An Empirical Model for Dynamic Friction in Microfabricated Linear Microball Bearings

Transcription

1 25 American Control Conference June 8-, 25. Portland, OR, USA TB6. An Empirical Model for Dynamic Friction in Microfabricated Linear Microball Bearings Xiaobo Tan, Alireza Modafe, and Reza Godssi Abstract Te frictional dynamics of microball bearings needs to be understood for effective design and control of micromacines using suc bearings. In te prior wor by te autors and coworers, a vision-based, non-intrusive measurement metod was developed for caracterizing te microtribological beavior of linear microball bearings. In tis paper a novel sceme is proposed for computing te frictional force and te relative velocity between bearing elements using noisy position information extracted from te video. It enables one to reveal essential frictional caracteristics despite te low capture rate (3 frames/second) of te camera. Experimental and numerical results sow tat te dynamic friction beavior is captured by a memoryless viscous model. I. INTRODUCTION Te field of microelectromecanical systems (MEMS) as witnessed rapid advances over te past two decades, and te trend of miniaturization could revolutionize virtually all aspects of our life []. Tis presents unique opportunities and callenges in modeling and control designs, since dominant players in system dynamics differ in micro/nano and macro domains [2]. Many pysical penomena are important yet not well understood at micro and nano scales, a well-nown example of wic is friction. In order to provide low-friction and robust support in micromacines, Godssi et al proposed a linear microball bearing structure [3] consisting of microfabricated V-grooves and commercially available stainless-steel microballs. Understanding of tribological beaviors of microball bearings is essential as it is directly related to te proper design and effective control of micromacines based on suc bearings. Experiments were conducted to measure te static coefficient of friction (COF) of a linear microball bearing [3]. Its dynamic COF was investigated wit a vision-based experimental setup [4], were a Coulomb friction model was used. Recently te autors and teir coworers upgraded te experimental system to capture motions of all bearing elements (te slider, te stator, and te microballs) wit infrared imaging. Tis led to caracterization of some important tribological beaviors suc as rolling vs slipping and effects of surface rougness [5]. However, a faitful matematical model was not available for te microball bearings since te Coulomb friction was a X. Tan is wit te Department of Electrical and Computer Engineering, Micigan State University, East Lansing, MI 48824, USA. xbtan@msu.edu A. Modafe and R. Godssi are wit te Institute for Systems Researc and te Department of Electrical & Computer Engineering, University of Maryland, College Par, MD 2742, USA. {modafe,godssi}@glue.umd.edu /5/$ AACC 2463 crude approximation, as evidenced by te scattering of data points on te friction vs. velocity plot [4]. Classical frictional models typically express te frictional force as a static function of te relative velocity between te contact surfaces, wic may include te static friction, Coulomb friction, Stribec friction, and viscous friction [6], [7], [8]. Te Dal model describes te friction in te stiction regime (also called presliding regime or microslip regime) in terms of te micro-displacement [9]. Te relationsip between te friction and te displacement can be ysteretic []. A dynamic friction model (called LuGre model) was presented by Canudas de Wit et al wit an internal state representing te average deflection of contacting asperities [], and was furter extended by oters [], [2]. Tis model demonstrated, among oter properties, te ysteresis between te friction and te sliding velocity for unidirectional sliding. Suc ysteresis was also reported for unidirectional, unsteady sliding velocities and modeled troug a time lag by Hess et al [3]. Similar ysteresis beavior was studied for a forced oscillator wit a compliant contact [4]. Te experimental results reported in [], [], [3], [4] were all based on macroscopic systems. Tis paper aims to model te dynamic friction in linear microball bearings. Te term dynamic friction (as opposed to te static friction) refers to te friction wen a relative motion between te slider and te stator is present. In te experiment te stator is fixed to a forced oscillator, and te motion of te slider is driven only by te friction. Hence te friction is directly lined to te acceleration of te slider, wic is determined based on successive video images. In general te bearing dynamics is very complex due to fabrication-related surface irregularities, ball-to-ball or ball-to-wall collisions, and oxide growt on contact surfaces [5], and te slider may occasionally demonstrate stic-slip, slow drift, or sudden impacts. However, te empasis of tis paper is on te frictional beavior wen te slider is (relatively) steadily sliding. Te reason is tat steady sliding is supposed to be te normal operating condition of tese bearings once te fabrication processes are refined. Due to te constraint of camera speed, te (relative) velocity regime is about [-.2,.2] m/s. Note tat tis limitation does not undermine te importance of te study ere since interesting friction-related dynamics typically taes place at low-velocities and during velocity reversals [5]. In particular, te friction beavior in tis regime will prove very important for some applications (e.g., sub-micron positioning) envisioned for micromotors based on te bearing.

