Adaptive Fuzzy Sliding Controller with Dynamic Compensation for Multi-Axis Machining

Transcription

1 J. Software Engneerng & Applcatons, 009, : do:10.436/jsea Publshed Onlne November 009 ( Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng Hu LIN 1, Rongl GAI 1 Shenyang Insttute of Computng Technology, Chnese Academy of Scences, Shenyang, Chna; School of Computer Scence and Technology, Unversty of Scence and Technology of Chna, Hefe, Chna. Emal: {lnhu,garl}@sct.ac.cn Receved June 31 st, 009; revsed August 30 th, 009; accepted September 10 th, 009. ABSTRACT The precson of mult-axs machnng s deeply nfluenced by the trackng error of mult-axs control system. Snce the mult-axs machne tools have nonlnear and tme-varyng behavors, t s dffcult to establsh an accurate dynamc model for mult-axs control system desgn. In ths paper, a novel adaptve fuzzy sldng model controller wth dynamc compensaton s proposed to reduce trackng error and to mprove precson of mult-axs machnng. The major advantage of ths approach s to acheve a hgh followng speed wthout overshootng whle mantanng a contnuous CNC machne tool process. The adaptve fuzzy tunng rules are derved from a Lyapunov functon to guarantee stablty of the control system. The expermental results on GJ-110 show that the proposed control scheme effectvely mnmzes trackng errors of the CNC system wth control performance surpassng that of a tradtonal PID controller. Keywords: Fuzzy Sldng Control, Adaptve, Compensaton Control, Trackng Error 1. Introducton Whle basc machne tool errors form one of the major sources of naccuracy n mult-axs machnng, achevng hgh precson n actual machne tool performance also crtcally depends upon dynamc performance of the ndvdual axs controllers [1]. Axs controllers n processng of CNC machnng need to synchronze mult-axs motons to generate the requred machned surface []. In general, each partcular machne tool axs has ts own poston and velocty, beng drven separately along the desred tool path generated by the nterpolator of the CNC system. When mult-axs machne tools are machnng lnear segments n step mode, the mpact of trackng error on machnng accuracy s not a serous problem. Otherwse, t would be dffcult to elmnate trackng errors whle trackng the axal poston command, whch then contrbutes to contour error formaton. Ths stuaton would be especally problematc on the condtons of contnuous processng of the mult-axs machnng. As one example, Fgure 1 provdes an llustraton ndcatng the nfluence of trackng error n a -axs system whle machnng n the auto mode. The nterpolator s responsble for generatng the deal poston commands for coordnated movement of the axes, whch are ndcated by postons B and B +1. Guded by the nterpolator, the axs controller manages the task of controllng the movement of a partcular axs. The actual machnng ponts of A and A +1 then correspond to nterpolaton ponts B and B +1 due to the trackng error. Accordngly, the trackng error s about B -A and B +1 -A +1, thus the machnng error s BD, whch s called contour error, and s the result of the trackng error of each axs. Over the past several decades, many advanced controllers have been desgned whch attempt to rectfy these trackng errors. Some examples of these nclude the self-tunng fuzzy PID type controller [3 4], fuzzy sldng controller [5], adaptve sldng controller wth self-tunng fuzzy compensaton [6 10], adaptve neural fuzzy control [11], fuzzy controller and turnng algorthm [1], and self-tunng fuzzy logc controller [13]. Fgure 1. The nfluence of trackng error Copyrght 009 ScRes

2 Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng 89 Overshootng and oscllaton of the mult-axs controller are specal concerned n the artcle. Snce t s dffcult to establsh an accurate dynamc model of the mult-axs machne tools, here, a novel adaptve fuzzy sldng model controller wth dynamc compensaton s proposed. The advantages of ths model nclude ts capacty to reduce trackng error, thus contour error and mprove mult-axs machnng precson.. Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton Adaptve fuzzy sldng controller wth dynamc compensaton s the core of the mult-axs CNC system. The basc functon of ths system s to generate the poston of servo axes of mult-axs machne tools. Dgtal encoder feedbacks servo to montor the poston of the servo motor to the CNC system referred to as the actual axs locaton of the machne tool. Interpolaton of the CNC system generates the deal trajectory of each axs, whch s called the deal locaton of axs of the machne tools. Whle avodng overshootng and oscllaton n CNC machnng, the man role of the adaptve fuzzy sldng controller wth dynamc compensaton s to mnmze the trackng error, as computed by the actual versus deal locatons (Fgure ). The adaptve fuzzy sldng controller wth dynamc compensaton s a steady-state control system that ncludes two components - the dynamc compensaton and adaptve fuzzy sldng control..1 Dynamc Compensaton Dynamc compensaton s calculated from an array of parameters ncludng computng of trackng error, deadzone of trackng error, parameters of dynamc compensaton and compensaton control. The purpose of these determnatons s to mnmze trackng error of the CNC machnng tools (Fgure 3). nterpolaton ponts R+ adaptve fuzzy sldng controller wth dynamc compensaton C- dynamc compensaton adaptve fuzzy sldng control software DAC encoder amplfers motor hardware Fgure. The servo axs control system usng adaptve fuzzy sldng controller wth dynamc compensaton Fgure 3. The frame of dynamc compensaton.1.1 Computng Trackng Error The method used for computng trackng error conssts of calculatng trackng error n each nterpolaton cycle based on the actual versus deal locatons of the CNC machnng tool, as expressed n the followng formula: e p c (1) where e ( e, e,,,, ) X e Y e Z e A e B s the trackng error n C the th nterpolaton cycle, p ( x, y, z, a, b, c ) s the poston of the deal locaton coordnate of the th nterpolaton cycle, c ( x, y, z, a, b, c ) s the poston of the actual locaton coordnate of the th nterpolaton cycle, as ndcated by the dgtal encoder..1. Deadzone of Trackng Error The deadzone of trackng error parameter s used to defne a range, n whch dynamc compensaton s not used. The calculaton of ths parameter s based on the assumpton that the maxmal dstance between the actual locatons of mult-axs and the correspondng nterpolaton postons of mult-axs s wthn an nterpolaton step when the program s runnng at the programmng velocty. That s: e v t () where e s defned as presented n Equaton (1), t s the nterpolaton cycle tme of the mult-axs CNC system, v s the programmng velocty of the partcular workpece program. It has been proven by Roger that the trajectory planner makes the maxmal nterpolaton error [14], expressed by e max, when t passes the adjacent program wth programmng velocty. Snce the maxmal nterpolaton error s accepted by the operator of CNC system, t can be appled to bound the dynamc compensaton of trackng error, and be referred to as the deadzone of the trackng error. If, and only f, the trackng error s greater than e max, the dynamc compensaton s actve. Practcal experence shows that the pursut of reducng the trackng error to zero easly leads to vbraton and overshootng n mult-axs machnng, so the ntroducton of a deadzone of trackng error s conducve to the stablty of mult-axs controller..1.3 Parameters of Dynamc Compensaton The overshootng of mult-axs machne tools wll result n oversze cuttng. Snce ths s unacceptable for hgh precson machnng, dynamc compensaton needs to avod overshootng and vbraton of the mult-axs CNC system. Parameters of dynamc compensaton nclude the followng prncples: 1) The parameters of dynamc compensaton are zero when the velocty of trajectory plannng s zero. ) The parameters of dynamc compensaton are zero when the trackng error acheves the constrant of the deadzone, defned by chapter.1.. 3) The parameters of dynamc compensaton can n- Copyrght 009 ScRes

