Component Swapping Modularity for Distributed Precision Contouring

Transcription

1 IEEE/ASME TRANSACTIONS ON MECHATRONICS 1 Component Swapping Moularity for Distribute Precision Contouring Aza Ghaffari, Member, IEEE, an A. Galip Ulsoy, Fellow, IEEE Abstract Prior works use numerical simulations to verify component swapping moularity (CSM for various systems. Moreover, current CSM algorithms o not investigate control parameter allocation for a realistic control configuration when the control consists of multiple parts. Thus, this work primarily focuses on presenting an experimentally valiating an empirical methoology to allocate an calibrate various parts of the controller to improve CSM in a precision multi-axis servosystem. As a seconary contribution, the concept of a moifie reference contour is introuce to simplify control allocation an to improve the moularity of the cross-coupling control (CCC which is utilize to achieve high-precision contouring. First, the unifie linear CCC algorithm is presente for multiaxis servo-systems. Initially, the sensitivity of the contouring algorithm versus the control parameters is stuie numerically. Then, empirical calibration an sensitivity analysis are conucte to obtain the optimal set of control parameters. Hence, the lowest feasible contour error is obtaine for possible configurations of the servo-system. It is shown that the results of the empirical analysis are consistent with those of the numerical analysis. Despite ramatic ifferences between the servo-system variants, experimental results show that full CSM is achieve with the same controller for the variants of the servo-system. Inex Terms Component swapping moularity, istribute control, precision contouring, cross-coupling control. I. INTRODUCTION The simplest controller configuration is esirable to satisfy a given set of esign criteria for all variants of a networke system compose of swappable components. Component swapping moularity (CSM is a esign measure to evaluate the flexibility of a moular control system to cope with the effect of variable system characteristics ue to swapping a component with a comparable counterpart. A istribute control structure as shown in Fig. 1 has been propose to improve CSM. The propose structure inclues a base an local controller. The base controller is the same for all variants of the system. The local controller, usually place in the swappable component, is calibrate to achieve the best feasible close-loop performance. Such component swapping moularity is achieve when esire performance, sufficiently close to a centralize controller optimize for each variant, is realize for the istribute controller structure in Fig. 1. Component swapping moularity is improve when a esire close-loop performance is achieve with a low-orer local controller. Various esign algorithms to achieve CSM have been propose, incluing the LMI-base Metho [4], Direct Metho [13], an 3-Step Metho [1]. Numerical simulations Authors are with Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI , aghaffar@umich.eu an ulsoy@umich.eu. Controller Base Controller Network Swappable Component Local Component Local Component Local Controller Component Harware Local Controller Component Harware Controller Harware Controller Harware Plant System Harware Fig. 1: Swappable smart component in a control network. have proven CSM esign to be effective in various applications, incluing engine ile spee control [4], [11], variable camshaft timing engine [2], an battery swapping moularity in plug-in hybri electric vehicles [12]. This work primarily focuses on eveloping an empirical methoology to allocate an calibrate control parameters in a realistic setting such that minimal control reesign is ensure when a component is swappe. Although conventional stability tools are use uring the control esign process, it shoul be note that the contribution of this work is focuse on experimental verification of CSM esign in istribute precision contouring systems [7], [14]. The experimental setup inclues a two-axis micro-precision servo-system with moular control boars an axes. The ultimate goal of a precision positioning application is to keep the actual contour ientical to the reference contour. So, the notion of contour error is introuce which refers to the shortest istance from the current position to the reference