Model based, nonlinear control of near-field acoustically levitated objects
D. Ilssar1, I. Bucher1, H. Flashner2 1 Technion – Israel Institute of Technology
Dynamics laboratory, Mechanical engineering
Technion 3200003, Haifa, Israel
e-mail: [email protected]
2 University of Southern California
The department of Aerospace and Mechanical Engineering
Los Angeles, CA 90089, USA
Abstract The near-field acoustic levitation phenomenon uses ultrasonic vibrations of a driving surface, to elevate
the average pressure in the thin gas layer trapped between this surface, and a freely suspended planar
object. This is done, exploiting the compressibility and the viscosity of the entrapped gas. Due to the
pressure elevation, the planar object can be manipulated vertically between dozens and hundreds of
microns, without any mechanical contact. This paper introduces a novel, simplified model, describing the
governing, slow dynamics of near-field acoustically levitated objects. This model replaces the commonly
used Reynolds equation based model, with a second order, ordinary differential equation, where the
stiffness and damping terms are given explicitly. Due to the simplicity of the model presented here, it sets
a convenient foundation for model based control algorithms. Indeed, based on this model, a continuous
gain scheduling controller is designed and implemented experimentally, showing good performance.
1 Introduction
When two closely placed surfaces, surrounded by gas, experience relative normal oscillations, a force,
consisting both oscillatory and constant terms, is acting on the two surfaces, by a thin layer of gas trapped
between them, commonly known as squeeze-film. This phenomenon occurs due to the viscosity and the
compressibility of the gas. Namely, due to its viscosity, the gas trapped between the surfaces cannot be
immediately squeezed out, causing the amount of gas leaking out of the film during compression, to be
equal to the amount of gas enters it during decompression [1]. Furthermore, since the gas is compressible,
the pressure distribution between the surfaces fluctuates around a nominal profile whose average value is
higher than the ambient pressure (e.g. [2–4]). Accordingly, the average force induced by the entrapped gas
is repulsive, thus by these means, a freely suspended planar object, can be levitated above a rapidly
oscillating driving surface, without any mechanical contact.
It was formerly shown that the abovementioned levitation mechanism, commonly known as near-field
acoustic levitation, is governed by the amplitude of the driving surface’s oscillations (e.g. [2,5,6]). Thus it
can be used for manipulating planar objects vertically, without contact. A possible application for near-
field acoustic levitation is handling silicon wafers throughout inspection and manufacturing processes (e.g.
[7]), replacing traditional chucks, that make mechanical contact with the substrates, and thus,
contaminating the highly controlled work environment.
The two prevailing approaches for modelling the behavior of the squeeze-film, are the radiation pressure
theory (e.g. [8–10]), and Reynolds equation (e.g. [3,4]). However, when dealing with the dynamic
response of near field acoustically levitated objects, only the latter can be used, since the radiation
pressure approach refer to the entrapped gas as inviscid, and thus cannot describe the effect of damping.
1579
As shown below, Reynolds equation is a nonlinear partial differential equation, thus it is too complex for
practical uses, such as serving as a basis for model based control algorithms. Therefore, Ilssar and Bucher
[2], formulated a simplified model, describing the governing, slow dynamics of near-field acoustically
levitated objects, using a single, second order ordinary differential equation. This model is based on
Reynolds equation, so it comprises both the conservative levitation force originating from the
compressibility of the gas, and the damping force acting due to its viscosity, in an explicit manner.
However, since the model derived by Ilssar and Bucher, is restricted to the case where the driving surface
oscillates uniformly, as a rigid piston, Ilssar et al. [6] Generalized it so it represent the dynamics of a
system, where the driving surface vibrates as a general standing wave. The abovementioned general model
is introduced in the present paper, alongside two empirical methods, adjusting this formulation to a
specific configuration. Next, after numerical and experimental verification, this model is exploited for
sake of formulating a height dependent, gain-scheduled PID controller, enabling rapid and accurate
vertical positioning of a levitated object. Finally, the necessity of a resonance tracking loop, operating
along the levitation height control loop, is demonstrated.
2 The theoretical model
The drawing appears in Figure 1, describes schematically the system discussed in the present paper. This
system consists a driving surface, oscillating as a pure standing, flexural wave, in a constant frequency
denoted , and according to a constant spatial profile a . The investigated system also includes a freely
suspended planar object, levitated above the driving surface, due to the pressure elevation produced by the
rapid oscillations of the driving surface. The system considered throughout this paper is axisymmetric,
implying that the levitated object moves only vertically, and the spatial profile of the driving surface’s
oscillations depends only on the radial coordinate, denoted r .
