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Digital Q-Control and Automatic Probe Landing in Amplitude Modulation Phase Imaging AFM Mode
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th World Congress Proceedings of the the 20th9-14, World Congress Available online at www.sciencedirect.com The International Federation Control Toulouse, France, July 2017 The International International Federation Federation of of Automatic Automatic Control The of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 10882–10888
Digital Q-Control and Automatic Probe Landing in Amplitude Modulation Phase Digital Q-Control and Automatic Probe Landing in Amplitude Modulation Phase Digital Probe Landing in Imaging Mode Digital Q-Control Q-Control and and Automatic Automatic ProbeAFM Landing in Amplitude Amplitude Modulation Modulation Phase Phase Imaging AFM Mode Imaging AFM Mode Imaging *AFM Mode ** * * Sergey Belikov*, John Alexander*, Marko Surtchev**, and Sergei Magonov* *, Marko **, and Sergei Magonov* Surtchev Sergey Belikov Belikov*, John Alexander **,, and Sergey Surtchev John* Alexander Alexander**,, Marko Marko Surtchev** and Sergei Sergei Magonov Magonov** Sergey Belikov**,, John SPM Labs, Tempe, AZ 85283 USA * * Labs, Tempe, 85283 USA *SPM Tel: 805-324-2872; e-mail: AZ [email protected] * SPM Labs, Tempe, AZ 85283 SPM Labs, Tempe, AZ 85283 USA USA ** Tel: 805-324-2872; e-mail: [email protected] NT-MDT America, Tempe, AZ 85284 Tel: e-mail: [email protected] ** Tel: 805-324-2872; 805-324-2872; e-mail: [email protected] ** America, Tempe, AZ 85284 ** **NT-MDT NT-MDT NT-MDT America, America, Tempe, Tempe, AZ AZ 85284 85284
Abstract: We present a new digital design of Q-control, i.e. controllable change of cantilever quality Abstract: We present aa new digital design of Q-control, controllable change of cantilever quality factor, targeted for digital implementation at an Thei.e. designed Q-controller the amplitude Abstract: We digital of Q-control, i.e. controllable change of quality Abstract: We present present a new new digital design design of FPGA. Q-control, i.e. controllable changechanges of cantilever cantilever quality factor, targeted for digital implementation at an FPGA. The designed Q-controller changes the amplitude and phase of cantilever excitation based on amplitude and phase of the deflection signal measured by a factor, targeted targeted for for digital digital implementation implementation at at an an FPGA. FPGA. The The designed designed Q-controller Q-controller changes changes the the amplitude amplitude factor, and phase of cantilever excitation based on amplitude and phase of the deflection signal measured digital lock-in amplifier. This significantly differs from the conventional design, where effective Q-factor and phase phase of of cantilever cantilever excitation excitation based based on on amplitude amplitude and and phase phase of of the the deflection deflection signal signal measured measured by by aaa and by digital lock-in amplifier. This significantly differs from the conventional design, where effective Q-factor is modified by adding a self-excitation force proportional to the cantilever deflection of an earlier time. digital lock-in amplifier. This significantly differs from the conventional design, where effective Q-factor digital lock-in amplifier. This significantly differs from the conventional design, where of effective Q-factor is modified by adding aa self-excitation force proportional the cantilever an earlier time. Our new digital Q-control algorithm is justified by flexibleto beam asymptoticdeflection models based on the Euleris modified by adding self-excitation force proportional to the cantilever deflection of an earlier time. is modified by adding a self-excitation force proportional to the cantilever deflection of an earlier time. Our new digital Q-control algorithm is justified by flexible beam asymptotic models based on the EulerBernoulli equation and the Krylov-Bogoliubov-Mitropolsky averaging technique. Many controversial Our new digital Q-control algorithm is justified by flexible beam asymptotic models based on the EulerOur new digital Q-control algorithm is justified by flexible beam asymptotic models based on the EulerBernoulli equation averaging technique. Many controversial features Q-controland canthe nowKrylov-Bogoliubov-Mitropolsky be verified by models. By contrast with conventional that Bernoulli equation and the Krylov-Bogoliubov-Mitropolsky averaging technique. Many Bernoulliof equation and the Krylov-Bogoliubov-Mitropolsky averaging technique. implementation Many controversial controversial features of Q-control can now be verified by models. By contrast with conventional implementation that requires additional analog electronics, our design is implemented at an FPGA in parallel with many other features of Q-control can now be verified by models. By contrast with conventional implementation that features of Q-control can now be verified by models. By contrastatwith conventional implementation that requires additional analog electronics, our design is implemented an FPGA in parallel with many other control signal processing algorithms. ability of Q-controlat decrease tip-sample interaction with requires additional analog our design is FPGA parallel many requiresand additional analog electronics, electronics, ourThe design is implemented implemented attoan an FPGA in in parallel with with many other other signal processing algorithms. ability Q-control to decrease tip-sample interaction with acontrol higherand effective and increase it with a The lower one isof landing (soft approach) of control and signal processing algorithms. The ability of Q-control to decrease interaction with control and signal Q processing algorithms. The ability ofillustrated Q-controlon to automatic decrease tip-sample tip-sample interaction with aa higher effective Q and increase it with a lower one is illustrated on automatic landing (soft approach) the tip with minimal indenting of the sample surface, or damage to the tip. higher effective effective Q Q and and increase increase it it with with aa lower lower one one is is illustrated illustrated on on automatic automatic landing landing (soft (soft approach) approach) of of a higher of the tip with minimal indenting of the sample surface, or damage to the tip. the tip with minimal indenting of the sample surface, or damage to the tip. the tip with minimal indenting of the sample surface, or damage to the tip. © 2017, IFAC (International Federation of AutomaticEmbedded Control) Hosting byNumerical Elsevier Ltd. All rights reserved. Keywords: Atomic Force Microscopy; Q-Control; Control; Methods. Keywords: Atomic Force Microscopy; Q-Control; Embedded Control; Numerical Methods. Keywords: Atomic Force Microscopy; Q-Control; Embedded Control; Numerical Methods. Keywords: Atomic Force Microscopy; Q-Control; Embedded Control; Numerical Methods.
1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The method of Q-control for Amplitude Modulation (AM) The for (AM) AFM after of introduction in 1993 Modulation [1] remains The method method ofitsQ-control Q-control for Amplitude Amplitude Modulation (AM)a The method for Amplitude Modulation (AM) AFM after of itsQ-control introduction in 1993 1993 [1]of the remains controversial control technique. Short reviews method AFM after its introduction in [1] remains aaa AFM after control its introduction in 1993 [1]of the remains controversial technique. Short reviews method and references up to 2010 can be found in [2, pp. 268-272] controversial control technique. Short reviews of the method controversial control technique. Short reviews ofpp. the268-272] method and references up to 2010 can be found in [2, [3, pp. 59-61]. Ashby in 2007 paper [4] provides the and references up to 2010 can be found in [2, pp. 268-272] and references up to 2010 can be found in [2, pp. 268-272] and [3, pp. 59-61]. Ashby in 2007 paper [4] provides the most critical remarks against the method used for gentle and [3, pp. 59-61]. Ashby in 2007 paper [4] provides the and [3, pp. 59-61]. Ashby in 2007 paper [4] provides the most critical remarks against the method used for gentle imaging of soft materials. Histhe final adviceused is “Itfor more most remarks against method most critical critical remarks against the method used foris gentle gentle imaging of softto materials. final advice is “It more advantageous use the His intrinsic Q factor andis imaging of His final is is more imaging of soft soft materials. materials. His final advice advice is “It “It is small more advantageous to use the intrinsic Q factor and small oscillation amplitude.” Nevertheless, the rate of Q-control advantageous advantageous to to use use the the intrinsic intrinsic Q Q factor factor and and small small oscillation Nevertheless, the Q-control publications remains intensive and most of of them claim oscillation amplitude.” amplitude.” Nevertheless, the rate rate of Q-control oscillation amplitude.” Nevertheless, the rate of Q-control publications remains intensive intensive and most most offollowing them claim significant advantages. Typical claims are the [5]: publications remains and of them publications remains intensive and most offollowing them claim claim significant advantages. Typical claims are the [5]: “Q factor can be increased to lower the maximum forces significant advantages. Typical claims are the following [5]: significant advantages. Typical claims are the following [5]: “Q factor can be increased to lower the maximum forces acting between tip and sample in air” and “the possibility to “Q factor can be increased to lower the maximum forces “Q factor can be increased to lower the maximum forces acting between tip and sample in air” and “the possibility to actively reduce the quality factor allows to increase the scan acting between tip and sample in air” and “the possibility to acting between tip and sample in allows air” andto“the possibility to actively reduce the quality factor increase the scan speed”. willthe evaluate on asymptotic actively reduce quality factor allows to the actively We reduce the quality these factorclaims allowsbased to increase increase the scan scan speed”. Weand will evaluate these claims based on asymptotic modelling experiments with automatic probe speed”. evaluate claims on asymptotic speed”. We We will will evaluate these these claims based based onlanding. asymptotic modelling and experiments with automatic probe landing. modelling and automatic Basic approach to activewith Q-control is probe usinglanding. cantilever modelling and experiments experiments with automatic probe landing. Basic approach to active Q-control is using cantilever deflection velocity feedback, because oscillator damping is Basic approach to active Q-control is using cantilever Basic approach tofeedback, active Q-control is using damping cantilever deflection velocity because oscillator is the coefficient of velocity in the Simpleoscillator Harmonicdamping Oscillator deflection velocity feedback, because is deflection velocity feedback, because oscillator damping is the coefficient of velocity in the Simple Harmonic Oscillator (SHO). However, there is in no the direct measurement velocity the coefficient of Simple Harmonic the coefficient of velocity velocity in the Simple HarmonicofOscillator Oscillator (SHO). However, there isneeded. no direct measurement of velocity and some estimation Among the difficulties are (SHO). However, there measurement of (SHO). However, thereis is is no no direct direct measurement of velocity velocity and some estimation is needed. Among the difficulties high oscillating frequency and significant noise. Ref. [6] and some some estimation estimation is is needed. needed. Among Among the the difficulties difficulties are are and high oscillating oscillating frequency and significant noise. However, Ref. are [6] proposes observer design to estimate velocity. high frequency and significant noise. Ref. high oscillating frequency and significant noise. However, Ref. [6] [6] proposes observer design to estimate velocity. analog Q-control implementations are usually based on an proposes observer design to estimate velocity. However, proposes observer design to estimate velocity. However, analog Q-control implementations are usually based on an estimation of the velocity by applying a phase shift to the analog Q-control implementations are usually based on an analog Q-control implementations are usually based on the an estimation of the velocity by applying a phase shift to displacement signal using a time delay circuit [2,3,5,7]. estimation of the velocity by applying a phase shift to the estimation of the velocity by applying a phase shift to the displacement signal using a time [2,3,5,7]. Unfortunately, the justification thisdelay phase circuit shift approach is displacement using time delay circuit [2,3,5,7]. displacement signal signal using aa of time delay circuit [2,3,5,7]. Unfortunately, the justification of this phase shift approach is based on an ansatz that the deflection signal is harmonic. This Unfortunately, is Unfortunately, the the justification justification of of this this phase phase shift shift approach approach is based on an ansatz that the deflection signal is harmonic. This is true only in steady state oscillation without tip-sample based based on on an an ansatz ansatz that that the the deflection deflection signal signal is is harmonic. harmonic. This This is is true true only only in in steady steady state state oscillation oscillation without without tip-sample tip-sample is true only in steady state oscillation without tip-sample