3 90 Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng crease n a step-wse manner when the mult-axs CNC system s runnng n an acceleraton phase. 4) The parameters of dynamc compensaton requre a step-wse reducton when the mult-axs CNC system s runnng n a deceleraton phase. 5) The parameters of dynamc compensaton acheve a peak value when the mult-axs CNC system s runnng n the programmng velocty. Based on the prncples of dynamc compensaton, the followng functon can be selected as the parameter of dynamc compensaton: v comp e le e max (3) v Where v s the velocty n the th trajectory cycle, le s the length of the trackng error of mult-axs n the trajectory cycle, defned by Equaton (4), comp ( comp X, comp, comp,,, ) Y comp Z comp A comp B f the parameter of C dynamc compensaton s n the trajectory cycle. X Y Z A B C le e e e e e e (4).1.4 Compensaton Control The output of the adaptve fuzzy sldng controller wth dynamc compensaton for mult-axs machnng proposed n ths paper can be expressed by the followng equaton: ucol uc u (5) Where u col s the total output of the poston controller of the mult-axs CNC system, u c s the output of the dynamc compensaton controller and u s the output of the adaptve fuzzy sldng controller. We can defne the rules for compensaton control as follows: uc k p comp le emax (6) uc 0 0 le emax Where k p s the proportonal gan of the CNC system, whle the meanngs of other varables are presented as Equaton (3), (4) and (5). The dynamc compensaton controller adds a compensaton varable to each axs accordng to the vector of the mult-axs trackng error. It can produce a rapd reducton n the trackng error whle smultaneously reducng the frequency of adjustng parameters of the adaptve fuzzy sldng controller. In addton, the dynamc compensaton controller can mprove the stablty of the numercal mult-axs control system.. Adaptve Fuzzy Sldng There exst a number of nonlneartes and uncertantes n the mult-axs control system. These result from structural or unstructured uncertantes, such as backlash, saturaton and frcton. It s very dffcult to establsh the boundary for these nonlneartes and uncertantes n the CNC system, partcularly n a mult-axs system. Dynamc compensaton parameters are changed accordng to the trajectory velocty and therefore can contrbute to an ncrease n the uncertanty of the mult-axs system. As an approach to rectfy these problems, an adaptve fuzzy sldng control structure s proposed. Ths structure s referred to as adaptve fuzzy sldng controller wth dynamc compensaton snce t ncorporated the dynamc compensaton algorthm dscussed above. Dynamc systems wth multple knds of nonlneartes and uncertantes, such as mult-axs machne systems, can be expressed as the followng Formula [15 17]: x () t f( X,) t f( X,) t u (7) ( n) ˆ ( n1) T Where X ( x, x,..., x ) s the state of the system n n and u R and x R are the control nputs and outputs of the system, respectvely. The nonlnear model system conssts of the reference model f ˆ( Xt, ) and multple nonlneartes and uncertantes f ( Xt, ), whch nclude the dynamc compensaton and nherent nonlnear characterstcs of the mult-axs CNC system. A hypothess can be proposed based on: f ( xt, ) F( xt, ) (8) The tme-varyng sldng surface s defned as: n1 d sxt (, ) e0. dt It s referred to as the sldng swtchng lne n D space, where λ s a postve constant and ( n1) T e X X d ( e, e,..., e ) s the trackng error vector. In general, when consderng second-order system as an example, the deal fuzzy sldng control law s: (9) u* fˆ( X, t) x d e k( x, x ) sat( s, ) (10) Where k( x, x ) and sat ( s, ) are defned as: kxx (, ) Fx ( ) (11) 1, s 1 sat(, s ) s, 1 s 1 1, s 1 (1) And η s a postve constant, whle φ s the thckness of the boundary layer. Suppose: ( x) 1 m1 m l A l11 l1 1 ( x ) ( ( x )) l A (13) Copyrght 009 ScRes

4 Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng 91 G61G05.1Q1F10000 X Y15.3 Z A86.46 C X-1.46 Y Z A C X Y Z A C X-1.53 Y Z A86.77 C X Y Z A C X-1.08 Y Z A C X-.99 Y0.635 Z A C X-.905 Y1.57 Z A87.7 C X- 8 Y.51 Z A C X-.733 Y3.458 Z A C X-.648 Y4.408 Z A C M Fgure 4. The program for test Where the adaptve law of fuzzy sldng controller can be desgned as [18]: T = e p ( x) (14) And γ s a postve quantty, P s a defnte symmetrc matrx and P s the fnal array of the defnte symmetrc matrx P. Stablty of the adaptve fuzzy sldng controller s guaranteed by the Lyapunov functon. 