ON MODEL-BASED CONTROL OF HYDRAULIC ACTUATORS

Transcription

1 Prceedings f RAAD 03, 12th Internatinal Wrksh n Rbtics in Ale-Adria-Danube Regin Cassin May 7-10, 2003 ON MODEL-BASED CONTROL OF HYDRAULIC ACTUATORS Panagitis Chatzaks and Evangels Paaduls Deartment f Mechanical Engineering Natinal Technical University f Athens Hern Plytechniu 9, Athens, Greece chatzak@central.ntua.gr, egaad@central.ntua.gr ABSTRACT- Electrhydraulic servsystems exhibit highly nnlinear behavir t the effect that classical linear cntrllers, e.g., PD, usually achieve a limited erfrmance. Lad static and dynamic arameters variatins are als cntributing t the limitatin f their sitin and frce tracking erfrmance. This aer resents a mdelbased cntrller alied t a fully detailed mdel f an electrhydraulic servsystem aiming at imrving its sitin and frce tracking erfrmance. Fluid, servvalve, cylinder and lad dynamics are taken int accunt. Simulatin results shw the strategy t be rmising in cntrlling hydraulic servactuatrs. The arach can be further extended t the cntrl f hydraulically driven maniulatrs and simulatrs. Keywrds: Hydraulics, Electrhydraulic Servsystem, Servvalve, Hydraulic Dynamics, Mdel-Based Cntrl. INTRODUCTION Many mbile, airbrne and statinary alicatins emly hydraulic cntrl cmnents and servsystems. Hydraulic servsystems can generate very high frces, exhibit raid resnses, and have a high wer-t-weight rati cmared t ther technlgies. On the ther hand, they exhibit a significant nnlinear behavir due t the nnlinear flw/ressure characteristics, il cmressibility, time varying eigenfrequency behavir, nnlinear transmissin effects, flw frces acting n sl and frictin, which is nt nly largely uncertain but is greatly influenced by external lad disturbances. In recent years, many cntrl schemes that d nt deend n erating int linearizatins have been rsed t imrve hydraulic servsystem erfrmance [1]. This arach sacrifices cntrller erfrmance in favr f its rbustness, while anther study uses a classical analysis t int ut that cmmn PID cntrllers are inadequate fr frce tracking due t hydraulic system fundamental limitatins, [2]. Pressure feedback has been used t imrve the erfrmance f classical PD cntrllers, [3]. Varius ther cntrllers frm neural t adative have maintained gd erfrmance f electrhydraulic servsystems, [4-6]. Hwever, the resnse f such a system heavily deends n the lad and its variatins, thus the erfrmance and rbustness f neural cntrllers is an issue. The nnlinear adative frce cntrl scheme f an active susensin achieves better erfrmance than cnventinal linear cntrllers, [5], where nly cylinder uncertainties were cnsidered. The same alies t [7] where the backsteing arach f [5] was extended frm frce t mtin cntrl using a hydraulic actuatr with a three-way valve. Lad uncertainty arameters were taken int accunt t result in a recise mtin f a single-rd hydraulic actuatr, [6]. A hybrid sitin/ frce cntrl scheme was rsed, using a time-delayed dynamic inversin and a Lyaunv analysis, resectively, [8. 9]. In these aers, either the servvalve leakages have been neglected r unreasnable acceleratin estimatins have been used in the feedback cntrl law by twice differentiating the actual ([8]) r desired ([9]) istn sitin. Als, the cylinder utut frce is calculated frm the ressure dr acrss the cylinder and therefre, nt at the lad. Anther time-delayed scheme is intrduced fr trque tracking in cnstrained and free mtin fr a hydraulic rbt with rrtinal valves, [10]. The dynamics f the hydraulic servsystem were relaced by a simle time-delay, while a frce sensr measured the actual trque. In cnstrained mtin, where the dynamics f the system are simlified due t the small istn rd mvement, frce tracking is quite gd. Hwever, the reductin f system dynamics t a simle timedelay results in r frce tracking in free mtin. A mdel-based cntrller f a hydraulically driven maniulatr was studied but the feedfrward cntrller terms are calculated using desired and nt actual sitins, leading t r results, [11]. In this aer, a fully detailed mdel f an electrhydraulic servsystem, which includes fluid, servvalve, servactuatr and lad dynamics, is resented and used fr evaluating the rsed mdel-based cntrller fr frce tracking cntrl, bth in free and cnstrained mtin. It als cmares its sitin tracking erfrmance t that f a classical linear cntrller, using intensive simulatins. Lad dynamic and static arameters are varied widely s as t test the rsed cntrller in varius lad cnditins. Simulatin results shw the technique t be rmising in rviding rbust sitin and frce cntrl and in extending the arach t hydraulically driven