interaction and cannot be justified in transition. Conventional interaction and cannot be justified in transition. Conventional Q-controller design based on oversimplified SHO model may interaction cannot be in Conventional interaction and and cannot be justified justified in transition. transition. Conventional Q-controller design based on oversimplified SHO model may also suffer from a “spill-over effect” due to unmodeled Q-controller may Q-controller design design based based on on oversimplified oversimplified SHO SHO model model may also suffer from a “spill-over effect” due to unmodeled dynamics [7]. Ref. [7] addressed this issue by using negative also suffer suffer from from aa “spill-over “spill-over effect” effect” due due to to unmodeled unmodeled also dynamics [7]. Ref. [7] [7] addressed this issue issue by that using negative imaginary approach to addressed design a Q-controller guarantees dynamics [7]. Ref. this using negative dynamics [7]. Ref. [7] addressed this issue by by that using negative imaginary approach to design aa Q-controller guarantees stability with unmodeled dynamics. Ref. [8] utilizes imaginary approach to design Q-controller that guarantees imaginary approach to design a Q-controller that guarantees stability with unmodeled dynamics. Ref. [8] utilizes modulated-demodulated approach with an LTI controller in stability with unmodeled dynamics. Ref. [8] stability with unmodeled dynamics. Ref. [8] utilizes utilizes modulated-demodulated approach with an LTI controller in between. The LTI controller is designed using a positive modulated-demodulated approach with an LTI controller in modulated-demodulated approach with an LTI controller in between. The LTI controller is designed using a positive position whichis roll-off of the between. The controller designed using aa positive between.feedback The LTI LTImethod controller is“ensures designedrapid using positive position feedback method which “ensures rapid roll-off of the controller response, minimizing modes”. position method which “ensures rapid roll-off of position feedback feedback method whichspillover “ensures into rapidhigher roll-off of the the controller response, minimizing spillover into higher modes”. A design similar to “modulated-demodulated control” was controller controller response, response, minimizing minimizing spillover spillover into into higher higher modes”. modes”. A similar control” was earlier implemented in Q-control at commercial AFMs A design design similar to to “modulated-demodulated “modulated-demodulated control” was A design similar to “modulated-demodulated control” was earlier implemented in LTI Q-control at There commercial AFMs [9,10], but without the controller. are other Qearlier implemented in Q-control at commercial AFMs earlier implemented in Q-control at commercial AFMs [9,10], but without the LTI controller. There are other Qcontrol approaches that use special electronics, including [9,10], but without the LTI controller. There are other [9,10], but without the LTI controller. There are including other QQcontrol approaches that use special electronics, capacitive coupling [11], Positive Position Feedbackincluding with the control approaches that use special electronics, control approaches that use special electronics, including capacitive coupling [11], Positive Feedback the use of an active impedance in the Position piezoelectric shuntwith control capacitive coupling [11], Position Feedback with the capacitive coupling [11], Positive Positive Position Feedback with the use of an active impedance in the piezoelectric shunt control method etc.impedance Recently, in tuned oscillator (with additional use active the shunt control use of of an an[12], active impedance in the piezoelectric piezoelectric shunt control method etc. Recently, tuned oscillator additional circuitry[12], to tune damping) was used(with for quantitative method [12], etc. Recently, tuned (with additional method [12], etc.effective Recently, tuned oscillator oscillator (with additional circuitry to tune effective damping) was used for quantitative measurements [13]. Q-control was extended to higher circuitry circuitry to to tune tune effective effective damping) damping) was was used used for for quantitative quantitative measurements [13]. Q-control was extended to flexural eigenmodes [14] using cantilevers with integrated measurements [13]. [13]. Q-control Q-control was was extended extended to to higher higher measurements higher flexural eigenmodes [14] using using layer). cantilevers with integrated integrated actuation (bounded piezoelectric flexural eigenmodes [14] cantilevers with flexural eigenmodes [14] using cantilevers with integrated actuation piezoelectric layer). actuation (bounded piezoelectric layer). According(bounded to the references, “Q-control actuation (bounded piezoelectric layer). of high-frequency According to the references, “Q-control of high-frequency microcantilevers typically requires the development of highAccording of According to to the the references, references, “Q-control “Q-control of high-frequency high-frequency microcantilevers typically requires the development bandwidth analog circuit implementation” [8]. However, microcantilevers typically typically requires requires the the development development of of highhighmicrocantilevers of highbandwidth analog in circuit implementation” [8]. asymptotic However, current advances FPGA technology and bandwidth analog circuit implementation” [8]. However, bandwidth analog circuit implementation” [8]. However, current advances FPGA technology and asymptotic modelling of AFM in dynamics developing digital current advances in FPGA technology and asymptotic current advances in FPGA allow technology and pure asymptotic modelling of AFM dynamics allow developing pure digital implementation at an FPGA. This is one of the contributions modelling of AFM dynamics allow developing pure modelling of AFM dynamics allow developing pure digital digital implementation at an FPGA. This is one of the contributions of our paper. at design especially critical for an FPGA. FPGA. Thisisis is one one of the the contributions contributions implementation implementation atDigital an This of of Digital design is commercial AFMs, where Q-control method critical should for be of our our paper. paper. Digital design is especially especially critical for of our paper. Digital design is especially critical for where Q-control method should be be commercial AFMs, implemented at an FPGA with other measurement, commercial AFMs, AFMs, where along Q-control method should commercial where Q-control method should be implemented at an FPGA along with other measurement, control, signal processing, and communication algorithms for implemented at at an an FPGA FPGA along along with with other other measurement, measurement, implemented control, signal signal processing, and and communication communication algorithms algorithms for for multiple applications. control, processing, control, signal processing, and communication algorithms for multiple applications. applications. multiple multiple applications.
Copyright © 2017, 2017 IFAC 11369 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 IFAC 11369 Peer review© of International Federation of Automatic Control. Copyright ©under 2017 responsibility IFAC 11369 Copyright © 2017 IFAC 11369 10.1016/j.ifacol.2017.08.2445
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Sergey Belikov et al. / IFAC PapersOnLine 50-1 (2017) 10882–10888
We demonstrate how AFM asymptotic dynamics, derived in [15] for single-resonant and in [16] for multi-resonant AFM modes, justify the digital design and FPGA implementation. These models can also provide a deeper analysis of Q-control controversies described in [4]. In the majority of literature, design of Q-control is based on simplified SHO models without tip-sample interactions. Publications demonstrate that quality factor can be varied in a wide range. AFM applications, however, are based on tipsample interaction that is coupled with Q-control. This coupling may be the main source of the controversies. This is especially true for viscoelastic samples. Indeed, viscoelasticity is a velocity dependent feature that contributes to the change of the linearized system effective quality factor, along with Q-control. In such situations, there are no general suggestions and Q-control should be studied individually for each application. Automatic probe landing is an important application of Qcontrol where a complete theoretical and experimental study can be done almost independently of the sample material. Indeed, before landing, the tip is not interacting with the sample. Interaction starts only at the last short moment when condition of landing is identified. At this short moment the sample material behaves elastically with instantaneous modulus [17] and does not affect the quality factor of the cantilever. Our earlier paper [18] describes automatic landing procedures in different AFM resonant modes including Amplitude Modulation with Phase Imaging (AM-PI) [19, Table 1]. In the current paper, we demonstrate how Q-control influences the landing procedures. 2. BACKGROUND
Z
Zc
The mechanical setup of a controllably oscillated AFM probe, i.e. cantilever with a sharp tip, is presented in Fig. 1. It can be described by the following Euler-Bernoulli type of equation with the additional terms ( and F): Z Z F Z , Z / t Z (1) 2 x l a 4
2
2
t
x
4
S
where Z=Z(x, t) [m] is the vertical position at the point x [m] at time t [sec]; (f) [sec-1] is the dissipative term defined as a linear operator acting on function fZ/t, (This linear operator will be identified below); a 2 EI / S with E [Pa] – the elastic modulus of the cantilever’s material, [kg m-3] – density, S [m2] –area of the cross-section of the cantilever, and I [m4] –the moment of inertia of the cross-section; the RHS of (1) is the acceleration due to applying the concentrated force F [N] at the point xl [m]. In an AFM this force is due to tip-sample interaction. The cantilever is excited by one or more harmonics that vibrate the base located at x0 (Fig. 1): ~ ~ ~ (2) Z 0 , t Z c A n cos n t n ~ n [rad]
~ An
[m], ~ n [rad/sec], and are the amplitude, frequency and phase of the nth
where Zc [m] is the central position;
Cantilever
𝐴𝐴𝑛𝑛 cos 𝜔𝜔𝑛𝑛 𝑡𝑡 + 𝜑𝜑𝑛𝑛 𝑡𝑡
Tip Sample x
Piezo Scanner
Fig. 1. Mechanical setup of an AFM, including the cantilever attached to a vibrating base and a sample on a piezo scanner.
number (usually not more than two) and parameters ~ ( A n , ~ n , ~ n ) of the generated harmonics. Other three boundary conditions of free-end cantilever are presented in [16, formula (3)]. Solution z(x,t) of (1)-(2) with no tip-sample interaction, i.e. F0, can be presented as
zx,t
z n t Z n x ,
where { Z nl x } is an orthogonal
l
n 1
Z n l 1
basis described in [16, formula (9)],
l
. A linear
damping operator is defined [16] as
n Z n x
l
n 1
n n Z n x l
n 1
with n
n 2
4Q n 1
n
(3)
2Q n
where n and Qn are resonant frequency and quality factor of the nth eigenmode Z nl x of the cantilever that can be
2
zn 2 n z n n z n ~ ~ ~ l ~ ~ ~ ~ e n A i i i cos i t i 2 n sin i t i
This section summarizes the necessary results derived in [16].
t
L l
identified by thermal tune [20]. Derivations and details can be found in [16]. Coefficients zn satisfy the following equations
2.1 Modelling of Multi-Resonant AFM Modes
2
10883
harmonics of the excitation. The instrument user defines a
where
l
en
(4)
are the constants calculated in [16, formula (17)].
