Estimation of the shear force in transverse dynamic force microscopy using a sliding mode observer

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Atomic Force Microscopy (AFM) has become a widespread and important technique for the study of nano-scale specimens since its inception in 1986 by Binnig et al. 1 Broadly speaking, an AFM can provide high resolution images in different settings including ambient, aqueous and vacuum environments.This makes it especially suitable for the investigation of biological specimens under physiological conditions.A key component in most AFMs is a micro-cantilever.The interaction between the tip of the cantilever and the sample creates bending/shear moments on the cantilever, which can be indirectly measured via a laser based sensor system.This is then used to create a high resolution topographical image via a raster scan over the specimen surface.In standard AFMs, the cantilever is mounted horizontally and the devices are operated in a contact 1 or intermittent-contact mode. 2,3In contrast, the Transverse Dynamic Force Microscope (TDFM) at Bristol addresses the problem of non-contact imaging of a sample.This is important for certain types of biological specimens.Under ambient room conditions, any sample will be covered by an ordered, thin water layer; hence, recorded changes in the cantilever resonant dynamics measure the short-range lateral forces between an oscillating vertically oriented cantilever (VOC) and a surface.It has been demonstrated that the visco-elastic response of the ordered water layer between the tip and the surface results in a contrast mechanism [4][5][6][7] for cantilever-specimen distance detection.The "snap-to-contact" behaviour is experienced by conventional AFMs.This is due to the gradient of the surface attractive force being larger than the spring constant of the cantilever. 8In the TDFM, this is prevented by the vertical orientation of the cantilever.
Because of the pivotal role of the micro-cantilever, it is important to understand its dynamic behaviour and to have knowledge of its physical parameters and characteristics.Unsurprisingly, the problem of estimating the cantilever parameters in AFM devices has been investigated for many years in the literature.A real-time methodology to determine the probe loss areas in a dynamic atomic force microscopy based image was established by De et al. 9 and an observer-based approach for estimating some unknown force affecting the dynamics of a cantilever in Electric Force Microscopy devices was proposed by Besancon et al. 10 More recently, the tip-sample interaction forces based on a two degree of freedom mathematical model of a tapping mode AFM was investigated by Xu et al. when the cantilever is immersed in liquid. 11However, all these results pertain to horizontally mounted cantilever arrangements.
This paper addresses the problem of estimating the tip-sample interaction forces with a realtime implementable sliding mode observer [12][13][14][15] for the TDFM, which operates in a non-contact scanning regime with a vertically oriented cantilever.Sliding mode observers have the advantage of robustly detecting unknown signals with finite time convergence guarantees. 13Comparisons with a quasi-static method (as often employed in beam analysis) are made.This approach is realized under additional assumptions which are not required for the sliding mode observer.
A schematic of the TDFM setup is shown in Figure 1.In the TDFM, the scattered evanescent field from the cantilever tip is gathered by a lens (B) with large numerical aperture (NA) and transmitted to the objective lens.The objective lens then focuses the light onto the photo-diode.The amplitude variations (both in x and y directions) and the height (z) of the cantilever tip relative to the cover slip can be measured via the signals coming from the four sectors of the photo-diode.(In reality the actual TDFM system is more complex and detailed than the schematic presented in Figure 1, which only shows the main components. 16) The available data for the estimation (and control) problems are the excitation signal at the top of the cantilever, the distance (z) of the cantilever from the cover slip (the summed light intensity of all photo diode sectors) and the horizontal position of the cantilever tip (obtained from suitably scaled light-intensity signals from the photo-diode sectors).
Since the cantilever specimen distance cannot be directly measured, other factors should be taken into account to construct the specimen topography.Hence, understanding the behaviour of the shear forces helps to improve the mechanism for detecting the probe/specimen distance.The dynamics of the cantilever in air, with the shear force interaction due to a thin water layer at the tip, can be presented in the form of the partial differential equation (PDE) 18,19 with boundary conditions In the above, E is Young's modulus, α is the internal damping constant of the cantilever, I is the secondary moment of area, S is the cross-sectional area, ρ is the density of the probe, L is the length of the cantilever, ζ denotes position along the probe axis, Y is the transversal displacement at any point along the probe during vibration, Ẏ and Ÿ are the first and second derivatives of Y with respect to time t, u(t) is the external excitation signal applied at the top of the cantilever, and finally f (t) is the tip-sample interaction force applied at the tip of the cantilever.The aim of this paper is to estimate the unknown shear force signal f (t), which will allow better interpretation and understanding of the scan result.