contour. The prevailing esign tren for such contouring algorithms, known as variable-gain cross-coupling control (CCC, was originally introuce by Koren [9], [1] an is shown in Fig. 2. A basic CCC is comprise of two parts: 1 a contour error estimate (CEE algorithm, compose of time-varying gains C 1 an C 2, which calculates the shortest istance from current position to the reference contour, an 2 an error compensator, G c (s, which is esigne to rive the contour error to zero. The wiely use CCC algorithms effectively improve contouring precision by compensating for the effect of uncertainty an by reucing contour error in multi-axis servosystems. The variable-gain CCC is a multi-input-multi-output compensator esigne to incorporate coupling between all the axes. Therefore, ue to the centralize nature of the variable-gain CCC, it is not feasible to fully istribute the parameters of the variable-gain CCC into local an base controllers [7], [14]. Thus, it is not possible to achieve full CSM for the variable-gain CCC. Ghaffari an Ulsoy [6] have introuce

2 IEEE/ASME TRANSACTIONS ON MECHATRONICS 2 r 1 (θ r 2 (θ Position control p 1 C 1 p 2 + G c (s C 2 Position control ˆ 1 Dist. Comp. u 1 + Axis 1 C 1 C 2 CCC 1 CCC 2 u 2 + Axis 2 Dist. Comp. ˆ 2 p 2 v 2 v 1 p 1 Contour error estimate & Cross-coupling control Fig. 2: Variable-gain CCC for a two-axis servo-system, originally propose by Koren an Lo [9]. the moifie error feeback concept to eliminate the nee for a separate cross-coupling loop by moifying the error signal use in each of the position feeback loops. In this work, it is shown that the moifie error feeback concept can be transforme into a moifie reference contour concept such that the contouring algorithm is further simplifie. In either case, the cross-coupling is achieve without changing the position control structure. The moifie reference is a convex combination of the leaing reference point an an estimate of the closest point of the reference contour to the current position which is obtaine using the Newton-base CEE algorithm [6]. An optimal moifie reference is obtaine by properly tuning the positive gain λ < 1, where λ remains the same for all the axes. So, G c (s, C 1, an C 2 are no longer neee which reuces the number of control parameters which in turn improves CSM. The propose contouring algorithm, shown in Fig. 3(b for one axis, inclues a position control, feeforwar compensation, isturbance compensation, an the Newton-base CEE algorithm [6]. The moifie reference contour reuces the compensator in the CCC algorithm to a proportional gain, λ, which further simplifies the tuning process of the control loop. A servo-system is comprise of a power amplifier, electric machine, an linear stage. Swapping an axis, partially or entirely, with a compatible counterpart may change the system ynamics which may compromise the performance of the positioning or manufacturing process unless the moifie system controller is re-calibrate. The propose contouring algorithm is fully moular an suitable for use in CSM esign. Therefore, it is convenient to conuct a sensitivity analysis of the control structure of each axis without consiering the coupling effect from the other axis. Moreover, the result of the sensitivity analysis is use to optimally istribute the control elements between the local an base controllers such that the best contouring performance is achieve with the simplest local controller. Given the propose structure of the integrate controller (see Fig. 3(b, first, a numerical sensitivity analysis is conucte to stuy the effect of each parameter on contouring performance. Then, an empirical methoology is evelope to evaluate the sensitivity of the close-loop performance with respect to the control parameters. It is shown that the empirical results are aligne with the numerical results. With careful esign of the controller full CSM can be achieve even when all the control parts are locate in the base controller. Thus, when an axis is swappe the control nee not be re-calibrate to ensure the lowest feasible contour error. The isturbance compensation ramatically attenuates the effect of friction an nonlinearity [3], [8]. An initial estimate of the axis ynamics is require to implement the isturbance compensation. So, the first step of the control esign is to ientify system ynamics of each axis to implement