Advanced Controllers for Feed Drives

Transcription

1 Keynote Paper Advanced Controllers for Feed Drives Y. Koren (1) and C. C. Lo SUMMARY To achieve the high precision reqired in ftre contoring machining applications. accrate servo-controllers for the feed drives are needed. For this prpose conventional P controllers. which are tilized on many CNC systems, are not adeqate. and more advanced control algorithms mst be implemented. This paper smmarizes existing servo-controllers for contoring applications and presents an evalation of three basic types of controllers: feedback controller, feedforward controller, and cross-copling controller. The evalation of servo-controllers incldes: ( l) their abilities in eliminating different error sorces and (2) their practical limitations in machine-tool control. The evalation is spported by simlation and experimental reslts. n addition, some directions for ftre servo-control algorithms are also sggested. KEY WORDS: Nmerical control, machine tool, servo-controller. NTRODUCTON A general block diagram of a control loop sed in machine tools is shown in Figre 1. n a typical machine tool system, each axis of motion is driven by a control loop. The controlled variable, which is the axial position, is fed back and compared with the reference inpt, which is the reqired axial position. The reslting error signal actates the drive and motor throgh a controller so as to minimize, or eliminate, the position error. The simplest possible controller is one where the otpt signal is proportional to the error signal and is called the Proportional or P-controller. There are two types of CNC systems: (1) point-to-point (PTP), and (2) contoring (or continos-path). For PTP systems, a good axial positioning accracy at the target points is reqired, and sally, a conventional P controller can satisfy this reqirement [ 17, 19] (note, however, that in high-friction machines, a P controller is reqired). However, a conventional P controller may reslt in significant contor errors in machining contoring systems [18,19,35]. The term "contor error" is sed to denote the error component orthogonal to the desired trajectory (i.e., the deviation of the ctter location from the desired path)_ The contor error in machining a desired contor on a two-?.xis system is shown in Figre 2. We shold emphasize that the contor error, rather than the axial positioning error, is the prime concern, althogh the latter is sally given as the specification of CNC systems. Redction of contor enors can be performed by three basic approaches: ( 1) applying more sophisticated axial controllers, (2) adding feedforward controllers, and (3) sing a cross-copling controller. n the first approach, more comprehensive controllers than the P-controller are tilized (e.g., PD controller, state-feedback controller, etc.), and conseqently, a redction in the position errors of each individal axis is achieved Reqired position R Distrbances L Position feedback f----.j Axial position p Figre 1. Block diagram of a control loop sed in machwe tools. Y-axis Lx-axis Desired path "i-.,...- Actal path E : Contor error E,, E y : Axial errors Reference point "-- R P\ Tool Ey R: Reference position P: Actal tool position Figre 2: Contor error in machining a contor. [4,l2,15,n, ,56]. The second approach is based on adding a feedforward controller (e.g., ZPETC, KF, etc.) to compensate for the axial position errors [14,16,2,29,3,32-34,4,42-48,5-52,54,55]. The above two approaches intend to redce the axial tracking errors (Ex and Ey in Fig. 2) and thereby redce the resltant contor error. By contrast, the philosophy of the cross-copling control method [8,11,18,22,24-27,31,5] is that the elimination of the contor error is the controller objective, rather than the redction of the individal axial errors. Therefore. the cross-copling concept calls for the constrction of a contor-error model in real time and its tilization in the determination of a control law that redces or eliminates the contor error. Besides these three basic approaches, other control algorithms sch as adaptive control [ 11,21,33,39-42,5], repetitive control [ 1,41,42,49,53], predictive control [5-7,13], and optimal control [!,25] may also be applied to improve the contoring performance of machine tool systems. These control algorithms cold be considered as variations (e.g., optimal control, predictive control) or accessories (e.g., repetitive control, adaptive control) of these three basic servo-controllers. The prpose of this paper is to discss the above position servo-controllers and present a comparison of the three basic approaches (i.e., feedback, feedforward, and cross-copling controllers). The evalation of servo-controllers incldes: (1) their abilities in minimizing or eliminating different error sorces and (2) their limitations in machine tool control applications. The evalation is spported by simlation and experimental reslts. 2 ERROR SOURCES Contor error sorces in machining may be classified into three categories: (1) mechanical hardware deficiencies (e.g., backlash, nonstraightness of the table, etc), (2) ctting process effects (e.g., tool deflection de to ctting force, tool wear, etc), and (3) controller and drive dynamics. The total dimensional error is a combination of all errors from the above sorces. The first and second sets of error sorces can be redced by improving the mechanical hardware or tilizing compensation techniqes, bt cannot be redced by the control techniqes discssed in this paper. The third set of error sorces, however, can be eliminated or redced by improving the servo-control algorithms and is the main concern of this paper. These error sorces are freqently overlooked by many machine tool bilders, bt they may be the dominant sorces, especially in high-speed machining. Here we deal only with the contor error sorces that are cased by the controller, drive dynamics, and external distrbances. These error sorces can be frther classified into three categories: (1) mismatch in axial-loop parameters, (2) external distrbances, and (3) the contor shape in nonlinear trajectories and corners: 2.1 Parameter Mismatch A mismatch in axial-loop parameters cases contor errors. For example, 111 tracking a linear path, when sing P controllers, a mismatch in the open-loop gains cases a steady-state contor error r 18, 19] E _ (fyj_y_) (Ky - Kx) ss-.f Kx Ky (1) Annals of the CRP Vol. 41/2/

