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Modeling and Control of Miniaturized ElectricLinear Drive with Two-phase Current Excitation
1st Bela Schulte WesthoffMechatronic Systems Lab
Technische Universitat BerlinBerlin, Germany
2nd Jurgen MaasMechatronic Systems Lab
Technische Universitat BerlinBerlin, Germany
Abstract—In this contribution the concept development, math-ematical modeling, realization and control of a two-phase minia-turized electric linear drive is presented. The objective of thedeveloped linear drive is the vertical positioning of a toolelectrode in the electrical discharge machining (EDM) process.Application-related, the installation space is severely limited,requiring the actuator to have an outer diameter of ≤ 5mm. Themain challenge in developing the actuator is to generate sufficientactuator force to ensure the implementation of a robust positioncontrol. This is realized using a Lorentz-force operated actuatorwith two-phase current excitation enabling a process stroke of30 mm and stationary positioning accuracy of less than 10µm.
Index Terms—linear drive, miniaturized, linear actuator, ana-lytical modeling, control, two-phase current
I. INTRODUCTION
In this paper, the concept development of a miniaturizedlinear drive with an outer diameter of ≤ 5mm is presented. Incertain applications, the outside diameter of a linear actuatoris restricted due to limited installation space. Nevertheless,the actuator is required to generate a high force density overa specified process stroke. Fig. 1 displays an applicationexample for the use of a miniaturized linear drive, in whichthe drive is used for the optimization and automation of thedie sinking EDM process. To dynamically adapt the discretegeometry of the countersink electrode, the displayed assemblydemands the vertical positioning of multiple electrode rods. Inthis work, the modeling and control of a miniaturized linearactuator is realized based on the force requirements to positiona tool electrode with a mass of m = 6 g.As summarized in [1], there are many different types of linear
drives to choose from. Due to high control qualities, easyinstallation processes and low maintenance electromagneticlinear direct drives are often used for the uni axial positioningof objects [2], [3]. In [4] Stephan Schrader designed a lineardirect drive with an outer diameter of 7.9 mm, which, however,only generated a stroke of a few millimeters. This paperdisplays how a two-phase current control is utilized andapplied to generate an actuator stroke of 30 mm at high forcedensity despite having an outer diameter of dA ≤5 mm.In section II, the mathematical modeling and the realization
This contribution is accomplished within the joined project VariSenk4EMD(project number 426311818), funded by the German Research Foundation(Deutsche Forschungsgemeinschaft, DFG).
Fig. 1. Bundled tool electrode assembly for electrical discharge machiningprocess.
of the designed actuator is discussed. The two-phase actuatorforce is derived analytically and it is presented that theactuator force can be increased linearly in dependence of thelength of the slider. Furthermore, the voltage equations ofthe two phases of the miniaturized actuator are derived. Itis shown that the voltage equations of the two-phase coilscan be considered as independent equations. In section III,we present the implementation of a cascaded position controlsystem based on the analytical model of the two-phase lineardrive. The control performance is experimentally evaluated andfindings of the presented work are finally summarized in theconclusion.
II. DEVELOPMENT AND MODELING OF THE TWO-PHASELINEAR DIRECT DRIVE
A. Actuator Requirements
Since the width of the square electrode segments is 5 mmas displayed in Fig. 1, the miniaturized linear actuator mustdispose an outer diameter of dA ≤5 mm. Furthermore, a staticprocess stroke of 30 mm is required to dynamically generatecomplex depth profiles for highly adaptable tool geometries.Application-related, the actuator is demanded to generate aharmonic oscillation of the electrode, for which the actuator isrequired to operate at a frequency ω ≤ 5 Hz with an amplitudeAe = 1000µm. Accordingly, the required force amplitude Fr
to accelerate the tool electrode with a mass m = 6 g againstthe acceleration of gravity g is estimated by
Fr = m(ω2Ae + g) = 0.07 N. (1)
2021 IEEE/ASME International Conference onAdvanced Intelligent Mechatronics (AIM)
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Energy-dissipating effects, like friction between the electrodesor counter forces due to the dielectric fluid are initiallyneglected in this concise calculation.
