A New Predictive Approach for Bilateral Teleoperation With Applications to Drive-by-Wire Systems

Transcription

1 1 A New Predictive Approach for Bilateral Teleoperation With Application to Drive-by-Wire Syte Ya-Jun Pan, Carlo Canuda-de-Wit and Olivier Senae Departent of Mechanical Engineering, Dalhouie Univerity Halifax, Nova Scotia, Canada B3J 2X4 Laboratoire d Autoatique de Grenoble, UMR CNRS 5528 ENSIEG-INPG, B.P. 46, 38 42, ST. Martin d Hère, FRANCE Abtract In thi paper, a new predictive approach i propoed for the ipedance control of bilateral teleoperation yte. The propoed control tructure include two irror predictor/oberver in both: the ater and lave ide. Thee predictor/oberver are ued to iultaneouly etiate the ater and the lave internal dynaic, and thereby to avoid the ue of the delayed tranitted inforation. A a conequence, the influence of the delay on the whole yte can be iniized and the perforance can be iproved. Under a et of uited hypothei, the propoed control tructure i hown to be uniforly ultiate table even in the preence of tie-varying delay. Siulation reult are preented to how the effectivene of the propoed approach. The behavior of the control tructure i alo experientally deontrated while perforing reote teering of a all autonoou vehicle. Index Ter Bilateral teleoperation yte, Predictive approach, Delayed yte, Network controlled yte, Tie varying delay. I. INTRODUCTION During the lat everal decade, any different teleoperation yte have been developed to allow huan operator to execute tak in reote or hazardou environent. They are widely applied in different circutance uch a in the outer pace, underea, nuclear plant, urgical operation, vehicle teering, etc. However, a tie delay incur in a long ditance traniion of inforation. In the cae of bilateral teleoperation 1, ignal (i.e. environent contact torque or force) of the lave yte are reflected back to the ater ide operator providing hi with a feel of what i reotely ened. Siilarly, operation coand are ent to the lave yte in order to execute the deired operation. According to the ued ignal traniion echani, the delay incurring during uch an operation, can be of different nature. For intance, traniion delay ay be contant, tie varying (low, or fat), all or large. Revied for IEEE Tranaction on Robotic Correponding author: Prof. Carlo Canuda-de-Wit, Laboratoire d Autoatique de Grenoble, UMR CNRS 5528, ENSIEG-INPG, B.P. 46, 38 42, ST. Martin d Hère, FRANCE. Tel: Eail: carlo.canuda-de-wit@inpg.fr. 1 Signal fro the ater and the lave yte are reciprocally tranitted in both direction. Bilateral teleoperation yte are by contruction feedback yte with delay traniion in the loop. If no particular attention i paid, the delay at bet degrade the cloed-loop perforance, at wort detabilize the bilaterally controlled teleoperator 1. A a conequence tability ha been the ain concern ince the beginning of tudie on teleoperation 2, 8. Nonethele, developent of controlled teleoperation yte with both tability and perforance aeent i till a oewhat open proble. The ai of thi paper i to contribute to the undertanding on the control deign allowing to iprove the teleoperation perforance while preerving cloed-loop tability. A. Previou work In the literature, there are any control chee propoed for dealing with the tie delay in the teleoperation yte: paivity baed analyi 14, 32; pole placeent control 2 3; oberver baed deign 4; H control 8, 9; liding ode control 1 11; the ater tate prediction approach 12, etc. A recent and quite coplete overview on the robotic teleanipulation yte can be found in 13. 1) Paivity baed control: The paivity control approach ha the ability to enforce the paivity of the whole yte enuring cloed-loop tability. However, yte perforance degradation occur when the delay i large, becaue enforcing cloed-loop paivity reult in controller with low gain 5, 33. In yte where the traniion delay i tie-varying and only an (oewhat conervative) upper bound i known, the paivity approach ay reult in conervative deign (if the axial delay value i taken into account) or paivity ay be lot for large delay. Other proble ay alo coe fro the digital ipleentation of the control law 14 that ay break the paivity of cloed loop tructure, unle oe particular care during the ipleentation (to cobat with packet lo) i taken 34, 7. 2) Robut control: In 8, 9, a tandard H control proble i forulated where the ater and lave ide are tabilized locally under the auption that there i no contact torque. In the cae that a contact torque i applied, a third controller i deigned to tabilize the cloed loop yte. The tie

2 2 delay i treated a a perturbation to the yte, which often reult in a conervative approach. The liding ode control provide another poibility for the tie delayed yte while any concern hould be on the practical ipleentation uch a the high witching gain and the chattering phenoena. The liitation of the SMC chee in 11 i that the delay till exit in the ipedance control target which ean that the tability of the teleoperation yte ay not be enured even though the target can be realized by the robut SMC chee. 3) Finite pectru aignent: Pole placeent approach for contant tie delay ha alo been ued in the context of tele-operation. The cloed-loop yte pole can be aigned arbitrarily according to finite pectru aignent (FSA) approach 3. In the FSA, the cloed-loop pole are placed uing an infinite dienional controller, reulting in a finite dienional cloed-loop yte. Soe extenion to the cae of tie-varying delay ha been recently reported in 21, 22. One iportant drawback of the FSA i that all variation in the etiated tie delay, ay reult in large deviation of the cloed-loop olution. 4) Prediction baed control: The prediction ethod propoed in 12 can only predict the tate in the ater ide while it cannot predict the tranitted data in bilateral channel. The yte perforance degrade epecially when contact torque i applied in the lave ide a a reult. B. Fact and contribution Mot of the approache entioned previouly work well a long a the tie delay i contant and not too big when copared to the cloed-loop deired bandwidth. However, in any real traniion edia and in oe application of practical interet, the traniion delay are not only tievarying, but they can alo vary in a large enough range o a to create tability and/or perforance degradation. Hence it i iportant to deal with the control proble in the teleoperation yte in the exitence of tie varying delay. Since perforance ha been the bottleneck of the teleoperation proble o far, it ee then natural to earch for control ethod having the ability to predict the behavior of the ater and lave dynaic, o a to copenate for the effect of the traniion delay of iportant agnitude. In other word, look for le conervative control deign, yielding the bet poible predictable perforance while preerving the cloedloop tability. Control tructure baed