Force Reflecting Bilateral Control of Master-Slave Systems in Teleoperation

Transcription

1 Force Reflecting Bilateral Control of Mater Syte in Teleoperation A. ALFI, M. FARROKHI Departent of Electrical Engineering, Center of Excellence for Power Syte Autoation and Operation Iran Univerity of Science and Technology, Tehran , IRAN eail: Abtract. In thi paper, a iple tructure deign with arbitrary otion/force caling to control teleoperation yte, with odel iatche i preented. The goal of thi paper i to achieve tranparency in preence of uncertaintie. The aterlave yte are approxiated by linear dynaic odel with perturbed paraeter, which i called the odel iatch. Moreover, the tie delay in counication channel with uncertaintie i conidered. The tability analyi will be conidered for two cae: ) tability under tie delay uncertaintie and 2) tability under odel iatche. For the firt cae, two local controller are deigned. The firt controller i reponible for tracking the ater coand, while the econd controller i in charge of force tracking a well a guaranteeing tability of the overall cloedloop yte. In the econd cae, an additional ter will be added to the control law to provide robutne to the cloedloop yte. Moreover, in thi cae, the local lave controller guarantee the poition tracking and the local ater controller guarantee tability of the inner cloedloop yte. The advantage of the propoed ethod are two fold: ) robut tability of the yte againt odel iatche i guaranteed and 2) tructured yte uncertaintie are well copenated by applying independent controller to the ater and the lave ite. Siulation reult how good perforance of the propoed ethod in otion tracking a well force tracking in preence of odel iatche and tie delay uncertaintie. Key word: Bilateral Teleoperation, Tranparency, Force Reflection, Tie Delay Syte.. Introduction Materlave teleoperation yte enable huan interaction with environent that are reote, hazardou, or inacceible to direct huan contact, uch a ining, ub ea exploration, and ore recently in health care []. The ain coponent of a teleoperation yte are: ) a et of two robotic anipulator, referred to a the local ater yte (or ater for hort) and the reotely located lave yte (or lave for hort), 2) the counication channel, 3) the huan operator, and 4) the tak environent.

2 In bilateral teleoperation, huan operator applie force to the ater in order to produce the deired output, which can be poition or velocity. The output of ater i tranitted to the lave through the counication channel to produce the deired force in the lave yte that generate the poition or velocity at the reote ide, repectively. Due to the interaction between the lave yte and it environent, a reaction force i generated, which i ent back a the reflecting force to the ater to coplete the cloedloop yte. By feeding the reflecting force back to the ater robot and then to the huan operator, which i called forcereflecting control in teleoperation yte, the overall perforance can be iproved [2]. Baed on the tiedelay in counication channel and uncertaintie in the tak environent, there are two ajor iue in teleoperation yte: tability and perforance. Tranparency i the ajor criterion for perforance of the teleoperation yte in preence of tie delay in counication channel a well a tability of the cloedloop yte. If the lave accurately reproduce the ater' coand and the ater correctly feel the lave force, then the huan operator experience the ae interaction a the lave would. Thi i called tranparency in teleoperation yte [3]. The counication channel becoe an iportant iue when the ditance between the ater and the lave i too long. In thi cae, a tie delay appear in inforation traniion that cannot be ignored. Due to thi tie delay, perforance of the bilateral teleoperation yte can be degraded, which can even lead to intability of the reotely controlled anipulator. For the firt tie, Ferrel howed the intability of a teleoperation yte with tie delay [4]. In 98, Vertut and Coiffet howed that tability of teleoperation yte with tie delay could be achieved by decreaing bandwidth of the cloedloop yte [5]. To overcoe the tie delay proble, variou ethod have been propoed in literature. In 989, Anderon and Spong tabilized the yte againt large, but contant tie delay by uing a paive odel for traniion channel [6]. In 99, Nieeyer and Slotine extended thi reult by introducing the concept of wave variable ignal to tabilize the teleoperation yte under unknown but contant tie delay [7]. Further, in 997, they ued a filter in the wave variable ignal that iproved the error between the poition of ater and lave by tranitting the integral of wave variable ignal [8]. Ki et al. ued force reflection and hared copliant control to iprove the teleoperation tak [9]. Buttolo et al. propoed odelbaed firtorder lidingode controller for both the local and the reote ite. Unfortunately, in addition to the high frequency ignal proble, the teleoperation yte becoe untable with inor delay []. In 25, Valdovino et al. propoed higherorder lidingode ipedance control to olve the chattering proble []. Leeraphan et al. propoed an adaptive gain to enure a paive yte for every tie delay that adapt characteritic ipedance. However, they did not invetigate the tranparency for their teleoperation yte [2]. Zhu and Salcudean propoed an adaptive otion/force controller for unilateral or bilateral teleoperation yte 2