B. Mathematical Modeling of the Linear Direct Drive
To accommodate all actuator requirements, the two-phaselinear direct drive shown in Fig. 2 was designed. The actuatoris based on the physical principle of Lorentz force. The fouraxially magnetized permanent magnets penetrate the statorcoils with a radial magnetic field and thus generate an axialforce FL on the slider [5], [6]:
FL =
˚Vac
j× BdV. (2)
To analytically calculate the force of the actuator, an ap-proximation of the permanent magnetic flux distribution isnecessary. In Fig. 2 the flux distribution is visualized. Themaximum axial magnetic flux Φax,max arises at the centerof the permanent magnet and is calculated by integrating theremanence flux density BRem over the cross-sectional area ofthe permanent magnet APM (Fig. 3):
Φax,max =
¨APM
BRem · dA. (3)
Demagnetization effects of the permanent magnet by meansof resistances in the magnetic circuit are neglected in thiscalculation. In the center of the ferromagnetic pole of theslider, the axial magnetic flux is zero. Between these points,the development of the axial magnetic flux is approximated
Fig. 2. Linear drive equipped with a two-phase coil winding and axiallymagnetized magnets inside the slider.
by a Fourier Transformation through the first fundamentaloscillation:
Φax(x) = −Φax,max sin
(x− xL
λ2π
). (4)
The variable x defines the coordinate of the stator fixed coor-dinate system and xL is the position of the slider as illustratedin Fig. 2. λ specifies the wavelength of the fundamentaloscillation of the axial magnetic flux and is a multiple of thepole length lp and the coil length lc:
λ = 2lp = 4lc. (5)
Using Maxwell’s law [7]‹B · dA = 0 (6)
the relationship between axial magnetic flux Φax(x) and theradial flux density Br(r, x) is determined. As illustrated inFig. 3 we obtain:
Br(r, x)2πrdx =
¨APM
Bax(x)dA
−¨APM
Bax(x+ dx)dA. (7)
Inserting
Φax(x) =
¨APM
Bax(x)dA (8)
into (7) yields:
Br(r, x) = − 1
2πr
dΦax(x)
dx. (9)
By substituting (4) into (9), it follows for a radius r largerthan the permanent magnet radius rpm:
Br(r, x) = Φax,max1
rλcos
(x− xL
λ2π
). (10)
To ensure that the Lorentz force acts in the same axial directiondespite the alternating radial magnetic flux density Br(r, x),a two-phase coil winding and current control is implemented.
Fig. 3. Magnetic flux distribution in a cross-section of the linear drive.
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The coil pairs α and β (Fig. 2) are driven with 90 degreephase-shifted currents, which are adjusted in dependence ofthe slider position xL and the current amplitude i:
iα(xL) = i cos(xL
λ2π
), (11)
iβ(xL) = i cos(xL
λ2π +
π
2
). (12)
By means of the according winding direction, a discretizedcurrent coverage is generated. Fig. 4 qualitatively illustratesthe current density for the slider position xL = 0 and xL = lc
2 .The fundamental oscillation of the discretized current densityof the stator jst is illustrated with the dotted line in Fig. 4.
The current density moves proportionally to the slider-bound radial magnetic field Br and formulates to
jst = j cos
(x− xL
λ2π
). (13)
Inserting (13) and (10) in (2) and integrating over the activecoil volume Vac yields the generated axial force FL of theactuator. Since the current and the magnetic field are perpen-dicular to each other, the scalar quantities are used:
FL =
ˆ ls+xL
xL
ˆ 2π
0
ˆ ro
ri
j cos
(x− xL
λ2π
)Φax,max
1
λrcos
(x− xL
λ2π
)rdrdφdx. (14)
ls denotes the slider length which is given as a function of thepole number P and the pole length lp = λ
2 by:
ls = pλ
2. (15)
By shifting the axial integration limit by xL the dependenceof the sliders position xL eliminates from (14). The relationof the current density amplitude j and current amplitude iis given in dependence of the coil surface area Ac and thenumber of windings per coil Nc:
j =Nci
Ac=
4Nci
λ´ roridr. (16)
Fig. 4. Current density distribution in the discrete coil pairs for two-phasecurrent feeding.