As shown in [16], asymptotically with the accuracy
2
O Qn
,
solutions of equations (4) can be approximated by solutions of the following equations l ~ 2 zn 2 n z n n z n e n A n ~ n ~ n cos ~ n t ~ n 2 n sin ~ n t ~ n
with a single ~ n n O Q n 1 and A n O Q n 1 . RHS of this equation is a harmonic function with frequency ~ n . Changing the time origin such that the phase of the harmonic function is nullified, we get the following equation with asymptotic accuracy O Q : 2
n
~ 2 zn 2 n z n n z n E n cos ~ n t
where
~ l ~ 2 E n e n An n
(5)
.
In most Q-control publications, some equivalent of Eq. (5) is a starting point of development without connection to Eq. (2). However, there are no natural forces applied to the eigenmodes, and RHS of Eq. (5) is just a calculated term related to the actual excitation in Eq. (2), applied to the base of the cantilever as shown in Fig. 1. An algorithm that calculates the amplitude E~ n in Eq. (5) should recalculate
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amplitude
~ An
and phase
~ n
of the excitation in Eq. (2) that is
physically applied to the base of the cantilever. Eq.(5) also suggests that equations for different eigenmodes are asymptotically (with accuracy O Q n 2 ) uncoupled. However, the equations derived from Eq. (1)-(2) with F 0 are coupled. In publication [16] we also derived the asymptotic dynamics of amplitude-phase (defined as van der Pole coordinates [16, Eq. (36)]) for AFM resonant modes, including AM-PI. This dynamics is critically important because the AFM controller works with amplitudes and phases measured by lock-in amplifiers. For the purpose of this paper we use asymptotic amplitudephase A– dynamics for a single-resonant frequency n. Asymptotic approximation for a conservative tip-sample interaction force F(z) and small amplitude A is the following [18]: n dA A A 0 sin dt 2 A0 n A 0 F z d cos z A 2 N dt
(6)
a. b. c. Fig. 2. Translation of a sample attached to the piezo scanner, driven by a stepper motor and piezo scanner.
where Ao is the amplitude of the oscillation with F0; N is the normalizing force defined in [18]; and Q . As expected, 1
n
speeds of A and are proportional to . AM-PI) cannot be reached, the stepper motor must move the piezo scanner up, i.e. closer to the tip (Fig. 2c).
2.2 Control of z-Positioning of the Sample with Stepper Motor and Piezo Scanner Comprehensive description of control for nanopositioning systems, including AFMs, can be found in [21]. For the purpose of this paper, i.e. Q-control and landing of the AFM probe, we concentrate only on vertical z-positioning of the sample controlled by the stepper motor and the piezo scanner. Fig. 2 is an illustration of the sample vertical translations. The stepper motor sets a rough position for the piezo scanner, which holds a sample. The piezo scanner can be extended or contracted, moving the sample respectively closer to the tip or farther away. In most applications, a combined use of a stepper motor and piezo scanner brings the probe into interaction with the sample. The stepper motor can be commanded to either move with a predefined velocity until “stop” command, or perform a move at a predefined distance. At the beginning of the AFM experiment, the sample is placed relatively far from the probe, and tip-sample interaction force is negligible. Vertical positioning of the tip relative to the sample uses a stepper motor for coarse translation and a piezo actuator for fine approach. In probe landing, both actuators are used. After the landing is accomplished, most subsequent actions are made using only the piezo scanner. The piezo scanner is extended or contracted by applying negative or positive voltage, respectively. Digital controller generates a normalized voltage u in the range [1,1] that after amplification is applied to the piezo scanner. With u1 the piezo scanner is fully compressed (Fig. 2a). It is always in this state when feedback is off, when sample is as far from the tip as the position of the piezo scanner allows (Fig. 2a). With u1 the piezo scanner is fully extended (Fig. 2b). If in this state the feedback is on but the set point (amplitude in
2.3 Review of Probe Landing in Amplitude Modulation Phase Imaging (AM-PI) AFM Mode As was formulated in [18], landing is characterized by a sharp temporal change in the force gradient F z z while the piezo scanner is extending and the sample is approaching the tip. Analyzing (6), we conclude [18] that the change of θ is a reliable indicator of landing in AM-PI. The use of phase as a reliable landing indicator is illustrated in Fig. 3c –see the jump in “Phase [deg]” plot. Meanwhile the amplitude trajectory, shown on “Normalized Amplitude [0,1]” plot in 3b, is smoothly approaching the set point and cannot indicate the landing event. Fig. 3 shows the process of landing without Q-control. In this example, as shown in the Bode plots (Fig. 3a) obtained by a frequency sweep, resonant frequency is 277 kHz and Q factor is 474. (Phase is plotted in NT-MDT convention, i.e. zero rather than 90o at resonance). When the probe approaches the sample in air, one should consider “squeeze-film-damping” [22], a phenomenon that emerges because the probe is near a flat surface: when the probe that is oscillating at its resonant frequency approaches a sample, there is a reduction of resonant frequency and amplitude. Reduction of these parameters can serve as an indicator that the tip, although not yet interacting, is very close to the sample. The Automatic Landing Procedure in AM-PI mode consists of three stages [18]. Stage I makes a rough approach based on “squeeze-film-damping” [22]; stage II compresses the piezo scanner to have margin to extend it during the final stage III that ends with the landing. Fig. 2b,c demonstrate that the same location of the sample can be obtained by different
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combinations of the piezo scanner position controlled by the stepper motor and the extension of the piezo scanner itself, controlled by the normalized voltage u. AM-PI Landing Stage I uses the stepper motor to move a piezo scanner, fully extended because feedback is on and sample is far from the tip, to the position where the amplitude set point is reached. This set point is equal to 90% of the initially measured amplitude. Stepper motor is moving continuously at high speed until measured amplitude reaches 97% of the initial amplitude, and then at slow speed until it reaches 90%. The blue horizontal line on the left part of the plot in Fig. 3b is 90% of the initial amplitude. Stage I ends when the normalized amplitude (red) crosses the 90% set point (blue horizontal line). Fast and slow motions can be easily recognized in Fig. 3b,c – they are separated by near vertical parts of the trajectories. AM-PI Landing Stage II is iterative. During each iteration, we keep the same amplitude set point (blue line in Fig. 3b). First we turn feedback off and wait for u=1 (fully compressed piezo scanner); then move to a predefined distance by the stepper motor (to place the sample closer to the tip); next, with feedback on, measure normalized control voltage u and stop the stage II when it is large enough (default u>0.4, i.e. piezo scanner is less than 30% of its full size). Otherwise – continue the loop. Five iterations of this stage are shown in Fig. 3 (see trajectories before the blue staircase in Fig. 3b). In Fig. 3d, u1 indicates the regions of trajectories where feedback is off. An increase of u, i.e. compression of piezo scanner, at each of four iterations can also be seen. AM-PI Landing Stage III is also iterative and stops when the AM-PI condition of landing has been met, i.e. phase or phase noise jump exceeds the threshold (default 2 o). Each iteration decreases the amplitude set point by default of 3% – see blue staircase in Fig. 3b. A new iteration starts by measuring u, followed by turning feedback off. If the measured u < 0 (extended), a stepper motor moves the piezo scanner (with the sample) closer to the tip to compress it and keep closer to neutral position u0. (For u>0, decreasing the amplitude set point makes the compressed piezo scanner extend and stepper motor move is not necessary). Fig. 3 shows 7 iterations of stage III. The stage stopped because the phase jumped. The piezo scanner in this moment is close to the neutral position (|u|<0.1).
a. Frequency sweep: Q=474.
b. Set point (blue) and z-servo controlled amplitude (red).
c. Phase trajectory. Sharp change at the end indicates landing.
d. Normalized position “u” of piezo scanner: 0 –neutral; -1 – fully extended; +1 –fully contracted. At landing the position is close to neutral (|u|<0.1). Fig. 3. Automatic control of landing in AM-PI mode without Q control: 7 steps on Stage III.
3. DESIGN OF DIGITAL Q-CONTROL FOR FPGA 3.1 Theory Similarly to most conventional Q-control designs, we start with Eq. (5). However, as mentioned in Section 2.1, this equation is linked with the boundary condition in Eq. (2) describing physical excitation of the base in Fig. 1. In ~ particular A n , ~ n for a single ~ n n must be recalculated back from Eq. (4)-(5) before applying to the base. Eq. (3) is the relationship between damping coefficient n and quality factor QQn. A naive way to modify in Eq. (5) would be to add hypothetical feedback control (7) u z 2 z where zzn, so that Eq. (5) becomes (index n is omitted)
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~ 2 z 2 z n z u z E cos ~ n t
(8)
~
and has the modified damping . Aside from the problems with estimating dz/dt, there are even more serious objections on how to transfer the hypothetical control u to the excitation in Eq. (2). Indeed, Eq. (2) assumes that ~ ~ ( A t , ~ t )( A n t , ~ n t ) vary much slower than cos ~ n t . (Otherwise one cannot define amplitude and phase.) Amplitude-phase dynamics in Eq. (6) satisfies this requirement, but feedback u z in Eq. (8) does not. Proper control should use slowly varying (A,) described by Eq. (6). They are measured by digital lock-in amplifier.
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a. Frequency sweep: Q=833.
a. Frequency sweep: Q=255.
b. Set point (blue) and z-servo controlled amplitude (red).
b. Set point (blue) and z-servo controlled amplitude (red).
c. Phase trajectory. Sharp change at the end indicates landing.
c. Phase trajectory. Sharp change at the end indicates landing.
d. Normalized position “u” of piezo scanner: 0 –neutral; -1 – fully extended; +1 –fully contracted. At landing the position is close neutral (|u|<0.05). Fig. 4. Automatic control of landing in AM-PI mode with increased Q: 14 steps on Stage III.
d. Normalized position “u” of piezo scanner: 0 –neutral; -1 – fully extended; +1 –fully contracted. At landing the position is close neutral (|u|<0.2). Fig. 5. Automatic control of landing in AM-PI mode with decreased Q: 4 steps on Stage III.
