State Convergence-based Control of a Multi- Master-Single-Slave Non-linear Teleoperation System

Transcription

1 Acta Polytechnica Hungarica Vol. 15, No. 4, 18 State Convergence-baed Control of a Multi- Mater-Single-Slave Non-linear eleoeration Syte Uar Farooq 1,3, Jaon Gu 1, Mohaed El-Hawary 1, Valentina E. Bala, Mariu M. Bala, Ghula Abba 4, Muhaad Uan Aad 1, Jun Luo 5 1 Deartent of Electrical and Couter Engineering Dalhouie Univerity, Halifax, N.S. B3H 4R, Canada Deartent of Autoatic and Alied Software, Aurel Vlaicu Univerity of Arad, Roania 3 Deartent of Electrical Engineering, Univerity of he Punab, Quaid-e-Aza Cau, Lahore, 5459 Pakitan 4 Deartent of Electrical Engineering, he Univerity of Lahore, Pakitan 5 School of Mechatronic Engineering and Autoation, Shanghai Univerity, China uar.farooq@dal.ca, aon.gu@dal.ca, elhawary@dal.ca, valentina.bala@uav.ro, ariu.bala@uav.ro, ghula.abba@ee.uol.edu.k, uan.aad@dal.ca, luoun@hu.edu.cn Abtract: hi aer reent the deign of a tate convergence baed control chee, for a ulti-ater-ingle-lave nonlinear teleoeration yte. he control obective i that the lave follow the weighted otion of the ater yte, in free otion, and the ater yte receive the caled force feedback, while the lave yte i in contact with the environent. o achieve the deired obective, extended tate convergence architecture i odified and aroriate control gain are choen following a Lyaunov baed tability analyi. MALAB iulation conidering a two-degree-of-freedo tri-ater-inglelave nonlinear teleoeration yte are rovided to how the validity of the rooed chee. Keyword: tate convergence ethod; ultilateral teleo-eration yte; nonlinear dynaic; MALAB/Siulink 7

2 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte 1 Introduction eleoeration refer to the control of a reote roce and for an iortant cla of robotic due to it wide range of alication [1]. yical unit of a teleoeration yte are a huan oerator, ater aniulator, counication channel, lave aniulator and the environent. he working of the yte i uch that the huan oerator initiate the tak through the ue of a ater aniulator which i intalled at the local ite. hi tak-related inforation i then tranitted over the counication channel to the reote environent where the lave aniulator erfor the deired tak. At the ae tie, lave aniulator rovide a feedback, which i rereentative of the reote environent condition, to the huan oerator through the ater aniulator who then can guide the lave aniulator by adating to the environent condition. hi for of teleoeration yte i known a bilateral, a the inforation flow both way a ooed to the unilateral verion where the flow of inforation i fro ater to lave ide only. Clearly, the bilateral chee i ore reliable becaue of the reence of the feedback connection which rovide the oerator with a feel of environent. On the other hand, the reence of the feedback connection in a bilateral yte oe a great challenge ainly becaue of the tie delay in the counication channel and the uncertaintie of different unit colicate the roble further. A variety of control law have been rooed in literature to tabilize the bilateral yte againt the tie delay of the counication channel while roviding the atifactory tak erforance []- [4]. he groundbreaking work on the ue of traniion line theory and the wave variable in bilateral teleoeration yte for the bai of any other algorith [5], [6]. Such a cla of algorith i collectively referred to a aivity aradig for the bilateral control of teleoeration yte and by far, i the ot oular choice for deigning bilateral yte due to their trong robutne to tie delay [7]. he other tye of bilateral algorith include liding ode control [8], H control [9], Lyaunov-Kraovkii functional baed control [1], diturbance oberver baed control [11], adative control [1], intelligent control [13]-[15] and the tate convergence baed control [16], [17]. In recent year, another cla of teleoeration yte, known a ultilateral teleoeration yte, ha eerged due to the need of erforing the reote tak in a cooerative fahion. In order to aintain the tability and to enure the atifactory tak erforance in uch yte, variou control algorith have alo been rooed. Sall gain theore baed aroach i reented in [1] for a ulti-ater-ingle-lave teleoeration yte where the lave yte i influenced by the ater yte according to a weighting criterion and a force feedback i rovided equally to all the ater yte. Baed on the theory of adative control and wave variable, the control of an uncertain ingle-aterulti-lave teleoeration yte i dicued in [] where each lave i ade to follow the ater coand in the reence of tie varying delay. A diturbance 8