on prediction repreent an intereting alternative to tackle thi proble. Neverthele, reader hould note that control prediction ethod are odelbaed control tructure requiring precie dynaic odel for both the ater and the lave yte; a price due by the well known fact that perforance and tability can be iultaneouly reached only if oe yte knowledge i available. In thi paper, a new predictive approach i propoed for the ipedance control of bilateral drive-by-wire teleoperation yte. The propoed control tructure include two irror predictor/oberver in both: the ater and lave ide. Thee predictor/oberver are ued to iultaneouly etiate the Huan Operator Fig. 1. G( ) v F r h Mater Counication Slave General Block diagra of the teleoperation yte. v r F e Environent ater and the lave internal dynaic, and thereby to avoid the ue of the delayed tranitted inforation. A a conequence, the influence of the delay on the whole yte can be iniized and perforance can be iproved. Under a et of uited hypothei, the propoed control tructure i hown to be uniforly ultiate table even in preence of tie-varying delay. Note that the propoed approach i developed in the cae of yetric tie-varying delay in both counication channel, but how good robutne property in the preence of ayetric one. Let u ention that ot of tudie dedicated to teleoperation aue yetric and contant delay while few work are devoted to tie-varying delay (ee for intance 7). Future work will focu on the predictive baed teleoperation approach with ayetric rando delay and packet dropout. Finally, robutne of the propoed control approach with repect to the odel uncertaintie i partially addreed via iulation, and upported via experiental trial done on a teleoperated all autonoou vehicle platfor. A particularity of the vehicle teleoperation cae, i that: it i a ingle degree-of-freedo (1 DOF) yte the environent odel i nonlinear due to the friction caued in the rotational operation of the vehicle wheel 16, 17, 18 it i iportant to have reflected force feedback. The contact friction force i then neceary to be tranitted back fro the lave ide to the the teering operation ide o that the huan operator could have better feeling of the environent during the teering. Thi i neceary for driving cofort, and to avoid vehicle overteering/underteering, which repreent the tranparency of the teleoperation control chee 15, 18. the teer-by-wire i a bilateral teleoperation yte (ee above), while the longitudinal teleoperation i a unilateral teleoperation yte (the vehicle peed i ent a a reference to the lave ide). Although the tudy hown here i particularly addreed to the proble of wheeled ground vehicle drive-by-wire teleoperation, the reult are general enough to be applied to yte of different nature. C. Notation, hypothei and definition 1) Notation: For iplicity reaon, the tie dependency in the ignal will, in general, be excluded, unle we would like to tre uch a dependency, a in the cae of delayed ignal where thi dependency will be explicitly indicated. For intance, yte of the for ẋ(t) = Ax(t) A d x(t T (t)),

3 3 will iply be noted a: ẋ = Ax A d x(t T ) where x = x(t), and T = T (t). The notation ˆx will indicate etiated ignal, wherea x will denote error ignal. c will indicate a generic contant. In general, the linear operator are written without it Laplace ignal dependence, i.e. the operator G() : u y, i iply denoted a y = Gu. Alternatively, and when needed, we will ue the notation y = G u. 2) Hypothei: With repect the general teleoperation chee hown in Figure 1, the following hypothee are ade: Auption 1: We aue that yte knowledge i available, in ter of odel for: the environental ipedance odel. In teleoperation of robot anipulator, the odel i aued to be a eoryle linear ap decribed by a pring-like force/diplaceent odel. In vehicular application, the environent i decribed by the interaction force between wheel and ground. Thi i, in general, decribed by a nonlinear odel with internal dynaic. And, the ater and the lave actuator. In our vehicle teering application, the actuator are of one degree of freedo, therefore actuator can be iply decribed by ingleinput ingle-output double integrator. Auption 2: The following ignal are aued to be eaured by the correponding enor: The huan torque F h interacting with the driving wheel i eaured by placing a torque enor in a proper location, Driving wheel poition angle x (eventually the rotational velocity v ay alo be aued to be eaurable, by tie-derivation of x ). The aggregate effect of the environent torque F e decribed by the wheel/ground interaction, can be eaured at the level of the wheel axi rotation. The wheel orientation (with repect to the chai of the vehicle) x (and eventually it tie-derivative, v ) are eaured by an encoder. Auption 3: The traniion delay T = T (t) i bounded a T T < and T T < 1, where T and T are poitive contant. Auption 4: We conider through the paper that the interaction between the operator and the teer wheel, a well a the one between the vehicle and the environent, i a copliant phenoenon 2. We thu aue that, for any delay T (t), the huan and contact torque, F h (t) and F e (t), cannot ecape in finite tie, i.e. there exit poitive contant ρ h, and ρ e, uch that: F i (t T (t)) F i (t) < ρ i, i = h, e 3) Definition: The following definition will be ued in thi paper. i the Euclidean nor and induced atrix nor when apply. Definition 1: Schur Copleent: For atrice Ω 1, Ω 2, Ω 3 where Ω 1 = Ω T 1 < and Ω 2 = Ω T 2 >, then Ω 1 Ω T 3 Ω 1 2 Ω 3 < if and only if, Ω1 Ω T 3 < or Ω 3 Ω 2 Ω2 Ω 3 Ω T <. (1) 3 Ω 1 Definition 2: 24 A linear atrix inequality (LMI) ha the for i= F (x) = F x i F i >, (2) i=1 where x R i the variable and the yetric atrice F i = Fi T R n n, i =,, are given. The LMI (2) i equivalent to a et of n polynoinal inequalitie in x, i.e., the leading principle inor of F (x) ut be poitive. In (2), F (x) < can alo be interpreted in a iilar way. Definition 3: H -Attenuation property : Given z(t) a controlled output and w(t) a diturbance, if there exit oe poitive calar γ and β uch that: z(t) 2 dt γ 2 w(t) 2 dt β. (3) then the H -attenuation of the diturbance ignal w on the controlled output z i aid to be atified. In other word, the yte will be finite-gain L 2 table w.r.t the diturbance input 25. Thu, a in 27, the following perforance index i then conidered throughout the paper: J(w) = (z T (t)z(t) γ 2 w T (t)w(t))dt. (4) Moreover, a we deal with tie-delay yte, the following nor will be ueful for any ignal v(t). v(t) c = Corollary 1: By (5), it then follow: li τ τ τ = li τ τ = li τ up v(t θ) (5) θ T (t), v(t) c dt up v(t θ) dt θ T (t), v(t) dt. The outline of thi paper i a follow. In ection II, we preent the conidered teleoperation odel, the auption on the traniion delay a well a the target ipedance odel. Section III i devoted to the predictor-baed control deign. Firt the control tructure that allow to reach the target odel i given. Then the propoed predictor (with pecific gain on the delayed prediction error) are given in order to etiate oe inttantaneou variable intead of uing eaured delayed one. It i hown that the prediction error atifie a delayed tate equation which ha to be tabilized. The way to deign the gain on the delayed prediction error, uch that the prediction error tie-delay yte i table, i preented in ection IV uing a Lypaunov type ethod. A three tep deign ethod i propoed : a firt tep to evaluate the

