Development of a novel dynamic friction model and precise tracking control using adaptive back-stepping sliding mode controller

Transcription

1 Mechatronic 6 (2006) Development of a novel dynamic friction model and precie tracking control uing adaptive back-tepping liding mode controller Jeong Ju Choi a, eong Ik Han b, Jong hik Kim c, * a Department of Mechanical and Intelligent ytem Engineering, Puan National Univerity, outh Korea b Department of Mechatronic, uncheon Firt College, outh Korea c chool of Mechanical Engineering and RIMT, Puan National Univerity, Puan , outh Korea Received 9 March 2005; accepted 28 October 2005 Abtract The LuGre model which ha been widely ued to decribe the friction phenomenon for mechanical ytem conit of tiffne and vicou term. The tiffne term of the LuGre model how the friction torque to act linearly for the internal tate of friction dynamic. Thu it cannot repreent the hyterei phenomenon of friction in the pre-liding phae. Epecially the hyterei ha the non-local memory characteritic. In thi paper, the non-local memory hyterei phenomenon i analyzed through experiment and the improved friction model uing the Preiach model i propoed. In order to implement the Preiach model, the neural network algorithm i ued to increae the efficiency of the Preiach algorithm. Baed on the improved friction model, the adaptive back-tepping liding mode controller (MC) i deigned to improve tracking performance in the liding and pre-liding phae. To evaluate the performance of the propoed friction control ytem, experiment are executed for a ball-crew ervo ytem and the atifactory reult are hown. Ó 2005 Elevier Ltd. All right reerved. Keyword: Friction model; liding mode; Back-tepping; Neural network. Introduction In the high-accuracy poition tracking control, a major ource of uncertainty i friction which include tatic, Coulomb and vicou friction, and tick-lip and treibeck effect []. To analyze friction phenomena, the variou friction model have been propoed [2,3]. The repreentative friction model i the LuGre model which can decribe the well-known friction phenomena. The LuGre model repreent a pre-liding diplacement which ha a pring like behavior near zero relative velocity [3]. The intereting motion in the pre-liding phae i hyterei. ome propertie of hyterei phenomena are wiping-out and homogeneity which are due to non-local memory characteritic [4].In real mechanical ytem, thi non-local memory hyterei phenomenon can be oberved. In cae of the LuGre model, * Correponding author. Tel.: ; fax: addre: jkim@puan.ac.kr (J.. Kim). the hyterei of friction can not how the non-local memory phenomena becaue the tiffne term of the LuGre model repreent the friction in the pre-liding phae to act linearly for the pre-liding diplacement. Actually, the motion of diplacement in the pre-liding phae how hyterei characteritic a the diplacement i occurred. In order to how the hyterei motion in mechanical ytem, the deformed Dahl model i applied [5,7]. However, thi model i not dynamic model but tatic relation linear diplacement and friction. In order to improve the friction model which include thi phenomenon, the Preiach model i ued. A the Preiach model i applied to the internal dynamic of the LuGre model, the behavior of friction motion in the pre-liding phae doe not act linearly for the internal tate anymore and how the non-local memory property of the friction. The friction model can how more actual friction motion. However, the Preiach model need tranition value to complete the Preiach plane and it take long time to /$ - ee front matter Ó 2005 Elevier Ltd. All right reerved. doi:0.06/j.mechatronic

