Iterative Control of Dynamics-Coupling-Caused ... - Semantic Scholar

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 6, NOVEMBER 2005

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Iterative Control of Dynamics-Coupling-Caused Errors in Piezoscanners During High-Speed AFM Operation Szuchi Tien, Qingze Zou, and Santosh Devasia

Abstract—This paper addresses the compensation of the dynamics-coupling effect in piezoscanners used for positioning in atomic force microscopes (AFMs). Piezoscanners are used to position the AFM probe, relative to the sample, both parallel to the sample surface ( and axes) and perpendicular to the sample surface ( axis). In this paper, we show that dynamics-coupling from the scan axes ( and axes) to the perpendicular axis can generate significant positioning errors during high-speed AFM operation, i.e., when the sample is scanned at high speed. We use an inversion-based iterative control approach to compensate for this dynamics-coupling effect. Convergence of the iterative approach is investigated and experimental results show that the dynamics-coupling-caused error can be reduced, close to the noise level, using the proposed approach. Thus, the main contribution of this paper is the development of an approach to substantially reduce the dynamics-coupling-caused error and thereby, to enable high-speed AFM operation. Index Terms—Atomic force microscope (AFM), inversion theory, iterative control, nanopositioning.

I. INTRODUCTION

T

HIS paper addresses the dynamics-coupling problem that arises during high-speed nanoprecision positioning using piezo tube-scanners (piezoscanners). It is noted that piezoscanners are extensively used in scanning-probe microscopes such as the atomic force microscope (AFM). For example, in AFM applications, the piezoscanner is used to position a probe over a sample, both parallel to the sample surface ( and axes) and perpendicular to the sample surface ( axis), as shown in Fig. 1. However, the scanning movement of the AFM-probe (parallel to the sample surface) can cause positioning error in the perpendicular ( axis) direction. This coupling effect, from the scan axes to the perpendicular axis, becomes significant as the AFMs operation speed (scan frequency) approaches the vibrational resonance frequencies of the piezoscanner and excites its vibrational dynamics. Thus, the dynamics-coupling-caused errors during high-speed scanning limits the operational speed

Manuscript received June 1, 2004. Manuscript received in final form May 24, 2005. Recommended by Associate Editor R. Moheimani. This work was supported in part by the National Institutes of Health (NIH) under Grant GM68103 and in part by the National Science Foundation (NSF) under Grant CMS 0301787. S. Tien and S. Devasia are with the Department of Mechanical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: devasia@ u.washington.edu). Q. Zou is with the Department of Mechanical Engineering, Iowa State University, Ames, IA 50011 USA. Digital Object Identifier 10.1109/TCST.2005.854334

of AFMs. This paper shows that the dynamics-coupling effect can be effectively modeled and compensated-for by using an inversion-based approach. Additionally, for applications where the and axes movement is repetitive (as in AFM scanning operations), an iterative inversion-based approach [1]–[3] is used to further reduce the dynamics-coupling-caused error. Convergence of the iterative approach is investigated and experimental results show that the coupling-caused error can be reduced, close to the noise level, with the proposed approach. Thus, the main contribution of this paper is the development of an approach to substantially reduce dynamics-coupling-caused error and thereby, to enable high-speed AFM operation. Compensating for the dynamics-coupling-caused error in the probe-sample distance (i.e., axis positioning) is important in AFM applications such as nanofabrication [4], [5] and imaging of soft biological samples [6], [7]. For example, nano-scale parts can be fabricated by using the AFM probe as an electrode to induce local-oxidation [4], [5] by applying a voltage between the AFM probe and the surface. The probe-sample distance has a dominant effect on the applied current (the probe-sample interaction in this application) and the formation of the current-induced oxide. Therefore, dynamics-coupling-caused error in the probe-sample distance leads to distortion in the size and shape of the nanofabricated parts [4]. Similarly, precision control of the probe-sample distance is needed to maintain small probesample forces when imaging soft biological samples such as living cells with an AFM [8], [9]. The probe-sample force (i.e., the probe-sample interaction in this application) depends on the probe-sample distance. Therefore, error in the probe-sample distance can lead to a large probe-sample force and cause the AFM-probe to protrude into the soft biological sample, leading to sample damage and tip contamination. Therefore, it is important to precisely control the probe-sample distance because large variations will result in the distortion of the fabricated parts during nanofabrication and cause sample damage when imaging soft biological samples. Accounting-for the dynamics-coupling-caused error in the probe-sample positioning is critical to increase AFMs throughput because the effect of the dynamics-coupling-caused error becomes significant as the operating speed is increased. In particular, this error becomes large when the scan frequency (at which the probe is moved relative to the sample surface) is increased and approaches the vibrational-resonant frequency of the piezoscanner. For a typical AFM, with a scan range in the tens of microns, the vibrational-resonant frequency tends to be in the hundreds of hertz. To avoid positioning errors, the

