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y^K 539.62
Contact mechanics modeling of piezo-actuated stick-slip microdrives
H.X. Nguyen, C. Edeler, S. Fatikow
University of Oldenburg, Oldenburg, 26121, Germany
This paper draws a line from early attempts of modeling stick-slip microdrives to open questions from today’s research. As a basis, it contains a collection of substantial investigations on piezo-actuated stick-slip microdrives for nanomanipulation purposes. Friction models showing special characteristics and their mathematical representations are reviewed. It is found that the working properties of stick-slip drives strongly depend on friction characteristics of the contact points between the guiding elements, which is known for years. However, numerous publications in the field of friction and remaining problems — which cannot be explained by known friction models — indicate that there is a demand for even more friction-related research.
Former attempts to model stick-slip drives are based on the so-called LuGre friction model, which is shortly presented. An empirical model called CEIM is also analyzed. It is an adaption of the elastoplastic model. The latter can cover not only the phenomenon “0-amplitude” (described by the authors in recent publications), but also stick-slip based force generation scenarios. Nevertheless, interesting friction characteristics such as the generation of pN forces with stick-slip drives, which are already proven, cannot be covered by known friction models. It is pointed out which characteristics have to be considered.
Keywords: stick-slip drive, actuator, contact mechanics, piezoelectric
1. Introduction
The stick-slip microdrives have been used widely and effectively in miniaturized micro- and nanoposition and manipulation systems. Their main advantages are the simplification in design and very good working characteristics which offer a theoretically unlimited smooth motion accompanying with a very high resolution of several nanometers working in stepping and scanning mode, respectively. These intrinsic properties have been exploited for many applications in the field of micro and nanorobotic.
The stick-slip drives were introduced for the first time in the literature by Pohl [1] for the purpose of micropositioning of an object inside the scanning tunneling microscopy. It is simply a one degree of freedom translational stage actuated by a piezoelectric tube using the motion principle called “stick-slip”. The typical components and the principle of the stick-slip drives are briefly illustrated in Fig. 1. The stick-slip principle is carried out by a sequence of a stick-phase and a slip-phase. During the stick-phase with a slow deformation of the piezoelectric element, the runner moves along with the actuator due to the static frictional forces appearing at the contacts between the touching elements of actuator and runner. This phase is followed
by the slip-phase, in that the piezoelectric element deforms rapidly in the opposite direction. Because of the inertial force, the runner cannot fully follow this movement and therefore, the runner slides on the guiding system; as a result, a step is performed after a small back-step.
In order to achieve a required simplification in design, a multiple number of piezoelectric actuators are combined to serve usually not only as the actuating elements but also the supporting and guiding system for the runner. However, the consequence of this integration is that the motion characteristics of the drive depend strongly on the dynamic behavior of the whole system, where contact mechanics between actuators and the runner plays an important part.
The contacts in stick-slip microdrives determine mostly their dynamical characteristics such as step length, back-step, and the amplitude, frequency and damping time of the microvibrations which are exerted after the slip-phase. Wear properties and lifetime of stick-slip microdrives are also influenced by the contact mechanics. It is known that stick-slip microdrives are quite mature and several prototypes developed into commercial products [2]. However, there has done only little research in the domain dealing with an increase of reliability or in deep analysis of tribology aspects.
© Nguyen H.X., Edeler C., Fatikow S., 2012
For this reason, the present paper focuses on several aspects governing the contact mechanics of stick-slip microdrives, in that the up-to-date friction models are deeply analyzed. After the introduction, mechanical models of such devices are presented. The paper reviews attempts to model stick-slip microdrives based on the so-called LuGre friction model, which is the most promising approach for a long time. The elastoplastic friction model, an extension of the LuGre model is also analyzed, which can cover the important phenomenon of 0-amplitude. Moreover, a new friction model called CEIM is discussed, based on the elastoplastic model and enlarging the model capabilities. Finally, the authors present latest results concerning the generation of p-scale forces using stick-slip drives. The results show that even the sophisticated models presented before fail in achieving similar results. Thus, contact mechanics modeling has to be improved in general.
