Research Article An Adaptive Regulator for Space Teleoperation System in Task Space

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1 Abtract and Applied Analyi, Article ID 586, 7 page Reearch Article An Adaptive Regulator for Space Teleoperation Syte in Tak Space Chao Ge, Weiwei Zhang, Hong Wang, and Xiaoyi Li Intitute of Inforation Engineering, Hebei United Univerity, Tanghan, Hebei 639, China Correpondence hould be addreed to Chao Ge; gechao365@26.co Received 26 June 24; Accepted 2 July 24; Publihed 6 Augut 24 Acadeic Editor: Zheng-Guang Wu Copyright 24 Chao Ge et al. Thi i an open acce article ditributed under the Creative Coon Attribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original work i properly cited. The proble of the gravity inforation which can not be obtained in advance for bilateral teleoperation i tudied. In outer pace exploration, the gravity ter change with the poition changing of the lave anipulator. So it i neceary to deign an adaptive regulator controller to copenate for the unknown gravity ignal. Moreover, to get a ore accurate poition tracking perforance, the controller i deigned in the tak pace intead of the joint pace. Additionally, the tie delay conidered in thi paper i not only tie varying but alo unyetrical. Finally, iulation are preented to how the effectivene of the propoed approach.. Introduction A teleoperation yte can extend the huan-ening and anipulative capabilitie to the reote environent. Now theteleoperationytehabeenuedinanyarea,for intance, outer pace operation, robotic teleurgery, handling of toxic, and harful aterial, and underwater exploration [ 3]. In bilateral teleoperation yte, the ater and the lave anipulator are connected via a long counication network. So the tie delay cannot be ignored. It ha been noticed that the preence of tie delay can influence the yte perforance. And it can even detabilize a table yte. It i recognized that Anderon and Spong [4] propoed the firt delay independent controller for contant tie delay via the cattering tranforation. Later, the wave variable ethod wa introduced [5]. Then the wave variable ethod had been iproved by any paper [6 8]. However,itinecearytorecognizethefactthattheclaic catteringtranforationayleadtopoitiondrift.thena proportional derivative plu daping (PD+d) controller wa propoed in [9] and the cattering tranforation approach wa dicarded. Later, a iple proportional plu daping (P+d) controller wa propoed in []. However, the tie delay conidered in thi paper wa contant. In [], the odel-free linear oberver-baed PD output feedback control deign wa ued to olve the output feedback tabilization for indutrial robotic yte. Reference [2]preented a hybrid control trategy for the trajectory tracking control proble of robotic anipulator. The ue of Internet for teleoperation counication provide obviou benefit. Neverthele, the tie delay caued by Internet counication channel i not only tie varying but alo unyetrical. Tie delay ha becoe an unavoidable factor affecting the tability of cloedloop yte [3 7]. Thank to [8, 9], a iple P+d controller can enure poition tracking and ayptotic convergence to zero of velocitieinthepreenceofvariabletiedelay. Coparing with other control ethod, the PD controller ha irreplaceable advantage. Owing to it iple tructure and eay ipleentation, it ha been ued extenively. However, in [9 2], it i eay to ee that an auption wa needed before deigning the controller, in which the gravity inforation can be achieved in advance. However, in any practical application, it i a eriou retriction for the application of PD controller in teleoperation yte. Moreover, thi iue arie for the teleoperation application oftherobotintheouterpace.inouterpace,different