3 that can be applied to both poition and rate control ethod [3]. Munir and Book ued the Kalan filter and a tieforward oberver to predict the wave variable and to copenate for the tie delay [4]. HahtrudiZaad et al. analyzed tranparency of the teleoperation yte in preence of tie delay uing threechannel control ethod [5]; neverthele, their ethod i difficult to realize. Yokokohji et al. propoed a control chee that provide the ideal kinethetic coupling uch that it doe not require any knowledge of the paraeter of the huan operator dynaic and the reote object [6]. Leung et al. odeled the tie delay a a perturbation to the yte, and deigned the yte to be robut againt uch perturbation uing H optial control a well a µ ynthei [7]. Baier and Schidt propoed a novel control trategy by uing twoport lole line theory for tabilization and tranparency of bilateral teleoperation yte [8]. Ching and Book propoed bilateral teleoperation baed on wave variable with adaptive predictor and direct drift control ethod [9]. Recently, Alfi and Farrokhi propoed a new tructure for bilateral tranparent teleoperation yte in preence of tie delay [222]. To achieve coplete tranparency and tability in the propoed ethod, directforce eaureentforce reflecting control i eployed at the local ite. In the directforce eaureentforce reflecting control, the poition/velocity ignal i tranitted fro the ater to the lave. Then, uing a force enor, the reflected force fro the environent i ent back in the oppoite direction to produce feedback ignal, which i applied to the ater in order to give the operator a ene of what i happening in the reote ite []. The goal of thi paper i two fold: ) developing a iple tructure for caled bilateral teleoperation yte with arbitrary otion/force caling, and 2) developing condition for robut tability of the propoed tructure in preence of uncertaintie in the dynaic of teleoperation yte. The tability analyi of the propoed tructure i conidered for two cae: ) tability againt the tie delay uncertaintie and 2) tability againt odel iatche. In the firt cae, two local controller are deigned. The firt controller i reponible for tracking the ater coand. The econd controller i in charge of force tracking a well a guaranteeing the tability of the overall cloedloop yte. In the econd cae, a robut control ter will be added to the control law to provide tability robutne to the cloedloop yte. Moreover, in thi cae, the local lave controller guarantee the poition tracking and the local ater controller guarantee tability of the inner cloedloop control yte. In thi way, the lave follow the ater with good accuracie depite odel iatche. In addition, in the core of the propoed control tructure, an identification algorith i developed, which, in addition to predicting the tiedelay paraeter, ake the cloedloop yte virtually independent of the tie delay. Moreover, a a reult, the forward and the backward tie delay need not to be identical in the propoed tructure. In the ret of thi paper, whenever it i referred to the tie delay, it ean the tie delay in counication channel. 3

4 Thi paper i organized a follow. In Section 2, the baic propoed control ethod i dicued. Section 3 decribe deign of the controller. Analytical work about the tability of the propoed tructure i given in Section 4. Section 5 how the iulation reult. Finally, concluion will be drawn in Section The Baic Propoed Control Schee The baic propoed control tructure i hown in Figure. In thi Figure, G, C and y denote the tranfer function of the local yte, the local controller, and the output, repectively. The ubcript and denote the ater and the lave, repectively. Moreover, T (i.e. fro the lave to the ater) tie delay, repectively. Ze i ipedance of the environent, and T indicate the forward (i.e. fro the ater to the lave) and the backward Fe i the force exerted on the lave fro it environent, F h i the force applied to the ater by the huan operator, K p and K f are the arbitrary otion and force caling factor, repectively, F i the input force applied to the lave, and Fr i the reflected force. The following auption have to be tated firt: ASSUMPTION. The lave yte act in a nonfree tak environent. ASSUMPTION 2. The forward and the backward tie delay are unknown and not identical. ASSUMPTION 3. The contact force i eaurable and available for the local controller. Moreover, directforce eaureentforce reflecting control i ued at the local ite. ASSUMPTION 4. In order to give the operator the force enation, the contact force i ent back, a the reflecting force, to the ater robot. The ain goal of the propoed control chee can be tated a follow: The cloedloop control yte ut be table with oe ild and eaytoachieve condition againt I) tie delay uncertaintie and II) odel iatche. 2 The lave output y ha to follow the ater output y with arbitrary otion caling tracking. Notice that the ater and the lave output can be conidered either poition or velocity. K p, which i called poition 3 The reflecting force F r ha to follow the operator force F h with arbitrary force caling K f, which i called force tracking. Thee goal, in thi paper, are achieved by deigning two local controller: one in the reote ite the local ite C and the other in C. Thee controller are naed the local lave and the local ater controller, repectively (Figure ). The local lave controller guarantee the otion tracking. The local ater controller guarantee the force tracking a well a the tability of the overall yte. In the next ection, deign of the local controller i decribed. 4

