A Sliding Mode Controller Design for Position Synchronization of Dual Spindle Servo Systems

Transcription

1 Available online at Procedia CIRP 1 (2012 ) th CIRP Conference on High Performance Cutting 2012 A Sliding Mode Controller Deign for Poition Synchronization of Dual Spindle Servo Sytem Burak Sencer* a, Tatuya Mori b and Eiji Shamoto a a Nagoya Univerity, Graduate School of Engineering Furo-cho, Chikua-ku, , Nagoya, Japan b Daido Amitar, , Oaka, Japan * Correponding author. Tel.: ; fa: addre: burak@upr.mech.nagoya-u.ac.jp. Abtract Motion ynchronization of ervo ytem carrie great importance in variou preciion manufacturing operation. For double-ided (duple) milling of thin plate, in particular, the cutting tool mut engage with both ide of workpiece in a preciely ynchronized manner to avoid forced vibration. A continuou liding mode controller (SMC) i developed to deal with the problem of poition ynchronization for dual pindle ervo ytem. Synchronization error i defined a the differential poition error between the two ervo ytem that follow ame reference motion command. Hence, the developed continuou SMC penalize 3 error tate; namely the individual ervo poitioning error, and the ynchronization error between them. A a reult, individual trajectory tracking a well a motion coupling objective i introduced. The final control law i implemented in the form of a PID controller in real time, and double-ided milling eperiment are performed to verify it effectivene The Publihed Author. by Publihed Elevier BV. by Elevier Selection B.V. and/or Selection peer-review and/or peer-review under reponibility under reponibility of Prof. Konrad of Profeor Wegener Konrad Wegener Open acce under CC BY-NC-ND licene. Keyword: Servo Control, Synchronization, CNC, Sliding mode control 1. Introduction Modern manufacturing method have been developed to achieve higher productivity, preciion and at the ame time to lower the manufacturing cot. Double-ided milling [1] ha been propoed for finihing thin teel plate with improved atne and urface quality (ee Fig. 1). In thi face milling proce, thin teel plate are clamped vertically and both ide of the plate are cut imultaneouly uing multiple face cutter placed on identical two pindle. The double-ided milling method relie on cutting both ide of the workpiece with preciion cutter poition ynchronization to cancel out the proce force. A a matter of fact, if the cutter on the left and right hand ide face mill do not enter the uncut urface at the ame intant, a force unbalance will occur in the normal direction to the feed motion. Thi reultant force can trigger forced vibration on thin teel plate and caue poor urface finih [1]. Thi phenomenon i hown in Figure. In the current tate, mater-lave control approach i utilized in the indutry for motion ynchronization of multiple ervo drive in computer numerical control (CNC) machine tool [2]. In double ided milling, Figure 1: Double Sided Face Milling Operation The Author. Publihed by Elevier B.V. Selection and/or peer-review under reponibility of Profeor Konrad Wegener Open acce under CC BY-NC-ND licene /j.procir

