HYBRID ADAPTIVE FRICTION COMPENSATION OF INDIRECT DRIVE TRAINS

Transcription

1 Proceedings of the ASME 29 Dynaic Systes and Control Conference DSCC29 October 12-14, 29, Hollywood, California, USA DSCC HYBRID ADAPTIVE FRICTION COMPENSATION OF INDIRECT DRIVE TRAINS Wenjie Chen Dept. of Mechanical Engineering University of California Berkeley, California 9472 Eail: Kyoungchul Kong Dept. of Mechanical Engineering University of California Berkeley, California 9472 Eail: Masayoshi Toizuka Dept. of Mechanical Engineering University of California Berkeley, California 9472 Eail: ABSTRACT This paper investigates the friction copensation topic for indirect drive trains in the absence of precise end-effector easureents. Friction, one of the ain factors that diinish control perforance, is investigated and copensated by anipulating the reference trajectory as well as the torque input. The otor side torque copensation utilizes the odified LuGre odel to design an adaptive friction observer, while the otor side reference is odified by injecting the estiated load side friction. A hybrid controller structure is proposed to engage or disengage the load side copensator. Both ethods are cobined to effectively reject the friction effects in indirect drive trains. The effectiveness of the proposed schee is experientally verified. INTRODUCTION Indirect drive trains, e.g., robotic joints that utilize gear transissions, are coonly used in industrial applications due to the high torque capacity. The indirect gear transissions, however, set challenges in high precision otion control because of gear copliances and nonlinearities such as friction and torque hysteresis. Particularly, the nonlinear friction effects on the load side are difficult to copensate for, because: 1) precise load side easureent is often unavailable, and 2) it is indirectly actuated through the gear transission. Since the load side perforance is critical in practical applications, a ethod that copensates for friction on the load side as well as the otor side is necessary. Over the past several decades, odeling and copensation of the friction effects have been intensively studied fro various viewpoints, including static/dynaic characterizations and feed- THIS WORK WAS SUPPORTED BY FANUC LTD., JAPAN. REAL-TIME CONTROL HARDWARE AND SOFTWARE WERE PROVIDED BY NA- TIONAL INSTRUMENTS, INC. back/feedforward copensation schees 1, 2]. For exaple, any odel-based friction copensation techniques have been developed to reject the friction effects by adaptively 3] and robustly 4] observing the real friction. Also, typical approaches, such as the disturbance observer 5] or repetitive control 6], have been used to copensate for friction. Most of these studies, however, verified their perforance based on siple one-ass systes or direct drives only. When a two-ass syste or an indirect drive is considered, the friction effects introduced fro the load side reain a challenge. For this reason, friction characteristics in haronic drive systes were studied in 7,8]. The perforance and stability issues due to friction forces (e.g., liit cycles) were addressed in 9]. Soe copensation schees were proposed to overcoe the friction effects in haronic drives. Most of the, however, focused on the otor side copensation only 8] or torque tracking control 1]. To iprove the load side position tracking perforance, the position easureents of both otor side and load side have been utilized in the controller design 11]. Such controllers, however, are difficult to ipleent in industrial robots, which are usually not equipped with sensors for load side position easureent. In this paper, a friction copensation schee for indirect drive trains is developed in the absence of precise load side easureents. The proposed ethod anipulates both the torque input and the reference trajectory, which is controlled by a hybrid syste schee based on the tracking perforance. The effectiveness of the ethod is verified by experiental results. SYSTEM OVERVIEW Single-Joint Indirect Drive Model Figure 1 shows the scheatic of a single-joint indirect drive echanis. The subscripts and l denote the otor side and 1 Copyright 29 by ASME

