A Robust Adaptive Friction Control Scheme of Robot Manipulators

Transcription

1 ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 A Robust Aaptive Friction Control Scheme of Robot Manipulators Jen-Shi Chen an Jyh-Ching Juang Department of Electrical Engineering National Cheng Kung University University Roa ainan aiwan juang@mail.ncu.eu.tw Abstract In this paper the tracing problem for robot manipulators in the presence of unnown friction an moeling uncertainties is consiere. A composite tracing control scheme is propose in which an aaptive friction estimation is use to estimate the etent of friction an a robust controller is then esigne to enhance the overall stability an robustness. It is shown that the esign of the friction estimator an that of the robust control gain can be conucte separately. Performance issues of the robust aaptive friction controller are illustrate in a simulation eample mae for a two egree of freeom planar robot manipulator. Keywors: Friction Compensation Robust Aaptive Control Robot Manipulators.. Introuction racing control of robot manipulators is the most common tas in robotic applications. However the resulting tracing performance is subject to nonlinearities such as frictions an moel uncertainties such as unnown loa. he presence of frictions an uncertainties imposes a severe performance limitation for robot manipulators leaing to phenomena such as tracing lags [] position errors [] sluggish response [3] stic-slip motions [3] [4] an in some cases limit cycles [5]. Although the ifficulty of moel uncertainty an friction force coul be somewhat mitigate by the inherent ynamic behaviors of manipulators [6] an eplicit account of these effects is epecte to result in improve performance. Compensation of moel uncertainties an frictions for robot tracing control has receive a consierable amount of attentions in the literature. he control of uncertain system is usually accomplishe using either an aaptive control or a robust control philosophy. In general the aaptive approach is applicable to a wie range of uncertainties but robust controllers are simpler to implement an easier to tune. Many aaptive controllers [7]-[3] for robot manipulators have been propose to eal with unnown parameters. However the use of an aaptive control algorithm may lea to a complicate implementation. On the other han the robust controllers evelope in [4]-[8] allow for unstructure time-varying uncertainties but they are unable to guarantee asymptotic tracing (even in the absence of isturbances). he nonlinear H controller esign relies on the solving of a positive efinite function from a nonlinear partial ifferential Hamilton-Jacobi inequality (HJI). In [7] the associate HJI equation for nonlinear H inverse-optimal control problem for Euler-Lagrange system is solve analytically. Similar approaches have been aopte in [9] []. However the aforementione robust control approaches o not aress the friction effect eplicitly. As a result the robust controller often results in a large feebac gain an ecessive control activity. he incorporation of friction moel in the esign process shoul allow the control gain to be ecrease. In [] a nonlinear controller synthesis proceure is evelope for position tracing control of friction mechanical manipulators a local solution is erive by means of a certain perturbation of the ifferential Riccati equation. In [] a nonlinear estimator with respect to Coulomb friction coefficient was propose. One salient feature of the approach is the use of a nonlinear observer which epens on the absolute value of the velocity to estimate the friction coefficient. he metho was then further analyze in []. he paper attempts to eten the Lyapunov function base algorithm as establishe in [] to nonlinear systems an aress the associate robustness issues. In particular nonlinear H control esign approach is evelope to aress the control esign problem for systems that are subject to parametric uncertainties Coulomb frictions an Stribec effects. he esign metho transforms the ifferential HJI equation into quaratic or linear matri inequalities. Conitions on matri inequalities are establishe for the synthesis of controller. he resulting controller is a composite controller that inclues aaptive friction estimation an robust feebac control. he propose scheme attempts