2 maniulatrs and mtin latfrms. The aer is rganized as fllws. Fllwing system hysical mdeling, a mdel-based cntrller fr bth cnstrained and free mtin is develed, and simulatin results are rvided. Cmarisns against classical cntrl are resented. PHYSICAL MODELING The develment f an accurate dynamic mdel fr a hydraulic servsystem is imrtant fr understanding the system and fr develing a rbust cntrller. T this end, a descritin f the dynamics fr the fluid subsystem, the servvalve, the cylinder and the lad is required. Fig. 1 shws a hydraulic servactuatr, including a servvalve, a cylinder, a frce sensr and an inertial lad. The dynamic mdel must als take int accunt the wer suly (a variable dislacement cnstant ressure um) and transmissin line dynamics. Small lses due t filters r ther hydraulic cmnents are lumed int line lsses. Fig. 1. Schematic mdel f hydraulic servactuatr. The mdel f the hydraulic subsystem was develed using Linear Grahs ([12]), which allw a systematic generatin f system state-sace equatins, using three sets f equatins, namely the elemental equatins and the cmatibility and cntinuity equatins. In hydraulic systems, element equatins describe the relatinshi between ressure and flw fr the elementary hydraulic elements such as the inertial, caacitr and resistive element. Cmatibility equatins result in ressure dr equatins alng a clsed circuit, while cntinuity equatins result in flw cntinuity at systems ndes r clsed surfaces. The mdel, in its detailed frm is useful in understanding the hysical henmena in the system and can be used t evaluate cntrller erfrmance in simulatin. Mdeling f several key cmnents is discussed next. Hydraulic Unit and Transmissin Lines The custm-made cnstant ressure wer unit, equied with a PARKER PVP41 Series variable dislacement istn um, is mdeled as a surce f cnstant ressure, while transmissin lines are mdeled as an inertance, a resistance and a caacitance cnnected in a T-cnfiguratin (lumarameter line mdel). This is a suitable descritin fr simulatin urses rvided that the frequency f scillatins in the system is significantly less than that crresnding t wave ragatin, [13]. Servvalve The MOOG G761 Series servvalve is a high erfrmance tw-stage design valve. The utut stage is a clsed center, fur-way, sliding sl, while the ilt stage is a symmetrical duble nzzle and flaer, driven by a trque mtr. Since its natural frequency is rders f magnitude higher than the desired clsed l bandwidth, nly its rifices resistive effects was taken int accunt, and is described by ([14]) PR = CR QR QR (1) where P R is the ressure dr acrss the rifice, Q R is the flw thrugh the rifice and the cefficient f C R is a functin f fluid density r, the rifice area A and the discharge cefficient C d, C = r 2 C A R ( ) 2 2 d (2) The valve sl sitin mdulates the rifice area, which in turn affects the magnitude f C R, and is cntrlled by an inut vltage cmmand. The square rt f the inverse f the rifice resistance, called hereafter servvalve rifice hydraulic cnductance G R is defined as GR = C -12 / R. The fur symmetric and matched servvalve rifices make u a fur-legged flw ath f fur nnlinear resistrs mdulated by the inut vltage and thereby the servvalve is mdeled as the hydraulic equivalent f a Wheatstne bridge. Tw schematic mdel versins f the servvalve are given in Fig. 2 where the rifice leakage is taken int accunt (Fig. 2.b) and neglected (Fig. 2.a). Fig. 2. Schematic mdel f servvalve, when rifice leakage is (a) neglected and (b) taken int accunt. In the latter case, nly tw nnlinear resistrs are required t build the servvalve mdel. Hwever, in this case and fr simulatin urses, tw dynamic mdels shuld be used fr the servvalve, in rder t allw fr bth frward and backward mtins f istn rd. In ther wrds, t have the sign f rd velcity changed, the servvalve must be able t reverse the il flw directin, as required by the headside and rearside istn gyratrs equatins, resectively, see Fig. 3.