There is a significant difference in increasing and decreasing Q. From (3), (7), (8), (omitting index n),
~ ~ Q n / 2 n / 2
To change
~ Q
(9)
from Q to infinity, should be changed from
zero to , that is not difficult for small . However, to ~ change Q from Q to zero, should be changed from zero ~
to , that is a more difficult task for small Q . ~
~
As long as ( A t , ~ t )( A n t , ~ n t ) in Eq. (2) change slowly, the harmonic steady state solution of Eq. (5) is a good approximation. The solution is, (index n omitted),
~ z ( t ) A ~ cos t
~
(10)
where amplitude and phase can be calculated by substituting Eq. (10) to Eq. (5). For ~ n and n<<1, A
n
l ~ en A Q ,
/2
(11)
n
From Eq. (10), z ( t ) A ~ ~ sin ~ t ~ , and we can temporarily, assuming ~ n , replace hypothetical control in Eq. (7) by the open-loop control ~ ~ ~ u , 2 A ~ sin t ~
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~ l 2 ~ A e n n sin t
~
,
/
(12)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Sergey Belikov et al. / IFAC PapersOnLine 50-1 (2017) 10882–10888
10887
This signal can be generated by digital lock-in amplifier that measures A ~ and ~ . Substituting u in (8) by RHS of Eq. (12) 2
z 2 z n
~ l 2 A e n n
~ M sin t
~
~ cos t
(13) where M1 for the open loop (to be introduced in section 3.2). Parameter in Eq. (13) was introduced in such a way that amplitude of harmonic function in square brackets is less than unity, i.e. 1 / M
Shaker Amplitude:
this, Q-control can be generated by applying the following waveform to the shaker that excites the base (Fig. 1): Q Control Generator ~ ~ A S G Q X sin t
(14)
This condition is important for an FPGA implementation that uses fixed-point arithmetic and scaling the numbers. Varying , the range of || can theoretically be from zero to infinity, and so is the range of Q. Smaller defines a larger range of . For implementation purposes, we define the following control parameters:
Fig. 6. User interface elements of Q-control.
G Q 1 S / M ,
~
S
S Q GQ
(15)
0 1
~ cos t
Q
S
~ cos t
(17)
“Shaker G0 Freq [kHz]” “Shaker Fraction Amp [0-1]” “Q Phase [-180:+180deg]” “Q Gain [-1:+1]”
Usually
~ cos t ,
Y
S
~
.
2 z 2 z n A S G Q M sin ~ t
Q
User interface elements for Q-control are shown in Fig. 6. The following table relates the elements to the parameters of Eq. (17). Table 1. Q-control parameters and related user interface elements ~ “Shaker G0 Range [0-10]” A
~ l 2 A S A e n n /
Q Gain: GQ Shaker Excitation Fraction: S With these notations, Eq. (13) can be written as
~ A S G Q M S
is kept constant when tip is far from
the sample, (it is the free resonant amplitude –see next section). 3.3 Q-Control Analysis
~ ~ A An
and phase ~ ~ n of the single excitations in Eq. (2) should be recalculated from these control parameters. Theoretically, phase ~ should be close to 90o
Mathematically, applying the generator in Eq. (17) is equivalent to the following equation (compare with (15)):
(see Eq. (11)), and ~ ~ n in Eq. (2) could be recalculated with this number. In practice, however, there may be a phase shift in the instrumentation that requires compensation in a real implementation.
However, in contrast to (18), the generator (17) implements feedback control because X and Y are measured in the loop. A practically suggested selection of Q is the following: when the cantilever is far from the sample and the base (Fig. 1) is oscillating with the free resonant frequency of the cantilever, select a positive gain, e. g. GQ0.1, and maximize the amplitude “Normalized Amplitude [0,1]” plotted in Fig. 3b, which is the amplitude of the deflection z(t) signal measured by the digital lock-in amplifier. According to Eq. (18), maximum amplitude of z(t) takes place at the maximum amplitude of the excitation
Amplitude
3.2 Remarks on an FPGA Implementation In an FPGA implementation, digital lock-in amplifier measures in-phase and quadrature components of the single nth mode of the deflection signal (16) X M cos ~ , Y M sin ~
2 z 2 z n A S G Q M sin ~ t
~ G Q M sin t
where M is a scaled unitless calculated magnitude (0
/2
Q
~
. In implementation,
however, Q is selected indirectly by maximizing the amplitude of the cantilever when it is far from the sample. Then G Q M sin ~ t Q ~ G Q M sin ~ t Q cos ~ G Q M cos t ~
Q
sin G X ~
Q
sin t ~
Y cos t .
In implementation, we do not recalculate
Q
~ ~ A An
in Eq. (2) ~
from AS, but directly select the related parameter A S as the amplitude of the excitation applied to the base in Fig. 1. With
G
Q
M sin
~
Q
~
Q
Q
S
sin ~ t cos S
Q
S
~ cos t
(18)
~ cos t
(19) ~
t
The square of the amplitude of RHS of (19) is 2 2 2 A Q G Q M S 2 G Q M S sin Q and for positive 1
GQ has its maximum at
1
Q
/2
. Then Eq. (18) is
2 z 2 z n A S G Q M S cos ~ t
~
Q
G Q M cos
~
~
(20)
Eq. (20) provides an important interpretation for majority of Q-control designs, including this digital implementation. Q-control is an adaptive regulator of the excitation amplitude applied to the base (Fig. 1) at one of the eigenfrequencies n.
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In fact, intrinsic Q of the cantilever has not been changed, although Bode diagrams in Fig. 3a, 4a, and 5a show a different Q. The reason is that the further the excitation frequency ~ is from resonant n, the smaller is the magnitude M. With a positive gain GQ in Eq. (20), reduction of M causes reduction of (now non-resonant) excitation amplitude and further reduction of M (positive feedback). Effectively this reduces non-resonant amplitudes, imitating increasing Q (Fig. 4a). With a negative gain GQ in Eq. (20), we have the opposite (negative feedback), i.e. reduction of M causes increase of (now non-resonant) excitation amplitude, causing an increase of M, which looks like a reduction of Q (Fig. 5a). As for Q-control controversies [4] and Ashby’s suggestion that “It is more advantageous to use the intrinsic Q factor and small oscillation amplitude”, we can offer the following. Qcontrol in fact does not change the intrinsic Q factor, but has the ability to adaptively decrease the oscillation amplitude. Thus, it follows Ashby’s advice, with addition of adaptive amplitude reduction. This is the true value of the method that would be better called “Adaptive regulator of the excitation amplitude”. Adaptive decrease of excitation amplitude lowers the forces acting on the tip, imitating the increase of “effective Q”. This causes Z-servo to behave slower. Adaptive increase of excitation amplitude acts in the opposite direction.
motor and piezo scanner, and automatic probe landing in AM-PI mode. Theory was used to design a Q-Control for digital FPGA implementation, and we view the method as adaptive regulator of the excitation amplitude applied to the base. This view allows reconciling some controversies of Q-control method. Applications of the digital Q-control to automatic probe landing validate the method in practice for these applications. REFERENCES [1]
[2]
[3] [4]
[5]
[6]
[7]
[8]
4. AUTOMATIC PROBE LANDING IN AM-PI MODE WITH DIGITAL Q-CONTROL We present screenshots of GUI panels of a commercial AFM (Fig. 3-6) to better illustrate on-line operator experience. Fig. 3, 4, and 5 demonstrate the process of landing respectively without Q-control (GQ=0), with increased “effective Q” (GQ>0), and with decreased “effective Q” (GQ<0). The automatic landing was reviewed in Section 2.3. Fig. 3a, 4a, and 5a shows Bode plots and calculated quality factors 474, 833, and 255, respectively. One can see that number of steps of Stage III before landing increases with Q. There are 7, 14, and 4 steps, respectively (Fig. 3b, 4b, and 5b) that are roughly proportional to Q. This behaviour has the following clear explanation. Without Q-Control (Fig. 3), decrease of the deflection amplitude at stage III (Fig. 3b) before the landing is caused only by natural “squeeze film damping” [22]. With increased “effective Q” (Fig. 4), natural decrease of deflection amplitude causes its further decrease due to a reduction of excitation amplitude. Artificial decrease of deflection amplitude causes Z-servo to extend the piezo scanner less. It takes more steps to land but the landing is softer. With a reduced “effective Q” (Fig. 5), natural decrease of deflection amplitude causes an increase of excitation amplitude, increasing the deflection amplitude. This forces the Z-servo to extend the piezo scanner more (closer to the probe). It takes less steps to land but the landing may be more invasive. 5. CONCLUSIONS We provide a background theory of asymptotic models of resonant AFM operation modes, positioning with stepper
[9] [10] [11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21] [22]
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J. Mentz, Q. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” Appl. Phys. Lett. 62 (19), pp. 2344-2346, 1993. A. Schirmeisen, B. Anczykowski, and H. Fuchs, “Dynamic Force Microscopy,” in B. Bhushan (editor) Nanotribology and Nanomechanics An Introduction, Springer, pp. 243-281, 2005. R. Garcia, Amplitude Modulation Atomic Force Microscopy, WILEYVCH&Co., 2010. P. Ashby, “Gentle Imaging of soft materials in solution with amplitude modulation atomic force microscopy: Q control and thermal noise,” App. Phys. Lett., 91, 254102, 2007. H. Holscher, U. Schwarz. “Theory of amplitude modulation atomic force microscopy with and without Q-Control,” Int. J. Of Nonlinear Mech. 42, pp. 608-625, 2007. D. Sahoo, T. De, and M. Salapaka, “Observer based imaging methods for Atomic Force Microscopy,” 44th IEEE Conf. On Decision and Control and European Contr. Conf., pp. 1185-1190, 2005. M. Fairbairn, and S. Moheimani, “Resonant control of an atomic force micro-cantilever for active Q control,” Rev. Sci. Instr., 83, 083708, 2012. K. Karvinen, and S. Moheimani, “Modulated-demodulated control : Q control of an AFM microcantilever,” Mechatronics, 24, pp. 661-671, 2014. Agilent MAC Mode III Data Sheet 2009. Agilent 5500 SPM User’s Guide, 2009. M. Huefner, A. Pivonka, J. Kim, C. Ye, M. Blood-Forsythe, M. Zech, and J. Hoffman, “Microcantilever Q control via capacitive coupling,” Appl. Phys. Lett., 101, 173110, 2012. Asha, Taranath H B, “Active Q Control of an AFM Micro Cantilever Using PPF Controller,” Int. J. Of Electrical and Electronics Research, vol.3, issue 1, pp. 341-345, 2015. Q. Dagdeviren, J. Gotzen, H. Holster, E. Altman, and U. Schwarz, “Robust high-resolution imaging and quantitative force measurement with tuned-oscillator atomic force microscopy,” Nanotechnology 27, 065703:1-10, 2016. M. Ruppert, and S. Moheimani, “Multimode Q Control in TappingMode AFM: Enabling Images on Higher Flexural Eigenmodes,” IEEE Trans. Contr. Sys. Tech., 2016. S. Belikov, and S. Magonov, “Classification of Dynamic Atomic Force Microscopy Control Modes based on Asymptotic Nonlinear Dynamics,” Proc. Am. Contr. Conf., pp. 979-984, 2009. S. Belikov, and S. Magonov, “Modeling of Cantilever Beam under Dissipative and Nonlinear Forces with Application to Multi-Resonant Atomic Force Microscopy,” 20th IFAC World Congress, 2017. S. Magonov, M. Surtchev, J. Alexander, I. Malovichko, and S. Belikov, “Mapping of Nanoscale Material Properties of Polymers in Quasistatic and Oscillatory Atomic Force Microscopy Modes,” MRS Advan, 2016. S. Belikov, J. Alexander, M. Surtchev, I. Malovichko, and S. Magonov, “Automatic Probe Landing in Atomic Force Microscopy Resonance Modes,” Am. Control Conf., 2017. S. Belikov, J. Alexander, M. Surtchev, and S. Magonov, “Implementation of Atomic Force Microscopy Resonance Modes based on Asymptotic Dynamics using Costas Loop,” Am. Contr. Conf., pp. 6201-6208. 2016. S. Belikov, J. Alexander, C. Wall, I. Yermolenko, S. Magonov, and I. Malovichko, “Thermal Tune Method for AFM Oscillatory Resonance Imaging in Air and Liquid,” Am. Control Conf., pp.1009-1014, 2014. A. Fleming, and K. Leang, Design, Modeling and Control of Nanopositioning Systems, Springer, 2014. A. Kumar, and R. Pratap, “Effect of flexural modes on squeeze film damping in MEMS cantilevers,” J. of Micromechanics and Microengineering, 17, pp. 2475-2484, 2007.