Figure 1: Schematic layout of the investigated system, consisting a driving surface, a squeezed gas layer,
and a levitated object
It was shown in the past (e.g. [2,11–13]), that the dynamic response of the levitated object consists of two
time scales; a fast time scale, associated with the oscillations of the driving surface and the resulting low
amplitude oscillations exhibited by the levitated object, and a slow time scale, describing the transient
dynamics of the object. As seen in Figure 4, presenting four typical dynamic responses of levitated
objects, the slow evolution of the levitated object, is 2-3 orders of magnitude slower than the excitation,
implying that the dynamic response of the levitated object can be decomposed to a slow process, denoted
h , and a fast process.
2.1 Formulating the governing equations
Before formulating the system’s governing equations, the following non-dimensional measures are
defined
2 2
0 0
2 2 2
0 0 0 0 0
12 12, , , , , , .
a a a
r rp h h r a aP H H R T t
p h h r p h p h H h h H
(1)
1580 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
Here, , ap p are the pressure distribution inside the squeeze-film, and the ambient pressure respectively, 0h
denotes a reference, typical clearance between the driving surface and the levitated object, and stands
for the dynamic viscosity of the entrapped fluid. Moreover, denotes the non-dimensional excitation
function, whose magnitude is much smaller than unity, and is the squeeze number, governing the
instantaneous behavior of the squeeze-film. It was formerly shown that , , prevail the instantaneous
behavior of the system (see [2]).
To formulate the coupled dynamics of the system illustrated in Figure 1, the behavior of the squeeze-film
should be coupled with the dynamics of the levitated object. The first equation considered here is
Reynolds equation, representing the behavior of the fluid residing inside the squeeze-film. Assuming the
film is axisymmetric and isothermal, this equation takes the following non-dimensional form [2]:
3 2 .PHP
RH P H RR R T
(2)
Reynolds equation originates in the Navier-Stokes momentum and continuity equations, referring to the
entrapped fluid as an ideal gas. An essential assumption made in the derivation of Reynolds equation is
that the clearance between the levitated object and the driving surface is much smaller than the system’s
lateral dimensions. This assumption leads to omission of all inertia terms [3,4], leaving the equation only
with viscous and pressure terms.
In order to fully define the pressure distribution inside the squeeze-film, assuming the clearance H and its
slow component H are known, boundary conditions need to be imposed. The first boundary condition
needs to be taken, demands pressure continuity at the center of the film. Moreover, it is customary to
assume pressure release boundary conditions, dictating the pressure at the boundaries of the film to be
equal to the ambient pressure. I.e., the boundary conditions are taken as following
0,
,0, 1, 1.
R T
P R TP R T
R
(3)
It was shown by Minikes and Bucher [14,15], and by Nomura et al. [16] that when the driving surface and
the levitated object do not fully overlap, they induce pressure radiation. The latter results in energy
leakage, reducing the accuracy of the pressure release conditions. However, since in the system discussed
in this paper, the levitated object and the driving surface overlap, the boundary conditions given in (3)
seem valid.
As mentioned above, in order to describe the overall dynamics of the system, Reynolds equation needs to
be coupled with the equation of motion, describing the vertical translation of the levitated object. This
equation considers the pressure acting on the levitated object by the surrounding fluid, and the gravity
force. For sake of consistency with Reynolds equation, the equation of motion is formulated in terms of
the pressure distribution and the thickness of the squeeze-film, thus it takes the following form [2]:
122
0
2 2 2
0 00
21 d sin ,ap rH g
R P R H TT mh h
(4)
where g denote the gravitational acceleration.
3 Simplified modelling
As seen in Figure 4, the slow time scale, related to the transient response of the system, is much more
significant than the fast time scale, associated with the rapid oscillations of both the levitated object and
the driving surface. Thus, for practical applications, such as parametric studies or controlling the dynamics
of near-field acoustic levitation based system, it is sufficient to find a model, describing merely the slow
dynamics. Ilssar and Bucher [2], formulated a second order, ordinary differential equation, representing
only the slow time scale, assuming the driving surface oscillates uniformly, and Ilssar et al. [6],
HIGH FREQUENCY ACTUATORS AND PROCESSES 1581
generalized it so it suits the more general case considered here, where the driving surface oscillates as a
pure standing wave. This model is indeed simple enough to set the foundation for a model based
controller. Therefore, since it is used in the upcoming section for control purposes, the generalized model
derived by Ilssar et al is introduced here. Next, in order to make the model applicable, a numerical method
and an experiment based method, used to adjust it to a specific system, are introduced. Finally, an
experimental setup is introduced and modelled using both methods, in order to compare them, and to set
the ground for a model based control.
3.1 Introducing the general simplified model
It was shown by Ilssar and Bucher [2] that first term on the right hand side of the equation of motion (4),
representing the influence of the surrounding pressure on the levitated object, can be decomposed into two
distinct forces. The first force is the conservative levitation force emanating from the compression of the
entrapped gas, and the second force is the dissipative damping force, originating from its viscosity.
Moreover, in order to describe the damping associated with slow evolution of the system, the terms
connected with the fast time scale can be ignored. The latter implies that since the oscillations of the
driving surface are related to the fast time scale, the excitation form (i.e. the mode shape of the driving
surface being excited) does not affect the dissipative force related to the slow dynamics of the system.