2 u θ Figure 1. Motor J f (d ) Reducer N, f h k j, d j f l (d l ) Load Inertia SINGLE-JOINT INDIRECT-DRIVE MECHANISM the load side quantities, respectively. d represents the viscous daping coefficient, and J is the oent of inertia. θ and u represent the angular position and otor torque output, respectively. k j and d j are the linear stiffness and the viscous daping coefficients of the reducer. The gear ratio of the reducer is denoted by N. f, f l, and f h represent the nonlinear friction forces at the otor side, the load side, and the reducer, respectively. Note that f and f l include the viscous daping effects d and d l, respectively. These friction forces are further discussed later. Equations of otion for the syste in Fig. 1 are J θ + f + f h = u 1 N J l θ l + f l = k j ( θ N θ l J l θ l ( ( )] θ θ N k j N θ l )+d j θ l ( ) θ N )+ d j θ l Friction Model In the syste shown in Fig. 1, the energy is dissipated ainly by three friction forces: the otor bearing friction, f, the load output bearing friction, f l, and the haronic drive gear eshing friction, f h. Although the gear eshing friction f h is doinant in a free load condition 7], the load side friction f l can becoe ore significant as the load side inertia increases. The cobination of syste odel (1)-(2) gives (1) (2) J θ + 1 N J l θ l = u F (3) where F = f + f h + f l N can be explained as the entire friction force iposed on the whole syste referred to the otor side. There are several reasons to odel the entire friction force F as reflected on the otor side. First of all, it is easier to identify the friction on the otor side for industrial indirect drives. Furtherore, fro the torque copensation point of view, the friction force is usually copensated at the otor side where the actuator is. Also it is not necessary to odel these friction forces separately, especially when they exhibit siilar behaviors. To describe the entire friction force F, this paper eploys the Lund-Grenoble (LuGre) odel 1]. The LuGre odel uses an internal friction state, z, governed by ż = v β v g(v) z (4) g(v) = F c +(F s F c )e ( v2 /v 2 s) (5) F = σ z+σ 1 ż+ σ 2 v (6) where v is the relative velocity between the two contacting surfaces at the otor side, i.e., v = θ. σ, σ 1, and σ 2 represent the icro-stiffness, icro-daping of the internal state z, and the acro-daping of the velocity v, respectively. The function g(v) is chosen to capture the Stribeck effect, where F C and F S are the levels of Coulob friction and stiction force, respectively. v s is the Stribeck velocity. The internal friction state z can be regarded as the deflection of bristles, which represents the asperities between the two contacting surfaces. In Eq. (4), note that an additional paraeter, β, is used to define the noinal icro-stiffness of the internal state z, which differs fro the standard LuGre odel. This odification will siplify the friction copensation algorith. Equations (4)-(6) give the entire friction force, F, as F = K T Φ (7) where K = σ 1 + σ 2 σ β σ 1 ] T, Φ = v z v g(v) z ] T. Notice that the odified LuGre odel still preserves the following property of standard LuGre odel 1]: Property 1. F c g(v) F s. z(t) F s β, t if z() F s β. In reality, it can be assued that the deflection of bristles starts at rest, i.e., z() =. Therefore, z(t) F s β, t. This property will be used in the subsequent controller design to ensure the bounded stability of the friction observer. In practice, friction characteristics ay vary due to the variations of the noral forces in contact, lubricant condition, teperature, aterial wear, etc. 3] Variations in the noral forces usually cause an ipact only on the static paraeters, i.e., F C, F S, v