Patent Application: US-77085704-A

Abstract:
methods and apparatus are described for control of friction at the nanoscale . a method of controlling frictional dynamics of a plurality of particles using non - lipschitzian control includes determining an attribute of the plurality of particles ; calculating an attribute deviation by subtracting the attribute of the plurality of particles from a target attribute ; calculating a non - lipschitzian feedback control term by raising the attribute deviation to a fractionary power ξ =/ where n = 1 , 2 , 3 . . . and m = 0 , 1 , 2 , 3 . . . , with m strictly less than n and then multiplying by a control amplitude ; and imposing the non - lipschitzian feedback control term globally on each of the plurality of particles ; imposing causes a subsequent magnitude of the attribute deviation to be reduced .

Description:
the invention and the various features and advantageous details thereof are explained more fully with reference to the nonlimiting embodiments that are illustrated in the accompanying drawings and detailed in the following description . descriptions of well known starting materials , processing techniques , components and equipment are omitted so as not to unnecessarily obscure the invention in detail . it should be understood , however , that the detailed description and the specific examples , while indicating preferred embodiments of the invention , are given by way of illustration only and not by way of limitation . various substitutions , modifications , additions and / or rearrangements within the spirit and / or scope of the underlying inventive concept will become apparent to those skilled in the art from this disclosure . within this application several publications are referenced by arabic numerals within brackets . full citations for these , and other , publications may be found at the end of the specification immediately preceding the claims after the section heading references . the disclosures of all these publications in their entireties are hereby expressly incorporated by reference herein for the purpose of indicating the background of the invention and illustrating the state of the art . the invention can include a method ( and / or apparatus based on the method ) to control a dynamic attribute of a plurality of structures toward a pre - assigned ( pre - determined ) value or variable behavior of that attribute . the control of the dynamic attribute can be based on the concepts of non - lipschitzian dynamics and the use of a non - lipschitzian global feedback control term . optionally , the invention can include maintaining the control until the deviation is reduced to zero whereupon the target has been reached . in a preferred embodiment , the invention can include a method ( and / or apparatus based on the method ) to control sliding and frictional properties ( such as friction coefficient , friction force , sliding velocity , slip time ) of a plurality ( e . g ., array ) of atoms and / or molecules towards a pre - assigned ( pre - determined ) value of a target ( average sliding velocity , slip time , friction coefficient and friction force ). the invention can also include a method ( and / or apparatus based on the method ) to control shear forces and static forces , viscosity , and adhesion forces towards a pre - assigned value of a target ( shear and static forces , viscosity , and adhesion forces ). the invention can also include a method ( and / or apparatus based on the method to control sliding trajectory , speed , direction and diffusion of atomic and molecular chains and polymers sliding on surfaces towards a pre - assigned value of a target ( sliding trajectory , speed direction and diffusion coefficient ). implementation of the non - lipschitzian friction control technique is applicable but not limited for slip time and velocity control in a quartz micro balance apparatus , friction coefficient and friction force control in an atomic force microscope , and friction forces , loss and elastic moduli control in a surface force apparatus . implementation of non - lipschitzian control algorithm can be achieved either through imposing controlled vibrations of the sliding surfaces and / or the afm tip ( normal and / or in - plane ) or electromechanical , electro - optical , or optical excitations applied to the sliding system and / or the lubricant according to the proposed algorithm . implementation of control algorithm can be also achieved by imposing controlled vibrations of the sliding surfaces with a surface force apparatus , a quartz microbalance and / or using cantilevers and / or cantilever arrays . in addition , electromechanical , electro - optical , and optical control can be utilized in conjunction with ( applicable for ) all the previously described friction measurement apparatuses . as in the generic case , this control can be