21 Linear Slide 2 IJMTM 2

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International Journal of Machine Tools & Manufacture 40 (2000) 10511064

High precision linear slide. Part II: control andmeasurements

Samir Mekid *, Olivier Olejniczak

Universite de Technologie de Compiegne, Division Systemes Mecaniques, 60206 Compiegne Cedex, France

Received 29 June 1999; accepted 12 November 1999

Abstract

Design and construction of a linear slide have been discussed in Mekid [S. Mekid, High precision linearslide. Part I: design and construction, Int. J. Mach. Tools & Manufact. 40 (2000) 10391050]. Very highprecision in nanometric scale depends on mechanical design and servo control with a very precise andadequate metrology. However, servo technology is employed as a method for going beyond mechanicalaccuracy limits. For that purpose the linear slide is controlled using two methods: ProportionalIntegralDerivative (PID) controller and Internal Model Controller (IMC) that would compensate automatically for

unmodeled mechanical behaviors such as prerolling phenomena.The paper focuses on the design of a numerical controller able to handle imprecision in the model of

behavior of mechanical and electromechanical components of the bench. The metrology frame includinglaser interferometer and optic linear encoder was used for the measurements. 2000 Elsevier Science Ltd.All rights reserved.

Keywords: Linear axis control; Brushless motor with permanent magnets; High precision motion; Numerical controllers;Internal Model Control

1. Introduction

For decades, the accuracy was limited by the performance of the mechanical design but recentlyservo-technology was employed as a method for going beyond mechanical accuracy limits. Bysetting up the linear slide project, the purpose was to get specialized knowledge and experiencein the field of very high precision machine design, small stroke and heavy carriage [1]. A previousconcern was for large stroke (3 m) and heavy carriages (220 kg) [2,3].

* Corresponding author. P.O. Box 9167, 31982 Hofuf, Saudi Arabia. Fax: +966-3-5940-514.E-mail address: [email protected] (S. Mekid).

0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 8 9 0 - 6 9 5 5 ( 9 9 ) 0 0 1 1 0 - 8


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1052 S. Mekid, O. Olejniczak / International Journal of Machine Tools & Manufacture 40 (2000) 10511064

Modeling and experimental validation carried out in order to conceive a single axis translationsystem, fitting into the following specifications:

Mass of the slide table=100 kg, Stroke=220 mm, Positioning accuracy=16 nm, Maximal translation speed=10 mm/s, Global geometrical accuracy 1 m.

Some mechanical phenomena that influence the accuracy are still quite difficult to model preciselysuch as prerolling friction: for which several attempts were made [3,18,19]. Some of them areunknown, such as electrical non-linearities. The Internal Model Control (IMC) controller will

compensate automatically and dynamically for the effect of unmodeled mechanical behaviors(prerolling resistance, electric non-linearities and magnetism) on the quality of the straight motion.It is used to decrease the dependence of the PID controller over the accuracy of this model. Thistechnique has been found superior because it deals with operating conditions.

The paper introduces some aspects relative to the development of a speed and position numeri-cal-controller for the linear slide. The first part describes the electrical and electronic structure ofthe linear slide bench to give an understanding of the validity of some approximation made inorder to develop a single variable and linear model of behavior suitable for the setting up of PIDcontroller. Because of the complexity equations involved, while studying the functioning of a testbench at nano scale, it was decided to develop a PID controller using a conventional structuredynamically compensated by a more global approach: the IMC.

The last part of this paper, which is based on the analysis of experimental results, discussesimprovements and limitations of the structure proposed for the controller.

2. Control design

Fig. 1 shows a global view of the physical structure of the control of the linear slide bench.The following paragraph presents the structure of the numerical-feedback control system includingan inner-speed control-loop with a PID corrector and an outside position control loop with a

Proportional (P) corrector. A model of behavior is developed to control the brushless motor andthen introduce the concept of Internal Model Control (IMC), which was used to decrease thedependence of the PID controller over the accuracy of this model.

Brushless servomotors are now a standard in the industry because of their very low cost andtheir need for minimal maintenance [4]. With permanent magnets on the rotor, they appear nowas an interesting option to replace step motors in applications where a high quality of positioningis required. Moreover, their electromechanical dynamic performance and reliability are farsuperior to that of AC motors and are simpler to control than asynchronous motors when currentis chosen to command the statoric torque. The use of linear operational amplifiers as powerconverters to feed the statoric coils in the motor offers a very dynamic and linear response.