s, and σ 2, while the changes in lubricant condition, teperature and/or aterials ay affect both static and dynaic paraeters. By fixing the noinal icro-stiffness, β, the adaptation of real icro-stiffness, σ, effectively changes the level of static paraeters, i.e., F C, F S, and v s in g(v). Thus, the adaptation of σ, σ 1, and σ 2 can account for ost paraeter variations in the odified friction odel, even if the dynaics of internal state z does not strictly follow Eq. (4). Therefore, σ, σ 1, and σ 2 are the paraeters to be adapted in the proposed algorith, i.e., K is to be updated in real tie, and Φ is the regressor. In the case that the load side friction effects are not negligible, it is necessary to consider the load side friction separately. Assuing that the load side friction shares the sae characteristics as the entire friction force, F, a ratio r l can be introduced to 2 Copyright 29 by ASME

3 θ ld Figure 2. F 1 θ d + F 2 C fb u fb u ff + + u P θl θ BLOCK DIAGRAM OF THE OVERALL CONTROL SYSTEM FRICTION COMPENSATION As shown in the controller structure (i.e., Eqs. (9)-(1)), the proposed friction copensator consists of two parts. The otor side friction copensation is designed into feedforward controller F 2 by injecting the torque input, and the load side friction effects are further copensated in feedforward controller F 1 by anipulating the otor side reference, θ d. derive the load side friction f l fro F, i.e. f l = r l F (8) This assuption is reasonable in ost cases, especially when the otor is spinning in one direction. The transient behavior during the velocity reversal ay not strictly follow the assuption. This velocity reversal case, however, is not of the interest in ost applications since the transient tie is usually sufficiently short to be negligible. Controller Structure Figure 2 shows the overall control structure of the singlejoint indirect drive syste. It consists of two feedforward controllers, F 1 and F 2, and one feedback controller, C f b. The feedforward controller, F 2, is designed as u f f (t) = J θ d (t)+ 1 N J l θ ld (t)+ ˆF (t) (9) where ˆF is the estiated entire friction force. θ ld is the desired load side acceleration. In the case that the acceleration is easured by sensors (e.g., load side acceleroeters), θ ld can be replaced with θ l. θ d is the otor side position reference generated by F 1 fro the desired load side position, θ ld, i.e. θ d (t) = N k j Jl θ ld (t)+ ˆf l (t) ] + Nθ ld (t) (1) where ˆf l is the estiated load side friction, and θ ld is the second derivative of θ ld. Equation (1) is derived fro Eq. (2) by neglecting the joint daping d j, since k j ( θn θ l ) d j ( θ N θ l ) usually holds within the syste bandwidth range. The feedback controller, C f b, which consists of a proportional position control loop and a proportional-integral (PI) velocity control loop, can be described as ( C f b (s) = k v + k ) i (s+k p ) (11) s where k p is the gain of the position loop, k v and k i are the gains of the velocity loop. The above controller is discretized by the Euler ethod for ipleentation. Motor Side Friction Copensation Feedback Friction Copensation Let e = θ θ d be the otor side position tracking error. Define a first order sliding surface, p, as p = ė + k s e (12) where k s is a positive constant. Note that e is sall or converges to zero if p is sall or converges to zero, since e (s) p (s) = s+k 1 s is asyptotically stable. Fro Eqs. (3) and (12), the otor side error dynaics is derived as J ṗ = u F 1 N J l θ l J θ d + J k s ė (13) The controller structure yields the following control law u(t) = u f f (t)+u f b (t) (14) = J θ d (t)+ 1 ] N J l θ t ] ld (t)+ ˆF(t) k v s (t)+ k i s (τ)dτ where s (t) = ė (t) + k p e (t) is the error ter in the velocity loop. By applying the adaptive control technique, an adaptive friction observer is designed to obtain ˆF(t), i.e. ˆF = ˆK T ˆΦ (15) ] T where ˆΦ = θ ẑ θ. ˆK, ẑ g( θ, and ẑ 1 are the estiates )ẑ1 obtained fro the following update laws: ˆK = Γ ˆΦp (16) ẑ = θ β θ g ( θ ) ẑ γ 3 p (17) ẑ 1 = θ β θ g ( θ θ ) ẑ 1 + γ 4 g ( θ ) p (18) where Γ = diag(γ,γ 1,γ 2 ), γ 3 and γ 4 are positive adaptation gains. Notice that the uneasurable internal friction state z is 3 Copyright 29 by ASME