Consequently, the non-dimensional form of this force, is taken as formulated by Ilssar and Bucher, who
derived it for the case of uniform excitation (see [2], equations 35-36)
2
0
2 3
0
3 d,
2 ddamping
a
r HF
p h H T
(5)
where dampingF is normalized by 2
0ap r .
Contrary to the damping force, the levitation force does depend on the excitation form, so it needs to be
modified according to a specific form of excitation. In order to calculate this force, Ilssar et al. [6]
considered a system where the driving surface oscillates arbitrarily, in proximity of a fixed wall whose
nominal distance from the driving surface is H . Moreover, in order to take nonlinear effects into account
they developed the pressure distribution into a second order asymptotic series, so they described the
degenerated system according to
2 3ˆ1 sin , , 1 , , .A BH T H H R T P R T R T R T (6)
where since the magnitude of is much smaller than unity, the small parameter is defined by
ˆ ˆ, , .H R H R R H R (7)
Obviously, since the parameter takes the small magnitude of the non-dimensional excitation function,
the norm of ̂ equals to unity, where for sake of consistency with the model derived in [2], the weight
function of the norm is chosen as 1.
The average force acting on the wall in the degenerated system, yields a good approximation for the
averaged levitation force acting on a levitated object whose nominal height is H . The latter is since the
levitation force is associated with the compression of the entrapped fluid which is hardly affected by the
levitated object’s motion. Namely, the evolution of the system is much slower than the rapid oscillations,
thus it hardly affects the compression of the gas. Moreover, the amplitude of the levitated object’s rapid
oscillations of the are 2-3 orders of magnitude smaller than the excitation magnitude (e.g. [2]), so the
levitated object’s fast motion can also be ignored.
Ilssar et al. [6] substituted (6) into Reynolds equation (2), and decomposed the obtained expression into
two equations, related to the terms of order and 2 . By harmonic balancing these equations, they
achieved approximated solutions for ,A B , depending on the excitation form ̂ . Once an approximated
expression for the pressure is found, the time averaged force acting on the wall, corresponding to the
1582 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
slowly varying levitation force acting on a freely suspended object, levitated at the same average height,
can be found. This is done by integrating the pressure acting on the wall from both its sides, over its area,
and averaging over a single excitation cycle, in order to eliminate terms associated with the fast time scale.
Thus, the non-dimensional, slowly varying levitation force is taken as following
2 2 1
0 0 0
211 d d d ,
2levitationF R P R T K
(8)
where levitationF is normalized by 2
0ap r .
The function K is an excitation form dependent calibration function, denoting the dependency of the
averaged levitation force on the squeeze number. This function captures the relatively complex features of
the levitation process, considering only the slow dynamics of the system. It should be noted that it is
usually impractical to find a closed form solution for the second order approximation of the pressure
distribution, implying that a closed form for the function K can also not be found. Therefore, in the
following sub-section, two empirical methods for finding this function, are presented.
The slowly varying damping and levitation forces found above, provide a second order approximation for
the slowly varying influence of the surrounding gas, meaning that they describe the first term in the
equation of motion (4), averaged over one period of excitation. Therefore, (4) is averaged over a single
excitation period, followed by substitution of the forces given in (5) and (8), providing the following
simplified equation, accounting for the slow dynamics of the system illustrated in Figure 1
* 2
2
2 2 3
d d.
d d
s dC u K H CH H
GT H H T
(9)
Here, u representing the excitation magnitude, *
sC denoting a coefficient of the stiffness term, dC
standing for the coefficient of the damping term and G symbolizing the gravity, are given as followings:
* 2 2 4 3 2
0 0 0 0 0 0, , 3 2 , .s a du a r h C r p m h C r m h G g h (10)
Obviously, since the levitated object is no longer fixed, the squeeze number is a functional of the slow
evolution H .
3.2 Approximating the calibration function, empirically
As stated above, in order to use (9)-(10) for modelling of the slow dynamics of an acoustically levitated
object, a closed form for the function K must be found. In the upcoming sub-sections, two empirical
methods, enabling to approximate this function, using either numerical simulations or experiments, are
presented.
3.2.1 The numerical approach
Since the motion of the levitated object was neglected when formulating the levitation force, it should also
be ignored when calculating the function K . Therefore, when calculating K based on numerical
simulations, the levitated object can be assumed fixed, so the numerical scheme consists of numerous
ordinary differential equations, achieved by discretizing Reynolds equation (2) as following:
HIGH FREQUENCY ACTUATORS AND PROCESSES 1583
2
2
2 22 2
2 2
13
1
n nn n
n n n nn
nn n n
n nnn
H PP P H PH P
T H R R R R
PP P HH H P
H R R H T
(11)
where 1,n N denotes the index of the radial node residing at 1 1R n N .