2 2 Abtract and Applied Analyi location have different gravity acceleration. However, with oreandoreadvancedtechnology,itipoibletodetect thegravity.itiexpeniveandaytakealongtieandit cannot adapt to the requireent of real-tie control. So an adaptive regulator i neceary. In the exiting literature about the teleoperation yte, ot controller were deigned in joint pace. And the objective of ot paper were to deign a controller to ake the joint poition error between the ater and the lave convergence to zero. However, the ultiate ai of uing teleoperation i to ake the lave coplete a certain job under the control of the huan operator. And the job i copleted by the terinal actuator. It i neceary to notice the phenoenon where the ater and the lave anipulator have different joint length and joint a and even different dynaic odel ay exit. Under thi phenoenon, the joint ynchronization cannot guarantee the terinal actuator ynchronization. Thi ituation can be avoided with the controller deigning in the tak pace [2]. In oe literature for pace teleoperation [2 24], the gravity ter wa directly neglected when the gravity acceleration wa zero. In thi paper, we conider the ituation that gravity inforation cannot be obtained. An adaptive regulator i added in the typical P+d controller to copenate for the unknown gravity ignal. Moreover, to ake the lave anipulator coplete it job accurately under doination of the huan operator, the controller i deigned in tak pace. Conider the Internet counication channel; the tie delay conidered in our paper i tie varying and ayptotical. Thi paper i organized a follow. Section 2 preent the odeling fraework and the related propertie and auption. The controller i propoed and the tability analyi i deprived in Section 3. InSection 4, oe iulation reult are applied to prove the effectivene of the approach. Finally, oe concluion will be provided and oe future work will be entioned. 2. Proble Forulation Conider a ater-lave bilateral teleoperation yte odeled a a pair of n-degree-of-freedo (DOF) erial link with revolute joint. The nonlinear dynaic are preented a M (q ) q +C (q, q ) q +G (q )=τ J T (q )F h, M (q ) q +C (q, q ) q +G (q )=J T (q )F e τ, where repreent the ater anipulator and repreent the lave anipulator. q (t), q (t) R n are the vector of the joint diplaceent; q (t), q (t) R n are the vector of joint velocitie; q (t), q (t) R n are the vector of joint acceleration; M (q ), M (q ) : R n R n n are the poitive definite inertia atrice; C (q, q ), C (q, q ) : R n R n R n n are the atrice of centripetal and corioli torque; G (q ), G (q ) : R n R n are the gravitational torque; F h, F e R n are the huan operator force and the environent force, repectively; J (q ), J (q ) R n n are the () Jacobian atrice for the ater anipulator and the alve anipulator; τ,τ R n aretheappliedtorque. Here the following auption for Jacobian atrix J i (q i )(i=,)i introduced. Auption. The Jacobian atrix J i (q i )(i=,)hould be noningular. Reark 2. A we all know, in reality indutry production, the ingularity proble ay exit in the Jacobian atrix. Then the J i (q i ) and the torque of yte will tend to infinity when there exit ingularity in Jacobian atrix. Thank to the new technology, thi ituation can be avoided by intitutional deign and trajectory planning. So thi auption in thi paper i reaonable. Itiwellknownthattheworkpaceofanipulatorcanbe tranfored fro joint pace to tak pace with a noningular Jacobianatrix.Sowehave x i =h i (q i ), x i =J i (q i ) q i, x i = J i (q i ) q i +J i (q i ) q i, where x i i the joint poition in tak pace and i=,. For the ake of iplicity, the arguent of function areeliinatedhereafter.andtheaterandlaverobot dynaic in the tak pace are preented a follow: where M t x +C t x +G t =J T τ F h, M t x +C t x +G t =F e J T τ, M t =J T M J, M t =J T M J, C t =J T C t =J T G t =J T G, G t =J T G. (C M J (C M J J )J J )J Below, fundaental propertie of the ater and lave anipulator needed in the ubequent analyi are preented. Property. The inertia atrix M ti (q) i poitive definite function, and there exit poitive contant and 2 uch that,, (2) (3) (4) I M ti (q) 2 I. (5) Property 2. The atrix M ti (q) 2C ti (q, q) i kew-yetric. Property 3. For all q i,x,y R n, there exit a poitive calar α i uch that C i (q, x)y α i x y.