2 ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 to eplore the aaptive estimation approach to account for parametric friction an utilize the robust control approach to boun the effects ue to moel uncertainties an estimation errors. hrough such a coherent approach the challenging tass towar frictions an uncertainties can be simultaneously aresse uner the same framewor. he paper is organize as follows. An aaptive friction compensation scheme for the tracing problem is propose in Section. he composite control scheme combines the merits of friction estimation an feebac control. he approach is etene to account for uncertainties an Stribec effect in the system ynamics is shown in Section 3 leaing to a more complete approach in ealing with frictions an uncertainties. In Section 4 performance issue of the controller is illustrate in a simulation stuy mae for a two egree of freeom planar robot manipulator. Finally conclusions are given in Section 5. Aaptive Friction Compensation Scheme he classical moel of friction is a retaring force that is a nonlinear memory-less o function of velocity. In its simplest form the Coulomb friction force that opposing the motion is the prouct of the friction coefficient an sign of the velocity. In this section the tracing problem of the manipulator will be consiere to account for this type of friction effect. Consier an n -lin mechanical manipulator whose ynamical equation is given by [3] M( q) q + V( q q ) + G( q) + F( q ) = τ () where { } q= q is the vector of generalize n coorinates with elements τ is the control torque input vector n q M ( q ) is the inertia matri which is n symmetric an positive efinite for all q V( q q ) is the centripetal/coriolis torque Gq ( ) is the gravitational torque an F( q ) is the friction torque with components F( q ) = n acting inepenently on each joint. he esign goal is to fin the control law of τ or u such that the response tracs the esigne trajectory. o this en a moification of the compute-torque controller [3] is employe. Here the moification refers to the use of an aaptive estimation scheme for friction compensation. By inspecting () it can be seen that if the control torque τ is chosen accoring to τ = V( q q ) + G( q) + M( q) u () for some u the combine system () an () becomes q = um ( q) F( q ) (3) Note that the inverse of the inertia matri M ( q) always eists. Let q an q be respectively the esire angular position an velocity in the joint space q q an efine the error vector as e =. From (3) the q q state space error ynamics can be epresse as e = Ae+ BuBM ( q) F( q ) Bq (4) I where A = I B = an q is the esire angular acceleration. It is note that the pair ( A B ) is controllable. When only Coulomb frictions are consiere each F ( q ) bears the following form where ( ) = sgn ( ) F q q is an unnown constant coefficient. Let { ( q )} Σ ( ) iag sgn q = the iagonal matri whose elements along the main iagonal are sgn ( q ). hen the friction torque can be epresse in the matri form F q =Σ q (5) ( ) ( ) with = { } being the coefficient vector of the Coulomb friction. o trac the error ynamics a composite control law is evelope in which the controller is assume to contain three parts: one is to ensure trajectory tracing another epens linearly on the state for nominal stability an the other attempts to compensate for the nonlinear friction base on the estimator. Let ˆ be the estimate of the friction coefficient vector then the control signal is represente as u = q + Ke+ M ( q) Σ ˆ ( q ) (6) for some state feebac gain matri K to be etermine. Substituting the control scheme (6) into (4) the closeloop state space equation can be written as e = ( A+ BK) ebm ( q) Σ( q ) (7) where is the friction estimation error which is efine as = ˆ. he friction estimate ˆ = { ˆ is } assume to be of the following form [] [] ˆ = ς g q = n ( ) for some auiliary variable ς an ifferentiable function ( ) g q. aing the erivative of the friction estimate leas to ˆ ' = ς g ( q ) q sgn( q ) = n. ' Here g ( q ) stans for the erivative of ( ) g q with respect to its argument q. In vector form the friction estimation scheme becomes ˆ = ς g q (8) { ( )} with ς = { ς }. Liewise the erivative of the friction estimate follows ˆ = ς G Σ q G Σ B ( A + BK) e + G Σ M Σ (9) ' where G is a iagonal matri with elements g ( q ). Note that the two matrices an Σ commute. From G