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 6, NOVEMBER 2005

AFM tends to be operated at 1/10th to 1/100th of the vibrational-resonant frequency; during high-precision positioning applications such as imaging of soft samples, the maximum achievable scan frequency can be as low as 0.1 Hz to 1 Hz [8]. However, high-speed operation of the AFM is necessary in applications such as nanofabrication and imaging of biological samples. For example, high-speed AFM operation (with scan frequencies in the hundreds of hertz) is needed to study the time evolution of fast biological processes such as the rapid dehydration and denaturation of collagen and the movement of human cells [8]–[11]. Lack of sufficiently-high-speed AFM operation results in image distortions during time-lapse imaging of rapid processes because the sample topography and properties can change significantly during the time needed to acquire each individual image. Low-speed AFM operation is also a problem in nanofabrication because it limits the maximum achievable throughput. Therefore, there is a need to address problems that limit high-speed AFM operation such as the dynamics-coupling-caused error. Compensating for the dynamics-coupling-caused error is challenging because the piezoscanner-based, AFM-probe-positioning system tends to have low gain margin. The dynamics-coupling-caused error can be reduced by using feedback control such as the standard proportional-integral derivative (PID) controller [12]. Recent applications of advanced feedback controllers have also led to substantial improvement in AFM-probe positioning [13]–[15]. In general, high-gain feedback can lead to substantial reduction of the positioning errors. However, feedback techniques have had limited success in compensating for positioning errors because the low gain margin of the piezoscanner-based positioning system limits the maximum applied feedback gain in order to maintain stability of the closed-loop system [16], [17]. The low gain-margin limitation of feedback controllers can be alleviated by augmenting the feedback controller with feedforward inputs [15], [18]. For example, in the inversion-based feedforward approach, the dynamics of the piezoscanner is modeled and then inverted to find an input that achieves precision positioning with the piezoscanner [19]–[22]. It is noted that the addition of this inversion-based feedforward input can improve the tracking performance of any feedback controller—even in the presence of computational error due to modeling uncertainties (the size of acceptable uncertainties has been quantified in [23]). Early works, on the use of feedforward input for precision positioning in scanning probe microscopes (such as the AFM), showed that substantial reduction in positioning error in the scan axes can be achieved during high-speed operation [19], [20]. This paper shows that the feedforward approach can also be used to reduce the dynamics-coupling-caused error and thereby, to reduce the error in the probe-sample distance and the probe-sample interaction. Additionally, we use an inversion-based iterative control technique [1]–[3] to further reduce the coupling-caused error in the probe-sample distance in positioning applications where this error is repetitive. For example, during AFM operation, the piezoscanner is used to repetitively move the AFM-probe relative to the sample, back and forth, resulting in a repetitive dynamics-coupling-caused error in the axis—we show that this repetitive coupling error can be

Fig. 1. Schematic diagram of AFM operation. The piezoscanner enables positioning of the AFM probe both parallel (along the x and y axes) and perpendicular (along the z axis) to the sample surface. The probe-sample distance (interaction) is controlled using feedback in the z axis.

substantially reduced through an iterative control approach. Moreover, we present a frequency domain interpretation for the convergence of the inversion-based iterative approach proposed in [1]–[3] and use the analysis to design the iteration procedure. The inversion-based iterative feedforward approach was implemented on an AFM piezoscanner and experimental results are presented to show that the dynamics-coupling-caused error can be significantly reduced by using the inversion-based iterative control method. The rest of the paper is organized as follows. The dynamics-coupling problem is formulated in Section II. The inversion-based iterative control algorithm and the convergence analysis are presented in Section III. Experimental results are presented with discussions in Section IV and our conclusions are in Section V. II. PROBLEM FORMULATION A. Use of Piezo in the AFM The schematic diagram of the piezoscanner-based positioning of the AFM-probe, relative to the sample surface, is shown in Fig. 1. The AFM operation process is initiated by gradually reducing the distance between the AFM-probe and the sample (by using the piezo) until a prespecified probe-sample interaction is achieved, i.e., the AFM-probe deflection reaches a specified set-point value . Note that the AFM-probe deflection (which depends on the probe-sample interaction) can be measured using an optical sensor as shown in Fig. 1. A controller is used to maintain a desired AFM-probe deflection as the AFM-probe is moved by piezos to scan the sample surface. The scanning movement is a rastering (back and forth) movement of the AFM-probe relative to the sample in the -direction while the position is incremented. Remark 1: In typical applications such as AFM imaging, the AFM-probe deflection is maintained at a constant value (the setpoint value ); large variations in the AFM-probe deflection can cause sample damage (or AFM-probe damage). Variations in the set-point value of AFM-probe deflection may be required, however, to manipulate or modify a sample, e.g., to indent a sample during nanofabrication.

