Multiestimation-Based Adaptive Control of Robotic Manipulators With Applications to Master Slave Tandems

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1 Multietimation-Baed Adaptive Control of Robotic Manipulator With Application to Mater Slave andem A. IBEAS and M. de la SEN Intituto de Invetigación y Dearrollo de Proceo Facultad de Ciencia Campu de Leioa Apdo. 644, Bilbao SPAIN Abtract: A multietimation-baed adaptive controller i deigned for robotic manipulator. Multietimation baed technique can improve the tranient repone of adaptive ytem by witching between the variou etimation cheme. he witching proce i ubject to a minimum dwell or reidence time in order to guarantee tability. hen, the multietimation cheme i applied to the teleoperation of robotic manipulator. he robot are controlled by uing an impedance type control, which i a typical architecture of control in teleoperation procee. Keyword: adaptive control, robot control, uperviion, witching, impedance control, teleoperation. Introduction Nowaday, robotic manipulator are enively ued in the indutrial field. he deire of a high-peed or a high-preciion performance for thi ind of mechanical ytem ha led to reearch into improved control ytem. hee high performance control ytem need, in general, the dynamical model of the robotic manipulator in order to generate the control input []. Robotic manipulator are highly nonlinear and coupled multivariable dynamical ytem. Modelling error may be preent due to the uncertainty in the inertia mae of the ytem or in the tatic and dynamic friction force at the manipulator joint. Moreover, the parameter of the ytem may not be nown in advance ince it may handle variable payload during the operation cycle. Adaptive control i an alternative for the deign of high performance robotic manipulator controller []. However, adaptive control cheme may lead to a poor tranient repone in term of reference output deviation due to a wrong initialization of the parameter. In thi wor, a multietimation-baed adaptive cheme i ued to improve the tranient repone of adaptive robotic manipulator controller by online electing the appropriate etimate and adaptive controller parameterization from a finite et of identification-adaptive controller parameterization pair. Multietimation-baed controller have been ued in [3,4] in order to improve the tranient performance of continuo and dicrete time linear adaptive ytem. In the general problem etting, there exit a et of parameter etimation algorithm running in parallel. At each time, each etimation algorithm give the etimate parameter vector of the ytem. A higher level witching tructure between the variou etimation cheme decide the etimation algorithm that parameterize the adaptive controller in real time. he witching tructure act a an intelligent upervior of the ytem deciding at each time the identifier that would lead to a performance improvement and parameteriing the adaptive controller with it. A minimum dwelling or reidence time between conecutive witching i tated in order to guarantee tability [3,4]. Multietimation-baed adaptive control improve the poor tranient repone of adaptive ytem due to an inadequate initialization of the etimation algorithm. In a econd tage, multietimation-baed technique are applied to teleoperation ytem. eleoperation ytem are widely tudied and implemented nowaday ince they can end the human reach and manipulation capabilitie to remote place or hazardou location. eleoperation can be found in a broad variety of application lie in pace and underea exploration, nuclear operation or medical application [5]. A typical teleoperation ytem i formed by two robot, namely the mater and the lave manipulator, the human operator and the remote environment. he aim of the teleoperation ytem i the remote control of the lave manipulator. hi ta ha two tage. he firt one i to decide the way in which the mater and the lave manipulator hare their information and the nature of thi, i.e., which variable are tranmitted between them: poition, force or both. Several approache have been given to deal with thi problem [6]. hu, the control ytem can be of poition-poition, force-force, or force-poition type. For example, the force-poition type cheme i uch that the mater robot end it poition to the lave robot and thi one end the force meaured in it end effector to the mater robot. he econd tage i the deign of the controller for each robot. Depite the diverity of approache given in the literature, it i poible to claify mot of then into two major categorie [7], namely:

2 Impedance Control [8,9]. Hybrid poition/force control [0]. In thi paper, an impedance control technique i choen for deigning the control ytem. A tranient repone improvement can be achieved by uing multietimation-baed technique a it i hown in computer imulation. l q Plant Decription A robotic manipulator with n degree of freedom i characterized by a et of n generalized coordinate q = [ q q q n ] which define the o-called joint pace. ypically, thee coordinate are defined by the angle between two conecutive lin of the inematic chain. he general dynamical equation of a generic robotic manipulator can be written in the joint pace a : τ = M( qq ) + V( qq, ) + Fq ( ) + Gq ( ) + J ( q) f () n n where M ( q) R i the o called inertia matrix, which it i n poitive definite, V( q, q ) R i a vector containing the Corioli n and centripetal term, Fq ( ) R repreent the tatic and n dynamic friction term, Gq ( ) R i the gravity vector, n r J R, where r i the number of degree of freedom of the manipulator in the ta pace, i the Jacobian of the manipulator, r f R ( r 6) are the ernal force (and torque) and n τ R i the joint torque vector. A uual, a upercript dot dq denote the time derivative, q. Equation () repreent a et dt of nonlinear differential equation. An important property of Equation () i that it can be repreented in a linear in the parameter form a: τ = W P + J f () n p where W ( qqq,, ) R i the o called regreion matrix, p P R, where p i the number of parameter, i.e., mae, length and friction parameter from Equation (). hi i an important and ueful property that will be ued in the adaptive control algorithm deign. he main purpoe of thi paper i to how how multietimation baed technique improve the tranient repone of adaptive ytem. hu, the two degree of freedom robotic manipulator hown in Figure i ued. Y l q Figure. Robot manipulator. hi manipulator i the implet one that ha all the element neceary for a repreentative decription of the problem. wo generalized coordinate are then needed q = [ q q] and are choen to be the angle defined in Figure. hi robotic manipulator alo ha two degree of freedom in the ta pace, i.e. x, y. hu, J R in thi cae. he dynamical model of the robot can be obtained by uing the Newton-Euler method or the Lagrange method from the analytical dynamic. he following aumption are made in order to obtain a dynamical model of the ytem: X Mae m, m of both element are concentrated in the dital point of each one. Inertia tenor are zero for both element, i.e., ma i uppoed to be concentrated and not ditributed. he robot bae i fixed to ground. here i no gravitational effect ince the manipulator i aumed to operate in a horizontal plane. Gravitational effect can be added to the model with minor difficulty although they have not been conidered in order to clarify the multietimation proce. Friction term will be modeled by a vicou friction term, proportional to q i, plu a Coulomb type friction, which i proportional to ign( q i ). By uing the above aumption, the dynamical model for the robot manipulator lead to the following dynamical matrice: lm + llmc + l ( m+ m) lm + llmc M ( q ) = lm + llmc lm V( q, q ) = mllq mllqq mllq [ ν q ign q ν q ign q ] Fq ( ) = + ( ) + ( ) where = in( q), c = co( q) and o on, i the typical notation in robotic. hee equation can be rewritten according () by chooing the following decription for the regreor matrix and the parameter vector:

3 [ m m ν ν ] P = lq W q 0 ignq ( ) 0 W = 0 W 0 q 0 ign( q ) where: W = l ( q + q ) + llc( q + q ) + lq llq llqq and W = ll c q + l ( q + q ) + ll q 3 Control Algorithm he control algorithm i baed on an impedance type control. hi algorithm conit on the following. he inematic model of a robot can be written in the form: y = hq ( ) where y define the vector of ta variable, containing the carteian poition and orientation, q denote the vector of joint variable and h ( ) i the direct inematic function that can be derived for any robot manipulator by uing well etablihed technique. he manipulator end-effector velocity in the ta pace i related to the joint rate via the Jacobian tranformation: y = J( q) q aing the time derivative, we obtain, at configuration at which the Jacobian i non-ingular that: y = J q + Jq q = J( q) ( a J q ) (3) where a y i the acceleration vector in the ta pace. he Jacobian i a quare matrix if the degree of freedom of the manipulator in the joint pace and in the ta pace are equal. If J i a non-quare matrix with full column ran with n r (non le articulation than degree of freedom in the ta pace), then the invere of the Jacobian ha to be replaced with the peudoinvere ( ) # J = J JJ in (3). Replacing the joint acceleration (3) in the robot dynamic (), we obtain: τ = M( q) J ( a J q) + V( q, q ) + F( q ) + G( q) + J f (4) If we chooe now the control law to be: τ = M( q) J ( w J q) + V( q, q ) + F( q ) + G( q) + J f (5) the carteian-pace robot dynamic ha been reduced to the decoupled form: y = a = w (6) hu, the manipulator mut avoid Jacobian ingularitie in order to maintain the control law (5) non-ingular. In [], a virtual pring baed method i ued to avoid ingularity proximity. It i typical in control algorithm involving teleoperation technique to develop an impedance baed control, [5,6]. he baic idea of thi method i to mae the robotic manipulator to behave a an impedance device. hi mean that under an ernal force f, we want the robot to behave a the ma-damped reference ytem decribed with the linear matrix differential equation: M y + By + K y = f (7) where the matrice M = M > 0, B = B 0, K = K 0 [ > 0 ( 0) tand for poitive definite (emidefinite) matrice] are choen to be diagonal in order to decouple the robot dynamic in the ta pace and mae the behavior of the ytem intuitive for the operator. In fact, a linear reference model for the manipulator i choen through Eq. (3) and (5) where the nonlinear effect are cancelled (6). aing the Laplace tranform in equation (7) with zero initial condition and maintaining the ame notation for function and their tranform except the argument t or, we obtain: K M + B + d() = () v F (8) which relate the deired velocity of the ytem to the ernal force. Equation (8) can be rewritten a: K - vd() = M+ B+ F() = Z () F () for all complex uch that the invere exit, where Z () i generically the o called impedance. he deired ta pace acceleration i obtained from (8) a: - ad() = vd() = Z () F () and taing the invere Laplace tranform: - ad() t = L Z () F () (9) Note that if given an ernal force f, we calculate the acceleration defined above while maing w = a d, then the robotic manipulator behave a the impedance device defined by the equation (7). hu, (9) i the reference acceleration for (6). 4 Single Etimation Algorithm When the robot parameter are unnown, parameter etimation ha to be ued. For the implementation of the control algorithm a well a the parameter etimation, a digital(dicrete) etimation algorithm will be ued. Denote the ampling period a. At each ampling time t = the joint torque, and joint poition, velocitie and acceleration are meaured. he etimated robot parameter value are initialized by a vector P ˆ0. At each ampling time, the parameter vector i updated by the following etimation algorithm: FW E P P (0) ˆ ˆ τ + = + c + WFW FW W F + = F ζ c + WFW F where ζ ( 0,] i the forgetting factor of the etimation algorithm, and F0 = F0 > 0. he calar free-deign parameter c ( 0, ) act modifying the adaptation rate. hi parameter 3

4 can be upervied by a cloed-loop performance algorithm in order to get a fine tuning of it []. he adaptation error E τ i defined a E = ˆ τ τ - τ where ˆτ i defined for all time a: ˆ ˆ ( ) ˆ(, ) ˆ( ) ˆ τ = M qq + V qq + Fq + Gq ( ) + J f () or ˆ ˆ τ = WP+ J f where ˆP i the etimate of P via (0) with analogy to the plant model (). Conidering Equation () and (), the adaptation error i reduced to: ˆ E ˆ M( ) J ( J ) Mˆ τ = τ τ = q a ( ) d q q q () which will be ued in the etimation algorithm. he adaptive control law i generated by ubtituting the etimated robot parameter in equation (5): ˆ ( ) ˆ(, ) ˆ( ) ˆ τ = M q J a J q + V q q + F q + G( q) + J f (3) ( d ) 5 Multietimation Scheme If the etimation algorithm tart running with an etimate parameter vector far away from the real plant parameter vector, then the tranient will have large deviation from the deired