3 Acta Polytechnica Hungarica Vol. 15, No. 4, 18 oberver-baed chee i reented in [3] to control a ulti-ater-ingle-lave teleoeration yte with different degree-of-freedo. In thi ethod, reaction force i etiated and a odal tranforation i introduced to accolih the oition and force tracking tak. A Lyaunov-baed aroach i reented in [4], [5] to deign a ultilateral controller for dual-ater-ingle-lave nonlinear teleoeration yte. he aroxiation ability of fuzzy logic control ha been ued in [6] to deign an adative controller for uncertain dual-ater-dual-lave teleoeration yte. hi aer reent the deign of a tie-delayed ulti-ater-ingle-lave nonlinear teleoeration yte baed on the ethod of tate convergence. In our earlier work [7], we have reented an extended tate convergence control architecture where k-ater yte can cooeratively control l-lave yte. However, thi extended tate convergence ethod i only alicable to linear teleoeration yte when the counication channel offer no tie delay. hee two liitation have been addreed in thi aer by conidering the nonlinear dynaic of ater/lave yte and contant ayetric counication tie delay. A Lyaunov-baed tability analyi i reented and control gain of the extended tate convergence ethod are elected to enure the tability of the ultilateral yte againt the counication tie delay and to achieve the zero tracking error of the lave yte. In order to validate the rooed chee, MALAB iulation are erfored on a tri-ater-inglelave nonlinear teleoeration yte. Modeling of Multilateral Nonlinear eleoeration Syte We conider a nonlinear ultilateral teleoeration yte which i coried of n-degree-of-freedo -ater and ingle-lave aniulator a: M, q q C q q q g q Fh, 1,,..., (1) M q q C q, q q g q Fe () nn Where M, M, nn n1 C, C and g, g rereent inertia atrice, corioli/centrifugal atrice and gravity vector for ater/lave yte n1 n1 reectively. Alo, q, q, q, q n1, q, q, n1 n1, and Fh, Fe rereent the oint variable of ater/lave 9

4 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte aniulator naely oition, velocity, acceleration, torque and external force ignal, reectively. In the equel, the following roertie of the ater/lave aniulator (1)-() will be utilized in roving the tability of the cloed loo teleoeration yte: Proerty 1: he inertia atrice are yetric, oitive definite and bounded i.e., there exit oitive contant l and u uch that I M q I. l u Proerty : A kew-yetric relation exit between the inertia and x M q C q q x x corioli/centrifugal atrice uch that,, n. Proerty 3: he corioli/centrifugal force vector are bounded i.e., there exit oitive contant f Proerty 4: If the oint variable q uch that C q, q q f q of corioli/centrifugal atrice i alo bounded.. and q are bounded, then the tie derivative In addition to the above roertie, we ake the following aution: Aution 1: he gravity force vector for the ater/lave aniulator are aued to be known. Aution : he oerator are aued to be aive i.e., there exit oitive contant, 1,,..., uch that Fh q dt. Alo, the environent i aued to be aive and i odeled by a ring-daer yte i.e., Fe Keq Be q nn nn where K e and B e are oitive definite diagonal atrice. In addition to the above roertie and aution, we will ue the following lea in roving the tability of the ultilateral teleoeration fored by (1), (): n Lea 1: For any vector ignal xy, and calar, tie delay oitive definite atrix K nn t f and, the following inequality hold over the tie t f t f t f interval [, t f]: Lea : For the vector ignal x K y t d dt x Kxdt y Kydt x 1/ inequality hold: x t x t x t d x n and the tie delay, the following 1