3 Abtract and Applied Analyi 3 Property 4. The gravity function G t and G t canbelinearizedandcanbeexpreedag t (q ) = Y a and G t (q ) = Y a,wherey and Y are known nonlinear function and uually are called regreor atrice, while a and a are unknown vector. The following tandard auption are ipoed on the teleoperation yte. Auption 3. The variable tie delay ha an upper bound T i ;thati, T i (t) T i, i=,. Auption 4. Following tandard conideration, we aue that the huan operator and the environent are paive (force to velocity) ap; that i, there exit calar κ i R + uch that for all t t x T F hdσ κ, t x T F edσ κ. (6) Before propoing the controller, the following lea i needed. Lea 5. For any vector ignal x, y, any variable tie delay T i (t) T i, and contant α>,oneha t 2 x T (σ) y (σ+θ) dθ dσ α x T2 i T i (σ) α y 2 2. (7) 3. Controller Deign In original reearche, little attention wa paid to the gravity ter; oreover, they even ade the auption that the gravity can be accurately obtained in advance. However, in any cae, the gravity not only cannot be accurately obtained directly, but alo cannot be eaured. Thi iue arie for the teleoperation application of the robot in the outer pace. It i well known that the gravity coefficient g i differentfortheearth,outerpace,andtheoon.theabove two iue ay render that the precie value of paraeter g i not known for teleoperation. In thi ection, we conider thi proble and propoe a new controller cobining a typical P+d controller and an adaptive regulator. Without the accurate gravity inforation, the new direct force-feedback controller cobined with the etiate gravity inforation i deigned a follow: τ =J T [K (x (t T (t)) x ) B x + G t ], τ =J T [K [x x (t T (t))]+b x G t ], where K, K, B,andB are poitive contant calar and the G t and G t are the etiation for G t and G t,repectively. In addition, they are decribed a follow: (8) where a (t) and a (t) are the adaptive atrice with the tuning law a a (t) = Γ Y x T, () a (t) = Γ Y x T, where Γ and Γ are poitive definite atrice. Theore 6. For the teleoperation yte () controlled by (8) with Auption 4,the cloed-loop yte i table with the tuning law () and the following condition hold: 2B B >K K (T 2 + T2 ). () Andthevelocityandpoitionerrorarebounded(i.e., x i,x x l )andx i l 2 ;oreover x x (t T (t)) l. Proof. Let u conider the Lyapunov function candidate: V = 2 xt M t x + K x T 2K M tx + K 2 (x x ) 2 t + ( x T F h K x T F e)dσ+k + K k, K K (2) V 2 = 2 at Γ a + K a T 2K Γ a, (3) where a = a a and a = a a. Property and Auption 4 guarantee that the function V i poitive definite and it i obviou that V 2 i alo poitive definite with the poitive Γ and Γ. The derivative of the above function V i preented a follow: V = x T ( C t x G t +J T τ F h )+ 2 + K K x T ( C tx G t +F e J T τ )+ 2 xt M t x xt M t x +K ( x x )(x x )+ x T F h K x T F e; K (4) with Property 2 and the new controller () the ubequent equationcanbegivena V = x T [K (x (t T (t)) x ) B x + G t ] + K x T K [K [x (t T (t)) x ] B x + G t ] +K ( x T xt )(x x ) = x T B x K x T B x + x T K (x (t T (t)) x ) K G t =Y a, G t =Y a, (9) + x T K (x (t T (t)) x )+ x T G t + K x T G t, K (5)