output reulting in a bad performance. We have choen a parallel multietimation cheme in order to improve the tranient repone of the adaptive ytem. he architecture of the adaptive ytem i decribed in Figure. Figure. Multietimation cheme. here exit N e etimation algorithm running in parallel. Each one, i different from another in what i concerned with the etimate parameter vector initialization and/or the ind of etimation algorithm and integrate the o called etimation cheme. It alo exit an adaptive controller uch that it i parameterized at each time intant by one etimation algorithm which i choen from the parallel multietimation cheme. Denote by c K = {,,..., Ne} the integer that define the etimation algorithm which parameterize the adaptive controller at the ampling time t =. A witching rule baed on the identification error E (i) τ ( i Ne ) of the N e identification algorithm, chooe at each ampling time the identifier that parameterize the adaptive control law defined in (3) via (0). he adaptation error at t = ξ, E i defined a: ξ (i) τξ i ( d ) ( Mˆ ˆ ˆ ˆ ) () i () ˆ (c ξ ) ˆ ( ) ˆ(c ξ ) (, ) ˆ (c ξ ) Eτξ = τ τ = M q J a J q + V q q + F ( q ) + ˆ(c ξ ) (i) (i) (i) (i) + G ( q) ( q) q+ V ( q, q) + F ( q) + G ( q) (4) where all the quantitie are taen in the ampling time tξ = ξ. A performance index baed on the above identification error, i propoed in order to evaluate the performance of each identifier with the aim of chooing through time the bet one, via appropriate witching, to parameterize the adaptive controller. he propoed performance index of the uperviion ha the form: () i ξ () i () i ( ) λ τξ ξ τξ ξ = NC J = E Q E (5) where N c > 0 i the horizon ize of the uperviory index (5) and Q ξ i an at leat poitive emidefinite matrix which allow to weight in a different way each robotic degree of freedom. Equation (5) i a meaure of the long term accuracy of each identification algorithm, where the forgetting factor of the performance index (5), λ ( 0,] (which may be in general ample dependent) etablihe the effective memory of the index. hi i the dicrete time verion becaue of the digital implementation of the etimation algorithm and control. Once the uperviory index ha been tated, the witching rule for the adaptive controller reparameteriation i obtained from the performance index (5) a follow. If the time interval (or the number of ample in the digital cae) pat ince the lat witching i maller than the reidence time (reidence number of ample), then the witching i not allowed and the ame etimation algorithm i ued to parameterize the adaptive controller. he reidence time interval, within no witching i allowed, ha to be repected becaue of tability conideration. he reidence time i ubject to a minimum lower-bound in order to guarantee cloed-loop tability, [4]. If the time interval paed ince the lat witching, i more or equal than the reidence time, then the witching i allowed. In thi cae, the identifier choen from the multietimation cheme i uch that it generate the minimum performance index (5) for all etimation algorithm available. If (eventually) two etimator poe the ame index (what i an event of zero probability), then that with the mallet poition in the ordered et K i choen. If π denote the lat ample in which a witching happened, the witching map c at a ample, ( > π ) can be tated a: c π < ND c = ( ζ ) ( r) j j = min { ζ J ( ) = min { J ( ) r K } π ND where N D (poitive integer) i the dwell (or reidence) number of ample where the witching proce between etimator i not allowed from the preceding witching. he reidence time i then given by D = N D S. Multietimation cheme allow to improve the tranient repone of the adaptive ytem a it i hown in the following imulation example. 4