2. Mechanical modeling
The first extensive attempt to model stick-slip drives was implemented by Breguet [3]. It is the model of a one degree of freedom translational drive using shear piezoelectric actuators. By using the so-called LuGre friction model, the author is successful to obtain essential results from the simulation and measurement. However, the presented model seems to be too simple and does not satisfy in many mechanical aspects, especially the tribology issues.
\ Load system
\___________________.
Runner
0 Time
Fig. 1. Drive configuration and stick-slip principle: initial position (a), stick phase with slow expansion of the piezoelectric actuator (b), slip phase with rapid contraction of the actuator (c), the deformation of the actuator (d)
In this paper and for further investigations, we present a new mechanical model which describes the dynamic behavior of a one degree of freedom transitional stick-slip manipulator developed in our division.
Figure 2 shows the actual structure of the manipulator which can be viewed as the runner. The manipulator is integrated into a mobile microrobot to form a four degree of freedom mobile microrobot for the purpose of micro- and nanohandling. A tool (here a gripper) for manipulation task is attached to the runner. In this design, the runner is actuated, supported and guided by six piezoelectric actuators which are arranged in a defined configuration. The preload, which is necessary for the function of the drive, is created by using flexible hinge structures integrated in to the runner.
Generally, the dynamic behavior of the drive depends on the dynamics of each individual actuator and the contact dynamics between actuators and runner. In fact, they exhibit deviations because of their fabrication tolerances and the asymmetric constraint of the runner. The latter makes the dynamic response of each actuator different. Therefore, we propose to model each actuator separately. As a result, the mechanical model of the drive has at least two parts: dynamic modeling of each individual actuator and the dynamic modeling of the runner (rigid-body kinematics).
2.1. Modeling of the actuator
The structure of one of the six actuators in 2-dimensions is shown in Fig. 3(d), which includes three main components: the ruby hemisphere, the glue layer and the laser-structured piezoceramic part. Figure 3(e) shows the mechanical model of the actuator in an actuated position. Here the ruby hemisphere is supported on two columns that act as highly symmetrical piezoelectric actuators. The nominal height of each of actuator is h0 and the skew symmetric displacements are yl on the left hand side and yr on the right hand side of actuator. The combined spring constants and the damping constants representing the glue layers, piezoelectric actuators and the ruby hemispheres are represented as the pairs Cl, Dl for the left hand side and Cr, Dr for the right hand side. The ruby hemisphere has a radius R
Fig. 2. The design of the vertically translational manipulator
Fig. 3. Mechanical modeling of the z-axis
and a mass of ma with the inertia Ia around its center of gravity. The vertical displacement of the hemisphere from its nominal position is y and the angular displacement of the hemisphere is 0. Normally, the ruby hemisphere is loaded by normal force Fn and friction force Ff.
The dynamic equations of the actuator are given by:
may = Fyr + Fyl - Fn - Fga, (1)
Ia0 = (Fyr - Fyl)r + (Fr + F - Fn)a sin 0-...
... - Ff (R - a cos 0), (2)
where Fga is the gravity force in y-direction; a is the distance from the center of the hemisphere to the center of gravity of the ruby hemisphere; Fyl and Fyr are the forces generated by the spring damper pairs, represented as:
Fyl =(yl- y + r0)C +(yx -y + r0)D[, (3)
Fyr = (yr - y - r0)Cr +(yr - y- r0)Dr. (4)
The normal force Fn is calculated by the geometry, the arrangement of actuators in the system and the interaction between runner and actuators. The friction force Ff is a
critical component and will be discussed in the next sec-
tion.
2.2. Modeling of the runner motion
As shown in Fig. 3(c), the runner is supported by six ruby hemispheres which are shown in Fig. 3(d) as the element that makes contact with the runner. There are six normal forces (A to F) leading to six friction forces (Af to Ff ), respectively. In Breguet’s model the runner is modeled as a lumped mass mr which has only one “virtual” stick-slip contact point for all actuators. The preload is applied on the runner as the sum of normal force of all contact points. In reality due to the geometry, there appear discrepancies between these six contact points leading to the differences in these six normal forces and therefore in six friction forces. This requires an adaption for the general model, either by adding guiding characteristics or by modeling each contact point with individual parameters. The latter seems to be a challenge and hard to model because of some unknown individual friction parameters. However, the model of each individual actuator addressed above can be used for a general dynamical model of the runner in three-dimensions [4].