5 Mater F h y C () G( ) e T K p y G( ) Z e F e F C( ) F r e T K f Figure. The baic propoed control chee (the firt for). 3. Controller Deign In thi ection, deign of the local controller i preented. It hould be entioned that the deigner can elect arbitrary value for otion/force caling. 3.. The Local Controller Deign of the local lave controller i baed on the copliance control ethod. Thi ean that force eaureent are ued at the reote ite []. According to Figure, if the output of the ater and the lave robot i poition, then the tranfer function fro the lave to the ater can be written a X () () () KC p G T = e. () X () ZG () C () G () e Since the forward tie delay doe not appear in the denoinator, the tranfer function in Equation () i finite dienional. Hence, the tie delay will not have any effect on the tability of the yte. In addition, one can ue the claical control ethod (uch a PD) to deign a local lave controller C for the reote ite uch that the yte in () i table. Therefore, the poition of the lave robot will follow the poition of the ater robot in uch a way that the tracking error for the poition converge to zero. The effect of the tie delay will be addreed in the next ection The Local Mater Controller The local ater controller i deigned baed on the direct forceeaureent forcereflecting control. Thi controller ut aure tability of the cloedloop yte a well a force tracking. The force tracking ean that the reflecting force fr () t ha to follow the huan operator force fh () t. Firt, the following variable are defined: ZC e () G() Gˆ() = ZG () C () G (), (2) e 5

6 F h C () G () e T Gˆ () F e F h C () Ge () T F r F r e T K f Figure 2. The econd for of the propoed control chee. Figure 3. New control chee (the third for). R () C () C () Plant P () = Ge () T Y () G () T P () = Ge () Model Figure 4. The Sith predictor control ethod. F () () e C Ge M() = =. (4) F ( T ) () () T C Ge h T G ()= Gˆ () G (). (3) Uing thee variable, the control chee, hown in Figure, can be iplified a in Figure 2. Notice that, the local lave controller C i deigned uch that pole of Gˆ are in the lefthand ide of the SPlane (i.e. C guarantee the poition tracking). Then, the tranfer function of the overall cloedloop yte can be written fro Figure 2 a Now, ince the force tracking i perfored by ending the contact force through the reflection path of the counication channel, a new output in the block diagra of Figure 2 can be defined a F r. Thi block diagra can be iplified a the block diagra in Figure 3. Uing thi block diagra, the tranfer function of the overall cloedloop yte can be written a T where T = T T and Fr() = KfFe() e. T F KC f () Ge () r M() = = (5) T F C () Ge () h Notice that, the role of C () are to provide tability to the overall yte and enure the force tracking. Fro Equation (5), it can be een that the tie delay exit in the denoinator of the cloedloop tranfer function yielding an infinite dienional tranfer function. That i, the tie delay can detabilize the yte by reducing it tability argin or degrading it perforance. A a reult, one cannot ue the claical control ethod (like PD) to deign a local ater controller C uch that the overall yte in (5) i table. Therefore, the fundaental proble in the propoed control ethod (i.e. the three different for in Fig., 2 and 3) i to cope with the tie delay properly. The ot popular and effective ethod to olve the tie delay proble in a table SISO proce i the Sith predictor [23]. Thi predictor can effectively cancel out tie 6

7 delay fro the denoinator of the cloedloop tranfer function. In thi way, one can ue the claical control ethod for the local ater controller. Figure 4 how the block diagra of the Sith predictor. In thi Figure, P() = Ge () T denote the tranfer function of the plant, where G i a table, trictly proper rational function, characterizing the delayfree part of the plant and T i a poitive real nuber repreenting the tiedelay. Moreover, P () = Ge () function of the noinal odel, where G () and T are the noinal verion of G () and T, repectively. T denote the tranfer The ain drawback of Sith predictor i it enitivity to the proce odel. In other word, in Sith predictor: ) the tie delay ut be contant and known a priori and 2) the odel ut be known preciely [24]. Hence, application of Sith predictor are liited in teleoperation yte. To overcoe thee liitation, it i neceary to find a echani to copenate for tie delay uncertaintie and odel iatche. In the next ection, the tability condition for the cloedloop yte will be dicued. 4. Stability Analyi Stability of linear tieinvariant (LTI) delay yte can be copletely deterined by the root of it characteritic equation [25]. In thi ection, the tability analyi i perfored for two cae: ) Stability analyi due to the tiedelay uncertaintie, 2) Stability analyi due the odel iatche. 4.. Stability Analyi for TieDelay Uncertaintie In thi part, tability of the propoed tructure againt the tiedelay uncertaintie i decribed. In order to avoid yte intability due to the tiedelay uncertaintie, an adaptive filter i eployed for predicting the tie delay. Figure 5 how the tructure of the local ater controller. According to thi Figure, the cloedloop tranfer function i T C() Ge () M() = C() G () C() G () e e T T [ ] (6) where G () i defined in Equation (3). It i obviou that tability of the cloedloop yte depend on the tie delay. If the actual tie delay T i equivalent to the etiated tie delay T, then the cloedloop yte i table. In other word, if T = T Equation (6) can be written a T C() G () e M() = (7) C () G () 7