5 6 Simulation Reult Simulation example in which the tranient repone i improved by uing multietimation compared with ingle etimation are now preented. hee example illutrate the uefulne of the propoed cheme. In order to compare the tracing efficiency performance of each control cheme, the following tracing performance index ha been propoed: [ ξ ξ ] [ ξ ξ ] J ( ) = q ( ) q ( ) K q ξ ( ) q ( ) (6) p de de ξ = where K ξ i a ymmetric (at leat) poitive emidefinite matrix. q ( ξ ) i the joint acceleration vector in the ampling intant tξ = ξ and q de ( ξ ) i the deired joint acceleration vector defined in the deired ta pace acceleration through equation (3). he parameter of the etimation algorithm are ζ = 0.95 and c = c = for all 0. Qξ = Kξ = I in the performance indice (5)-(6). N D = ample while = 0.00ec. he real plant parameter vector i P = [ ] and l = l = meter. he multietimation cheme i formed in thi cae by two etimator running in parallel. hee etimator are () initialized by P ˆ 0 = [ ] and () P ˆ 0 = [ ] while the adaptation 0 matrix i initialized by F0 = 0 I 6 6 in both cae. he controller i initialized by the firt etimation algorithm. All value are given in MKS unit. he conventional adaptive cheme compoed by a () ingle etimation algorithm i initialized by ˆP 0. he reference input, i.e., the ernal force f R, ha been choen to be a ine ignal for each degree of freedom in the firt example and a random ignal paed through a low-pa filter for the econd example. he reference model i the ame for each degree of v () f () = for freedom in the ta pace, i.e., di i ( ) i = x, y. he following imulation are obtained. Example : Figure 4. Multietimation cheme output. Figure 5. Performance index for both cheme. Figure 3. Single etimation output. Figure 6. Multietimation cheme witching map. 5

6 Example : he witching map i the ame in both example (ee Fig. 6) becaue the ytem witche to the better controller when the retriction impoed by the reidence time i fulfilled. Comparing imulation in Figure (3)-(4) and (7)-(8) above, it can be oberved that a tranient repone improvement can be achieved by uing thi ind of method. Firt, the performance index propoed (6) i maller in the multietimation cheme than in the conventional ingle-etimation one (ee Figure 5 and 9). Second, the plant output deviation from the deired output during the tranient are alo maller in the multietimation cae. he tracing performance improvement i achieved by the witching to a better controller parameterization when the reidence time allow the witching proce, a it can be oberved from the witching map (Fig. 6) and the tracing performance index (Fig. 5 and 9). Figure 7. Single etimation output. Figure 8. Multietimation cheme output. 7 Application to eleoperation a In thi ection, the multietimation-baed adaptive control cheme i applied to teleoperation procee. he robotic ytem i formed by two robot, namely, the mater and the lave robot. If only the mater motion and/or force are tranmitted to the lave, then the teleoperation ytem i called unilateral. If, the lave motion and/or force are tranmitted to the mater a well, then the teleoperation ytem i called bilateral [3]. A traditional claification of the teleoperation control architecture i baed on the information tranmitted between both manipulator. hu, the control ytem can be either of poition-poition, force-force, or force-poition type. From a theoretical point of view, it i alo poible to implement a poition-force control but it ha been little ued. here are two important propertie of the teleoperation ytem namely tability and tranparency. A every control algorithm, the cloed loop ytem mut remain table. he tranparency property i referred to the fidelity with which the mater manipulator reproduce force meaured in the lave robot end effector. he above teleoperation control algorithm have different propertie in what i concerned with tability and tranparency condition. he poition-poition type controller i the mot table one but it ha a poor tranparency. On the other hand, the force-force type algorithm i the leat table but it can achieve a greater tranparency. In thi wor, a force-force control cheme ha been choen to implement the teleoperation architecture [4]. he Figure 0 repreent the control algorithm chema. Figure 9. Performance index for both cheme. 6

