Flexure design for precision positioningusing low-stiffness actuators
Abstract: This paper analytically investigates flexures to mechanically guide the moving partof a precision positioning system actuated by Lorentz actuators, where the control bandwidthis typically restricted by the second or higher mechanical resonances that can include internalmodes of the positioning mass. Based on analytical models, mechanical resonant frequenciesare derived for a given set of parameters to determine flexure dimensions and material. Asa result of the derivation and analysis, a model is proposed to predict this second resonantfrequency for a given first resonant frequency to achieve better control design and performance.As the verification, the effectiveness of the proposed model is confirmed by using Finite Element Analysis ,as well as an experimental setup for frequency response.
1. INTRODUCTION
For positioning systems, a variety of components are avail-able to guide motion, such as roller bearings [Fujii et al.(2010)], air bearings [Hou et al. (2012)] and magneticbearings [Choi and Gweon (2011)]. Among them, flexures[Yong et al. (2012)], elastic components supporting themoving parts, are commonly used in compact precisionpositioning systems for their relatively simple mechanism,without nonlinear disturbances such as backlash and fric-tion present in bearings. Particularly by using piezoelectricor Lorentz actuators (voice coil actuators), flexure-guidedsystems can even achieve positioning with nanometer res-olution within a short stroke [Schitter et al. (2007); Tumaet al. (2014)]. Dependent on applications and actuatortypes, many methods are available to design flexures.
In the case of flexure-guided systems with piezoelectricactuators, the first mechanical resonance that deformsthem typically limits the the control bandwidth of theresulting closed-loop system [Schitter et al. (2007)]. Forthis reason, the mechanical structures are designed tohave the resonance at a high frequency for a high controlbandwidth [Fantner et al. (2006)]. In the mechanicaldesign, the first resonant frequency can be approximatedby the undamped natural frequency ωn=√k/m usingthe moving mass m and the stiffness k [Kenton and Leang(2012)]. In this calculation, it is typically assumed thatthe moving mass is sufficiently solid. The stiffness k isinfluenced by the combined stiffness of the piezos and theflexures. The stiffness of the flexures can be obtained fromtheir dimension and material properties, for example byusing the Castigliano’s second theorem [Kenton and Leang(2012); Lin et al. (2013)]. For complicated mechanicalstructures, stiffness or compliance matrices are used withtransformation matrices to calculate the natural frequencyfor each rotational and translational axis, treating themoving masses as a solid body [Lai et al. (2011); Xiaoand Li (2013)]. To achieve a high control bandwidth, thestructures are tuned, such that the natural frequenciesof uncontrollable axes including the rotational modes arehigher than those of controllable actuation axes [Yonget al. (2012)]. In addition to these well-studied flexuredesign strategies, there is a clear criterion to select theflexure materials that the ratio of the Young’s modulus Eand the density ρ (i.e. E/ρ) is desired to be maximized forhigh resonant frequencies [Kenton and Leang (2012)].In contrast to the piezo-actuation systems, using Lorentzactuators can have a closed-loop control bandwidth that ismuch higher than the first resonant frequency, dependenton the mechatronic system design. These systems canbe categorized as low-stiffness actuators [Ito and Schitter(2016)] and are applied to optical disk drives (ODDs)[Chaghajerdi (2008); Heertjes and Leenknegt (2010)] oran atomic force microscope [Ito et al. (2015a)]. The me-chanical structures of these systems are typically designed,such that the first resonance is the suspension modealong the actuation axis. Its resonant frequency is setto a relatively low frequency by designing flexures for alarge actuation range or for better disturbance rejection[Ito et al. (2015b)]. The low stiffness is also beneficial todecrease the power consumption when positioning withan offset is required.
In contrast to the first, the secondor higher resonances, including structural internal modesof the positioning mass, may limit the achievable closed-loop control bandwidth [Munnig Schmidt et al. (2014)].Therefore, the second and higher resonant frequencies aretypically targeted for high frequencies in the structuredesign [Lee et al. (2002, 2007)]. Particularly for ODDs tosatisfy such requirements, finite element analysis (FEA) isoften utilized in the design [Zhang et al. (2008)]. In suchdesign, FEA can be applied iteratively to achieve desiredperformances by varying design parameters [Song et al.(2009)].
So far no clear guideline exits to design flexures for low-stiffness actuators in contrast to the well-studied me-chanical design strategies of flexures for piezo-actuationsystems. However, design guidelines of a low-stiffness-actuation system may allow an initial design that is al-ready close to an optimum, which might be further im-proved by applying FEA. Therefore, this paper analyti-cally investigates mechanical resonances that may limit theachievable closed-loop bandwidth. As a result, the paperproposes a simple model to predict the second mechanicalresonant frequency, yielding design guidelines to achieve ahigh control bandwidth.
