Mechatronics 6 (6) 5 6 An aaptive switching learning control metho for trajectory tracking of robot manipulators P.R. Ouyang a, W.J. Zhang a, *, Maan M. Gupta b a Avance Engineering Design Laboratory, Department of Mechanical Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, SK, Canaa S7N 5A9 b Intelligent Systems Research Laboratory, Department of Mechanical Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, SK, Canaa S7N 5A9 Receive March 4; accepte August 5 Abstract In this paper, a new aaptive switching learning control approach, calle aaptive switching learning PD control (ASL-PD), is propose for trajectory tracking of robot manipulators in an iterative operation moe. The ASL-PD control metho is a combination of the feeback PD control law with a gain switching technique an the feeforwar learning control law with the input torque profile. The torque profile is upate by the previous torque profile (which makes sense for learning). Furthermore, in this new control metho, the switching control scheme is integrate into the iterative learning proceure; as such, the trajectory tracking converges very fast. The ASL-PD metho achieves the asymptotical convergence base on the LyapunovÕs metho. The ASL-PD metho possesses both aaptive an learning capabilities with a simple control structure. The simulation stuy valiates this new metho. In particular, both position an velocity tracking errors monotonically ecrease with the increase of the number of iterations. The convergence rate with the ASL-PD metho is faster than that of the aaptive iterative learning control metho propose by others in literature. Ó 5 Elsevier Lt. All rights reserve. Keywors: Aaptive control; Iterative learning; Switching gain control; PD control; Trajectory tracking; Robot manipulator. Introuction The control of robot manipulators has attracte a great eal of attentions ue to their complex ynamics an wie applications in inustrial systems. Basically, the control methos can be classifie into the following three types. The first type is the traitional feeback control (proportional integral erivative (PID) control or proportional erivative (PD) control [ 4]) where the errors between the esire an the actual performance are treate in certain ways (proportional, erivative, an integral), multiplie by gains, an fe back as the correct input torque. The secon type is the aaptive control [5 ] * Corresponing author. Tel.: + 36 966 5478; fax: + 36 966 547. E-mail aresses: chris.zhang@usask.ca, chris_zhang@engr.usask.ca (W.J. Zhang). where the controller moifies its behaviour in response to the changes in the ynamics of the robot manipulator an the characteristics of the isturbances receive by the manipulator system. The thir type is the iterative learning control (ILC) [ 7] where the previous torque profile is ae to the current torque in a certain manner. Some other control methos, incluing the robust control, moel base control, switching control, an sliing moe control, can be in one or another way reviewe either as specialization an/or combination of the three basic types, or are simply ifferent names ue to ifferent emphases when the basic types are examine. The use of traitional PD control is very popular not only because of its simple structure an easy implementation but also its acceptable performance for inustrial applications. It is known that the PD control can be use for trajectory tracking with the asymptotic stability if the 957-458/$ - see front matter Ó 5 Elsevier Lt. All rights reserve. oi:.6/j.mechatronics.5.8.