7 Figure 0. eleoperation architecture. he force made by the operator i tranmitted to the lave robot and the force meaured by the lave robot in it end effector i tranmitted to the mater one, defining a bilateral control algorithm. For each robot, there exit a multietimation-baed adaptive controller a that defined in the above ection. Each controller i independent for each robot and they can be tuned eparately. Alo, the mater and the lave robot may potentially be of different type, i.e., they may have different power, different inematic configuration or even a different number of degree of freedom lie in the cae of teleoperation with redundant robot. are the ame a before for both manipulator. here are two etimation cheme for each robot initialized by ˆ () P0 m = [ ] ˆ () P0 m = [ ] ˆ () P0 = [ ] ˆ () P 0 [ = ] while the ingle () etimation cheme i initialized by ˆ () P 0 m and P ˆ 0 repectively. Note that ince the controller deign and the etimation parameter are the ame for the above Example and and for the mater manipulator of thi example, the above imulation can be interpreted a the mater robot behavior for thi cheme. hu, the above imulation in Section 6 are completed now with imulation concerning the lave robot a well. Multietimation adaptive controller i independent from one robot to another and thu, reidence time, number of etimation cheme and free deign parameter can be choen independently for each one. he following imulation reult are obtained: Example 3: 8 eleoperation Simulation Reult Simulation example for the teleoperation ytem in which the tranient repone i improved by uing multietimation compared with ingle etimation are now preented. he ame tracing performance index a above (6) i ued for evaluating the reult. he lave movement i uppoe to be free in the pace, i.e. it doe not intercept any obtacle. Alo, the operator impedance i aumed to be much maller than the mater impedance. In the preented imulation both robot (mater and lave) and their reference model are the ame, although in a real teleoperation ytem each robot may be of different power or even have a different inematic configuration. A multietimation-baed adaptive controller i deigned for each one. he parameter for each adaptive controller can be choen independently. In thi cae, the etimation algorithm parameter are the ame a before for both robot. he real parameter for each robot are (a ubcript m i ued for denoting the mater device and a ubcript i ued for denoting the lave robot) for the mater robot P and for the lave, m = [ ] [ ] P =. Note that the m parameter i different from one robot to another. hi fact can be interpreted a that the lave robot i puhing a ma load of value 35g guided by the mater robot. he reference input i the ame ine ignal a ued in the above Example and the reference model Figure. Single etimation output for the lave robot. 7

8 Figure. Multietimation output for the lave robot. Figure 3. Performance index for both cheme, lave manipulator. previou one. In the cae when the parameter of any of the manipulator change their value, a reet mechanim for the etimation algorithm mut be included in the multietimation cheme. If the input ignal to the plant i peritently exited, then all the etimate parameter vector converge to the ame limit which i the real plant parameter vector. hu, all the etimation algorithm become identical and the multietimation cheme degenerate to the ame identification algorithm repeated N e time. he reet mechanim carrie the etimate parameter vector to their original location and the adaptive matrix gain to their original value. It i aumed that change between different plant parameterization are far enough from each other in order to efficiently apply a time invariant identification algorithm between change to the globally time varying ytem. he reet algorithm i formed by two tage. he firt one conit on deciding when the ytem ha reached to a table regimen. When the ytem reache a tationary tate, the identification performance indice (5) do not change ignificantly. hu, teting whether thi indice are almot invariant, the permanent regimen can characterized. he econd tage conit on detecting when the parameter ha changed. For thi purpoe, the ame quality indice (5) are ued. When the ytem detect an abrupt change in the indice (5), once a table regimen ha been reached, it activate the reet mechanim. he following imulation have been obtained for the lave manipulator (the mater manipulator behavior i the ame a in Section 6 ince no force reflection i conidered ). Example 4: Figure 4. Switching map for the lave multietimation cheme. he multietimation baed adaptive controller improve the tranient repone of the adaptive ytem by the witching to a better controller during the tranient repone (Fig. and ). hu, the lave tracing performance i improved for the teleoperation