8 F h e T C () G () G () Identification algorith e T ˆ G () F e F h C () G () e T F r F r e T Figure 5. Structure of the local ater controller. Figure 6. The deired controlloop configuration. Figure 6 how the block diagra of the equivalent cloedloop yte for T error T T. Hence, the cloedloop yte can becoe untable. = T. But due to the etiation Suppoe, there exit etiation error for the tie delay and let T = T δ denote the etiated tie delay. Then, the cloedloop tranfer function, given in Equation (6), can be written a T C() G () e Mδ() =. (8) T C () G () δ C () G () e ( e ) By conidering the characteritic equation of (8), it i obviou that tability of the cloedloop yte depend on the tie delay. Thi fact can be hown a T () = C () G () C () G () e ( e δ ). (9) δ Now, the proble i to find the tability condition uch that the cloedloop yte becoe table. To do thi, conider the nodelayed verion of G () and C () a Then, the tranfer function in (8) can be rewritten a where polynoial D () and N () are equal to where it i aued that deg( D ()) deg ( N ()) Ng () Nc () G () =, C () =. () D () D () g c T N () e Mδ () =, () T δ D () Ne () ( e ) D () = N () N () D () D () (2) g c g c N () = N () N () (3) g c > and polynoial D () i Hurwitz. In the following theore, the condition for tability of the cloedloop yte without any upper bound on the tie delay will be hown. THEOREM. The cloedloop yte in Figure 5 i independent of the etiated tie delay T, if N () <, ω. D () 2 = jω 8

9 where D () and N () have are given in (2) and (3), repectively, () D i Hurwitz, deg( D ()) deg ( N ()) >, Ng () and Dg () are the nuerator and the denoinator of the nodelay tranfer function G (), repectively, and Nc () and Dc () are the nuerator and the denoinator of C (), repectively. Proof. The ethod of twodienional tability (2D) tet i ued here for proof [26]. In thi teting ethod, the yte ut be table for T = (i.e. table for no tie delay), which i true for the cloedloop yte in Figure 5, ince polynoial D () i aued to be Hurwitz. Now, uing the characteritic equation of the cloedloop yte in (), the equation for the 2D tet can be written a () = D () N ()( z z ), (4) δ where z = e T and z = e T, (, z, z ) =, (5) δ = = (6) δ(, z, z) δ(, z, z ). When no olution for (5) and (6) exit, then, according to the 2D tet, the cloedloop yte ut be table. Uing the characteritic equation, we have and (, z, z ) = D () N ()( z z ) =, (7) δ = δ(, z, z) δ(, z, z ) = D N z z ( ) ( )( ) = zzd ( ) N( )( z z) =. (8) Fro (7) we have D () z = z. (9) N () Subtituting (9) into (8) give = δ(, z, z) δ(, z, z ) D () = δ, z, z N () D () D () = z z D( ) N( ) =. N () N () (2) Hence, = =. (2) 2 δ(, z, z) znd () ( ) zd() D( ) N( D ) () Fro there, it yield 9

10 N () N ( ) D () D( ) 2 z z =, (22) j T The root of (22) for = jω and z = e ω = ut lie in the lefthand ide of the plane. Subtituting j T e ω z = and = jω into (22) give N( jω) 2 jωt jωt N( jω) e e =. (23) D( jω) D( jω) Factoring out j T e ω j T and noting that e ω, reult into jωt N( jω) jωt N( jω) jωt e e e = D( jω) D( jω), (24) which give N( jω) jωt N( jω) jωt e e =. (25) D( jω) D( jω) Uing the polar for, we have N( jω) N( jω) coωt jinωt coωt jinωt D( jω) = D( jω) (26) Noting that in polar for, the agnitude i an even function, while phae i an odd function, one can conclude N( jω) N( jω) 2 co ωt Dj ( ω = ) D( jω (27) ) Hence, the condition N( jω) D( jω ) < 2 i atified for all frequencie. Therefore, the characteritic equation of the cloedloop tranfer function, given in (8), will not have any root with poitive real part, which yield a table yte independent of T. Reark. In the proof of the above theore, it wa hown that the condition for tability of the cloedloop yte, without any liit on the tie delay, i N( jω), ω. D( jω ) < 2 (28) It i obviou that for a linear yte it i alway poible to deign the local controller uch that: ) they are table, 2) the cloedloop yte i tranparent, and 3) the inequality (28) hold. Hence, uing the propoed control ethod, tability can be aured. Reark 2. In order to increae the robutne of the overall yte, with uncertaintie in tie delay, one can deign the local controller uch that (28) i alway valid. However, it hould be noted that there exit a