This paper is organized as follows. Section 2 introducesa flexure-guided positioning system actuated by Lorentzactuators. In Section 3 the resonant frequencies of theflexures are analytically derived. Using the results, designguidelines of low-stiffness actuators are discussed in Sec-tion 4. Section 5 and Section 6 present FEA and experimen-tal results for verification. Section 7 concludes this paper.
2. SYSTEM DESCRIPTION
A positioning system to be considered in this paper isillustrated in Fig. 1, where the moving mass in the centermoves along the Z axis. In the same manner as piezo-actuation systems, it is assumed that the moving massitself is sufficiently rigid. (i.e. the structural internal modesoccur at high frequencies.) For a large actuation rangeand high disturbance rejection of the system, the stiffnessbetween the moving mass and the fixed frame is desired tobe sufficiently low. To satisfy the requirements, Lorentzactuators (i.e. voice coil actuators) are supposed to beused to generate the force Fzfor the Z actuation. Sincethese actuators utilize the Lorentz force, they have nomechanical stiffness between the moving mass and thefixed frame [Munnig Schmidt et al. (2014)], which is idealto construct a low-stiffness actuator, unlike piezoelectricactuators.
To mechanically guide the moving mass along the Z axis,several types of flexures are available, such as hinges[Lin et al. (2013)]. Among them, leaf-spring flexures areselected in this paper because they are ideal to realizea low stiffness [Yong et al. (2012)]. For simplicity ofimplementation, all the leaf-spring flexures are identical,having dimensions of length L, width w and height h, asshown in Fig. 1. In the next sections, the influences of thesedimensions and the flexure material on the mechanicalresonances are discussed, in order to increase the secondmechanical resonance with respect to the first. This isdesirable to achieve a high closed-loop control bandwidth.
3. MECHANICAL ANALYSIS
3.1 First resonant frequency
In the case of a low-stiffness actuator, the structure isdesigned to have the suspension mode at a relatively lowfrequency as the first mechanical resonance. To derive itsfrequency in the flexure design, can be modeled as shown in Fig. 2, where a coordinateframe is attached to the clamped end. On the other end,the force Fflex represents the actuation force distributedto each flexure. The bending moment M (y) of the flexureat y can be given as
M (y) =−Fflex(L−y)−Mx, (1)
where Mx is the moment resulting from the guide due tothe layout of the multiple flexures. Using (1), the deflectionz(y) of the flexure at y can be derived by solving thefollowing equation [da Silva (2006)]
d2z(y)dy2=−M (y)EIx, (2)
under the following conditions
z(0) = 0,dz(y)dy����y=0= 0,dz(y)dy����y=L= 0, (3)
where E and Ix are the Young’s modulus and the secondmoment of inertia (i.e. Ix= wh3/12), respectively. Thesolutions are given as
Mx=−FflexL/2, (4)
z(y) =Fflexy26EIx(32L−y) . (5)
Since the deflection at y = L is the resulting displacementof the moving mass, the stiffness of a single flexure kflexcan be obtained as
kflex= Fflex/z(L) = 12EIx/L3= Ewh3/L3. (6)
By assuming that the flexure damping and weight aresufficiently small, the undamped natural frequency ap-proximates the first resonant frequency ω1of the low-stiffness actuator as follows
ω1= √nkflexmm= √n12EIxmmL3, (7)
where mm is the weight of the moving mass and n is thenumber of the flexures.
3.2 High resonant frequenciesWhile the first resonance of the positioning system corre-sponds to the suspension mode along the actuation axis(Z axis), the second resonance might result from a reso-nance along uncontrollable axes, such as rotational modes around the X or Y axis. For these modes, there are somecountermeasures. For example, the moving mass rotationcan be suppressed by widening the distance of the parallelflexures. This also increases the respective resonant fre-quency, and it can be derived from the natural frequencyaround the corresponding axis [Yong et al. (2012)]. Moreimportantly in the case that Lorentz actuators are used,the excitation of the rotational modes can be prevented byadjusting the actuator position or by using over-actuation[Ronde et al. (2014); Ito et al. (2015a)].
While the aforementioned options are available to copewith resonances along the uncontrollable rotational axes,they are not applicable to the second and higher modesalong the controllable axis (i.e. the vertical Z axis). Thisis because they cannot be approximated by the naturalfrequency, and the actuation along the Z axis may excitethem. Therefore, it is important to derive their frequencyin order to increase it in the mechanical design, enablinga high control bandwidth.
To calculate the higher resonant frequencies in the direc-tion of the Z axis, the positioning system can be modeledas shown in Fig. 3, where zi(y, t) represents the deflectionof the i th flexure at position y and time t. An equationof motion of the i th flexure for free vibration is given as[Chakraverty (2008)]
公式8
where A is the cross-section area and ρ is the densityof the flexure material. The solution of (8) is given as[Chakraverty (2008)]
公式9
where κ has the relation of
公式10
The frequency is denoted by ω, and¯Ai,¯Bi,¯Ci,¯Di,¯Eiand¯Fiare parameters to be determined under given boundaryconditions.