Inserting (16) and (15) into (14) yields the relation of currentamplitude and actuator force:
FL = NcΦax,max4
λ2i
ˆ 2π
0
dφ
ˆ pλ2
0
cos(xλ
2π)2
dx
= kmi, (17)
with
km =pNcΦax,max2π
λ. (18)
Given the number of turns per coil Nc = 18 and the rema-nent flux density of the permanent magnet BRem = 1.2 T itfollows:
km = 0.28 Tm. (19)
Following (1) to achieve the desired slider acceleration, currentamplitudes of approx. ir = 0.25 A are required:
ir =Fr
km=
0.07 N
0.28 Tm≈ 0.25 A. (20)
A huge advantage of the developed two-phase linear drive isthat the actuator force can be increased linearly in dependenceof the number of build-in permanent magnets p in the slider.If the force requirements on the actuator increase due toneglected EDM-manufacturing influences, the actuator designcan be adapted by installing additional magnets.
C. Mathematical Modeling of the Electrical Circuits
In order to design the current control, an analytical modelof the electrical circuits is derived. To analytically calculatethe induced voltage of the circuit, it is assumed that the coilwindings of the stator are distributed continuously in the axialdirection. Following, nc represents the axial winding densityof the coil:
nc =Nc
lc=
4Nc
λ. (21)
The voltage equations of the stator coil windings α and β areobtained as a function of the resistances Rα,β , the currents iα,βand the time variation of axial magnetic flux passing throughthe coils Φax,α,β(x). In the following, the voltage Uα of theα winding is considered representatively:
uα = Rαiα +
ˆlc,α
dΦax,α
dtncdx. (22)
The inductive voltage dΦax,α
dt of the α coil is generated by fourcomponents:ˆ
lc,α
dΦax,α
dtncdx = Lα
diαdt
+ Lα,βdiβdt
+ iαdLαdt
+
ˆlc,α
dΦPM
dtncdx. (23)
The first and second term of (23) are induced voltages dueto the change over time of the magnetic energy stored in theα winding. Lα is the inductance of the α coil and Lα,β isthe mutual inductance of the α and β coil. The second term
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of (23) is thus induced by the β coil and dependent of iβ .Since the mutual inductance Lα,β is significantly smaller thanthe inductance Lα, this term can be neglected in the voltageequation (23).
Lα,β << Lα. (24)
The last two terms of (23) represent induced voltages due tothe movement of the slider. The terms can be reformed as afunction of the slider speed vL:
uv,α = iαdLαdxL
vL + vL
ˆlc,α
dΦPM
dxLncdx. (25)
The first part of (25) originates from the local change of theinductance Lα. Since dLα
dxL≈ 0 this term can be neglected. The
second term is induced by the permanent magnets. The voltageequation simplifies to
uα = Rαiα + Lαdiαdt
+ vL
ˆlc,α
dΦPM
dxLncdx. (26)
To classify the magnitude of the induced voltage of thepermanent magnets, the maximum local change of magneticflux dΦPM
dxL|max and the maximum velocity of the slider vL,max
at a frequency of 5Hz and maximum amplitude of 1000µmare inserted into the last term of (26):
uα,pm,max = vL,max
ˆlc,α
dΦPM
dxL|maxncdx ≈ 0.0022V. (27)
Compared with the resistor voltage, this term can be neglectedas well. The ohmic resistance and the inductance of the α andβ coils are measured to be:
Ri ≈ 3.6Ω, (28)Li ≈ 160µH, with i = α, β. (29)
Accordingly, the two differential equations for the voltageequations of the α and β coil can be approximated as in-dependent:
ui = Riii + Lidiidt, with i = α, β. (30)
D. Construction of the Linear Drive
In Fig. 5 the manufactured linear actuator with the bondedtool electrode is displayed. The construction drawing of theactuator is shown on the right. To minimize friction, the sliderbearing was manufactured of PTFE. To provide access tothe electrical contact of the stator coils and to prevent eddycurrents, a slot was milled into the soft magnetic stator ofthe actuator. Next to approximation in the derivation of theanalytical model, there are minor design-related differencesbetween the constructed and modeled actuator which mightresult in small parameter deviations of the force factor km.Due to the integrated slot for the current supply of the coilsin the ferromagnetic stator (Fig. 5), the magnetic flux throughthe coils and thus, the resulting axial Lorentz force is reduced.Moreover, the difficulty of winding coils in this dimensionallows deviations in the number of turns per coil, which maybe reflected by a reduced km factor.