3 (9) suppose that the auiliary variable ς is esigne in accorance with ς = GΣ q + GΣ B ( A+ BK) e = GΣ ( q + Ke) () then the friction estimation error ynamics can be written as =G ΣM Σ () he following theorem provies a criterion for close-loop stability. heorem : With respect to the system that is subject to unnown Coulomb friction the controller (6) an the friction estimator (8) can be esigne to lea to global asymptotic tracing performance provie that there eists a positive efinite matri R such that G in the friction estimation scheme (8) is esigne such that M ( qg ) ( q ) + G( q ) M( q) > R () for all q an q. Proof: he close-loop system is the combination of (7) an (). Consier the Lyapunov function caniate Ve ( ) = epe+ for some symmetric positive matri P. he erivative of Ve ( ) along the state trajectory is Ve ( ) = e Σ S where ( ) ( ) PA+ BK + A+ BK P PBM S = M B P GM M G Applying the Schur complement technique [4] an observing that the matri M is positive efinite the conition for Ve ( ) to be negative is that MG + GM > (3) an e Σ ( ) PA ( + BK) + ( A+ BK) P+ PBGM+ MG BP < (4) Let R be a positive efinite matri that satisfies () then (3) is clearly satisfie an (4) is also satisfie when there eist P an K to the following equation ( ) ( ) PA+ BK + A+ BK P+ PBR BP < (5) Solutions to (5) always eist since the pair ( A B ) is controllable. Inee by setting the feebac gain matri K as K = R B P (6) he matri P can be obtaine by solving the following algebraic inequality PA A P PBR B P + < (7) Ve ( ) is negative for nonzero Hence the erivative of e an as long as () is satisfie. he global asymptotic tracing property is thus ensure. he operation of the controller is epicte in Figure. he controller contains a compute torque controller in the inner loop an error feebac in the outer loop. he ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 3 esire acceleration q is use to as a part of the feeforwar comman an the tracing error e is amplifie through the control gain K to constitute the feebac control signal Ke. For the friction estimator the acceleration error Ke + q is integrate to give an estimate of the auiliary variable ς which in turn provies an estimate of the friction ˆ in accorance with (8). he control signal then combines the feebac term an friction estimation term to achieve stability an compensate for the influence of friction forces as escribe in (6). As the Coulomb friction torque enures a sign change in the presence of velocity reversal the friction estimator also aopts a relay-type esign so that the friction effect can be compensate in a timely manner. he resulting control signal u will eperience some jumps at the instants of velocity sign changes. q q e K G Σ q s M Σ ˆ u τ qq g() Figure. Aaptive friction compensation bloc iagram It is also note that since the Lyapunov function being employe is a quaratic function once the controller is esigne to satisfy the conitions in heorem eponential stability of the close-loop system is achieve []. A ey in the esign of the aaptive controller is the etermination of G or such that () is verifie. he selection of q g 's g is rather arbitrarily as long as () is satisfie. A representative is µ q µ q µ µ q e g( q ) = ( e ) q ν e or µ q + e for some positive scalar µ an ν. he esign of the friction compensation scheme involves two steps. One is etermine the scalar function g ( q ) an positive efinite matri R such that () is satisfie for all q an q. Another is to solve the positive efinite solution P to the inequality (7). Since () an (7) are inequalities to be satisfie there remains freeom in the esign of the estimator an controller. In practice the parameters g ( q ) R an P can be tune to satisfy other esign goals such as goo transient response an quic parameter convergence. Physical systems are subject to some egrees of uncertainty. his is especially true for a robot manipulator when it is carrying unnown loas. In the net section the effects of uncertainty an Stribec effect will be accounte for base on the propose friction controller scheme.

4 ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 3 Robust Aaptive Control Scheme for Friction an Uncertainty It is nown that moel uncertainty ue to unnown loa or unmoele ynamics may egrae the system performance an even cause instability. Moreover the Coulomb friction moel in (5) is not capable of representing more complicate friction behaviors such as Stribec effects. In this section to ensure robust performance an stability the aaptive friction compensation scheme of the previous section is augmente with some robust control scheme to enhance the overall robustness. A robust aaptive control scheme is propose to account for uncertainties an more complicate friction behaviors. he ynamic equation uner consieration remains the same as () ecept that the inertia matri centripetal/coriolis torque gravitational torque an frictional torque are subject to perturbations. M( q) = M ( q) + M( q) V( q q ) = V ( q q ) + V( q q ) Gq ( ) = G( q) + Gq ( ) F( q ) = F ( q ) + F( q q ) (8) where M ( q ) V ( q q ) G ( q) an F ( q ) are