4 estiated by a dual-observer structure 12] with the two state estiates, ẑ and ẑ 1. Equations (17) and (18) result in the following observer error dynaics: z = β θ g ( θ ) z + γ 3 p (19) z 1 = β θ g ( θ θ ) z 1 γ 4 g ( θ ) p (2) where z = z ẑ and z 1 = z ẑ 1. Stability Analysis The following theore proves the stability of the closed-loop syste under additional assuptions as stated therein. Theore 1. For the syste odel described by Eqs. (1)-(6), global asyptotic tracking perforance on the otor side can be achieved by the proposed controller in Eqs. (9)-(11) and adaptive friction observer in Eqs. (15)-(18), if the integrator ter of the feedback controller, C f b, is suppressed, i.e., k i =, and the real load side acceleration easureent, θ l, is used in Eq. (9). Proof. Suppose that the feedback controller gains are k p = hk s J k s + h, k v = J k s + h (21) where h is a positive constant. By the assuption in the theore, k i =. Then the control law in Eq. (14) can be rewritten as u = J θ d + 1 ] N J l θ ld + ˆF J k s ė + hp ] (22) Fro Eqs. (13) and (22), the control law results in the following closed loop dynaics J ṗ = hp J l ) ( ( θ l θ ld F ˆF ) N = hp J l N ël K T ˆΦ K T Φ (23) where ë l = θ l θ ld is the load side acceleration tracking error, K = K ˆK and Φ = Φ ˆΦ are the estiation errors. Let a Lyapunov function candidate for the syste be V = 1 2 J p K T diag(γ,γ 1,γ 2 ) 1 K + 1 k 1 z k 2 z 2 1 2γ 3 2γ 4 (24) where k 1 = σ and k 2 = β σ 1 are the last two entries in K. In ost applications, it is reasonable to assue that the actual friction paraeters, K = ] T, σ 1 + σ 2 σ β σ 1 are unknown but constant. It follows that K = and K = ˆK. Therefore, differentiating V in Eq. (24) and substituting the update laws fro Eqs. (16)-(18), the derivative of V is obtained as V = hp 2 J l N p ë l k 1 β θ γ 3 g ( θ ) z 2 k 2 β θ γ 4 g ( θ ) z 2 1 (25) Recall that h is a positive constant. Thus, every ter on the right hand side of Eq. (25), except for the second ter, is negative sei-definite. Moreover, if the load side acceleration is easured by sensors (i.e., θ ld is replaced with θ l in u f f in Eq. (9)), the second ter becoes zero. It follows that V = hp 2 k 1 β θ γ 3 g ( θ ) z 2 k 2 β θ γ 4 g ( θ ) z 2 1 hp 2 (26) Fro Lyapunov stability theory, it is concluded that the otor side tracking error, p, the friction paraeter estiation errors, K, and the internal friction state estiation errors, z and z 1, are globally uniforly bounded. Since the actual friction paraeters K are constant during one trajectory task, and actual internal friction state, z, is bounded as shown in Property 1, it follows that the estiates, ˆK, ẑ, and ẑ 1, are globally uniforly bounded as well. In addition, the actual otor side position and velocity, θ and θ, are globally uniforly bounded since the generated otor side references are bounded by the characteristics of the feedforward controller, F 1. Equations (24) and (26) also iply that p as t. Therefore, the otor side tracking error e asyptotically converges to zero. The above theore provides a sufficient condition to achieve global asyptotic tracking perforance on the otor side. Two assuptions were introduced to theoretically guarantee the stability of the proposed ethod. These assuptions, however, ay be relaxed without jeopardizing the closed loop stability of the syste in practice. The relaxation is deonstrated in the experients with preserved integrator in C f b to aintain the basic feedback controller structure (i.e., k i ). Also, the desired reference θ ld is used in Eq. (9) instead of the real easureent θ l to attenuate the dependence on the sensors fro