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IFAC PapersOnLine 50-1 (2017) 10882–10888
Digital Q-Control and Automatic Probe Landing in Amplitude Modulation Phase Digital Q-Control and Automatic Probe Landing in Amplitude Modulation Phase Digital Probe Landing in Imaging Mode Digital Q-Control Q-Control and and Automatic Automatic ProbeAFM Landing in Amplitude Amplitude Modulation Modulation Phase Phase Imaging AFM Mode Imaging AFM Mode Imaging *AFM Mode ** * * Sergey Belikov*, John Alexander*, Marko Surtchev**, and Sergei Magonov* *, Marko **, and Sergei Magonov* Surtchev Sergey Belikov Belikov*, John Alexander **,, and Sergey Surtchev John* Alexander Alexander**,, Marko Marko Surtchev** and Sergei Sergei Magonov Magonov** Sergey Belikov**,, John SPM Labs, Tempe, AZ 85283 USA * * Labs, Tempe, 85283 USA *SPM Tel: 805-324-2872; e-mail: AZ [email protected] * SPM Labs, Tempe, AZ 85283 SPM Labs, Tempe, AZ 85283 USA USA ** Tel: 805-324-2872; e-mail: [email protected] NT-MDT America, Tempe, AZ 85284 Tel: e-mail: [email protected] ** Tel: 805-324-2872; 805-324-2872; e-mail: [email protected] ** America, Tempe, AZ 85284 ** **NT-MDT NT-MDT NT-MDT America, America, Tempe, Tempe, AZ AZ 85284 85284
Abstract: We present a new digital design of Q-control, i.e. controllable change of cantilever quality Abstract: We present aa new digital design of Q-control, controllable change of cantilever quality factor, targeted for digital implementation at an Thei.e. designed Q-controller the amplitude Abstract: We digital of Q-control, i.e. controllable change of quality Abstract: We present present a new new digital design design of FPGA. Q-control, i.e. controllable changechanges of cantilever cantilever quality factor, targeted for digital implementation at an FPGA. The designed Q-controller changes the amplitude and phase of cantilever excitation based on amplitude and phase of the deflection signal measured by a factor, targeted targeted for for digital digital implementation implementation at at an an FPGA. FPGA. The The designed designed Q-controller Q-controller changes changes the the amplitude amplitude factor, and phase of cantilever excitation based on amplitude and phase of the deflection signal measured digital lock-in amplifier. This significantly differs from the conventional design, where effective Q-factor and phase phase of of cantilever cantilever excitation excitation based based on on amplitude amplitude and and phase phase of of the the deflection deflection signal signal measured measured by by aaa and by digital lock-in amplifier. This significantly differs from the conventional design, where effective Q-factor is modified by adding a self-excitation force proportional to the cantilever deflection of an earlier time. digital lock-in amplifier. This significantly differs from the conventional design, where effective Q-factor digital lock-in amplifier. This significantly differs from the conventional design, where of effective Q-factor is modified by adding aa self-excitation force proportional the cantilever an earlier time. Our new digital Q-control algorithm is justified by flexibleto beam asymptoticdeflection models based on the Euleris modified by adding self-excitation force proportional to the cantilever deflection of an earlier time. is modified by adding a self-excitation force proportional to the cantilever deflection of an earlier time. Our new digital Q-control algorithm is justified by flexible beam asymptotic models based on the EulerBernoulli equation and the Krylov-Bogoliubov-Mitropolsky averaging technique. Many controversial Our new digital Q-control algorithm is justified by flexible beam asymptotic models based on the EulerOur new digital Q-control algorithm is justified by flexible beam asymptotic models based on the EulerBernoulli equation averaging technique. Many controversial features Q-controland canthe nowKrylov-Bogoliubov-Mitropolsky be verified by models. By contrast with conventional that Bernoulli equation and the Krylov-Bogoliubov-Mitropolsky averaging technique. Many Bernoulliof equation and the Krylov-Bogoliubov-Mitropolsky averaging technique. implementation Many controversial controversial features of Q-control can now be verified by models. By contrast with conventional implementation that requires additional analog electronics, our design is implemented at an FPGA in parallel with many other features of Q-control can now be verified by models. By contrast with conventional implementation that features of Q-control can now be verified by models. By contrastatwith conventional implementation that requires additional analog electronics, our design is implemented an FPGA in parallel with many other control signal processing algorithms. ability of Q-controlat decrease tip-sample interaction with requires additional analog our design is FPGA parallel many requiresand additional analog electronics, electronics, ourThe design is implemented implemented attoan an FPGA in in parallel with with many other other signal processing algorithms. ability Q-control to decrease tip-sample interaction with acontrol higherand effective and increase it with a The lower one isof landing (soft approach) of control and signal processing algorithms. The ability of Q-control to decrease interaction with control and signal Q processing algorithms. The ability ofillustrated Q-controlon to automatic decrease tip-sample tip-sample interaction with aa higher effective Q and increase it with a lower one is illustrated on automatic landing (soft approach) the tip with minimal indenting of the sample surface, or damage to the tip. higher effective effective Q Q and and increase increase it it with with aa lower lower one one is is illustrated illustrated on on automatic automatic landing landing (soft (soft approach) approach) of of a higher of the tip with minimal indenting of the sample surface, or damage to the tip. the tip with minimal indenting of the sample surface, or damage to the tip. the tip with minimal indenting of the sample surface, or damage to the tip. © 2017, IFAC (International Federation of AutomaticEmbedded Control) Hosting byNumerical Elsevier Ltd. All rights reserved. Keywords: Atomic Force Microscopy; Q-Control; Control; Methods. Keywords: Atomic Force Microscopy; Q-Control; Embedded Control; Numerical Methods. Keywords: Atomic Force Microscopy; Q-Control; Embedded Control; Numerical Methods. Keywords: Atomic Force Microscopy; Q-Control; Embedded Control; Numerical Methods.