3][14][15] Given the measured tip position Y (ζ = L), the shear force f (t) and the excitation signal u(t) , it is necessary to approximate the dynamic relationship between those three variables via a finite dimensional ordinary differential equation (ODE).This is achieved by the method of lines. 20The idea is to divide the cantilever probe into n − 1 equal sections and to consider n nodes distributed along the probe.Denote Y j as the displacement at node j and δζ as the distance between two consecutive nodes.Using a finite difference formula the boundary condition (3) for the approximate model becomes Thus, for the boundary condition (4), the relation is obtained The right hand side of ( 5) can be approximated as Equations ( 2), ( 6), ( 7) and (8) imply that the values of Y 1 , Y 2 , Y n , and Y n−1 are 'known', i.e. depend on the dynamics of the remaining nodes.Hence, understanding the dynamics of the remaining 'middle' n − 4 nodes is key.From (1), the fourth partial derivative of Y with respect to the spatial variable ζ can be approximated using finite differences as follows for j = 3, . . ., n − 2. Using the boundary conditions ( 6), ( 7) and (8), and the approximate model ( 9), the dynamics of nodes Y j for j = 3, . . ., n − 2 are obtained.A linear time invariant system based on the dynamics of Y j for j = 3, . . ., n − 2 is used.The relations between Y j , Ẏj , f , and u allow the establishment of an ODE of order 2(n − 4).Using a matrix-vector representation, this ODE is written as a state-space system 21 with a dynamic state variable vector x p (t) of dimension 2(n − 4), in the form where the matrices A p , B p , D p , and C p are of appropriate size.Crucially, ( 10) is a good approximation of the PDE providing δζ is small enough.In (10) the output y(t) which represents the cantilever tip position is taken as Y n−2 since it is assumed that Y n ≈ Y n−2 for large enough n.Furthermore the shear force can be approximated by a linear combination of the state variables x p in (10) without the knowledge of the derivative of Y n .Further details can be found in Nguyen et al. 17 Clearly the larger the number of nodes, the more accurately (10) approximates the real PDE (1).However the greater the number of nodes, the higher the order of the state-space and the greater the computational burden.For this reason, a lower order (approximate) model of ( 10) is more desirable.
To create such a model, a broad range of methods for model order reduction, available in the systems theory literature, can be employed: see for example Refs.22 and 23 and the references therein.
Here the 'balanced truncation' method by Moore, 22 has been used.Hence, for design purposes, a model of the form is used with the property that the input/ouput behaviour (u, f )  → y of ( 11) closely matches that of (10).The key property of (11) compared to (10) is that the dimension of the vector x(t) is orders of magnitude lower than x p (t).Although the elements of x(t) no longer have physical meaning, this system is now ideal for design of the shear force estimator.In (11) the inputs u(t) and y(t) are known and measured.However the state, x(t), and in particular the shear force, f (t), are unknown.Here a so-called sliding mode observer (see for example Refs.12-15 and the references therein) will be used to estimate f (t) from the known quantities y(t) and u(t) and knowledge of the model in (11).In this paper, a design proposed in Edwards et al. 12 is employed.The sliding mode observer has the following form: where k > 0 is a scalar gain, sgn(•) represents the signum function and the output estimation error is the difference between the output of the observer ŷ(t) and the measured value y(t).The gain G represents design freedom and must be selected to ensure the matrix (A − GC) is Hurwitz stable (i.e. the eigenvalues of the matrix (A − GC) have negative real part, e.g.Ref. 14).It can be shown that for an appropriate choice of G and the scalar gain k, the output estimation error e y (t) is driven to zero in finite time and a so-called sliding motion takes place. 12,14During the sliding motion, on average, the high frequency switching term −ksgn(e y (t)) must replicate f (t) for sliding to be maintained.The average value of ksgn(e y (t)) necessary to maintain sliding is known as the equivalent injection 14,24 and can be approximated by low-pass filtering the actual injection signal ksign(e y (t)).
Here a simple first-order low-pass filter for the fast-switching value of ksgn(e y (t)) is used to obtain the estimate, f (t), of the shear force from the equivalent injection.Let f (t) satisfy where τ is a small positive scalar which represents the filter time constant.Since f (t) approximates the equivalent injection, 24 it follows f (t) ≈ f (t).Note that f (t) is available in real time from (15)  and as a result, changes in the shear force f (t) can be estimated in real time.A diagram of the observer and further explanations are presented in Figure 2.
The experiments which follow were carried out at ambient conditions, which were 20 • C and a room humidity of 60%.The external excitation u(t) = d 0 sin(ωt) is sinusoidal with frequency ω and FIG. 2. TDFM and nonlinear sliding mode observer based estimator of the shear force f given an excitation signal u(t) at the top of the cantilever and the measured position y(t) of the tip of the cantilever.The error signal e y (t) (14) creates the drivers for the estimator, the observer injection term ksgn(e y (t)) and Ge y (t), to guarantee fast and robust convergence.The low pass filtering of the switching observer injection ksgn(e y (t)) provides the shear force estimate f (t).amplitude d 0 .Furthermore, it is assumed that the tip sample interaction force at the tip can be split into a viscous and an elastic force: 18 where ν is the dissipative interaction constant and κ is the elastic interaction constant.The cantilever is made of Silicon Nitride (Si3N4) and the parameters are given as follows: Young's modulus E = 210 GPa, ρ = 3100 kg/m 3 , length L = 28 µm, width w = 2 µm, thickness t c = 200 nm.The cantilever was brought within a distance of 10 nm to a quartz cover slip (No. 0, i.e. 85-130 µ m thick) to initially determine an excitation-to-cantilever model in free air.This yielded the necessary data to obtain a reduced-order state space model (11).Then, fourteen data sets X 1 , X 2 , X 3 , . . .X 14 , including input and output signals, were collected from experiments.The tip-to-surface distances corresponding to X 1 , X 2 , X 3 , . . .X 14 are linearly increasing for 1.5 nm to 8 nm respectively.The top of the cantilever was excited at its resonance frequency of 352.75 kHz.The amplitude of the excitation signal is 1.8 nm.