the isturbance compensation. Then, the compensate loop can be moele using a more accurate linear moel. Thus, the axis moel after isturbance compensation is obtaine to esign the rest of the controller. Boe plots of the close-loop system plus extensive experimental simulations show that the closeloop system is robust with respect to the parameters of the isturbance compensation loop. So, a fixe set of reasonably large gains for the position control loop guarantees highprecision contouring. Thus, swapping moularity is improve. The work of Ghaffari an Ulsoy [] evelops the empirical sensitivity analysis an provies the necessary steps for moel ientification an control esign. This paper extens the results of Ghaffari an Ulsoy [] in two irections: 1 a numerical sensitivity analysis is provie, an it is shown that the numerical results are in line with the experimental results. 2 After esigning the CCC for CSM, extensive experiments are conucte to verify that the obtaine controller inee requires no tuning when one axis is swappe. Thus, emonstrating experimentally for the first time that full CSM is achieve. The rest of the paper is organize as follows: Section II presents the integrate contouring algorithm using the moifie reference concept. Section III explains the system configuration an the ientification process. Section IV presents the numerical an empirical sensitivity analysis an provies experimental results an numerical analysis to support the obtaine control parameter set. Section V conclues the paper. II. MODIFIED REFERENCE AND INTEGRATED CONTOURING ALGORITHM The ynamic equations of a single servo-axis can be represente as: p=v (1 t v=bv +a(u+, (2 t where u is comman, is isturbance incluing loa an friction, v is velocity, p is position, k = a/b is the DC gain, an τ = 1/b is the time constant of the axis. Assume the reference contour map is r(θ, where θ is a real number an inicates the current position along the reference path. Position an velocity error vectors of a multiaxis servo-system are efine as p = r(θ p an ṽ = r(θ/t v, where bolface letters inicate vectors, e.g.,

3 IEEE/ASME TRANSACTIONS ON MECHATRONICS 3 r(θ 1 λ + λ r(ˆθ from CEE r = (1λr(θ+λr(ˆθ p 2 r(ˆθ r(θ r s + Feeforwar comp. s 2 + b s a K V ˆ Disturbance z = ω comp. z v s + ω z ξ = ω z (z + v +ˆbv u â ˆ = ω ξ s + ω ǫ (a p p p 1 + K I s + K P + + (b u c u + + Servo-axis (inclues friction an loa v p Fig. 3: (a Moifie reference improves contouring precision. (b Integrate contouring control for one axis. Contour error estimate is calculate using the Newton metho [6]. The reference contour, shown in (a as r(θ, is a vector an its projection along one axis is r(θ as shown in (b. f = [f 1 f 2 f n ] T for a n-axis system. The contour error, ǫ, is the shortest istance from the current position, p, to the reference contour, r. Let ǫ occur at θ, i.e., ǫ = r(θ p. Denote the contour error estimate as ǫ = r(ˆθp, where ˆθ is an estimate of θ which is obtaine using the Newton-base CEE algorithm [6]. A. Moifie Reference Ghaffari an Ulsoy [6] have shown that the moifie feeback concept facilitates CCC esign by removing the contour error compensator from the control loop. Instea, the position control loop of each axis acts on the moifie feeback along the relevant axis. Contouring precision is improve by properly tuning the ratio of the contour error an the position error in the moifie error. Intuitively speaking, the moifie error vectors etermine in which irection the current point will progress which spans the area between p an ǫ. Please see Fig. 3(a. The moifie error vectors are introuce as follows: ˇp = (1λ p+λǫ = r p (3 ˇv = (1λṽ +λ t ǫ = t r v, (4 where the moifie reference contour is introuce as: r = (1λr(θ+λr(ˆθ, λ < 1. ( Without incluing the effect of contour error in the control loop, accurate position tracking is not feasible ue to the wie range of reference contours, uncertainties, an external isturbances. The moifie reference is another interpretation of the moifie feeback concept which shows that one can irectly inclue the effect of contour error in the moifie reference instea of changing the feeback. As shown in Fig. 3(a, the moifie reference reefines the leaing point of the contour on a straight line between r(θ an r(ˆθ. Parameter λ is tune such that esire contouring precision is achieve for a given set of reference contours. If