the load side. Feedforward Friction Copensation The above friction copensation schee is in the feedback for, since the actual physical easureents are used in the regressor estiate, ˆΦ. When doing tracking control, feedforward control often iproves the perforance in the sense that it allows anti-causal ters in the controller. It recovers the liitation of feedback controller due to the dynaic lag of the plant. Recall that the feedback controller ust see an error first before it can take any corrective action. In particular, this liitation affects the perforance by the quick direction reversal of Coulob friction force. 4 Copyright 29 by ASME

5 By replacing the easureents in the friction observer with corresponding desired trajectory profile, excluding the otor side error ter, p, a feedforward version of the proposed copensation schee is obtained, where ˆΦ and ẑ are odified as ] T ˆΦ d = θ d ẑ d θ d, ẑ g( θ d = θ d β θ d d)ẑd g ( θ ) ẑ d (27) d In this for, the dual estiations of internal friction state z are replaced by single desired (noinal) internal state ẑ d as in Eq. (27). Note that Eq. (27) uses desired trajectory without the feedback correction ter, p. In this feedforward for, the friction observer structure is siplified by reducing the nuber of adaptive variables fro 5 to 3, and the noises and disturbances fro feedback easureents are also avoided. Load Side Friction Copensation Due to the characteristics of indirect drives, perfect tracking perforance on the otor side does not guarantee perfect load side tracking perforance. Since the load side perforance is of the ost interest in practical applications, it is necessary to apply friction copensation as in Eq. (1) to achieve desired perforance objectives on the load side. In Eq. (8), the load side friction, f l, is odeled as a scaled quantity of the entire friction force, F. Thus, once friction paraeters are adapted successfully on the otor side, the load side friction f l can be estiated as ˆf l = ˆr l ˆF (28) where ˆF is the estiated entire friction force fro the otor side friction copensator, and ˆr l is the estiated scaling factor obtained by the following update laws: ˆr l = γ rl ˆFê l, ê l = ˆθ l θ ld (29) where γ rl > is the adaptation gain, ˆθ l is the estiated load side position produced by soe estiation algorith, such as the extended kineatic odel based Kalan filter with acceleroeters and gyroscope (KKF-AG) described in 13]. Hybrid Copensator Structure Since the load side friction estiate, ˆf l, is injected into the otor side reference, θ d, it is desired that the copensated reference is still sooth and continuously differentiable, since the otor side velocity reference, θ d, is needed in the feedback controller, C f b. To ensure this, the load side copensation should be active only when the friction paraeters in the otor side copensator have converged to soe sub-optial values and then set fixed (i.e., Γ = ). This leads to the following hybrid copensator structure. Θ,k+1 2 Θ,k 2 Figure 3. q ˆF ˆf l = (δ, δ) Θ,k+1 2 Θ,k 2 q 1 ˆF ˆf l = ˆr l ˆF / (δ, δ) HYBRID COMPENSATOR STRUCTURE Figure 3 shows the hybrid copensator structure. q and q 1 are the two discrete controller states of the hybrid copensator. In industrial applications, the sae task is usually executed repeatedly. Define Θ,k 2 as the two-nor of the otor side position tracking error in the k-th execution of the sae task, i.e. ( n Θ 2,k = θ,k (i) ) (3) i=1 where θ,k (i) = θ,k (i) θ d (i) is the otor side position tracking error at the i-th tie step in the k-th execution of the sae task, and n defines the duration of each execution. The controller starts fro the state q, where the adaptive otor side copensator is activated, and the load side copensator is disabled (i.e., ˆf l = ). If Θ,k+1 2 Θ ( δ, δ ), where (,k 2 δ, δ) is the convergence boundary, the copensator schee is switched to the state q 1. This eans that the friction adaptation in the