TIEN et al.: ITERATIVE CONTROL OF DYNAMICS-COUPLING-CAUSED ERRORS

Fig. 2. AFM-probe-deflection dynamics. The AFM-probe deflection affected by the z axis input u and the x axis input u .

z

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is

a) Dominant coupling effects arise from the axis scanning: Dynamics-coupling-caused error in the achieved AFMprobe deflection in the axis can arise due to the movements in, both, the axis and the axis. However, the effect of the axis movement on the axis tends to be less prominent because the scan frequency in the axis is substantially smaller than the scan frequency in the axis—thus, the higher scan frequency (in the axis) is closer to the vibrational resonance frequencies in comparison to the axis. In particular, the scan frequency in times smaller than the scan frequency the axis tends to be in the axis, where the number of lines, , in each scan tends be 100 or more. Thus, the scan frequency in the axis tends to be at least 100 times slower than the scan frequency in the axis. Thus, the dominant coupling effect arises from the scan axis movements. b) Control objective: The objective is to compensate-for the coupling effects and achieve a desired AFM-probe deflecduring AFM operation. Although the proposed tion inversion-based approach can be used to reduce movement-induced vibrations (coupling effects) from both the scan axes ( and ), we focus this paper on the reduction of the prominent axis coupling effect to illustrate the proposed approach; the method can also be extended to reduce the coupling-effect from the axis (if needed).

Fig. 3. AFM-probe deflection control scheme.

B. Dynamics Model

and the AFM-probe deflection is represented by

The main contributions to the AFM-probe deflection are and 2) the axis input as shown in 1) the axis input Fig. 2. Therefore, the AFM-probe-deflection dynamics can be modeled as

) of the closedThe output (i.e., the AFM-probe deflection loop system can be found from the block-diagram in Fig. 3 as

(2) where is the desired AFM-probe deflection and the effect of this reference input ( ) on the AFM-probe deflection is represented by (3) and

is the sensitivity of the closed-loop system (4)

generated by the Additionally, the AFM-probe deflection axis feedforward input is represented by (5) caused by the

axis input

(6) which is referred to as the -to- dynamics-coupling effect.

where , , are the Fourier transform of , and , respectively, represents the open-loop dynamics of the AFM-probe deflection due to the axis input , and represents the open-loop, coupling-effect dynamics due to the axis input . C. Control Scheme An inversion-based, feedforward-plus-feedback approach is used to compensate for the -to- dynamics-coupling effect and to achieve the desired AFM-probe deflection as shown in Fig. 3. The control scheme is shown in Fig. 3, where the axis conis the combination of the feedback input trol input and the feedforward input (1)

D. Compensation for Errors in AFM-Probe Deflection The goal is to augment current feedback control schemes with an inversion-based feedforward input to significantly reand achieve the duce the -to- dynamics-coupling effect desired AFM-probe deflection . Therefore, rather than focusing on the feedback controller design, this paper focuses on to achieve this goal; the proposed the feedforward input approach can be implemented with any currently available feedis chosen such back controller . The feedforward input compensates-for that its effect on the AFM-probe deflection the -to- dynamics-coupling effect and achieves the desired AFM-probe deflection , i.e., from (2)

(7)

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 6, NOVEMBER 2005

III. INVERSION-BASED CONTROL In this section, we present an inversion-based approach to obtain the feedforward input. Next, this inverse input is used as an initial input in an iterative approach, and the convergence of the iterative approach is analyzed and used to design parameters of the iteration algorithm. Additionally, the effect of noise on the convergence of the iterative approach is studied. A. Inversion-Based Feedforward Control The inverse (feedforward) input can be obtained by using (7) as

(8) The desired AFM-probe deflection can be achieved, if there , and . is no error in the models An advantage of this frequency domain inversion method is that the approach is also applicable to nonminimum phase systems as shown in [24]. It is noted that modeling error can result in error when computing the inverse input needed to reduce the error in the AFM-probe deflection. Modeling error can only be accounted-for using feedback. However, the use of feedforward improves the performance when compared to the use of feedback alone—even in the presence of modeling error (provided the modeling error is not too large). The amount of acceptable modeling error is quantified in [23].

dynamics. Let the error system’s dynamics be described by

between the model of the and the actual system’s dynamics

(10) where, for any given frequency , represents the magnitude error and represents the phase error, reand the itspectively. The conditions on the phase error [in (9)] for convergence of the iterative control eration gain law (9) are given by the following lemma. and Lemma 1: Let the actual system dynamics be stable, and let be a frequency where its model and do not have zeros on the imaginary both is finite and greater than zero in (10)]. axis [therefore, Then, the AFM-probe deflection converges to desired value, if and only if i.e., 1) the magnitude of phase variation is less than , i.e., , at frequency ; and in (9) is chosen as 2) the iteration coefficient (11) Proof: The iterative control law (9) can be rewritten using (2) as