11 trade off between N( jω) D( jω ) and the preciion of the tranparency. That i, aking thi agnitude too all ight coproie the tranparency. Thi fact i true in practice a well; due to the exiting delay in counication channel and uncertaintie in the environent dynaic, there i a coproie between tability and tranparency [2, 22]. In other word, tranparency of the overall yte can be iproved without coproiing the tability of the yte, if the tie delay i all enough. Baed on thi, Alfi et al. have propoed a theore to relax the tability condition given in (28) for all tie delay [2]. Firt, all tie delay ut be defined for teleoperation yte. PROPOSITION : The tie delay in counication channel in teleoperation yte i conidered all for T. ec. Proof: Brook [27] propoed a bandwidth between 4 and Hz for teleoperation yte. Conequently, by uing the following firtorder approxiation for tie delay in Laplace tranfor e T = T (29) and alo noting that T( j ) 2 2 e ω = T( jω) = Tω, it give T =. ec. Therefore, when it i referred to all tie delay in counication channel, it ean a tie delay approxiately le than or equal to. ec. THEOREM 2. Let δ = T T denote the etiation error for the tie delay. Then, the propoed control yte, hown in Figure 6, i table for all tie delay if N () D () = j ω <, ω, where D () and N () have are given in (2) and (3), repectively, () D i Hurwitz, deg( D ()) deg ( N ()) >, Ng () and Dg () are the nuerator and the denoinator of the nodelay tranfer function G (), repectively, and Nc () and Dc () are the nuerator and the denoinator of C (), repectively. Proof. Uing the firtorder approxiation for T e and T e yield T e = T, T e = T. (3) Subtituting (3) into the characteritic equation (9) give

12 T T δ() = C() G () C() G ()( e e ) = C() G () C() G ()( T T ) = C() G () C() G ()( T T ) = C () G ()[ T ( T )] ( T T ) C() G () e. = (3) Subtituting G () = N () D () and C () = N () D () into (3) give g g c c () = C () Ge () δ ( T T) T ( T ) () N () e. = D (32) Uing (3), (32) and δ = T T, Equation (8) can be rewritten a δ N () e Mδ () = e δ D () N () e T ( δ ) (33) Since e T ( δ ) doen't play any role in the tability of the cloedloop yte, according to the Typkin theore [28] the condition for the cloedloop tability i N () D () = j ω <, ω. (34) Reark 3. In the proof of the above theore, it wa hown that the condition for tability of the cloedloop yte with all tie delay in counication channel i N( jω), ω. D( jω ) < (35) Obviouly, for a linear yte, it i alway poible to deign the local controller uch that: ) they are table, 2) the cloedloop yte i tranparent, and 3) the inequality (35) hold. Hence, uing the propoed control ethod, tability can alway be aured for all tie delay in counication channel [2]. o Reark 4. It hould be noted that the condition given in (35) for all tie delay doe not have any conflict with the reult of Theore, given in (28). In other word, if condition (28) i atified, then (35) i guaranteed a well. Thi i becaue in the proof of Theore 3, the tie delay wa aued to be all, which ipoe ore retriction on the tie delay but le retriction on the deign of the local controller. Reark 5. Baed on reark 2, if (35) i conidered with all tie delay, then, the robutne of the cloedloop yte will be iproved coniderably againt all tie delay, but the tranparency of the yte ight be coproied. In the next ection, the effect of the odel iatch on the tability of the propoed ethod will be addreed. 2

