Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 1, pp. 93-106, Warsaw 2014 TM-AFM NONLINEAR MOTION CONTROL WITH ROBUSTNESS ANALYSIS TO PARAMETRIC ERRORS IN THE CONTROL SIGNAL DETERMINATION José Manoel Balthazar Universidade Estadual Paulista – UNESP, Rio Claro-SP, Brazil; e-mail: jmbaltha@rc.unesp.br Angelo Marcelo Tusset Universidade Tecnológica Federal do Paraná – UTFPR, Ponta Grossa-PR, Brazil; e-mail: a.m.tusset@gmail.com Átila Madureira Bueno Universidade Estadual Paulista – UNESP, Sorocaba-SP, Brazil; e-mail: atila@sorocaba.unesp.br Nonlinear motion of the microcantilever probe in the Atomic ForceMicroscope (AFM) has been extensively studied consideringmainly the van derWaals forces. Since the behavior of the microcantilever is vital to quality of generated images, the study of control strategies that force the probe to avoid undesired behavior such as chaoticmotion, is also of significant importance. A number of published works has shown that the microcantilever is subject to chaoticmotion for a certain combination of parameters. For such a parameter combination, the control system must suppress the chaotic motion. Here, an study of the AFM mathe- matical model is presented, aiming to find a region of operation of the AFM where the motion is chaotic. In order to suppress the chaotic motion, a periodic orbit of the system is obtained, and the controller forces the system to that periodic orbit. Two control strategies are used, namely: The State Dependent Riccati Equation (SDRE) and the Optimal Linear Feedback Control (OLFC). Both control strategies consider the complete nonlinearities of the system, and theOLFCguarantees the global stability.Thenumerical simulations carried out showed the efficiency of the control methods as well as the sensitivity of each control strategy to parametric errors. Without the parametric errors, both control strategies were effective inmaintaining the system into the desired orbit.On the other hand, in the presence of parametric errors, the SDRE technique wasmore robust than the OLFC. Key words: AFM, SDRE control, optimal linear feedback control, uncertainty 1. Introduction The invention of theScanningTunnelingMicroscope (STM)andof theAtomicForceMicroscope (AFM)byGerdBinnig, in the1980s, started surface investigation in theatomic scale. Since then, many improvements and developements had been made achieving important results by simple contactmeasurements.Nevertheless, the contactAFMcannot generate true atomic resolution in stable operation (Mestron et al., 2007;Morita et al., 2009;Bhushan, 2004). TheAFMsystemhas become a popular and useful instrument to measure intermolecular forces that can be applied in electronics, biological analysis, materials, semiconductors, etc. A typical AFM consists of a microcantilever with a sharp tip mounted to a piezoelectric actuator, as shown in Fig. 1. The microcantilever displacement is determined by a position sensitive photodetector, from a laser beam reflected off themicrocantilever end-point, providing feedback signal to the control system (Morita et al., 2009; Bhushan, 2004; Jalili et al., 2004). In themid1990s, noncontactAFMtechniquesachived trueatomic resolutionunderattractive regimeat roomtemperature.ThenoncontactAFMoperates in the staticmodeordynamicmode, i.e., static AFM or dynamic AFM, respectively. In the static AFM, the force Fts interacting 94 J.M. Balthazar et al. between the tip and sample translates into deflection of the cantilever, and the image is a map (x,y,Fts) with Fts = const. Different techniques provide several opportunities to take pictures from different types of samples, generating a wide range of information. The different methods of generating images – also called scanning modes or modes of operation – mainly refer to the distance between the probe tip and the sample at the time of scanning, and to theways the tipmoves over the sample surface. The tip displacement due to the tip and sample interaction forces are translated into images, and since the tip and sample forces strongly depend on the tip and sample distance, differentmodes of operation generate different images (Frétigny, 2007;Moritaet al., 2009; Bueno et al., 