5 Acta Polytechnica Hungarica Vol. 15, No. 4, 18 3 Modified Extended State Convergence Architecture he author recently rooed an extended-tate convergence architecture [7] for k-ater-l-lave delay-free linear teleoeration yte, which can be odeled on tate ace. he ai of the reent tudy i to exlore the alicability of the extended tate convergence architecture for nonlinear ultilateral teleoeration yte in the reence of ayetric contant counication delay. For ilicity, a ulti-ater-ingle-lave nonlinear teleoeration yte i conidered in thi aer which can be oberved in literature. Further, the extended tate convergence architecture i lightly odified by eliinating the gain ter which are reonible for direct traniion of oerator force to lave G i yte. However, all other control gain ter are ket the ae. Since the gravity force vector are aued to be known, they are included in torque inut and will therefore becoe art of the extended tate convergence architecture. he odified tate convergence architecture i hown in Figure 1 and variou araeter defining the architecture are decribed below: F h : It rereent the force exerted by the th oerator onto the th ater aniulator. Figure 1 Modified extended tate convergence architecture for ulti-ater-ingle-lave teleoeration 11

6 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte K K K 1 aniulator. K 1 : It rereent the tabilizing feedback gain for the th ater nn and K nn ater oition and velocity ignal reectively. K K K 1 aniulator. are the tate feedback gain for the th : It rereent the tabilizing feedback gain for the lave K 1 nn and K nn lave oition and velocity ignal reectively. are the tate feedback gain for the : It rereent the contant tie delay in the counication link connecting the th ater yte to the lave yte. : It rereent the contant tie delay in the counication link connecting the lave yte to the th ater yte. R R 1 R : It rereent the influence of the otion ignal generated by the th ater aniulator (when oerated by the th huan oerator) into the lave aniulator. he atrice nn R 1 and nn R weight the th ater aniulator oition and velocity ignal reectively. R R 1 R : It rereent the effect of lave otion into the th ater aniulator. he lave oition and velocity ignal are weighted by the atrice nn R 1 and nn R to influence the th ater aniulator otion. 4 Lyaunov-baed Stability Analyi and Control Deign Uing the odified extended tate convergence architecture of figure 1, we intend to achieve the following control obective: Control Obective 1: he lave aniulator otion i the weighted effect of the ater aniulator otion i.e., liq t q t t 1 Where (3) i a weighting factor which i ued to cale the th ater aniulator otion and obey the roerty:

7 Acta Polytechnica Hungarica Vol. 15, No. 4, 18 Control Obective : he tatic force reflected onto the th oerator i a function of the environental force and the other oerator alied torque i.e., i, i, i 1,,..., h e h F f F, F (4) o achieve the above obective and to how the yte tability, tate convergence-baed cloed loo ulti-ater-ingle-lave nonlinear teleoeration yte will be analyzed uing a Lyaunov-Kraovkii functional technique. oward thi end, we firt write the control inut for the ater/lave yte uing Figure 1 a: g q K 1q K q R 1q t R q t, 1,,..., (5) 1 1 (6) g q K q K q R q t R q t 1 1 By lugging (5) in (1) and (6) in () and uing the aution and 3, the cloed loo ater/lave yte can be given a: M q C q K 1q K q R 1q t R q t F, 1,,..., h (7) (8) M q C q K q K q R q t R q t F 1 1 e Poition Coordination Behavior We now how the tability analyi of the cloed loo teleoeration yte fored by (7) and (8) by introducing the following theore: heore 4.1: By electing the control gain of the ulti-ater-ingle-lave teleoeration yte (7), (8) a in (9) and on the atifaction of the 1 inequalitie a in (1), the tability of the cloed loo teleoeration yte of (7), (8) can be deontrated a in (11). K K K, K K 3 K, 1,,..., R R K, R R K, 1,,..., (9) Where K nn nn and K1 are oitive definite diagonal atrice. 13

8 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte 3 K1 K K, 1,,..., K1 K K 1 1 (1) Where, are oitive calar contant. li q li q li q li q, 1,,..., (11) t t t t Proof: Conider the following Lyaunov-Kraovkii functional: V q q q q q q q M q q M q q K q 1,,,, e 1 1 h e 1 1 t 1 q q K q q q K q d t 1 t t q Kq q F d q F d t q K 1 q d t (1) By taking the tie-derivative of (1) along the yte traectorie defined by (7) and (8), and uing the aivity aution along with the roerty of the robot dynaic, we have: V q K1q K q R 1q t R q t 1 q K1q K q R 1q t R q t Keq Be q e 1 1 q K q q Kq q K q q q K q q q K1q q t K1q t q K1q q t K1q t 1 1 (13) After ilifying and grouing the ter in (13), we get: 14