2 In modeling te first question would be weter te friction-velocity relationsip sows ysteresis (memory) or is static. And ten one would lie to now wat exactly is te relationsip. In te vision-based measurement system bot te friction and te (relative) velocity are derived from te position information available troug te camera. Te camera captures 3 frames/second, and tis (relatively) slow capture rate presents two serious callenges. First te images are typically blurred leading to errors in position identification. Te second problem is tat, witout proper care, te derived velocities and acceleration will deviate significantly from te true values. In tis paper a tree-stage data processing algoritm is presented to best extract te information of interest: a) approximation of derivatives (acceleration, velocities) using te center-difference metod to minimize te pase error; b) amplitude-based filtering to reduce measurement noises; and c) frequency-weigted compensation to reconstruct te derivatives. Bot numerical and experimental results sow tat te algoritm can effectively recover te derivative quantities based on te position snapsots. A muc cleaner friction-velocity plot obtained troug te algoritm sows tat, (at least for te velocity range considered in te paper) te friction is a static function of te velocity. At low relative velocities te friction is approximately linear in te relative velocity tus demonstrating a viscous beavior; as te relative velocity increases, te friction approaces saturation. A Langevin function is used to approximate te relationsip. Te predictions based on te model agree well wit te experimental results. Te remainder of te paper is organized as follows. Te microball bearing and te experimental setup are described in Section II. Te tree-stage algoritm is presented in Section III. Identification of te friction model is discussed in Section IV. Finally conclusions are provided in Section V. tese mars, and position information of te slider and te stator can be extracted troug image processing (National Instruments, IMAQ Vision Builder TM ). Wen switced to te nigt-sot mode (infrared), te camera can trac te slider, te stator, and te microballs simultaneously [5]. Fig.. Scematic of a linear microball bearing [4]. Positioning slide Microball bearing Fig. 2. Camera Servo system Vision-based experimental setup. II. MICROBALL BEARINGS AND EXPERIMENTAL SETUP A scematic of a linear microball bearing is sown in Fig.. Te bearing consists of two silicon plates (slider and stator) and stainless-steel microballs of diameter 285 µm. Two parallel V-grooves, wic ouse te balls, are etced on te plates using potassium ydroxide (KOH). Fig. 2 sows te picture of te experimental setup. Te stator of te bearing is mounted on an oscillating platform driven by a servomotor troug a cran-slider mecanism (for more details, see [4]). Te platform (and ence te stator) moves only in te direction of te underneat guiding rails, so does te slider of te bearing. Te position x(t) of te stator is approximately x stat (t)=x stat sin(ωt), () were ω is te angular velocity of te servomotor, and te oscillating amplitude X stat can be adjusted. Distinct mars are placed and also etced on te stator and slider of te bearing, respectively. A CCD camera (Sony DCR- TRV22) wit a 24 magnification lens captures motions of 2464 III. THREE-STAGE DATA PROCESSING ALGORITHM Te equation of motion for te slider is: M slid (t)=f fric (t), (2) were M slid denotes te mass of te slider, = ẍslid (t) denotes its acceleration, and F fric is te friction between te stator and te slider. It is clear tat te friction is equal to normalized by M slid. Te slider experiences periodic motion during steady sliding. To determine weter tere is ysteresis or oter memory effect in te friction vs. te relative velocity relationsip, one can cec te pase difference between and v rel, were v rel denotes te relative velocity: v rel (t)=v slid (t) v stat (t)=ẋ slid (t) ẋ stat (t). If te pase difference is a multiple of π, would be a static (memoryless) function of v rel, and it would be a dynamic function of v rel oterwise. Note tat due to te nonlinearity of friction, eac of, v slid, v rel, and will typically contain more tan one frequency components.