As the first effort, in [5] the authors tried to model the preload as the sum of all normal forces, which is consistent with measurements using a force sensor. In this paper, the motion equation of the runner with the above mentioned approach is rewritten. The nominal motion of the runner in x- direction is represented as follows: mr x = Af + Bf + Cf + Df +
+Ef + Ff - Fgr + Fe, (5)
where Fgr is the gravity force in movement direction of the runner and Fe is the external force applying on the runner. It is assumed that the six friction forces are identical; therefore they can be represented by Ff as a sum of them. Finally, the total value of friction force appearing in the contact points is calculated to:
Ff = Af + Bf + Cf + Df + Ef + Ff. (6)
In a simple case, the gravity force and the external force are neglected, which leads to dynamic equation of the runner:
mrx = Ff- (7)
It can be seen that the runner movement strongly depends on friction force as well as the dynamic property of the piezoelectric actuator(s). In fact, besides the main motion of the runner, there appear erratic motions in the directions which are orthonormal to the runner motion. These motions are theoretically supposed to be perfectly constrained. The actuator dynamic properties, the actuator arrangement and the characteristics of the guiding system determine the characteristics of the runner motion in three dimensions and they should therefore be considered as well. However, this issue is out of the scope of this paper, therefore modeling friction and contact mechanics in focus. This means that the dynamic property of actuators is neglected and an ideal model is assumed.
3. Friction modeling — the state of the art
3.1. The LuGre model
The LuGre friction model is well known as a dynamic model for friction invented by scientists from Lund and Grenoble [6]. It is a single state friction model capturing most of the friction behavior that has been observed experimentally including Stribeck effect, hysteresis, springlike characteristics, and undesired stick-slip motion. Breguet used the LuGre friction model for the simulation of a linear stick-slip actuator. The fundamental equations are repeated here for further discussion.
The model is based on the average deflection z of the asperities and its derivative z and modeled by set of Eqs. (8)—(10):
dz v
— = v-^-*— z, dt g(v)
g(v) = Fcoul + (Fstat -FcoulV"(V/Vs)2
Ff = g0 z + G1 z + g2 V,
(8)
(9)
(10)
where g0, g1? g2 are stiffness, damping and vicious coefficient respectively; FCoul is Coulomb friction level; Fstat is level of stiction force; v is the relative velocity between two surfaces, and vS is the Stribeck velocity.
Simulation results using the LuGre friction model cover essential characteristics of the stick-slip actuator. However, revealed by Fig. 4, a phenomenon appears (called “0-amplitude” by the authors), which cannot be simulated by the LuGre friction model. The 0-amplitude is understood in such a way that under a certain applied actuator voltage the runner oscillates around its initial position. In this case, no net displacement is generated. This reveals the question of the causation of 0-amplitude. Breguet comments that 0-amplitude is the problem of “limits of contact deformation — presliding”. He brings in a material-related estimation of the value.
3.2. The elastoplastic model
The elastoplastic model is a single state model possessing both presliding and stiction, in which presliding is elastoplastic. That means, under loading, frictional displacement is first purely elastic and then passes to plastic [7, 8]. This model can be obtained from the LuGre friction model by replacing Eq. (8) by Eq. (11). In that, a(z, x) is a transition function determining the state of frictional displacement depending on breakaway distance zba and the maximum average deflection of the asperities for constant v in a steady state zss
dz
~dt
- = v -
a (z, x) |v g(v)
with
a( z,x) =
0 |z| < zba,
0 <am (.) < 1 zba 1, |z| > ^
< z < ze,
(11)
(12)
a m (.) is a smooth transition function defined by
am (•) = 2Sin
+ z.