4 4 Abtract and Applied Analyi where G t = G t G t =Y ( a a ) and G t = G t G t = Y ( a a ). With the following equation: we can get x i (t T i (t)) x i (t) = T i (t) V = B x 2 K B K x 2 K x T K x T T (t) x i (t+θ) dθ, (6) T (t) x (t+θ) dθ x (t+θ) dθ + x T G t + K x T G t. K (7) The derivative of the above function V 2 i given a follow: V 2 = a T Γ a + a T Γ with the adaptive tuning law () yielding a (8) V 2 = a T Γ ( Γ Y x )+ K a T K Γ ( Γ Y x ) = (Y a ) T x K (Y K a ) T x, Tie delay () Tie () 3 2 T (t) T (t) Figure : The tie delay. V= V + V 2 = B x 2 K B K x 2 K x T K x T T (t) T (t) x (t+θ) dθ x (t+θ) dθ. (9) Huan inert force (N) 2 Now let u invoke Lea 5 to obtain a bound on the integral of V. Towardthiend,integratetheaboveequation fro to t, yielding V (t) V() B x 2 K B K x Force Force 2 Tie () Figure 2: The huan inert force. + K β 2 x 2 + K T 2 2β x 2 + K β 2 = (B K β 2 x 2 + K T 2 2β x 2 K T 2 ) 2β x 2 ( K B K K β K T 2 ) 2 2β x 2. (2) Reark 7. In thi paper, we conider the ituation that gravity inforation cannot be obtained. An adaptive regulator i added in the typical P+d controller to copenate for the unknown gravity ignal. Reark 8. In order to ake the lave anipulator coplete it job accurately under doination of the huan operator, the controller i deigned in tak pace. Conider the Internet counication channel; the tie delay conidered in thi paper i tie varying and ayptotical. Thu, atifying condition (2) with the nonnegativity of V, x i l 2 canbeproved;oreover, x i,x x l and x x (t T (t)) l canbeproved. Reark 9. Fro the above proof, we can ee that the adaptive tuning law i derived in the paper. Therefore, we can obtain le conervative reult by optiizing thee paraeter.

5 Abtract and Applied Analyi 5 Poition () Tie () Poition () Tie () x x x 2 x x x 2 Figure 3: The poition of the ater and the lave in tak pace with the known gravity ter. Figure 5: The poition of the ater and the lave in tak pace with adaptive regulator Poition ().4.2 Poition () Tie () Tie () x x x 2 x x x 2 Figure 4: The poition of the ater and the lave in tak pace without adaptive regulator. Figure 6: The poition of the ater and the lave in tak pace with the known gravity ter. The nuerical exaple given in the next part how the effectivene of our reult. 4. Siulation In order to how the effectivene of the propoed chee, in thi ection oe iulation are preented in which the local and reote anipulator are odeled a a pair of 2 DOF erial link. where M (q) = M (q) = [ M M 2 M 22 ], C (q, q) = C (q, q) = [ C C 2 ], C 2 C 22 G (q) = G (q) = [ G G 2 ] (22) M (q ) q +C (q, q ) q +G (q )=τ J T (q )F h, in which M (q ) q +C (q, q ) q +G (q )=J T (q )F e τ, (2) M = (2l co (q 2 )+l 2 )l 2 2 +l 2 ( + 2 ), M 2 =l l l 2 2 co (q 2 ),

6 6 Abtract and Applied Analyi.2 The anipulator Jacobian atrix J i (q i ) apping fro joint pace to tak pace i given a Poition () Tie () in which J i =[ J J 2 J 2 J 22 ] (24) J = l in (q ) l 2 in (q +q 2 ), J 2 = l 2 in (q +q 2 ), J 2 =l co (q )+l 2 co (q +q 2 ), J 22 =l 2 co (q +q 2 ). (25) x x x 2 Figure 7: The poition of the ater and the lave in tak pace without adaptive regulator. Poition () x Tie () x x 2 Figure 8: The poition of the ater and the lave in tak pace with adaptive regulator. M 2 =M 2, M 22 =l 2 2 2, C = l l 2 2 in (q 2 ) q 2, C 2 = l l 2 2 in (q 2 )( q + q 2 ), C 2 =l l 2 2 in (q 2 ) q, C 22 =, G =g( 2 l 2 co (q +q 2 )+( + 2 )l co (q )), G 2 =g 2 l 2 co (q +q 2 ). (23) Moreover, with the detailed definition of a T, at,y,and Y, G t and G t can follow thi definition: G t =J T [ co (q +q 2 ) co (q ) g ][ 2 l 2 ]. (26) co (q +q 2 ) gl ( + 2 ) For iulation, in the ater anipulator we chooe the paraeter = kg, 2 = 7kg, l =.7, and l 2 =.5 andtheateranipulatoruuallyplacedin the laboratory; therefore, we