2 98 J.J. Choi et al. / Mechatronic 6 (2006) accomplih to obtain value through experiment. Therefore, the deformed Dahl model i ued to reduce the time for experiment to identify the value of the tranition function of the Preiach model and the neural network algorithm i ued to interpolate the value of tranition function. Baed on the propoed friction model, the precie tracking control ytem i applied to a ball-crew ytem. In order to deign the controller for the propoed friction model, the back-tepping liding mode control method i ued and unknown friction parameter are updated baed on the Lyapunov tability theorem. To how the effectivene of the deigned control ytem, the MC and propoed control ytem are compared and the experimental reult are hown. 2. Friction behavior in the pre-liding phae 2.. Analyze friction model through experiment The propertie of the friction in mechanical ytem have complex motion uch a pre-liding diplacement, break-away and tick-lip etc. Regarding thee behavior, many reearcher have tudied through experiment. The repreentative reult wa the LuGre friction model and it ha been applied to the friction compenation control ytem. The LuGre model i derived from the britle model in which it i aumed that the friction interface between two urface i thought a a contact between britle [3]. The friction force can be repreented with non-linear internal friction dynamic a follow [,3]: F LuGre ¼ r 0 z þ r _z þ r 2 _q _z ¼ _q j_qj gð_qþ z r 0 gð_qþ ¼F C þðf F C Þe ð_q=_q Þ 2 where z and q are the internal tate of friction model and the diplacement of the ytem, repectively, r 0 and r are the tiffne and vicou term of the britle model, repectively, r 2 i the vicou friction coefficient, F C, F and _q are Coulomb friction, tatic friction and treibeck velocity, repectively, and gð_qþ how the treibeck behavior. ðþ ð2þ ð3þ Fig.. Friction againt velocity. Thi model allow a good decription of the teady tate friction and mooth tranition at the velocity reveral a hown in Fig.. However, thi model cannot be enough conidering the complex behavior of friction in the pre-liding phae, that i, around zero velocity. Epecially thi friction model imply conidered that the motion of friction around zero velocity region i like a fixed tiffne. Therefore it cannot decribe the non-local memory hyterei which i affected not only by the current poition but alo by the extremum value of the pat diplacement. In order to clarify the motion in the pre-liding phae, the imulation and experimental reult are compared uing a ball-crew ytem. The layout of the ball-crew ytem and chematic diagram are hown in Fig. 2. For the experiment, P-III 800 MHz with A/D and D/A board which i manufactured by N.I. Corp. i ued and 40,000 pule/rev encoder i ued to obtain the rotation angle and more detail pecification i repreented in Table. To how the nonlocal memory characteritic and friction drift of the LuGre model, the ocillated command input wa excited to make low motion and the tail end inuoidal ignal ha 0.2 Hz frequency. imulation and experimental reult were compared againt diplacement. At thi time, command input i hown in Fig. 3 5 how the imulation reult for the LuGre model and the experimental reult repectively. Fig. 2. Layout of the ball-crew ytem.

3 J.J. Choi et al. / Mechatronic 6 (2006) Table pecification of the ytem Device Model pecification DC motor FMD-E30EA 300 W, 75 V, 4.8 A, 3000 rpm Encoder D ,000 pule/rev Torque enor DR-2 Max 5 Nm DAQ board PCI6024E oftware LabView Verion 6. The LuGre model how that the friction force i drifted to the poition but the experimental reult how the elliptical orbit of the friction againt the ocillated input torque. Thi elliptical orbit how when the command input return to the pat extremum value, the friction torque againt the input poition i initialized that i the non-local memory property. To how more carefully the effect of the non-local memory property, the reference ignal which ha the different pat extremum wa excited. Fig. 7 and 8 were obtained by the imulation baed on the LuGre model and experimental reult againt the reference poition given by Friction torque (Nmm) Poition (mm) Fig. 5. Experimental reult. 0.4 Command input (Nmm) Time (ec) Fig. 6. Reference poition. Fig. 3. Command input torque. 0.5 Friction torque (Nmm) Poition (mm) Fig. 4. imulation reult of the LuGre model. Fig. 7. Relation between the poition and the friction torque baed on the imulation reult by the LuGre model. Fig. 6, repectively. The output friction force hown in the experimental reult were initialized by the pat extremum which i the wipe-out phenomenon, but the LuGre

4 00 J.J. Choi et al. / Mechatronic 6 (2006) f p (z) f f z Fig. 9. Changing the bai from z f p to a b plane. Fig. 8. Relation between the poition and the friction torque baed on the experimental reult. model could not account it [2,6]. Through experimental reult, it i found that the LuGre model cannot how enough the complex friction motion in the pre-liding phae. Therefore a friction model to conider the motion in thi phae i required. To upplement it, we propoed the novel friction model uing the Preiach model which i uually ued in electro-magnetic ytem [8,9]. Thi model well how the non-local memory hyterei and it can be ubtituted for the linear tiffne term in the friction and internal dynamic Propoed friction model uing the neural network Preiach model The Preiach model i introduced to account for the hyterei phenomenon and the modified LuGre model i propoed a follow: F p ¼ r 0 hðzþþr _z þ r 2 _q ð4þ _z ¼ _q hðzþ gð_qþ j_qj ð5þ hðzþ ¼ f pðzþ z 6 z ba ð6þ z z > z ba where q i the diplacement of the ytem, z ba i the diplacement of the break-away and f p (z) which i ued in the Preiach model i the dynamic of the tate z before the break-away and z i the teady tate of z. In the modified friction model, the behavior h(z) i divided into two part which are before and after the break-away friction a hown in Eq. (6). Before the break-away friction, f p (z) i ued the Preiach model to repreent the non-local memory hyterei behavior. In the Preiach model for the modified friction model, it i repreented a follow: Z Z f p ðzþ ¼ lða; bþc ab zdadb ð7þ apb where l(a,b) i the ditributed function, c ab i the hyterei operator, z i the diplacement in the pre-liding phae and a and b correpond to up and down witching value of the pre-liding poition, repectively. The Preiach model hown in Eq. (7) can be numerically implemented for computing the ytem output. In order to evaluate the double integral of the Preiach model, a finite different method wa propoed by Mayergozy a follow [4]: f p ðzþ ¼ XN k¼ ½F ða k ; b k Þ F ða k ; b k ÞŠ where F ða k ; b k Þ¼f ak f ak b k, f ak i the output value on the branch which correpond to the input value a k, f ak b k i the output on the tranition curve attached to the limiting acending branch at the point f ak a hown in Fig. 9. Therefore, the relationhip between z and f p (z) can be repreented by a and b hown in Fig. 9. The numerical output of the Preiach model can be calculated by the ummation of the trapezoid Q k in Fig. 0. A the terminal input i monotonically increaed or decreaed, Eq. (8) can be rewritten a Eq. (9) and (0), repectively [8,9]. f p ðzþ ¼ XN k¼ ½F ða k ; b k Þ F ða k ; b k ÞŠ þ F ðz; b N Þ f p ðzþ ¼ XN ½F ða k ; b k Þ Fða k ; b k ÞŠ k¼ þ F ða N ; b N Þ F ða N ; zþ ð8þ ð9þ ð0þ In order to verify the induced friction model, the ocillated input i applied to the Preiach model and the imulation ( M k, m k ) ( M k, m k ) Q k Fig. 0. Geometric analyi of the Preiach model.