the nominal moels an M ( q) V( q q ) G( q) an F( qq ) are unnown perturbations. In aition to parameter uncertainties changes of loa an configuration may result in perturbations on M ( q) V( q q ) an Gq ( ). he uncertainty F( qq ) associate with the friction torque may be a function of angular velocity an position [3]. In particular when Stribec effect is consiere the friction toque can be epresse as F ( q ) sgn( q ) e ν sgn( q ) q α β = + (9) In the above the term α sgn( q ) is the Coulomb friction an β e ν q sgn( q ) stans for the Stribec effect. he latter is note for its behavior of the ecrease of friction torque when the relative velocity increases. he friction moel incluing Stribec effects can be rewritten as ν q = β F ( q) sgn( q ) ( e )sgn( q ) where = α + β. he first term in the above is treate as the nominal term an the secon term is the perturbe term. he inner-loop compute torque with respect to the perturbe system becomes τ = M ( qu ) + V( qq ) + G( q) () Substituting (8) into the ynamic equation () yiels q u M = F ( q ) + w () where w stans for the eogenous input to the system which is inee a representation of the perturbations: ( ( ) ) w=m M u M F + V + G It can then be shown that the tracing error satisfies the equation e = Ae+ Bu+ BwBM Σ Bq () Suppose that the control signal is esigne as u = q + Ke+ M ˆ ( q) Σ( q ) (3) where ˆ is as in (8). he resulting error ynamics become e = ( A+ BK) e+ BwBM Σ (4) In orer to enhance robustness so that the effect ue to the eogenous input w can be compensate the auiliary variable in the friction estimator ς is selecte in accorance with ( ς = GΣ q + Ke ) + Ne (5) In the above the matri N is to be etermine. With the selection (5) the ynamic of friction error follows = NeG Σ M Σ + G Σ w (6) Let z = Ce be a fictitious output the robust tracing control esign problem can be reformulate as the fining of a control u for the system such that the L -norm gain from w to z is less than γ for any permissible frictions. he close-loop system representation becomes e = f ( e ) + gew ( ) t (7) z = h( e ) A BK BM e where f( e + Σ ) = N GΣM Σ B ge ( ) = G Σ an he ( ) [ C ]. e = he above close-loop system has an L -norm gain from w to z less than γ if an only if there eists a n+ n + scalar C function V : with V () = such that the following inequality is satisfie [5] [6] V f( e ) + V g( e ) g ( e ) e e Ve γ (8) + h ( e )( h e ) < where is the partial erivative of V with respect to Ve the state vector e. Consier the following Hamilton-Jacobi function caniate Ve ( ) = epe+ Π (9) for a positive efinite matri P an a positive iagonal matri Π. Substituting this caniate into (8) it can be shown that a sufficient conition for robust stability is that the following matri inequality is satisfie for nonzero e an : 4

5 e P( A+ BK) + PBB P+ C C e γ + Π N + Π G ΣB P e γ + e PBM Σ + PBΣ GΠ γ + Π ΣGM Σ + Π Σ GGΣΠ < γ he esign is greatly simplifie by setting the matri N as N =ΠΣ M B P+ γ ΣG B P (3) In this case the conitions for robust tracing can be state as PA ( + BK) + ( A+ BK) P+ γ PBBP + CC < (3) an GM ΠΠ M G + γ GG < (3) heorem : he system uner unnown Coulomb/Stribec frictions an parametric uncertainties is robustly trace provie that there eist G an a positive iagonal matri Π such that (3) is satisfie for all q an q an there eists a positive efinite matri P such that the following algebraic Riccati matri inequality is satisfie PA + A P + C C γ PBB P < (33) Proof: It has been shown that once (3) an (3) are solvable the system can be robustly trace. he matri inequality (3) involves two unnowns P an K. Selecting the control gain matri as K =γ B P (34) leas to the equation to be solve in (33). he bloc iagram of the robust aaptive friction controller is epicte in Figure. he resulting control signal is synthesize by combining the esire acceleration comman q error feebac Ke aaptive friction estimation ˆ an robustness enhancement Ne. Ecept for the path Ne the architecture is similar to that of the aaptive friction compensator in Figure. he aition term Ne epens on the robustness/performance level γ an can be regare as a signal for robustness enhancement. hrough the auiliary variable ς an the esign of N it appears that there eists a separation principle in the robust tracing control of structure uncertainty an friction. Inee the esign of the control gain is achieve by solving (3) while the esign of the friction estimator is accomplishe by solving (3). he two esigns can be conucte separately. he common variable to (3) an (3) is the robustness/performance level γ which serves to guie the traeoff between friction estimator an robust controller. ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 q q e N K q G Σ M Σ ˆ s u τ qq g() Figure. Robust aaptive control system bloc iagram he conition (3) is more stringent than (). Inee when the matri Π is set as the ientity matri the conition (3) becomes MG + GM > γ MGGM which is more restrictive than (). It is also observe that if one relaes the robustness requirement i.e. bouneness of the map from w to z an allows γ to approach to infinite the equations associate with the robust aaptive control esign boil own to those of the aaptive control esign in the previous section. Hence one can regar the robust aaptive friction control as a generalization of the aaptive friction control. he steps for the etermination of the control an estimation parameters for the robust aaptive controller are a. Select a positive iagonal matri Π g () an b. Select the friction estimation function { } obtain the matri G c. For all q an q etermine the achievable boun γ in accorance with (3). Solve P to the matri inequality (33) e. Determine K an N from (34) an (3) respectively. 4 Simulation Eample In orer to verify the resulting performance an robustness of the propose friction compensation schemes the simulation of a two-lin planar manipulators is performe. Consier a two-lin planar manipulator as epicte in Figure 3. he two lin lengths are l an l an the two masses are m an m respectively. It is assume that the lin masses are concentrate at the ens of the lins. he angular positions of the two lins are enote by q an q respectively. he ynamic equation of manipulator is given by () where the inertia matri M ( q ) is given by ml + mllc + ( m+ m) l ml + mllc M( q) = ml + mllc ml the centrifugal/coriolis torque V( q q ) is given by qq q V( q q ) = m ll s q 5 q the gravity Gq ( ) is

6 ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 mglc + ( m+ m) glc given by Gq ( ) = an the mglc friction torque F( q ) is moele as ν ( q ) sgn( q ) β( e )sgn( q ) Fq ( ) =. In the ν ( q ) sgn( q ) β( e )sgn( q ) above c s an c represent cos( q ) sin( ) an i cos( q + q ) respectively. i j i ij i q i Position Velocity.5 racing (High Gain Control) Comman Figure 4. Position an velocity tracing response of Joint (high gain control).. racing (High Gain Control) Figure 3. wo lin robotic manipulator. he following parameters are use in the simulation: m = g m = g l =.5m an l =.m. he esire commans are of sinusoial type with q =.sin(. t) ra an q =.5sin(.5 t) ra. he coefficients associate with the friction are =.5 =.3 β =. β =. ν =.5 an ν =.4. hese coefficients are selecte so that representative friction behaviors can be simulate. In the following three ifferent controllers will be esigne an simulate. he first one is a traitional high gain feebac controller. he secon one employs the aaptive friction estimation scheme to mitigate friction effects. he thir one is the robust aaptive control scheme that contains friction estimation an robustness enhancement features. All the simulations are conucte using matlab/simulin toolboes. o assess the effect of frictions a controller that employs the compute torque an high gain feebac is use. In this high gain control scheme the compute torque is as given in () an the control signal u is synthesize in accorance with u = q + Ke. he control gain matri K is selecte so that the eigenvalues of the matri A + BK are locate at the esire locations. In the 4 8 simulation K is selecte as K = 4 8 an the resulting eigenvalues are locate at ± j an ± j. Figures 4 an 5 epict the commans an responses of the two joints respectively. Essentially both the position an velocity can be trace. However the response is subject to lag an istortion which are very pronounce uring velocity reversals. Figure 6 further shows the position tracing error after sec. he maimal position error for Joint an Joint are respectively.5 ra (.85 eg) an.37 ra (.43 eg). Position Velocity Comman Figure 5. Position an velocity tracing response of Joint (high gain control). Position racing Error Position racing Error (High Gain Control) Joint Joint Figure 6. Position tracing errors (high gain control). he propose aaptive friction compensation scheme in Section is applie to the robot manipulator in orer to mitigate the effects ue to frictions. he objective of the esign is to achieve stability an trac the esire trajectory. o this en let the nonlinear function in the friction estimator in (8) be q g q = µ µ e + η q (35) ( ) 6