2 where K, and Kv are the open-loop gains of the X- andy-axis: fx and fv are the reqired veloci;ies along the X- andy-axis: and f i= --J fx2 fy 2 ) rs <he reqired feedrate (i.e., velocity! along the linear path. For example, if Kx = 1 s-1, Ky/Kx = 1.3, fx = fy = 1.8 m/min (= 71 ipm), then, Ess = 62 -tm (= 2.4 minch). The mismatch in time constants (i.e.. -r, ;< -ry) in practical systems is sally more significant than that in the open-loop gains and. conseqently, may case a significant transient contor error. especially in high-speed ctting [39]. For example, if Kx = Ky = 1 s-1, Tx = 4 s, 1:y =5s, fx = fy = 1.8 m/min (= 71 ipm) then, Emm = 15 -tm. (Calclated by a simlation in which the sampling period=.1 sec). 2.2 Distrbances A distrbance in control loops means an external action to the loop that changes or distrbs the desired operation of the controlled variable. The controlled variable in or case is the tool position relative to the part. The load distrbances inclde the friction of the machine gides and the ctting forces. There are two types of friction: Colomb friction and viscos friction. The former is sally modeled as a step distrbance where its direction is always opposite to the motion. The latter is a fnction of the feedrate and increases as the feedrate increases [9]. Only Colomb friction is considered in this paper. The ctting force F is a fnction of the feeds and the depth of ct a [3,23]. Typical vales are a=.73 and = 1. The relationship between the feed s and the feedrate f is f=psn (3) where N is the spindle speed and p is the nmber of teeth in the milling ctter (in trning p = ). The ctting force is a vector. Eqation 2 shows the magnitde of the ctting force. The direction of the ctting force, however, is not determined by Eqation 2. n slot milling or in end milling with constant depth of ct, the ctting force has a constant angle to the tangent of the direction of the ctting force (see Fig. 3), and this determines the direction of the ctting force [3]. Combining the Colomb friction and the ctting forces, the distrbance loads consist of a step component, which is always opposite to the motion, and a trajectory-dependent component with a magnitde that is a fnction of the feedrate (assming that N is constant), The axial load distrbances sed in the simlations are described in Figre 3 and given in the following eqations: Dx = Djx sign(fx) De cos(- 8) Dy = Dfy sign(fy)- De sin(- 8) where De = Kc fl.73 where sign() is the signm fnction, Djx and Djy are the distrbances cased by Colomb friction (in the simlation we assme Dfx = Dfy). De is the distrbance cased by the ctting force, 8 is the angle between the tangent of the desired trajectory and the X-axis, and is the angle between the resltant ctting force and the tangent of the desired trajectory. Y-axis Lx-axis D 1yCFriction force) Desired contor Ctting direction Ctting force De Figre 3: The distrbance model sed in simlations. For machining a linear contor when tilizing a P controller, the steady-state contor error cased by the above distrbances is given by the following eqation [!8]: (2) (4) " = fydx- Dvfx.. 1(!-e T/tiKpf where 1: is the open-loop time constant, Tis the sampling period, and Kp is the P eon troller gain. 2.3 Contor Shape When prodcing a linear contor_ each axis receives a posi6on ramp inpt as a reference. The slope of the ramp is proportional to the reqired axial velocity. For ramp inpts, the control loop does not need special tracking abilities. When prodcing nonlinear contors, however, the inpts to the control loops are also nonlinear. For example, in order to prodce the circle ( X _ p )2 y2 = p2, where pis the radis of the circle, the inpts to the control loops are x = p ( 1 - coswt) y = p sinwt where w = f/p and f is feedrate along the circle. t is known that in response to these inpts each axial motor will have initial transients and then move in a sinsoidal motion with the same anglar velocity w, bt with a phase shift. Both the phase shift and motion amplitde of the motors are fnction of w. Therefore the contor error is also a fnction of w. For a matched second order system, Poo and Bollinger [35] formlate the steady-state contor (or radial) error in a circlar motion in the following: where and w 11 are the control loop damping factor and natral freqency, respectively. (Note that f = pw.) Similar errors occr for other types of nonlinear shapes sch as ellipses, parabolas or high-order splines. Another type of contor error occrs in ctting sharp corners [ 19]. For example, if the table moves only in the X-direction before the corner, and then in the Y -direction, the X -axis will still move when the Y motion starts, which cases a contor error. 2.4 Conclsions As shown in Eqations 1 6, the error de to all sorces, except the Colomb friction, increases as the feedrate increases. Or intermediate conclsions are the following. (1) The contor error is amplified by the feedrate. Since the feedrate is proportional to the spindle speed (see Eq. 3), and the latter is proportional to the ctting speed, the contor error is amplified by the ctting speed. Therefore, a more effective servo-controller (in terms of error redction) is needed for systems that tilize high-speed machining. (2) For the machine tool systems with large Colomb friction, an effective servo-controller is needed even at low ctting speeds. 3 FEEDBACK CONTROLLERS All the controllers of machine tools have feedback. The term "feedback controller" here means a controller that ses only basic feedback principles. n CNC these controllers may be classified into three basic classes: P, PD, and state-feedback controllers. 3.1 P Controller Proportional (P) controller is the most conventional feedback controller in CNC. t sends correction signals proportional to the difference between the reference position and the actal position. The proportional gain, is sally designed so that the closed loop damping ratio (closed! is eqal to.77. For this damping ratio, the following eqation can be sed for the selection of the proportional open-loop gain [!7]. 1 Kopen = T 2T where Tis the sampling period, 1: is the open-loop time constant, and!<open is the open-loop gain, which is the prodct of the P-eon troller gain (Kp) and the other gains in the system sch as motor's gain (Km). gear ratio (Kg). encoder gain (Ke), etc. (5) (6) (7) 69