5 J.J. Choi et al. / Mechatronic 6 (2006) Fig.. imulation of the wiping-out phenomenon. reult i compared with the experimental reult in Fig. 8.A hown in Fig., the imulated reult how the wiping-out characteritic well. The Preiach model for the propoed friction model need the value of f ak b k which can be calculated in the tranition curve. In order to obtain them, lot of experiment hould be executed and it take a long time. Furthermore, without enough the tranition curve for the Preiach plane, the hyterei curve i repreented with the dicrete curve a hown in Fig.. In order to compenate thi dicrepancy, the neural network algorithm wa uggeted in thi paper. The neural network update the weighting value off-line to interpolate the tranition curve for the Preiach model. A ued the neural network algorithm with the updated weighting value, the experimental number of time to acquire the tranition curve could be reduced. In order to obtain the target data for learning, the deformed Dahl model i ued. Thi friction model repreent the hyterei loop baed on the width and the lope in accordance with the poitive and negative velocity friction, repectively, a hown in Eq. () and (2) [5]. f _q<0 ¼ f rðf c þ f r Þ f c ðq r qþ ðþ f c þ f r þ ðq r qþ f _q>0 ¼ f rðf c þ f r Þþf c ðq r þ qþ ð2þ f c þ f r þ ðq r þ qþ where f r i the friction force at the velocity reveral, f c i the teady tate rolling Coulomb friction force, q and q r are the diplacement of the ytem and the linear poition at the velocity reveral, repectively, and repreent the initial and reveral lope of the hyterei loop, and the friction force at the velocity reveral f r i calculated a follow: f r ¼ f n o =2 p ð þ kþ 2 þ 4k ðþkþ ð3þ 2 where k ¼ q r f c. Alo, to make the target data for learning, five time experiment are performed which are hown in Fig. 2. Baed on the experimental reult and the deformed Dahl model, the target loop are completed a hown in Fig. 3. Fig. 2. Hyterei curve obtained by experiment. Fig. 3. Interpolated hyterei curve baed on the deformed Dahl model. To contruct the neural network for the etimation tranition curve, two input neuron and 50 hidden layer neuron were ued. The chematic of neural network i hown in Fig. 4. The neuron of the input conit of the value of the extremum for the pat and current poition. z(t) Pat extremum z(t) Etimated f k k Fig. 4. chematic of the neural network for etimating the tranition curve for the Preiach plane.