4 4 perforance on the chee without prediction, the econd one overcoe oe nonconvexity proble and olve the deign of predictor gain (in the for of oe LMI to be olved), and the thrid one evaluate the perforance of the obtained olution. Section V coplete the cloed-loop tability proof by the analyi of the error dynaic between the cloed-loop yte and the target one. The final perforance i then qualified in relation with the predictor deign. In ection VI iulation reult are provided and the experiental fraework i decribed in ection VII. Concluding reark end the paper. II. PROBLEM FORMULATION Thi ection decribe the odel of the teleoperation yte addreing in particular the proble of vehicle teleoperation, and the forulation of the control objective. A. Teleoperation odel The following dynaic of the ingle DOF ater and lave yte i conidered { M v = u F h, (6) M v = u F e, where v i and u i (i =, ) are the actuator otor angular velocitie and torque repectively. The ubcript and tand for ater and lave repectively. M i (i =, ) i the a. F h i the torque applied at the ater ide by the huan operator and F e i the torque exerted on the lave by it environent. The tructure of the teleoperation yte i hown in Fig.2, with G = 1 M and G = 1 M. C and C are the controller to be deigned. H denote the huan operator ipedance, and Fh ext i external torque applied by the operator. Fig. 2. ext Fh H Fh G v u Mater C T (t) T (t) C u Slave G v Fe EV ext F e Block diagra of the teleoperation controlled yte In vehicle teleoperation, the environent force are due to the wheel/ground interaction. Thee force can be odelled by any of the exiting tire/road friction odel, ee for exaple 16. The environent force are due to: the torque produced by the interaction between the longitudinal and the lateral otion of the vehicle, known a the auto-alignent torque. They are ainly preent at high peed. the torional torque due to the tire rotational friction. They are doinant when the vehicle peed i low. In our vehicle teleoperation etup, where ainly low peed are operated, we will then neglect the auto-aligneent torque, = and only conider the torional torque a the unique ource for F e. Thi torque can be approxiated a a nonlinear function of the for 16, 17: ( v ) F e = σ 1 v f n arctan, (7) δ where σ 1, f n and δ are contant depending on the tire and road adheion, v i the wheel rotational angular peed. When uing (7), it i iportant to reark that the ipedance ap for the environent EV, i not anyore a linear ap, a in any claical forulation. Thi iue will introduce additional technical difficultie pecific to our vehicle teleoperation proble. Finally let u note that thi paper deal with bilateral teleoperation proble for the linear plant (6) with nonlinear environental force (7): thi i not a claical robot teleoperation, a thi conider vehicle teleoperation. A well known in autootive control, the ot iportant nonlinearity i in that cae the tire/road contact friction, odelled here by (7) 16, 17 (and thu taken into account in the theoretical developent). Moreover, in a vehicle, all the non linear force (a Corioli one) are tranitted through the tire/road interaction. In our chee, they could be captured by F e. B. Traniion delay etiation In network controlled yte, it i uual to denote a T = T (t), the half of the tie delay reulting fro the round trip inforation traniion. In wirele traniion yte, the incurred delay i ore likely to be deterinitic and better known when an ad hoc protocol a UDP i choen. Unlikely thi cae, in wire-baed TCP traniion edia T i badly known and need to be etiated. In addition, the tie delay between ater-to-lave, i not necearily the ae than the one fro lave-to-ater. Although thi latter proble i never really treated pecifically, it i worth to be entioned. Delay etiation ha been addreed in the literature in variou anner. The poibility to get an etiate, ˆT of T, rate fro odel-baed protocol etiation 21, to the ue of direct eaureent. An exaple of the latter, the ethod conit in uing tie-taping of the tranitted variable. That i, in ending a local (ater) tie ignal t to the reote (lave) ide, and ent it back to the ater ide. Thi allow to get the round-trip tie, half of which being denoted ˆT. Of coure thi delay will be updated at each eaureent aple (o not for all tie t). Thi ethod, often ued to evaluate the traniion delay, i intereting becaue it i eay to ipleent and it add oe filtering action on the delay etiation. The delay i aued to fulfil Auption 3. Reark 1: Note that with thi ethod only the round trip delay can be eaured. To eaure the one direction tie delay it i required that both coputer (the one in the ater and the lave ide) have a ynchronized tie bai. In thi paper the ean of the round trip tie will be conidered. Variation around thi value can be conidered a noie. C. Target ipedance The control objective i to deign a control law uch that, in preence of traniion delay, the following delay-free

5 5 target ipedance i obtained, { x = Y (1 α)f h N F e, x = Y W x k f F e. where Y (), N(), Y (), and W () are linear deign operator for the ater and lave ide, repectively. α and k f are alo deign contant. Thee pecification, can alo be written only in ter of velocity and torque variable (ee alo F ig.1), F = F h, F e T and v = v, v T, a where Y ()(1 α) Y ()W ()Y ()(1 α) (8) v = G F (9) G = G() = Y ()N() (Y ()k f Y ()W ()Y ()N()) The target ipedance atrix G i choen according to the uer pecification ubject to the paivity contraint of the ap F v for the yte (9), given by F T (t)v(t) >, or equivalently, λ in { G(jω) G(jω) }, where λ in ( ) denote the iniu eigenvalue, and G(jω) = G( jω) T. In thi paper, the pecification are elected a: Y () = 1/ ( M 2 B K ) Y () = 1/ ( M 2 B K ) W () = B w K w N() = β α and k f are all contant. Note that the previou choice i very iilar to thoe ued for ot of the well-known ipedance control approache. Alo it correpond in our cae to the tranparency objective, i.e. the driving cofort 15. III. PREDICTOR-BASED CONTROL DESIGN In thi ection, we introduce the predictor-baed control chee deigned to perfor the target ap (or reference odel) decribed by (8). To thi ai we firt aue that both poition and velocity (in the ater and the alve ide) are available for feedback, a decribed in Auption 2. In the lat part of thi ection, in thi ection an extenion including a velocity oberver a well will be preented. The