ba
n-
zss zba
1
+ —. 2
(13)
Fig. 4. Measurement and simulation of 0-amplitude using the LuGre friction model [5]
V J
The transition function a(z, x) in the model expresses
a smooth transition between the elastic and plastic deformation. If |z| < zba, a(z, x) equals zero, the pure elastic deformation is obtained. If |z| > zss, a(z, x) equals unity, we have a pure plastic deformation as in the LuGre friction model. If zba < |z| < zss, 0 < a(z, x) < 1, a smooth transition is applied by using am (zba, zss). zba and zss are critical values determining the behavior of the model as pure elastic, mixed or pure plastic deformation which covers effects such as the 0-amplitude. It is therefore important to know the parameters contributing to the values of zba and zss. The solution for this problem is to combine the empirical gathered values of 0-amplitude and the elastoplastic progress of the elastoplastic model. By choosing empirical coefficients for the transition function, the first simulation is implemented with the values zba = 50 nm and zss = 50 nm leading to the simulated step length as in Fig. 5(2). The following simulations exert to different tendencies by changing the ratio between zba and zss as shown in Fig. 5(1, 3). It can be seen that the 0-amplitude is an empirical value and equates to the breakaway distance zba. The elasto-plas-tic model can cover the 0-amplitude phenomenon. However, the comparison between Figs. 4 and 5 shows that the quality of the simulation is still comparative.
3.3. The CEIM model
As analyzed above, although the elastoplastic model can cover the phenomenon 0-amplitude, the value of zba and zss were chosen empirically and the quality of simulation is only comparative. The 0-amplitude is supposed to be
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_ 2 - ratio 50 % /' /-
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Amplitude, %
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Fig. 5. 0-amplitude simulation using the elastoplastic model [5]
equal to breakaway distance relating closely with the normal force applied on stick-slip drives; the sum of this normal forces is further called preload. The simulations of dependence of 0-amplitude expressed by step length on preload using both the LuGre friction model and the elasto-plastic model were implemented as shown in Fig. 6. It is obvious that neither the elastoplastic model nor the LuGre friction model cover the preload dependency trend. Therefore, some adaptions to the elastoplastic model were introduced.
In [9], the authors proposed an empirical model, called CEIM, to simulate the dependency of 0-amplitude and stick-slip based generated force on preload Fpreload. This is an adaption from the elastoplastic model and an empirical model. It is validated by numerous measurements using a special test bed.
The calculation of zba and zss is as follows:
zba = (aE + b) Fpreload + cE + d,
fba ,
(14)
(15)
E" = o ln( E) + p, (16)
where E* is the equivalent Young’s modulus; a, b, c, d, o, p and fba are empirical coefficients.
The simulation result is shown in Fig. 7. It fits well with the measurement data.
For the stick-slip based force generation simulation, the adaptions for the elastoplastic model are determined by: dz = a(z, x) | V g0 z
dt ^preload M
(17)
Ff =g0 z + g1 z, (18)
G0 = (asE + bs)Fpreload + csE + d s, (19)
where as, bs, cs and ds are empirical coefficients. The comparison between simulation and measurement is showed in Fig. 8. It can be seen that the simulation result with material of wolfram fits well with the experimental data. However, with aluminum the simulation shows different trend. This suggests that the model should be improved in any case. Obviously, friction characteristic strongly depends on the actual material pair. The CEIM model was mainly optimized for steel and hard metal, therefore other materials such as aluminum or ceramics do not show a high accordance with simulation (although there exist numerous measurements).
There are some commentaries on the empirical model CEIM:
- The relation between zba and zss defined by Eqs. (14), (15) has the advantage of reducing the number of parameter of the model. It can be seen that the above mentioned modeling of zba is a very easy but effective approach, although it is based only on empirical data. In this case, zba increases with preload, which results in a larger breakaway distance and shorter step length. That means, preload defines zba and therefore 0-amplitude.
- The results of the authors show that the difference of the absolute values of friction coefficients m static and m kinetic does not influence the simulation result significantly. Therefore, they should be treated identically and replaced by a common jw. This also reduces the amount of parameters in the model.