λ =, there is no cross-coupling. Moreover, λ = 1 is not possible to achieve because controllability is lost between the current position an the reference point. B. Integrate Control Algorithm The experimental setup consists of linear stages riven by ball screw mechanisms. The ball screws exhibit consierable friction with stiction levels up to 1% of the comman signal. So, it is esire to alleviate the effect of friction an external isturbance using the following isturbance estimator [3], [8]: where t ˆ = ω ˆ+ω ξ, (6 ξ= ω z(z +v+ˆbv u, â (7 t z=ω zz ω z v, (8 where â an ˆb are estimates of a an b, respectively. Also, ω z an ω represent filter cut-off frequency for acceleration an isturbance estimators, respectively. An estimate of acceleration, v, is obtaine as v +z. Thus, ω z shoul be large enough in comparison to b such that an accurate estimate of acceleration is obtaine which is use to calculate ξ, which is an un-filtere estimate of the isturbance. So, to obtain a smooth estimate of isturbance, one can filter ξ using a first orer low-pass filter with cut-off frequency ω. A reasonable ω falls between b an ω z, i.e., b ω ω z. Denote u = u c ˆ. Then one can convert (2 to v ( t = bv +a u c ˆ+. (9 Then, the transfer function of the compensate velocity loop, from u c to v in Fig. 3(b, can be calculate using Laplace

4 IEEE/ASME TRANSACTIONS ON MECHATRONICS 4 sbrio 9632 Axis-2 control NI 914 u 2 AKD Drive Ethernet hub Host PC Axis-3 control NI 914 r 2 r 3 sbrio 9632 Contour error estimate r = (1λr(θ+λr(ˆθ Encoer 2 Encoer 3 RS-48 to TTL u 3 AKD Drive ABC ABC Hall sensor+ Encoer 2-Axis table Hall sensor+encoer (a (b (c Fig. 4: (a The servo-system is comprise of three linear axes, enote Axis-2, Axis-3, an Axis-1, respectively from bottom to top. (b Control box, incluing (1 Ethernet hub, (2 & 3 control boars, (4 interface circuit, an ( & 6 servo-rives. (c Schematic of the experimental setup. Axis-2 is swappable with Axis-1. transformation of (6 (9 an basic arithmetic operations which gives the following where V(s = B u(s A(s U c(s+ B (s D(s, (1 A(s B u (s=a(s+ω z (s+ω (11 B (s=as(s+ω z (12 (( A(s=s 3 +(ω z +bs 2 a + ω z +ˆb ω â +ω zb s+ +ω ω zˆba â, (13 where V(s,U c (s, an D(s are Laplace transformations of v(t,u c (t, an (t, respectively. From (1 (13 one can observe that the steay state gain from the isturbance to velocity output is zero. Thus, the stiction level can be ramatically attenuate which enables the esigner to fit a linear moel to the compensate loop fairly precisely. In fact, the experiments show that after isturbance compensation one can fit a secon orer moel to each axis with an accuracy above 9%. However, it shoul be note that the DC gain from u c to v equals â/ˆb. Also, ω z an ω are selecte such that b ω ω z. Thus, the location of the ominant pole is largely etermine by ˆb. So, consiering the physical limitations of the servo-axis, the esigner can reshape the isturbance compensation loop to meet a esire transient performance. Moreover, care has to be taken that the upper level of isturbance attenuation, â/(ˆbω, stays as low as possible. Since b ω ω z, then it is reasonable to simplify (1 as a first orer system free of isturbance. The resulting reuce moel, which will be use to esign the position control algorithm, is given as follows G (s = P(s U c (s = a s(s+b, (14 where a an b can be obtaine using system ientification. Using the following control law u c = K V ˇv +K K PK I Pˇp+ ˇp+ s2 +b s s a r (1 one can transform the position loop into P(s R (s = a K P K I s 3 +(b +a K V s 2 +a K P s+a, (16 K P K I where R (s is the Laplace transformation of the moifie reference along one of the axes r = (1 λr(θ + λr(ˆθ. As shown in Fig. 3(a, instea of tracking the lea point of the reference contour, r(θ, the propose contouring control tracks the moifie reference, r, which incorporates contour error in the tracking process. The shape an feerate of the reference contour affect selection of λ. Also, when λ = there is no cross-coupling between the axes of the servo-system. III. SYSTEM CONFIGURATION AND IDENTIFICATION The experimental setup an its schematic are shown in Fig. 4. Three linear stages are use to incorporate swapping moularity in the system. Axis-1 an Axis-2 are structurally ientical with length an mass of 1 mm an 4.46 kg, respectively. Stage length of Axis-3 is 1 mm an stage mass is 