2012). The two dinamic AFMbasic methods are AmplitudeModulation (AM-AFM) and Frequen- cyModulation (FM-AFM), in both AM-AFM andFM-AFM themicrocantilever is deliberately vibrated at a predetermined amplitude and frequency, near the microcantilever eigenfrequency. The tip-sample interaction forces cause changes in the amplitude, phase and frequency of the microcantilever oscillation, i.e., the tip-sample interaction forces are modulated in the micro- cantilever motion (Morita et al., 2009; Bueno et al., 2012; Polesel-Maris and Gauthier, 2005; Couturier et al., 2001; Bueno et al., 2010; Zhong et al., 1993). Initially, the AM-AFM was used only in noncontact mode, but later it was also used at a closer distance involving repulsive tip-sample interactions in the intermittent contact mode AFM, or TappingMode (TM-AFM). In the tapping mode, the microcantilever amplitude of oscillation is modulated as the mi- crocantilever tip scans the sample. This modulation causes the microcantilever tip to only tap on the sample surface near the extreme of the oscillation cycle, minimizing the frictional forces that are present in the contact mode and reducing the damage on soft samples, providing high resolution topographic images even for sample surfaces that are easily damaged or dificult to image by other AFM techniques (Morita et al., 2009; Zhong et al., 1993; Hansma et al., 1994). Under certain physical conditions the AFM system is subject to undesirable behaviors such as bifurcations and chaotic motion, due to the nonlinear effects of the tip-sample interaction forces. This type of irregularmotion impairs the AFMperformance since it degrades the atomic forces measurements, generating poor resolution and inaccurate images. In the TM-AFM the chaotic motion often occur during the transition from noncontact to tapping mode indicating the presence of complex dinamics (Morita et al., 2009; Bhushan, 2004; Ashhab et al., 1999a; Hu and Raman, 2006; Raman et al., 2008). The nonlinear behavior of themicrocantilever has also been used to improve sensitivity and material contrast bymodulating high frequency distortions on themicrocantilever motionwhen themicrocantilever is harmonically driven very close to the sample surface. In principle, higher harmonics contain detailed information about the tip-sample potential (Morita et al., 2009; Raman, 2008) In the literature, a number of control methods have been proposed aiming to suppress or mitigate undesirable irregular motion and its effects. In (Yabuno, 2008), the microcantilever amplitude of vibration is controlled by applying an additional nonlinear damping via nonlinear feedback. In (Hornstein and Gottlieb, 2008), the model differs from standard lumped mass models by the inclusion of nonlinear elastic terms yielding a consistent set of system parameters that incorporates the influence of the modified microcantilever dispersion and the controller, which is introduced as a part of the generalized force, and affects both the equilibrium and the time dependent solution of the microcantilever equation. In (Yamasue and Hikihara, 2006), a chaotic microcantilever in AM-AFM is stabilized using the Time-Delayed Feedback Control (TDFC) method, forcing themicrocantilever to a periodic orbit of the system. In (Salarieh and Alasty, 2009), a chaotic TM-AFM is controlled using theTDFCmethod. The feedback gain is obtained and adapted according to aminimun entropy algorithm. In (Ashhabet al., 1999a), theAFMismodeled considering thevanderWaals potential TM-AFM nonlinear motion control with robustness analysis... 