In the analyses made in this study, the spatial derivatives in (11) were calculated using central differences
formulas, while the needed values at the external nodes 1,n N were determined according to the
boundary conditions (3). Moreover, assuming the excitation form and magnitude are known, the radial
dependent, instantaneous clearance H is known, leaving the pressure as the only unknown function. In
order to find the instantaneous values of the pressure distribution, the coupled system (11) is solved using
numerical integration.
It should be noted that the derivation of the levitation force presented above, ignores the transient behavior
of the pressure, which is a valid assumption since the time constants of the pressure are much lower than
those of the levitated object's evolution. Hence, according to (8), the levitation force of a specific
configuration, is obtained by integration of the pressure distribution obtained from a numerical simulation,
over the system’s area, and averaging over an integer number of excitation periods, after convergence to
steady state. Calculating the levitation force for various squeeze numbers in the desired work range, and
division by 2 , results in the desired values of K , where should be kept small in the simulations,
in order to comply with the asymptotic assumptions, stating that this parameter is much smaller than unity.
Finally, in order to approximate a closed form expression for K , curve fitting on the obtained discrete
values should be done.
3.2.2 The experiment-based approach
As in the numerical approach, the dynamics of the levitated object can also be disregarded when
approximating the function K , based on experimental data. Thus, approximating K
experimentally can be done by measuring the time averaged levitation heights at steady state eqH ,
obtained under various constant excitation magnitudes equ . The steady state measures ,eq eqH u are related
by the following transcendental relation, obtained from (9), when nulling the time derivatives of H
* 2 2 .s eq eq eqC u K H GH (12)
This equation, obtained for every pair ,eq eqH u , provides a single value of K . Curve fitting on these
discrete values, yields a closed form approximation that can be used in (9).
Obviously, since it is based on experimental data, the experiment-based approach helps correcting
inaccuracies in the model, such as pressure release boundary conditions, thus it is more reliable than the
numerical approach. Moreover, the experiment-based approach can take additional static forces, such as
magnetic forces acting on the system, into account. These forces are taken as supplementary terms on the
right hand side of (12), thus when using the experiment-based approach, K is calculated considering
the additional static forces (see sub-section 3.3).
3.3 Modelling the experimental setup
Figure 2 shows the setup on which all the experiments discussed in this paper were performed. This setup
consists a piezoelectric actuator, composed of a Langevin transducer, and a structure used to magnify the
1584 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
displacements at its base, in order to produce significant excitation amplitudes. This actuator was used to
supply the required excitation, for levitating a planar object whose weight and radius are 134.7 g and 50
mm respectively. For sake of high efficiency, all components comprising the actuator are designated to
operate at approximately 28.5 kHz, where all axial elements resonate at their first elastic longitudinal
mode, and the upper plate whose top surface functions as the driving surface, resonates at its second
axisymmetric flexural mode. Obviously, the system cannot be not perfectly leveled, thus undesired
degrees of freedom such as lateral translation and tilting can arise (e.g. [17,18]). Therefore, as seen in
Figure 2, a magnetic centering array, using magnetic repulsion forces between three centering elements
and a series of magnets placed on the perimeter of the levitated object, was implemented.
In order to preserve high performance, and to meet the model's assumption, referring to the excitation as a
pure standing wave, in all of the experiments discussed throughout this paper, the excitation frequency
was determined according to a Phase-Locked loop, resonance tracking algorithm (e.g. [19,20]).
Nevertheless, in all of the experiments the frequency changed by less than 0.1%. Hence, these variations
are not taken into account in the analyses.
Figure 2: Right: A photograph of the experimental rig, showing the piezoelectric actuator in the middle
and the levitated object with centering the passive magnets (set aside for visibility). Left: Schematic
layout of the piezoelectric actuator, and a sample levitated object.
In order to model the experimental setup, using both approaches suggested above (see sub-sections 3.2),
three sets of experiments were performed. In the first experiment set, the excitation form ̂ , denoting the
mode-shape of the driving surface being excited, was measured using a laser Doppler sensor (PolytecTM,
OFV-303), under different excitation magnitudes, when the system was loaded with the mass mentioned
above. According to these experiments, the excitation form does not change with the levitation height,
thus ̂ is taken as the measured, levitation-height independent function, complying with (7). Next, the
relation between the amplitude of the voltage supplied to the system, denoted as V , and the amplitude at
the center of the loaded driving surface, was measured using a laser Doppler sensor (PolytecTM, OFV-
551), under excitation with several intensities in the allowable range. Using this experiment set, and the
measured excitation form, relating the amplitude at the center of the driving surface to a , the relation
between V and the excitation magnitude u was found. Finally, the steady-state height of the levitated
object was measured several times under numerous constant excitation magnitudes, using a laser
triangulation sensor (KeyenceTM, LK-H008). These experiments, combined with the data from the former
experiments, relate the steady-state levitation height to the excitation magnitude, enabling the calculation
of K , according to the experiment-based approach. It should be noted that although all three
experiment sets are essential in order to apply the experiment-based approach, the first experiment set is
sufficient for the numerical approach.