can et the g = 9.8 / 2.The lave anipulator paraeter are = 7kg, 2 = 5kg, l =.5, l 2 =.4, and g =.2 / 2.Thecontroller paraeter are et a K = 2, B = 4, K = 4, B = 6, andγ =.. The initial joint configuration of the ater and lave are et a x () = [ ] T and x () = [ ] T. The iulation reult are ued to verify the following: when we ove the ater robot, doe the lave anipulator follow the ater anipulator? Baed on Theore 6, the delay are et a T (t) = in(t) and T (t) =. +.2 in(t). Figure how the tie delay we et in thi paper. The huan operation inerting force are preented in Figure 2. Figure 3 how the poition of x and x when the gravity ter can be obtained in advance and g = 5/ 2. Figure 4 how the poition of x and x without the adaptive regulator and g=5/ 2. Figure 5 how the poition of x and x with the adaptive regulator and g = 5/ 2. Figure 6 how the poition of x and x when the gravity ter can be obtained in advance and g=/ 2. Figure 7 howthepoitionofx and x without theadaptiveregulatorandg = / 2. Figure 8 how the poition of x and x with the adaptive regulator and g=/ 2. Fro Figure 3, wecaneethatthelaveovewith the ater when the gravity ter i known in advance. However, in Figure 4, withouttheadaptiveregulator,the lave doe not ove with the ater, pecially, in the x 2 direction. In Figure 5, the proble i olved by applying the new controller we propoed in thi paper. By coparing Figure 7 and 8, we alo can get the ae concluion.

7 Abtract and Applied Analyi 7 5. Concluion In thi paper, the cae that the gravity ter cannot be obtained i conidered in outer pace exploration. An adaptive regulator i propoed to copenate for the unknown gravity ter. By propoing the proper Lyapunov function, the adaptive tuning law i alo derived in our paper. Moreover, the iulation reult prove the effectivene of our controller. In thi paper, we conider that the Jacobian atrix i accurately known. However, in oe cae, the Jacobian atrix cannot be known. Therefore, in the future, we will conider uingadaptivetheorytoetiatetheunknownjacobian atrix. Conflict of Interet The author declare that there i no conflict of interet regarding the publication of thi paper. Reference [] C. Mitantiuk, K. Ohihi, and S. Katura, Etiation of action/reaction force for the bilateral control uing Kalan filter, IEEE Tranaction on Indutrial Electronic, vol.59,no., pp , 22. [2] I. G. Poluhin, X. P. Liu, and C. H. Lung, Stability of bilateral teleoperator with generalized projection-baed force reflection algorith, Autoatica, vol. 48, no. 6, pp. 5 6, 22. [3] Z. J. Li, X. Q. Cao, and N. Ding, Adaptive fuzzy control for ynchronization of nonlinear teleoperator with tochatic tie-varying counication delay, IEEE Tranaction on Fuzzy Syte,vol.9,no.4,pp ,2. [4] R. J. Anderon and M. W. Spong, Bilateral control of teleoperator with tie delay, IEEE Tranaction on Autoatic Control, vol.34,no.5,pp.494 5,989. [5] G. Nieeyer and J.-E. Slotine, Stable adaptive teleoperation, IEEE Oceanic Engineering, vol.6,no.,pp.52 62, 99. [6] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Tie Delay Syte, Control Engineering, Birkhäuer, Bael, Switzerland, 23. [7] P. Arcara and C. Melchiorri, Control chee for teleoperation with tie delay: a coparative tudy, Robotic and Autonoou Syte,vol.38,no.,pp.49 64,22. [8] G. Nieeyer and J. E. Slotine, Teleanipulation with tie delay, International Robotic Reearch, vol.23,no. 9, pp , 24. [9] D.LeeandM.W.Spong, Paivebilateralteleoperationwith contant tie