3.67 kg. The rest of the electrical an mechanical technical characteristics are the same for the three axes an are liste in Table I. The CSM experiments are conucte on a servo-system compose of axes (2,3. Then, Axis-2 is swappe with Axis-1. Also, to highlight the ifferences cause by swapping Axis-2 with Axis-1, the three axes are installe in the following orer, from bottom to top: Axis-2, Axis-3, an Axis-1. Please see Fig. 4(a. So, Axis-1 works as an external isturbance for axes (2,3 which causes consierable variation between ynamics of axes (1,3 an those of axes (2,3. The objective is to esign an optimal contouring algorithm with the highest level of CSM for the given servo-system. As explaine in Section II, isturbance compensation plays a major role in improving close-loop performance. Thus, the

5 IEEE/ASME TRANSACTIONS ON MECHATRONICS Parameter Value Unit Drive Groun ball screw Motor Brushless servo 3-phase Encoer.12 µm rotary Encoer output A qua B, inex Linear accuracy 12 µm Linear Repeatability 2 µm Max. linear velocity 1 mm/s Screw lea 2 mm Comman signal (V Axis1 Axis2 Data Axis3 Average Comman signal (V TABLE I: System parameters for the linear stages. steps to esign the isturbance compensation are explaine first. The isturbance estimate is given by (6 (8. So, the values of â anˆb, which are estimates of a an b, respectively, are require to implement the isturbance estimator. Stiction is iscontinuous an a major source of isturbance in ball screw mechanisms, consuming up to 1% of the control power signal. The stiction information for each axis is obtaine experimentally an shown in Fig. (a. On average, the stiction consumes.%, 8.8%, an 7.7% of the control signal for Axis- 1, Axis-2, an Axis-3, respectively. Also, the friction level is slightly higher for Axis-2 because Axis-2 is installe uner axes (3,1. Thus, linear moels cannot fit the system ynamics precisely, unless the friction is compensate using a static or ynamic compensator. The ynamic isturbance compensation in Fig. 3(b relies on the estimates of the system parameters a an b, an ramatically affects contouring performance. So, â an ˆb are initialize using the result of an open-loop system ientification. The results obtaine are â {169, 648, 1116} an ˆb {1.1,6.,.8}. As expecte, the response time is % faster, an the DC gain is 7% larger for Axis-1 in comparison to those of Axis-2. After isturbance compensation, one can fit a linear moel to each axis with an accuracy of more than 9% an use the linear moel to esign the position control loop. So, a close-loop system ientification is conucte to obtain the ynamic moel of each axis after isturbance compensation. A schematic of the close-loop system ientification is shown in Fig. (b. The exact moel of the compensate velocity loop is given by (1. However, it is assume that b ω ω z. So, one can accurately replace (1 with a first-orer system: V(s U c (c = a s+b. (17 The equivalent transfer function of Fig. (b is G i (s = P(s R(s = a K P s 2 +(b +a K V s+a K P. (18 The controller gains are selecte as K P = 4 an K V =.1. The ientification results obtaine are a = {873,74,81} an b = {19.7,24.7,19.9}. The variation in the values of a an b is relatively small. As note previously, except for mass an length, the three axes have the same physical characteristics. Thus, it can be conclue that the isturbance compensation effectively attenuates external isturbances an axis friction. So, the resulting linear moels are relatively close Velocity(mm/s Velocity(mm/s Velocity(mm/s (a K V Dist. Comp. u ˆ c u r + K P + + Servo-axis G (s (b Fig. : (a Stiction ientification. (b Close-loop system ientification with isturbance compensation. IV. CONTROL SENSITIVITY ANALYSIS AND EXPERIMENTAL VERIFICATION OF SWAPPING MODULARITY An overview of the control algorithm for one servo-axis is shown in Fig. 3(b. Tunable control parameters are selecte as { Ω = â,ˆb,k V,K P,K I,a,b }. (19 Assume a part of the multi-axis servo-system is swappe with a compatible counterpart. The worst case scenario is to achieve CSM by re-calibrating all the control parameters. The objective, however, is to reuce the contour error to its minimum feasible level by tuning the fewest number of control parameters. Apart from the moifie reference loop which uses the CEE algorithm, each axis consists of four loops: 1 isturbance compensation, 2 