3 For conventional feedrates e.g.,.2s m/min lo ipm) and small distrbance loads, this controller gives reasonable contor errors (e.g.,. l mm), and the dominate sorces for dimensional errors on the parts are other than the servo errors (e.g., machine geometry. machine temperatre. and tool deflection). 3,2 PD Controller n a PD controller. the correction signal is a combination of three components: a proportional, an integral, and a derivative of the position error. The task of the integral (!) controller is to eliminate the steady-error when position ramp inpts are the references, as in the case of linear cts, and to reject the external distrbctnces. However, implementing an!-controller by itself will case instability, and it mst be combined with a proportional action to enable a stable system. The derivative (D) controller aids in shaping the dynamic response of the system. The combination is known as a PD controller. Since a compter is tilized as the controller, a digital PD is implemented. There are different ways to design digital PD controllers. We can, for example, formlate the digital PD controller law by approximating the continos-time PD controller with backward difference or Eler's or Tstin's methods [ 1]. n the following analyses, the PD controller law (Hx(z) and Hy(z) for X- and Y-axis, respectively) is formlated based on backward difference approximation, sing z-transform Tz z-1 Hx(z) Hy(z) Kp K1 Z-f Ko Tz (8) where Kp, KJ, Ko are the proportional, integral, and derivative gains, respectively. The integral gain K1 is chosen large enogh to garantee a good ability in distrbance rejection, and the derivative gain Ko is designed to garantee small overshoot. Usally, the controller gains can be designed based on root locs or freqency domain methods. The two main problems with PD controllers in contoring application are (l) poor tracking of corners and nonlinear contors (see Section 7.1 below) and (2) significant overshoots. To redce the effect of these problems, the gain K1 shold be small and the implementation of the controller reqires ;carefl preprograming of acceleration and deceleration periods, whereas these are not needed with a P controller. n addition to the basic PD shown in Figre 4a, some different FlOstrctres, shown in Figres 4b, 4c, and 4d, were applied in practical servocontrollers [1,28,36-38,56]. However, since Figre 4a represents the most common strctre, it is sed in the analyses and simlations in this paper. extra state representing the integral of the otpt error. The state-feedback controller then is similar to a PD controller, and therefore. it will not be frther discssed in this paper. 4 FEEDFORW ARD CONTROLLERS To decrease the tracking errors, feedforward controllers might be added to the control loops. We discss below two basic types and two improved feedforward controllers. 4, Basic Feed forward Controllers There are two principal types of feedforward controllers, which are shown in Figre 5. The principle of the design in Figre Sa [14.16,3,4,42,4S- 48,S,S2,S4,55] is simple: implement in the control compter a transfer fnction Go- 1(z) that is the exact inverse of the one of the real control loop, G(z), i.e., Go 1(z)G(z) l, and then the actal position becomes eqal to the reqired position. The design in Figre Sb has the same objective [2,29,32,34]. f we implement an inverse transfer fnction of the drive nit in the feedforward controller block, as shown in Figre Sb, we obtain the following closed-loop eqation:!'_ (z _ Do- 1 (zjd(z) H(z)D(z) R J- H(z)D(z) where H(z) and D(z) represent the transfer fnctions of the software controller and the drive nit, respectively. f Do(z) D(z), the overall relation between the reqired position and the actal position becomes l:. Nevertheless, there are some differences between these two types of feedforward controllers. () The first feedforward controller is the inverse of the feedback control loop, which consists of the controller and the drive, and therefore, it becomes more complicated if a more comprehensive controller (e.g., PD controller) is tilized. Whereas, the latter is the inverse of the drive only, and therefore, the design of its corresponding feedforward controller is simple and independent of the design of the feedback controller. (2) f the feedforward controller (Go l(z) in Fig. Sa or D-l(z) in Fig. Sb) incldes poles located on or otside the nit circle in the z-domain, the design of the feedforward controller mst be modified. The modification of the design in Figre Sa, that will be discssed below, is easier than that in Figre 5b. n the contination of this paper, only the first type of feedforward controller is discssed. (9) (a) R (b) G(z) E:p - H(z) D(z) ' Feedforward controller t Feedforward controller L (a) ' p (c) Figre 4: Different ways to implement controllers with PD fnctions. (d) R 3,3 State-Feedback Controller The most common measrable states in CNC are the position and the velocity. The axial position is measred by an incremental encoder and an associated conter, and the velocity may be measred either by a tachometer [ly,28] or by differentiating the measred position [36-38,56]. The velocity closed loop, however, can be either closed in hardware [ Y,28j or in the compter [3ti-38,S6j. f the feedback states are the position and the velocity, the state-feedback controller is in fact a PD-controller (shown in Figre 4b withot the integral action). However, to eliminate the steady-state error, the state-feedback controller can also implement the integral control by agmenting the state vector with an (b) H(z) Controller, D(z) Drive nit Figre 5: Two principle types of feedforward controllers. 4,2 Zero Phase Error Tracking Controller (ZPETC) A feedforward controller entitled "Zero Phase Error Tracking Controller (ZPETC)" was proposed by Tomizka [47]. Since we will se this controller as a representative of feedforward controllers, we will elaborate on its principles. The concept of the ZPETC is based on pole/zero cancellation, i.e., Go 1(z)G(z) l. 691