3. Experment The expermental platform s a mult-axs CNC system, referred to as GJ-310 CNC. Ths system s based on the PC archtecture, n whch the servo board and the I/O board are connected by the SSB bus. The adaptve fuzzy sldng controller wth dynamc compensaton descrbed above s used as an axs poston controller n the moton control component of the GJ-310 CNC. We selected a program wth a smultaneous 5-axs movng to test the proposed adaptve fuzzy sldng controller as shown n Fgure 4. The program conssted of a mcro-lne segment, whose length was approxmately 1mm. Corners whch were present between the small lne segments enabled for an easy detecton of the trackng error effect upon machnng accuracy. 3.1 Expermental Parameters The parameter settngs of the GJ-310: the axs number s 6, the encoder nput equvalent s 16384, the servo perodcal tme s ms, the nterpolaton cycle tme s ms, the maxmal error s 0.mm, the maxmal shape error s 0.05mm, and the other parameters are shown as table 1. Accordng to establshed gudelnes of adjustng numercal control machnes, we obtaned the control rules, as shown n table. Trangle membershp functons of the nput varables are shown n Fgure Experment Results When machnng the program (Fgure 4) wth the GJ-310 CNC system, whle separately usng the adaptve fuzzy sldng controller wth dynamc compensaton and the PID controller, we can get the poston of each axs, as shown n Fgure 6. In Fgure 6, seres 1 s the deal poston of each axs generated by the trajectory planner (ndcated wth a dot), seres s the actual poston of each axs generated by the PID controller (ndcated wth a dash) and seres 3 s the actual poston of each axs generated by the adaptve fuzzy sldng controller wth dynamc compensaton (ndcated wth a sold). We obtaned the trackng errors of each axs as shown n Fgure 7, where seres 1 s the trackng error generated by the PID controller (ndcated wth a dot) and seres s the trackng error generated by the adaptve fuzzy sldng controller wth dynamc compensaton (ndcated wth a sold). Note, the startng poston of the program s ndcated by -1mm, 15mm, -150mm, 86mm, -91mm. 3.3 Expermental Analyss From Fgure 6 t s clear that the adaptve fuzzy sldng Table 1. The parameters of GJ-310 CNC category Parameters axs X Axs Y Axs Z unts Axs A Axs B Axs C unts D/A channel Encoder channel Proportonal gan Integral gan Dfferental gan Maxmum velocty mm/mn deg/mn Maxmum error mm arcmm Output offset mm arcmm Jog velocty mm/mn deg/mn Transfer velocty mm/mn deg/mn Maxmum voltage v v Jog acc-tme ms ms Transfer acc-tme ms ms Copyrght 009 ScRes

5 9 Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng The output u The Trackng Error e Table. The control rules Error change rate e LN MN SN ZZ SP MP LP LP LP LP LP LP LP LP LP MP SP SP SP MP MP LP LP SP ZZ SP SP SP MP MP LP ZZ ZZ ZZ ZZ ZZ ZZ ZZ ZZ SN LN MN MN SN SN SN ZZ MN LN LN MN MN SN SN SN LN LN LN LN LN LN LN LN 4. Conclusons Strctly speakng, the CNC system typcally has more than one axs. Therefore, an deal lnear system s not avalable. In ths way, research s drected at resolvng nonlnearty and uncertanty control problems of multaxs machnng tools, and reducng trackng errors of mult-axs. CNC systems have mportant mplcatons for Fgure 5. The trangle membershp functons of the nput varables controller wth dynamc compensaton and the PID controller produce nearly dentcal endng ponts. Moreover, use of the adaptve fuzzy sldng controller wth dynamc compensaton results n smooth control, thereby producng an accurate trackng of the trajectory wth no overshootng or vbraton n the machnng. From Fgure 7 t s apparent that the adaptve fuzzy sldng controller wth dynamc compensaton can reduce trackng errors of each servo axs. As one example of the program, an examnaton of axs Z whch has the longest movng trajectory, provdes for a means of comparson of the effects between the two controllers. The adaptve fuzzy sldng controller wth dynamc compensaton reduced the trackng error by almost 44% of the PID controller and the trackng error of axs C, whose movng trajectory was the shortest n the example of program, was reduced by about 19% of the PID controller. From the above smulaton results, the proposed adaptve fuzzy sldng controller wth dynamc compensaton demonstrates a perfect performance, whch can abolsh effects of the system trace performance caused by trackng nonlneartes and uncertantes dsturbances. In ths way, the control system can produce a hgh degree of precson n mult-axs machnng. Fgure 6. The poston of each axs Copyrght 009 ScRes