Since each flexure is clamped to the fixed frame in Fig. 3,two boundary conditions are obtained as follows
公式11
Similarly,as the flexures are fixed to the moving mass, two more boundary conditions are given as
公式12
By considering that the shear forces of the flexures aty = L accelerate the moving mass, the last boundarycondition can be derived from Newton’s first law (cf. [Miuand Temesvary (1992)]) as follows
公式13
Using the boundary conditions (11)-(13) for (9),the following equation can be derived
公式14
图3
where mfis the total mass of the flexures (i.e. nρAL) (cf.[Miu and Temesvary (1992)]). By defining κjL as the j thsolution of (14) in the range of [0,∞), the j th resonantfrequency can be calculated by using (10) as follows
ωj= κ2j√EIxρA. (15)
Note that the first resonant frequency ω1obtained from(15) is different from (7) in that the flexure weight isconsidered for the derivation.
4. DESIGN OBJECTIVE AND PREDICTION MODEL
For the positioning system as a low-stiffness actuator, thesecond resonant frequency ω2is desired to be maximizedwhile ω1is set to a relatively low frequency, as discussed.Therefore, the resonant frequency ratio ω2/ω1may beselected as the objective function to be maximized in themechanical design. From (15) the ratio is given as
ω2ω1= (κ2Lκ1L)2. (16)
Since κ1L and κ2L are solutions of (14) and are influencedonly by the mass ratio mm/mf, the frequency ratio ω2/ω1can be evaluated as a function of the mass ratio. Bynumerically solving (14), the blue solid line in Fig. 4 isobtained.
Although the numerical simulation allows to draw thewhole graph of Fig. 4 to predict ω2/ω1, eventually it isdesirable to describe the relation of the frequency andmass ratios by a simple equation for accurate prediction.In order to do so, it is assumed that the flexures aresufficiently lighter than the moving mass (i.e. mf ≪mm).Under this assumption, (7) can be used for ω1togetherwith (15) for ω2to derive the resonant frequency ratio,leading to the following relation
公式17
To derive κ2L, mf/mm= 0 is substituted in (14), resultingin
公式18
By computing the above equation, κ2L = 4.73 is obtained.By inserting this number into (17), it is approximated as
公式19
To evaluate the effectiveness of the above approximation,(19) is plotted as the red dashed line in Fig. 4. The graphclearly shows that the simplified model (19) is effective
图4
表1
to approximate the ratio of the first and second resonantfrequencies in the range of about mm/mf> 3 without anycomputation, which holds for most flexure-guided systems.
In mechanical design, (7) can be used to calculate thefirst resonant frequency ω1. With the result, the simplifiedmodel (19) can be applied to predict the second resonantfrequency ω2for a given mass ratio. More importantly,the simplified model (19) and Fig. 4 lead to a flexure-design guideline that the mass ratio mm/mfneeds tobe maximized for a high control bandwidth. In otherwords, the flexure weight mfneeds to be minimized for agiven moving mass mm, and the dimensions and materialproperties of the flexures are regarded as parameters totune the weight mfand the stiffness kflex(6) in themechanical design.
Notice that the simplified model (19) and the discussionin this section hold with any flexures with an uniformcross section for a constant Ixalthough the equations areoriginally derived for leaf-spring flexures.
5. FINITE ELEMENT ANALYSIS
In this section, the analytical model developed in Section 4that is Equation (19) is verified by means of Finite ElementAnalysis (FEA), carried out with a software (Workbench,ANSYS, Canonsburg, US).
The FEA model consists of an aluminum block as themoving mass guided by four flat-spring flexures made ofbrass. Table 1 lists the design parameters of the flexures,determined according to realistic dimensional constraints.
To analyze the mechanical resonances, the first two modeshapes and their eigenfrequencies are obtained by FEA.Fig. 5 shows the simulated mode shapes of the model witha mass ratio mm/mf= 3.78. The first mode in Fig. 5(a) is the suspension mode, the frequency of which corresponds(a) First mode (suspension) at 66.3 HzZXYZXY(b) Second mode at 890 HzFig. 5. Modes shapes simulated by FEM for a mass ratiomm/mf= 3.78. The first and the second mode shapesare shown in Plot (a) and Plot (b), respectively.Table 2. Frequency ratios obtained by FEAand by simplified model (19) for a mass ratioof 3.78.Method ω2/ω1FEA 13.4Model (19) 12.6Error -6.0 %to the natural frequency in (7). Fig. 5(b) shows the secondmechanical resonance, the frequency of which correspondsto (15). Because the simulated first and second modesoccur at 66.3 Hz and 890 Hz, respectively, the simulatedfrequency ratio ω2/ω1is 13.4.