B. Iterative Inversion-Based Control To account for modeling errors in the computation of the inverse input in (8), we use an iterative inversion-based procedure proposed in [1]–[3]. The iterative control law is (12) (9) for , where is the initial input, is is the Fourier transform of the desired the iteration-gain, is the Fourier transform of AFM-probe deflection , is the axis feedforward input at the th iteration, and the Fourier transform of the AFM-probe deflection measured at th iteration step. the It is noted that the iterative approach in (9) assumes that the output (AFM-probe deflection) can be measured without noise; the effect of noise on the convergence of the iterative approach is studied at the end of Section III. from (8) Remark 2: The inverse feedforward input for the iteration can be chosen as the initial inverse input process to speed the convergence of the iteration procedure because the initial inverse input is then close to the desired . feedforward input C. Convergence Analysis for the Iterative Control Law In the following, the convergence of the iterative control law (9) is quantified in terms of the error in modeling the system’s

Similarly, the input at the next iteration step as

can be written

(13) The change in the input, from one iteration step to another, can be found by subtracting (12) from (13) and then using (10) to obtain

(14) Repeated application of (14) leads to

(15)

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Therefore, the iterative control law converges, i.e. or (16) if and only if the mapping-coefficient (in (15)) is less than one in magnitude, i.e. (17) Equivalently, the iteration procedure converges if the square of the mapping-coefficient’s magnitude given by

Fig. 4. Geometric interpretation for convergence analysis: the iteration algorithm converges if the magnitude of the mapping coefficient is less than one [see (17)]. The shaded region represents the phase error. (a) Small phase error. (b) large phase error.

(18) is less than one, i.e.

or (19) Since the error in the magnitude is assumed to be nonzero 0 , (19) is satisfied if and only if 1) , i.e., and its multiples, and 2) the iteration gain is chosen as for for

(20)

However, since the actual system dynamics is unknown, the phase variation with a magnitude larger than will result (i.e., from in the sign-change of the function 0 to 0, or vice versa). If the sign-change is known a priori (or if it can be estimated experimentally), then it can be accommodated by changing the . If this sign change is not known sign of the iteration gain a priori, then it is necessary for the phase variation to be less to ensure convergence of the iteration with a fixed than . (constant sign) iteration gain Additionally, the fixed point of the iterations (the desired ) satisfies (12); therefore feedforward input (21) (22)

or where the AFM-probe deflection with the input

is (23)

This completes the proof of Lemma 1. c) Geometric interpretation of Lemma 1: The iteration algorithm converges if the magnitude of the mapping coefficient is less than one [see (17)]. If the phase variation is small, i.e., [see shaded region in Fig. 4(a)] then the

mapping-coefficient’s magnitude can be made less than one by choosing a sufficiently small iteration gain . Therefore, the mapping coefficient will lie inside the unit circle, and its magnitude is less than one. On the other hand, if there is large phase variation [see shaded region in Fig. 4(b)] then a fixed (constant sign) choice of the iteration gain is not sufficient to keep the mapping coefficient within the unit circle for all possible phase errors. For example, the figure shows the case where any positive iteration-gain leads to a mapping coefficient with magnitude greater than one. Thus, the iteration will not converge with any for fixed (constant sign) choice of the iteration gain all possible phase errors in the shaded region [Fig. 4(b)]. d) Acceptable phase uncertainty: The iterative approach can be used to reduce the tracking error if the phase uncertainty ), i.e., the model should provide informais small (less then tion about the direction in which the input should be changed to correct the output error. If this direction is uncertain at a specific frequency, then the iterative algorithm may not converge at that frequency (with a constant-sign iteration gain). This restriction on the acceptable phase uncertainty is reflected in the following Lemma. be a set of frequencies where the modeling Lemma 2: Let error satisfies the following: and then the iterative control law converges if the iteration gain is chosen as for otherwise

(24)

Proof: The convergence of the iteration for frequency in follows from Lemma 1; and the iteration converges the set is zero. in the first step if the iteration gain e) Effect of measurement noise: Noise affects the convergence of the iterative approach; the effect of measurement noise ( as shown in Fig. 3) is studied in the following—the effect of other noise (such as input noise) can be investigated in a similar manner.

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Lemma 3: Let the conditions of Lemma 1 be satisfied and let at each iteration step be bounded, i.e. the noise (25) [see Then, the error in computing the feedforward input (16)] can be made arbitrarily small by choosing a sufficiently and a large enough number of iteration small iteration gain steps. Proof: The measurement noise affects the update law used in the iterative approach. In particular, the AFM-probe-deflection is replaced by the noisy measurement in the iterative control law, (9) to obtain the input at the th iteration step as

tends to zero from Lemma 1. The second term and can result [in (30)] depends on the size of the noise in a nonzero error (in the output) even if the number of iteration steps is increased. However, this second term can be made small ; to show this, we rewrite by choosing a small iteration gain [using (18)] the second term in (30) as (31) where

(26) , the measured AFM-probe deflection at the where th iteration step, is given by (32) (27) and is as in (2), the actual system dynamics is the additional term due to measurement noise at the th [see iteration step. Moreover, the desired feedforward input (16)] satisfies (12) because it is the fixed point of the iterative algorithm in Lemma 1, i.e.