cheatic diagra i a hown in Fig.3. A. Control tructure The propoed control tructure i iilar to the Anderon- Spong approach but include oe predicted ter a: u = αf h B v K x β ˆF e, u = (1 k f )F e B v K x B w K w ˆx (1) where ˆF e = ˆF e (ˆv) = σ 1ˆv f n arctan ( ˆv δ ) i the predicted contact torque at tie t, uing the prediction of ˆv. F ext h H Fh G x u Ob z Mater C ˆ ( t) F e x, x, F Predictor T (t) T (t) h x, x, F zˆ ( t) Predictor e C Slave Ob u x G Fe x EV ext F e Fig. 3. The cheatic diagra of the bilateral teleoperation yte with the propoed control chee including the bi-predictor. (Ob: Oberver) Siilarly ˆx, and ˆv = ˆx are the predicted ater poition and velocity, repectively. Note that ˆF e, and ˆx need to be predicted becaue they are ued reotely, and hence they are not locally available 2. With the control law (1), the cloed loop dynaic of the teleoperation yte reult in x = Y {(1 α)f h β ˆF } e, (11) x = Y {W ˆx k f F e }. (12) The control goal clearly appear to be the atch of the target ipedance tructure (8). The control deign difficulty i thu reued to the proble of how to predict the contact torque F e (t) at the ater ide, and how to predict the ater poition x (t) at the lave ide. Predictor hould only ue current and/or delayed eaureent. Since the predictor will for part of the local controller on each ide of the teleoperation yte, it becoe neceary to have a irror reproduction of thoe predictor. In other word, ater and lave controller will have two predictor at each ide a hown below. Predictor are deigned on the bai of the cloed-loop equation (11), and (12). Thi choice i otivated for the reaon that will appear clearly when the predictor delayed tructure will be preented in forthcoing ection. B. State pace repreentation A a preliinary to the predictor deign, the following tatepace repreentation for the Equation (11)-(12) i introduced. For (11), we have with A = b 1 = and, for (12) ż = A z b 1 F h b 2 ˆFe, (13) y = c T z = 1 z = x 1 K M (1 α) M B M, b 2 = x, z = β M. v, ż = A z b 1ˆx b 2 F e, (14) y = c T z = 1 z = x. 2 A ignal i locally available when it i eaurable at the ide (ater or lave) that i ued. =

6 6 where A = b 1 = 1 Kw M B w M with the tranforation: K M B M, z =, b 2 = z1 k f M z 2,. x = z 2 (15) v = z 1 1 M (B z 2 B w ˆx ) (16) Reark 2: Note that by contruction the A, A atrice have table eigenvalue. Thi will be ueful later during the tability analyi of the predictor. Note alo that the eigenvalue of thee atrice, are not copletely free; they are table, but pecified by the deired target ipedance. C. Predictor deign We firt aue that both poition and velocitie et (x, v ), (x, v ) are eaurable at their repective ater and lave ide. The velocity eaureent can be relaxed, a explained later in thi ection, by the ue of additional oberver. For iplicity, we firt then preent the predictor deign on the bai of the fully eaured tate z, and z, which can be coputed by a linear tranforation fro the previouly entioned poition and velocity ignal. The predictor tructure will be deigned on the bai of the table tate-pace repreentation given previouly. The ain purpoe of the prediction contruction, i to predict F e (or equivalent to predict v, ince F e depend on v ) at the ater ide, and iultaneouly to predict x at the lave ide. In other word, the proble i to recontruct x in the ater coputer, and vie vera to recontruct x in the lave coputer. The difficultie in doing thi lay at the following point: firt note that F h (repectively F e ) i not acceible at current tie at the lave ide (repectively at the ater ide). Thee ignal play the role of input, and without the a proper predictor can not be contructed. Thee ignal can be eaured, and could be ent throughout the counication network. Therefore, their delayed verion F h (t T ), and F e (t T ) could be ued for control purpoe. However, a torage of F e and F h during T econd would be neceary to ipleent uch a control chee. To avoid a eory buffer we have preferred to artificially delay F h and F e. econdly, the tate-pace equation are coupled with each other through the etiate: ˆFe (ˆv), and ˆx e, therefore it i andatory to have a irror reproduction of predictor for both equation at each ide of the teleoperated yte. Thu, we need to build four predictor: two at the ater ide and two at the lave ide. They have however the ae tructure, and need to be initialized with the ae initial condition at the ae tie. 3 3 If both clock of ater and lave coputer are not ynchronou, thi will generate oe jitter (noie) in the traniion ignal. The propoed predictor have the following tructure: ẑ = A ẑ b 1 F h (t T ) b 2 ˆFe (ẑ, ẑ ) E ẑ (t T ) z (t T ), (17) ẑ = A ẑ b 1 F e (t T ) b 2ˆx E ẑ (t T ) z (t T ) (18) where ˆx = c T ẑ, ˆFe (ẑ, ẑ ) = ˆF e (ˆv ), with ˆF e ( ) a pecified before in (1), and ˆv = ẑ 1 B w M (ẑ 2 ẑ 1 ). Note that, a F e and F h are artificially delayed the ae prediction equation are ipleented in both ide. Thi tructure i then iple and preerve the feaibility of the propoed oberver. Thereby, we aue that tie delay ha been etiated accordingly to the ethod preented in the previou ection 4, i.e. ˆT = T. Finally, E and E are the gain atrice to be deigned. Note that they are only operating over the delayed error prediction. Let z i (t) = ẑ i (t) z i (t), be the obervation error variable, then fro the tate-pace repreentation (13)-(14), and the predictor equation (17)-(18), the error dynaic are given by z = A z E z (t T ) b 1 F h (t T ) F h (t), (19) z = A z E z (t T ) b 1 F e (t T ) F e (t), (2) which can be cated in the following general for z(t) = A z(t) E z(t T (t)) d(t), (21) where A, E, and z follow obviou notation fro (19), and (2). The vector d(t) collect all the coponent of the lat vector in the above expreion. It can be alo rewritten a d(t) = b F (t), with F (t) = F (t T (t)) F (t). Thi vector will be een here a a diturbance having the following property. Lea 1: Aue that the tie delay atified auption 3 and that the huan and contact torque, F h (t), and F e (t) atify auption 4. Then there exit a contant c(t, ρ h, ρ e ) uch that d(t) ha the following bound, d(t) c(t, ρ h, ρ e ) (22) with c( ) being proportional to all it arguent. Proof: The proof i traightforward uing auption 3 and 4 and the definition (21) of d, i.e.: d(t) = b F (t), with F (t) = F (t T (t)) F (t). Reark 3: The auption on continuity ued in the previou lea hould not be undertood a an a priori boundedne (circular arguent) of the ignal involved in the cloed-loop. Indeed, thi hypothei only iplie that finite ecape-tie of the future yte cloed-loop olution will not occur. Note that thi hypothei doe not avoid ignal to diverge. For intance F i (t) and F i (t T ) ay be radially unbounded, while it difference, F (t) i kept bounded during 4 A it will be dicued latter, it i poible to account for inaccuracie on ˆT, but for reaon of iplicity, the cae ˆT (t) = T (t) i now conidered.