Fig. 6. Simulation and measurement of preload-dependent progress using the LuGre friction model and elastoplastic model [5]
Fig. 7. Simulation and measurement of step length dependent progress using CEIM model [9]
Fig. 8. Measurement and simulation of force generation preload trend using CEIM model [9]
30
20
10
I I I I 1 - average of 5 measurements
_ 2 - standard deviation
3-simulation ..
_
3/*'
-
2^ —^ —'
10 15
Amplitude, %
20
25
Fig. 9. Measurement and simulation of wN force generation
- The Stribeck curve has no importance for stick-slip drives and it is therefore neglected.
*
The tangential stiffness g0 is modeled as a linear function of preload. The rising preload leads to an increased stiffness. This could be interpreted as follows: When a higher level of preload is applied, shorter asperities (or an increased number of interconnecting asperities) are obtained, resulting in increased Young’s modulus and therefore stiffness, respectively.
4. Contact mechanics
The analysis of existing friction models addressed above shows that dynamic contact modeling should be considered in any case. The origin of the discussed friction models is in the macroscopic world where they are effectively applied for control of machines afflicted with friction. On the other hand, contact geometry on microscopic scale is mostly neglected. In stick-slip drives the flat-sphere contact type is typically used where the contact radius of the contact is in the range of few micrometers or even less.
In the particular case when contact forces have very small values, important questions come up: Can the contact still be described as a homogeneous contact (with quite a number of interacting asperities per contact) or does friction characteristic transit towards a single asperity contact per macrocontact? Existing friction models fail to describe the dynamics behavior of the contacts. In [9], it is shown that forces in the range of single pN can be generated using a drive preload of 70 pN. Considering six macrocontacts, the individual preload will be in the range of ~0-30 pN. This is the upper working range of commercial atomic force microscopes. However, the presented models cannot cover pN force generation characteristics. Figure 9 shows our newest result for the simulation of wN force. It is seen that there is a total difference between the simulated force and the measured force. The measured data shows that the generated force is always exist, even for the case of very small amplitude and this force can be reached to the maximum of
4 pN at the amplitude of 7 %. Whereas the simulation result shows that there always exists the 0-amplitude status at the beginning where no force is generated and when the amplitude is about 5 %, the force is generated and increases proportionally with the increase of amplitude. This deviation shows a drawback of the existing friction models.
Possibly, the key of future research is in closing the gap between top-down friction models (macrofriction) and bot-tom-up friction models (such as AFM-related friction). For both areas, numerous friction models exist, but there is still no all-embracing approach.
5. Conclusion and outlook
In this work the problem of modeling stick-slip microdrives is studied. An analysis of the existing friction models has been given. It has been shown that — although stick-slip microdrives are quite mature — there is little understanding of the theory behind such kind of devices, especially the contact mechanic problem. Modeling of stick-slip microdrives has been proven to be a challenging task. This is mainly due to the difficulties of friction force relating closely to the contact mechanics of the contacts between actuators and runner. Our recent results show that the generation of w-scale force cannot be covered by the existing friction models. This is not only the existence issue of the current research but also the motivation for future research, where the contact mechanics in microscopic and macroscopic point of views should be absolutely considered. A new contact mechanics model is therefore actually necessary.
Otherwise, it is shown that several contacts exist between runner and actuators and the type of these contacts is usually a sphere-flat one. In fact, there are discrepancies between these contacts. The question is coming up, whether one should combine these contacts as a represented contact or simulate them separately? The contact mechanics of stick-slip microdrives should not be considered as a local problem but as a global one in that the precision of the model is
affected by various parameters in such a way that the guiding system plays an important role. The use of an integrated model combining contact model and mechanical model offers a good possibility for future research.
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Поступила в редакцию 10.06.2012 г.
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Nguyen Ha Xuan, Dipl.-Phys., University of Oldenburg, Germany, xuan.ha.nguyen@uni-oldenburg.de Edeler Christoph, Dipl.-Ing., University of Oldenburg, Germany, christoph.edeler@uni-oldenburg.de Fatikow Sergej, Prof., University of Oldenburg, Germany, sergej.fatikow@uni-oldenburg.de