HIGH FREQUENCY ACTUATORS AND PROCESSES 1585
Figure 3 compares the numerical and the experimental forms of K , obtained according to the
experiments detailed above. This figure shows that the experimental values of K , achieved over 20
experiments, in which more than 20 steady state excitation magnitudes and their corresponding steady
state heights were measured, are quite repetitive. However, there are significant deviations between the
values obtained from the different approaches, indicating an error in the numerical approximation of
K . These deviations can strongly be attributed to the fact that the numerical approach does not take
the force induced by the magnetic array presented in Figure 2, into account, where the experiment based
approach does consider it. Namely, as mentioned above, when using the experiment based approach, the
magnetic force is taken as an additional term in the right hand side of (12), thus it is included in K .
Figure 3: Comparison between the values of K(σ) obtained using the different methods. (a) The curve
fitted to the measurements, (b) The area enclosing all of the measurements, (c) The curve fitted to the
numerically obtained values (d) The values calculated according to the numerical approach, (e) The
difference between fitted curves.
Since the disagreement between the experimental and the numerical values of K , originate vastly
from the magnetic forces, it seems appropriate to validate the simplified model (9) with the numerically
obtained K , ignoring the effect of these forces. For this sake, in Figure 4, a few typical dynamic
responses, obtained using this model are compared to their corresponding responses achieved using the
original, Reynolds equation based model (2)-(4). The latter was calculated numerically by solving the
equation of motion (4) together with the ordinary differential equation system (11), using finite differences
in space and numerical integration in time.
Curve Mass [g] Excitation
frequency [kHz] ‖a‖ [μm]
δ at steady
state
Steady-state
error
a 134.7 28.5 7 0.0696 0.1659%
b 134.7 28.5 2.5 0.0360 0.0779%
c 1000 4 2 0.0870 0.3976%
d 20000 2 4 0.3244 2.3425%
Table 1: Significant characteristics of the dynamic responses presented in Figure 4
From Figure 4 and from Table 1, presenting important properties of the compared responses, it can be
seen that the simplified model calculated utilizing the numerical approach, accurately describes the slow
dynamics of the system formulated by the Reynolds equation based model, when is small. However,
since the simplified model assumes that 1 , as this condition wakens, the correlation between the
simplified model and the original model deteriorates. Nevertheless, in most practical uses the small
1586 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
parameter does not reach such high values. The good correlation between both models, supports the
hypothesis that the major cause of the deviations between the numerical approach and the experiment-
based approach is the force applied by the magnetic array.
Figure 4: Comparison between typical dynamic responses whose properties are given in Table 1. In solid
black curves: the results obtained from the original model. In dashed orange curves: the results obtained
from the simplified model utilizing the numerically obtained values of K(σ).
Figure 5: Solid black curve: a measured dynamic response of the levitated object under excitation with
several constant magnitudes. Dashed orange curve: The corresponding responses obtained using the
simplified model (9) utilizing the experimentally obtained K(σ).
Figure 5 compares between a measured and its corresponding theoretical dynamic response, obtained
under excitation with several constant magnitudes. Here, the theoretical results were obtained using the
simplified model (9), utilizing the experimentally obtained K . It can be seen from Figure 5, that the
deviations between the theoretical and the measured steady-state heights do not exceed 2%, over the entire
work range. Moreover, at low levitation heights, the theoretical model accurately predicts the dynamic
HIGH FREQUENCY ACTUATORS AND PROCESSES 1587
behavior of the system as well. The latter implies that beside the stiffness term found empirically, the
damping term calculated theoretically almost perfectly predicts the actual damping. However, as the
levitation height increases, the theoretical prediction of the transient response deteriorates. For instance,
when the levitation height is around 190 µm, the predicted rising time is about 9 times shorter than its
experimental value. These deviations can be explained by the fact that the simplified model (9) is based on
Reynolds equation, derived assuming the clearance between the levitated object and the driving surface is
very small. Thus, as this clearance grows, the validity of (9) deteriorates.
Obviously, the theoretical damping term can be adjusted empirically, as shown by Ilssar and Bucher [2].
However, the simplified model (9) seems adequate for purposes of designing a controller for the system
presented in Figure 2, assuming the controller is sufficiently robust. Thus in the next section, the model (9)
is utilized as a basis for a model based controller, governing the slow dynamics of the system.
4 Closed loop control
Both the conservative and the dissipative terms, comprising the simplified model (9), were derived under
the assumption that the excitation magnitude and frequency are constant, considering terms up to order 2 . The latter implicitly suggests that (9) is valid if the slow dynamics of the system is sufficiently slow
such that
2d d .T (13)
From the definition of , that includes both the slow evolution of the system, and the excitation
magnitude, it is implied that if the excitation magnitude varies slowly so the latter is met, the simplified
model still holds. Thus, (9) can function as a basis for a model based controller, governing the slow
dynamics of the system, by changing the excitation magnitude sufficiently slow. Indeed, in the
forthcoming sub-sections, a gain-scheduled controller, governing the slow dynamics of the experimental
setup is designed based on (9). Next, after numerical simulations used to assess the performance of the
controller, the latter is evaluated experimentally.