7 7 any receding and finite tie-interval. Reark 4: By Lea 1, it i alo poible to treat the cae when ˆT T. In the error equation (19)-(2), thee ter will appear a follow z = A z E z (t ˆT ) b 1 F h (t ˆT ) F h (t), (23) z = A z E z (t ˆT ) b 1 F e (t ˆT ) F e (t), (24) and hence they are till in the general for (21), but with ˆT, intead of T. Reark 5: The robutne iue w.r.t paraeter variation can be conidered to oe tand, a the yte decription (21) ay capture odel uncertaintie a diturbance (a long a thee uncertaintie can be ebedded in a diturbance ter atifying (22)). D. Relaxing velocity eaureent Due to the obervability for of the choen tate-pace repreentation, it i alo poible to derive a predictor on the bai of the poition eaureent only, y, and y. For thi, we need to include an additional oberver, in the following way. At the ater ide, we add the following full-order oberver defined with the new variable z i z = A z b 1 F h b 2 ˆFe l y c T z. Defining ɛ = ɛ (t) = z (t) z (t), we have ɛ = (A l c T )ɛ, where l R 2 1 i deigned uch that A l c T i table. Hence ɛ L 2 L, ˆɛ L. Which iplie that ɛ, or equivalently z z, a t. The ignal z (t) i delayed artificially, o that z (t T ) can be ued locally at the place of z (t T ). Note that z (t) need alo to be ent to the ater ide a well, uch that when it arrive delayed by the network, z (t T ) can be alo ued in the irror predictor, a decribed previouly. Siilarly, the oberver in lave ide i defined a z = A z b 1 F e b 2ˆx l y c T z. A before, defining ɛ = ɛ (t) = z (t) z (t), we have ɛ = (A l c T )ɛ where l R 2 1 i deigned uch that A l c T i table. Siilar procedure a in ater oberver i applied for the delayed ue, and the tranfer of the oberved vector z (t). Reark 6: Note alo that the ignal ˆFe, and ˆx are obtained fro the repective predictor that are till running in parallel to thee additional oberver. In total, we need to ipleent two predictor, and one oberver at each ide of the teleoperated yte, in cae that only poition eaureent are ued. However thi i not andatory. Reark 7: It can be eaily hown, that the general for of the error equation becoe in thi cae z = A z E z(t T ) d Eɛ, Being ɛ the vector concatenating the coponent of ɛ, and ɛ. It i traightforward to how, that the tability propertie of yte with ɛ =, are fundaentally the ae than the yte above with ɛ bounded, and converging to zero exponentially. For thee reaon, thi cae will not be treated anyore in what follow. IV. FEEDBACK GAIN - E DESIGN In thi ection, we will focu on the proble of how to deign the feedback gain E for the yte, z = A z E z(t T ) d (25) z(t) = z (t), t T (t), (26) with the diturbance d bounded a decribed in the previou ection and aued to belong to L 2 (, ), and T i the tie-varying delay. Convergence and tabilitie iue of thi equation are next analyzed. The control deign for E for yte of the for z = A z E z(t T ), with poible untable atrix A do exit. In 26 a ufficient condition by olving a non-convex atrix inequality i derived to deign the E atrix. The coputation procedure for E involve the ue of the algorith propoed in 28. However, thi algorith cannot be applied to the yte when there are diturbance and the tie delay i tie varying, a it i the cae here. On the other hand, if A and E are given, then it i poible (a hown in 27) to verify the tability and attenuation property of the above equation, but when the ai i to deign the feedback atrix E, thi approach canot be ued (due to non convexity). In thi ection, we propoe a new procedure to olve the proble forulated above. It i iportant to note that, by contruction, the atrice A and A are both table, a dicued in Reark 2. A a conequence atrix A i alo table. Since d i bounded (in the ene decribed by Equation (22)), the tability of the predictor can be enured even if E =. Neverthele, it i poible to deign a delayed feedback gain E uch that the perforance of the predictor can be iproved in the ene of the attenuation property given in Definition 3. Propoition 1: The propoed procedure for deigning E i baed on the following 3 tep: Step 1. Copute the iniu attenuation gain, γin, for E =. Thi provide a eaure of the baeline attenuation cae. Step 2. Deign a gain E baed on a Lyapunov (delay-free) analyi. Thi yield attenuation bound that i rather conervative, Step 3. Tet the attenuation bound obtained with gain E deigned in Step 2, uing le conervative Lyapunov-Kraovkii (delay-dependent) functional,

8 8 until obtain a better attenuation gain, than the one obtained in Step 1. Each of the three tep are decribed in detail next. A. Step deign No.1 Find the iniu diturbance attenuation gain γin when E = according to the following theore. Theore 1: Under the auption that A i table, there alway exit P = P T >, γ > atifying A L 1 = T P P A I P P γ 2 <. (27) I Therefore, the etiation ignal z(t) atifie an H attenuation property w.r.t the diturbance ignal d(t), according to Definition 3, i.e.: z(t) 2 dt γ 2 d(t) 2 dt β. The iniu diturbance attenuation gain γin i obtained by olving the convex optiiation proble: γ in = in P γ (28).t. L 1 <, P >. Proof: See Appendix A. B. Step deign No.2 Obtain E. The error dynaic in (21) can be rewritten a z = A z E z(t T ) d = (A E) z E z(t T ) z d = (A E) z Eg d, (29) where g = g(t) = z(t T (t)) z(t). A a conequence of the boundedne of d, and the linearity of the above equation, olution z(t) do exit and are uniquely defined. Therefore, they can not ecape in finite tie, and thereby there exit a poitive contant ν uch that g ν (3) for t, ). The deign of E i now done along the following theore. Theore 2: Aue that A i table. Given γ 1 > and γ 2 >, if there exit P = P T > and S uch that : L 2 = <. (31) A T P P A S T S I S P S T γ 2 1 I P γ 2 2 I Then the atrix E obtained a: E = P 1 S, (32) enure that yte (29) i table and that the etiation ignal z(t) atifie an H attenuation property w.r.t the diturbance ignal g(t), d(t), according to Definition 3, i.e.: z(t) 2 dt γ 2 1 γ 2 2 g(t) 2 dt d(t) 2 dt β (33) Note that γ 1 and γ 2 are choen to iniize firt the delayed ter, i.e. γ 1 i choen a all a poible. Thi tep could be further iproved by iniizing both γ 1 and γ 2, but poibly leading to a large gain E. Therefore a trade-off ha been ought between attenuation level and prediction gain. Proof: See Appendix B. Note that forally, the above reult do not give an etiation of the attenuation ratio copatible (or coparable) with the one obtained in the firt tep. Thi i due to conervativene of the delay-free tability approach ued for thi tep. Therefore, it i advied to ue another le conervative ufficient condition (taking into account the delay) to tet whether the obtained olution E can achieve a better attenuation perforance. The following tep i then propoed to tet the perforance of the predictor after introducing the gain E in the yte. C. Step deign No.3 Tet the perforance for the following yte z = A z E z(t T ) d, (34) with E deigned in Step 2. Theore 3: Aue that A E i table 5. Given γ >, if there exit P = P T >, r 1 >, r 2 > uch that L 3 = Ξ P E P E P E P E T P ψ 1 E T P ψ 2 E T P γ 2 T 1 I P γ 2 I <, (35) where Ξ = (A E) T P P (A E) T r 1 A T A ( T 1 T T )r 2 E T E I, ψ 1 = r 1 (1 T )T 1 I and ψ 2 = r 2 (1 T ) 2 T 1 I, then, yte (34) i table and the etiation ignal z(t) atifie an H attenuation property w.r.t the diturbance ignal d(t), according to Definition 3, i.e.: z 2 dt γ 2 (1 T ) d(t) 2 dt β The iniu diturbance attenuation gain γ i obtained by olving the convex optiization proble: γ 3 = in P,r 1,r 2 γ (36).t. L 3 <, P >, r 1 >, r 2 > γ = γ 3 (1 T ), (37) Proof: See in Appendix C. Reark 8: Note that L 3 could alo be written a L 3 = E T P µt 1 I <, (38) Ξ P E P P γ 2 I 5 Thi i iplicitly obtained in the Step 2, due to the firt upper left ubatrix in inequality (31), i.e. If there exit, a olution for E fulfilling (31), it i neceary that thi olution atify (AE) T P P (AE)I <. Fro the well known neceary and ufficient condition of the Lyapunov inequality, (A E) can only be table.