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1053S. Mekid, O. Olejniczak / International Journal of Machine Tools & Manufacture 40 (2000) 10511064

Fig. 1. Structure of the linear bench control structure.

2.1. Mathematical model

The equations involved in the control of current driven brushless electrical motor with perma-nent magnets on the rotor are linear, and consequently easy to program in the numerical calcu-lators [5].

Eq. (1) describes the relation between the angular acceleration of the rotor and the torquedeveloped by the motor [6]:

J.d2

qdt2Cem(t)Cr(t) (1)

where

d2q

dt2acceleration of the rotor (m/s2)

J inertia of the mechanical system (kg/m2)Cem(t) electromotive torque of the motor (Nm),Cr(t) resistive torque (loading torque, friction...) (Nm)

and Eq. (2) gives the value of the electromotive torque Cem(t) as a function of the statoric current:

Cem(t)Km.i(t).siny(t) (2)

where

Km constant given by the constructor characterizing the motor (Nm/A)i(t) amplitude of the statoric current (A)y(t) angular difference between the statoric and the rotoric magnetic fields angle (rad)


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To simplify the control of the linear slide, the resistive torque Cr(t) can be considered as anearly static external disturbance, whose influence will be compensated for by the integral termof the corrector. In addition, we assume Cr(t)=0 in Eq. (1). The impact of this approximation willbe lowered by the great deal of attention paid to the design of the bench and pertaining to theavoidance of all sources of static friction. With this hypothesis, Eqs. (1) and (2) will be combinedleading to Eq. (3) which summarizes the behavior of a current-driven brushless motor:

J.d2q

dt2Km.i(t).siny(t) (3)

In order to optimize the functioning of the motor, from the power consumption point of view,

the angular difference between the statoric and the rotoric magnetic fields y(t) can be set top

2.

In view of the position of the rotoric field the optimizing is easily achieved by feeding each statoriccoil individually in order to form an homogeneous magnetic field. In this particular application, arotary incremental encoder with a 12-bit resolution gives the position of the rotor.

Ify=p

2, the sinus term of Eq. (3) disappears. Eq. (4), which describes the behavior of a current

driven brushless AC motor (with permanent magnets on the rotor) becomes a second order, linearand mono-variable equation suitable for the setting up of a PID controller [9].

J.d2q

dt2Km.i(t) (4)

2.2. Proportional integral derivative (PID) controller

The structure of the numerical control system is presented in Fig. 2 including an inner feedbackspeed control with a PID corrector and an outside position control loop with a P corrector [8].

The Process Model block figures the implementation of the control law deduced from Eq. (4).It determines the value of the current, which commands the motor.

Fig. 2. Structure of the linear slide control system.


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The block called Mechanical Load describes the process, namely the totality of mechanical andelectromechanical phenomena. This represents the real behavior of the linear slide. Eq. (1)assumes that the load constituted by the mechanical system is mostly inertial [7]. When studyingthe output of the linear slide, while working in nearly static conditions, or at extremely low speeds[3,11], it appears that various and complex phenomena such as electromagnetic reluctance in themotor, eccentricity of the motorization roller, recirculation noise in the rotors bearings, surfaceroughness of the motorization bar, ground vibrations transmitted to the slide table through thegranite base, oil pressure pulsation in the hydrostatic bearings...etc, affect considerably thebehavior of the bench. These mechanical behavior effects were ignored in the implementation ofthe controller because the complexity of the equations involved is incompatible with the powerof most real-time calculators. This led to the implementation of an Internal Model Control.

The speed calculus-block includes the real-time, first-order derivation algorithm used to evaluatethe translation speed of the slide table from the information furnished by the linear incremental

encoder.The numerical controller was developed on a 12 MHz 32 bits 68020 microprocessor system

with a VME bus, working under PDOS, a multitasking operating system and programmed in Clanguage. Statoric currents sources are commanded by two analogical outputs over the range 6A with a 16 bit resolution. In accordance with the dynamic of the system, the digital/analogicalconverters frequency was set to Fe=330 Hz [7].

2.3. Internal model control (IMC)

To compensate dynamically for the effect of unmodeled mechanical behaviors on the qualityof the straight motion, an Internal Model Control scheme (whose global approach of the notionof precision of the model avoids focusing on any specific phenomena or mechanical behavioreffects) was added on top of the PID controller [10,12,13]. Basically, it consists of introducinginto the control a variable characterizing the difference between the output of the process andmodeling [14,15]. Its value is then converted into an image of the extra torque required to compen-sate unmodeled loads and is added to the motors command calculated by the PID controller.