4.1 Controller design
A gain-scheduled controller consists of a family of linear controllers of the same form, designed for small
deviations around different operating points. The latter leads to a nonlinear controller whose coefficients
vary with a scheduling variable, indicating on the appropriate operating point and thus also on the suitable
coefficients. Thus, in order to devise a gain-scheduled controller, relying on the simplified model (9), the
latter is linearized around a family of equilibrium points, using first order Taylor expansion, providing the
following LPV (linear parameter-varying) model
*
2* 2
2 2 3 3 2
2 2dd 1 d.
d d deq
eq s eq eqd
s eq
eq eq eq eqH
K H C u K HK H CH HC u H u
T H H H H T H
(14)
Here, , 1u H are small deviations around the equilibrium point ,eq equ H denoting a constant
excitation magnitude and its resulting steady state levitation height respectively, where the relation
between ,eq equ H is obtained from (12).
The controller chosen to be implemented on the system is of a PID form, based on the velocity algorithm
suggested by Kaminer et al. [21]. Here, the proportional and the integral actions act on the error between
the reference height r and the levitation height H , whereas the derivative action acts on H [22].
Moreover, the three controller parameters , ,p i dk k k , denoting the proportional, integral, and derivative
gains respectively, vary with the levitation height H , implying that H is the scheduling variable. For
1588 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
causality, and in order to avoid amplification of high frequency noise, the differentiator is connected in
series to a low-pass filter whose cut-off frequency is denoted LPF , as illustrated in Figure 6. According to
the above, the gain-scheduled controller is given by
p du k H r H k H (15)
where
d d d
, .d d d
i LPF LPF
Hk H r H
T T T
(16)
As seen in Figure 6, the measured signal goes through an analog low-pass filter, that was not taken into
account in (15), (16) since its cut-off frequency is chosen to be much higher than those of the closed loop
and the filter connected to the differentiator. Another component that is disregarded in the controller
design is an anti-windup loop, also appear in Figure 6.
Figure 6: Schematic layout of the controlled system.
It should be noted that under the assumption of zero steady-state error, the controller given by (15), (16)
has no hidden coupling terms, meaning that around each equilibrium point, the gain-scheduled controller
is similar to the linear controller designed to achieve the desired performance at this point [23]. Therefore,
the controller coefficients at every operating point were chosen in a way that satisfies the required
performance locally, according to the LPV (14). Here, in order to achieve rapid convergence to steady
state with reduced oscillations, the controllers were designed to have two couples of identical and thus real
valued poles, implying on critical damping. These requirement reduces the controller designs into two
degrees of freedom, determined by setting the cut-off frequencies of the closed loop and the
differentiator's low-pass filters to desired values, chosen to attenuate high frequency noise without
affecting the system’s performance.
Ilssar et al. [6] stated that if the scheduling variable of the closed-loop system discussed above, is slowly
varying and the initial state is sufficiently close to the initial equilibrium point, than the response of the
system is uniformly bounded. Furthermore, if the scheduling variable H converges to a constant steady
state value, than the error vanishes as T . The abovementioned properties can be proven following
Khalil [24], who verified these properties for the velocity algorithm given in [21].
Cut-off frequency of the closed loop 10 Hz
Cut-off frequency of the differentiator's filter (ωLPF) 300 Hz
Cut-off frequency of the analog Low-pass filter 2000 Hz
Sampling rate 2000 Hz
Table 2: The parameters of the closed loop system.
HIGH FREQUENCY ACTUATORS AND PROCESSES 1589
In order to assess the behavior of the chosen gain-scheduled controller, its performance was examined
utilizing a numerical scheme, describing the closed-loop system discussed above. Here, the levitation
system was modelled based on the simplified model, where K was approximated experimentally using
the data, collected in the preliminary experiments described in the sub-section 3.3.
Figure 7 shows the step responses of the simulated closed loop system, to a reference signal, varying
between 60 µm and 180 µm, obtained utilizing the abovementioned numerical scheme, with the values
appear in Table 2. In order to describe a more realistic situation, measurement noise was added to the
abovementioned simulation, where, since the simplified model is valid only if the condition (13) is met,
the measurement noise was taken as colored noise with maximal magnitude of 2 µm, whose cut-off
frequency is 300 Hz. As seen in Figure 7 (orange curves), the gain-scheduled controller is expected to
provide a rapid, and non-oscillatory convergence to steady state, with no steady-state errors, in the entire
work range.
Figure 7: A theoretical (orange curves), and its corresponding measured (blue curves) dynamic response
of the closed loop, to an input reference, varying in steps between 60 µm and 180 µm.