based on the concepts of non - lipschitzian dynamics and the use of a ( terminal attractor based ) non - lipschitzian global feedback control term . extensive numerical simulations , some of which are described below , have actually proven the robustness , efficiency , and convenience of the invention applied in the context of controlling friction . non - lipschitzian ( terminal attractor based ) global control feedback is an important aspect of the invention and provides several advantages . first , the presence of a terminal attractor in the control term provides robustness and ensures very fast approach to target . second , the global control turns out to be more efficient and easier to implement compared to non - global control . fast time scales and ease of implementation make the invention a very suitable tool for phenomena in nanoscale systems where accessibility is an issue ( as in friction , for instance ). however , the applicability of the invention is quite general . this preferred embodiment of the invention can include an algorithm to control friction of sliding nano - arrays . this algorithmic control can be based on the concept of a terminal attractor and is global in that : ( i ) it can require only knowledge of the velocity of the center of mass and ( ii ) it can be applied globally to the all members of the plurality of particles ( e . g ., the whole array ). the inventors have already demonstrated the efficiency and robustness of the control by reaching a broad spectrum of target velocities — both close to or far from natural attractors — in very short transient times . extensive numerical simulations have been performed on arrays of different sizes ( 3 & lt ; n & lt ; 256 ) in order to verify that size effects are not critical for the inventive control . the numerical and graphical results of some of these numerical simulations are presented in fig1 a - 1d and fig2 for a typical one - dimensional nano - array of n = 15 particles . in this preferred embodiment , the velocity of the particles ( e . g ., average sliding , center of mass velocity of an array of nanoparticles ) can be measured using a quartz crystal microbalance . the control term can be imposed on each of the plurality of particles via an optical pulse ( e . g ., from a tuned laser ). the optical pulse can define a spot ( having a size and flux density ) that is sufficiently large and uniform to evenly impose the control term on each of the plurality of particles . in this case , the optical pulse intensity and its duration should be controlled electronically via the control term which can be provided as an input signal to the electronics . a plurality of such optical pulses over time can in - turn define a duty cycle . in an alternative embodiment , the invention can include a method ( and / or apparatus based on the method ) to control intensity , phase , ( e . g ., synchronized array ) of lasers towards a pre - assigned ( pre - determined ) value of a target intensity and / or target phases . again , this control can be based on the concepts of non - lipschitzian dynamics and the use of a non - lipschitzian ( terminal attractor based ) global feedback control term . in this alternative embodiment , the intensity and / or phases of the lasers can be measured using a charge coupled device . the control term can be imposed on each of the plurality of lasers via electronics or optics that are provided with the control term as an input signal to the electronics . it is important to appreciate that the invention can address fundamental issues related to targeting and control of an attribute of a dynamic system ( e . g ., friction in nanoscale driven nonlinear particle arrays , synchronization of laser arrays , etc . ), by using the global feedback control approach that is based on the properties of terminal attractors . it should be appreciated that the invention can include the application of terminal attractors to second order systems ( e . g ., friction control , laser synchronization , etc .). it should also be appreciated that the invention can include the feed back of such a non - lipschitzian feedback control in the context of a second order system , simultaneously , into all state equations , thereby defining a non - lipschitzian feedback global feedback control . when applying the control to the nano - array , the inventors &# 39 ; objectives were to : ( i ) provide the ability to reach a targeted value of the average sliding velocity using only small values of the control ; ( ii ) significantly reduce the transient time needed to reach the desired behavior . to that effect , the invention can include a global feedback control algorithm that uses the concept of a terminal attractor , which is usually associated to non - lipschitzian dynamics . the equations of motion in the presence of the terminal attractor based control