7 ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 for some positive µ an η. he erivative of g( q ) with respect to q is foun to be ( ) g q e. Selecting µ = µ = an ' q = µ + η η = = it can be shown that for () to be vali it η suffices to select the matri R as R = ηλmin( M) I where λ ( M ) min is the minimal eigenvalue of M. For the parameters given above λ ( M ) min is compute as.336 an the matri R is selecte as R =.5I. A feebac control gain can thus be etermine by solving (7) for P. he feebac control gain use in the simulation is K =. he eigenvalues of the matri A+ BK are.64 ± j.464 an.64 ± j.464. Figures 7 an 8 respectively epict the tracing responses of the two joints with the use of the aaptive friction compensation. As the friction is estimate asymptotically tracing is achieve. he estimate friction coefficients for the two joints are shown in Figure 9. he estimate friction coefficients converge to the true Coulomb friction coefficients which are.5 an.3 respectively. he position tracing errors after sec of the aaptive friction control are shown in Figure. he maimal position error for Joint is.5 eg an that for Joint is.574 eg. Note that the position errors are much smaller than the previous case. It is also note that the position error of Joint is significantly reuce. Position Velocity.5 racing (Aaptive Control) Comman Figure 7. Position an velocity tracing response of Joint (aaptive control). Position Velocity.5 racing (Aaptive Control) Comman Figure 8. Position an velocity tracing response of Joint (aaptive control). Friction Estimates Friction Estimation (Aaptive Control) Joint Joint Figure 9. Estimate friction coefficients (aaptive control) Position racing Error racing Error (Aaptive Control) Joint Joint Figure. Position tracing errors (aaptive control). 7 In the robust aaptive control esign two equations namely (3) an (33) neee to be solve. o compare the result with that of the aaptive friction controller the tuning parameter Π is set as the ientity matri. A performance level γ = 3.5 is then etermine as such a selection will verify (3) for the same friction estimator in (35). he feebac control gain is then esigne as K = he resulting eigenvalues of the matri A+ BK are.593± j.6

8 ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 an.8 ± j.33. he position an velocity responses of the esign are shown in Figures an respectively. Satisfactory tracing responses are achieve. In comparison with the aaptive controller (Figures 7 an 8) the transient responses appear to be better for the robust aaptive controller. his improvement is attribute partly to the increase of feebac gain K an partly to the injection of Ne to the friction estimator. he estimate friction coefficients are shown in Figure 3. Again parameter convergence is achieve. he maimal position errors after sec as epicte in Figure 4 for Joint an are.679 eg an.577 eg respectively. Position Velocity racing Respon se (Robust Aaptive Control) Comman Figure. Position an velocity tracing response of Joint (robust aaptive control). Position Velocity racing (Robust Aaptive Control) Comman Figure. Position an velocity tracing response of Joint (robust aaptive control). In the robust aaptive control esign the resulting control gain K is a full matri since the matri is selecte as C =. he first two rows of C are use to govern the response of each joint. hir row is to weight the overall position error. As a result the position errors of the two joints in Figure 4 appear to be the reflection of each other. In summary in the robust aaptive control scheme friction estimation function g weighting matri on parameter error Π robustness C level γ an performance output function C can all be selecte for the tuning of the resulting controller so that not only asymptotical tracing performance is assure but also transient response an robustness can be achieve. Friction Estimates Friction Estimation (Robust Aaptive Control) Joint Joint Figure 3. Estimate friction coefficients (robust aaptive control) Position racing Error racing Error (Robust Aaptive Control) Joint Joint Figure 4. Position tracing errors (robust aaptive control). 5 Conclusion In this paper a robust aaptive friction control algorithm is evelope to achieve robust tracing control of the manipulator that is subject to both unnown friction an parameter perturbation. he esign metho relies on the use of a friction estimation moel to estimate the friction coefficient an a robust tracing control scheme to ensure stability of the perturbe system uner both parametric uncertainty an estimation error. More importantly it is shown that the esign of the friction estimator an that of the robust control gain can be conucte separately or sequentially. hat is there seems to eist a separation principle in friction observer an robust state feebac control esign. Effectiveness of the esign proceure has been supporte by the simulations mae for a two egree-of-freeom robot manipulator. 8