13 4.2. Stability Analyi Againt Model Miatche In thi ection, the tability of the cloedloop yte againt odel iatche i invetigated. It i well known that robot anipulator have nonlinear dynaic. On the other hand, ince teleanipulator are odeled with linear dynaic in thi paper, there i a odel iatch between the reality and the propoed ethod. Moreover, there are uncertaintie in the tak environent. In addition, the tie delay uncertainty, which wa addreed in the previou ection, can alo be conidered a a odel iatch. According to Figure 4 C () C () =, (36) C () G () T T P () = Ge (), P () = Ge (). (37) Y() C () P () M() = = R () C ()[ G () P () P ()] (38) It i obviou that tability of the cloedloop yte depend on the odel iatch. Thi fact can be hown by conidering the characteritic equation of the cloedloop yte a Auing that there i no odel iatch, i.e. P () = P (), we have () = C ()[ G () P () P ()] (39) δ Y() C() P() M() = = R () C () G () (4) Now, uppoe there are odel iatche and let the odel uncertainty be repreented by oe ultiplicative perturbation δ (). Then, the tranfer function of the plant with uncertaintie can be written a P () = P ()( δ ()) (4) Without any lo of generality, if the nor of the odel uncertaintie i bounded, then, according to the following Theore, the cloedloop yte in Figure 4 i table. THEOREM 3. The controller C () provide robut tability if and only if δ( jω) M( jω) <, ω A neceary and ufficient condition for robut perforance i δ( jω) M ( jω) F ( jω) M ( jω) <, ω c w 3

14 F h Cr () U r C( ) P ( ) F P ( ) Figure 7: Equivalent of the Sith Predictor Schee. where M c i the copleentary enitivity tranfer function, M i the enitivity function of the cloedloop yte, M ( jω) = M ( jω), and Fw i the weighting function (the perforance weight). c Proof. See reference [29]. A it wa entioned earlier, the ater controller guarantee tability and full tranparency, when there i only tie delay uncertainty. Now, ince there are odel iatche a well (i.e. both the tie delay uncertaintie and the odel paraeter uncertaintie exit), a robut ter will be added to the control law. Firt, a procedure hould be developed to convert the block diagra of Figure 4 to the block diagra in Figure 7 and vice vera a follow: C ( ) C( ) = C ( ) G () () C ( C ) = C ( ) G () (42) (43) In Figure 7, the cloedloop tranfer function can be written a Y( ) CP () ( ) Gry () = =. (44) R ( ) [ P ( ) P ()] C () According to the control yte theory, the ideal control would be realized if Y() = R (). (45) A a reult, the following condition ut be atified to have an ideal control yte: The iple deign procedure for C () can then be tated a [3] P () = P () (46) CP ( ) () = (47) C () = P () F (), (48) where F () = ( τ ) n, (49) in which F () i a low pa filter, and τ and n denote the filter tie contant and the order of the filter, repectively. 4

15 F h Cr () U r C( ) P ( ) F P ( ) Figure 8. New robut control tructure. The order of the filter ut be elected uch that the controller C () i realizable. In addition, theoretically, τ i elected uch that an optial coproie between the perforance and the robutne i reached [3]. Zhang et al. have propoed a iple quantitative tuning procedure for thi purpoe [3]. In their procedure, τ can be elected a α T, where T = T T. The larger the paraeter α, the wore the noinal perforance and the better the robutne. The range for α i fro. to.2. By conidering Equation (48) for deigning the controller, a precie odel i needed. But in practice, it i hard to get the precie odel for teleoperation yte. Hence, it i iportant to conider the odel iatche in tability analyi. The new tructure for robutne againt the odel iatche i hown in Figure 8. In thi Figure, the cloedloop tranfer function fro G uy F r to U can be written a Y( ) C () P () () = = (5) U() [ P () P ()] C () Notice that, the above equation i equal to the cloedloop tranfer function given in Equation (44). The goal i to deign the robut controller C () to cope with the odel iatche in Figure 8. The following Theore provide the required condition for the robut controller. r THEOREM 4. Conider the cloedloop control yte in Figure 8. Suppoe the nor of uncertaintie i defined a P( jω) δ( jω) = γ, ω P( jω) where P () and P () are the plant and the odel given in (37), repectively. Then, according to the allgain theore, the controller C () in Figure 8 provide robut tability if r δ ( jω) M( jω) < 5

16 F h Cr ( ) F ( ) = ( δ ( )) F( ) F r δ ( ) M ( ) Figure 9. The equivalent block diagra of Figure 8. Figure. The equivalent of Figure 9. Fh Cr ( ) U C( ) G ( ) e T G ˆ ( ) F e G ( ) e T Identification algorith Figure. The tructure deign againt odel iatch. F r e T where C () and F( ) are defined in (48) and (49), repectively, Cr () F () M() = C () F () i Hurwitz, (i.e. the δ ()[ F ()] cloedloop yte i table without odel iatche), andδ () =. δ ()[ F ()] Proof. Subtituting (48) into (5) give r G uy () = F () P() [ P () P()] F () P () P () (5) If we conider the odel uncertaintie by oe ultiplicative perturbation δ () a P () = P ()( δ ()), (52) then, (5) can be rewritten a Let u define the following new bounded variable: Uing (54), (53) can be iplified a G uy F () () = δ ()[ F ()]. (53) δ ()[ F ()] δ () =. (54) δ ()[ F ()] G () = F ()( δ ()) (55) uy It i apparent that if there exit any odel iatch, then δ () and conequently δ (). Hence, if δ () i bounded, δ () i bounded a well. On the other hand, () δ α yield δ () β, where α and β are oe poitive nuber. Equation (55) deontrate that the odel iatch can be repreented by uncertaintie in the 6