3 ÈFA A P Í v Î = È Î Í 0 1 A1 È Î ÍQA 1 (3) ÈFA A P Í Î v = È - Î Í 0-1 A2 È Î ÍQA2 (4) where A 1 and A 2 are the istn areas at headside and rearside, resectively (see Fig. 1). This reversal f flw can be btained either by using tw ressure surces, ne set always at um ressure, while the ther set always at tank ressure, r by assuming that the larity f the ressure surce changes accrding t the sign f the inut vltage sent t the servvalve. Cntrarily, mdeling the servvalve as a hydraulic bridge the tlgy f the real servvalve is resected and the servvalve mdel is the same fr any inut cmmand. The directin f il flw, i.e., the directin f rd mtin, is btained by fllwing the arriate flw ath, which is cntrlled by the inut cmmand sign. Hwever, this mdeling chice intrduces numerical stiffness rblems, because tw f the fur resistances f the bridge are always very large, [15]. On the ther hand, it als intrduces the hydraulic daming f the real system, caused by the residual flws leaking thrugh the rifices when the servvalve is clsed. This mdeling technique fr the servvalve is fllwed in this aer. Due t symmetry and match f the rifices the magnitude f C R f any rifice can be btained frm ne rifice fr any inut cmmand. The ther rifice resistances are calculated as CR 1( isv) = CR3( isv) = CR3( - isv) = CR, PÆA (5) CR2( isv) = CR4( isv) = CR 1( - isv) = CR, PÆB where C RP, Æ A and C RP, Æ B are the rifice resistances f each flw ath, see Fig. 2. As shwn in revius wrks, exerimental data fr the servvalve rifice cnductances can be btained and a lynmial reresentatin f G R can be fund using curve fitting algrithms, [16], [17]. alicatin f cntinuity and cmatibility laws, alng with individual elements equatins, leads t a set f nine nnlinear first rder differential equatins as fllws, ṖL 1 = (( Ps - PCL 1) RL 1 - QL 1) CL 1 (6) Q = P - P - P I ( ) (7) L1 CL1 R1 1 L1 ( ) (8) ( R R L in L ) (9) Ṗ = Q - Av -Q -Q - R P C 1 L1 1 R2 R4 in L 1 Ṗ = Q Q A v - Q R P C Q = P - P - P I ( ) (10) ( ) (11) L2 2 R3 CL2 L2 ṖCL2 = QL 1 - PCL2 RL2 CL2 v = ( AP 1 1- A2P2- Bv - Bs( v -vl) (12) -FKs - F( v )- Fg ) M ḞKs = Ks( v -vl ) (13) vl = ( FKs Bs( v -vl)- FgL ) ML (14) where P = L P - 1 P2 is the lad ressure dr and the remaining variables are defined in Table I. Table I. Nmenclature. Variable Definitin R L, C L, I L Resistance, caacitance and inertance f lines Q L, Q sv, Q A Flw in lines, t/frm cylinder, due t mtin P s, P P, P T Suly and servvalve inlet/utlet ressures C 1, C 2 Fluid caacitance in cylinder chambers P 1, P 2 Pressure in cylinder chambers M, M L Pistn and lad inertia F(v ), B, Culmb frictin & viscus frictin ceff. v, v L Pistn and lad center f mass velcity F g, F gl Gravity frce n istn and lad B s, K s Daming and stiffness f frce sensr θ Angle with resect t the hrizntal F, F s Frce acting n cylinder and lad (measured) V sv, i sv Inut vltage and current t servvalve K sv, K sv, Servvalve gain and its value fr i sv = 0 x d, v d, F d Desired sitin, velcity and frce e, e F Psitin and frce errr K P, K V, K F Psitin, velcity and frce errr gain Servcylinder The servvalve drives a custm-made dubleacting single-ended MOOG servcylinder with an internal analg R-Series MTS linear dislacement transducer and secial lw frictin seals and glands. Because f the latter, the stick-sli effect is minimal and the istn