otor side copensator has reached its sub-optial stage, and cannot significantly iprove the otor side tracking perforance. At this stage, the load side copensator is enabled while the friction paraeters in the otor side copensator are fixed. If Θ,k+1 2 Θ,k 2 / ( δ, δ ), the otor side tracking perforance can be further iproved by adapting the paraeters in the otor side copensator. Thus, the copensator schee is switched back to the initial state q. EXPERIMENTAL RESULTS Experiental Setup The proposed friction copensation ethod is applied to a single-joint indirect drive syste shown in Fig. 4. The experiental setup consists of 1) a servo otor with a 2, counts/revolution encoder, 2) a haronic drive with a 8:1 gear ratio, 3) a load-side 144, counts/revolution encoder, 4) and a payload. The anti-resonant and resonant frequencies of the setup are about 11Hz and 19Hz. A MEMS gyroscope (Analog, Type: ADXRS15) is installed at one end of the payload and two acceleroeters (Kistler, Type: 833A3) are installed at the ends of 5 Copyright 29 by ASME

6 Motor Torque Coand (N) Experient Curve Fit Figure 4. SINGLE-JOINT INDIRECT DRIVE SYSTEM SETUP Figure Motor Side Velocity (rad/sec) STATIC FRICTION IDENTIFICATION RESULT the payload syetrically as shown in Fig. 4. The load side encoder is only for perforance evaluation. Also, the acceleroeters and gyroscope are only for load side inforation estiation, not directly for friction copensation algoriths; e.g., θ ld is used in Eq. (9), since it does not practically influence the stability in the experients. Finally, the controller is ipleented in a LabVIEW real-tie target installed with LabVIEW Real-Tie and FPGA odules. The sapling rate is selected as 1 khz. Friction Identification Friction identification is conducted to set initial values in the adaptive friction observer. The steady state friction regie (gross otion) is characterized by ż =, θ =, and θ l = 2]. Thus, the odel in Eqs. (1)-(6) is reduced to the static relation u = F ( θ ) = F c +(F s F c )e ( θ 2 /v 2 s) ] sgn ( θ ) + σ2 θ (31) This nonlinear function properly describes a static velocitytorque (friction force) characteristic, as shown in Fig. 5. Each point in Fig. 5 is obtained by keeping the otor side velocity constant for the sae aount of distance in a closedloop anner. The average value of the steady state torque (friction force F ( θ ) ) at the corresponding velocity is recorded. The experient is repeated at various velocities for the sae path to obtain the whole static velocity-torque ap. The nonlinear least squares ethod in the Optiization Toolbox of MATLAB is applied to obtain the static friction paraeters, F C, F S, v s, and σ 2. It is shown in 2] that the dynaic paraeters, σ and σ 1, can be identified using an ARX (AutoRegressive with extra input signal) or a BJ (Box Jenkins) odel structure with pre-sliding otion data. The resolution of the otor encoder, however, liits the ipleentation of this ethod. Thus, the ethod in 3] is eployed here to obtain the rough estiate of σ, using rap and step input response data, which gives ˆσ = 33.34(rap) and 57.3(step) N rad 1, respectively. Note that, the identification of σ 1 is still not available due to the lack of a high resolution encoder. This proble can be solved by using the proposed copensation schee, since σ and σ 1 are adapted in real tie. Table 1. Position Velocity (rad/sec) Acceleration (rad/sec 2 ) THE IDENTIFIED FRICTION PARAMETERS (SI UNITS) ˆF c ˆF s ˆv s ˆσ ˆσ 1 ˆσ Figure 6. DESIRED LOAD SIDE TRAJECTORY Thus, for siplicity, it is set that ˆσ = 4 N rad 1 and ˆσ 1 = N rad 1 sec. All the identified paraeters are listed in Tab. 1. Motor Side Friction Copensation Fig. 6 shows the desired load side trajectory in this experient, which is designed as a fourth-order tie optial trajectory. Feedback Friction Copensation To show the effectiveness of proposed adaptive algorith, the estiates of friction paraeters were initialized to one half of the identified values. The feedback controller gains are set as k p = 3, k v =.3, and k i = 1. (one relaxation of Theore 1). The adaptation gains in the feedback adaptive friction copensator (FB-A) are selected as γ,γ 1,γ 2,γ 3,γ 4 ] =.1,5,,.1,.1]. For coparison, a feedforward Coulob friction copensator (FF-C) is designed, i.e., ˆF = ˆF c sgn ( θ ) d, using the sae initial guess, ˆF C. Figure 7 shows the otor side and the load side tracking 6 Copyright 29 by ASME