9 One limitation of the propose (robust) aaptive friction compensation schemes is the reliance on the velocity an position measurements for feebac control an friction estimation. In the future the esign of observer-base schemes will be investigate to rela the requirement on sensors. 6 Acnowlegements he wor was supporte by the National Science Council aiwan uner Grant NSC 94-3-E References [] A. Yazizaeh K. Khorasani Aaptive Friction Compensation Base on the Lyapunov Scheme. Proceeings of the IEEE Conference on Control Application pp September 996. [] B. Frielan an Y.-J. Par On Aaptive Friction Compensation. IEEE ransactions on Automatic Control Vol. 37 No. pp [3] A.-H. Brain E. D. Pierre C. Canuas e Wit. A Survey of Moels Analysis ools an Compensation Metho for Control of Machines with Friction. Automatica Vol. 3 No. 7 pp [4] L. Cao an H. M. Schwartz Stic-Slip Friction Compensation for PID Position Control. Proceeings of American Control Conference pp June. [5] H. Olsson an K. J. Aström Friction Generate Limit Cycles. IEEE ransactions on Control System echnology Vol. 9 No. 4 pp July. [6] R. Ortega A. Loria P. J. Niclasson an H. Sira- Ramirez Passivity-Base Control of Euler-Lagrange Systems. Springer-Verlag 998. [7] M. Feemster P. Veagarbha D. M. Dawson an D. Haste Aaptive Control echniques for Friction Compensation. Proceeings of the American Control Conference pp [8] A. Visioli R. Aamini G. Legnani Aaptive Friction Compensation for Inustrial Robot Control. IEEE/ASME International Conference on Avance Intelligent Mechatronic Proceeings pp [9] G. Liu Decomposition-Base Friction Compensation Using a Parameter Linearization Approach. Proceeings of the IEEE International Conference on Robot an Automation pp May. [] G. ao On Robust Aaptive Control of Robot Manipulators. Automatica Vol. 8 pp [] K. K. an. H. Lee S. N. Huang an Xi Jiang Friction Moeling an Aaptive Compensation Using a Relay Feebac Approach. IEEE ransactions on Inustrial Electronics Vol. 48 No. pp February. [] S. S. Ge. H. Lee an S. X. Ren Aaptive Friction Compensation of Servo Mechanisms. International Journal of System Science Vol. 3 No. 4 pp ICGS- ARAS Journal Volume (5) Issue (I) Jan. 6 9 [3] G. Song R. W. Longman L. Cai Integrate Aaptive-Robust Control of Robot Manipulators with Joint Stic-Slip Friction. Proceeings of the IEEE International Conference on Control Application pp [4] G.-W. Lee an F.-. Cheng Robust Control of Manipulators Using the Compute orque Plus H Compensation Metho. IEE Proceeings Control heory Application Vol. 43 No. pp January 996. [5] Z. Qu an J. Dorsey Robust racing Control of Robots by a Linear Feebac Law. IEEE ransactions on Automatic Control Vol. 36 No. 9 pp September 99. [6] E. Gortcheva A. Poznya an V. Diaz De Leon On Nonlinear Robust Control for Robot Manipulator with Unnown Friction. Proceeings of the IEEE Conference on Decision & Control pp [7] J. Par an W. K. Chung Analytic Nonlinear H Inverse-Optimal Control for Euler-Lagrange System. IEEE ransactions on Robotics an Automation Vol. 6 No. 6 pp December. [8] Hongrui Wang Liin Wei Dejun Mu Ying Li Robust Aaptive Control of X-Y Position able with Uncertainty. Proceeings of the 5 th Worl Congress on Intelligent Control an Automation pp June 4. [9] S. Li Nonlinear H Controller Design for a Class of Nonlinear Control System. International Conference on Inustrial Electronic Control an Instrumentation pp [] L. Acho Y. Orlov an L. Aguilar Global H Control Design for racing Control of Robot Manipulators. Proceeings of the American Control Conference pp [] L. Aguilar Y. Orlov an L. Acho Nonlinear H racing Control of Friction Mechanical Manipulators. Proceeings of the American Control Conference pp [].-L. Liao an.-i Chien An Eponentially Stable Aaptive Friction Compensator. IEEE ransactions on Automatic Control Vol. 45 No. 5 pp May. [3] F. L. Lewis C.. Aballah an D. M. Dawson Control of Robot Manipulators. Macmillan Publishing Company 996. [4] S. Boy L. El Ghaoui E. Feron an V. Balarishnan Linear Matri Inequalities in System an Control heory. SIAM 994. [5] A. J. Van er Schaft -gain Analysis of Nonlinear L Systems an Nonlinear H Control. IEEE ransactions on Automatic Control Vol. 37 pp [6] A. Isiori an A. Astolfi Disturbance Attenuation an H Control via Measurement Feebac in nonlinear systems. IEEE ransactions on Automatic Control Vol. 37 pp