For comparison, the corresponding frequency ratio is alsocalculated by using the simplified model (19) with the massratio of 3.78. The obtained ratios are listed and comparedin Table 2. The difference of the values between the FEAthe simplified model is -6.0 %. This negative value may bepartially due to the fact that the simplified model (19)tends to provide a small frequency ratio as discussed withFig. 4. Overall, the FEA demonstrates the effectiveness ofthe derived model (19) to calculate the frequency ratio.For further analysis, an experimental setup is prepared tomeasure a frequency response.
6. EXPERIMENTAL FREQUENCY RESPONSE
In this section, the frequency response of an experimentalsetup is measured to analyze the influence of the second mechanical resonance, which (19) predicts.
A setup similar to the model used in the FEA of theprevious section is prepared. For manufacturing flexureswith the dimension in Table 1, four flexures are madefrom brass sheets by laser cutting. These flexures areused to mechanically guide an aluminum block as themoving part. For actuation, a Lorentz actuator (AVM12-6.4, Akribis Systems, Singapore) is installed on the centerof the moving part (cf. Fig. 1), such that the rotationalmodes are not excited. A voltage amplifier of a 2 kHzbandwidth is used to drive the Lorentz actuator. Themoving part displacement is measured by a laser Dopplervibrometer (OFV534, Polytec, Irvine, US), which typicallyhas a delay of several microseconds. The sensor is installedto measure the motion of the mass center, intending tohave the rotational modes unobservable.
The mass of the total flexures and the moving part includ-ing the actuator is measured by a scale (Acculab VIC303,Sartorius, G¨ottingen, Germany) as mf= 0.055 kg andmm= 0.2 kg, which results in a mass ratio of mm/mf= 3.64.From the mass ratio, the simplified model (19) predicts thefrequency ratio of the system as ω2/ω1= 12.3.
To discuss the influence of the predicted resonances onfrequency response and control design, a Bode plot ismeasured by using the amplifier reference as the inputand the displacement signal as the output. The resultsare shown in Fig. 6, where the first and second resonancesare visible at 67.6 Hz and 851 Hz, respectively. From thesevalues, the ratio is given as ω2/ω1= 12.6 and is comparedwith the analytical value in Table 3. The error of thesimplified model is only -2.3 %, confirming the predictedvalue. Furthermore, the measured results are added inFig. 4, where the plotted point is in good agreement withthe theoretical relation of the mass and frequency ratios.The measured frequency response together with the FEAresults verify the model presented in (19) to calculate thefrequency ratio only from a given mass ratio.
The magnified gain around the second resonance in Fig. 6shows that it is paired with an anti-resonance at a lowerfrequency, which would be due to the collocation of theactuator and the sensor [Preumont (2011)]. Because sucha pair advances the phase, itself alone does not typicallyrestrict the control bandwidth [Munnig Schmidt et al.(2014)]. However when such dynamics occur together witha phase lag or delay, for example due to amplifiers and sen-sors, the resonance may restrict the achievable bandwidth,dependent on control [Matsubara (2008)]. Therefore, thesecond mechanical resonance is targeted to occur at a highfrequency. Additionally for model-based motion controldesign, a system needs to be well-modeled, particularly atlow frequencies. When a frequency response is utilized formodeling, a pair of resonance and anti-resonance increasesthe model order [Yamaguchi et al. (2011)], which typicallyalso increases the resulting control order [Skogestad andPostlethwaite (2005)]. Thus, in order to simplify the mod-eling as well as the control design and implementation, itis desired that the resonance is shifted to a high frequencyin mechanical design.
For such a desired frequency response with the secondmechanical resonance at a high frequency, the simplifiedmodel(19) can be used to predict the second mechanicalresonance for a given first resonant frequency to guide the system design. Furthermore, the model also provides adesign guideline of the flexures that their total weight mfis minimized for a given moving mass mm.
图6
7. CONCLUSION
For better control design and performance of a flexure-guided positioning system with Lorentz actuators, leaf-spring flexures are investigated based on an analyticalmodel. As a result, a simple design guideline is proposed topredict the second mechanical resonance, which may be alimiting factor in the control design, for a given first reso-nant frequency. The proposed design rule is further verifiedby FEA and experimentally validated by a measured Bodeplot. The new method proposes a clear guideline to designflexures that their total weight needs to be minimized incomparison to the moving mass. By doing so, the secondresonant frequency can be maximized with respect tothe first resonance, enabling a high control bandwidth orsimplifying control design and implementation.
ACKNOWLEDGEMENTS
This work has been supported in part by the AustrianResearch Promotion Agency (FFG) under project number836489.2016 IFAC MECHATRONICSSeptember 5-8, 2016. Loughborough University, UK204