velocity, 3 position, an 4 integral action. Sensitivity analysis is unertaken numerically an empirically to stuy the effect of each parameter on contouring precision. It is shown that the experimental results are aligne with the numerical results. The isturbance compensation is initialize as explaine in Section III. Then, the close-loop ientification is conucte to obtain the compensate moel for each axis; the results of which are use to tune the rest of the control loops. The outermost loop is tune base on the following reference moel P(s R (s = αω 2 n (s 2 +2ζω n +ω 2 n(s+α. (2 The control gains are calculate as αω n K I = (21 2αζ +ω n K P = 2αζω n +ω 2 n a (22 K V = α+2ζω n b a (23 v p

6 IEEE/ASME TRANSACTIONS ON MECHATRONICS 6 (s b 4 (e b 4 (s b 2 (e b 2 Contour error ( m Contour error ( m Contour error ( m Contour error ( m (s a 8 (e a 8 (s a 2 (e a 2 Contour error ( m Axis-1 Axis-2 Axis Contour error ( m Contour error ( m 1 1 Axis-1 Axis-2 Axis Contour error ( m (a Fig. 6: Variation of contour error versus (a isturbance compensation parameters, â an ˆb, an (b feeforwar compensation parameters, a an b. (s b an (s a simulations, an (e b an (e a experiments. (b which show that the integral gain is inepenent of axis ynamic moel. Table II lists the K V,K P, an K I values for the esire ω n = 2 ra/s, ζ =.7, an α = 3 ra/s. After initializing the contouring control, the cross-coupling is achieve with λ =.7. The control parameters are tune sequentially, i.e., all parameters are fixe except one which is calibrate to achieve the smallest contouring error. The parameters are calibrate from innermost loop to the outermost loop in the following orer: ˆb3,â 3,ˆb 2,â 2,K V3,K P3,K I3,K V2,K P2,K I2,b 3,a 3,b 2, an a 2. When Axis-2 is swappe with Axis-1, the control parameters of Axis-1 are tune in the following sequence: ˆb1,â 1,K V1,K P1,K I1,b 1, an a 1. However, the control parameters of Axis-3 are not re-calibrate to highlight the level of CSM achieve only using the control parameters of the swappe axis. The results of the system ientification are use to initialize the control parameters as liste in Table II. The results of the numerical simulations an experiments are shown in Fig. 6(a (b an 7. The numerical simulations closely follow the experimental results. Minor mismatches are because of moeling error. Contouring performance shows the lowest level of sensitivity to variation of control parameters of Axis-1. Also, except for the isturbance compensation Axis â ˆb KV K P K I a b TABLE II: Initial values of the control parameters obtaine from moel ientification. parameters, the contouring precision shows similar sensitivity to the rest of the control parameters of Axis-2 an Axis-3. As Fig. 6 shows, the contouring precision has a local minimum versus variation in the parameters of the feeforwar an isturbance compensators. Moreover, as Fig. 7 shows, the position control gains, if chosen large enough, have a negligible effect on contouring performance, the reason for which lies in the system configuration explaine by (16. Assuming K V =.1,K P = 4, an K I = 11.8, an incorporating 2% variation in a = 8 an b = 2, Fig. 8 shows that the variation in the close-loop Boe plot is negligible which inicates low sensitivity of the close-loop system with respect to large K P an K I. The sensitivity analysis of K p an K I, shown in Fig. 7, suggests an increase in K P an K I to make them as large as feasible. However, excessively large values of K P an K I push the ominant poles towars the imaginary axis an cause oscillations in contouring an consequent instability of the close-loop system. There are slight ifferences between sensitivity curves obtaine from the simulations an experiments in Fig. 6(a because of the effect of unknown nonlinear ynamics in the isturbance compensation loop. Nevertheless, Fig. 8 shows that the close-loop system is relatively robust to the variations of the linear moel of the isturbance compensation loop. Moreover, the system ientification reveals that the isturbance compensation loop fits a secon orer linear moel with accuracy more than 9%. Thus, one can overlook the ifferences between the results of the experiments an simulations in Fig. 6(a. Hence, the isturbance compensation parameters of Axis-1 an Axis-2 are selecte ientical. The feeforwar sensitivity analysis is shown in Fig. 6(b.