3 Terefore, one needs to loo at teir fundamental frequency components wen calculating te pase difference. A callenging problem ere is tat all derivative quantities, (t),v slid (t),v stat (t) and v rel (t), can only be obtained troug finite differencing of noisy position snapsots, [n] and x stat [n] (note in practice, x stat [n] is also extracted troug image processing instead of being calculated from ()). Witout proper care, suc differencing operations may introduce spurious pase sift in addition to te usual approximation error. It is sown next tat te widely used forward-difference metod is inferior to te centerdifference metod in terms of pase-preservation. A. Comparison of Finite-Difference Metods Te analysis is conducted for a generic sinusoidal function x(t)=xsin(ωt + φ). For signals wit more tan one frequency component, te analysis olds for eac component. Calculate v(t) = ẋ(t)=ωxcos(ωt + φ), (3) a(t) = ẍ(t)= ω 2 X sin(ωt + φ). (4) Sample x(t) wit time step, and denote x[n] = x(n). One is ten interested in approximating {v(n)} or {a(n)} based on te sequence {x[n]}. Note tat to connect tis to te microball bearing problem, x(t) plays te role of (t) in calculation of te slider acceleration, and it plays te role of (t) x stat (t) in calculation of te relative velocity. Two metods, forward-difference and center-difference, are compared for approximation of bot te velocity (3) and te acceleration (4). Under te forward-difference metod, te estimates are x[n] x[n ] v f [n] =, a f [n] = v f [n] v f [n ] x[n]+x[n 2] 2x[n ] = 2. Under te center-difference metod, te estimates are x[n + ] x[n ] v c [n] =, (5) a c [n] = 2 x[n + ]+x[n ] 2x[n] 2. (6) ) Forward-difference metod: Let t = n and θ = ωt + φ. It ten follows tat were m = sin(ω/2) ω/2, and φ = tan ( cos(ω) sin(ω) ). Comparing wit (3), one can see tat not only te magnitude of v f [n] is scaled from tat of v(n) by m, but also its pase differs from tat of v(n) by φ. Similarly, one can compute a f [n]: a f [n]= x[n]+x[n 2] 2x[n ] 2 = X [sin(θ)+sin(θ 2ω) 2sin(θ ω)]. (8) 2 Write sin(θ 2ω) =sin(θ)cos(2ω) cos(θ)sin(2ω), write sin(θ ω) in a similar fasion, and plug tem into (8). After some manipulations, one gets a f [n]= m 2 ω 2 X sin(ωt + φ φ 2 ), (9) were m 2 =( sin(ω/2) ω/2 ) 2, and φ 2 = ω. Terefore, te magnitude of a f [n] is scaled from tat of a(n) (compare (4)) by m 2, and its pase differs from tat of a(n) by φ 2. 2) Center-difference metod: For te center-difference metod, v c [n] = x[n + ] x[n ] 2 = X [sin(θ + ω) sin(θ ω)]. 2 It can be sown tat v c [n]= sin(ω) ω and v c [n] is in pase wit v(n). Furtermore, one calculates a c [n] = ωx sin(ωt + φ)= sin(ω) v(n), () ω x[n + ]+x[n ] 2x[n] 2 = X [sin(θ + ω)+sin(θ ω) 2sin(θ)] 2 2( cos(ω))x = 2 sin(ωt + φ) = ( sin(ω/2) ) 2 ω 2 X sin(ωt + φ) ω/2 = ( sin(ω/2) ) 2 a(n), () ω/2 and again, a c [n] is in pase wit a(n). From te above discussions, altoug bot difference metods introduce magnitude scaling, a signal generated troug te center-difference metod carries te correct pase wile te one generated troug te forwarddifference metod does not. v f [n]= (x[n] x[n ]) = [X sin(θ) X sin(θ ω)] = X [cos(θ)sin(ω)+sin(θ)( cos(ω))] B. Te Data Processing Algoritm 2( cos(ω))x = cos(θ φ ) Te analysis in Subsection III-A is based on te assumption tat x[n] is sampled from a noiseless sinusoidal = m ωx cos(θ φ )=m ωx cos(ωt + φ φ ), (7) signal x(t) of single frequency. Tis is typically not te One sould tin of tis as only an analogy - true and x stat can ave case wit te position data extracted from video images. multiple frequency components, as one will see in Section IV. Te treestage algoritm to be presented in Subsection III-B deals wit multiple- Hence te proposed data processing algoritm consists of frequency case. Te same remars will apply for te simulation example tree stages (Fig. 3): a) calculating a crude estimate of in Subsection III-B, were two sinusoidal signals are used. velocity or acceleration (depending on te context) using 2465