(28) at the th Then, the deviation of the feedforward input can be obtained iteration from the desired feedforward input by: 1) substituting the measured AFM-probe deflection from (27) into (26); and 2) subtracting (26) from (28) to obtain

Although this term (32) may not be zero, it can be made sufficiently small by choosing a small enough iteration gain because (33) from L′Hôpital’s rule. Note that conditions to apply L′Hôpital’s rule are met since, for small positive values of the iteration gain near the origin , and denominator are difboth the numerator ferentiable polynomials, and the derivative of the denominator

(34)

(29) By repeatedly using (29), the error computing the feedforward input can be rewritten as

in

is sufficiently small. is nonzero when the iteration gain In particular, the denominator of the derivative is positive from (18), and the numerator of the derivative is positive when the iteration gain is smaller than . This completes the proof of Lemma 3. Remark 3: The iteration gain should be small to reduce the effect of noise. In contrast, the iteration gain needs to be large to achieve fast convergence. These conflicting requirements on the iteration gain can be reconciled by choosing the iteration gain to be frequency dependent; small at frequencies where significant noise is present (such as very high frequencies) and large at frequencies where the noise is small (typically at low frequencies). In this sense, the frequency-dependent iteration gain tends to act as a low-pass filter. IV. EXPERIMENTAL RESULTS AND DISCUSSION

(30) It is noted that the first term in the previous equation becomes small as the number of iteration steps increases because

The inversion-based approach was applied to an AFM [Molecular Imaging (MI) PicoSPM II] piezoscanner, and comparative experimental results are presented with and without the proposed dynamics-compensation.

TIEN et al.: ITERATIVE CONTROL OF DYNAMICS-COUPLING-CAUSED ERRORS

Fig. 5. Experimental AFM system. The measured error in the AFM-probedeflection, e (t) and the x-displacement x(t) were collected using a personal-computer-based (PC-based) data collection system. This PC-based system was also used to apply the x axis input u (t) and to apply the inversion-based feedforward input u (t) in the z axis.

A. Experiment Setup and Modeling Experiment Setup: The AFM-probe deflection (error signal) was measured using a photodetector (Advanced Photonix, Inc.) and the lateral position, , of piezoscanner was measured with an inductive sensor (Kaman measuring systems) while the AFM was operated in the contact mode [8]. For feedback control, the built-in proportional-integral (PI) controller (MI PicoSPM II system) was used to control the AFM-probe deflection. As shown in Fig. 5, the error in the AFM-probe-deflection, , and the -displacement, , were collected using a personal-computer-based (PC-based) data collection system. This PC-based system was also used as feedforward controller to apply the axis input and the inversion-based feedforward input in the axis. Isolating the Dynamics-Coupling Effect: Dynamics-coupling is the significant cause of errors in the AFM-probe deflection when working with relatively flat samples, e.g., to create nanometer scale features on silicon surfaces [4] and to image DNA condensation for gene therapy and cancer research [25]. To isolate and quantify the dynamics-coupling effect, we imaged a relatively flat area of a calibration sample (NanoDevices NGR-21 010). The setpoint value of the AFM-probe deflection was 29 nm (which corresponds to a reading of 0.5 V at the AFM’s sensor display); note that the goal is to reduce the deviation of the AFM-probe-deflection from this setpoint value of 29 nm. The experiments aim to show the compensation for dynamics-coupling effects by comparatively evaluating the AFM-probe deflection with and without the proposed, dynamics-coupling compensation. Remark 4: Another effect that can cause error in the AFMprobe deflection is variation in surface topography, e.g., due to sample features and tilt in the sample relative to the scanning plane. The sample tilt was minimized in our experiments by adjusting the sample’s orientation (using tilt-adjustment motors available in typical AFM systems); the residual sample tilt was

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Fig. 6. Experimental bode plots of the AFM-probe-deflection dynamics: x-to-z coupling dynamics G^ (!) (solid line) and the nominal z axis dynamics G (!) (dotted line).

reduced by the proposed iterative feedforward approach. Similarly, feedforward approaches can also be used to account for AFM-probe deflection due to sample features [26]. Experimental Modeling: The model of the -to- coupling dynamics, , and the axis dynamics, (which are needed to implement the inversion-based approach, see (8), (9), were measured experimentally by using a digital signal analyzer (DSA). For example, to obtain the model of the -tocoupling dynamics, , a sinusoidal signal generated by the DSA was applied as input to the piezoscanner in the axis direction, and the AFM-probe-deflection signal was measured. This input–output data was fed back to the DSA and used to obtain the frequency response. The axis model was obtained in a similar way, using a sinusoidal (50 mv amplitude) input. The Bode plots of the -to- coupling dynamics, , and the nominal axis dynamics, are shown in Fig. 6. To estimate range-dependent magnitude and phase variations was obin the dynamics, the axis dynamics model tained with different input amplitudes (25 mv and 100 mv). The phase and magnitude variations, from the nominal model obtained with the 50 mv input, for two other input amplitudes are shown in Fig. 7 and these are used to estimate the phase and magnitude variations ( and ) and used to design the iterative control law. In particular, the maximum acceptable value of the iteration gain for guaranteed convergence of the algorithm (computed at different frequencies according to Lemma 2) is shown in Fig. 7 (bottom plot). B. Dynamics-Coupling Effect The dynamics-coupling effect becomes significant when the scan frequency becomes close to the piezoscanner’s vibrational resonance frequency (462 Hz). For example, at 10 Hz (which is low compared to resonance frequency), the magnitude of the -to- coupling dynamics is small compared to the magnitude of the axis dynamics as shown in Fig. 6.