4.2 Implementing the control algorithm
Following the satisfying theoretical performance of the controller, the latter was implemented
experimentally, using the system illustrated schematically in Figure 6. The system consists of the
experimental setup presented in Figure 2, whose excitation frequency was determined by a resonance-
tracking algorithm, an analog low-pass filter, reducing high frequency noise, and the control loop designed
above. As seen in Figure 6, the controller closes a feedback on the levitation height, measured using a
laser triangulation sensor (KeyenceTM, LK-H008), where since the controller’s input is non-dimensional,
the physical levitation height yielded by the sensor, was converted to a non-dimensional form using (1).
Moreover, the controller outputs the excitation magnitude u , but in practice it commands the amplitude of
the input voltage V . Thus, the former was converted into the desired values using the relation found in the
second preliminary experiment set, discussed in sub-section 3.3.
In addition to the theoretical results discussed above, Figure 7 also presents the corresponding measured
results, filtered such that only frequencies lower than 35 Hz are taken into account. It can be seen that
there is a good correlation between the theoretical and the measured results, however, at significant
levitation heights, the agreement between the theoretical and the measured transient responses, weaken.
These deviations are due to the fact that as shown above, the theoretical model overestimates the actual
damping at significant heights, resulting in an oscillatory measured response where the theoretical model
predicts a critically damped response. Moreover, it can be seen from both the theoretical and the
1590 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
experimental responses that at low altitudes, the controller is more sensitive to noise. Thus around 60 µm,
the experimental gain-scheduled controller did not keep the desired steady-state height. Yet, around 70 µm
and beyond, it tracked the commended steady-state height, accurately.
Finally, although it is not the focus of this paper, the importance of the phase-locked loop, determining the
excitation frequency of the experimental system, should be emphasized. Figure 8 shows the measured
dynamic response of the levitated object, as presented in Figure 7, and the instantaneous excitation
frequency, determined by a phase-locked loop resonance tracking algorithm. From this figure one should
notice that the resonance frequency of the system depend on the levitation height. This phenomenon
occurs since the thickness of the squeeze-film affects the stiffness and damping of the system (e.g. [3]).
Moreover, additional causes such as temperature and moisture variations, change the resonance frequency
as well. Therefore, as seen in Figure 8, the frequency variation also has a time scale, much slower than the
slow dynamics of the system. Obviously, since the system discussed in this paper is a high-Q system,
slight deviations from resonance can reduce the excitation magnitude, significantly.
In order to enable the controller designed above to work properly, the excitation magnitude should depend
directly on the controlled amplitude of the input voltage. Thus, working with a constant excitation
frequency seems impractical, and so, a resonance-tracking algorithm must be an integral part of any
control loop, governing the dynamics of near-field acoustically levitated objects.
Figure 8: Blue curve: the measured levitation height of the system (as presented in Figure 7). Black
curve: the excitation frequency, determined by the phase-locked loop, resonance tracking algorithm.
5 Conclusions
It was shown in the past that the dynamic behavior of near-field acoustically levitated objects consists of
two time scales. A slow time scale, associated with the transient response of the levitated object and a fast
time scale, related to the excitation, and the resulting rapid oscillations of the levitated object. It was also
shown that the slow time scale is much more significant than the fast time scale. Thus, in order to control
the dynamics of near-field acoustically levitated objects, it is usually sufficient to consider merely their
slow dynamics, implying that a relatively slow control system is adequate. Nevertheless, due to the
nonlinear behavior of the system, originating in the gas trapped between the driving surface and the
levitated object, to achieve satisfactory performance, the control algorithm should be nonlinear as well.
In this paper, a novel semi-analytical model, capturing the slow dynamics of a near-field acoustically
levitated object, using a single, second order ordinary differential equation, was derived. Based on this
model, a continuous gain-scheduled PID controller was designed, and provided adequately good
performance, although the simplified model on which it is based underestimates the real damping of the
system at significant air-gaps. Thus, it is the authors' opinion that based on the performance presented
above, the simplified model developed in this paper is suitable to serve as the basis for more sophisticated
HIGH FREQUENCY ACTUATORS AND PROCESSES 1591
model based controllers. Moreover, it is safe to assume that an empirically corrected damping term would
have resulted in a superior controller.
Finally, it was demonstrated that since the thickness of the squeeze-film affects the stiffness and damping
of the system, and because temperature and moisture variations may change the properties of the
entrapped gas, the resonance frequency is constantly varying. Moreover, due to the low damping of the
piezoelectric actuator, small deviations from the resonance, can reduce the excitation magnitude
dramatically. Thus, in order to implement a control loop, governing the dynamics of near-field
acoustically levitated objects, the latter must include a resonance-tracking algorithm.
Acknowledgements
This research was funded by the Israeli Ministry of commerce under the Metro 450, Magnet program and
the Ministry of Science, Technology and Space.