Fig. 5. Picture and construction drawing of the miniaturized electric lineardrive.
III. CONTROL OF THE TWO-PHASE LINEAR DRIVE
To control the position of the electrode, a cascaded control-with two loops is implemented [8]. This control structure isillustrated in Fig. 6. The current control builds the inner loop ofthe cascade and the position control the outer loop. The currenttransformation has no dynamic influence on the system, butdescribes the transformation of the current amplitude i into αand β coordinates iα and iβ . The goal of the electrode controlis to generate a steady state accuracy of less than 20µm.Moreover, a time constant for the step response behaviour ofthe electrode of TP ≤ 0.2 s should be generated to achieve ahighly dynamic position control.
A. Design of the Current Control
To design the current control, (30) is transformed into theLaplace domain, resulting in the transfer function:
Gc(s) =I(s)
U(s)=
1Ri
LiRis+ 1
, with i = α, β. (31)
A sampling frequency of fs = 20 kHz is chosen for the currentcontrol. By approximating the sampling element using Taylorseries expansion, its influence on the control behaviour can
Fig. 6. Control structure of the cascaded position control of the tool electrode.
335
be approximated by a first order lag element with the timeconstant TE = 2
fs. The extended controlled system results in
Gc(s) =1Ri
LiRis+ 1
1
TEs+ 1, with i = α, β. (32)
The current is controlled with a PI-controller, which is de-signed using the magnitude optimum design method [9]. Inthe design method, the optimization objective is to match themagnitude of the close loop transfer function to the desiredmagnitude of m = 1 for a wide range of frequencies.The designed control loop is illustrated in Fig. 7. The reference
Fig. 7. Implementation design of the current control for the electric circuit.
filter is used to compensate the stable numerator polynomial ofthe controlled system and optimize the disturbance rejection.The control parameters result in
Kp,c = 4.051V
A, Tn,c = 3.63 · 10−5 s. (33)
The measured step response for a current step of 0.5 A isillustrated in Fig. 8. Due to the highly dynamic design ofthe current control, a rise time of approx. 100µs is achieved.The closed loop current control behaviour can therefore besummarized by a first order lag element with the time constantTc ≈ 70µs:
Gw,c(s) =1
Tcs+ 1. (34)
B. Design of the Position Control
The differential equation of the mechanical system wasderived in (1) following Newton’s second law. The mass m is
Fig. 8. Current step response of 0.5A of the designed current control loop.
composed of the slider- and the tool electrode-mass and totals6 g. The acting force F is composed of the actuator force kmiand the counteracting gravitational force mg:
mx = kmi−mg. (35)
Since the dynamics of the current control with a time constantof Tc ≈ 100µs are more than 103 times faster than the desiredtime constants of the mechanical system of Tp = 100 ms, thetime delay caused by the current control can be neglected whencontrolling the mechanical system.For the design of the controller, it must be taken into accountthat in the derivation of the mechanical model, several simpli-fications have been made. In section II, parameter inaccuraciesdue to assumptions made in the derivation of the parameterkm have already been discussed. Moreover, the influence offriction forces and other dissipating forces (e.g., air resistanceon the electrode) are neglected in the mechanical model whichmight result in structural deviations.For these reasons, a robust extended state controller is im-plemented [10]. Transforming (35) into the state space modelyields
x =
[0 10 0
]x +
[0kmm
]i−
[0g
]. (36)
The control structure of the extended state controller is illus-trated in Fig. 9. To counteract the electrode weight mg, theconstant current ig is applied to actuator:
ig =mg
km≈ 0.22A. (37)
Moreover, a Lueneberger observer is implemented to mon-itor the state vector x of the electrode, consisting of the ob-served electrode position xL,o and electrode velocity vL,o. Us-ing the feedback control vector rT, the poles of the controlledsystem are shifted from pa 1,2 = 0 to the negative real axis atpb 1,2 = −100. Thereby, a fast binomial behavior is aspired,which prevents the electrode from overshooting. Moreover, thestate controller is extended with a PI-controller in order to
Fig. 9. Implementation design of the extended PI state controller for theposition control of the tool electrode.