7 IEEE/ASME TRANSACTIONS ON MECHATRONICS 7 Contour error ( m (s v 1 (s p Axis-1 Axis-2 Axis (e v (e p Contour error ( m Magnitue (B Phase (eg Frequency (ra/s Fig. 8: Close-loop Boe for 2% variation in a = 8 an b = 2. 1 (s i (e i 1 6 A3-A1 A3-A2 1 1 Fig. 7: Variation of contour error versus position control gains. (s v, (s p, an (s i simulations, an (e v, (e p an (e i experiments. The simulations an experimental sensitivity results are very close because the isturbance compensation efficiently alleviates the effect of uncertainties, friction, an external isturbances. The contour error shows the same level of sensitivity with respect to variation in b an a of Axis-2 an 3. The simulations suggest the same sensitivity curves for the feeforwar parameters of Axis-1. Moreover, since the experiments show that the contour error only changes slightly when feeforwar parameters of Axis-1 vary, then the feeforwar parameters can be the same for all three axes. The result of the empirical sensitivity analysis is given in Table III. Various experiments have been conucte, an it is conclue that swapping Axis-1 with Axis-2, or vice versa, oes not affect the control parameters which inicates that full CSM is achieve without re-calibrating any of the control parameters. The experimental verification of CSM is shown in Fig. 9. The reference contour is a circle with raius 2 mm an frequency. Hz implemente as r 1 = r 2 = 2cos(πt an r 3 = 2sin(πt. Because each axis changes irection two Axis â ˆb KV K P K I a b TABLE III: Control parameters obtaine from the empirical sensitivity analysis. 1 Contour error ( m Time (s Fig. 9: Contouring performance when Axis-1 is swappe with Axis-2. Full CSM is achieve. The integral of contour error is µm an µm (i.e., nearly ientical for A3-A1 an A3-A2 configuration, respectively. times uring a full perio, the friction then changes sign at the beginning of each quarant which causes error spikes at times 2, 2., 3, an 3. s. The integral of contour error is µm an µm for A3-A1 an A3-A2 configuration, respectively. Hence, the obtaine control parameters from the empirical analysis result in almost ientical contouring performance with the same control for the axis combinations A3-A1 an A3-A2, which inicates full swapping moularity. Further experiments, not shown here, reveal that the obtaine controls are inee optimal an contouring is achieve with highest possible precision. Achieving full CSM reuces time in tuning an calibration of control in istribute manufacturing systems. The conucte empirical analysis reveals that the linear moel of the isturbance compensation loop provies an accurate basis for esigning the propose contouring algorithm. In other wors, instea of esigning the control loop using the original ynamics of the servo-system, one can use the results of the close-loop system ientification to guarantee near optimal contouring performance an full CSM. To achieve optimal

8 IEEE/ASME TRANSACTIONS ON MECHATRONICS 8 contouring, the empirical analysis can be conucte in a ball aroun the controller gains initialize using the results of the close-loop system ientification. [14] Z. Wang, D. Yu, Y. Hu, an Y. Tao, Design of istribute cross-couple controller base on motion control bus, in Proc. of Chinese Control an Decision Conference, 29, pp V. CONCLUSIONS The unifie contouring framework presente improves CSM in networke multi-axis servo-systems. The moifie reference concept eliminates the nee for a separate controller to ensure cross-coupling. Thus, the close-loop system remains ientical to the position control loop. So, the control can be esigne using conventional methos. Flexible allocation an the moular structure of the moifie reference contour improves CSM esign an reuces tuning an calibration in istribute manufacturing systems. An empirical sensitivity analysis was conucte to optimally istribute the control parameters between the local an base controller. Extensive experiments were conucte, an the results showe that espite ramatic changes cause