delay, IEEE Tranaction on Robotic,vol.22,no. 2, pp , 26. [] E. Nuño, R. Ortega, N. Barabanov, and L. Baañez, A globally table PD controller for bilateral teleoperator, IEEE Tranaction on Robotic,vol.24,no.3,pp ,28. [] S. Ila and P. X. Liu, PD output feedback control deign for indutrial robotic anipulator, IEEE/ASME Tranaction on Mechatronic,vol.6,no.,pp.87 97,2. [2] S. Ila and P. X. Liu, Robut adaptive fuzzy output feedback control yte for robot anipulator, IEEE/ASME Tranaction on Mechatronic, vol. 6, no. 2, pp , 2. [3] R. Q. Lu, H. Y. Wu, and J. J. Bai, New delay-dependent robut tability criteria for uncertain neutral yte with ixed delay, the Franklin Intitute. Engineering and Applied Matheatic,vol.35,no.3,pp ,24. [4]R.Q.Lu,H.Li,andY.Q.Zhu, QuantizedH Filtering for Singular Tie-varying Delay Syte with Unreliable Counication Channel, Circuit, Syte, and Signal Proceing,vol. 3, no. 2, pp , 22. [5] B. Zhou, H. Gao, Z. Lin, and G. Duan, Stabilization of linear yte with ditributed input delay and input aturation, Autoatica,vol.48,no.5,pp ,22. [6] B. Zhou, Z.-Y. Li, and Z. Lin, Oberver baed output feedback control of linear yte with input and output delay, Autoatica,vol.49,no.7,pp ,23. [7] E. Y. Chen and W. X. Zheng, Stability analyi of tiedelay neural network ubject to tochatic perturbation, IEEE Tranaction on Cybernetic,vol.43,no.6,pp ,23. [8] E. N uo, L. Baãez, R. Ortega, and M. W. Spong, Poition tracking for non-linear teleoperator with variable tie delay, International Robotic Reearch,vol.28,no.7,pp.895 9, 29. [9] C. C. Hua and X. P. Liu, Delay-dependent tability criteria of teleoperation yte with ayetric tie-varying delay, IEEE Tranaction on Robotic,vol.26,no.5,pp ,2. [2] T. Naerikawa, Bilateral control with contant feedback gain for teleoperation with tie varying delay, in Proceeding of the 48th IEEE Conference on Deciion and Control held jointly with 29 28th Chinee Control Conference (CDC/CCC 9),pp , Deceber 29. [2]T.Iaida,Y.Yokokohji,T.Doi,M.Oda,andT.Yohikawa, Ground-pace bilateral teleoperation experient uing ETS- VII robot ar with direct kinethetic coupling, in Proceeding of the IEEE International Conference on Robotic and Autoation (ICRA ),vol.,pp.3 38,May2. [22] M. Nohi, M. Ando, and T. Bock, Contact tak by pace teleoperation uing force reflection of counication tie delay, in Proceeding of the IEEE International Sypoiu on Coputational Intelligence in Robotic and Autoation,pp.93 98, June 25. [23] X. Wang, W. Xu, B. Liang, and C. Li, General chee of teleoperation for pace robot, in Proceeding of the IEEE/ASME International Conference on Advanced Intelligent Mechatronic (AIM '8), pp , Xian, China, Augut 28. [24]N.Y.Lii,Z.Chen,B.Pleintinger,C.H.Bort,G.Hirzinger, and A. Schiele, Toward undertanding the effect of viualand force-feedback on robotic hand graping perforance for pace teleoperation, in Proceeding of the 23rd IEEE/RSJ 2 International Conference on Intelligent Robot and Syte (IROS ),pp ,October2.

8 Advance in Operation Reearch Advance in Deciion Science Applied Matheatic Algebra Probability and Statitic The Scientific World Journal International Differential Equation Subit your anucript at International Advance in Cobinatoric Matheatical Phyic Coplex Analyi International Matheatic and Matheatical Science Matheatical Proble in Engineering Matheatic Dicrete Matheatic Dicrete Dynaic in Nature and Society Function Space Abtract and Applied Analyi International Stochatic Analyi Optiization