term c ( t ) read : { umlaut over ( φ )} j + γ { dot over ( φ )} j + sin ( φ j )= ƒ + κ ( φ j + 1 − 2φ j + φ j − 1 )+ c ( t ) ( 3 ) c ( t )= α ( v target − v cm ) ξ ( 4 ) is the non - lipschitzian control term based on the concept of terminal attractor . in equation ( 3 ), the first term on the left represent the an acceleration of a particle j , the second term on the left represents a velocity of the particle j , the third term on the left represents a position of the particle j , the first term on the right ƒ is a ( e . g ., ambient ) force applied to the particles , the second term on the right represents the interaction between the particle j and its two nearest neighbors j − 1 and j + 1 ( κ is a strength of interaction between a particle of interest and its two nearest neighbors ) and the third term on the right represents the non - lipschitzian feedback ( terminal attractor based ) control term . in equation ( 4 ), v cm = ( 1 / n ) ⁢ ∑ n = 1 n ⁢ ⁢ ϕ . n and represents the average ( e . g ., center of mass ) velocity of the plurality of particles , v target is the targeted ( pre - determined ) velocity ( e . g ., for the center of mass of the plurality of particles ), α is the control amplitude , ξ = 1 /( 2n + 1 ), and n = 1 , 2 , 3 . . . . more generically , the fractional power can be of the form ξ = 1 /( 2m + 1 )/( 2n + 1 ), where n = 1 , 2 , 3 . . . and m = 0 , 1 , 2 , 3 . . . , with m strictly less than n . preferred embodiments of the invention utilize the fractional power form where the numerator is 1 since these provide enhanced efficiency in practical dynamic implementations . while most dynamical systems of interest do satisfy the lipschitz condition , the terminal attractor dynamics that the inventors have discovered is so useful for controlling friction violates it by design . as a result , trajectories reach the terminal attractor in finite time . to illustrate this phenomenon , consider a simple example of a terminal attractor , namely the equation { dot over ( φ )}=− φ 1 / 7 . at the equilibrium point , φ = 0 , the lipschitz condition is violated , since ∂{ dot over ( φ )}/∂ φ =−( 1 / 7 ) φ − 6 / 7 tends to minus infinity as φ tends to zero . thus , the equilibrium point φ = 0 is an attractor with “ infinite ” local stability . p this is precisely the effect realized with the control term c ( t ). indeed : ⅆ c ⅆ v cm = - ( 1 / 7 ) ⁢ α ⁡ ( v target - v cm ) - 6 / 7 , ( 5 ) i . e ., dc / dv cm →−∞ as v cm → v target . it is important to note that the determination ( calculation ) of the non - lipschitzian feedback control term requires only knowledge of the average velocity of the plurality of particles ( e . g ., array ), which is an readily ( experimentally observable ) available quantity . it is also important to note that the non - lipschitzian feedback control term can be applied identically and concomitantly to all the particles ( e . g ., in the array ) upon which it acts as a uniform force proportional to ( v target − v cm ) ξ . to assess the performance of the invention for more “ realistic ” interaction potentials , the inventors replaced the linear interaction in equation ( 3 ) by the morse interaction : f j = γ β ⁢ { exp ⁡ [ - β ⁡ ( ϕ j + 1 - ϕ j ) ] - exp ⁡ [ - 2 ⁢ β ⁡ ( ϕ j + 1 - ϕ j ) ] } - γ β ⁢ { exp ⁡ [ - β ⁡ ( ϕ j - ϕ j - 1 ) ] - exp ⁡ [ - 2 ⁢ β ⁡ ( ϕ j - ϕ j - 1 ) ] } . the inventors &# 39 ; simulations indicate that the control algorithm remains robust and efficient . as already mentioned , the inventors also performed preliminary simulations for arrays as large as n = 256 . the outcome is comparable to the results presented here , which suggests that the invention remains efficient in systems larger than the atomic size . experimental results are presented in fig1 a - 1d and fig2 for ξ = 1 / 7 , but the invention performs equally well for other values such as 1 / 3 , 1 / 5 and 1 / 9 , 1 / 11 . . . . fig2 plots the center of mass velocity as a function of the maximum control amplitude α . the inventors chose three values of the target velocity , namely 0 . 1 ( bottom ), 1 . 0 ( middle ), and 3 . 0 ( top ). the triangles show the velocity of center of mass for control defined by equation 6 . all the parameters are the same as in fig1 and initial conditions were chosen randomly . the inventors performed extensive testing of the embodiment of the invention represented by ( equations 3 - 4 ) by choosing numerous values of the target velocity . at the target itself , the non - lipschitzian terminal attractor has “ infinite attraction power ”, which endows the invention with excellent efficiency and robustness , as illustrated in fig1 a - 1d for four values of the target velocity , namely : v target = 0 , 0 . 