6 02 J.J. Choi et al. / Mechatronic 6 (2006) Friction torque (Nmm) For the learning rule of the neural network the error back propagation algorithm wa ued a follow: W kj ðt þ Þ ¼W kj ðtþþgd pk O pj d pj ¼ f _ j ðnet pj Þ XM d pk W kj ¼ XM d pk W kj O pj ð O pj Þ f ðnetþ ¼ -0.6 Target data -0.8 Etimated k¼0 k¼0 ð4þ ð5þ þ e net ð6þ where M and g are the quantity of the output and the learning ratio, repectively and d pk and W kj are the error and the weighting factor of the output layer, repectively and O pj i the output of the output layer. The target data and the learned tranition curve are hown in Fig Adaptive back-tepping liding mode controller The back-tepping deign and liding mode control method were ued for the precie tracking control for a ball-crew ytem with non-linear friction [0,2]. To compenate the friction force, the modified LuGre friction model wa ued. Moreover, dynamic parameter r 0, r and r 2 of the friction model were conidered a unknown and the update law wa obtained by uing the Lyapunov tability theory. The chematic of the control ytem i hown in Fig. 6. The implified econd-order differential equation for the ball-crew ytem with friction i conidered a follow: J q ¼ u F p Poition (mm) Fig. 5. Target and etimated tranition curve. ð7þ where J i the moment of inertia, and F p i the friction torque. The ytem dynamic with friction can be written a follow: J q ¼ u r 0 hðzþ r f_q hðzþf ð_qþg r 2 _q ¼ u r 0 hðzþ r 2 _q r hðzþf ð_qþ ð8þ q r u where f ð_qþ ¼ j_qj gð_qþ ; r 2 ¼ r þ r 2 To deign a back-tepping controller, tate n and n 2 are defined a follow: n ¼ q r q n 2 ¼ _q r _q a ð9þ ð20þ where a = cn i the virtual control a a tabilizing function. From Eq. (9) and (20), the dynamic of tate n and n 2 are derived a follow: _n ¼ _q r _q ¼ n 2 þ a ¼ n 2 cn _n 2 ¼ n _a ð2þ ¼ q r J fu r 0hðzÞ r 2 ð_q r n 2 aþþr hðzþf ð_qþg _a ð22þ In order to tabilize tate n and n 2, the liding mode control method i ued and the liding urface i defined a follow: ¼ kn þ n 2 ð23þ And the dynamic of the liding urface can be obtained by the derivative of a follow: _ ¼ k _ n þ _ n 2 ¼ kðn 2 cn Þþ q r J fu r 0hðzÞ r 2 ð_q r n 2 aþ þ r hðzþf ð_qþg _a ð24þ In order to elect the liding mode controller, the Lyapunov candidate functional i defined a follow: V ¼ q r, q r - ξ MC Adaptive Backtepping Controller ˆ σ, ˆ σ, σ 0 ˆ2 u, ξ2, Ball-crew ytem Adaptive Law Fig. 6. chematic diagram of the precie tracking control ytem. ð25þ To atify the Lyapunov tability theory and the reaching law of the liding mode control, the derivative _ i defined a follow:. q q

7 J.J. Choi et al. / Mechatronic 6 (2006) _ ¼ D K gnðþ ð26þ where D and K are poitive definite deign parameter. From Eq. (24) and (26), the back-tepping liding mode controller i elected a follow: u ¼ Jfkðn 2 cn Þþ q r _a þ D þ K gnðþg þ r 0 hðzþþr 2 ð_q r n 2 aþ r hðzþf ð_qþ ð27þ To obtain the adaptive law of unknown parameter r 0, r 2 and r, let u define a Lyapunov candidate functional for the cloed loop ytem a follow: V ¼ V þ ~r 2 0 2c þ ~r c þ ~r 2 ð28þ 2 2c where ~r 0 ¼ r 0 ^r 0, ~r 2 ¼ r 2 ^r 2, ~r ¼ r ^r and c 0, c 2 and c 2 are deign parameter. Uing Eq. (24), the derivative of V i written by _V ¼ _ þ ~r _^r þ ~r 2 _^r2 þ ~r 2 _^r2 c 0 c 2 c 2 ¼ k n _ þ q r kn q r þ _a D K gnðþ 4. Experimental reult To evaluate the performance of tracking repone for the ball-crew ytem with friction, the propoed control ytem and the MC uing the reaching law method are compared. The propoed and MC control ytem are implemented by the LabView oftware which i produced by N.I. Corporation. Firtly, we compared the tracking performance for the friction compenator uing the LuGre model. Fig. 7 and 8 how the tracking and error repone for the MC with/without feedforward friction compenator. At thi time, the ame gain for the MC are ued to compare the effect of the friction compenator. A may gue, the MC with feedforward friction compenator i better than without it. However, repone of both cae are hifted a little poitive ide becaue the parameter of friction dynamic are different from poitive and negative direction. To evaluate the performance of the propoed control algorithm, we compared the tracking and error repone for the MC with friction compenator againt the reduced reference ignal a hown in Fig. 9. For the experiment, J ð^r 0 r 0 ÞhðzÞ J ð^r 2 r 2 Þð_q r n 2 aþ þ J ð^r r ÞhðzÞf ð_qþ _a þ ~r 0 _~r 0 þ ~r 2 _~r 2 þ ~r _~r c 0 c 2 c ¼ ~r 0 J hðzþþ _^r þ ~r 2 c 0 J ð_q r n 2 aþþ _^r2 c 2 ~r 2 J hðzþf ð_qþ _^r D 2 K gnðþ ð29þ c To atify the Lyapunov tability theory, we can chooe adaptation law a follow: _^r 0 ¼ c 0 J hðzþ _^r 2 ¼ c 2 J ð_q r n 2 aþ ð30þ _^r ¼ c hðzþf ð_qþ J And the final adaptive back-tepping liding mode control law i elected a follow: u ¼ Jfkðn 2 cn Þþ q r _a þ D þ K gnðþg þ ^r 0 hðzþþ^r 2 ð_q r n 2 aþ ^r hðzþf ð_qþ ð3þ In order to evaluate the performance of the propoed control ytem, the MC uing the reaching law method which wa propoed by Gao and Hung [] and the propoed control ytem were compared. The tructure of the MC uing the reaching law i imply repreented by u ¼ u eq þ D þ K gnðþþf friction ð32þ where u eq i the equivalent control law and F friction i the feedforward friction compenation term. Fig. 7. Tracking repone for the reduced reference ignal. Fig. 8. Error repone of the tracking ignal.