9 Acta Polytechnica Hungarica Vol. 15, No. 4, V q K K q q R q t Kq q K1 K q q R1q t Kq 1 1 q K K1 q q K K1 q q Be q q R q t q R q t q t K q t q t K q t 1 1 (14) By lugging the control gain of (9) in (14), and by adding and ubtracting the ter q K1 q, q K1 q fro (14), we have: V q K q t q q K q t q e q K q q K q q B q 1 q K1 q q t K1 q t q K1 q t 1 q K1 q q t K1 q t q K1 q Now, we define the following error ignal: q q e q q t e q q t t (15) (16) By re-writing (15) in ter of the tie-derivative of the error ignal of (16) and uing the relation q t q t q t d, we have: 15

10 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte 1 1 V q K q t d q K q t d q 3 K1 q q K1 q q Be q e K1 e q q e K1 e q q (17) By integrating (17) over the tie interval [, t f] and uing lea 1, we get: V d q K q d q K q d t f t f t f 1 q K q d q K q d tf tf 1 t t t e 1 f f f q K q d q K q d q B q d tf tf e K q 1 e q 1 q q 1 1 d e K e d (18) By grouing the ter in (18), we can ilify the bound on the tie-derivative of the Lyaunov- Kraovkii functional a: V t V 3 K K K q f in 1 1 in K1 K K q in K1 e q 1 1 K e B q in 1 q in e 1 (19) Now, if the inequalitie in (1) are atified, the right hand ide of (19) will reain negative. Since V() and V(t f) are oitive and right hand ide of (19) i negative, it can be concluded that V(t f)-v() reain bounded enuring that V(t f) will reain bounded. aking the liit a t f and uing the robot roertie, it can be aid that the ignal q, q, q q, q, q L and q, q, e, e L q q. o rove the yte tability in the ene of (11), we have to how that the acceleration ignal of ater/lave yte and their tie derivative reain bounded. o thi end, we rewrite (7) and (8) without external force (ince they are aued to be aive and bounded) a: 16

11 Acta Polytechnica Hungarica Vol. 15, No. 4, 18 1 q M C q K 1q K q R 1q t R q t () (1) 1 q M C q K 1q Kq R 1q t R q t 1 1 By conidering the control gain of (9) along with () and (1), it i now required to how that the ignal q q t, q q t L. hee 1 ignal can be written a: q q t q q q q t q q t q q q q t he firt et of arenthee on the right hand ide of () are bounded by virtue of q, q, q q L while the econd et of arenthee are bounded by virtue of lea and q, q L. hi ilie that the left hand ide of () are alo bounded. By uing the roertie 1 and 3 of the robot dynaic and the reult q, q, q q, q, q, q q t, q q t L, it can 1 be concluded that the ignal q, q are bounded. Since the ignal q, q alo belong to, then by Barbalat lea: L li q li q li e li e. Now, it i left to how the boundedne of t t t q t q the tie derivative of () and (1) to colete the roof. By taking their tie derivative, we have: dq d M C q K q K q R q t R q t dt + dt d dt 1 M C q K 1q K q R 1q t R q t () (3) 17

12 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte dq d M C q K q K q R q t R q t dt dt M d C q K q K q R q t R q t dt 1 1 By uing the roertie 3 and 4 of the robot dynaic and the earlier reult q, q, q q, q, q, q q t, q q t, q, q L, 1 it can be concluded that the econd derivative ter in (3) and (4) are bounded. he boundedne of the firt derivative ter in (3) and (4) follow fro the roertie 1 and of the robot dynaic, the boundedne of the ignal q, q, q, q M M M M M C C M 1. hu and conidering the right hand ide of (3) and (4) reain bounded ilying that the ignal q, q L are uniforly continuou. herefore, we have: li q li q. t t hi colete the roof. heore 4.: he deired oition of the lave yte, a entioned in (3), i achieved under the control gain of (9) and the condition that the ignal q, q, q, q, e, e q q converge to zero a t. Proof: he convergence of the ignal q, q, q, q, e, e q q ha been hown in heore 4.1. hu, by uing the reult fro heore 4.1 and by ubtituting the control gain of (9) in (8), we have: 1 (4) li q q t (5) t By uing the relation we can write (5) a: t t q t q q d, and the reult li q, q 1 t li q. hu, lave oition coordination i t achieved in the abence of environental force a tie goe to infinity and control obective 1 i achieved. hi colete the roof. 18