1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The method of Q-control for Amplitude Modulation (AM) The for (AM) AFM after of introduction in 1993 Modulation [1] remains The method method ofitsQ-control Q-control for Amplitude Amplitude Modulation (AM)a The method for Amplitude Modulation (AM) AFM after of itsQ-control introduction in 1993 1993 [1]of the remains controversial control technique. Short reviews method AFM after its introduction in [1] remains aaa AFM after control its introduction in 1993 [1]of the remains controversial technique. Short reviews method and references up to 2010 can be found in [2, pp. 268-272] controversial control technique. Short reviews of the method controversial control technique. Short reviews ofpp. the268-272] method and references up to 2010 can be found in [2, [3, pp. 59-61]. Ashby in 2007 paper [4] provides the and references up to 2010 can be found in [2, pp. 268-272] and references up to 2010 can be found in [2, pp. 268-272] and [3, pp. 59-61]. Ashby in 2007 paper [4] provides the most critical remarks against the method used for gentle and [3, pp. 59-61]. Ashby in 2007 paper [4] provides the and [3, pp. 59-61]. Ashby in 2007 paper [4] provides the most critical remarks against the method used for gentle imaging of soft materials. Histhe final adviceused is “Itfor more most remarks against method most critical critical remarks against the method used foris gentle gentle imaging of softto materials. final advice is “It more advantageous use the His intrinsic Q factor andis imaging of His final is is more imaging of soft soft materials. materials. His final advice advice is “It “It is small more advantageous to use the intrinsic Q factor and small oscillation amplitude.” Nevertheless, the rate of Q-control advantageous advantageous to to use use the the intrinsic intrinsic Q Q factor factor and and small small oscillation Nevertheless, the Q-control publications remains intensive and most of of them claim oscillation amplitude.” amplitude.” Nevertheless, the rate rate of Q-control oscillation amplitude.” Nevertheless, the rate of Q-control publications remains intensive intensive and most most offollowing them claim significant advantages. Typical claims are the [5]: publications remains and of them publications remains intensive and most offollowing them claim claim significant advantages. Typical claims are the [5]: “Q factor can be increased to lower the maximum forces significant advantages. Typical claims are the following [5]: significant advantages. Typical claims are the following [5]: “Q factor can be increased to lower the maximum forces acting between tip and sample in air” and “the possibility to “Q factor can be increased to lower the maximum forces “Q factor can be increased to lower the maximum forces acting between tip and sample in air” and “the possibility to actively reduce the quality factor allows to increase the scan acting between tip and sample in air” and “the possibility to acting between tip and sample in allows air” andto“the possibility to actively reduce the quality factor increase the scan speed”. willthe evaluate on asymptotic actively reduce quality factor allows to the actively We reduce the quality these factorclaims allowsbased to increase increase the scan scan speed”. Weand will evaluate these claims based on asymptotic modelling experiments with automatic probe speed”. evaluate claims on asymptotic speed”. We We will will evaluate these these claims based based onlanding. asymptotic modelling and experiments with automatic probe landing. modelling and automatic Basic approach to activewith Q-control is probe usinglanding. cantilever modelling and experiments experiments with automatic probe landing. Basic approach to active Q-control is using cantilever deflection velocity feedback, because oscillator damping is Basic approach to active Q-control is using cantilever Basic approach tofeedback, active Q-control is using damping cantilever deflection velocity because oscillator is the coefficient of velocity in the Simpleoscillator Harmonicdamping Oscillator deflection velocity feedback, because is deflection velocity feedback, because oscillator damping is the coefficient of velocity in the Simple Harmonic Oscillator (SHO). However, there is in no the direct measurement velocity the coefficient of Simple Harmonic the coefficient of velocity velocity in the Simple HarmonicofOscillator Oscillator (SHO). However, there isneeded. no direct measurement of velocity and some estimation Among the difficulties are (SHO). However, there measurement of (SHO). However, thereis is is no no direct direct measurement of velocity velocity and some estimation is needed. Among the difficulties high oscillating frequency and significant noise. Ref. [6] and some some estimation estimation is is needed. needed. Among Among the the difficulties difficulties are are and high oscillating oscillating frequency and significant noise. However, Ref. are [6] proposes observer design to estimate velocity. high frequency and significant noise. Ref. high oscillating frequency and significant noise. However, Ref. [6] [6] proposes observer design to estimate velocity. analog Q-control implementations are usually based on an proposes observer design to estimate velocity. However, proposes observer design to estimate velocity. However, analog Q-control implementations are usually based on an estimation of the velocity by applying a phase shift to the analog Q-control implementations are usually based on an analog Q-control implementations are usually based on the an estimation of the velocity by applying a phase shift to displacement signal using a time delay circuit [2,3,5,7]. estimation of the velocity by applying a phase shift to the estimation of the velocity by applying a phase shift to the displacement signal using a time [2,3,5,7]. Unfortunately, the justification thisdelay phase circuit shift approach is displacement using time delay circuit [2,3,5,7]. displacement signal signal using aa of time delay circuit [2,3,5,7]. Unfortunately, the justification of this phase shift approach is based on an ansatz that the deflection signal is harmonic. This Unfortunately, is Unfortunately, the the justification justification of of this this phase phase shift shift approach approach is based on an ansatz that the deflection signal is harmonic. This is true only in steady state oscillation without tip-sample based based on on an an ansatz ansatz that that the the deflection deflection signal signal is is harmonic. harmonic. This This is is true true only only in in steady steady state state oscillation oscillation without without tip-sample tip-sample is true only in steady state oscillation without tip-sample
interaction and cannot be justified in transition. Conventional interaction and cannot be justified in transition. Conventional Q-controller design based on oversimplified SHO model may interaction cannot be in Conventional interaction and and cannot be justified justified in transition. transition. Conventional Q-controller design based on oversimplified SHO model may also suffer from a “spill-over effect” due to unmodeled Q-controller may Q-controller design design based based on on oversimplified oversimplified SHO SHO model model may also suffer from a “spill-over effect” due to unmodeled dynamics [7]. Ref. [7] addressed this issue by using negative also suffer suffer from from aa “spill-over “spill-over effect” effect” due due to to unmodeled unmodeled also dynamics [7]. Ref. [7] [7] addressed this issue issue by that using negative imaginary approach to addressed design a Q-controller guarantees dynamics [7]. Ref. this using negative dynamics [7]. Ref. [7] addressed this issue by by that using negative imaginary approach to design aa Q-controller guarantees stability with unmodeled dynamics. Ref. [8] utilizes imaginary approach to design Q-controller that guarantees imaginary approach to design a Q-controller that guarantees stability with unmodeled dynamics. Ref. [8] utilizes modulated-demodulated approach with an LTI controller in stability with unmodeled dynamics. Ref. [8] stability with unmodeled dynamics. Ref. [8] utilizes utilizes modulated-demodulated approach with an LTI controller in between. The LTI controller is designed using a positive modulated-demodulated approach with an LTI controller in modulated-demodulated approach with an LTI controller in between. The LTI controller is designed using a positive position whichis roll-off of the between. The controller designed using aa positive between.feedback The LTI LTImethod controller is“ensures designedrapid using positive position feedback method which “ensures rapid roll-off of the controller response, minimizing modes”. position method which “ensures rapid roll-off of position feedback feedback method whichspillover “ensures into rapidhigher roll-off of the the controller response, minimizing spillover into higher modes”. A design similar to “modulated-demodulated control” was controller controller response, response, minimizing minimizing spillover spillover into into higher higher modes”. modes”. A similar control” was earlier implemented in Q-control at commercial AFMs A design design similar to to “modulated-demodulated “modulated-demodulated control” was A design similar to “modulated-demodulated control” was earlier implemented in LTI Q-control at There commercial AFMs [9,10], but without the controller. are other Qearlier implemented in Q-control at commercial AFMs earlier implemented in Q-control at commercial AFMs [9,10], but without the LTI controller. There are other Qcontrol approaches that use special electronics, including [9,10], but without the LTI controller. There are other [9,10], but without the LTI controller. There are including other QQcontrol approaches that use special electronics, capacitive coupling [11], Positive Position Feedbackincluding with the control approaches that use special electronics, control approaches that use special electronics, including capacitive coupling [11], Positive Feedback the use of an active impedance in the Position piezoelectric shuntwith control capacitive coupling [11], Position Feedback with the capacitive coupling [11], Positive Positive Position Feedback with the use of an active impedance in the piezoelectric shunt control method etc.impedance Recently, in tuned oscillator (with additional use active the shunt control use of of an an[12], active impedance in the piezoelectric piezoelectric shunt control method etc. Recently, tuned oscillator additional circuitry[12], to tune damping) was used(with for quantitative method [12], etc. Recently, tuned (with additional method [12], etc.effective Recently, tuned oscillator oscillator (with additional circuitry to tune effective damping) was used for quantitative measurements [13]. Q-control was extended to higher circuitry circuitry to to tune tune effective effective damping) damping) was was used used for for quantitative quantitative measurements [13]. Q-control was extended to flexural eigenmodes [14] using cantilevers with integrated measurements [13]. [13]. Q-control Q-control was was extended extended to to higher higher measurements higher flexural eigenmodes [14] using using layer). cantilevers with integrated integrated actuation (bounded piezoelectric flexural eigenmodes [14] cantilevers with flexural eigenmodes [14] using cantilevers with integrated actuation piezoelectric layer). actuation (bounded piezoelectric layer). According(bounded to the references, “Q-control actuation (bounded piezoelectric layer). of high-frequency According to the references, “Q-control of high-frequency microcantilevers typically requires the development of highAccording of According to to the the references, references, “Q-control “Q-control of high-frequency high-frequency microcantilevers typically requires the development bandwidth analog circuit implementation” [8]. However, microcantilevers typically typically requires requires the the development development of of highhighmicrocantilevers of highbandwidth analog in circuit implementation” [8]. asymptotic However, current advances FPGA technology and bandwidth analog circuit implementation” [8]. However, bandwidth analog circuit implementation” [8]. However, current advances FPGA technology and asymptotic modelling of AFM in dynamics developing digital current advances in FPGA technology and asymptotic current advances in FPGA allow technology and pure asymptotic modelling of AFM dynamics allow developing pure digital implementation at an FPGA. This is one of the contributions modelling of AFM dynamics allow developing pure modelling of AFM dynamics allow developing pure digital digital implementation at an FPGA. This is one of the contributions of our paper. at design especially critical for an FPGA. FPGA. Thisisis is one one of the the contributions contributions implementation implementation atDigital an This of of Digital design is commercial AFMs, where Q-control method critical should for be of our our paper. paper. Digital design is especially especially critical for of our paper. Digital design is especially critical for where Q-control method should be be commercial AFMs, implemented at an FPGA with other measurement, commercial AFMs, AFMs, where along Q-control method should commercial where Q-control method should be implemented at an FPGA along with other measurement, control, signal processing, and communication algorithms for implemented at at an an FPGA FPGA along along with with other other measurement, measurement, implemented control, signal signal processing, and and communication communication algorithms algorithms for for multiple applications. control, processing, control, signal processing, and communication algorithms for multiple applications. applications. multiple multiple applications.
Copyright © 2017, 2017 IFAC 11369 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 IFAC 11369 Peer review© of International Federation of Automatic Control. Copyright ©under 2017 responsibility IFAC 11369 Copyright © 2017 IFAC 11369 10.1016/j.ifacol.2017.08.2445
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Sergey Belikov et al. / IFAC PapersOnLine 50-1 (2017) 10882–10888
We demonstrate how AFM asymptotic dynamics, derived in [15] for single-resonant and in [16] for multi-resonant AFM modes, justify the digital design and FPGA implementation. These models can also provide a deeper analysis of Q-control controversies described in [4]. In the majority of literature, design of Q-control is based on simplified SHO models without tip-sample interactions. Publications demonstrate that quality factor can be varied in a wide range. AFM applications, however, are based on tipsample interaction that is coupled with Q-control. This coupling may be the main source of the controversies. This is especially true for viscoelastic samples. Indeed, viscoelasticity is a velocity dependent feature that contributes to the change of the linearized system effective quality factor, along with Q-control. In such situations, there are no general suggestions and Q-control should be studied individually for each application. Automatic probe landing is an important application of Qcontrol where a complete theoretical and experimental study can be done almost independently of the sample material. Indeed, before landing, the tip is not interacting with the sample. Interaction starts only at the last short moment when condition of landing is identified. At this short moment the sample material behaves elastically with instantaneous modulus [17] and does not affect the quality factor of the cantilever. Our earlier paper [18] describes automatic landing procedures in different AFM resonant modes including Amplitude Modulation with Phase Imaging (AM-PI) [19, Table 1]. In the current paper, we demonstrate how Q-control influences the landing procedures. 2. BACKGROUND
Z
Zc
The mechanical setup of a controllably oscillated AFM probe, i.e. cantilever with a sharp tip, is presented in Fig. 1. It can be described by the following Euler-Bernoulli type of equation with the additional terms ( and F): Z Z F Z , Z / t Z (1) 2 x l a 4
2
2
t
x
4
S
where Z=Z(x, t) [m] is the vertical position at the point x [m] at time t [sec]; (f) [sec-1] is the dissipative term defined as a linear operator acting on function fZ/t, (This linear operator will be identified below); a 2 EI / S with E [Pa] – the elastic modulus of the cantilever’s material, [kg m-3] – density, S [m2] –area of the cross-section of the cantilever, and I [m4] –the moment of inertia of the cross-section; the RHS of (1) is the acceleration due to applying the concentrated force F [N] at the point xl [m]. In an AFM this force is due to tip-sample interaction. The cantilever is excited by one or more harmonics that vibrate the base located at x0 (Fig. 1): ~ ~ ~ (2) Z 0 , t Z c A n cos n t n ~ n [rad]
~ An
[m], ~ n [rad/sec], and are the amplitude, frequency and phase of the nth
where Zc [m] is the central position;
Cantilever
𝐴𝐴𝑛𝑛 cos 𝜔𝜔𝑛𝑛 𝑡𝑡 + 𝜑𝜑𝑛𝑛 𝑡𝑡
Tip Sample x
Piezo Scanner
Fig. 1. Mechanical setup of an AFM, including the cantilever attached to a vibrating base and a sample on a piezo scanner.