This control scheme is presented in Fig. 3 where block M represents the model of the Process

Fig. 3. Principle of the IMC applied to the PID speed control loop.


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and M1 its reverse. M1 exists and is stable because the model of behavior of the bench (Eq.(4)) was kept single variable and linear and all zeros were eliminated from it through variousapproximation as detailed in [16]. Moreover, the calculus of the modeling error is achieved atthe acceleration level of the control in order to avoid the presence of integral terms in the modelM. It means that the modeling error is considered as the difference between the slide tablestranslation acceleration predicted by the model and the one measured. The real acceleration isobtained by a double-numerical derivation of the information provided by the linear incrementalencoder. A low-pass numerical filter was used to reduce the amount of high frequency noiseintroduced by the derivations and focus the action of the IMC loop on very low frequency disturb-ances. Indeed, if the modeling error calculated at the sample time n contains a lot of numericalnoise, when used to modify the value of the motors command at the time n+1, it would act asa powerful noise generator which goes against the initial purpose of the IMC.

The order of the filter was limited to reduce the dephasing. The very low cut-off frequency of

the filter does not affect the natural frequency of the control as the IMC loop acts independentlyof the PID control which determines the dynamic of the servo system (Figs. 4 and 5).

3. Experiments

The following paragraph presents experimental results obtained with the PID controllerincluded, and IMC loop for speed and position control of the linear slide.

3.1. Dynamic of response

The dynamic of the system is mostly set by the systems mechanical components (magneticsaturation in the motor, sliding between the motorization bar and roller) and by the tuning of thecorrectors. The natural frequency of the PID speed control loop was estimated at around 110 Hzand only 30 Hz for the position-feedback loop which means that the dynamic of the linear slidestranslation control will be very low. This has been confirmed by experimentation. Indeed, a timeresponse of 1 s is measured in Fig. 6 which shows the position output in response to a 25 nm com-mand.

It also appears that the IMC contributes to slowing down the system. However, the IMC loopacts independently of the PID control. When the bench is nearly static, the accuracy on the

Fig. 4. Graphical representation of the separate action of the IMC loop and PID servo control.


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Fig. 5. Compensation of prerolling friction by IMC.

Fig. 6. Position output of the controller including the IMC to a 25 nm command.

measure of the modeling error is poor due to the noise induced by the double-numerical derivationrequired to calculate the acceleration of the slide table. Therefore, during the dynamic phase ofthe positioning, the IMC compensation of the motors command may act against the action calcu-lated by the PID control which consequently slows down the response of the system. The samebehavior has been predicted in [12].


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The time response reaches 15 s for a displacement of the slide table over the totality of thestroke as shown in Fig. 7.

In this case, the dynamic of the system is set by the shape of the command, which was calcu-lated in order to avoid the magnetic saturation of the motor.

3.2. Dynamic error

Important dynamic errors were measured both on the speed and position output of the system.A dynamic error of nearly 25% is shown in Fig. 8(a), which presents the response of the controllerto a 0.001 m/s command.

In Fig. 6, which presents the position output of the system to a 25 nm command, the slidetable overtakes its optimal position by two resolutions of the linear incremental encoder.

The high amplitude of the dynamic error is partly due to the presence of a static load not

included in the model and which must be compensated for by the integral term of the speedcorrector, introducing a delay in the system. This load results from mechanical behaviors suchas the crushing of the motorization bar under the pre-load exerted on the roller [3]. The delayintroduced by the IMC also disturbs the positioning of the slide table and therefore is seen bythe PID control as an external disturbance, which contributes to the overstepping [12].

As an illustration of the phenomena, the stabilization time to a command of amplitude 0.001m/s is 40% compared to only 25% with the IMC control (Fig. 8) measured on the speed outputof the PID servo control alone.

Nevertheless, the dynamic error can be restrained to fit the resolution of the linear incrementalencoder by using a dynamic command as displayed in Fig. 9 instead of a step command.

The gain on the precision of the output is counterbalanced by a loss in the dynamic as theresponse time measured in Fig. 9 is about 3 s while it is only 1 s in Fig. 6.