13 Acta Polytechnica Hungarica Vol. 15, No. 4, Force Reflection Behavior Let u now invetigate the force exerienced by the oerator in teady tate when the lave i in contact with the environent. oward thi end, the teady tate behavior of the cloed loo ater yte (when the velocity and acceleration ignal converge to zero) i firt found fro (7) a: K 1q R 1q t Fh, 1,,..., (6) By uing the relation tability analyi can be given a: li t q t t q t q q d and the earlier reult fro, and the control gain of (9), oerator force in (6) Fh K q q, 1,,..., (7) Siilarly, the behavior of the lave yte in teady tate including the environental force can be obtained fro (8), (9) and two earlier theore a: e 1 F Kq Kq (8) By adding and ubtracting 1 Kq force in ter of oerator force a: fro (8), we can write the environental F F Kq e h (9) Fro (9), it i evident that the environental force i indeed roortional to the weighted effect of the oerator force. hu, the econd control obective of (4) i alo achieved. 5 Siulation Reult he rooed tate convergence baed chee for ulti-ater-ingle-lave teleoeration yte i verified in MALAB/Siulink environent uing a two degree-of-freedo three ater and one lave aniulator with the dynaic of (1), (). he correonding inertia atrice, corioli/centrifugal atrice and gravity vector are given in (3)-(33): 19

14 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte 11 1 c c g M q, C q, q 11 1, g q 1 c c g 1 1 co co l l l q 11 1 l l q 1 1 l (3) (31) c q l q c q q l q 11 in, 1 1 in 1 1 c q l in q, c g a l in q q a l in q, g a l in q q (33) 1 g 1 g 1 1 g 1 Where 1, are the ae of link 1 and reectively; l1 l l are the length of link and ag i the acceleration due to gravity. By etting the araeter for the ater yte a: 1 kg, l 1 and the lave yte a: 1 1 kg, l 1 and by defining the alha influencing factor for the ater yte a: 1.5,.3, 3. and by etting the gaa value a unity and by auing the tie delay in the counication channel a: 1,.5,, we olve the inequalitie of (1) and ubequently elect the deciive oitive definite atrice a: K diag,1, K diag 4,. he control gain for the ulti-ateringle-lave nonlinear teleoeration yte can now be found uing (9). We firt iulate the tri-ater-ingle-lave nonlinear teleoeration yte under the control of contant oerator force and in the abence of environent force. 1 3 F t F, t 15, F 5 N, F N, F 3N which he huan force are h o o o o vanih after 15 ec. he reult for thi iulation are hown in Figure. It can be een that the ultilateral yte reain table, owing to the boundedne of the ignal and both the oint oition of the lave yte converge to the deired reference. We alo conider the realitic cae where the oerator force vary linearly with tie. he iulation reult for thi cae are hown in Figure 3. It can again be een that the lave oition ignal indeed follow the deired reference oition. Finally, we iulate the nonlinear teleoeration yte in the reence of environent force conidering the contant oerator force. he iulation reult for thi cae are deicted in Figure 4 where the lave coe in contact with the environent at t=15 ec. After the contact i ade, ater oition are reduced and the lave i unable to follow the et reference. hi i in line with the theoretical reult. he reduction in ater oition i due to the force reflected onto the ater yte by the lave yte when it i in contact (3)

15 Acta Polytechnica Hungarica Vol. 15, No. 4, 18 with the environent while the error in lave deired traectory i the reult of it direct interaction with the environent a can be een fro (8) Joint 1 Poition Signal of Maniulator(rad) Deired Slave Po Mater 1 Po Mater Po Mater 3 Po Slave Po ie(ec) (a) Joint Poition Signal of Maniulator (rad) Deired Slave Po Mater 1 Po Mater Po Mater 3 Po Slave Po ie(ec) (b) Figure ri-ater-ingle-lave nonlinear teleoeration yte with contant oerator force in free otion (a) Joint 1 oition ignal (b) Joint oition ignal 1