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Fig. 7. Phase and magnitude variations, from the nominal z axis model, at two other input amplitudes: 25 mv (dotted line) and 100 mv (solid line). The bottom plot shows the allowable maximum value of the iteration gain (! ) for guaranteed convergence of the algorithm, computed according to Lemma 2.

In particular, the magnitude of the -to- coupling dynamics is 3.9 db, which is 6% of the magnitude of the axis dynamics (which is 20.3 db). However, when the scan frequency is at the resonance frequency of 462 Hz, the magnitude of the coupling dynamics has increased significantly from 3.9 db (at low frequency) to 19.5 db. Thus, at the higher scan frequency of 462 Hz, the 19.5 db magnitude of the coupling dynamics is larger than the 14.7 db magnitude of the axis dynamics . Dynamics-Coupling Effect Can Cause Large Deviations in the AFM-Probe Deflection: The effect of the relatively large magnitude of the -to- dynamics-coupling effect at high scan frequency is shown in Fig. 8, which shows that the couplingcaused error

(i.e., deviation of the AFM-probe deflection from the setpoint value 29 nm) can be as large as 50.49 nm when the scan range was 30 m and the scan rate was 100 Hz. This relatively-large 50.49 nm variation in the AFM-probe deflection implies that large forces are being applied on the sample. Thus, these experimental results show that significant dynamics-coupling-caused error is generated in the AFM-probe deflection due to the scanning movement in the axis. Therefore, to avoid the unwanted excessive probe-sample interactions (e.g., forces that can lead to sample damage), the -to- dynamics-coupling effect on the AFM-probe deflection must be compensated-for during high-speed AFM operation. Suppression of Vibration in the Lateral Axis: In order to suppress movement-induced vibrations in the axis (which can also adversely affect the AFM-probe deflection), a relatively smooth (filtered) input was applied in the axis. Specifically,

Fig. 8. Experimental results: x axis displacement with a scan range of 30 m (upper plot) and the corresponding x-to-z dynamics-coupling-caused error in the AFM-probe deflection.

was a 100 Hz sinusoid, with a gradthe input in the axis ually increasing (and decreasing) amplitude, which was also numerically filtered (1 KHz cutoff frequency) before being applied on the piezoscanner. Therefore, the axis movement does not have significant vibrations as shown in Fig. 8. C. Implementation and Results Inversion-based Feedforward Control: The model of the -to- coupling dynamics and the model of the axis were used to find the axis inverse feeddynamics in the frequency domain (see (8) and forward input Fig. 3). Then the inverse Fourier transform was used to find the . When computing corresponding time domain signal using (8), the measured frethe inverse control input quency response data was used to directly compute the inverse in the frequency domain. An alterfeedforward input nate approach would be to use a finite-order transfer function and representation of the measured frequency response via curve-fitting method before using them to compute the inverse input [20]. However, such method can introduce additional error in computing the inverse input due to the modeling error between the transfer function and the measured frequency response, especially, when using low-order transfer function models (to reduce computational effort). The resulting error in AFM-probe deflection, without the inverse feedforward input, is shown in Fig. 9(a) and the error in AFM-probe deflec, is shown in tion, with the inverse feedforward input Fig. 9(b). Implementation of the Iterative Inversion-Based Control: The coupling-caused error in the AFM-probe deflection was further reduced by using the iterative inverse input; the iterations were performed after scanning each line. The maximum allowable iteration gain for guaranteed convergence of the iterations was computed [see Fig. 7(c)] using the estimates of the phase and magnitude deviation from our nominal axis model,

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Fig. 10. Computation of iterative inversion-based control involves: 1) resetting the measured error in the AFM-probe deflection, ( ), to eliminate the jump between the initial and final values and 2) adding pre- and postcomputation time [ ( )]. Data is provided for the third interval before finding the inverse, iteration step.

e t

Fig. 9. Coupling-caused error in the AFM-probe deflection: with feedback control alone without inversion-based feedforward input [plots (a1),(a2)]; with inversion-based feedforward input [plots (b1),(b2)]; and at the 13th iteration step with the iterative inverse input [plots (c1),(c2)].