frictin is nly due t Culmb and viscus frictin. The istn areas are nt equal, thus tw gyratrs are used t describe the cnversin f ressure t frce, see Fig. 3. The istn gyratr equatins are given by Eqs. (3) and (4). System Full Mdel The grah f the full mdel f the hydraulic servsystem is shwn in Fig. 3. This als includes an inertial lad and a frce sensr between the lad and the rd, which is mdeled as a first rder system with high stiffness and daming. The Fig. 3. Electrhydraulic servsystem full mdel. The servvalve ressure dr and flw variables P R1, Q R2, P R3 and Q R4 are assciated with the state

4 variables thrugh ressure dr and flw equatins as result f alicatin f the cmatibility and cntinuity laws, resectively and f algebraic maniulatins. It has been assumed that the transfer f wer is exclusively frm the hydraulic system t the lad. Other assumtins related t hysical limitatins are resented in [18]. CONTROL Having the detailed servsystem mdel, several sitin and frce cntrl laws were set u and evaluated using MATLAB/ Simulink. Cntrl System Setu The custm-designed benchmark setu shwn in Fig. 4 was built at the NTUA t test the rsed cntrller. The articular design f the setu allws fr easy changes in the static and dynamic cmnents f the inertial lad, driven by the hydraulic actuatr. This is achieved by varying the angle f the cylinder with resect t the hrizntal and by changing the cylinder-see inertia, by adding r remving weight. Fig. 4. Schematic f cntrl system setu and benchmark. A MOOG G A1 Series cntrller is used t read the servcylinder headside and rearside ressure (frm tw ressure sensrs n the valve manifld), and the istn rd sitin and velcity (frm the built-in analg LDT). A frce cell at the end f the rd will rvide the lad frce. The cntrller card will be interfaced t a PC running the QNX real-time erating system. T use nnlinear and mdel-based cntrllers, the PID cntrl sectin f the card will be by-assed and the card will be used nly fr reading sensrs measurements and fr sending the arriate cntrl vltages t the servamlifier. The servamlifier in turn will send arriate inut currents t the servvalve. Hydraulic Servactuatr Descritin fr Cntrl In rder t rvide an equatin sufficient fr cntrl urses Eqs. (8) and (9) are cmbined by taking the difference f ressure derivatives P 1 and P 2 between the cylinder chambers multilied by the headside and rearside sides, resectively, Ḟ = ( QR 1 -QR4) A1 C1 ( QR3 -QR2) A2 C2 (15) -Av - R ( A C A C) P in L where by definitin, the first art f the resultant equatin is the derivative f cylinder utut frce 2 2 and A = A1 C1 A2 C2, while the flws t and frm the servvalve, frm the alicatin f cntinuity law, have been written as QL 1 = QR 1 QR2 (16) QL2 = QR3 QR4 Fr deriving the cntrl law, the servvalve rifice cnductances are estimated as a linear functin f the inut current. Fr sitive inut cmmands these are given by -12 / GRP, ÆA( isv) = Km isv CR, (17) -12 / GRP, ÆB( isv) =-Kl isv CR, where C R, has a very large value, which refers t the resistance f the rifice at servvalve clsure, and K m and K l are sitive cnstants, which crresnd t the main and leakage flw ath, resectively. Fr negative inut cmmands K m and K l in Eq. (17) are reversed. Using Eq. (15), flw/ressure thrugh an rifice equatins, Eq. (1), and Eqs. (17), it can be written F Av - Ksv, = Ksv isv (18) where the cylinder internal hydraulic lsses have been neglected due t servcylinder design and the use f secial seals and glands that minimize these lsses. Thus, the servvalve gain K sv and K sv, are defined by the equatin Ksvi Ksv, = ( QR 1 -QR4) A1 C1 ( QR3 -QR2) A2 C2 (19) The terms K sv and K sv, deend n rifice cnductance, the sitin f the istn rd, which mdifies the caacitance f cylinder chambers, the servvalve inlet and utlet ressure and the cylinder chambers ressures. Prvided that the um and tank ressures can arximate the inlet and utlet ressure f the servvalve, resectively, K sv and therefre K sv, are simlified. Hwever, three variables have t be measured t calculate estimates f K sv and K sv,. Using cnductances, Eq. (19) results in K i K G P -P -G P A C ( ) sv sv sv, = R, PÆA s 1 R, PÆB ( GRP, ÆA P2 -GRP, ÆB Ps -P2) A2 C2 (20) Relacing the G R accrding t Eq.(17), (20) yields Ksv( P1, P2, P3) = K m( A1/ C1 PS - P1 A2/ C2 P2) Kl( A1/ C1 P1 A2/ C2 PS -P2) (21) -12 K SV, ( PPP,, S ) CR, / 1 2 = [ A1/ C1( PS -P1 - P1) A2/ C2( P2 - PS -P2)] Cylinder frce F in Eq. (12) is exressed as the sum f the required frce t accelerate the istn F M, the viscus and Culmb frictin F fr, the gravity frce acting n istn and the lad alied frce F s measured by a frce sensr F = FM Ffr Fg Fs (22)

5 The measured frce F s in Eq. (14) is the sum f the required acceleratin frce F ML and the gravitatinal frce F gl Fs = FML FgL (23) Since the natural frequency f the frce sensr is rders f magnitude higher than any ther exhibited in the system it is assumed that vl. Substituting Eqs. (22) and (23) in Eq. results in ( M ML) v ( M ML) gcsq (24) Bv Av - Ksv, = Ksv isv which can be used t describe the electrhydraulic actuatr when the resulting mtin is f cncern. In mst cases, the lad mass is rders greater than the istn mass, i.e., ML ML. This assumtin, tgether with vl, and Eqs. (23), (24), lead t a useful descritin f the hydraulic servactuatr dynamics fr frce cntrl, i.e. F s Bv Av - Ksv = K i, sv sv (25) where the whle inertia is assigned at the lad. Frce Cntrl The frce tracking erfrmance f the rsed mdel-based frce cntrller is evaluated n the full hydraulic servsystem, described by Eqs. (6)- (14), using MATLAB/Simulink. A cnstrained task and a free mtin task are investigated next. In the first case, where the mtin f the istn rd is cnsidered cnstrained by a hysical bstacle, e.g., a wall, it is assumed that bth the velcity and the acceleratin f the istn rd are zer. This assumtin simlifies Eq. (25) and reveals that the inut cmmand mdifies nly the magnitude f the derivative f the measured frce F s - Ksv, = Ksv isv (26) By setting the inut current cmmand fr mdelbased frce cntrl f the cnstrained cylinder as MBFC isv = ( F d KF( Fd - Fs)- Ksv, ) Ksv (27) where the desired frce F d is given by Fd = FdL FgL (28) where F dl is the net desired frce alied by the cylinder n the lad and F gl is an estimate f the gravitatinal frce n the lad. Defining the frce errr as ef = Fs - Fd and rvided that the estimates f K sv, K sv, and F gl are accurate, Eq. (27) becmes ėf KF ef = 0 (29) which guarantees exnential frce cnvergence. With K F = 100, and taking int accunt that the time cnstant in Eq. (29) is t = 1 K F, cnsistency between exected and simulated resnses results (see Fig. 5.a). The versht in Fig. 5.a is