4 te center-difference metod; b) performing Fast Fourier Transform (FFT) and ten filtering based on te amplitude of individual frequency components; and c) reconstructing te velocity or acceleration by amplitude compensation for remaining frequency components. x[ n ] Center- Diff. Fig. 3. vc[ n] or a [ n] c FFT & Filtering vˆ c[ n] or aˆ [ n] c Ampl. Scaling Te tree-stage data processing algoritm. vc[ n] or a [ n] ) Stage. Center-difference: Te estimate v c [n] (or a c [n]) is obtained troug te center-difference metod (5) (or (6)). Note tat eac frequency component of v c [n] (or a c [n]) carries te same pase as te original v(n) (or a(n)) but te amplitude undergoes certain scaling. 2) Stage 2. FFT and filtering: FFT is performed for two purposes: filtering and preparing for amplitude compensation (Stage 3). It is tempting to implement low-pass filtering for noise reduction. However, low-pass filtering is not appropriate ere since te signals may carry igorder armonics due to te nonlinear frictional dynamics. Filtering out all ig-frequency components, terefore, can distort te true dynamics. Furtermore, plain low-pass filtering introduces additional pase sift for signals. In tis paper amplitude-based filtering is used instead. Let M max be te maximum amplitude among all frequency components. Ten a frequency component will be eliminated if its amplitude M ε M max, were ε is some small constant. 3) Stage 3. Amplitude compensation: Let be te index of remaining frequency components after filtering. Ten (see Fig. 3) ˆv c [n] = V sin(ω n + α ), â c [n] = A sin(ω n + β ). Based on () and (), amplitude scaling is performed to compensate for te frequency-dependent amplitude distortion introduced in te difference metods: ω v c [n] = sin(ω ) V sin(ω n + α ), ω /2 ā c [n] = ( sin(ω /2) )2 A sin(ω n + β ). Simulation is conducted to compare te tree-stage algoritms wit te forward-difference algoritms. For illustration, two sinusoidal trajectories are specified first: x (t)=x cos(ωt + π), x 2 (t)=x 2 sin(ωt), were x (t) mimics te relative position of te slider to te stator, and x 2 (t) mimics te slider position (again, it is not implied ere tat true or x stat is purely sinusoidal; c 2466 refer to te footnote in Subsection III-A). One would lie to calculate v [n] (analogy of relative velocity) and a 2 [n] (analogy of acceleration) based on snapsots x [n] and x 2 [n] wit sampling time. Note tat x (t) and x 2 (t) are specified in suc a way tat tere is a pase difference of π between v (t) and a 2 (t) (ence no loop in te a 2 (t) vs. v (t) plot). In simulation, X =., X 2 = 3. 4, ω = 6π, =.33, and te filtering level ε =.2 (altoug x (t) and x 2 (t) are purely sinusoidal, one will get extra frequency components in FFT due to te finite number of data points). Fig. 4 compares te a 2 vs. v plots obtained troug te tree-stage algoritm and te forward difference algoritm, respectively. It is clear tat te forward-difference algoritm leads to a spurious loop, introduced by te pase errors in approximating v and a 2. Te pase difference between a 2 and v in Fig. 4 (a) is numerically calculated to be 2.83, wic is consistent wit te analytical value obtained troug (7) and (9). On te oter and, te tree-stage algoritm is able to sow te pase difference of π between a 2 and v correctly (Fig. 4). a 2 a (a) Finite difference metod v (b) Tree stage algoritm v Fig. 4. Comparison of te tree-stage algoritm wit te forwarddifference algoritm. Fig. 5 demonstrates te difference te amplitude-scaling step maes. Te top plot sows te error in te reconstructed v (wit respect to te true v (t)) witout te scaling step, wile te bottom one sows te error wen scaling is adopted. One sees tat te approximation accuracy is significantly improved by amplitude compensation. IV. IDENTIFICATION OF FRICTION MODEL Te tree-stage algoritm is ten used to process te actual position data collected from te microball bearing troug imaging. Fig. 6 sows te trajectories of te stator, te slider, and teir relative positions, respectively. Note tat altoug te slider experiences slow, random drift due to fabrication-related surface irregularities, its periodic oscillation is dominant. Te stator oscillates wit frequency 2.95 Hz and amplitude.2 mm.