G e

TABLE I DYNAMICS-COUPLING-CAUSED ERROR

shown in Fig. 7(a),(b). Note that convergence is guaranteed in the frequency range [0, 1000] Hz if the iteration gain was is sufficiently small because the phase variation as shown in Fig. 7(a) (according to Lemma 2). In less than the experiments, the iteration gain was chosen to be a conin the frequency range Hz and stant chosen to be zero outside this frequency range to compute the input at each iteration step [according to (9)]. Moreover, the ingiven by (8) was applied as the verse feedforward input , i.e., . input at the initial iteration Discrete Fourier-Transform Computations: The frequency domain computations were implemented with discrete Fourier transform (DFT) with 2048 data points over the frequency range 3 Hz–1000 Hz. The input was iteratively updated after each scan. During each iteration step, the time-domain signal of the error (in the AFM-probe deflection) was extended to provide sufficient pre- and post computation time as shown in Fig. 10(a) before the input was computed with DFT. The amount of time needed for pre- and post-actuation can be quantified in terms of the zeros of the system dynamics [22]. Moreover, the final value of the signal value was reset, as shown in Fig. 10(a), so that there is no jump between the initial and final values. The error caused by this resetting on the computation of the inverse is small if the resetting is done far from the time interval ([0,0.5] s) during which the inverse input is applied to the AFM. For example, the resetting of the error signal around time 0.6 s in Fig. 10(a) does not significantly affect the inverse input during the time interval [0,0.5] s as shown in Fig. 10(b). The effect of the reset can also be quantified in terms of the zeros of the system [22]. Results With the Iterative Inversion-Based Control: The iterative approach resulted in significant reduction of the coupling-caused error in the AFM-probe deflection by the 13th iter-

yThe noise level was estimated by collecting the probe-deflection signal when the AFM-probe was not scanning the sample. This noise includes the effect of ambient vibrations, which can be reduced by using dedicated vibration isolation systems.

ation step as shown in Fig. 9(c). To quantify the reduction of the coupling-caused error in AFM-probe deflection, the root-meanand maximum error are tabulated square (RMS) error in Table I, where

(35) denotes the total number of sample points obtained at time belonging to the time interval s during which the inverse inputs were applied (and outputs were measured).

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(from 10.32 nm to 1.96 nm) and the maximum error was reduced by 86% (from 50.49 nm to 7.23 nm). Thus, the application of inversion-based iterative control substantially reduced the coupling-caused error in the AFM-probe deflection during highspeed positioning of the AFM-probe over the sample surface. This results in better control of the AFM-probe-to-sample forces during high-speed scanning—thereby, enabling high-speed operation of the AFM. V. CONCLUSION

Fig. 11. Power spectrum. (a) Scanning input in the x axis u (t). (b) Error in the AFM-probe deflection e (t) when feedback control was applied alone (solid line) and after the 13th iteration step (dotted line).

D. Discussion Dynamics-Coupling Effect: Significant coupling effect is present at high frequencies even though the scanning input (in the axis) has relatively-low frequency content. For example, [28] of the dynamics-coupling caused the power spectrum error in the AFM-probe deflection is shown in Fig. 11, where (36) represents the DFT of the error signal , is the number of data points in the signal and denotes the complex conjugate. The power spectrum of the scanning input is computed similarly, and shown in Fig. 11. It can be seen from Fig. 11(a), that the prominent component of the scanning is at the scan frequency of 100 Hz and the compoinput nents of higher frequencies (e.g., 200 Hz and 300 Hz) are relatively small due to the filtering discussed in Section IV-B. However, this low-frequency scanning input does result in significant error in the AFM-probe deflections at the higher frequencies [e.g., 200 Hz and 300 Hz as shown in Fig. 11(b)]. This error in the AFM-probe deflection can be reduced by using the iterative inversion-based control; the reduction of error is shown in Fig. 11(b) which compares the error with the use of feedback alone and with the use of the input at the 13th iteration. Reduction of Dynamics-Coupling Caused Error: Experimental results show that the iterative inversion-based feedforward control can significantly reduce the coupling-caused error in the AFM-probe deflection. For example, the reduction in the dynamics-coupling-caused error with the model-based inverse input (without iterations) is shown in Fig. 9(b) and Table I. was reduced by 25% (from In particular, the RMS error 10.32 nm to 7.74 nm) and the maximum error was reduced by 22% (from 50.49 nm to 39.25 nm). Additionally, the iterations can be used to further reduce the dynamics-coupling-caused due to coupling error. For example, the maximum error at the 13th iteration step (see Table I) is comparable to the noise was reduced by 81% level. In particular, the RMS error