16 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte Joint 1 Poition Signal of Maniulator(rad) Deired Slave Po Mater 1 Po Mater Po Mater 3 Po Slave Po ie(ec) (a) Joint Poition Signal of Maniulator (rad) Deired Slave Po Mater 1 Po Mater Po Mater 3 Po Slave Po ie(ec) (b) Figure 3 ri-ater-ingle-lave nonlinear teleoeration yte with tie varying oerator force in free otion (a) Joint 1 oition ignal (b) Joint oition ignal

17 Acta Polytechnica Hungarica Vol. 15, No. 4, Joint 1 Poition Signal of Maniulator(rad) Deired Slave Po Mater 1 Po Mater Po Mater 3 Po Slave Po ie(ec) (a) Joint Poition Signal of Maniulator (rad) Deired Slave Po Mater 1 Po Mater Po Mater 3 Po Slave Po ie(ec) (b) Figure 4 ri-ater-ingle-lave nonlinear teleoeration yte in free lu contact otion (a) Joint 1 oition Concluion ignal (b) Joint oition ignal In thi work, the deign of a ulti-ater-ingle-lave nonlinear teleoeration yte in the reence of ayetric contant counication tie delay i reented, baed on the extended tate convergence theory. A Lyaunov-baed tability analyi i carried out to find the control gain for the odified extended 3

18 U. Farooq et al. State Convergence-baed Control of a Multi-Mater-Single-Slave Non-linear eleoeration Syte tate convergence architecture. he efficacy of the rooed chee i finally verified through iulation in the MALAB/Siulink environent by conidering a two degree-of-freedo, tri-ater-ingle-lave robotic yte. Future work involve the deign of tate convergence baed ultilateral nonlinear teleoeration yte in the reence of tie varying delay with exeriental validation. Acknowledgeent hi work wa uorted by Natural Science and Engineering Reearch Council of Canada and by Nova Scotia Graduate Scholar Progra. Reference [1] M. Ferre, M. Bu, R. Aracil, C. Melchiorri, C. Balaguer: Advance in elerobotic, Sringer, 7 [] P. F. Hokaye, M. W. Song: Bilateral teleoeration: an Hitorical Survey, Autoatica, Vol. 4, 6, [3] R. Muradore, P. Fiorini: A Review of Bilateral eleoeration Algorith, Acta Polytechnica Hungarica, Vol. 13, No. 1, 16, [4] J. Artiga, G. Hirzinger: A Brief Hitory of DLR Sace elerobotic and Force Feedback eleoeration, Acta Polytechnica Hungarica, Vol. 13, No. 1, 16, [5] R. Anderon, M. W. Song: Bilateral Control of eleoerator with ie Delay: IEEE ranaction on Autoatic Control, Vol. 34, No. 5, 1989, [6] G. Nieeyer, J. J. E. Slotine: Stable Adative eleoeration, IEEE Journal of Oceanic Engineering, Vol. 16, No. 1, 1991, [7] J. Ryu, D. Kwon, B. Hannaford: Stable eleoeration with ie-doain Paivity Control, IEEE ranaction on Robotic and Autoation, Vol., No., 4, [8] A. Hace, M. Franc: FPGA Ileentation of Sliding Mode Control Algorith for Scaled Bilateral teleoeration, IEEE ranaction on Indutrial Electronic, Vol. 9, No. 3, 13, [9] M. Boukhnifer, A. Ferreira: H Loo Shaing Bilateral Controller for a wo-fingered ele-microaniulation Syte, IEEE ranation on Control Syte echnology, Vol. 15, No. 5, 7, [1] S. Ila, P. X. Liu, A. El-Saddik: Bilateral Control of eleoeration Syte, Vol., No. 1, 15,. 1-1 [11] A. Suzuki, K. Ohnihi: Frequency Doain Daing Deign for ie Delayed Bilateral teleoeration Syte Baed on Modal Sace Analyi, 4

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