5 P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 control gains are carefully selecte [ 4]. However, the PD control is not satisfactory for applications which require high tracking accuracy. This limitation with the PD control is simply ue to the inherent mismatch between the nonlinear ynamics behaviour of a manipulator an the linear regulating behaviour of the PD controller. Such a limitation is also true for the PID control. The aaptive control can cope with parameter uncertainties, such as the link length, mass, inertia, an frictional nonlinearity, with a self-organizing capability. Having such a capability, however, requires extensive computation an thus compromises its application for real-time control problems (especially in high-spee operations). In aition, since the aaptive control generally oes not guarantee that the estimate parameters of the manipulators converge to their true values [8], the tracking errors woul repeately be brought into the system as the manipulators repeat their tasks. Robot manipulators are usually use for repetitive tasks. In this case, the reference trajectory is repeate over a given operation time. This repetitive nature makes it possible to apply ILC to improve the tracking performance from iteration to iteration. It shoul be note that ILC can be further classifie into two kins: off-line learning an on-line learning. In the case of off-line learning control, information in the controlle torque in the current iteration oes not come from the current iteration but from the previous one. Philosophically, the learning in this case is shifte to the off-line moe. This then releases a part of the control workloa at real-time, which implies the improvement of real-time trajectory tracking performance. In the case of the on-line learning control, the feeback control ecision incorporates ILC at real-time. Another active area of research in the control theory is the switching control [9 ]. In the switching control technique, the control of a given plant can be switche among several controllers, an each controller is esigne for a specific nominal moel of the plant. A switching control scheme usually consists of an inner loop (where a caniate controller is connecte in close-loop with the system) an an outer loop (where a supervisor ecies which controller to be use an when to switch to a ifferent one). As such, the switching of controllers is taken place in the time omain. This unerlying philosophy may be moifie to perform such a switching with respect to the iteration of learning. In this paper, we present a new control metho. The basic concept of this new control metho is to combine several control methos by taking avantage of each of them into a hybri one. The architecture of this hybri control metho is as follows: () the control is a learning process through several iterations of off-line operations of a manipulator, () the control structure consists of two parts: a PD feeback part an a feeforwar learning part using the torque profile obtaine from the previous iteration, an (3) the gains in the PD feeback law are aapte accoring to the gain switching strategy with respect to the iteration. This new control metho is calle the aaptive switching learning PD (ASL-PD) control metho. The remainer of the paper is organize as follows. In Section, the ASL-PD control metho is escribe, an its features are iscusse. Section 3 is evote to the analysis of the asymptotic convergence of the ASL-PD control metho using the LyapunovÕs metho. In Section 4, simulation stuies are presente in which the ASL-PD metho is compare with others. Conclusions are given in Section 5.. Aaptive switching learning PD control scheme.. Dynamic moel of a robot manipulator Consier a robot manipulator with n joints running in repetitive operations. Its ynamics can be escribe by a set of nonlinear ifferential equations in the following form []: Dðq j ðtþþ q j ðtþþcðq j ðtþ; _q j ðtþþ_q j ðtþþgðq j ðtþ; _q j ðtþþ þ T a ðtþ ¼ T j ðtþ ðþ where t [, t f ] enotes the time an j N enotes the operation or iteration number. q j ðtþ R n, _q j ðtþ R n, an q j ðtþ R n are the joint position, joint velocity, an joint acceleration vectors, respectively. Dðq j ðtþþ R nn is the inertia matrix, Cðq j ðtþ; _q j ðtþþ_q j ðtþ R n enotes the vector containing the Coriolis an centrifugal terms, Gðq j ðtþ; _q j ðtþþ R n is the gravitational plus frictional force, T a ðtþ R n is the repetitive unknown isturbance, an T j ðtþ R n is the input torque vector. It is common knowlege that robot manipulators have the following properties []: (P) D(q j (t)) is a symmetric, boune, an positive efinite matrix; (P) The matrix _Dðq j ðtþþ Cðq j ðtþ; _q j ðtþþ is skew symmetric. Therefore, x T ð _Dðq j ðtþþ Cðq j ðtþ; _q j ðtþþþx ¼ 8x R n Assume that all parameters of the robot are unknown an that: (A) The esire trajectory q (t) is of the thir-orer continuity for t [,t f ]. (A) For each iteration, the same initial conitions are satisfie, which are q ðþq j ðþ ¼; _q ðþ_q j ðþ ¼; 8j N... ASL-PD controller esign The ASL-PD control metho has two operational moes: the single operational moe an the iterative operational moe. In the single operational moe, the PD control feeback with the gain switching is use, where information from the present operation is utilize. In the iterative operational moe, a simple iterative learning control is applie as feeforwar where information from pre-