95 force, and the cantilever is vibrated by a sinusoidal input.The forced dynamics is analysed using theMelnikovmethod determining the regions of the parameter space inwhich chaotic motion is possible. Then, using aPI controller, theMelnikovmethod is computed again, and the obtained relation is used to suppress the chaotic motion. In (Yamasue et al., 2009), an experimental stabilization of irregular and non-periodic mi- crocantilever oscillation in the AM-AFM with the TDFC technique is demonstrated using a magnetic excitation instead of typical piezoelectric excitation. In (Balthazar et al., 2013), the TM-AFMoperating in liquid ismodeled, and chaoticmotion is identified for awide range of the parameter values, and two control techniques used to suppress the chaoticmotion are compared, namely, the Optimal Linear Feedback Control (OLFC), proposed by Rafikov et al. (2008) and the TDFC, showing that the OLFC presents a faster transient response than the TDFC. In this work, the TM-AFM operating in vacuum is modeled based on the forced dynamical system suggested by Ashhab et al. (1999b), and the chaotic motion is observed for a certain parameter combination. In order to suppress the microcantilever chaotic motion, two control techniquesareproposed, theOptimalLinearFeedbackControl (OLFC)andtheState-Dependent Riccati Equation (SDRE). Additionally, the robustness of both control techniques is tested considering parameter uncertainties on the TM-AFMmodel and on the control signal. In Section 2, themathematical model of theTM-AFM is described. In Section 3, the control of chaoticmotion by the application of theOLFCandSDREmethods is presented. In Section 4, the robustness of the control techniques is tested by including parameters uncertainties on the control signal determination.Thefinal remarks and theacknowledgments are inSections 5and6, respectively. 2. TM-AFM mathematical model The physical model of the TM-AFM is shown in Fig. 1. The basis of the microcantilever is excited by a dither-piezo generating a displacement ψcos(wt). Themicrocantilever is controlled by a piezo-actuator. Fig. 1. Model of an AFM The first mode of vibration of the TM-AFM microcantilever can bemodeled as a vibrating mass-spring-damper system (Morita et al., 2009; Bhushan, 2004; Zhang et al., 2009) vibrating close to the surface of the sample, as shown in Fig. 2. The tip is considered as a sphere of radius R. In the equilibrium position (when only the gravity acts on it), the distance between the cantilever tip and the sample is given by Z0. The position of the cantilever measured from the equilibrium position is given by x. Themass of the microcantilever is given by m, and the spring and damping coefficients are given by k and c, respectively. Fu is the control system force used to control the microcantilever displacement by means of the Piezo-actuator. 96 J.M. Balthazar et al. Fig. 2. Model representative of AFM through amass-spring-damper According to Rutzel et al. (2003), the interaction between the tip of the cantilever and the surface of the sample can be modeled as being the interaction between a sphere and a flat surface, as ULJ(x,z0)= A1R 1260(z0 +x)7 − A2R 6(z0+x) (2.1) where ULJ(x,z0) is the Lennard-Jones (LJ) potential, A1 and A2 are theHamaker constants to the attractive and repulsive potentials, respectively. Then, the potential forces are represented by a sumof the attractive and repulsive forces (van derWaals force) (Rutzel et al., 2003), given by FLJ =− ∂ULJ ∂(x+z0) = A1R 180(z0+x)8 − A2R 6(z0+x)2 (2.2) In the TM-AFM, the tip only touches the surface of the sample in themaximum amplitude of oscillation. The contact between the tip and the sample is complicated and delicate, this the main reason to use this operation mode in fragile samples. Since the microcantilever must be driven to periocally oscilate during the scanning process, the microcantilever is excited by a harmonic force (Ashhab et al., 1999b; Yamasue and Hikihara, 2006; Salarieh and Alasty, 2009; Morita et al., 2009; Giessibl, 1995) F = ψcoswt (2.3) The conservative spring force is given by Fk = kx (2.4) The dissipative force is given by: Fc = cẋ (2.5) From the equilibrium of forces acting on the microcantilever, it results that mẍ = −Fk − Fc + FLJ + F + Fu and considering the foregoing relations (Eqs. (2.2)-(2.5)), the governing equation of motion becomes mẍ+ cẋ+kx = A1R 180(z0 +x)8 − A2R 6(z0+x)2 +ψcoswt+Fu (2.6) TM-AFM nonlinear motion control with robustness analysis... 