7 e l = θ l θ ld Load Side Position Error 4 x 1 4 (No Cop.) 3 (FF C) Load Side Friction Effect (FB A) e = θ θ d NO Cop. FF C.15 FB A Motor Side Position Error (No Cop.) (FF C) (FB A) Estiate of F (N) Estiate of z, z x 1 3 Internal Friction State Estiate Total Friction Estiate z z1 Figure 7. PERFORMANCE OF THE MOTOR SIDE COMPENSATORS (No-Cop.) Green-Solid]: Without Friction Copensator; (FF-C) Grey- Dash]: Feedforward Coulob Friction Copensator; (FB-A) Blue-Solid]: Feedback Adaptive Friction Copensator Table 2. IMPROVEMENTS COMPARED TO THE NO-COMP. CASE FF-C FB-A FF-A Hybrid Θ 2 l 18.3% 35.26% 37.61% 66.5% Θ l 12.69% 32.81% 35.23% 65.77% Θ % 74.41% 78.53% 75.74% Θ 41.74% 68.3% 73.31% 74.78% perforance of these copensators. It is clearly seen that, otor side tracking perforance is significantly iproved by the FB-A ethod, converging to a sub-optial stage in about 3 executions. The load side perforance is slightly iproved with reduced peak error. Let Θ l 2, Θ l, Θ 2, and Θ denote the two-nors and -nors of the load side and the otor side position errors in the last repeated execution, respectively. Table 2 shows the perforance indices iproved by these algoriths copared to the case without copensation (No-Cop.). Figure 8 shows the friction estiates by FB-A, where ˆF is fully adapted in about 3 executions. Figure 9 illustrates the adaptive friction paraeter estiation process. Note that, the converged values are not exactly twice the initial values. This is because the identified values ay not be exactly the actual values. Also the friction odel structure is odified to adapt only three paraeters. Besides, ˆk 2 is kept zero since ˆσ 1 = and γ 2 =. Feedforward Friction Copensation The odification of the proposed adaptive friction observer fro the feedback for to the feedforward for is also applied in the experient. The sae controller gains, k p, k v, and k i, and adaptation gains, γ, γ 1, and γ 2, fro FB-A, are used in the feedforward copensator (FF-A). Table 2 shows that the feedforward for slightly iproves the perforance of the feedback for. k = σ 1 +σ 2 k 1 = σ Figure FRICTION ESTIMATES BY FB-A OBSERVER 12 x Friction Paraeter Estiations (SI) Figure 9. FRICTION PARAMETER ESTIMATIONS Load Side Friction Copensation The otor side copensation results (e.g., Fig. 7) show that, even if the otor side achieves good tracking perforance, large position tracking error can still be observed at the load side. This indicates the necessity to ipleent the load side friction copensation. In Fig. 7, the oscillation in the load side position tracking error is ainly due to the transission error 14], while the offset in the tracking error is due to the load side friction effects, which is explained in Eq. (1). In 13], it is shown that this tracking error offset can be successfully estiated by the KKF-AG ethod. Therefore, this ethod is eployed here to estiate the load side position. To show the effectiveness of the proposed load side copensation algorith, ˆr l was initialized to 1.N where N is the reducer gear ratio. Figure 1 illustrates the perforance of friction copensator enabled on both the otor side and the load side (Hybrid), i.e., the state q 1 in Fig. 3. It can be seen that the load side position tracking error offset is significantly reduced by the hybrid copensator schee, while the otor side tracking perforance is aintained at about the sae level. This is also confired by the perforance indices in Tab. 2. The load side friction copensator converges to a sub-optial stage in less 7 Copyright 29 by ASME