4 However. if Go-1(z) incldes nstable poles. it cannot be implemented as a feed forward controller and. therefore. mst be modified. Let s assme that the closed-loop discrete-time transfer fnction. which incldes the controlled plant and feedback controller, is expressed as z-db(z-1 )B-cz-1) Gc1osect(z- 1 ) = A(z-1) where z-d represents a delay of d steps (sampling periods) normally cased by the control loop, A incldes the closed-loop poles, s incldes the acceptable closedloop zeros. and B- incldes the nacceptable closed-loop zeros. The "acceptable" zeros here mean the zeros that are located inside the nit circle and can be taken as the poles in the feedforward controller. By contrast, nacceptable zeros are located on or otside the nit circle and cannot be the poles of the feedforward controller since they will case instability. n practical application, a stable zero that is close to the nit circle and is located on or close to the negative real axis (e.g., z = -.97) may introdce oscillatory mode and, therefore, is regarded as an nacceptable zero. (1) f nacceptable zeros exist, Go-1(z) cannot be implemented in the feedforward controller becase it will case a significant oscillation in the control signals. Tomizka modified the feedforward controller strctre as shown in Figre 6. The modified feedforward controller Gj(z) has the following form: Mltiplying Eqations 1 and 11, yields the overall transfer fnction P(k) B-(z)B-(z-1) R(kd) = [B-(1 )]2 The freqency transfer fnction is given by Sppose that B-(e-iw) = Re jlm, where Re and m represent the real and imaginary parts respectively, the freqency transfer fnction becomes s-ce-jw) s-ceiw) 1.. [Bil)] [s:(j)] = [B-(1)]2 [(Re J lm)(re- J lm)] Re2 [m2 [B-(1)]2 = (B-(eiW) 112 B-(1) As can be seen in Eqation 14, the phase angle of the freqency transfer fnction is zero. Therefore, a zero phase error tracking is provided by the ZPETC method. z-ct B'(z-1) B(z -1) A(z-) ---: (11) (12) (13) (14) reslts in a better performance in corner tracking. The block diagram of the KF method is shown in Figre 7_ The low-pass filter filters ot the high-freqency inpt signals needed in comer tracking and conseqently smooths the resltant contoring path. Figre 7: Tracking control system with the inverse compensation filter. Some other feedforward controllers based on ideas similar to ZPETC and KF were proposed [ 14, 16,3,4,51,52,54]. They are not discssed here, since their principles are similar. 5 CROSS-COUPLNG CONTROLLER (CCC) The cross-copling control architectre was first proposed by Koren [18]. The main idea of cross-copling control is to bild in real time a contor error model based on the feedback information from all the axes as well as the interpolator, to find an optimal compensating law, and then to feed back correction signals to the individal axes. The cross-copling controller incldes two major parts: (1) the contor error model, and (2) a control law. Conseqently, the differences between the varios CCCs that were proposed by many other researchers who followed the original work are in the contor error model or in the control law [8,11,22,24-27,31], bt all of them are based on the original concept in [18]. A recent version of the CCC, the variable-gain cross-copling controller proposed by Koren and Lo [22], demonstrated excellent tracking ability on an experimental system and is smmarized below. The contor error model is shown in Figre 8. We see that the principle of the CCC is to place the tool P on the contor at point A rather than R, as done with feedback and feedforward controllers. The contor error mathematical model is given in the following eqations: E = - Ex C, Ey Cy where C, and Cy are C, = f(sin8, Ex, p) Cy =.f(cos8, Ey, p) where 8 is the angle between X-axis and the instantaneos tangent to the desired (15) (16) (17) trajectory and p is the instantaneos radis of crvatre (Note that p =constant for a circlar contor and p ==for a linear contor). The signals Ex and Ey are the axial position errors and are measred in real time; 8 and p are calclated at each sampling period based on the interpolator data. Figre 6: Zero phase error tracking control system. A zero phase error is achieved by the ZPETC method. The gain error. however, is not eliminated and may case large tracking errors. Some extra feedforward terms, which can be added to compensate for the gain error, were proposed in [14,51,52]. The major drawback of the ZPETC method is that it reqires precise knowledge of the dynamic behavior of the axial drive system. However, there might be a difference between the real drive system and the model sed in the compter (i.e., modeling error), and therefore, an additional error sorce is introdced to the controlled system. Another drawback is that the inverse transfer fnction in the feedforward controller will case large control signals (see Section 7.1 below). These signals in practice will be limited by the permissible maximm otpt of the digital-to-analog converter and the maximm voltage of the motor, and therefore, the performance of the controlled system is degraded. 4,3 nverse Compensation Filter (KF) t was fond (16,55] that the ZPETC method fails in corner tracking. To remedy this deficiency, Week proposed [55] an "inverse compensation filter (KF)" method, which adds a low-pass filter to the feedforward controller and Desired...- contor Y-axis p ( Radis of crvatre ) nstantaneos tangent R (Reference position) Ey P (Actal tool position) 8 '------'---' X-axis Figre 8: The modeling of the contor error. 692

5 Based on the above contor error model a cross-copling controller with a PD control law W(z) = Wp Wr Tzl Wo zt-l was implemented on a milling z- z machine and gave error redction of 5: to 1:1 compared with a system with axial P-controllers. A comparison with other controllers is presented below in Section 7. The block diagram of a two-axis cross-copling control system is shown in Figre 9. Axial error Ex --r--1_.. Control command (To servo drive) Cross-copling controller artificially and. thereby eliminate the contor error cased by mismatched loop parameters. n addition. the adaptive control algorithm can also be sed as an assistant or parallel algorithm for the crms-copling controller [11]. 6.4 Repetitive Control n addition to the general servo controilers, a special-prpose controller entitled "repetitive.. controller may be considered for handling the repetitive tasks with the presence of periodic references and periodic distrbance inpts. Tomizka proposed a repetitive controller and applied it to a disk-drive system and a trning machine [1,41,42.49,53]. When tilizing a repetitive controller, the axial errors measred when ctting a part are sed to compensate the errors in the next part. This algorithm m:,t be applied together with a conventional P controller [1.49] or other servo-control algorithms (e.g.. adaptive ZPETC in [41,42]). t cannot be applied independently and can only be applied to repetitive tasks. Moreover, tilizing the data in previos cts reqires a large memory if a complex part is being machined. 