ytem (ee Fig. 3). he etimation controller parameterization witching map i the ame a in the above example concerning the mater robot (Fig. 6) ince the witching logic witche to the bet controller parameterization according to the bet identification performance (5) when the reidence time contraint i fulfilled. Since the reidence time i the ame for both multietimation cheme, the witching proce occur at the ame ampling intant. In the following Example 4, the lave m parameter change abruptly in time. At the time t = 0.5ec the real plant parameter vector change to P = [ ]. hi can be interpreted a that lave robot tart puhing a new ma load added to the Figure 5. Single etimation output for the lave with abrupt change. 8

9 between parameterization far enough one from each other, the multietimation cheme improve the ytem behavior by accommodating the current controller to the new ituation. Acnowledgement: he author are grateful to MEC and to the UPV-EHU by it partial upport of thi wor through project DPI and UPV /UPV/EHU 00I06.I06.30 EB 835/000. A. Ibea i very grateful to MECD by it upport of thi wor through the FPU grant P Figure 6. Multietimation cheme output for the lave with a reet mechanim included. Figure 7. Switching map for the lave multietimation cheme. When the parameter of the robot manipulator change, the reet mechanim i activated and the multietimation cheme witche to the adequate controller (Fig. 7) improving the ytem behavior (ee Fig. 5 and 6). 9 Concluion In thi paper, a multietimation-baed adaptive controller ha been deigned for robotic manipulator. hi ind of technique can improve the tranient repone of the adaptive ytem by the appropriate witching between controller parameterization form the parallel multietimation cheme. he witching proce typically reduce plant output deviation during the tranient repone through the implementation of the witching operation between appropriate identifier and aociate controller parameterization. hen, the multietimation cheme ha been applied to the cae of teleoperation ytem a well.an impedancetype control ha been deigned in order to mae the robotic mechanical ytem to behave a an impedance type device. When the plant parameter change abruptly through time with change Reference: [] M.W. Spong, F.L. Lewi and C.. Abdallah, Robot Control, IEEE Pre, 993. [] J.J. Craig, Adaptive Control of Mechanical Manipulator. Addion-Weley Reading, MA. [3] K.S. Narendra, J. Balarihnan, Improving tranient repone of adaptive control ytem uing multiple model and witching. IEEE ran. on Aut. Contr. Vol. 39, nº 9, pp , 994. [4] A. Ibea, M. de la Sen and S. Alono-Queada, Adaptive Stabilization of dicrete linear ytem via a multietimation cheme. Inter. Math. J. Vol. 3,nº, 4, pp , 003. [5].B. Sheridan. elerobotic, Automatica, vol. 5, no. 4, pp ,989. [6] D.A. Lawrence. Stability and ranparency in Bilateral eleoperation. IEEE ran. Robotic and Automation, vol. 9, no.5, pp , 993. [7] C.C. Cheah and D. Wang. Learning Impedance Control for Robotic Manipulator. IEEE ran. Robotic and Automation, vol. 4, no. 3, pp , 998. [8] N. Hogan, Impedance Control: An Approach to Manipulation, Part I-III, Jour. of Dynamic Sytem, Meaurement and Control, Vol. 07, pp. -4, March, 985. [9] K. Hatrudi-Zaad and S.E. Salcudean, Analyi of Control Architecture for eleoperation Sytem with Impedance/Admittance Mater and Slave Manipulator. Inter. Jour. of Rob. Re. Vol. 0, nº 6, pp , 00. [0] M.H. Raibert and J.J. Craig, Hybrid poition/force control of manipulator. ASME J. Dyn. Syt. Mea. Contr. vol. 0, pp. 6, 98. [] E. Sánchez, A. Rubio and A. Avello. An Intuitive Force Feedbac to Avoid Singularity Proximity and Worpace Boundarie in Bilateral Controlled Sytem Baed on Virtual Spring. Proc. Of the IEEE Inter. Conf. on Intelligent Robot and Sytem. Vol., pp , 00. [] M. de la Sen and A. Almana, Adaptive control of manipulator with uperviion of the ampling rate and free deign parameter of the adaptation algorithm. Cybernetic and Sytem, Vol. 33, pp , 00. [3] W. H. Zhu and S. E. Salcudean. Stability Guaranteed eleoperation: an Adaptive Motion/Force Control Approach. IEEE ran. on Automatic Contr. vol. 45, no., pp , 000. [4] H. Kazerooni,. ay and K. Hollerbach. A Controller Deign Framewor for elerobotic Sytem. IEEE ran. on Cont. Sy. ech. Vol., no.,