number (usually not more than two) and parameters ~ ( A n , ~ n , ~ n ) of the generated harmonics. Other three boundary conditions of free-end cantilever are presented in [16, formula (3)]. Solution z(x,t) of (1)-(2) with no tip-sample interaction, i.e. F0, can be presented as
zx,t
z n t Z n x ,
where { Z nl x } is an orthogonal
l
n 1
Z n l 1
basis described in [16, formula (9)],
l
. A linear
damping operator is defined [16] as
n Z n x
l
n 1
n n Z n x l
n 1
with n
n 2
4Q n 1
n
(3)
2Q n
where n and Qn are resonant frequency and quality factor of the nth eigenmode Z nl x of the cantilever that can be
2
zn 2 n z n n z n ~ ~ ~ l ~ ~ ~ ~ e n A i i i cos i t i 2 n sin i t i
This section summarizes the necessary results derived in [16].
t
L l
identified by thermal tune [20]. Derivations and details can be found in [16]. Coefficients zn satisfy the following equations
2.1 Modelling of Multi-Resonant AFM Modes
2
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harmonics of the excitation. The instrument user defines a
where
l
en
(4)
are the constants calculated in [16, formula (17)].
As shown in [16], asymptotically with the accuracy
2
O Qn
,
solutions of equations (4) can be approximated by solutions of the following equations l ~ 2 zn 2 n z n n z n e n A n ~ n ~ n cos ~ n t ~ n 2 n sin ~ n t ~ n
with a single ~ n n O Q n 1 and A n O Q n 1 . RHS of this equation is a harmonic function with frequency ~ n . Changing the time origin such that the phase of the harmonic function is nullified, we get the following equation with asymptotic accuracy O Q : 2
n
~ 2 zn 2 n z n n z n E n cos ~ n t
where
~ l ~ 2 E n e n An n
(5)
.
In most Q-control publications, some equivalent of Eq. (5) is a starting point of development without connection to Eq. (2). However, there are no natural forces applied to the eigenmodes, and RHS of Eq. (5) is just a calculated term related to the actual excitation in Eq. (2), applied to the base of the cantilever as shown in Fig. 1. An algorithm that calculates the amplitude E~ n in Eq. (5) should recalculate
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amplitude
~ An
and phase
~ n
of the excitation in Eq. (2) that is
physically applied to the base of the cantilever. Eq.(5) also suggests that equations for different eigenmodes are asymptotically (with accuracy O Q n 2 ) uncoupled. However, the equations derived from Eq. (1)-(2) with F 0 are coupled. In publication [16] we also derived the asymptotic dynamics of amplitude-phase (defined as van der Pole coordinates [16, Eq. (36)]) for AFM resonant modes, including AM-PI. This dynamics is critically important because the AFM controller works with amplitudes and phases measured by lock-in amplifiers. For the purpose of this paper we use asymptotic amplitudephase A– dynamics for a single-resonant frequency n. Asymptotic approximation for a conservative tip-sample interaction force F(z) and small amplitude A is the following [18]: n dA A A 0 sin dt 2 A0 n A 0 F z d cos z A 2 N dt
(6)
a. b. c. Fig. 2. Translation of a sample attached to the piezo scanner, driven by a stepper motor and piezo scanner.
where Ao is the amplitude of the oscillation with F0; N is the normalizing force defined in [18]; and Q . As expected, 1
n
speeds of A and are proportional to . AM-PI) cannot be reached, the stepper motor must move the piezo scanner up, i.e. closer to the tip (Fig. 2c).
2.2 Control of z-Positioning of the Sample with Stepper Motor and Piezo Scanner Comprehensive description of control for nanopositioning systems, including AFMs, can be found in [21]. For the purpose of this paper, i.e. Q-control and landing of the AFM probe, we concentrate only on vertical z-positioning of the sample controlled by the stepper motor and the piezo scanner. Fig. 2 is an illustration of the sample vertical translations. The stepper motor sets a rough position for the piezo scanner, which holds a sample. The piezo scanner can be extended or contracted, moving the sample respectively closer to the tip or farther away. In most applications, a combined use of a stepper motor and piezo scanner brings the probe into interaction with the sample. The stepper motor can be commanded to either move with a predefined velocity until “stop” command, or perform a move at a predefined distance. At the beginning of the AFM experiment, the sample is placed relatively far from the probe, and tip-sample interaction force is negligible. Vertical positioning of the tip relative to the sample uses a stepper motor for coarse translation and a piezo actuator for fine approach. In probe landing, both actuators are used. After the landing is accomplished, most subsequent actions are made using only the piezo scanner. The piezo scanner is extended or contracted by applying negative or positive voltage, respectively. Digital controller generates a normalized voltage u in the range [1,1] that after amplification is applied to the piezo scanner. With u1 the piezo scanner is fully compressed (Fig. 2a). It is always in this state when feedback is off, when sample is as far from the tip as the position of the piezo scanner allows (Fig. 2a). With u1 the piezo scanner is fully extended (Fig. 2b). If in this state the feedback is on but the set point (amplitude in
2.3 Review of Probe Landing in Amplitude Modulation Phase Imaging (AM-PI) AFM Mode As was formulated in [18], landing is characterized by a sharp temporal change in the force gradient F z z while the piezo scanner is extending and the sample is approaching the tip. Analyzing (6), we conclude [18] that the change of θ is a reliable indicator of landing in AM-PI. The use of phase as a reliable landing indicator is illustrated in Fig. 3c –see the jump in “Phase [deg]” plot. Meanwhile the amplitude trajectory, shown on “Normalized Amplitude [0,1]” plot in 3b, is smoothly approaching the set point and cannot indicate the landing event. Fig. 3 shows the process of landing without Q-control. In this example, as shown in the Bode plots (Fig. 3a) obtained by a frequency sweep, resonant frequency is 277 kHz and Q factor is 474. (Phase is plotted in NT-MDT convention, i.e. zero rather than 90o at resonance). When the probe approaches the sample in air, one should consider “squeeze-film-damping” [22], a phenomenon that emerges because the probe is near a flat surface: when the probe that is oscillating at its resonant frequency approaches a sample, there is a reduction of resonant frequency and amplitude. Reduction of these parameters can serve as an indicator that the tip, although not yet interacting, is very close to the sample. The Automatic Landing Procedure in AM-PI mode consists of three stages [18]. Stage I makes a rough approach based on “squeeze-film-damping” [22]; stage II compresses the piezo scanner to have margin to extend it during the final stage III that ends with the landing. Fig. 2b,c demonstrate that the same location of the sample can be obtained by different
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combinations of the piezo scanner position controlled by the stepper motor and the extension of the piezo scanner itself, controlled by the normalized voltage u. AM-PI Landing Stage I uses the stepper motor to move a piezo scanner, fully extended because feedback is on and sample is far from the tip, to the position where the amplitude set point is reached. This set point is equal to 90% of the initially measured amplitude. Stepper motor is moving continuously at high speed until measured amplitude reaches 97% of the initial amplitude, and then at slow speed until it reaches 90%. The blue horizontal line on the left part of the plot in Fig. 3b is 90% of the initial amplitude. Stage I ends when the normalized amplitude (red) crosses the 90% set point (blue horizontal line). Fast and slow motions can be easily recognized in Fig. 3b,c – they are separated by near vertical parts of the trajectories. AM-PI Landing Stage II is iterative. During each iteration, we keep the same amplitude set point (blue line in Fig. 3b). First we turn feedback off and wait for u=1 (fully compressed piezo scanner); then move to a predefined distance by the stepper motor (to place the sample closer to the tip); next, with feedback on, measure normalized control voltage u and stop the stage II when it is large enough (default u>0.4, i.e. piezo scanner is less than 30% of its full size). Otherwise – continue the loop. Five iterations of this stage are shown in Fig. 3 (see trajectories before the blue staircase in Fig. 3b). In Fig. 3d, u1 indicates the regions of trajectories where feedback is off. An increase of u, i.e. compression of piezo scanner, at each of four iterations can also be seen. AM-PI Landing Stage III is also iterative and stops when the AM-PI condition of landing has been met, i.e. phase or phase noise jump exceeds the threshold (default 2 o). Each iteration decreases the amplitude set point by default of 3% – see blue staircase in Fig. 3b. A new iteration starts by measuring u, followed by turning feedback off. If the measured u < 0 (extended), a stepper motor moves the piezo scanner (with the sample) closer to the tip to compress it and keep closer to neutral position u0. (For u>0, decreasing the amplitude set point makes the compressed piezo scanner extend and stepper motor move is not necessary). Fig. 3 shows 7 iterations of stage III. The stage stopped because the phase jumped. The piezo scanner in this moment is close to the neutral position (|u|<0.1).
a. Frequency sweep: Q=474.
b. Set point (blue) and z-servo controlled amplitude (red).
c. Phase trajectory. Sharp change at the end indicates landing.
d. Normalized position “u” of piezo scanner: 0 –neutral; -1 – fully extended; +1 –fully contracted. At landing the position is close to neutral (|u|<0.1). Fig. 3. Automatic control of landing in AM-PI mode without Q control: 7 steps on Stage III.