2 Burak Sencer et al. / Procedia CIRP 1 ( 2012 ) mater-lave approach ha alo been utilized for cutter poition ynchronization of the left right pindle o that they can enter the workpiece at the ame time. In mater-lave control cheme, the actual poition of the mater pindle i fed back a the reference poition command to the lave pindle. Hence, the lave pindle follow the actual poition of the mater. Thi method i eay to implement and generally give atifactory reult. However, becaue of the nature of the materlave approach, any diturbance acting on the lave i not compenated by the mater drive. Furthermore, ince the poition of the mater i fed to the lave a an input, there i an unavoidable delay between the ervo making it practically impoible to achieve perfect ynchronization. In thi reearch we addre the problem of poition ynchronization i.e. motion ynchronization in CNC ervo drive in double-ided milling. A robut liding mode controller (SMC) i deigned to achieve both etpoint tracking and motion ynchronization between two ervo drive. Wide variety of other motion ytem uch a dual motor driven feed drive and preciion motion ytem can benefit from the developed robut ynchronization controller for drive ynchronization. 2. Controller Deign The objective of the controller i to atify poition tracking objective and at the ame time actively minimize the ynchronization error between two drive. The ynchronization error i defined a the differential poition error between two drive, uch a X and Y a: ε = e e y (1) Figure 2: Forced Vibration in Double Sided Milling e 1 0 e e y = 0 1 ε 1 1 e y F 3 2 e (2) The overall objective i to minimize all the 3 error tate, namely e,e y and ε to achieve both tracking a well a ynchronization on a dual ervo ytem. In order to achieve all of thoe objective, the Sliding Mode Controller (SMC) framework [3] i ued in thi reearch. The firt tep of deigning a SMC i to elect a table liding urface. In order to penalize both tracking and the ynchronization error, a 3 dimenional liding urface (S) i defined a: S 3 1 = λ 3 3 F 3 2 e 2 1 +F 3 2 e 2 1, ds dt = S= λfe+fe (3) where λ = diag(λ,λ y,λ ε ) repreent the deired bandwidth of error tate on the liding urface. In other word, if the error are on the liding urface, they ought to lide aymptotically to origin with 1 t order dynamic at a decay of λ. Hence, λ i a deign variable that need to be tuned for each error tate individually depending on the ervo dynamic. Becaue of diturbance and un-modeled dynamic, error mut however be puhed onto the liding urface. Thi i achieved by penalizing the deviation of the error from the liding urface uing the following Lyapunov energy function (E), where e = ref i et-point tracking error of X ai and e y = y ref y i for the Y ai. The ynchronization error (ε) can be calculated from individual tracking error a: E = 1 2 ST S (4) If the derivative of the Lyapunov function ( E) i trictly negative, the energy will decreae, tate will be pulled onto the liding urface, and thu aymptotic tability can be achieved. Thi condition i atified by etting the derivative of the energy function in Eq. (4) to, E =S T S=-S T KS (5)

3 252 Burak Sencer et al. / Procedia CIRP 1 ( 2012 ) where K = diag(k, K y, K z ) contain poitive definitive feedback gain. Above condition impoe tability of the liding mode controller, and derivative of the liding urface i obtained from Eq. (5) a: S= -KS (6) Combining Eq. (3) and (6), overall error dynamic of the liding mode controller can be obtained a: e=-f -1 KλFe- F -1 (KF - λf)e. (7) The final tep i to include the dynamic of the plant i.e. the ervo ytem into the controller to derive the control law. Conidering only the rigid body dynamic of the pindle [4], ervo drive can be modeled uing equivalent inertia M = diag( M, M y ) and vicou friction B = diag(b, B y ) a; u 2 1 -d 2 1 =M B (8) where ρ i poitive oberver gain matri and κ = diag(κ,κ y κ ε ) i ued to impoe limit on the integral control action againt the diturbance o that the obervation are kept within bound (d d d + ), ρ 2 3 = ρ 0 ρ ε 0 ρ y ρ ε 0 if ˆd d and S 0,κ = 0 if ˆd d + and S (12) 1 otherwie ρ,ρ y,ρ ε are diturbance oberver gain aociated with each error tate tunable by the uer. Oberving the diturbance from Eq.(11), the final liding mode ynchronization control law can be written from Eq. (10) by replacing the diturbance with the oberved one a: [M ref +B] + KλF + ρf] [MF-1 e+ feed forward proportional gain MF -1 [KF - λf] e+ρλf e=u (13) derivative gain integral term where u 21 i the control voltage end to the ervo 3. Controller Synthei amplifier d 21 i the eternal diturbance acting againt the motion and = [, y] T, = [, y] T, = [, y] T repreent A given in Eq.(13), the propoed liding mode drive actual poition, velocity and acceleration control law i linear and conit of feed forward term, kinematic. Eq. (8) can be re-organized a: proportional, derivative a well a integral gain. The controller gain are tuned by the uer to achieve both e= ref -M -1 ( u-d-b ) (9) tracking a well deired ynchronization. Guideline for tuning the propoed ynchronization liding mode control parameter are ummarized below: where ref contain reference poition command, λ = diag(λ,λ y,λ ε ) repreent the deired bandwidth and the control law of the liding mode controller, i.e. of the error tate. Since all the error are deired to torque command to the drive, i derived by combining be minimized by the ame decay on the liding Eq. (7) and (9): urface, bandwidth can be et identical if individual drive dynamic do not differ dramatically. M r +B+MF -1 KλFe+ MF -1 [KF-λF]e+ d=u. K = diag(k (10), K y, K z ) i the feedback gain matri. It Eq.(10) repreent the overall SMC control law for the drive ytem. It hould be noted that d repreent the actual diturbance uch a cutting, or the non-linear friction force that are acting on the drive. In order to enure zero teady tate error a well a improve the robutne of the liding mode controller againt unmodeled dynamic, actual diturbance need to be replaced by the oberved one, ˆd [4]. The following diturbance oberver i defined a: ( ) ˆd=ρκ Sdt ρκ λfe+ Fe dt (11) determine how much control action i produced to penalize deviation of error tate from the liding urface. Feedback gain can be et individually for each ai. It hould be noted that etting higher gain only for the tracking error (e,e y ) will emphaize more tracking in final objective lowering the amount of ynchronization between drive. The lat parameter i the diturbance oberver gain, ρ,ρ y,ρ ε. Diturbance oberver integrate liding urface (Eq. (13)), and hould be tuned a an integral gain. Higher gain will diminih the teady tate tracking error while adding tiffne to the ytem. A balance between the gain hould be given to