On the other hand, for a position command of 0.5 m, the IMC controller shows a significantimprovement. The settling time and the rise time are improved by a factor of 1/4 (Fig. 10) com-

Fig. 7. Position output of the controller including the IMC to a 0.22m command.


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Fig. 8. Speed output of the controller including: (a) IMC (b) PID to a 0.001 m/s command.

Fig. 9. Position output of the controller including the IMC to a 25 nm dynamic command steps.


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Fig. 10. (a) PID controller (b) IMC controller.

pared to the PID controller which lose time when trying to compensate for the mechanicalbehaviors by using several control cycles.

3.3. Static error and accuracy

The static error measured in the speed output to a command of amplitude 0.001 m/s shown inFig. 8 is about 0.003% of the optimum speed value and the accuracy about 10%. The firstmeasure results from the quality of the mechanical design of the bench, which eliminated anyclearance or hysteresis effect. The relatively low accuracy of the output reflects the presence ofnoise sources which are inherent to the mechanic of the bench (geometrical errors, influence ofrugosity) but also to the design of the servo controller (derivation noise) and which induce errorsin the calculus of the translation speed. Environmental factors (ground vibrations, magneticdisturbances) also disturb the motion of the slide table.

Nevertheless, IMC proves here its validity as it improves the accuracy of the speed-responseby 20% compared to the one measured with only the PID controller [Fig. 8(b)].

The static error in positioning is greatly inferior to the resolution of the linear-incremental

encoder as it was measured equal to 0.004% of the optimum value of the position on the responseof the IMC controller to a command of amplitude 25 nm (Fig. 6). The accuracy is more or lesstwo resolutions of the linear incremental encoder, as the slide table occasionally moves one resol-ution away from its optimum position, under the action of disturbance loads external to the sys-tem [3].

3.4. Compensation of unmodeled prerolling friction

As an example of an unmodeled mechanical behavior: prerolling friction that constitutes themajor problem for our case has been taken into account by IMC as follows. The rolling friction


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value depends on the preload between the roller and the rod. Consequently, it reduces the responsestability of the carriage. In fact, the PID term should firstly, compensate for the prerolling frictionand then start to move the carriage as a next step. So, it appears from Fig. 5 that PID controlneeds around 0.8 s to compensate for the resistant torque. However the IMC controller takes intoaccount the resistant torque almost immediately. This behavior shows that IMC could take intoaccount phenomena that are not included in the model because their mathematical formulationsare sometimes quite difficult to establish.

4. Measurements on precision movement

4.1. Metrology frame and measuring system

An axiom 2/20 laser interferometer from ZYGO has been used to measure straightness, pitchand yaw angle. Since refractive index is influenced by environment parameter, a correction isapplied; the initial value of the refractive index can be calculated from the values obtained, usingEdlens general formula [3]. A standard paroscientific pressure transducer with a digital interfaceboard has been used with 1.2 bar measure range and 108 as precision. Output pressure is fullycompensated for internal temperature effects over the calibrated temperature range.

An environment temperature transducer is used with 1% precision and 103C resolution withinan intelligent device. The two transducers with the axiom 2/20 laser interferometer were connectedvia RS-232 serial ports.

4.2. Movement precision

The friction drive was centered with the carriage to provide high linearity. Fig. 11 shows thestraightness of controlled motion of the system at very low speed (1 mm/s).

Fig. 11. Straightness of the slide table.


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Fig. 12. Reverse step motion.

4.3. Angle measurements

As for the pitch angle, the measurements have been done by the laser interferometer with theangle device. It was observed that a variation of pitch angle of 2.5 arcsec and roll angle of 3.5arcsec knowing that the noise announced by ZYGO Company is 0.06 arcsec.

4.4. Positioning precision

The step response determines the smallest step size that can be achieved with the drive systemwithout major disturbances. The minimum acceptable step response will be between two resol-utions (e.g. 1632 nm). Fig. 12 shows the response of the stage position which precisely followedthe reference. No backlash was observed.

To analyze the feed characteristic of a reversing motion, the table performs a square path witha 25 nm step and 2 s period.

Fig. 13 shows a continuous step positioning of the linear slide. A 50 nm step is negatively andpositively repeated five times and a 300 nm displacement is obtained overall.

Fig. 13. Continuous step response.


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5. Conclusion

A single axis stage mechanism with a heavy carriage, driven by a brushless motor, is dedicatedto small stroke. The accuracy is improved by using the internal model control (IMC) as comparedto PID controller.