6 Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng Acknowledgment Ths paper was supported by Natonal Technology Support Program of Mnstry of Scence and Technology under Grant 007BAP0B01 and Chnese Academy of Scences Knowledge Innovaton Program under Grant KGCX-YW-119. Fgure 7. The trackng error of each axs both theoretcal consderatons and practcal applcatons. Ths paper combnes dynamc compensaton control wth adaptve fuzzy sldng control for the desgn of an adaptve fuzzy sldng controller wth dynamc compensaton. The results of experment on the GJ-310 system show that ths controller can elmnate overshootng and vbraton n a CNC machnng control system, and mprove the precson of mult-axs machnng. REFERENCES [1] R. Ramesh, M. A. Mannan, and A. N. Poo, Trackng and contour error control n CNC servo systems, Internatonal Journal of Machne Tools and Manufacture, Vol. 45, No. 3, pp , 005. [] C.-C. Lo, A tool-path control scheme for fve-axs machne tools, Internatonal Journal of Machne Tools and Manufacture, Vol. 4, No. 1, pp , 00. [3] E. Yeşl, M. Güzelkaya, and I. Eksn, Self tunng fuzzy PID type load and frequency controller, Energy Converson and Management, Vol. 45, No. 3, pp , 004. [4] L. Wang, W. Du, H. Wang, and H. Wu, Fuzzy self-tunng PID control of the operaton temperatures n a two-staged membrane separaton process, Journal of Natural Gas Chemstry, Vol. 17, No. 4, pp , 008. [5] B. Wu, H. Ln, D. Yu, R. F. Guo, and R. L. Ga, Desgn of fuzzy sldng-mode controller for machne tool axs control, The 33 rd Internatonal Conference on Computers and Industral Engneerng, CIE70, 004. [6] Y. Zhang, F. Wang, T. Hesketh, D. J. Clements, R. Eaton, Fault accommodaton for nonlnear systems usng fuzzy adaptve sldng control, Internatonal Journal of Systems Scence, Vol. 36, No. 4, pp. 15 0, 005. [7] S.-J. Huang and W.-C. Ln, Adaptve fuzzy controller wth sldng surface for vehcle suspenson control, IEEE Transactons on Fuzzy Systems, Vol. 11, No. 4, pp , 003. [8] S.-J. Huang and H.-Y. Chen, Adaptve sldng controller wth self-tunng fuzzy compensaton for vehcle suspenson control, Mechatroncs, Vol. 16, No. 10, pp , 006. [9] R. J. Wa, C. M. Ln, and C. F. Hsu, Adaptve fuzzy sldng-mode control for electrcal servo drve, Fuzzy Sets and Systems, Vol. 143, No., pp , 004. [10] R. Shahnaz, H. M. Shanech, and N. Parz, Poston control of nducton and DC servomotors: A novel adaptve fuzzy PI sldng mode control, IEEE Transactons on Energy Converson, Vol. 3, No. 1, pp , 008. [11] C.-S. Chen, Dynamc structure adaptve neural fuzzy control for MIMO uncertan nonlnear systems, Informaton Scences, Vol. 179, No. 15, pp , 009. [1] D. Q. Truong and K. K. Ahn, Force control for hydraulc load smulator usng self-tunng grey predctor - fuzzy PID, Mechatroncs, Vol. 19, No., pp , 009. [13] J. Jamaludn, N. A. Rahm and W. P. Hew, Development Copyrght 009 ScRes

7 94 Adaptve Fuzzy Sldng Controller wth Dynamc Compensaton for Mult-Axs Machnng of a self-tunng fuzzy logc controller for ntellgent control of elevator systems, Engneerng Applcatons of Artfcal Intellgence, pp. 1 1, 009. [14] R. S. Blom, Desgn and development of a real-tme trajectory planner for the enhanced machne controller, Intellgent Systems Dvson Gathersburg, Maryland Unted States of Amerca : Natonal Insttute of Standards and Technology Manufacturng Engneerng Laboratory, [15] J. E. Slotne, Sldng controller desgn for nonlnear systems, Internatonal Journal of Control, Vol. 40, No., pp , [16] C. L. Hwang and C. Y. Kuo, A stable adaptve fuzzy sldng mode control for affne nonlnear systems wth applcaton to four-bar lnkage systems, IEEE Transactons on Fuzzy Systems, Vol. 9, No., pp. 38 5, 001. [17] I. Eker, S. A. Aknal, Sldng mode control wth ntegral augmented sldng surface: Desgn and expermental applcaton to an electromechancal system, Electrcal Engneerng, Vol. 90, No. 3, pp , 008. [18] R. L. Ga, H. Ln, X. P. Qn, D. Yu, and R. F. Guo, Adaptve fuzzy sldng control desgn for the axs system wth dynamc multple knds of nonlneartes and uncertantes, Proceedngs of the 38th Internatonal Conference on Computers and Industral Engneerng. Vol. 3, pp , 008. Copyrght 009 ScRes