due t the earlier hythesis f luming istn mass t the lad. Small transients near the steady state are due t small istn mtins. Nte that the istn is nt still, since the servvalve scillates arund its midint in rder t let the ressures in cylinder chambers t change, s as t rvide the desired lad ressure dr, see Fig. 5.b. In the case f free mtin, velcity and acceleratin in Eq. (25) cannt be neglected, and have t be cmensated. Setting the inut current fr mdelbased servactuatr frce cntrl in free mtin as MBF isv = ( B v Av F d (30) KF( Fd - Fs)- Ksv, ) Ksv where the desired frce is defined as in Eq. (28) and rvided that the estimatin f viscus frictin cefficient B and f the term A are accurate, Eq. (25) becmes a first rder system, as in Eq. (29). Fig. 5. a) Simulated and desired frce and errr in time segment s, with mdel-based cntrl fr cnstrained rd and resnse f an equivalent 1 st rder system, b) Simulated cylinder ressure and inut vltage cmmand, c) Simulated & desired frce and errr with mdel-based cntrl in free mtin. The need f measuring the acceleratin f the istn rd can be tackled by calculating the acceleratin frm Newtn s secnd Law and frce sensr measurements as vl = ( Fs - FgL) ML (31) In Fig. 5.c tracking f a desired randm frce is quite satisfactry and the simulatin results shw that the rsed mdel-based cntrller is rmising in rviding frce cntrl f hydraulic servsystems, given that the hysical key systems f the hydraulic servsystem are accurately mdeled thrugh intensive exerimental rcedure. Frce errr is nt zer rbably due t the arximatins and assumtins made regarding the system dynamics and due t uncertainties in estimating varius arameters that have t be cmensated and/ r calculated. Mtin Cntrl Next, the frce cntrller is used t aly the frce necessary t accelerate the lad alng a desired trajectry. Briefly, the desired frce, defined in Eq. (28), is nw set as Fd = M Lxd ML( KV e KP e) FgL (32)

6 Maniulating the dynamic equatin and the inut current cmmand fr frce cntrl, Eqs. (25) and (30), substituting the desired frce as given in Eq. (32), and assuming that the measured frce tracks the desired frce fast and recisely, which can be cnsidered valid as evidenced by the results in Fig. 5.c, then the resulting errr system dynamics are arximated by a hmgeneus 2 nd rder system e K e V KP e = 0 (33) which guarantees tracking errr cnvergence t zer, as lng as the gains f sitin and velcity errr, K P and K V, resectively, are sitive. The articular errr resnse is cntrlled by the selectin f K P and K V. The rsed mdel-based cntrller was cmared t a fixed gain classical PD cntrller. Gain selectin was made s as t ensure a 2 nd rder system resnse with critical daming and natural frequency f w n = 10 rad / s. The erfrmance f the rsed mdel-based cntrller is satisfactry regardless f the static and dynamic lad variatins, see Fig. 6.a. The resnse f the hydraulic system is similar t the ne exected theretically, i.e. it results in a resnse f an equivalent 2 nd rder system fr the same sitin errr inut. Small scillatins and discreancies frm the exected sitin errr resnse is due t differences between the real and estimated values in the feedfrward terms, see Eq. (30). Mrever, as shwn in Fig. 6.a, a cmarisn between a classical PD and the rsed mdelbased cntrller rves the suremacy f the latter in abslute sitin