9 9 with µ = { r 1 (1 T ) 1 r 2 (1 T ) 2 1 γ 2} 1. It i obviou that the LMI in (35) becoe non-convex if E i not known, due to the ter E T E. A direct deign of E fro thi tep, i therefore ore involved. Neverthele, for a given E, and if (35) ha a olution, we can check if the new obtained attenuation level i aller than the one obtained with E = in Step 1, i.e. if γ < γin. If o we keep thi olution, ele we go back to Step 2 and obtain another E with different γ 1 and γ 2 value, and repeat the tet in Step 3. Alternatively, tep 3 ay alo be done by uing aller bound on T, and T, o a to ooth down the conervativene of the ued analyi. Although, there i not full warrantee to reach a finite nuber of iteration before get a uited reult, it i very likely that the etiation in Step 3, will give a quite refined etiated, becaue the analyi i baed on a delay-dependent Lyapunov- Kraovkii functional, which are le conervative than the Lyapunov function ued in Step 1 and 2. V. CLOSED-LOOP STABILITY In thi ection we preent the cloed-loop tability propertie reulting fro the propoed control law together with the predictor. The analyi i baed on the contruction of an error dynaic on the variable e i (t) = z i (t) z r i (t) (i =, ), reulting fro the difference between the cloed-loop tate z i, and the tate of the deired target (8) dynaic, z r i, which can be expreed a: ż r = A z r b 1 F h b 2 F e, (39) ż r = A z r b 1 x b 2 F e (4) In Section II, the paivity of the target yte can be enured by the yte pecification, and thereby the tability of the tate z r i. A long a the huan and environent dynaic are decribed by paive operator, their interconnection with the repective port (F h, v h ), and (F e, v e ), will inherit cloed-loop tability. Equation (39)-(4) can thu be een a a reference odel for the the cloed-loop yte (13)-(14). A. Error dynaic The tracking error between the cloed-loop yte (13)-(14) and the target yte (39)-(4) write a: ė = A e b 2 ( ˆF e (ˆv ) F e ) (41) ė = A e b 1 (ˆx x r ), (42) The following relation hold: ˆx x r = (ˆx x ) (x x r ) = c T (ẑ z ) (z z r ) = c T z e Fro another hand, we have c T e c T z (43) ˆF e (ˆv ) F e (v ) = σ 1 (ˆv v ) f n arctan( ˆv δ ) arctan(v δ ) noticing that, fro (15)-(16), we have ˆv v = B w M c T z Fro the Lipchitz characteritic of arctan( ), and the ean value theore, there exit a analytic bounded function Φ(ζ), with bound Φ(ζ) 1, ζ,, uch that hence, arctan( ˆv δ ) arctan(v δ ) = Φ(ζ 1)(ˆv v ), ζ 1 ˆv, v ˆF e (ˆv ) F e (v ) = σ 1 f n Φ (ˆv v ) = σ 1 f n Φ B w c T M z c 1 z (44) with c 1 = (σ 1 f n ) Bw M c. Here we have ued the fact that Φ(ζ) 1, ζ,. The error dynaic (41) and (42) can be rewritten in copact for, a with, A = η( z ) = ė(t) = Ae(t) η( z ), (45) = Ae(t) D z (46) A 2 1 e (t) b 1 c T, e(t) = A e (t) b2 σ 1 f n Φ Bw M c T z b 1 c T z. Note that the yte pecification enure that A i a table atrix. The above yte decribe an table linear yte perturbed by the bounded ignal η( z ). B. Stability reult The tability of the yte i concluded in the following theore. Theore 4: The tracking error nor of the cloed loop yte (46) with the propoed control law (1), i uniforly bounded and tend, in finite tie, to a ball B r B r = {e(t) : e(t) r(γ, ρ )}, where γ > i the contant obtained by the olution of A T P P A I P D D T P γ 2 <, (47) I with P R 4 4, P = P T >. ρ > reult fro the bound of z ρ (a z i table). Proof: Firt note that due to the tability of A, there i alway a P and a γ uch that the LMI hold. Secondly, fro the boundedne of obervation error z hown in previou theore, and the bound (43)-(44), we can conclude the exitence of a contant ρ, uch that z ρ.,

10 1 Define the Lyapunov candidate V = e T P e. The derivative of V along the olution of the error equation give, V = e T (A T P P A)e e T P D z z T D T P e = e T (A T P P A γ 2 P DD T P I)e ( γ z 1 ) T ( γ DT P e γ z 1 ) γ DT P e e T e γ 2 z 2 e T (A T P P A γ 2 P DD T P I)e e T e γ 2 z 2. If the following inequality, with P being a poitive definite yetric atrix, A T P P A γ 2 P DD T P I <, or equivalently 6, the following LMI hold, then A T P P A I P D D T P γ 2 <. I V e T e γ 2 z T z. e 2 γ 2 ρ 2. That i to ay that the error e(t) i bounded, i.e. the etiation error nor e(t), tend, in finite tie, to a ball B r defined a B r = {e(t) : e(t) γ ρ κ(p ) = r}. with κ(p ) = λinp λ axp 1, being the conditioning nuber of P. Note that by virtute of the tability of atrix A, there alway exit P and γ atifying the above inequality. An alternative proof probably yielding le conervative etiated r for the et B r, can be derived by introducing the function (or functional) V (e, z) = V 1 ( z) e T P e with P being the atrix of the LMI of thi theore, and V 1 ( z) being, either the the function given in Theore 2, or the function given in Theore 3. For intance, if function V 1 in Theore 2 i ued, we get a expreion for V of the for V z 2 γ 2 1γ 2 3 γ 2 2c 2 e 2 γ 2 z 2 which how, that if γ < 1, the new etiate for r, i r = κ(p ) γ 2 1 γ 2 3 γ2 2 c2 1 γ which ay yield le conervative etiate. 6 Thi equivalence follow fro the Schur copleent, a defined in Definition 1 VI. ILLUSTRATIVE EXAMPLE: SIMULATION RESULTS The purpoe of the iulation preented in thi ection i to copare the new propoed predictor deign to the cae when no prediction i ued, and to evaluate the robutne of prediction control chee with repect to traniion delay uncertaintie. Data ued in iulation are given in Appendix D. Deign operator and contant: Y, Y, N, W, k f, are elected o a to fulfill the pecification in ter of driveability, and paivity of the operator G : F (t) v(t) in the target ipedance a defined in (9). The huan force F h i odeled a 15, F h = M hn 2 B hn K hn M hd 2 B hd K hd } {{ } H() x F ext h, (48) where H denote the huan operator ipedance. M hn, M hd, B hn, B hd, K hn, and K hd are contant, and Fh ext i external torque applied by the operator. The odel ha been experientally validated in 15. A. Coparion with the chee without predictor Here the propoed chee i copared with the conventional cae when there i no prediction applied, in which the cloed loop yte i { x = Y (1 α)f h NF e (t T ), x = Y W x (t T ) k f F e. with the propoed data given in Appendix D. A expected, for certain value of T, the paivity of the ap F (t) v(t), of the above cloed-loop equation doe no longer hold (note that thi ap i different to the target one becaue the delay preence in oe entrie of the reulting operator F v). Hence the yte under thi conventional chee ay be untable for certain tie delay in the yte. In the propoed chee, the delayed feedback gain atrice in the predictor, E and E are deigned according to the three tep decribed in Section IV. E i achieved according to Theore 2 when γ 1 = 1 and γ 2 = 18. Fro Theore 3, a better diturbance attenuation level γ = < = γ in 1.6 i achieved, where γin = i obtained according to Theore 1. Siilarly, E i achieved when γ 1 =.1 and γ 2 =.771 according to Theore 2. A better diturbance attenuation level γ =.295 <.6149 = γ in 1.6 i achieved, where γin =.7778 i obtained according to Theore 1. In F ig.4, the profile of the poition x and x are hown. x r and x r are the reference ignal fro the deired odel in (8). Note that the tracking error of the propoed chee are aller than that of the chee without prediction, in which the yte i till table becaue the paivity of the yte with the delay in thi iulation i till not broken. Note that the tracking error of the ater ide are aller than thoe of the lave ide. Thi i becaue the iniu diturbance attenuation level of the prediction error of the lave tate (γ =.2762) i aller than that of the ater tate (γ = ), hence the etiation error of the predictor

11 11 are different, which are reflected on the tracking error of the whole yte. Fig.5 how the evolution of the force F h and F e. A hown in F ig.6, the tracking error of the yte are copared between the two cae: with and without the delayed feedback control E i x i (t T i ) in the predictor. During the teady tate, the bound of the tracking error of the yte are around 32% aller with the incorporation of the delayed feedback gain in the predictor due to the aller prediction error bound. Thi further how the advantage of applied predictor in the teleoperation yte and another deign freedo for the tie delay controlled yte. B. Robutne with repect to tie delay uncertaintie We further tudy the robutne of the propoed chee on the tie delay uncertaintie. Two cae are conidered here. In the firt cae, the eaured delay are different a the real one, i.e. it i artificially et to be ˆT (t) =.5.1 in(t) ec, while the real tie delay i T (t) =.4.2 in(t) ec. A in Fig.7 (a) and (b), the propoed chee contain the robut property on the tie delay etiation. In the econd cae, the real tie delay fro lave to the ater ide i et to be.7.3 in(t) ec which i different with the tie delay fro ater to lave ide. A in Fig.7 (c) and (d), the yte i till table and hence the propoed chee ee to exhibit a certain degree of robutne with repect to tie delay uncertaintie (a pointed out in reark 4). A hown in Fig. 7 the robutne property i iproved thank to the predictor. e x x (a) propoed propoed deired deired no prediction no prediction Tie(ec).5.5 (b) propoed propoed no prediction no prediction Tie(ec) x e x (c) propoed propoed deired deired no prediction no prediction Tie(ec) (d) propoed propoed no prediction no prediction Tie(ec) Fig. 4. The evolution of the ater and lave poition: (a) x (t); (b) e x(t) = x (t) x r (t); (c) x (t); (d) e x(t) = x (t) x r (t). VII. EXPERIMENTAL APPLICATION The propoed predictive control chee i ipleented to the teleoperation control of the Pekee all autonoou vehicle in lave ide by the Drive-by-Wire yte in the ateride a hown in the experiental etup in Fig.8. F h F e Tie(ec) Fig. 5. e x e x (a) (b) Tie(ec) The evolution of the huan operation force and environental force Tie(ec) (a) (b) with E without E with E without E Tie(ec) Fig. 6. The coparion on the evolution of the tracking error: (a) e x (t) = x (t) x r (t); (b) e x (t) = x (t) x r (t). (Solid line: with delayed feedback gain E; Dah dotted line: without delayed feedback gain E) A. Syte decription The Drive-by-Wire yte conit of a yte that teer the wheel, a device that i controlled by the driver (teering wheel unit) and a controller that interact between thee two yte. The teering wheel unit can be conidered a the ater ide yte and conit of a teering wheel which i connected to a torque enor and a DC otor. With the torque enor, the huan applied torque on the teering wheel can be eaured. With the DC otor, the ater controller can give force feedback to the driver. The Pekee i an open robotic platfor deigned by Wany Robotic copany in france. Pekee i driven by two independent otorized wheel, with a free rotating cater wheel at the back. Thi layout offer ubtantial obility - for exaple the unit can do an on-the-pot U-turn. In Pekee, a video caera

12 12 e x (a) propoed no prediction Tie(ec) (b) propoed no prediction e x (c) propoed no prediction Tie(ec) (d) propoed no prediction Round trip delay uing UDP e x.5 e x Tie(ec) Tie(ec) Tie(ec) Fig. 9. Meaured Round Trip Tie delay uing UDP traniion protocol Fig. 7. The evolution of the tracking error: (a) e x(t) and (b) e x(t) when ˆT (t) T (t); (c) e x(t) and (d) e x(t) when the real delay fro ater to lave ide i.7.3in(t) ec. ext F h u x, v DSP PC DSP e Mater x, v, F x F, x, v, x ext h Web Server UDP Protocal High peed traniion line Web Server x, v, Wirele Traniion Fig. 8. Experiental etup e F ext h F, x, v, x x DSP PC DSP Slave u x, v i alo intalled uch that the iage in front of it can be tranitted to the coputer by wirele traniion. Thi ake it poible for teleoperation. The wirele counication i applied by the BeWAN Wi-Fi Acce Point. There are two odoeter (18 ipule/wheel-turn) in the two wheel for eauring the wheel rotational velocity. The etting of the coputing i 16-Mhz Mitubihi icro-controller (16-bit), with 256 KB Flah-ROM and 2KB RAM. The internet protocol UDP (Uer Datagra Protocal) i applied in the data traniion in thi experient. By uing UDP, the delay will be horter copared that of TCP/IP though oe data ay be lot. Here UDP i elected becaue the delay agnitude i a key point