17 filter F. () Conequently, Figure 8 i equivalent to Figure 9, and then equivalent to Figure. Hence, according to the allgain theore [32], the cloedloop yte in Figure i robut table if and only if δ M(), (56) Cr () F () where M() = C () F () i Hurwitz. I.e. the cloedloop yte i table without any odel iatche.o r < Reark 6. By conidering (56), it i apparent that the aller the value of δ ( jω) M( jω), the higher the robutne for the overall yte againt odel iatche. However, there exit a trade off between δ ( jω) M( jω) and the tranparency. That i, aking thi agnitude too all ight coproie the tranparency. Figure how the odified block diagra of the propoed tructure, hown in Figure, againt odel iatche. 5. Siulation The dynaic equation of the ater and the lave yte are conidered here a a inglelink anipulator by [33] J B M gl θ = u, 2 ( ) 2 ( J B MgL ) θ = u where J, M and L are the oent of inertia, the a, and the length of the anipulator link, repectively; B i the vicou friction coefficient, θ i the rotational angle, g i the gravity acceleration, and u i the input; indice and are for the ater and the lave, repectively. The yte paraeter are et to 2 J = 2 kg., B = 3 N/, L =.2, and M =.6 kg for the ater, J = kg.2, B = 5 N/, L =.3, and M = 2 kg for the lave, and Z e = for the environent ipedance. Moreover, the poition and the force caling are et equal to.2 and 4, repectively. In order to deontrate perforance of the propoed ethod, iulation are carried out for two cae: ) the tie delay uncertaintie (ection 4.) and 2) the odel iatche (ection 4.2). In addition, norally ditributed rando ignal are ued a uncertaintie in the tie delay. The tie delay i etiated with an FIR filter [34]. Order of the FIR filter i elected a P = 6, becaue a it i entioned in [34] for P 6 the etiation error of the tie delay will be very all and negligible. In iulation, three different controller are deigned. The firt controller i a PD controller, called the local lave controller C, which i ued for the reote ite. The econd controller i a PD controller, called the local ater controller C, which i ued for the local ite. The third controller i a robut controller, which i ued for the local ite. Notice that the latet controller i only required for odel iatche. The lave controller i deigned uch that 7

18 G ˆ( ) i table, and the ater controller i deigned uch that the goal of the cloedloop teleoperation yte i atified. Moreover, the ater controller i deigned uch that the robut tability condition againt tie delay uncertaintie, given in (28) and (35), i alo achieved. Finally, the robut controller i deigned uch that the robut tability condition againt odel iatche, given in (56), i achieved. 5.. Tie Delay Uncertaintie Siulation for the tie delay uncertaintie are carried out for different value of the tie delay. In cae I, the tie delay i all with oe perturbation, while in cae II the tie delay i relatively large with coniderable perturbation. Siulation for the tie delay uncertaintie are hown in Figure 22. Figure 2 and 6 repreent the tie delay for the cae I and II, repectively, wheret = T T. In addition, Table and 2 how the type of the local controller and their typical coefficient ued in cae I and II, repectively. Notice that, thee value are not unique and are elected only to atify the tability condition given in Equation (28) and (35). The tability condition can be checked uing the Bode plot given in Figure 3 and 7. The dahed line in Figure 3 and 7 repreent the righthand ide of Equation (28) and (35) (i.e. 2log(.5) 6db and 2log()=db ), repectively. Figure 45 and 89 repreent the tranparency repone for the tep and the inuoidal input. A thee figure how, the propoed ethod can cope very well with tie delay uncertaintie. Notice that, the tability condition in thee cae, given in (28) and (35), cannot cope with odel uncertaintie (Figure 2). Thi Figure confir the intability of the teleoperation yte againt odel uncertainty, where 2% and 25% paraetric uncertainty are aued for the vicou friction in the aterlave and the environent ipedance, repectively. Table. Type of the ater controller Cae Local Controller K P K D Cae I PD.25.5 Cae II PD 2.5 Table 2. Type of the lave controller K K D Cae Local Controller P Cae I PD Cae II PD