P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 53 vious operations is use. Together with these two operational moes, all information from the current an previous operations is utilize. Specially, the ASL-PD control metho can be escribe as follows. Consier the jth iterative operation for system () with properties (P an P) an assumptions (A an A) uner the following control law: T j ðtþ ¼K j p ej ðtþþk j _ej ðtþ þ T j ðtþ fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflffl{zfflfflffl} feeback feeforwar with the following gain switching rule 8 K >< j p ¼ bðjþk p K j ¼ bðjþk j ¼ ; ;...; N >: bðj þ Þ > bðjþ j ¼ ; ;...; N where T (t)=, e j (t)=q (t) q j (t), _e j ðtþ ¼_q ðtþ_q j ðtþ, an K p an K are the initial PD control gain matrices that are iagonal positive efinite. The matrices K p an K are calle the initial proportional an erivative control gains, while matrices K j p an Kj are the control gains of the jth iteration. b(j) is the gain switching factor where b(j) > for j =,,...,N, an it is a function of the iteration number. The gain switching law in (3) is use to ajust the PD gains from iteration to iteration. Such a switching in the ASL-PD control metho acts not in the time omain but in the iteration omain. This is the main ifference between the ASL-PD control metho an the traitional switching control metho (where switching occurs in the time omain). Therefore, the transient process of the switche system, which must be carefully treate in the case of the traitional switching control metho, oes not occur in the ASL-PD control metho. From () an (3) it can be seen that the ASL-PD control law is a combination of feeback (with the switching gain in each iteration) an feeforwar (with the learning scheme). The ASL-PD control metho possesses an aaptive ability, which is emonstrate by the aoption of ifferent control gains in ifferent iterations; see (3). Such a switching takes place at the beginning of each iteration. Therefore, a rapi convergence spee for the trajectory tracking can be expecte. Furthermore, in the ASL-PD control law, the learning occurs ue to the memorization of the torque profiles generate by the previous iterations that inclue information about the ynamics of a controlle system. It shoul be note that such learning is irect in the sense that it generates the controlle torque profile irectly from the existing torque profile in the previous iteration without any moification. Because of the introuction of the learning strategy in the iteration, the state of the controlle object changes from iteration to iteration. This requires an aaptive control to eal with those changes, an the ASL-PD has such an aaptive capability. ðþ ð3þ In the next section, the proof of the asymptotic convergence of the ASL-PD control metho for both position tracking an velocity tracking will be given. 3. Asymptotic convergence with the ASL-PD metho Eq. () can be linearize along the esire trajectory ðq ðtþ; _q ðtþ; q ðtþþ in the following way: DðtÞ e j ðtþþ½cðtþþc ðtþš_e j ðtþþfðtþe j ðtþ þ nð e j ; _e j ; e j ; tþt a ðtþ ¼HðtÞT j ðtþ ð4þ where D(t) =D(q (t)) CðtÞ ¼Cðq ðtþ; _q ðtþþ C ðtþ ¼ oc o_q _q ðtþþ og q ðtþ;_q ðtþ o_q F ðtþ ¼ od q oq ðtþþ oc oq q ðtþ q ðtþ;_q ðtþ q ðtþ;_q ðtþ _q ðtþþ og oq q ðtþ HðtÞ ¼Dðq ðtþþ q ðtþþcðq ðtþ; _q ðtþþ_q ðtþþgðq ðtþþ The term nð e j ; _e j ; e j ; tþ contains the higher orer terms e j ðtþ, _e j ðtþ, an e j (t), an it can be negligible. Therefore, for the jth an j + th iterations, Eq. (4) can be rewritten, respectively, as follows: DðtÞ e j ðtþþ½cðtþþc ðtþš_e j ðtþþfðtþe j ðtþt a ðtþ ¼ HðtÞT j ðtþ ð5þ DðtÞ e jþ ðtþþ½cðtþþc ðtþš_e jþ ðtþþf ðtþe jþ ðtþt a ðtþ ¼ HðtÞT jþ ðtþ For the simplicity of analysis, let K p ¼ KK for the initial iteration, an efine the following parameter: y j ðtþ ¼_e j ðtþþke j ðtþ ð7þ The following theorem can be prove. Theorem. Suppose robot system () satisfies properties (P, P) an assumptions (A, A). Consier the robot manipulator performing repetitive tasks uner the ASL-PD control metho () with the gain switching rule (3). The following shoul hol for all t [,t f ] q j ðtþ j!! q ðtþ _q j ðtþ j!!_q ðtþ provie that the control gains are selecte so that the following relationships hol: l p ¼ k min ðk þ C KDÞ > l r ¼ k min ðk þ C þ F =K _ C =KÞ > l p l r P ð6þ ð8þ ð9þ kf =K ðc þ C KDÞk max ðþ where k min (A) is the minimum eigenvalue of matrix A, an kmk max = maxkm(t)k for 6 t 6 t f. Here, kmk represents the Eucliean norm of M.