8 04 J.J. Choi et al. / Mechatronic 6 (2006) Fig. 9. Tracking repone for the reduced reference ignal. account for the non-local memory hyterei. In order to evaluate of the propoed friction model in the pre-liding phae, the wipe-out phenomenon i compared with experimental and computer imulation. The identification for tranition curve of the Preiach model i executed by the neural network method. A ued the neural network algorithm, we can reduce the time for experiment. The developed friction model i incorporated into a ball-crew ytem. The back-tepping liding mode control ytem i propoed and applied to the precie tracking ytem. The unknown parameter for friction model i adapted by the updating rule which are induced by the Lyapunov tability condition. The tracking performance of the propoed control ytem i compared with that of the tandard liding mode control ytem and the good effectivene of the propoed control ytem ha hown through experimental reult. Acknowledgment Thi paper preent reearch reult of the Korea cience and engineering foundation (R ). Reference ame gain of both the control ytem are ued in thi experiment. Actually, thee gain were obtained by the trial and error method becaue repone for large gain of the MC howed the table repone in the liding phae but the tracking repone blew up for mall reference ignal around the pre-liding phae. A hown in Fig. 9 and 20, it i found that the propoed MC how better tracking performance than the MC with feedforward friction compenator. Each root mean quare (RM) value for the error ignal i 0.08 for the propoed MC and for the MC with feedforward friction compenator. 5. Concluion Fig. 20. Error repone of the tracking ignal. The improved friction model i developed baed on the LuGre model in thi paper. Preiach model i propoed to [] Brian AH, Dupont P, Canuda De Wit C. A urvey of model, analyi tool and compenation method for the control of machine with friction. Automatica 994;30(7): [2] Dupont P, Armtrong B, Hayward V. Elato-platic friction model: contact compliance and tiction. In: Proceeding of the American control conference, Chicago, Illinoi, June p [3] Canuda De Wit C, Olon H, Atrom K, Lichinky P. A new model for control of ytem with friction. IEEE Tran Automat Contr 995;40: [4] Mayergoyz ID. Mathematical model of hyterei. pringer-verlag; 99. [5] Awaddy BA, hih WC, Aulander DM. Nanometer poitioning of a linear motion tage under tatic load. Tran IEEE/AME Mech 998;3(2):3 9. [6] Dupont P, Hayward V, Armtrong B, Altpeter F. ingle tate elatoplatic friction model. IEEE Tran Automat Contr 2002;47(5): [7] Dahl PR. A olid friction model. The Aeropace Corporation, EL egundo, CA, Tech. Rep. TOR-58(307-8)-, 968. [8] Cincotti, Daneri I. A non-linear circuit model of hyterei. IEEE Tran Magn 999;35(3): [9] Ge P, Jouaneh M. Modeling hyterei in piezoceramic actuator. Preci Eng 995;7(3):2 2. [0] Lin FJ, hen PH, Hu P. Adaptive backtepping liding mode control for linear induction motor drive. IEE Proc Electron Power Appl 2002;49(3): [] Gao W, Hung JC. Variable tructure control of nonlinear ytem: a new approach. IEEE Tran Ind Electron 993;40(): [2] Li Y, Wikander J. Model reference dicrete-time liding mode control of linear motor preciion ervo ytem. Mechatronic 2004;4(7):