97 Taking into account the following relationship between the variables in (2.6) T = wt y = x zs ẏ = ẋ wzs w2 = √ k m a = z0 zs b = c mw h = A1R 180mw2z9 s d = A2R 6mw2z3 s Zs = 2 3 3 √ A2R 3k U = Fu mw2zs the dimensionless equation of motion is given by y′′+ by′+y−F(y)−f cos(T)=U (2.7) where F(x1)= h (a+x1)8 − d (a+x1)2 (2.8) Defining the state variables as x1 = y and x2 = y ′, the dimensionless equation of motion can be transformed into state space equations, given by x′1 = x2 x ′ 2 =−bx2−x1+F(x1)+f cos(T)+U (2.9) According toYamasue andHikihara (2006), Salarieh andAlasty (2009), for someparameters, the system in equation (2.9) presents chaotic behavior. Considering the parameters in Salarieh and Alasty (2009), h = (9/25) · 10−5, f = 2, b = 0.04, d = 4/27, a = 0.8. For the initial conditions x1(0) = 0.8 and x2(0) = 0, and for U = 0, the behavior of the system can be seen in Fig. 3. From the simulation results, it can be seen that the system presents chaotic behavior. 3. Chaos control in TM-AFM Chaotic oscillations undermines the quality of the AFM images reducing the resolution and the operating range of the AFM. Stabilizing the system in a periodic orbit and suppressing the chaotic motion is essential for the accurate tip and sample interaction forces measurement (Morita et al., 2009; Bueno et al., 2012). 3.1. Obtaining periodic orbits Considering thenonlinear characteristics of the vanderWaals force – seeEqs. (2.2) and (2.8), in order to simplify the mathematical reasoning when applying the perturbation techniques, it is approximated by a Taylor series expansion resulting in linear quadratic and cubic terms, simplifying the mathematical reasoning. Then, from Eqs. (2.8) and (2.9), F(y) is expanded in a Taylor series at the point y =0, resulting F(y)≈−0.2315+0.5785y −1.0839y2 +1.8032y3 (3.1) Replacing Eq. (3.1) in Eq. (2.7) and considering U =0, results y′′+εy′+ c1y+ c2y 2− c3y3+p−f cos(T)= 0 (3.2) where ε =0.04, c1 =0.4215, c2 =1.0839, c3 =1.08032 and p =0.2315. 98 J.M. Balthazar et al. Fig. 3. (a)Microcantilever displacement; (b) phase portrait; (c) Lyapunov exponents λ1 =0.093741, λ2 =−0.133749; (d) Poincaré map; (e) frequency spectrum In order to obtain a periodic solution to Eq. (3.2), an expansion using the multiple scales method is done, considering T0 = T and T1 = εT , and solutions of the following formare looked for y = εµ1+ε 2µ2 (3.3) where ε is the parameter responsible for the balance (Nayfeh, 1981), with d dτ = D0+εD1+ε 2D2+ . . . d2 dτ2 = D20 +2εD0D1+ε 2(D21 +2D0D2)+ . . . (3.4) then, the derivatives become D20µ1+ c1µ1 =0 D20µ2+ c1µ2 =−2D0D1µ1−D0µ1− c2µ 2 1 (3.5) One possible solution to the system in Eq. (3.5) is y = εa1cos( √ c1T +β1)+ε 2a2cos(ε √ c1T +β2)+ ε2c2a1 6c1 cos(2 √ c1T +2β1) (3.6) TM-AFM nonlinear motion control with robustness analysis... 99 For the initial conditions y0 and y ′ 0, it results that β1 = β2 = 0, a1 = y0/ε and a2 =−y0/(6εc1), and replacing them into Eq. (3.6) periodic orbits of the form y = y0cos( √ c1T)− εy0 6c1 cos(ε √ c1T)+ εc2y0 6c1 cos(2 √ c1T) (3.7) are obtained. In Fig. 4, the periodic orbit for Eq. (3.7), for y0 =0.8 and y ′ 0 =0, is shown. Fig. 4. (a) Microcantilever displacement; (b) phase portrait for solution (2.9) 3.2. Application of the OLFC TheOLFCmethodwas developed byRafikov et al. (2008). Thismethod obtains an optimal linear feedback for a class of nonlinear systems ensuring the stability of the problem. In this section, the OLFC is applied attempting to drive the TM-AFM to the periodic orbit obtained in the previous section. TheTM-AFMequation ofmotionwith the control law is describedby the followingnonlinear equation x′1 = x2 x ′ 2 =−bx2−x1+F(x1)+f cos(T)+U (3.8) where U = ũo+uof (3.9) and uof is the feedback control. The feedforward optimal control ũo is given by ũo = ˙̃x2+ bx̃2+ x̃1−F(x̃1)−f cos(T) (3.10) where x̃ is thedesiredperiodic orbit (Eq. (3.7)). ReplacingEq. (3.10) intoEq. (3.8), anddefining the deviations from the desired orbit by e= [ e1 e2 ] = [ x1− x̃1 x2− x̃2 ] (3.11) results in e′1 = e2 e ′ 2 =−be2−e1+F(x1)−F(x̃1)+uof (3.12) The system of Es. (3.12) is written in the following form e′ =Ae+G(e, x̃)+Buof (3.13) where A= [ 0 1 −1 −b ] G(e, x̃)= [ 0 F(e1, x̃1)−F(x̃1) ] B= [ 0 1 ] 100 J.M. Balthazar et al. According toRafikov et al. (2008), Tusset et al. (2012b), if there existmatrices Qand Rpositive definite, being Q symmetric, such that the function Q̃ =Q−GT(e, x̃)P−PG(e, x̃) (3.14) is positive definite for the bounded matrix G, then the linear feedback control uof is optimal and