2 , 1 and 3 . referring to fig1 a - 1d , the bottom traces ( red color lines ) indicate the time series of the control ( equation 4 ), while the top traces ( blue color lines ) show the time series of the velocity of the center of mass . in all cases , the inventors reached and sustained the ( arbitrarily chosen ) target values for rather small values of the control . thus , fig1 a - 1d illustrate performance of the control algorithm . the inventors picked four values of the target velocities : v target = 0 ( fig1 a ), 0 . 5 ( fig1 b ), 1 . 0 ( fig1 c ), and 3 . 0 ( fig1 d ) for an n = 15 particle array . control was initiated at t = 2000 . in all of fig1 a - 1d , the top traces ( blue lines ) show time series of the center of mass velocities while the bottom traces ( red lines ) show the control . it is significant and important to note that in all cases , the desired behavior was achieved . the other parameters are : ƒ = 0 . 3 , γ = 0 . 1 , κ = 0 . 26 , and ξ = 1 / 7 . all the units are dimensionless and the initial conditions can be chosen randomly . the inventors applied the control at the time t = 2000 . all the results shown in fig1 a - 1d clearly indicate that with a very short transient time : convergence is very fast and the strength of the control is small . fig2 illustrates the performance of the algorithm for different values of the target velocities as a function of the parameter α ( see equation 3 ). the inventors chose random set of initial conditions for each value of the parameter α . indeed , fo r most target values the convergence to the target value is straightforward ( see upper and middle curves ). however , for a few values of v target , the dependence of the center of mass velocity , v cm on α turned out to be more irregular . these are the cases where the targeted values of the average velocities are in close proximity with those values without control ( i . e . the desired behavior is in the vicinity of natural attractors of the uncontrolled array ). thus , the inventors modified the control as follows : c ( t )= α ( v target − v cm ) ξ − β ( v av − v cm ) ξ sgn [( v av − v cm )( v cm − v target )] h [ r −| v target − v av |]. ( 6 ) the second term in equation ( 6 ) represents a repelling from a possible natural attractor of system ( 3 ) that would deflect the trajectory towards itself and away from the target velocity , v target . in general , the natural attractors are not known analytically and / or a priori . their presence is indicated only by the behavior of the system and accounted for by v av , which is the “ running ” ( time dependent ) average velocity and represents the moving run - time average of v cm . h (.) denotes a heaviside function , defined as h ( z )= 1 for z & gt ; 0 , and h ( z )= 0 for z & lt ; 0 . the heaviside function can be further defined as h ( z )= 1 for z = 0 or as h ( z )= 0 for z = 0 . the role of this heaviside function is to activate the terminal repeller only within a neighborhood of radius r from the natural attractor . the radius r can be termed a threshold . the coefficients α and α are positive numbers that represent the weights of the non - lipschitzian attractor and repeller , respectively . the inventors applied the algorithm to the target the value of v = 0 . 1 ( see the bottom curve in fig2 ). here , the inventors are close to the static solution ( stable fixed point ) v = 0 . therefore , for some values of the control amplitude α , the outcome average velocity is v = 0 ( instead of the desired velocity v = 0 . 1 ). the triangles in fig2 shows the center of mass velocity as a function of a but using control defined in equation 6 . this control will repel the fixed point of v = 0 , therefore the inventors observe even better performance of the invention . a practical application of the invention that has value within the technological arts is as an efficient tool for controlling friction between a plurality of particles and a surface , between sliding surfaces and between sliding surfaces and a lubricant . the invention is applicable to quartz microbalance , atomic force microscope , and surface force apparatus - type experiments . the invention is also applicable to cantilevers and arrays of cantilevers , and in particular to micro - electro - mechanical systems ( mems ) where frictional contact and resulting wear are important factors in their design . the invention is also applicable to fast controls such as optical , or usage of micro / nano cantilevers . the invention is also applicable to implementations at time scales slower than the characteristic times of the dynamical system . indeed , numerical simulations show that the control can be applied at much slower rates , while still maintaining the average value of the velocity close to the target . the “ price ” of such relaxed requirements are that longer times are needed to reach the target and larger fluctuations from the averaged value are observed . another practical application of the invention is as a tool for synchronizing a plurality of lasers . there