for the application. The UDP erver i applied in the Pekee local coputer and the UDP client i applied in the drive-by-wire ater ide yte. B. Modelling tep The yte paraeter are: Dive-by-Wire yte - 1 G () = M b, where where M =.276kg 2 and b =.11N are identified. In the experient, according to the phyical tructure of the Drive-by-Wire yte, the calculation of the Fh ext in the experient can be a the follow: F ext h = M wheel( u F ea ) M F ea, where M wheel =.1kg 2 i the a of the teering wheel, and F ea = F h u i the eaureent torque fro the encoder. In the lave ide yte, G () = 1 M, where M =.2 kg 2. The contact torque i a ae a in (7) a F e σ 1 v f n arctan ( v δ ), where σ1 =.2, f n =.1 and δ =.2 are elected. C. Control deign The propoed predictive controller i deigned according to the theory in Section III, the deigned paraeter of the ater and the lave ide i hown a the follow: for the ater ide, B =.3529 b, K = 2.76, α =.5, β = 1, l = 1, 1 T and for the lave ide, B =.5, K = 4, B w =.2, K w =.5, k f = 2, l = 1, 1 T. All the initial value of the predictor and oberver are et to be zero. Hence, the atrice A, A, and A atrix are all table. Alo fro thee data, the deired target reult to be paive. The feedback gain E and E are alo deigned according to the three tep with three Theore in the ubection IV. E = E = (49) By uing UDP protocol, the delay i oewhat a contant and not o varied according to the tie. So here in thi experiental approach ˆT (t) i treated by a contant, which i etiated by taking the half value of the round trip tie delay. A hown in Fig.9, the delay i tie varying and with the average to be around 2 3 illiecond. So in our application, ˆT =.1 ec, which i the half value of the round trip delay. D. Experiental Reult In the technical part and iulation reult it i hown that the prediction give degree of freedo to iprove the,

13 13 Slave poition Mater poition x Deired x Tie(ec) x Deired x Tie(ec) Fig. 1. The evolution of the ater and lave poition copared with the deired target. perforance of the tracking error between the cloed-loop yte and the target. We have therefore choen to ipleent only the predictive chee only. A hown in Fig.1, the yte perforance i copared with the deired target. Thi how that the lave poition i controlled in an efficient way. The experiental reult alo how that the propoed predictive approach i robut with repect to tie delay uncertaintie. Note that the cale of the tracking error i not that all a in the iulation reult. Thi ay be caued by the odel paraeter, tie delay uncertaintie and by the very iple enor and real-tie architecture of the Pekee. Another factor ay be caued by the data eaureent error both in the teering wheel and in the all autonoou vehicle otor. Fig.11 how the evolution of the force F h and F e. Soe further tudie ay concern the robutne analyi and deign w.r.t uch odelling error. Environental force F e Huan operator force F h (a) Tie(ec) Fig. 11. (b) Tie(ec) The evolution of the huan operation and environental force. VIII. CONCLUSIONS A new predictive approach ha been propoed for bilateral torque reflecting teleoperation yte. The ae predictor in both ide are deigned uch that the tranitting ignal at current tie can be etiated according to the knowledge of the teleoperation yte odel. At the ae tie, the deigned predictor convergence i enured and a better prediction accuracy can be achieved through a delayed feedback deign. Copared with the chee without predictor, the propoed approach can achieve better yte perforance while enuring the yte tability in the exitence of tie varying delay. Siulation reult, a well a experiental one, ephaize the efficiency and applicability of the propoed approach, which i a new fraework of teleoperation yte. Even if the robutne of the propoed approach to uncertaintie i not theoretically tackled here, the propoed chee how intrinic robutne propertie a decribed in iulation by changing the delay (leading to non yetric delay) and alo a the experiental reult are atifactory while the conidered odel ay different fro the real Pekee robot. Soe further tudie ay of coure concern the robutne analyi and deign w.r.t uch odelling error. Thi approach can be ued for the deign of drive-bywire yte, and alo in particular for thoe application in which the traniion delay are large and tie-varying, i.e. underwater vehicle and pace teleoperation. ACKNOWLEDGEMENTS Thi tudy wa realized within the fraework of the NECS ( /nec)-cnrs project. The author would like to thank the CNRS for funding the project. The author alo would like to thank P. Belleain and S. Mocanu fro LAG for their help during the experiental period. REFERENCES 1 W. R. Ferrell, Delayed force feedback, IEEE Tranaction on Huan Factor in Electronic, vol. HFE-8, pp , October R. Oboe and P. Fiorini, A deign and control environent for internet beaed telerobotic, International Journal of Robotic Reearch, vol. 17, pp , A. Fattouh and O. Senae, Finite pectru aignent for teleoperation yte with tie delay, in Proceeding of the IEEE Conference on Control and Deciion, Hawaii, USA, T. J. Tarn and K. Brady, A fraework for the control of tie delayed telerobotic yte, in Proceeding of the IFAC Robot Control, SYROCO, Nante, France, 1997, pp Chopra, N., Spong, M., and Lozano, R. Adaptive coordination control of bilateral teleoperator with tie delay in Proceeding of the IEEE Conference on Control and Deciion, Bahaa, 24, pp Lee, D., Spong, M., Paive Bilateral Control of Teleoperator under Contant Tie-delay 16th IFAC World Congre, Prague, Czech Republic, July 4-8, 25 7 Bereteky, P., N. Chopra and Spong M., Dicrete Tie Paivity in Bilateral Teleoperation over the Internet, Proceeding of the 24 IEEE International Conference on Robotic & Autoation, New Orlean, LA, April G. M. H. Leung, B. A. Franci, and J. Apkarian, Bilateral controller for teleoperator with tie delay via µ-ynthei, IEEE Tranaction on Robotic and Autoation, vol. 11, pp , O. Senae, and Fattouh, A. Robut H control of a Bilateral Teleoperation Syte under Counication Tie-Delay. In: 16th IFAC World Congre, 4-8 july 25, Prague, Czech republic