5 Error in v Error in v x stat x stat.5 x (a) Witout amplitude scaling x (b) Wit amplitude scaling Time (s) Fig. 5. Effect of amplitude compensation Time (s) Fig. 6. Experimentally measured trajectories of te slider (top), te stator (center), and teir relative position (bottom). Fig. 7 sows te relationsips between te slider acceleration (namely, friction normalized by te slider mass) and te relative velocity v slid v stat wen tree different scemes are used to compute tese quantities. Direct application of te forward-difference metod results in a noise-corrupted loop (Fig. 7(a)). If low-pass filtering (cutoff frequency 6 Hz) is used followed by te forward-difference, te loop trend is more evident (Fig. 7(b)). However, tere is no loop in te acceleration vs. relative velocity plot wen te tree-stage data processing algoritm (ε =.8) is applied (Fig. 7). Te pase difference between te fundamental frequency components of and v slid v stat is in Fig. 7(a), in Fig. 7(b), and 3.57 (very close to π) in Fig. 7 (c). Since analysis in Section III as sown tat te tree-stage algoritm conserves te pase of te original signal, one concludes tat te friction is well approximated by a static but nonlinear function of te relative velocity, 2467 and te ysteresis loops in Fig. 7(a) and (b) are artifacts of data processing. From Fig. 7(c), te friction is almost linear wit respect to te relative velocity wen te latter is low; and as te magnitude of te relative velocity increases, te friction approaces a saturation level. Hence a modified Langevin function L ( ) is proposed to model te friction were for any z, F fric (t) = M slid L (v rel (t)), (2) L (z) = A [ αz eαz + e αz e αz ]. (3) e αz To identify te parameters A and α, one calculates dl (z) 4 = αa [ dz (e αz e αz ) 2 ]. (4) (αz) 2 By evaluating te derivatives at v rel = and v rel =.5 in Fig. 7(c) and solving (4), te parameters are determined to be A =.2, α = Fig. 8 plots te friction vs. velocity relationsip obtained troug tis model, and it matces te experimental data well (Fig. 8). Simulation is furter conducted to validate te above model. Combining (), (2), and (3), te equation of motion (2) is integrated using te fourt order Runge-Kutta metod wit a time step of. second. Te oscillation amplitude of te stator X stat =.2mm and ω = 2π f wit f = 2.95Hz, bot estimated from te experimental data. Fig. 9 sows te simulated trajectories of te slider, te stator, and teir relative position. Note te absolute position values are of little relevance since te reference points are defined arbitrarily. Te oscillation amplitude of te slider in Fig. 9 is.2 mm, wile te amplitude of te fundamental frequency component of [n] in te experiment (Fig. 6) was.26 mm. Furtermore, te pase difference between and x stat in Fig. 9 is.384 radians, comparing to.46 radians measured in te experiment (Fig. 6). Tese agreements strongly support tat te proposed model is capable of capturing te friction beavior. V. CONCLUSIONS Tis paper was focused on te modeling of dynamic friction in MEMS-based linear microball bearings. Te only available information was te noisy position data of te slider and te stator extracted from videos captured at a low rate. To recover velocity and acceleration wit ig fidelity from tese noisy data, a tree-stage data processing algoritm was proposed. Te signals (derivatives of position) generated troug tis algoritm preserve te pase as well as te amplitude of te true signals. An enabling assumption of te algoritm is tat signals are periodic, wic olds true wen te slider oscillates and slides steadily. Te friction - (relative) velocity plot obtained troug te data-processing algoritm sowed tat ysteresis or oter memory effects, if existing at all, were insignificant at least for te velocity range examined in te paper. Te