7 SMULATON AND EXPERME"'TAL ANALYSES Figre 9: The cross-copling controller. 6 OTHER ALGORTHMS n addition to the basic three classes discssed above, there are other control algorithms that are discssed in this section. 6.1 Optimal Control The optimal control algorithm can be applied as an efficient controller law of servo-controllers. The principle of the optimal control is minimizing of a performance index sbject to the system differential eqations. The optimal control has a closed-form soltion for a linear system and a qadratic perfonnance index. n machine tool applications if the performance index incldes the axial position errors (and their incremental changes), the servo controller becomes a feedback controller ll]. f the contor eitor is sed in the performance index, the servo controller then becomes an optimal cross-copling controller [25]. 6.2 Predictive Control The basic idea of the Generalized Predictive Control (GPC) proposed by Bocher [6] is to make the plant's predicted otpt coincide with a setpoint or desired known trajectory. Three stages are perforrned to achieve this goal: ( 1) first the plant otpt is predicted, (2) then the ftre control vales that minimize the errors between the predicted otpts and the setpoint vales are calclated, and (3) finally only the first optimal control vale is applied. The process is iterative and repeated at every sampling period. The GPC method, however, can be regarded as a particlar case of optimal control, in which the performance index incldes the errors between the predicted otpts and the desired vales. 6.3 Adaptive Control Adaptive control in machining sally means adapting the ctting variable (feed and speed) to the variation in the ctting process [21]. Adaptive control techniqes, however, can be applied for completely different prpose - to adjst in real time the parameters of the servo-control loops. t can be sed as an accessory algorithm for feedback. feedforward, or cross-copling controllers. The feedforward controller is very sensitive to the modeling eitor and the variations of system parameters. And therefore, an adaptive algorithm is reqired if the feedforward control approach is applied to machine tool systems [4,42,5]. An adaptive control algorithm can be applied to a mismatched system [33,39]. This adaptive algorithm makes all the individal axial control loops match Uy The simlations and experiments were performed on a two-axis system. (The block diagram with feedback controllers is shown in Figre 1.) The motor is shown as a first-order model with gain Kmi and time constant -r;; H, and Hy are the axial controllers; Kc is the gain of the digital-analog converter; Kg is the gear ratio, K1 is the leadscrew pitch, and Ke is the encoder gain; Rx and Ry are the reference positions generated by the interpolator: P, and Py are the measred positions spplied by the encoders: fx and fy are the axial velocities; Ex and Ey are the axial position errors; U, and Uy are the compter-generated control commands to each moving axis; Dx and Dy ilie the distrbance loads on each axis. reslting from the slide friction and the ctting process forces. Three basic servo-controllers are analyzed in the following simlations and experiments. ( l) Feedback controllers: P controller and PD controllers (2) Feedforward controllers: ZPETC and KF (3) Cross-copling controller: variable-gain CCC. Note that both the feedforward controller and the cross-copling controller need axial feedback controllers. A P-controller is tilized as the axial feedback controller in the following analyses. Compter T' PWM interface Tl Motor gear leadscrew Hardware Figre 1: The block diagram of a two-axis machine. 7.1 Simlation Analyses Encoder Circlar and corner contors were simlated to analyze the controllers' ability to reject different error sorces. The system parameters sed in the simlations are listed in Table l. As was stated above, contor errors de to the control loop strctre are cased by three factors: ( 1) trajectory tracking disability, (2) axis mismatch, and (3) external distrbances. n addition. feedforward controllers have modeling errors and satration errors. 693

6 Table l: The System Parameters L1 sed in Simlations Open loop gains: KcKgKKeKrnx 1. s 1, KcKgKKeKrny 1. s l ( 11.5 s l in the mismatched system) Time constants: tx = 45 msec, 'ty 45 msec ( 65 msec in the mismatched system) Distrbances: Df z 1 bits (de to friction), De z 16 bits (de to ctting force) Uiimit maximm allowable control command (i.e., satration) Controller gains: Kp 1 (for P controller) Kp 7, Kr 5, Ko.5 (for PD controller) Wp 8, W1 6, Wo.6 (for CCC) By contrast, the CCC, P. and PD controllers are sally not inflenced by the satration constraints. 15,--!OOl E' :i 5f 5 t:: 1 c: > _./_P:o controller '.2" P-controller -.2" -! 5 OL 2C-l ;18 Anglar position (deg) Figre 11: Examining tracking ability by simlation of a circlar motion for different servo-controllers. Conditions: Feedrate 1.88 m/min ( 74 ipm). Radis of circle 2 mm (.79 inch). No distrbances, no mismatch in axial parameters. No modeling error in ZPETC. 1, , Tracking. f there are no distrbances and the axial parameters of the axes are identical, the contor error is cased only by trajectory tracking deficiency. Under these conditions (i.e., withot distrbances and with matched axial parameters), a circlar contor was simlated for different servo-controllers (see Fig. ). As expected the axial errors of the PD controller were mch smaller than those with the P controller (for f 1.88 tn/min, the PD error is within±.35 mm and the P error is within ± 3.2 mm). Bt srprisingly the contor error with the PD controller is larger than that with the P controller (see Fig. 11). The simlation reslts show that (l) the PD controller reslts in poor trajectory tracking (especially for a high anglar speed and small radis of crvatre), (2) the ZPETC (a feedforward algorithm) can achieve the best tracking performance if there is no modeling error and no distrbances, and (3) the CCC provides a mch better tracking ability, compared to P and PD controllers. Mismatch. Under the circmstance of iw distrbances, small feedrate, and large mismatch in axial