336
Fig. 10. Measurement setup to examine position.
compensate for structure and parameter inaccuracies in themodeled system. The control parameters are set to
Kp,m = 2.19 · 103 A
m, Tn,m = 3.18 · 10−3 s, (38)
r1 = 4.38 · 103 A
m, r2 = 20.94
As
m. (39)
The experimental setup for the actuator control system isshown in Fig. 10. For position measurement, a laser trian-gulation sensor with a resolution of ±2µm was used. Thestep response with the implemented binomial position controlbehavior of the tool electrode is displayed in the upper partof Fig. 11. A steady state accuracy of about ±10µm wasachieved. The desired harmonic excitation of the electrodeis shown in lower part of Fig. 11. The harmonic positionmeasurement displays that the system has non-modeled non-linearities, resulting in slight harmonics. The most significantnonlinear effect in the mechanical system is the position andspeed dependent friction acting on the slider.
The dynamics of the controller is limited by the imple-mented current saturation of imax = 0.8 A. This is illustratedin Fig. 12 which shows the current amplitude i for the step
Fig. 11. Measurement results of the position control for a large signal stepresponse a) and a sinusoidal reference b).
response. Nevertheless, a time constant of approx. Tp = 70 msis achieved.
Fig. 12. Measurement results of the position and the current amplitude forthe step response of the linear drive.
IV. CONCLUSION
By designing a two-phase miniaturized linear drive, werealized a force generation that allows the 6 g tool electrodeto be positioned in the vertical direction against the forceof gravity. We demonstrated that for the miniaturized linearactuators with an outer diameter of ≤ 5mm, the voltageequations of the alpha and beta coils can be considered andcontrolled independently. Based on the analytical model, asteady-state position control accuracy of±10µm was achievedusing the cascaded control structure. The force excitation wassufficiently large to generate a harmonic excitation with afrequency of 5 Hz and an amplitude of 1 mm. Furthermore,the modeled actuator design allows a linear increase of theactuator force by axial slider extension.
In future work, the friction forces exerted on the slidershould be studied in more detail to compensate for thenonlinear behaviour and thereby improve the dynamic positioncontrol even further. Moreover the laser sensor needs to besubstituted by an integrated sensor or by sensorless positionestimation in order to use the miniaturized actuator within inan embedded control system.
REFERENCES
[1] Boldea, Tutelea, Xu, Pucci: Linear Electric Machines, Drives andMAGLEVs: an Overview, IEEE Transactions on Industrial Electronics,2017.
[2] Dierk, S.: Elektrische Antriebe-Grundlagen, Springer-Verlag Berlin Hei-delberg, 2015.
[3] Isermann, R: Mechatronische Systeme, Springer-Verlag Berlin Heidel-berg, 2008.
[4] Schrader, S.: Entwicklung von elektromagnetischen Linearantrieben undAutofokusoptiken fur endoskopische Systeme. Ph.D. Thesis, 2005.
[5] John david Jackson: Classical Electrodynamics - 3rd ed., 1999.[6] Julius Adams Stratton: Electromagnetic Theory (IEEE Press Series on
Electromagnetic Wave Theory)-Wiley-IEEE Press, 2007.[7] Hermann A. Haus, James R. Melcher: Electromagnetic fields and energy,
Prentice Hall, 1989.[8] J. Lunze: Regelungstechnik1, Springer, 2006.[9] W. Leonhard: Control of Electrical Drives (Power Systems), Springer,
2003.[10] G. C. Goodwin, S. F. Graebe, M. E. Salgado: Control System Design,
University of Newcastle, 2000.
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