54 P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 Proof. Define a Lyapunov function caniate as V j ¼ e qs y jt K yj s P ðþ where K > is the initial erivative gain of PD control, an q is a positive constant. Also, efine y j = y j+ y j an e j = e j+ e j. Then, from (7) y j ¼ _e j þ Ke j an from () ðþ T jþ ðtþ ¼K jþ p e jþ ðtþþk jþ _e jþ ðtþþt j ðtþ ð3þ From (5) (7), (), (3), one can obtain the following equation: D_y j þðc þ C KD þ K jþ Þy j þðf KðC þ C KDÞÞe j ¼K jþ y j ð4þ From the efinition of V j, for the j + th iteration, one can get V jþ ¼ e qs y jþt K yjþ s Define DV j = V j+ V j. Then from (), () an (4), we obtain DV j ¼ e qs ðy jt K yj þ y jt K yj Þs ¼ bðj þ Þ ¼ bðj þ Þ e qs ðy jt K jþ e qs y jt K jþ y j s y j þ y jt K jþ y j Þs e qs y jt ððc þ C KD þ K jþ Þy j þðf KðC þ C KDÞÞy j Þs e qs y jt D _ y j s Applying the partial integration an from (A), we have e qs y jt D_y j s ¼ e qs y jt Dy j t ¼ e qs y jt ðtþdðtþy j ðtþþq e qs y jt D_y j s From (P), one can get y jt _Dy j s ¼ ðe qs y jt DÞ y j s y jt Cy j s e qs y jt Dy j s e qs y jt _Dy j s Then DV j ¼ e qt y jt ðtþdðtþy j ðtþq bðj þ Þ From (3), we have e qs y jt ðf KðC þ C KDÞÞe j s e qs y jt ðk jþ þ C KDÞy j s e qs y jt K jþ y j s ¼ bðj þ Þ P e qs y jt Dy j s e qs y jt K yj s e qs y jt K yj s ð5þ ð6þ Substituting () into (5) an noticing (6), we obtain Z DV j t 6 e qt y jt ðtþdðtþy j ðtþq e qs y jt Dy j s bðj þ Þ K e qs _e jt ðk þ C KDÞ_e j s e qs e jt ðk þ C KDÞ_e j s e qs _e jt ðf KðC þ C KDÞÞe j s K e qs e jt ðk þ C KDÞe j s K e qs e jt ðf KðC þ C KDÞÞe j s Applying the partial integration again gives e qs e jt ðk þ C KDÞ_e j s ¼ e qs e jt ðk þ C KDÞe j j t þ q þ e qs e jt ðk þ C KDÞe j s e qs _e jt ðk þ C KDÞe j s e qs e jt ðk _D _C Þe j s Therefore, DV j 6 e qt y jt Dy j q bðj þ Þ Ke qt e jt ðk þ C KDÞe j e qs y jt Dy j s qk e qs e jt ðk þ C KDÞe j s 6 e qt y jt Dy j Ke qt e jt l p e j bðj þ Þ q e qs y jt Dy j s qk e qs ws e qs e jt l p e j s e qs ws ð7þ
P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 55 where w ¼ _e jt ðk þ C KDÞ_e j þ K_e jt ðf =K ðc þ C KDÞÞe j þ K e jt ðk þ C þ F =K _C =KÞe j Let Q = F/K (C + C KD). Then from (8) an (9), we obtain w P l p k_e k þ K_e T Qe þ K l r kek Applying the Cauchy Schwartz inequality gives _e T Qe P k_e kkqk max kek From (8) () w P l p k_e k Kk_e kkqk max kekþ K l r kek ¼ l p k_e k K kqk l max kek p þ K l p kqk max kek P ð8þ l r From (P) an (8), base on (7), it can be ensure that DV j 6. Therefore, V jþ 6 V j ð9þ From the efinition, K is a positive efinite matrix. From the efinition of V j,v j >, an V j is boune. As a result, y j (t)! when j!. Because e j (t) an _e j ðtþ are two inepenent variables, an K is a positive constant. Thus, if j!, then e j (t)! an _e j ðtþ! for t [,t f ]. Finally, the following conclusions hol 8 < q j ðtþ j!! q ðtþ for t ½; t : _q j ðtþ j! f Š ðþ!_q ðtþ From the above analysis it can be seen that the ASL-PD control metho can guarantee that the tracking errors converge arbitrarily close to zero as the number of iterations increases. The following case stuies base on simulation will emonstrate this conclusion. h 4.. Trajectory tracking of a serial robot manipulator A two egrees of freeom (DOF) serial robot is shown in Fig., which was iscusse in [6] with an aaptive ILC metho. The physical parameters an esire trajectories are the same as in [6] an liste as follows. Physical parameters: m = kg, m = 5 kg, l =m,l =.5 m, l c ¼ :5 m, l c ¼ :5 m, I =.83 kg m an I =.3 kg m. Desire trajectories an the repetitive isturbances: q ¼ sin 3t; q ¼ cos 3t for t ½; 5Š ðtþ ¼a:3 sin t; ðtþ ¼a:ð e t Þ for t ½; 5Š where a is a constant use to examine the capability of the ASL-PD control to eal with the repetitive isturbances. The control gains were also set to be the same as [6] K p ¼ K ¼ iagf; g In the ASL-PD control metho, the control gains