by swapping one of the axes, the highest level of CSM is achieve. In other wors, it is experimentally shown for the first time that there exists an ientical optimal configuration for the position control loop such that the lowest contour error is obtaine for the available configurations of the two-axis servo-system. Moreover, the numerical simulation was proven to be fairly accurate. Thus, in future esigns, one can opt out of the empirical analysis an conuct numerical sensitivity analysis to tune the control for CSM. REFERENCES [1] M. Cakmakci an A. G. Ulsoy, Improving component swapping moularity using biirectional communication in networke control system, IEEE/ASME Transactions on Mechatronics, vol. 14, pp , 29. [2], Swappable istribute MIMO controller for a VCT engine, IEEE Transactions on Control Systems Technology, vol. 19, pp , 211. [3] W.-H. Chen, Disturbance observer base control for nonlinear systems, IEEE/ASME Transactions on Mechatronics, vol. 9, pp , 24. [4] A. Ghaffari an A. G. Ulsoy, Design of istribute controllers for component swapping moularity using linear matrix inequalities, in Proc. of IEEE International Conference on Avance Intelligent Mechatronics, 216. [], Experimental verification of component swapping moularity for precision contouring, in Proc. of American Control Conference, 217. [6], Dynamic contour error estimation an feeback moification for high-precision contouring, IEEE/ASME Transactions on Mechatronics, vol. 21, pp , 216. [7] C.-C. Hsieh, A.-P. Wang, an P.-L. Hsu, CAN-base motion control esign, in Proc. of SICE Annual Conference, 23, pp [8] M. Iwasaki, T. Shibata, an N. Matsui, Disturbance-observer-base nonlinear friction compensation in table rive system, IEEE/ASME Transactions on Mechatronics, vol. 4, pp. 3 8, [9] Y. Koren an C.-C. Lo, Variable-gain cross-coupling controller for contouring, CIRP Annals - Manufacturing Technology, vol. 4, pp , [1] Y. Koren, Cross-couple biaxial computer control for manufacturing systems, Journal of Dynamic Systems, Measurement, an Control, vol. 12, pp , 198. [11] S. Li, M. Cakmakci, I. V. Kolmanovsky, an A. G. Ulsoy, Throttle actuator swapping moularity esign for ile spee control, in Proc. of American Control Conference, 29, pp [12] S. Li, I. V. Kolmanovsky, an A. G. Ulsoy, Distribute supervisory controller esign for battery swapping moularity in plug-in hybri electric vehicles, ASME Journal of Dynamic Systems, Measurement an Control, vol. 134, article no. 4113, 212. [13], Direct optimal esign for component swapping moularity in control systems, IEEE/ASME Transactions on Mechatronics, vol. 18, pp , 213. Aza Ghaffari is a Postoctoral Research Fellow in the Mechanical Engineering Department at the University of Michigan, Ann Arbor. He receive the B.S. an M.S. egrees in electrical engineering from the K. N. Toosi University of Technology, Tehran, Iran, an the Ph.D. egree in mechanical an aerospace engineering from the Joint Doctoral Program between San Diego State University an the University of California, San Diego, CA, USA. His current research interests inclue istribute supervisory controller esign for swapping moularity, cross-coupling control of multi-axis servo-systems, eman response in power systems, extremum seeking an its application to maximum power point tracking in photovoltaic an win energy conversion systems, inuction machines, power electronics, an sliing moe control. A. Galip Ulsoy is C.D. Mote, Jr. Distinguishe University Professor Emeritus of Mechanical Engineering an W.C. For Professor Emeritus of Manufacturing at University of Michigan, Ann Arbor. He receive the Ph.D. from University of California at Berkeley (1979, M.S. from Cornell University (197, an B.S. from Swarthmore College (1973. His research interests are in ynamics an control of mechanical systems. He has receive numerous awars, incluing the 28 Rufus T. Olenburger Meal an the 213 Charles Russ Richars Awar from ASME, is a member of the National Acaemy of Engineering an is a Fellow of ASME, SME, IEEE an IFAC.