Using the model ( 11), a sliding mode estimator was designed.An example of a shear force estimate is shown in Figure 3(a) which exhibits sinusoidal signals with the same frequency as the excitation signal u(t).The inherent bandwidth limit of the sliding mode observer design is given by the low pass filter time constant τ = 5µs from (15). Figure 3(b) provides the RMS values fe f f (RMS -root mean square values) of the shear force estimates for each data series.It is clear that the shear forces corresponding to the data sets recorded close to the cover slip exhibit the largest interaction, i.e. the first data sets have the biggest amplitudes among the shear forces.It is now also possible to estimate the elastic and viscous parameters κ and ν (e.g. using a recursive least squares process).This introduces a further time constant of 5µs associated with the recursive least squares process.This time constant and the filter time constant τ in (15) are independent of a quasi-static harmonic balance assumption and are the result of well-understood linear filtering processes.It was observed that the elastic component −κY (L,t) contributes less than 1.3 % to the overall shear force energy so that this term can be disregarded.In contrast, the estimates of the viscous coefficient (see Figure 3(c)) follow, within the tip-to-surface range of 2-8 nm, a generally decreasing relation.This confirms the viscous effect of the thin water layer above the cover slip, which increases the closer the cantilever tip is to the cover slip.Note the significant difference in magnitude change for ν and  fe f f .This is due to uncorrelated sensor noise in the output signal y(t) = Y (L,t).Whilst the least squares methods for ν remain largely unaffected by this, the RMS-value fe f f is.
An important aspect of the sliding mode observer is its ability to work without the requirement of harmonic excitation and measurement signals, including a steady state assumption of the overall system under this condition.However, the TDFM adheres to these assumptions at least within some quasi-steady state.For comparative reasons, given the sinusoidal characteristics of the excitation u(t) and the cantilever tip position y(t), it is possible to compute the sinusoidal components of f (t), and subsequently the viscous coefficient ν of f (t) (see Refs. and 25 for a homogenous beam as used here).The derivation of the amplitude and phase for y(t) in relation to u(t) is done via a real-time implementable phase-locked loop (PLL) technique. 26It is well-known that a phase-locked loop technique has at its very best an inherent settling time of one oscillation period (see for example Chapter 3 26 ), i.e. 2.83 µs = 1/(352.75kH z).However, realistically the pull-in process of a PLL is several multiples of one oscillation period -the direct result of a nonlinear process of the PLL.Thus, for the estimation of the viscous component ν, the PLL relative to the sliding mode technique provides in Figure 4  for tip-to-surface distances between 1.5 nm to 6.5 nm were used.The PLL-results for ν appear to be 'low-pass-filtered' in relation to the sliding mode estimate, i.e. they are of lower frequency content and delayed.Considering the requirement for high-speed scanning and subsequent high-speed data processing, the sliding mode observer shows an advantage over the PLL-method in terms of speed and the basic underlying technical assumptions.
In conclusion, a constructive scheme to obtain a real time estimate of the shear forces affecting the VOC of a TDFM has been presented.A parametric representation of the shear force presents a scaled measure of the cantilever-specimen distance.An approximate ODE model of the cantilever dynamics was derived using the method of lines. 20Based on this ODE model, a reduced order model was used to construct a sliding mode observer to estimate the unknown shear force.Practical examples illustrate that the proposed real-time implementable scheme can reconstruct the unknown shear forces using measurement signals subject to noise.The model confirms an increase in viscous damping as the distance between the cantilever and the cover slip decreases.The method shows faster response times and higher bandwidth than a phase-locked loop based technique which operates under the assumption of a quasi-steady state balance of harmonic excitation and output signals.

FIG. 3 .
FIG. 3. Shear force estimate characteristics.(a)Estimate of shear force for data X1 for a cantilever/cover slip distance of 1.5 nm.(b)RMS value fe f f of shear force f as a function of the cantilever-cover slip distance.(c)Estimate of viscous coefficient ν as a function of the cantilever-cover slip distance.
18 and 19 for a generic tapered beam All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/Downloaded to IP: 137.222.120.226On: Fri, 25 Sep 2015 12:12:50 FIG.4.Shear force estimate characteristics as function of time for sliding mode (SMO) and PLL-technique.(a)Estimate of shear force for data varying between 1.5 nm-6.5 nm.(b)Estimate of shear force for data varying between 1.5 nm-2 nm.(c)Estimate of shear force for data varying between 5.5 nm-6.5 nm.