3. DESIGN OF DIGITAL Q-CONTROL FOR FPGA 3.1 Theory Similarly to most conventional Q-control designs, we start with Eq. (5). However, as mentioned in Section 2.1, this equation is linked with the boundary condition in Eq. (2) describing physical excitation of the base in Fig. 1. In ~ particular A n , ~ n for a single ~ n n must be recalculated back from Eq. (4)-(5) before applying to the base. Eq. (3) is the relationship between damping coefficient n and quality factor QQn. A naive way to modify in Eq. (5) would be to add hypothetical feedback control (7) u z 2 z where zzn, so that Eq. (5) becomes (index n is omitted)
10885
~ 2 z 2 z n z u z E cos ~ n t
(8)
~
and has the modified damping . Aside from the problems with estimating dz/dt, there are even more serious objections on how to transfer the hypothetical control u to the excitation in Eq. (2). Indeed, Eq. (2) assumes that ~ ~ ( A t , ~ t )( A n t , ~ n t ) vary much slower than cos ~ n t . (Otherwise one cannot define amplitude and phase.) Amplitude-phase dynamics in Eq. (6) satisfies this requirement, but feedback u z in Eq. (8) does not. Proper control should use slowly varying (A,) described by Eq. (6). They are measured by digital lock-in amplifier.
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a. Frequency sweep: Q=833.
a. Frequency sweep: Q=255.
b. Set point (blue) and z-servo controlled amplitude (red).
b. Set point (blue) and z-servo controlled amplitude (red).
c. Phase trajectory. Sharp change at the end indicates landing.
c. Phase trajectory. Sharp change at the end indicates landing.
d. Normalized position “u” of piezo scanner: 0 –neutral; -1 – fully extended; +1 –fully contracted. At landing the position is close neutral (|u|<0.05). Fig. 4. Automatic control of landing in AM-PI mode with increased Q: 14 steps on Stage III.
d. Normalized position “u” of piezo scanner: 0 –neutral; -1 – fully extended; +1 –fully contracted. At landing the position is close neutral (|u|<0.2). Fig. 5. Automatic control of landing in AM-PI mode with decreased Q: 4 steps on Stage III.
There is a significant difference in increasing and decreasing Q. From (3), (7), (8), (omitting index n),
~ ~ Q n / 2 n / 2
To change
~ Q
(9)
from Q to infinity, should be changed from
zero to , that is not difficult for small . However, to ~ change Q from Q to zero, should be changed from zero ~
to , that is a more difficult task for small Q . ~
~
As long as ( A t , ~ t )( A n t , ~ n t ) in Eq. (2) change slowly, the harmonic steady state solution of Eq. (5) is a good approximation. The solution is, (index n omitted),
~ z ( t ) A ~ cos t
~
(10)
where amplitude and phase can be calculated by substituting Eq. (10) to Eq. (5). For ~ n and n<<1, A
n
l ~ en A Q ,
/2
(11)
n
From Eq. (10), z ( t ) A ~ ~ sin ~ t ~ , and we can temporarily, assuming ~ n , replace hypothetical control in Eq. (7) by the open-loop control ~ ~ ~ u , 2 A ~ sin t ~
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~ l 2 ~ A e n n sin t
~
,
/
(12)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Sergey Belikov et al. / IFAC PapersOnLine 50-1 (2017) 10882–10888
10887
This signal can be generated by digital lock-in amplifier that measures A ~ and ~ . Substituting u in (8) by RHS of Eq. (12) 2
z 2 z n
~ l 2 A e n n
~ M sin t
~
~ cos t
(13) where M1 for the open loop (to be introduced in section 3.2). Parameter in Eq. (13) was introduced in such a way that amplitude of harmonic function in square brackets is less than unity, i.e. 1 / M
Shaker Amplitude:
this, Q-control can be generated by applying the following waveform to the shaker that excites the base (Fig. 1): Q Control Generator ~ ~ A S G Q X sin t
(14)
This condition is important for an FPGA implementation that uses fixed-point arithmetic and scaling the numbers. Varying , the range of || can theoretically be from zero to infinity, and so is the range of Q. Smaller defines a larger range of . For implementation purposes, we define the following control parameters:
Fig. 6. User interface elements of Q-control.
G Q 1 S / M ,
~
S
S Q GQ
(15)
0 1
~ cos t
Q
S
~ cos t
(17)
“Shaker G0 Freq [kHz]” “Shaker Fraction Amp [0-1]” “Q Phase [-180:+180deg]” “Q Gain [-1:+1]”
Usually
~ cos t ,
Y
S
~
.
2 z 2 z n A S G Q M sin ~ t
Q
User interface elements for Q-control are shown in Fig. 6. The following table relates the elements to the parameters of Eq. (17). Table 1. Q-control parameters and related user interface elements ~ “Shaker G0 Range [0-10]” A
~ l 2 A S A e n n /
Q Gain: GQ Shaker Excitation Fraction: S With these notations, Eq. (13) can be written as
~ A S G Q M S
is kept constant when tip is far from
the sample, (it is the free resonant amplitude –see next section). 3.3 Q-Control Analysis
~ ~ A An
and phase ~ ~ n of the single excitations in Eq. (2) should be recalculated from these control parameters. Theoretically, phase ~ should be close to 90o
Mathematically, applying the generator in Eq. (17) is equivalent to the following equation (compare with (15)):
(see Eq. (11)), and ~ ~ n in Eq. (2) could be recalculated with this number. In practice, however, there may be a phase shift in the instrumentation that requires compensation in a real implementation.
However, in contrast to (18), the generator (17) implements feedback control because X and Y are measured in the loop. A practically suggested selection of Q is the following: when the cantilever is far from the sample and the base (Fig. 1) is oscillating with the free resonant frequency of the cantilever, select a positive gain, e. g. GQ0.1, and maximize the amplitude “Normalized Amplitude [0,1]” plotted in Fig. 3b, which is the amplitude of the deflection z(t) signal measured by the digital lock-in amplifier. According to Eq. (18), maximum amplitude of z(t) takes place at the maximum amplitude of the excitation
Amplitude
3.2 Remarks on an FPGA Implementation In an FPGA implementation, digital lock-in amplifier measures in-phase and quadrature components of the single nth mode of the deflection signal (16) X M cos ~ , Y M sin ~
2 z 2 z n A S G Q M sin ~ t
~ G Q M sin t
where M is a scaled unitless calculated magnitude (0
/2
Q
~
. In implementation,
however, Q is selected indirectly by maximizing the amplitude of the cantilever when it is far from the sample. Then G Q M sin ~ t Q ~ G Q M sin ~ t Q cos ~ G Q M cos t ~
Q
sin G X ~
Q
sin t ~
Y cos t .
In implementation, we do not recalculate
Q
~ ~ A An
in Eq. (2) ~
from AS, but directly select the related parameter A S as the amplitude of the excitation applied to the base in Fig. 1. With
G
Q
M sin
~
Q
~
Q
Q
S
sin ~ t cos S
Q
S
~ cos t
(18)
~ cos t
(19) ~
t
The square of the amplitude of RHS of (19) is 2 2 2 A Q G Q M S 2 G Q M S sin Q and for positive 1
GQ has its maximum at
1
Q
/2
. Then Eq. (18) is
2 z 2 z n A S G Q M S cos ~ t
~
Q
G Q M cos
~
~
(20)
Eq. (20) provides an important interpretation for majority of Q-control designs, including this digital implementation. Q-control is an adaptive regulator of the excitation amplitude applied to the base (Fig. 1) at one of the eigenfrequencies n.
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In fact, intrinsic Q of the cantilever has not been changed, although Bode diagrams in Fig. 3a, 4a, and 5a show a different Q. The reason is that the further the excitation frequency ~ is from resonant n, the smaller is the magnitude M. With a positive gain GQ in Eq. (20), reduction of M causes reduction of (now non-resonant) excitation amplitude and further reduction of M (positive feedback). Effectively this reduces non-resonant amplitudes, imitating increasing Q (Fig. 4a). With a negative gain GQ in Eq. (20), we have the opposite (negative feedback), i.e. reduction of M causes increase of (now non-resonant) excitation amplitude, causing an increase of M, which looks like a reduction of Q (Fig. 5a). As for Q-control controversies [4] and Ashby’s suggestion that “It is more advantageous to use the intrinsic Q factor and small oscillation amplitude”, we can offer the following. Qcontrol in fact does not change the intrinsic Q factor, but has the ability to adaptively decrease the oscillation amplitude. Thus, it follows Ashby’s advice, with addition of adaptive amplitude reduction. This is the true value of the method that would be better called “Adaptive regulator of the excitation amplitude”. Adaptive decrease of excitation amplitude lowers the forces acting on the tip, imitating the increase of “effective Q”. This causes Z-servo to behave slower. Adaptive increase of excitation amplitude acts in the opposite direction.
motor and piezo scanner, and automatic probe landing in AM-PI mode. Theory was used to design a Q-Control for digital FPGA implementation, and we view the method as adaptive regulator of the excitation amplitude applied to the base. This view allows reconciling some controversies of Q-control method. Applications of the digital Q-control to automatic probe landing validate the method in practice for these applications. REFERENCES [1]
[2]
[3] [4]
[5]
[6]
[7]
[8]
4. AUTOMATIC PROBE LANDING IN AM-PI MODE WITH DIGITAL Q-CONTROL We present screenshots of GUI panels of a commercial AFM (Fig. 3-6) to better illustrate on-line operator experience. Fig. 3, 4, and 5 demonstrate the process of landing respectively without Q-control (GQ=0), with increased “effective Q” (GQ>0), and with decreased “effective Q” (GQ<0). The automatic landing was reviewed in Section 2.3. Fig. 3a, 4a, and 5a shows Bode plots and calculated quality factors 474, 833, and 255, respectively. One can see that number of steps of Stage III before landing increases with Q. There are 7, 14, and 4 steps, respectively (Fig. 3b, 4b, and 5b) that are roughly proportional to Q. This behaviour has the following clear explanation. Without Q-Control (Fig. 3), decrease of the deflection amplitude at stage III (Fig. 3b) before the landing is caused only by natural “squeeze film damping” [22]. With increased “effective Q” (Fig. 4), natural decrease of deflection amplitude causes its further decrease due to a reduction of excitation amplitude. Artificial decrease of deflection amplitude causes Z-servo to extend the piezo scanner less. It takes more steps to land but the landing is softer. With a reduced “effective Q” (Fig. 5), natural decrease of deflection amplitude causes an increase of excitation amplitude, increasing the deflection amplitude. This forces the Z-servo to extend the piezo scanner more (closer to the probe). It takes less steps to land but the landing may be more invasive. 5. CONCLUSIONS We provide a background theory of asymptotic models of resonant AFM operation modes, positioning with stepper
[9] [10] [11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
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