8 e l = θ l θ ld x 1 4 Load Side Position Error Figure 1. (Hybrid) (No. Cop.) (FF A) e = θ θ d NO Cop. FF A.15 Hybrid Motor Side Position Error (FF A) (Hybrid) (No. Cop.) PERFORMANCE OF THE HYBRID COMPENSATOR (No- Cop.) Green-Solid]: Without Friction Copensator; (FF-A) Grey- Dash]: Feedforward Adaptive Friction Copensator at Motor Side Only; (Hybrid) Blue-Solid]: Hybrid Copensator at the State q 1 Estiated r l /N Estiated Friction (N) Estiation of Friction Scaling Factor r l Total Friction F Friction Estiations Load Side Friction f l /N Figure 11. LOAD SIDE FRICTION ESTIMATES than 1 execution. The convergence of estiated friction ratio ˆr l and estiated load side friction ˆf l is also confired in Fig. 11. CONCLUSION Friction copensation in indirect drive echaniss was investigated by separating the proble into two parts, i.e., otor side and load side. The otor side copensator eployed the idea of an adaptive friction observer based on the odified Lu- Gre odel, while the load side copensator was ipleented by injecting the load side friction estiate into the generated otor side reference. A hybrid copensator schee was proposed to engage or disengage the load side copensator. It has been shown that additional care should be placed to the load side perforance due to the load side friction in addition to the friction effects on the otor side. Experiental results deonstrated the effectiveness of the proposed schee. REFERENCES 1] Canudas de Wit, C., Olsson, H., Astro, K. J., and Lischinsky, P., A new odel for control of systes with friction. IEEE Transactions on Autoatic Control, 4(3), pp ] Altpeter, F., Friction odeling, identification and copensation. PhD thesis, École Polytechnique Fédérale de Lausanne. 3] C. Canudas de Wit, P. L., Adaptive friction copensation with partially known dynaic friction odel. International Journal of Adaptive Control and Signal Processing, 11(1), pp ] Xu, L., and Yao, B., 2. Adaptive robust control of echanical systes with nonlinear dynaic friction copensation. In Proceedings of the 2 Aerican Control Conference, Vol. 4, pp ] Ohnishi, K., A new servo ethod in echatronics. Transaction of Japanese Society of Electrical Engineering, 17-D, pp ] Tung, E. D., Anwar, G., and Toizuka, M., Low velocity friction copensation and feedforward solution based on repetitive control. Journal of Dynaic Systes, Measureent, and Control, 115(2A), pp ] Taghirad, H. D., and Belanger, P. R., Modeling and paraeter identification of haronic drive systes. Journal of Dynaic Systes, Measureent, and Control, 12(4), pp ] Gandhi, P. S., Ghorbel, F. H., and Dabney, J., 22. Modeling, identification, and copensation of friction in haronic drives. In Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 1, pp ] Soo, J., and Toizuka, M., 25. Liit cycles due to friction forces in flexible joint echaniss. In Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp ] Teghirad, H. D., and Belanger, P. R., Robust friction copensator for haronic drive transission. In Proceedings of the 1998 IEEE International Conference on Control Applications, Vol. 1, pp ] Wen-Hong, Z., and Doyon, M., 24. Adaptive control of haronic drives. In Proceedings of the 43rd IEEE Conference on Decision and Control, Vol. 3, pp ] Tan, Y., and Kanellakopoulos, I., Adaptive nonlinear friction copensation with paraetric uncertainties. In Proceedings of the 1999 Aerican Control Conference, Vol. 4, pp ] Chen, W., 29. Hybrid adaptive friction copensation of indirect drive trains using joint sensor fusion. Master report, University of California at Berkeley. 14] Cheng-Huei, H., Chun-Chih, W., and Toizuka, M., 28. Suppression of vibration due to transission error of haronic drives using peak filter with acceleration feedback. In Proceedings of the 1th IEEE International Workshop on Advanced Motion Control, pp Copyright 29 by ASME