transfers the nonlinear systems from any initial state to the final state e(∞)=0 (3.15) minimizing the functional J = ∞∫ 0 (eTQ̃e+uTofRuof ) dt (3.16) The control uof can be found by solving the equation uof =−Ke (3.17) where K=R−1BTP and the symmetricmatrix P canbedetermined fromthe algebraicRiccati equation given by PA+ATP−PBR−1BTP+Q=0 (3.18) Defining the desired trajectory as the periodic orbit in equation (3.7), which was obtained from theMultiple Scales method, and considering the matrices A and B, given by A= [ 0 1 −1 −0.04 ] B= [ 0 1 ] (3.19) Choosing Q=104 [ 1 0 0 0.1 ] R= [0.01] (3.20) and solving equation (3.18), results P= [ 3193.7138 9.99 9.99 3.1933 ] K= [ 999 319 ] (3.21) Replacing equation (3.21) into equation (3.17), the optimal feedback control law is given by uof =−999e1−319e2 =−999(x1− x̃1)−319(x2 − x̃2) (3.22) For the optimal control verification, the function in equation (3.14) is numerically calculated by L(T)= eTQ̃e. The sufficient criterion to guarantee that the control signal in equation (3.22) is optimal is that L(T) is positive definite (Rafikov et al., 2008). Figure 5 shows theapplication of theOLFCto theTM-AFMproblem.Asobserved inFig. 4b, the function L(T) is positive definite for e→ 0. It can be concluded that the control signal of equation (3.22) is optimal andmoves the system of equation (3.8) to the desired orbit given by equation (3.7) in less than two seconds, as observed in Fig. 4a. TM-AFM nonlinear motion control with robustness analysis... 101 3.3. State-Dependent Riccati Equation (SDRE) The SDRE strategy is an effective algorithm for synthesizing nonlinear feedback controls by allowing the nonlinearities in the system states and, additionally, offering great design flexibility through state-dependent weighting matrices (Tusset and Balthazar, 2012; Tusset et al., 2012a; Mracek and Cloutier, 1998). 3.3.1. Application of SDRE control The dynamic system defined by Eq. (2.9) can be parameterized in first order equations and written in the state-dependent coefficient (SDC) and the non state-dependent coefficient in the following way (Tusset et al., 2012c) X′ =A(X)X+BUs+Φ(X,T) (3.23) where X = [x1,x2] T is the time dependent state vector and X′ ∈ R2 is the derivative of the state vector. Us = usf + ũs, where usf is the feedback control, ũs is the feedforward control and Φ(X,T) is the nonlinearities vector. The initial and final conditions are given by X(t0) =X0, X(∞)=0, respectively. Fig. 5. (a) Deviations from the desired orbit; (b) L(T) calculated in optimal trajectory; (c) displacement of TM-AFMwith and without OLFC control Considering F(x1) as the expansion F(x1)= −dx1(6a5+15a4x1+20a3x21+15a 2x31+6ax 4 1+x 5 1) (a+x1)8 − −h+da6 (a+x1)8 (3.24) the coefficient dependentmatrices are given by A(x1)=   0 1 −1− dx1(6a 5+15a4x1+20a 3x21+15a 2x31+6ax 4 1+x 5 1) (a+x1)8 −b   Φ(x1,T)=    0 − −h+da6 (a+x1)8 +f cos(wT)    B= [ 0 1 ] (3.25) 102 J.M. Balthazar et al. Astate feedback instead of an output feedback is adopted to enhance the control performan- ce. The cost function for the regulator problem is given by J = ∞∫ t0 [XTQ(X)X+uTsfR(X)usf ] dT (3.26) where Q is semi-positive-definite matrix and R positive definite. Assuming full state feedback, the control law is given by usf =−R−1(X)BT(X)P(X)X (3.27) The estate-dependent Riccati equation to obtain P(X) is given by AT(X)P(X)+P(X)A(X)−P(X)B(X)R−1(X)BT(X)P(X)+Q(X)=0 (3.28) Defining the feedforward control as ũs =−Φ(x1,T)= −h+da6 (a+x1)8 −f cosT (3.29) and replacing equation (3.29) into equation (3.23), the system of equation (3.23) can be repre- sented as in equation (3.11) in the form of deviations e′ =Ae+Busf (3.30) In Fig. 6, the application of the SDRE control technique is shown considering the matrices in Eq. (3.20). As it can be observed in Fig. 6, the SDRE technique drives the system to the desired periodic orbit of equation (3.7) in less than 2 seconds (see Fig. 6a). Fig. 6. (a) Deviations from the desired orbit (3.11); (b) isplacement of AFMwith and without SDRE control 4. The effect of parameter uncertainties Since mathematical models are subject to parametric inaccuracies or bad estimatives, in this Section, the robustness of theOLFCandSDREstrategies is tested considering the control signal determination sensitivity to parameter uncertainties (Shirazi et al., 2011) and, in order to test the ability of controllers, the perturbed control laws are applied to the nominal dynamic system. In Section 4.1, the robustness of both control strategies is tested for parameter uncertainties in the determination of the control signal. On the other hand, in Section 4.2, the control signal determination considers the uncertainties in each parameter isolated. TM-AFM nonlinear motion control with robustness analysis... 