6 (a) Forward difference (b) Low pass filtering & forward difference (c) 3 stage algoritm v v (m/s) slid stat x stat x stat Time (s) Fig. 9. Simulated trajectories of te slider, te stator, and teir relative positions. Fig. 7. Slider acceleration (friction) vs. relative velocity under different data processing scemes. (a) Forward-difference; (b) Low-pass filtering followed by forward-difference; (c) Tree-stage algoritm Fig. 8. Langevin approx. Experimental data v slid v stat (m/s) Comparison of te model wit te experimental data. friction demonstrates a viscous feature at low velocities, and ten approaces saturation. Suc beavior was captured well troug te Langevin function. Simulation predictions based on tis model acieved good agreement wit experimental measurements. Note tat te finding is interesting in tat classical friction models are typically discontinuous at zero velocity [5], [8]. Future wor involves understanding of te observed friction beavior. It is of particular interest to study te viscoelastic dynamics at te contact surfaces during te rolling motion of microballs. ACKNOWLEDGMENT Tis wor was supported in part by te NSF grant ECS Te first autor would lie to acnowledge useful discussions wit Professors P. S. Krisnaprasad, Ben Sapiro, and Brian Feeny. REFERENCES [] M. J. Madou, Fundamentals of Microfabrication: Te Science of Miniaturization. Boca Raton, FL: CRC Press, [2] W. S. N. Trimmer, Microrobots and micromecanical systems, Sensors and Actuators, vol. 9, no. 3, pp , 989. [3] R. Godssi, D. D. Denton, A. A. Seireg, and B. Howland, Rolling friction in a linear microstructure, Journal of Vacuum Science and Tecnology A, vol., pp , 993. [4] T.-W. Lin, A. Modafe, B. Sapiro, and R. Godssi, Caracterization of dynamic friction in MEMS-based microball bearings, IEEE Transactions on Instrumentation and Measurement, vol. 53, no. 3, pp , 24. [5] X. Tan, A. Modafe, R. Hergert, N. Galicecian, B. Sapiro, J. S. Baras, and R. Godssi, Vision-based microtribological caracterization of linear microball bearings, in Proceedings of ASME/STLE International Joint Tribology Conference, Long Beac, CA, 24, paper TRIB [6] B. Armstrong-Helouvry, Control of Macines wit Friction. Boston, MA: Kluwer Academic Press, 99. [7] N. Leonard and P. S. Krisnaprasad, Adaptive friction compensation for bidirectional low velocity tracing, in Proceedings of te 3st IEEE Conference on Decision and Control, Tucson, AZ, 992, pp [8] B. Armstrong-Helouvry, P. Dupont, and C. C. de Wit, A survey of models, analysis tools and compensation metods for te control of macines wit friction, Automatica, vol. 3, no. 7, pp , 994. [9] P. R. Dal, A solid friction model, Te Aerospace Corporation, El Segundo, CA, Tec. Rep. TOR-58(37-8), 968. [] J. Swevers, F. Al-Bender, C. G. Ganseman, and T. Prajogo, An integrated friction model structure wit improved presliding beavior for accurate friction compensation, IEEE Transactions on Automatic Control, vol. 45, no. 4, pp , 2. [] C. C. de Wit, H. Olsson, K. J. Astrom, and P. Liscinsy, A new model for control of systems wit friction, IEEE Transactions on Automatic Control, vol. 4, no. 3, pp , 995. [2] P. Dupont, V. Hayward, B. Armstrong, and F. Altpeter, Single state elasto-plastic friction models, IEEE Transactions on Automatic Control, vol. 47, no. 5, pp , 22. [3] D. P. Hess and A. Soom, Friction at a lubricated line contact operating at oscillating sliding velocities, Journal of Tribology, vol. 2, pp , 99. [4] J. W. Liang and B. F. Feeny, Dynamical friction beavior in a forced oscillator wit a compliant contact, Journal of Applied Mecanics, Transactions of te ASME, vol. 65, pp , 998. [5] N. Leonard, An investigation of control strategies for friction compensation, Master s tesis, University of Maryland, 99.