 25, 28, 30, 32 and 34 into eqn. 49 and using the popety in Section 2.2 yields 'V = -EKkaL + ;Kql - qtkbqfl + qtq, - qzk,q, + EY6 + [Rk-'Y$ + L, + qw, - qtv, + qt w2 - qtv2 + L, (50) In Appendix 8, we show that the second line of eqn. 50 is always less than o equal to zeo; theefoe, we can place an uppe bound on 'V in the following manne VG -k~m~~{~hlllz - kbll k,11~2il2 + Lax{R)IILII IItllII + IIq1II IIqzII (51) which can be witten in the matix fom V G -x:qx, (52) kail - +E2 Q=[ + -:& :4] X" = ClILll lla1ll ll~2111t : and (53) It is easy to show by the Geschgoin Theoem [I41 that if the contolle gains ae selected accoding to eqn. 45 then the matix Q defined in eqn. 52 will be positive definite. Theefoe, by Rayleigh's coefficient [I41 and the tiangle inequality [14], we can place a new uppe bound on Vas ' - ~miniq}llll12 (54) We now detail the type of stability fo the tacking eo. Fist note that eqn. 54 can also be witten as V(4 do G - Li.{Q) li~.(o)llz do (55) which can also be witten as V(0) - V(a) 2 Li.{QI Il~(4ll~ d~ (56) Because P is negative semi-definite as delineated by eqn. 52, we can state that V is a noninceasing function that is uppe bounded by V(0). Because V is noninceasing, uppe bounded by V(O), and lowe bounded by zeo, we can wite eqn. 56 as (57) IEE PROCEEDINGS-D. Vol. 140, No.3, MAY 1993

5 The bound delineated by eqn. 57 infoms us that L E L; 13 SPONG, M., and VIDYASAGAR, M.: Robot dynamics and [I, 121. To establish a stability esult fo the link position contol (John Wiley and Sons, New Yok, 1989) 14 BARNETT, S.: Matices in contol theoy (Robet E. Kiege tacking eo defined in eqn. 12, we establish the tansfe Publishing Company, Malaba Ra, 1984) function elationship between the position tacking eo 15 BENEALLEGUE, A., and MSIRDI, N.: Passive contol fo obot and the filteed tacking eo L. Fom eqn. 13, we can manipulatos with elastic joints, Poc. IMACS MCTS 91, Modeling state that and Contol of Technological Systems, Lille, Fance, Vol. 1, pp. Aft-OR1 (58) 16 CHEN, K., and FU, L.: Nonlinea adaptive motion contol fo a manidulato with flexible ioints. Poc. IEEE Int Conf. on Robotics s is the Laplace Tansfom vaiable [7] and and Automation, Scottsdaie, Az., Vol. 2, pp DE LUCA, A.: Dynamic contol of obots with joint elasticity. (7s) = ( ~ 1 + ainxj1 ~ ~ (59) Poc. IEEE Int. Conf. on Robotics and Automation, Apil 1988, pp Because W is a stictly pope, asymptotically stable 18 FICOLA, A., MARINO, R., and NICOSIA, s.: A singula petutansfe function and L E L;, we can state that [12] bation appoach to the contol of elastic obots. Poc. Alteton lim JJeLJJ = 0 (60) Conf. on Communication, Contol, and Computing Univ of IIL, m 19 KHORASANI, K., and SPONG, M.: Invaiant manifolds and thei audications to obot manbulatos with flexible ioints. Poc. IEEE In;. Conf. on Robotics and Automation, St. Louis, MO., SLOTINE, J., and HONG, S.: Two time scale sliding mode contol of manidulaton with flexible ioints. Poc. Ameican Contols Conf ~ state that the signals 4, 4, qm, Qm, d,, d,, 4 and K-I all 21 TOMEI, P., NICOSIA, S., and FICOLA, A.: An appoach to the we can show that the contol adaptive contol of elastic joint obots. Poc. IEEE Int. Conf. on Robotics and automation, Remak It is impotant to note that because the LYaPunov function is bounded above by Y(0) and below by zeo y e can emain bounded. H ~ ~ ~ ~, designed at the input uf will emain bounded. 