This paper showed that iterative, inversion-based, feedforward control can be used to compensate for the -todynamics-coupling error in piezoscanners during high-speed positioning. Convergence of the iterative approach was investigated and used to design the iteration algorithm. Experimental results were presented to demonstrate 1) the presence of significant -to- dynamics-coupling effect in the AFM-probe deflection during high-speed AFM operation; and 2) substantial (more than 80%) reduction of such coupling-caused error, in the AFM-probe deflection, by using the iterative, inversion-based, feedforward approach. REFERENCES [1] J. Ghosh and B. Paden, “Pseudo-inverse based iterative learning control for plants with unmodeled dynamics,” in Proc. Amer. Control Conf., 2000, pp. 472–476. [2] , “Iterative learning control for nonlinear nonminimum phase plants,” ASME J. Dyn. Syst. Meas. Control, vol. 123, pp. 21–30, Mar. 2001. [3] X. Wang and D. Chen, “Robust inversion-based learning control for nonminimum phase system,” in Proc. IEEE Int. Conf. System, Man, and Cybernetics, vol. 3, Oct. 2002, pp. 489–494. [4] E. Dubois and J. L. Bubendorff, “Kinetics of scanned probe oxidation: space-charge limited growth,” J. Appl. Phys., vol. 87, no. 11, pp. 8148–8154, 2000. [5] R. García, M. Calleja, and H. Rohrer, “Patterning of silicon surfaces with noncontact atomic force microscopy: field-induced formation of nanometer-size water bridges,” J. Appl. Phys., vol. 86, no. 4, pp. 1898–1903, 1999. [6] C. Rotsch, F. Braet, E. Wisse, and M. Radmacher, “AFM imaging and elasticity measurements on living rat liver macrophages,” Cell Biol. Int., vol. 21, no. 11, pp. 685–696, 1997. [7] J. Domke and M. Radmacher, “Measuring the elastic properties of thin polymer film with the atomic force microscope,” Langmuir, vol. 14, no. 12, pp. 3320–3325, 1998. [8] C. L. Grimellec, E. Lesniewska, M. C. Giocondi, E. Finot, V. Vie, and J. P. Goudonnet, “Imaging of the surface of living cells by low-force contact-mode atomic force microscopy,” Biophys. J., vol. 75, pp. 695–703, Aug. 1998. [9] M. B. Viani, T. E. Schaffer, A. Chand, M. Rief, H. E. Gaub, and P. K. Hansma, “Small cantilever for force spectroscopy of single molecules,” J. Appl. Phys., vol. 86, no. 4, pp. 2258–2262, 1999. [10] F. El Feninat, T. H. Ellis, E. Sacher, and I. Stangel, “A tapping mode AFM study of collapse and denaturation in dentinal collagen,” Dental Mater., vol. 17, pp. 284–288, 2001. [11] G. W. Marshall, I. C. Wu-Magidi, L. G. Watanabe, N. Inai, M. Balooch, J. H. Kinney, and S. J. Marshall, “Effect of citric acid concentration on dentin demineralization, dehydration, and rehydration: atomic force microscopy study,” J. Biomed. Mater. Res., vol. 42, pp. 500–507, 1998. [12] O. M. El Rifai and K. Youcef-Toumi, “Coupling in piezoelectric tube scanners used in scanning probe microscopes,” in Proc. Amer. Control Conf., 2001, pp. 3251–3255. [13] G. Schitter, P. Menold, H. F. Knapp, F. Allgöwer, and A. Stemmer, “High performance feedback for fast scanning atomic force microscopes,” Rev. Sci. Instrum., vol. 72, no. 8, pp. 3320–3327, 2001. [14] S. Salapaka, A. Sebastian, J. P. Cleveland, and M. V. Salapaka, “High bandwidth nano-positioner: a robust control approach,” Rev. Sci. Instrum., vol. 73, no. 9, pp. 3232–3241, Sep. 2002.

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Qingze Zou received the B.S. degree in automatic control from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1994, the M.S. degree in mechanical engineering from Tsinghua University, Beijing, China, in 1997, and the Ph.D. degree in mechanical engineering from the University of Washington, Seattle, in 2003. He was a Research Associate in the Department of Mechanical Engineering, University of Washington, from 2003 to 2004. He joined the Department of Mechanical Engineering, Iowa State University, Ames, in 2004 as an Assistant Professor. His primary research interests are in high-precision positioning, and controls of nanopositioning devices with applications to SPM-based combinatorial science, nanomaterial analysis and synthesis, biomedical imaging, and high-throughput nanomanufacturing. Dr. Zou is a Member of the American Society of Mechanical Engineers (ASME).

Szuchi Tien received the B.S. degree in naval architecture from the National Taiwan University (NTU), Taiwan, in 1995 and the M.S. degree in mechanical engineering from the National Cheng Kung University (NCKU), Taiwan, in 1997. He is currently working toward the Ph.D. degree in the Department of Mechanical Engineering, University of Washington, Seattle.

Santosh Devasia received the B.Tech. degree (Hons.) from the Indian Institute of Technology, Kharagpur, India, in 1988 and the M.S. and Ph.D. degrees in mechanical engineering from the University of California at Santa Barbara in 1990 and 1993, respectively. He is currently a Professor in the Mechanical Engineering Department, University of Washington, Seattle. Previously, he had taught in the Mechanical Engineering Department, University of Utah, Salt Lake City. His current research interests include inversion-based control theory, high-precision positioning systems for nanotechnology and biomedical applications. These projects are funded through NSF and NIH grants. Dr. Devasia is a Member of the American Institute of Aeronautics and Astronautics (AIAA) and the American Society of Mechanical Engineers (ASME). Currently, he is an associate editor for the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY and for the ASME Journal of Dynamic Systems, Measurement and Control.