parameters (15% mismatch in open-loop gains and 44% mismatch in time constants), the mismatch of the two axes will dominate the other error sorces. Figre 12 shows the simlation reslts for machining a circlar contor nder the above circmstances. The reslts show that all the controllers, except the P-controller, can achieve a very small contor error. f no modeling errors are assred, the feedforward controller (ZPETC) can provide the best tracking performance. Modeling error. Two limitations of the ZPETC method that were not considered in the above simlations are considered here. First, as stated in Section 4, the design of a feedforward controller reqires knowledge of the exact model of the drive dynamics. n practice, the control designer does not have a perfect knowledge of the drive model, and the model sed in the feedforward controller will be different than the actal drive dynamics. Conseqently, the practical system will reslt in a position error. Figre 13 demonstrates the sensitivity of the ZPETC method to the modeling error. t is shown that even a small modeling error can degrade the tracking performance of the ZPETC method. As compared to the CCC method (which does not reqire knowledge of plant parameters), ZPETC can provide a better tracking performance only when the modeling error is less than.5%. This limitation, however, is not realistic. Modeling errors of 2% to 5% or even higher in some parameters are more realistic. Satration. The other limitation of the ZPETC method is the satration of the control commands that the compter generates. For proper operation, the ZPETC method needs a very large control command in the transient state to overcome the axial error cased by the drive inertia. The large control command will satrate in a practical CNC system. Figre 14 depicts the inflence of satration constraints in the compter otpt on the ZPETC and CCC methods. n or system, the compter otpt U is a binary word sent to the PWM amplifier. For example, if the otpt line has 8 bits, the connand is limited to -128 to 127, and a command exceeding the above limitation will be trncated. As can be seen in the simlation reslts. the ZPETC is strongly limited by the the satration constraint. 5 E 3 E :; c: -5 P-controller Anglar position (de g) Figre 12: Examining the effect of mismatch in axial parameters by simlation of a circlar motion for different servo-controllers. Conditions: Feedrate.754 tn/min ( 29.7 ipm). Radis of circle 2 mm. No distrbances. No modeling error in ZPETC , 5 / ' 1 > _, ZPETC (5 % modeling error) 1. /ZPETC (2% modeling error) -::_-<.. /CCC, < 4-..Tl , ZPETC (no modeling error), ' / -lool_ W 1 W Anglar position (de g) Figre 13: The effect of modeling error on ZPETC method (simlation of a circlar motion). Conditions: Feedrate.754 tn/min ( 29.7 ipm). Radis of circle 2 mm. No distrbances. No mismatch in axial parameters. 1, , 5 E' 6 ZPETC (Ulimit =) 3 8 g , \'.. K 11 '-\"'-.. CCC.(Ulimit 4nits),\ ZPETC (UJimit 8 nits) '-..,_. ' ZPETC (Uiimit 4 mts) Anglar position (de g) Figre 14: The effect of satration on performance (simlation of a circlar motion). Conditions: Feedrate.754 tn/min ( 29.7 ipm). Radis of circle 2 mm. No distrbances, no mismatch in axial parameters. No modeling error in ZPETC 694

7 Distrbances. Figres 5 and 16 show the distrbance rejection ability for each of the for control algorithms. The simlation in Figre 15 cons1ders only the friction in the table gide-ways and that in Figre l o considers only the effect of the ctting force (see Eqations 3 and 4). Both simlation reslts show that (l) the CCC and PfD controller have a good ability in distrbance rejection (CCC is the best). and (2) P controller and ZPETC have very poor performances in distrbance rejection. (To achieve a good distrbance rejection. the ZPETC shold be combined with a properly designed feedback controller. Only then it can work in a realistic environment.) 7.2 Experimental Analyses The experimental tests were condcted on a 3HP two-axis milling machine, the system parameters of which are listed in Table 2. The machine was interfaced with or compter, which allows s to write or own control software and test it on this real system. As can be seen in Table 2, this machine has the following characteristics: (!J small mismatch in axial parameters, (2) high friction loads. and (3) low maximm feedrate limitation. According w the simlation analyses, we may expect that only the CCC and Pro controllers can perform well becase of the large friction distrbances existing on this machine. For this paper. linear and circlar motions were tested on this CNC system and the experimental reslts (shown in Figres 19 and 2) confirm this expectation [ 5,' P-controller ZPETC 1l 1 ' <; c !SOL W W 1 W W Anglar position (de g) Figre 15: Rejection ability of friction distrbance (simlation of a circlar motion for different servo-controllers). Conditions: Feedrate =.754 1n/min (= 29.7 ipm). Radis of circle = 2 mm. No mismatch in axial parameters. No modeling error in ZPETC. ' '. s g c: 21.4 : / P-controller X-position (mm) Figre 17: Simlation of a corner motion for different servo-controllers. Conditions: Feedrate =.9 rn/min (= 35.4 ipm). No distrbances, no mismatch in axtal parameters. No modeling error in ZPETC. 22, , s t: 3 t: " 21.81" 21.6 g 21.4 : 21.2 c.. 21 U!imil = 4 BLUs ZPETC..--- Anglar position (de g) Figre 16: Rejection ability of ctting force distrbance (simlation of a circlar motion for different servo-controllers). Conditions: Feedrate =.754 rn/min (= 29.7 ipm). Radis of circle = 2 mm. No mismatch in axial parameters. No modeling error in ZPETC. Corners. Corner tracking is a special, bt important case in contoring applications. Corner tracking has been simlated with each control algorithm (see Fig.!7). The reslts show that (1) a P controller can provide average qality in corner tracking, (2) a basic PD controller reslts in a significant overshoot at the corner, (3) both ZPETC and CCC methods can provide good accracy in tracking a corner. The ZPETC method, however, can provide good corner tracking only nder the conditions of very small modeling error and no satration constraint; both conditions are impractical in most systems. As can be seen in Figre 18, the ZPETC method fails at corner tracking de to the satration limitation. As stated in Section 4, the KF method, in which a low-pass filter precedes the feedforward controller, can be sed to provide a better performance in corner tracking. The simlation reslts in Figre 18 prove the effectiveness of the KF method. The KF method, however, cannot provide a good performance with a significant modeling error !.5 X-position (rnm) Figre 18: Examining the effect of satration on ZPETC and KF methods by stmlatwn of a corner motion. Conditions: Feedrate =. 9 rn/min ( = 35.4 ipm). No distrbances, no mismatch in axial parameters. No modeling errors. Table 2: The Parameters of the Experimental System 1 Kg= 32, Kt = 5.8 mm (.2 inch), Ke = 1 plses/mm KcKmx =.181 rev/(sec bit), 1:x = 44 msec, 1:y = 47 msec DJ 12. bit (for X-axis), U limit 127 bits Cfmax 2 rn/min), KcKmy =.185 rev/(sec-bit) DJ 13.5 bit (For Y-axis) BLU =.1 mm. 695