6 Conclusion In this pape, we have designed a hybid obust tacking contolle fo RLFJ obot manipulatos. The contolle was shown to adapt fo paametic uncetainties while compensating fo the uncetainties associated with the actuato dynamics (i.e. mechanical flexibilities). In spite of these detimental uncetainties and effects, we wee still able to ensue that the obot link position follows a desied tajectoy in the GAS sense. We also give specific sufficient conditions on the contolle gains that guaantee that the link position tacking eo is GAS. The implementation of the contolle only equies measuement of the link position, link velocity, actuato position, and actuato velocity. 8 Appendix In this appendix, we give some calculations elated to the poof of Theoem 5.1. Specifically, we show that the tem VI + V2 (61) PI = IY$ + :Kk- Y$ + i, (62) and ~2=qlw1-q~v1+q~w2-q~v*+i2 (63) is less than o equal to zeo. Fist, fom eqns. 62 and 47, we have VI = FYqj + qjtl;j + ;Kk-IYcj 7 Refeences 1 ORTEGA, R., and SPONG, M.: Adaptive motion contol of igid obots: a tutoial. Poc. IEEE Conf on Decision and Contol, Austin, TX., READMAN, M., and BELANGER, P.: Analysis and contol of a flexible joint obot. Poc. IEEE Conf on Decision and Contol, Honolulu HI., 1990, pp MRAD, F., and AHMAD, S.: Adaptive contol of flexible joint obots with stability in the sense of Lyapunov. Poc. IEEE Conf on Decision and Contol, Honolulu HI., 1990, pp GHORBEL, F., and SPONG, M.: Stability analysis of adaptively contolled flexible joint obots. Poc. IEEE Conf. on Decision and Contol, Honolulu HI., 1990, pp SPONG, M.: Modeling and contol of elastic joint obots, J. Dynamic Systems, Measuement, and Contol, 1987,109, pp CRAIG, J.J.: Adaptive contol of mechanical manipulatos (Ann Abo: UMI dissetation Infomation Sevice, 1986) 7 VIDYASAGAR, M.: Nonlinea systems analysis (Pentice Hall, New Jesey, 1978) 8 SLOTINE, J.J., and LI, W.: Theoetical issues in adaptive contol. Fifth Yale Wokshop on Applications of Adaptive Systems Theoy, Kale Univesity, KOKOTOVIC, P.V., KHALIL, H., and OREILLY, J.: Singula petubation methods in contol: analysis and design (Academic, New Yok, 1986) 10 CORLESS, M., and LEITMANN, G.: Adaptive contol of systems containing uncetain functions and unknown functions with uncetain bounds, J. Optim. Theoy Applic., Januay 1983, pp CORLESS, M.: Tacking contolles fo uncetain systems: application to a manutec R3 obot,.i. Dynamic Systems, Measuement, and Contol, Decembe 1989,111, pp VIDYASAGAR, M.: Input-output analysis of lage-scale inteconnected systems. Lectue Notes in Contol and Infomation Sciences, No. 29, (Spinge-Velag, New Yok, 1981) Substituting eqns. 19, 20, 22 and 24 into eqn. 64 shows that VI = 0. Secondly, fom eqns. 63 and 48, we have 2 V2 = q; wj - q;uj + e,t,lgj + k, Ej j=1 (65) Substituting fo ij and jj fom eqns. 37 and 40, espectively, into eqn. 65 yields Using eqns. 36 and 38, we can obtain a uppe bound fo V2 given in eqn. 66 as Substituting eqns. 35, 38 and 41 into eqn. 67 yields Obtaining a common denominato fo the fist two tems in eqn. 68 yields It is now obvious fom eqn. 69 that V2 < 0, and hence VI + V2 <