are virtually innumerable uses for the invention , all of which need not be detailed here . the terms a or an , as used herein , are defined as one or more than one . the term plurality , as used herein , is defined as two or more than two . the term another , as used herein , is defined as at least a second or more . the terms “ comprising ” ( comprises , comprised ), “ including ” ( includes , included ) and / or “ having ” ( has , had ), as used herein , are defined as open language ( i . e ., requiring what is thereafter recited , but open for the inclusion of unspecified procedure ( s ), structure ( s ) and / or ingredient ( s ) even in major amounts . the terms “ consisting ” ( consists , consisted ) and / or “ composing ” ( composes , composed ), as used herein , close the recited method , apparatus or composition to the inclusion of procedures , structure ( s ) and / or ingredient ( s ) other than those recited except for ancillaries , adjuncts and / or impurities ordinarily associated therewith . the recital of the term “ essentially ” along with the terms “ consisting ” or “ composing ” renders the recited method , apparatus and / or composition open only for the inclusion of unspecified procedure ( s ), structure ( s ) and / or ingredient ( s ) which do not materially affect the basic novel characteristics of the composition . the term coupled , as used herein , is defined as connected , although not necessarily directly , and not necessarily mechanically . the term approximately , as used herein , is defined as at least close to a given value ( e . g ., preferably within 10 % of , more preferably within 1 % of , and most preferably within 0 . 1 % of ). the term substantially , as used herein , is defined as largely but not necessarily wholly that which is specified . the term generally , as used herein , is defined as at least approaching a given state . the term deploying , as used herein , is defined as designing , building , shipping , installing and / or operating . the term means , as used herein , is defined as hardware , firmware and / or software for achieving a result . the term program or phrase computer program , as used herein , is defined as a sequence of instructions designed for execution on a computer system . a program , or computer program , may include a subroutine , a function , a procedure , an object method , an object implementation , an executable application , an applet , a servlet , a source code , an object code , a shared library / dynamic load library and / or other sequence of instructions designed for execution on a computer or computer system . all the disclosed embodiments of the invention disclosed herein can be made and used without undue experimentation in light of the disclosure . the invention is not limited by theoretical statements recited herein . although the best mode of carrying out the invention contemplated by the inventor ( s ) is disclosed , practice of the invention is not limited thereto . accordingly , it will be appreciated by those skilled in the art that the invention may be practiced otherwise than as specifically described herein . it will be manifest that various substitutions , modifications , additions and / or rearrangements of the features of the invention may be made without deviating from the spirit and / or scope of the underlying inventive concept . it is deemed that the spirit and / or scope of the underlying inventive concept as defined by the appended claims and their equivalents cover all such substitutions , modifications , additions and / or rearrangements . all the disclosed elements and features of each disclosed embodiment can be combined with , or substituted for , the disclosed elements and features of every other disclosed embodiment except where such elements or features are mutually exclusive . variation may be made in the steps or in the sequence of steps defining methods described herein . although the global feedback system described herein can be a separate module , it will be manifest that the global feedback system may be integrated into the meta - system with which it is associated . the appended claims are not to be interpreted as including means - plus - function limitations , unless such a limitation is explicitly recited in a given claim using the phrase ( s ) “ means for ” and / or “ step for .” subgeneric embodiments of the invention are delineated by the appended independent claims and their equivalents . specific embodiments of the invention are differentiated by the appended dependent claims and their equivalents . y . z . hu and s . granick , tribol . lett ., 5 , 81 ( 1998 ). h . fujita in proceedings ieee , tenth annual international workshop on micro electro mechanical systems , published by : ieee robotics and control division div ., new york , n . y . 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