19 Delay Counication Line.2 x Tie(ec) Magnitude(db) Frequency(rad/ec) Figure 2. Tie delay in counication channel (cae I), Figure 3. Bode plot for preenting tability condition in where T = T T. Equation (45), (cae I). Poition Tracking (rad) Mater Mater Tie(ec) 3 Huan Force Force Reflection.5 6 Force Tracking (N.) Huan Force Force Reflection Tie(ec) Figure 4. Tranparency repone for tep input (cae I). Mater 2 Poition Tracking (rad).5 5 Mater 5 5 Tie (ec) Huan Force 2 Huan Force Force Tracking (N.) Force Reflection 8 Force Reflection Tie(ec) Figure 5. Tranparency repone for inuoidal input (cae I). 9

20 Delay in Counication Line(ec) Tie(ec) Magnitude(db) Frequency(rad/ec) Figure 6. Tie delay in counication channel (cae II), Figure 7. Bode plot for verifying the tability condition in Equation (28), where T = T T (cae II). Poition Tracking (rad) Mater.2 6 Mater Tie(ec) Huan Force.5 Force Tracking (N.) Huan Force Force Reflection.5 Force Reflection Tie(ec) Figure 8. Tranparency repone for tep repone, cae (II). Poition Tracking (rad) Mater Mater Tie(ec) Force Tracking (N.) Huan Force Force Reflection.5 6 Huan Force Force Reflection Tie(ec) Figure 9. Tranparency repone for inuoidal repone, cae (II). 3 2 Mater Poition Tracking Tie(ec) Figure 2. Untable repone, when oe odel iatche are introduced. 2

21 5.2. Model Miatche In order to evaluate the effectivene of the propoed ethod againt odel iatche, paraeter uncertaintie in the ater, the lave and the tak environent, a well a the tie delay uncertaintie are conidered here. The following nonlinear equation repreent the dynaic equation for the ater and the lave robot J θ Bθ Kinθ = u. Moreover, 2% and 25% are added a extra uncertaintie to the vicou friction of the aterlave and the environent ipedance, repectively. In addition, the tie delay uncertainty, hown in Figure 2, i conidered in the counication channel. The elected value for the controller are K P = 2.5 and K = for the lave controller, and K.25 D P = and K =.5 for the ater controller. Moreover, the robut controller i elected a C = /( ). Notice that, thee D nuerical value are not unique and are elected only to atify the perforance and the tability condition. The order of the low pa filter, F () in (49), i equal to three. Recall that, the order of the low pa filter ut be elected uch that the robut controller i realizable. Furtherore, for the tie contant of the low pa filter ( τ = αt ), wheret = T T, the paraeter α i et to.5 (ee Section 4.2 for detail). Figure 2 how the Bode plot to verify the tability condition given in (56). The dahed line in thi Figure repreent the righthand ide of Equation (56) (i.e. 2log()=db ). Notice that, Reark 6 i conidered here. Figure 22 and 23 illutrate the tranparency repone of the propoed control ethod for the tep and the inuoidal input, repectively. A thee figure how, the propoed tructure exhibit good perforance for both the tability a well a the tranparency of the caled teleoperation yte in preence of odel iatche. It hould be noticed that the tie delay in force tracking i ore than the tie delay in poition tracking. Thi i due to the defined variable for the force tracking, in which the tie delay i the u of the forward and the backward tie delay. r Magnitude(db) Frequency(rad/ec) Figure 2. Bode plot for verifying the tability condition in (49). 2

22 .. Poition Tracking (rad) Mater Mater Tie(ec) Huan Force.5 Force Tracking (N.) Huan Force Force Reflection.5 Force Reflection Tie(ec) Figure 22. Tranparency repone for tep input. Poition Tracking (rad) Mater Mater Tie(ec) Force Tracking (N.) Huan Force Force Reflection.5 6 Huan Force Force Reflection Tie(ec) Figure 23. Tranparency repone for inuoidal input 6. Concluion To achieve tranparency and tability robutne againt odel iatche in caled teleoperation yte with arbitrary otion/force caling, a iple control chee wa propoed in thi paper. The odel iatche included uncertaintie in the tie delay in counication channel and odel paraeter uncertaintie. In the propoed approach, three controller were deigned: Two local controller, one for the ater ide, one for the lave ide, and a robut controller. Thee controller were deigned uch that the lave controller guarantee the otion tracking and the ater controller guarantee the tability of the inner cloedloop yte. The robut controller guarantee tability robutne againt odel iatche. The ajor advantage of the propoed ethod wa that one could ue the claical control ethod to deign the local controller a well a the robut controller. Furtherore, the controller deign would be traightforward; thee controller only need to atify one iple condition. In iulation, for a onedegreeoffreedo anipulator exaple, it wa hown that the propoed ethod i a viable choice for the teleoperation yte with odel iatche. Reference. Melchiorri, C. and Euebi, A.: Teleanipulation: Syte apect and control iue, in Proc. of the Modeling and Control of Mechani and Robot, World Scientific, Singapore, 996, pp

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