103 4.1. Control signal determination with parameter uncertainties and measurement noise Inorder to consider the effect of parameter uncertainties on theperformanceof the controller, the real parameters of the systemare supposed tohave uncertainties as follows: f =1.6+0.8r(t), b =0.032+0.016r(t), a =0.72+0.36r(t), d =(3.2+1.6r(t))/27 and h =0.288 ·10−5+0.144 · 10−5r(t), where r(t) are normally distributed random functions, as proposed in Shirazi et al. (2011). The simulation results are shown in Fig. 7. Fig. 7. Error parameters uncertainties; (a) OLFC subject to parameter uncertaintes, (b) OLFCwith parameter uncetainties for 3.5¬ T ¬ 3.75, (c) SDREwith parameter uncertainties for noise function: 0.02sinT , (d) SDREwith parameter uncertainties for 5¬ T ¬ 40 (steady state) It can be seen in Figs. 7a and 7b that the OLFC error increase over time when the control signal determination is subject to parameters uncertainties. Additionally, Figs. 7c and 7d show the robustness of the SDRE technique when the parameters have random uncertainties and measurement noise. 4.2. Control with uncertainty Considering that the control signal U is subject to parametric errors, the influence of each indivual parameter in the robustness of the control system is analyzed considering a random error of 20% of the nominal value. The sensitivity of the control to each parameter individually can be seen in Fig. 8. Figures 8a and 8b show that the OLFC error is unstable to uncertainties in the parameter a. In Figs. 8c and 8d, it can be seen that, except for the parameter a, the error is never larger than 2.0 ·10−6. In that case, the SDRE strategy is more robust in the presence of parameter uncertainties in the control signal determination. 5. Conclusion In order to suppress the chaotic behavior keeping the system in a controlled periodic orbit obtained fromMultipleScalesmethod, two control strategieswere considered,namely: theOLFC and SDRE strategies. Considering the results, it can be concluded that both controls are robust 104 J.M. Balthazar et al. Fig. 8. Errors for individual parameter uncertainties in the control signal determination; (a) OLFC individual parameter uncertainties, (b) OLFC individual parameter uncertainties for 3.5¬ T ¬ 3.8, (c) SDRE individual parameter uncertainties, (d) SDRE individual parameter uncertainties for 2¬ T ¬ 10 to parametric errors except for the parameter a where the error increases over time for the OLFC. Additionally, the SDRE strategy appears to have better robustness performance, even for theparameter a.TheOLFC is therefore not indicated in the case of errors in theparameter a since the error becomes unstable, as shown in Figs. 7a,b, and 8a,b. Acknowledgements The authors would like to acknowledge Conselho Nacional de Desenvolvimento Cient́ıfico – CNPq, São PauloResearch Foundation – FAPESP (grant: 2013/04101-6), and Coordenação de aperfeiçoamento de Pessoal de ńıvel superior – CAPES, for the financial support. References 1. Ashhab M., Salapaka M.V., Dahleh M., Mezic I., 1999a, Dynamical analysis and control of microcantilevers,Automatica, 35, 10, 1663-1670 2. Ashhab M., Salapaka M., Dahleh M., Mezić I., 1999b, Melnikov-based dynamical analysis of microcantilevers in scanning probemicroscopy,Nonlinear Dynamics, 20, 197-220 3. Balthazar J.M., TussetA.M., Souza S.L.T.D., BuenoA.M., 2013,Microcantilever chaotic motion suppression in tapping mode atomic force microscope, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 227, 8, 1730-1741 4. Bhushan B., 2004, Springer Handbook of Nanotechnology, Springer, Berlin 5. 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