The authors would like to express their gratitude to Dr. Nadav Cohen who designed the experimental
setup, and Mr. Ran Shaham who built the resonance-tracking loop and helped implementing the control
algorithm.
References
[1] M. Wiesendanger, Squeeze film air bearings using piezoelectric bending elements, Ecole
Polytechnique Fdrale de Lausanne, Lausanne: EPFL. (2001).
[2] D. Ilssar, I. Bucher, On the slow dynamics of near-field acoustically levitated objects under High
excitation frequencies, Journal of Sound and Vibration. 354 (2015) 154–166.
[3] W.E. Langlois, Isothermal squeeze films, DTIC Document, 1961.
[4] W.A. Gross, L.A. Matsch, V. Castelli, A. Eshel, J.H. Vohr, M. Wildmann, Fluid film lubrication,
John Wiley and Sons, Inc., New York, NY, 1980.
[5] A. Minikes, I. Bucher, S. Haber, Levitation force induced by pressure radiation in gas squeeze films,
The Journal of the Acoustical Society of America. 116 (2004) 217–226.
[6] D. Ilssar, I. Bucher, H. Flashner, Modelling and closed loop control of near-field acoustically
levitated objects, arXiv. (2016).
[7] G. Reinhart, J. Hoeppner, Non-contact handling using high-intensity ultrasonics, CIRP Annals-
Manufacturing Technology. 49 (2000) 5–8.
[8] B.-T. Chu, R.E. Apfel, Acoustic radiation pressure produced by a beam of sound, The Journal of the
Acoustical Society of America. 72 (1982) 1673–1687. doi:10.1121/1.388660.
[9] Y. Hashimoto, Y. Koike, S. Ueha, Acoustic levitation of planar objects using a longitudinal vibration
mode., Journal of the Acoustical Society of Japan (E). 16 (1995) 189–192. doi:10.1250/ast.16.189.
[10] Y. Hashimoto, Y. Koike, S. Ueha, Transporting objects without contact using flexural traveling
waves, The Journal of the Acoustical Society of America. 103 (1998) 3230–3233.
doi:10.1121/1.423039.
[11] A. Minikes, I. Bucher, Coupled dynamics of a squeeze-film levitated mass and a vibrating
piezoelectric disc: numerical analysis and experimental study, Journal of Sound and Vibration. 263
(2003) 241–268.
[12] D. Ilssar, I. Bucher, N. Cohen, Structural optimization for one dimensional acoustic levitation devices
– Numerical and experimental study, ISMA 2014 International Conference on Noise and Vibration
Engineering, Leuven Belgium, 2014.
[13] Y. Wang, B. Wei, Mixed-Modal Disk Gas Squeeze Film Theoretical and Experimental Analysis,
1592 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
International Journal of Modern Physics B. 27 (2013).
[14] A. Minikes, I. Bucher, Noncontacting lateral transportation using gas squeeze film generated by
flexural traveling waves—Numerical analysis, The Journal of the Acoustical Society of America. 113
(2003) 2464–2473. doi:10.1121/1.1564014.
[15] A. Minikes, I. Bucher, Comparing numerical and analytical solutions for squeeze-film levitation
force, Journal of Fluids and Structures. 22 (2006) 713–719.
[16] H. Nomura, T. Kamakura, K. Matsuda, Theoretical and experimental examination of near-field
acoustic levitation, The Journal of the Acoustical Society of America. 111 (2002) 1578–1583.
[17] E. Matsuo, Y. Koike, K. Nakamura, S. Ueha, Y. Hashimoto, Holding characteristics of planar
objects suspended by near-field acoustic levitation, Ultrasonics. 38 (2000) 60–63.
[18] S. Yoshimoto, H. Sekine, M. Miyatake, A non-contact chuck using ultrasonic vibration: Analysis of
the primary cause of the holding force acting on a floating object, Proceedings of the Institution of
Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 224 (2010) 305–313.
doi:10.1243/09544062JMES1557.
[19] R.E. Best, Phase-Locked Loops, McGraw Hill Professional, 2003.
[20] G.-C. Hsieh, J.C. Hung, Phase-locked loop techniques. A survey, Industrial Electronics, IEEE
Transactions on. 43 (1996) 609–615.
[21] I. Kaminer, A.M. Pascoal, P.P. Khargonekar, E.E. Coleman, A velocity algorithm for the
implementation of gain-scheduled controllers, Automatica. 31 (1995) 1185–1191. doi:10.1016/0005-
1098(95)00026-S.
[22] K.J. Aström, R.M. Murray, Feedback systems: an introduction for scientists and engineers, Princeton
university press, 2010.
[23] W.J. Rugh, J.S. Shamma, Research on gain scheduling, Automatica. 36 (2000) 1401–1425.
[24] H.K. Khalil, Nonlinear systems, Prentice hall New Jersey, 2002.
HIGH FREQUENCY ACTUATORS AND PROCESSES 1593
1594 PROCEEDINGS OF ISMA2016 INCLUDING USD2016