were switche from iteration to iteration base on the following rule: K j p ¼ jk p ; Kj ¼ jk ; j ¼ ; ;...; N First, consier a =. In that way, the repetitive isturbances were the same as [6]. Fig. a shows the tracking performance for the initial iteration, where only the PD control with small control gains was use, an no feeforwar is use. It can be seen that the tracking performance was not acceptable because the errors were too large for both joints. However, at the sixth iteration where the ASL-PD control metho was applie, the tracking performance was improve ramatically as shown in Fig. b. At the eighth iteration, the performance was very goo (Fig. c). The velocity tracking performance is shown in Fig. 3. From it one can see that the velocity errors reuce from.96 (ra/s) at the initial iteration to.657 (ra/s) at the sixth iteration, an further to.385 (ra/s) at the eighth iteration for joint. The similar ecreasing tren can be foun for joint. From Figs. an 3 it can be seen that 4. Simulation In orer to have some iea about how effective the propose ASL-PD control metho woul be, we conucte a simulation stuy; specifically we simulate two robot manipulators. The first one was a serial robot manipulator with parameters taken irectly from [6] for the purpose of comparing the ASL-PD metho with the metho propose in Ref. [6] calle the aaptive ILC. It is note that the result for the serial manipulator may not be applicable to the parallel manipulator. Therefore, the secon one is a parallel robot manipulator for which we show the effectiveness of the ASL-PD control metho both in the trajectory tracking error an the require torque in the motor. l l c l l c m,l m,l q Fig.. Configuration of a serial robot manipulator. q
56 P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 Angular Error of Actuator Angular Error of Actuator Angular error (ra).5.5 Angular error (ra).5 -.5 3 4 5 (a) - 3 4 5 Angular error (ra) 4 x -3 Angular Error of Actuator 3 - - -3-4 -5 3 4 5 (b) Angular error (ra).5 x -3 Angular Error of Actuator.5 -.5 - -.5 3 4 5 x -3 Angular Error of Actuator 3 x -4 Angular Error of Actuator Angular error (ra).5 -.5 - Angular error (ra) - - -3 -.5 3 4 5 (c) -4 3 4 5 Fig.. Position tracking errors for ifferent iterations uner ALS-PD control. (a) Angular errors for two joints in the initial iteration, (b) angular errors for two joints at the sixth iteration an (c) angular errors for two joints at the eighth iteration. the tracking performances were improve incrementally with the increase of the iteration number. As the gain switching rule was introuce at each iteration, the convergence rate increase greatly compare with the control metho evelope in [6]. Table shows the trajectory tracking errors from the initial iteration to the eighth iteration. From Table it can be seen that the tracking performance was consierably improve at the sixth iteration. The maximum position errors for joints an were.4 ras an.4 ras, respectively, while the similar results were achieve after 3 iterations using the aaptive ILC in [6]. (The maximum position errors for
P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 57 Velocity Error of Actuator Velocity Error of Actuator Angular velocity error (ra/s).5.5 -.5 Angular velocity error (ra/s).5.5 -.5 - - 3 4 5 Time omain (sec.) (a) -.5 3 4 5.8 Velocity Error of Actuator. Velocity Error of Actuator Angular velocity error (ra/s).6.4. -. -.4 -.6 Angular velocity error (ra/s).5..5 -.5 -. -.5 -.8 3 4 5 -. 