errr, esecially when the inertia f lad changes. Nevertheless, nt nly the tracking accuracy f the desired trajectry is higher with the rsed mdel-based cntrller ver a classical cntrller, but als the hase shift is smaller as the frequency f the desired mtin is getting higher. In Fig. 6.b, the rsed cntrller is cmared against a classical PD cntrller fr frequencies frm 1 Hz t 10 Hz. The sueririty f the mdel-based cntrller is nce again bvius. Fig. 6. (a) Simulated and desired istn sitin and errr with mdel-based cntrl and PD cntrl and sitin errr, (b) Linear PD and rsed mdel-based cntrller tracking erfrmance fr varius frequencies. CONCLUSIONS A mdel-based cntrller was develed fr a high erfrmance electrhydraulic servsystem t imrve its sitin and frce tracking erfrmance. The cntrller was tested n a fully detailed mdel f the system using MATLAB/ Simulink. Psitin and frce tracking was excellent desite variatins f lad static and/r dynamic cmnents, making the strategy t be rmising in rviding rbust cntrl t hydraulic maniulatrs and mtin simulatrs. REFERENCES 1. Merritt, H. E., Hydraulic Cntrl Systems, J. Wiley, Alleyne, A. et al., On the Limitatins f Frce Tracking Cntrl fr Hydraulic Active Susensins, Prc. American Cntrl Cnference, Philadelhia, Pennsylvania, June Li, D. and Salcudean, S. E, Mdeling, Simulatin and Cntrl f a Hydraulic Stewart Platfrm, Prc. IEEE Int. Cnf. On Rbtics and Autmatin, Aril Daachi, B. et al., A Stable Neural Adative Frce Cntrller fr Hydraulic Actuatr, Prc. IEEE Int. Cnf. On Rbtics and Autmatin, Seul, Krea, May Alleyne, A. and Hedrick, J. K., Nnlinear Adative Cntrl f Active Susensin, IEEE Trans. Cntrl Systems Technlgy, Vl. 3, N. 1, Bu, F. and Ya, B., Integrated Direct/Indirect Adative Rbust Mtin Cntrl f Single-Rd Hydraulic Actuatrs with Time-Varying Unknwn Inertia, Prc. IEEE/ASME Int. Cnf. Adv. Intelligent Mechatrnics, Cm, Italy, Sirusur, M. and Salcudean, S. E., On the Nnlinear Cntrl f Hydraulic Servsystems, Prc. IEEE Int. Cnf. On Rbtics & Autmatin, San Francisc, CA, Aril Six, Klaus, et al., A Time Delayed Dynamic Inversin Scheme fr Mechatrnic Cntrl f hydraulic Systems, Prc. IEEE/ASME Int. Cnf. On Advanced Intelligent Mechatrnics, Cm, Italy, Shl, G. A. and Brrw, J. E., Exeriments and Simulatins n the Nnlinear Cntrl f a Hydraulic Servsystem, IEEE Trans. On Cntrl Systems Technlgy, Vl. 7, March Zhu, W. H., et al., Emulatin f a Sace Rbt Using a Hydraulic Maniulatr n Grund, Prc. IEEE Int. Cnf. On Rbtics and Autmatin, Washingtn, DC, May Hnegger, M. and Crke, P. Mdel-Based Cntrl f Hydraulically Actuated Maniulatrs, Prc. IEEE Int. Cnf. On Rbtics and Autmatin, Seul, Krea, May Rwell, Derek and Wrmley, David N., Systems Dynamics: An Intrductin, Prentice Hall, Wattn, Jhn, Fluid Pwer Systems, Prentice Hall, Blackburn, J. F., et al., Fluid Pwer Cntrl, Technlgy Press f MIT & Jhn Wiley, Paaduls, E., and Gnthier, Y., On the Develment f a Real-Time Simulatr Engine fr a Hydraulic Frestry Machine, Int. J. f Fluid Pwer, Vl. 3, N. 1, Aril Paaduls, E., et al., Mdeling and Identificatin f an Electrhydraulic Articulated Frestry Machine, IEEE Int. Cnf. n Rbtics & Autm., Albuquerque, NM, Aril MOOG Inc., Techn. Bull. 117, Cntrl Div., E. Aurra, NY. 18. Chatzaks Panagitis, Design, Mdeling and Cntrl f a High-Perfrmance Electrhydraulic Servsystem, Master Thesis, in Greek, NTUA, 2002.