8 As can be seen in Figre 19, the ZPETC method reslts in the worst performance in linear cts. The ZPETC, as well as the P controller, cannot provide a good distrbance rejection ability (see Section 7.1). Moreover, the ZPETC reslts in an additional error de to the modeling error, which doesn't exist with a P controller. As expected, the advantage of the ZPETC is its ability in tracking nonlinear contors sch as circles, rather than linear contors. The P!D controller, as can be seen in Figre 19, reslts in a significant overshoot at the transient state (withot programmed acceleration). n addition, it also cases large overshoots at the end of the cts (for both linear and circlar cts). To remedy this drawback, pre-programming of acceleration and deceleration periods is reqired, whereas they are not needed with CCC, P, and ZPETC methods. Adding acceleration and deceleration periods will increase the ctting time. Overall, the CCC method provides the best performance in both linear and circlar cts. A circlar motion with a higher feedrate ( 1.5 m/min) has also been tested on the milling system (see Figre 21). According to Eqation 6, the increase in the feedrate makes the contor error de to trajectory tracking more significant. At high feedrates, the PD controller failed: it cannot provide good tracking. The oo 2o... t: 8 c \J -5,/ZPETC P-controller r -1.5 CCC t.---zpetc '.'- PD-controller / /.2" ' " Time (sec) Figre 19: Experimental reslts of a straight-line motion for different servocontrollers. Conditions: Feedrate 1.2 m/min ( 47.2 ipm). Angle between straight line and the X-axis 26.6o. 6 2o ti " ".4" -.8" -3l j W W 1 W 1 1W Anglar position (de g) Figre 2: Experimental reslts of a circlar motion for different servocontrollers. Conditions: Feedrate.377 m/min ( 14.8 imp), Radis of circle 2mm. ZPETC method provides better tracking compared to the P and PD controllers. (Note that the ZPETC method is expected to provide mch better tracking performance than P and PD controllers, when a relatively high feedrate, e.g., 1m/min, is tilized.) Nevertheless, the CCC is the only method that can provide a small contor error nder sch circmstances. 8 C(')NCLUSONS AND FUTURE DRECTONS Existing servo-controllers for contoring applications have been classified and tested in this paper. The control algorithms have been groped into three basic classes and have been analyzed throgh simlations and experiments. According to the simlations and experimental reslts, a comparison of these servocontrollers is smmarized in Table 3. Based on the comparison, the selection of servo-controllers for different machine tool contoring applications and ctting conditions is sggested in the following: () The P controller works well only when ctting a contor on a machine with small friction, small ctting loads, small mismatch in axial parameters, and conventional feedrates (e.g.,.25 m/min 1 ipm). (2) The PD controller has a good distrbance rejection ability and is more (3) robst to mismatched axial parameters. ts drawbacks are poor tracking ability of nonlinear contors and sharp corners, and in addition, it may reslt in an overshoot at stopping. Therefore, the PD controller is preferred on low-speed machines. Usally, a deceleration is needed at the end of every contor segment in order to avoid the overshooting problem. However, this increases the total ctting time. A special algorithm can be tilized to perform corner tracking: the component of the PD controller can be trned off before the corner. This, however, can case nderct and contor errors becase of the friction and other distrbances. The ZPETC and KF methods are preferred when ctting nonlinear contors at high-speed machining if the system model is well known and no varying or nonlinear characteristics exist. n addition, a properly designed feedback controller is needed to provide the distrbance rejection ability. (4) The CCC method provides a good contoring accracy nder any condition and, therefore, is the best choice for a servo-control algorithm. ts drawback, however, is a higher comptation load, since in addition to the conventional servo loop algorithm it has to also rn the CCC algorithm. With the existing comptation power (e.g., NTEL-8486) this does not impose a limitation on a 3-axis system, bt deteriorates the performance of 5-axis systems. Nevertheless, with the increase in comptation speed of microprocessors, this drawback will shortly disappear. Table 3: The Evalation of Servo-Controllers P control PD control Feed forward (with P controller) CCC Tracking nonlinear Fair Fair (low f) Excellent () Good trajectories Poor (high f) Fair (2) Sharp corners Axis mismatch ZPETC: Fair (,2) Fair Poor KF: Excellent(!) Excellent Fair (2) Fair Good Excellent (l) Fair (2) Excellent 2 / P-controller Distrbances: (a) Friction Poor Good Poor Excellent 6 too 3 ' '. (b) Ctting Poor Good Poor Excellent force Performance Reqires Special Overshoot is sensitive to a fast problems at stopping modeling error processor and satration Anglar position (deg) Figre 21: Experimental reslts of a circlar motion for different servocontrollers. Conditions: Feedrate 1.51 m/min ( 59.4 ipm). Radis of circle 2 mm. Comments:. Assme no difference between theoretical model and real system. 2. Assme 2% difference between theoretical model and real system..f feedrate Grading: Excellent, Good, Fair, Poor n addition to the basic control approaches, some other modified methods may also be considered as algorithms for servo-controllers. They are smmarized 696