The analysis of the results shows the ability of IMC control to deal with unmodelled mechanicalbehaviors such as prerolling friction even if the mathematical formulation is not known. The useof IMC improves the robustness of the PID servo systems speed output. When positioning aroundthe resolution of the linear incremental encoder, the imprecision on the error of modeling limitsthe advantages that can be gained from it.

With regard to the future improvement of IMC functioning, it seems appropriate to reduce theamount of noise included in the evaluation of the error of modeling. To that effect several optionsare available:

1. use of higher resolution speed or acceleration captors to compensate easily the unmodelledmechanical disturbances.

2. use of laser interferometer which permits a resolution of 1.25 nm (i.e. ZYGO) in linear positionbut the technology presents some difficulties in mastering temperature, pressure togetherwith hygronomy.

3. use of interpolation card for high linear incremental encoder resolution (HEIDENHAIN offersa 4 nm resolution instead of 16 nm)

The scheme of the model could also be completed in order to compensate for phenomenaappearing at nanoscale. By developing a new controller based on a state space control which

is better in controlling non-linear multi-variable control laws [12,17].The estimation of the error of modeling could be used, not to modify directly the command,

as in IMC, but to elaborate dynamically a model of this difference as in some predictive controlschemes. This approach would reduce considerably the dependence of the performances of theservo control on the quality of the measurement of modeling error.

References

[1] S. Mekid, High precision linear slide. Part I: design and construction, Int. J. Mach. Tools & Manufact. 40 (2000)10391050.

[2] S. Mekid, M. Bonis, Conceptual design and study of high precision translational stages: application to an opticaldelay line, Journal of ASPE 21 (1) (1997) 2935.

[3] S. Mekid, Conception et modelisation de systemes mecaniques de translation de haute precision. Influence de lamicro-dynamique de contact. Thesis report, Universite de Technologie de Compiegne, 1994.

[4] Y. Dote, S. Kinshita, Brushless Servo MotorsFundamentals and Applications, Clarendon Press, Oxford, 1990.[5] Y. Sakae, AC Motors for High Performance ApplicationAnalysis and Control, Marcel Dekker Inc, New

York, 1986.[6] W. Leonhard, Control of Electrical Drives, Springer-Verlag, Berlin, 1985.

[7] D. Lahmar, Etude dun systeme de commande numerique dun moteur synchrone autopilote pour application enpositionnement de haute precision. Thesis report, Universite de Technologie de Compiegne, 1993.

[8] J.C. Gille, P. Decaulne, J. Pelegrin, Dynamique de la commande lineaire, Dunod, 1981.


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[9] D.K. Anand, Introduction to control system, Pergamon Press, Oxford, 1984.[10] O. Olejniczak, M. Bonis, D. Lahmar, Control of permanent magnets brushless electrical motor: performances of

an internal model control algorithm, Proceedings of the 7th international precision engineering seminar, Kobe,

Japan, May 1993.[11] O. Olejniczak, Asservissement numerique de haute precision dun axe lineaire par un moteur synchrone a aimants

permanents. Ph.D. thesis, Universite de Technologie de Compiegne, 1994.[12] C. Foulard, S. Gentil, J.P. Sandraz, Commande regulation par calculateur numeriques, Eyrolles, 1987.[13] D. Lahmar, L. Loron, M. Bonis, Asservissement en tres basse vitesse dun moteur autosynchrone, Conference

canadienne sur lAutomatisation Industrielle, Montreal, juin 1992.[14] A. Rault, D. Jaume, Commande par modele interne: presentation et applications, Congres International S.E.E.

Automatique Appliquee, Nice, mai 1984.[15] C. Canudas, K.J. Astrom, K. Braun, Adaptive friction compensation in DC motor drives, Proceedings of IEEE

International Conference on Robotics and Automation 3 (1987) 681685.[16] Y. Okazaki, T. Bispink, M. Weck, Proceedings of the 3rd International Conference on Ultraprecision in Manufac-

turing Engineering, Aachen, Germany, May 1994.[17] P. Borne, G. Dauphin-Tanguy, J.P. Richard, F. Rotella, I. Zambettakis, Commande et optimisation des processus,

TECHNIP (1990).[18] S. Hiroshi, K. Tadayoshi, Journal of Japan Society for Lubrication Engineering 29 (1984) 1.[19] K. Tadayoshi, K. Osamu, Analysis of damped vibration of a system with rolling friction, Japanese Journal of

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