3 4 5 (b).4 Velocity Error of Actuator.5 Velocity Error of Actuator Angular velocity error (ra/s).3.. -. -. -.3 Angular velocity error (ra/s)..5 -.5 -. -.4 3 4 5 (c) -.5 3 4 5 Fig. 3. Velocity tracking errors for ifferent iterations uner ASL-PD control. (a) Velocity errors for two joints in the initial iteration, (b) velocity errors for two joints at the sixth iteration an (c) velocity errors for two joints at the eighth iteration. Table Trajectory tracking errors from iteration to iteration Iteration 4 6 8 max e j ðraþ.6837.4493.433.4. max e j ðraþ.5833.75..4 3.E4 max _e j ðra=sþ.9596.7835.9.657.385 max _e j ðra=sþ.5646.534.53.9. joints an were.4 an.46 (ra), respectively.) Therefore, the comparison of their metho an our metho emonstrates a fast convergence rate with the ASL-PD control metho. It shoul be note that the comparison of the velocity errors was not one as such information was not presente in Ref. [6]. It is further note that there were repetitive isturbances at each iteration in the simulation. To examine the capacity of the ASL-PD uner the repetitive isturbance conition, ifferent levels of the repetitive isturbances were applie in
58 P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 Maximum position errors.5.5.5 Effect of Repeatable Disturbances for Joint a= a= a=5 a= Maximum position errors.8.6.4. Effect of Repeatable Disturbances for Joint a= a= a=5 a= 4 6 8 Iteration number 4 6 8 Iteration number Maximum velocity errors.5.5.5 Effect of Repeatable Disturbances for Joint a= a= a=5 a= Maximum velocity errors.8.6.4..8.6.4 Effect of Repeatable Disturbances for Joint a= a= a=5 a=. 4 6 8 Iteration number 4 6 8 Iteration number Fig. 4. Effect of the repetitive isturbance on tracking errors. the simulation. Fig. 4 shows the maximum tracking errors from iteration to iteration for ifferent repetitive isturbances which are expresse by a constant a. A larger constant a means a more isturbance. It shoul be note that in Fig. 4 a = means there is no repetitive isturbance in the simulation, an a = means a large repetitive isturbance inclue in the simulation (specifically, the isturbance acte in joint was about % of the require torque an the isturbance acte in joint was about 4% of the require torque). From this figure, one can see that although the tracking errors for the initial iteration increase with the increase of the isturbance level, the final tracking errors of both the position an the velocity were the same for the ifferent repetitive isturbance levels at the final two iterations. Therefore, we conclue that the ASL-PD control metho has an excellent capability in terms of both rejecting the repetitive isturbance an robustness with respect to the isturbance level. y m r o m r l θ θ 3 q o l θ q l 5 P En-effector x o 3 l 3 m 3 r o 4 3 m 4 q 5 q 3 l 4 θ 4 r 4 o 5 q 4 4.. Trajectory tracking of a parallel robot manipulator A two DOFs parallel robot manipulator is shown in Fig. 5. Table lists its physical parameters. The robot system can be viewe as two serial robotic systems with some constraints; that is, the two en-effectors of these two serial robotic systems reach the same position. Because of this constraint, the ynamics is more complex than that of its Fig. 5. Scheme of a two DOFs parallel robot manipulator. serial counterpart. The etails about the ynamics of the parallel robot manipulator can be foune in [3]. The en-effector of the robot was require to move from point A (.7,.3), to point B (.6,.4), an to point C (.5,.5). The time uration between two nearby points
P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 59 Table Physical parameters of the parallel robotic manipulator Link m i (kg) l i (m) r i (m) I i (kg m ) h i (ra).4..5.5.6.3 3.5.8.3 4.6..5 5.6 was.5 s. The control was carrie out at the joint level where the inverse kinematics was use to calculate the joint position an velocity associate with the specific path of the en-effector. The path was esigne to pass through these three points with the objective of meeting the positions, velocities, an accelerations at these three points using the motion planning metho [4]. In this example, the control gains were selecte as follows: K p ¼ iagf; g; K ¼ iagf; g The gain switching rule was set to be K j p ¼ jk p ; Kj ¼ jk for j ¼ ; ;...; N Fig. 6 shows the position tracking performance improvement for the two actuators from iteration to iteration. From it one can see that, at the initial iteration, the maximum position errors were about. an.38 ra; only after four iterations, the maximum position errors were reuce to.8 an.5 ra; finally, after eight iterations, the maximum errors were reuce to.3 an.8 ra. Fig. 7 shows the velocity tracking performance improvement for the two actuators. At the initial iteration, the maximum velocity errors were about.7 an.68 ra/ s in the two actuators, respectively. But after four iterations, the maximum values were reuce to.5 an.4 ra/s. After eight iterations, the maximum errors in the two actuators became.46 an. ra/s for velocity, respectively. It shoul be note that, while the tracking performance was improve from iteration to iteration, the torques require to rive the two actuators were nearly the same from iteration to iteration after a few iterations. This can be seen from Fig. 8, especially from the fifth iteration to the eighth iteration. It can be seen also from Fig. 8 that the profiles of the require torques were very smooth even as the control gains become larger as the iteration number is increase. Such a property is very useful for the safe use of the actuators an the attenuation of vibration of the controlle plant. It is note that this property was misse in the switching technique in the time omain. Position error (ra)...8.6.4. -. Position Error of Actuator Initial Iteration n Iteration 4 th Iteration Position error (ra).4.3.. Position Error of Actuator Initial Iteration n Iteration 4 th Iteration Position error (ra ) -.4...3.4.5.5 -.5 - x -3 Position Error of Actuator 7 th Iteration 8 th Iteration -.5...3.4.5 6 th Iteration Position error (ra) -....3.4.5.5.5.5 x -3 Position Error of Actuator 8 th Iteration -.5...3.4.5 6 th Iteration 7 th Iteration Fig. 6. Position tracking performance improvement from iteration to iteration.
6 P.R. Ouyang et al. / Mechatronics 6 (6) 5 6 Velocity Error of Actuator 4 th Iteration 3 Velocity Error of Actuator Initial Iteration.5 n Iteration Velocity error (ra/s) -.5 - n Iteration Initial Iteration Velocity error (ra/s) - 4 th Iteration -.5...3.4.5 -...3.4.5 Velocity error (ra/s).3.. -. -. Velocity Error of Actuator 6 th Iteration 8 th Iteration 7 th Iteration Velocity error (ra/s).4.3.. -. -. Velocity Error of Actuator 6 th Iteration 7 th Iteration 8 th Iteration -.3...3.4.5 -.3...3.4.5 Fig. 7. Velocity tracking performance improvement from iteration to iteration. 3 Torque of Actuator 8 Torque of Actuator n iteration 5 th iteration 6 n iteration Torque T (NM) - - -3 8 th iteration Torque T (NM) 4 - -4 8 th iteration 5 th iteration -4...3.4.5-6...3.4.5 Fig. 8. The require torque profiles for iteration j =,5,8. 5. Conclusion In this paper, a new aaptive switching learning PD (ASL-PD) control metho is propose. This control metho is a simple combination of a traitional PD control with a gain switching strategy as feeback an an iterative learning control using the input torque profile obtaine from the previous iteration as feeforwar. The ASL-PD control incorporates both aaptive an learning capabilities; therefore, it can provie an incrementally improve tracking performance with the increase of the iteration number. The ASL-PD control metho achieves the asymptotic convergence base on the LyapunovÕs metho. The position an velocity tracking errors monotonically ecrease with the increase of the iteration number. The concept of integrating the switching technique an the iterative learning scheme works very well; especially with the achievement of a fast convergence spee. The simulation
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