9 below: (l) An adaptive algorithm can be added to a ZPETC method to cope with the modeling error and parameter variation. f the adaptive algorithm tilizes parameter estimation, it may be limitd by the richness of the inpt signals. (The most poplar contors in machine tools are lines and circles, which enable s to estimate accrately a second-order closed-loop system.) Therefore, a carefl design of the adaptive algorithm is reqired if the closed-loop system incldes higher order strctre. (2) A combined algorithm of PD control and ZPETC may have the ability to reject all the error sorces (see Table 3), becase of the ZPETC ability in trajectory tracking and the PD control in distrbance rejection. However, a deceleration at end of each contor is needed for redcing the overshooting cased by PD control. (3) A combined algorithm of the CCC and the feedforward controller, which is sggested in Figre 5b, can reject all the error sorces withot casing the overshooting (see Table 3). Based on the above analyses and discssions, combinations of the three types of servo-controllers for different environments are sggested. For example, a combined algorithm of the cross-copling controller and the feedforward controller (with a small modeling error) is sggested for tracking a nonlinear contor at high feedrates (see Figre 23). An important direction in ftre servo-controllers will be a design to accommodate high-speed machining, which, in trn, reqires minimizing contor errors dring transient periods rather than steady-state. High-speed machining reqires a proportional increase in the axial velocities. Therefore, for a certain transient period, the transient distance increases in proportion to speed. For example, if the transient period is O.l sec and the feedrate is.3 m/min (= 12 ipm), the transient period is.5 mm. However, if the feedrate is increased by 4 times to 12m/min, the transient distance becomes 2 mm (.8 inch). That means that the entire ctting will be performed dring transients, which reqires new control design. Another direction in ftre servo-controllers will be a design to accommodate precision machining of complex part srfaces. Machining of complex part srfaces keeps increasing for the following reasons: (l) the trend toward near-net shape manfactring reqires complex dies and (2) simplified assembly calls for fewer bt more complex parts. To cope with the above two trends, ftre position servo-drives for machine tools will be based on a combination of the cross.copling control and the feedforward control. As shown in Figre 22, the concept of the cross-copling control is that the tool has to be in the right place, i.e., minimizing the contor errors. By contrast, traditional and feedforward controllers are based on the strategies of bringing the tool to the reference at the right time. Note that withot the cross-copling controller, redcing the axial errors, which is the objective of the axial feedback and the feedforward controllers, does not necessarily redce the contor error. For example, as can be seen in Figre 22, the tool is broght closer to the reference point (from P to P'). This, however, reslts in a larger contor error (E < E'). Therefore, the core of the servo-drive feed control will be the cross-copling controller, bt it will be agmented by feedforward algorithms sch as ZPETC or K.F, which are strongly recommended for nonlinear cts at high feedrates (see Figre 23). This combination will minimize the contor error with a minimm time lag between reference command and the actal tool position. The concept of this ftre combined controller will be to bring the tool to the right place at the right time. CCC objective: Redce ----Q R (Reference point) Feedback & Feedforward objectives: Redce E by redcing Ex and Ey Figre 22: Controller objectives of the CCC, feedback, and feedforward controllers. P*D CCC P*D CCC Adaptive FF P: P controller P*D: PD controller with small gain CCC: cross-copling controller FF: feedforward controller Figre 23: Sggested combinations for the servo-controllers. CCC FF An additional direction of the design of servo-control algorithms will be a flexible selection of the combined controllers according to the variations of the environments sch as types of contors, modeling knowledge, feedrates, etc. (See Figre 23) As stated in Section 2, the error sorces cased by the hardware deficiencies (e.g., backlash, geometric errors) and the ctting process effects (e.g., thermal error, tool wear) cannot be redced by the control techniqes discssed in this paper. Therefore, in addition to the servo-control algorithm, compensation techniqes for the backlash, tool wear, friction, thermal and geometric errors, etc. will be introdced as an accessory algorithm in ftre CNC systems. Acknowledgements The following CRP colleages contribted to this paper: P. Bocher, H. Makino, H. B. Roelofs, H. K. Tonshoff, M. Week, and K. Yamazaki. We especially appreciate the contribtions of Professors G. Pritschow, M. Tomizka, A. G. Ulsoy, and H. Van Brssel; each of them sent a few pages with comments on the original draft that sbstantially innproved the qality of this paper. Parts of this research have been spported by the NSF ndstry/university Research Center at the University of Michigan and by NSF grant# DDM REFERENCES [l] Astrom, K. J. and Wittenmark, B., 1984, Compter Controlled Systems: Theory and Design, Prentice-Hall, nc. [2] Astrom, K. J. and Wittenmark, B., 1989, Adaptive Control, Addison Wesley Pblishing Company. [3] Ber, A., Rotberg, J., and Zombach, S., 1988, "A Method for Ctting Force Evalation of End Mills," Annals of the C!RP, Vol. 37/l/88, pp [4] Bin, H. Z., Yamazaki, K., and DeVries, M. F., 1983, "A Microprocessor Based Control Scheme for the mprovement of Contoring Accracy," Annals of the CRP, Vol. 32/l/83, pp [5] Bocher, P., Dmr, D., Kapsta, Z., and Dchaine, P. J., 1989, "mprovement of Prodctivity with Predictive Control in Speed and Position," Annals of the CRP, Vol. 38/l/89, Ag, pp [6] Bocher, P., Dmr, D., and Faissal Rahmani, K., 199, "Generalized Predictive Cascade Control (G.P.C.C.) for Machine Tool Drives," Annals of the CRP, Vol. 39/l/9, Ag, pp [7] Bocher, P., Dmr, D., and Damller, S., 1991, "Predictive Cascade Control of Machine Tools Motor Drives," Proceeding EP E 91, Florence, Vol. 2, Sep., pp [8] Brhoe, J. C, and Nwokah,. D., 1989, "Mltivariable Control of a Biaxial Machine Tool," Proceedings of the Symposim on Dynamic Systems, Measrement, and Control, ASME Winter Annal Meeting, San Francisco, California, Dec., pp