4 Burak Sencer et al. / Procedia CIRP 1 ( 2012 ) minimize ynchronization error at low frequency range. Individual element of the SMC in Eq.(13) are grouped a follow: M ref +Bi ma and friction feed-forward term to widen the overall control bandwidth MF -1 KλF + ρfi the 2 2 overall proportional (P) control gain matri, MF -1 [KF-λF] i the derivative (D) gain matri and ρλf i the integral (I) gain matri and the propoed ynchronization SMC control law can be implemented a a multi-input multi-output PID a: Kp Kp = Kp y Kp y Kp,Kd = Kd Kd y Kd y Kd,Ki = Ki Ki y Ki y Ki Figure 4: Eperimental Comparion againt Tandem Control 4. Eperimental Reult (14) Performance of the propoed controller ha been validated eperimentally on a dual motor etup a well a on the actual double-ided milling machine tool [1] that i in production. Eperimental tet platform i in hown in Figure 3. Motor ma and vicou friction parameter are identified from time domain identification method and cloed loop control i implemented on a dspace 1103 DSP board at 1[kHz]. In the eperiment, the propoed ynchronization control i compared againt the tandem control where each motor i controlled individually and ynchronization i achieved by the reference trajectory. Motor are rotated at 100 [rpm] and a repeating diturbance i given to the motor 2. For fair comparion, tandem controller parameter are obtained by elimination of the cro coupling term in the propoed liding mode controller: Figure 3: Eperimental Tet Platform u u y u u y = e e y T an d e m = e e y P r o p o e d (15) The eperimental tracking error are hown in Fig. 2, a-b. It can be noted from individual motor tracking that in tandem control motor 1 doe not help motor 2. However, in the propoed control both motor are ynchronized, and ynchronization error (Figure 4c) are reduced by roughly half a compared to the tandem approach. At lat, propoed SMC control i compared againt the mater-lave type controller in actual double-ided milling cutting. The propoed controller torque command i end to the amplifier of the conventional milling machine. Spindle are equipped with 4096[count/rev] encoder with quadrature decoding, and engagement of the teeth i detected with an accelerometer mounted on the workpiece. Synchronization error during are preented in Figure 5. A hown, mater-lave controller how maimum ynchronization error of 30 [count] after cutterworkpiece engagement. In contrat, the propoed SMC reduce the ynchronization error down to 8[count]. Thi reulted in a better force balance, le vibration and better urface finih.

5 254 Burak Sencer et al. / Procedia CIRP 1 ( 2012 ) Figure 5: Actual Double Sided-milling Cutting 5. Concluion A new liding mode motion ynchronization controller i propoed in thi reearch. It performance i validated againt both tandem and mater-lave control approache in eperiment and in the actual cutting. Variety of other motion ytem uch a dual motor driven feed drive or preciion motion ytem can benefit from developed liding mode controller. Reference [1] Mori T., Hiramatu T., Shamoto E., 2011, Simultaneou doubleided milling of fleible plate with high accuracy and high efficiency, Preciion Engineering, Iue 35, pp [2] G. Pritchow, On the influence of the velocity gain factor on the path deviation, Annal of CIRP 45/1 (1996) [3] V.I. Utkin, Variable tructure ytem with liding mode, IEEE Tranaction of Automatic Control AC- 22/2 (1977) [4] Altinta Y., Erkorkmaz K., Zhu W. H., 2000, Sliding Mode Controller Deign for High Speed Feed Drive, Annal of CIRP, Vol. 49, No.1, pp