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Modeling and robust control strategy for a
control-optimized piezoelectric microgripper.
Mathieu Grossard, Mehdi Boukallel, Nicolas Chaillet, Christine
Rotinat-Libersa
To cite this version:
Mathieu Grossard, Mehdi Boukallel, Nicolas Chaillet, Christine Rotinat-Libersa. Modeling and
robust control strategy for a control-optimized piezoelectric microgripper.. IEEE/ASME Transac-
tions on Mechatronics, Institute of Electrical and Electronics Engineers, 2010, PP (99), pp.1-10.
�10.1109/TMECH.2010.2050146�. �hal-00504464�
1
Modeling and robust control strategy for a
control-optimized piezoelectric microgripper
Mathieu Grossard, Mehdi Boukallel, Nicolas Chaillet, Member, IEEE, and Christine Rotinat-Libersa
Abstract—In this paper, modeling and robust control strat-
egy for a new control-optimized piezoelectric microgripper are
presented. The device to be controlled is a piezoelectric flexible
mechanism dedicated to micromanipulation. It has been previ-
ously designed with an emphasis to control strategy, using a
new topological optimization method, by considering innovative
frequency-based criteria. A complete non-linear model relating
the voltage and the resulting deflection is established taking
into account hysteresis as a plurilinear model subjected to
uncertainties. The approach used for controlling the actuator
tip is based on a mixed High Authority Control (HAC) / Low
Authority Control (LAC) strategy for designing a wide-band
regulator. It consists of a Positive Position Feedback (PPF) damp-
ing controller approach combined with a low-frequency integral
controller which is shown to have robustness performances as
good as a RST-based robust pole placement approach for the
microgripper. The rejection of the vibrations, naturally induced
by the flexible structure, and the control of the tip displacement
have been successfully performed. Because we had taken into
account frequency-based criteria from the first designing step of
our device, we demonstrate that the tuning of the HAC/LAC can
be easily performed and leads to low regulator order.
Index Terms—Flexible structure, hysteresis, microrobotics,
piezoelectric actuator, positive position feedback, robust control,
vibrations control.
I. INTRODUCTION
T
O achieve micromanipulation tasks, microgripper
devices are often compliant mechanisms, i.e. single-
bodies, elastic continua flexible structures that transmit a
motion by undergoing elastic deformation [1]. They are
opposed to jointed rigid body motions of conventional
articulated mechanisms. These compliant mechanism-based
micromanipulators are often coupled with smart materials
for actuation, such as PZT (Lead Zirconate Titanate)
piezoceramic. Piezoelectric actuation has become widespread
in micromanipulation systems where high positioning
accuracy is needed [2].
Manuscript received on May 5th, 2010.
M. Grossard and C. Rotinat-Libersa are with the Interactive Robotics
Laboratory, LIST, CEA, F-92265 Fontenay aux Roses, France ;
M. Boukallel is with the Sensory and Ambient Interfaces Laboratory, LIST,
CEA, F-92265 Fontenay aux Roses, France ;
N. Chaillet is with the Automatic Control and Micro-Mechatronic Sys-
tems Department, FEMTO-ST Institute, Unité Mixte de Recherche CNRS
6174 -University of Franche-Comté (UFC)/Ecole Nationale Supérieure de
Mécanique et des Microtechniques (ENSMM)/University of Technology of
Belfort-Montbéliard (UTBM), F-25000 Besançon, France.
Corresponding author: Mathieu Grossard, CEA, LIST, Interactive Robotics
Laboratory, 18 route du Panorama, BP6, FONTENAY AUX ROSES, F-92265
France ; E-mail: mathieu.grossard@cea.fr
phone: + 33 146 549 188, fax: + 33 146 548 980
Paper type: regular paper.
Recently [3], we have developed a prototype of a new
compliant piezoelectric monolithic gripper (Fig.1). It has been
designed using a global systematic approach, based on the
multidisciplinary topology optimization of the flexible struc-
ture. This method is based on the flexible building block
method called FlexIn ("Flexible Innovation"), which uses a
multidisciplinary genetic algorithm to optimize flexible struc-
tures [4] [5] [6]. Details on the design procedure can be found
in [3], and are reminded in section II of this article. When
the active beams of the symmetric jaws of the microgripper
are supplied by voltage, it results in a deformed shape that
produces a symmetric stroke of δ = ±10.51µm and a gripping
force of about 0, 84N along x-axis under ±100V . These static
mechanical characteristics of our device are of the same order
of magnitude of other well-known actuation schemes such as
unimorph or bimorph PZT actuators, widespread in the design
of microrobotic manipulators [1].
Fig. 1. 3D CAD model of the piezoeletric device with top face electrode
patterns (V
left
and V
right
are the controlled inputs for actuating the left and
right arms).
However, when the amplitude of the applied voltage reaches
high values (about 40V), the linear approximation between the
deflection and the voltage is not valid anymore, and hysteresis
phenomenon particularly arises [7]. It exists several methods
to deal with such a nonlinearity : linearization methods (using
charge compensation [8] for example), Preisach model [9]
(but, due to its complexity, the real time implementation of
this technique is difficult), linear or polynomial approximation
models [10]. Let us note that an another cause of precision
loss at the tip of the piezoelectric actuators is the drift due
to creep effects [11]. Another major drawback in using
flexible structure in micromanipulation tasks is the loss of
position control accuracy due to vibrations. Methodologies
used for synthesizing sophisticated robust controllers are not
so intuitive and often lead to high order regulator.
2
For the design of our microgripper (Fig.1), both purely
mechanical criteria (i.e. stroke and force at the tip) and
innovative frequency-based criteria [4] [3] have been used.
These last criteria are useful tools to ensure the efficient
control of flexible structures afterwards. These criteria allow
to take into account resonance amplitude modulations and
optimal pole/zero placement in the frequency spectrum
of the device, so that the designer can fit its open-loop
frequency response function in a desired way [12] [13]
[14]. As a consequence, because our device has been
frequency-optimized, it appears that the synthesis of a simple
robust HAC/LAC regulator is easy afterwards. Indeed, this
methodology, which combines damping controllers with low-
frequency integral controller, is particularly interesting for
the control of flexible structure. It can guarantee interesting
stability margins and lead to low order regulator compared
with classic robust pole placement approaches.
This paper is organized as follows. In the next section, we
briefly remind the underlying ideas of the control-oriented
optimization strategy that lead to the specific design of our
gripper (Fig.1). The non-linear voltage-deflection model in-
cluding hysteresis is presented in section III. In section IV,
the hysteresis model is identified and approximated by a
quadrilateral linear model subjected to uncertainty and with
a varying static gain. In the fourth part, a High Authority
Control (HAC) / Low Authority Control (LAC) controller
is synthesized and implemented to ensure the performances
required in micromanipulation. Finally, in section VI, we show
that our controller can have robustness as good as a RST-based
robust pole placement approach for the microgripper.
II. P
RESENTATION OF THE CONTROL-OPTIMIZED
COMPLIANT PIEZOACTUATOR PROTOTYPE SYNTHESIZED
BY
FLEXIN
Reader can find detailed descriptions of the FlexIn optimiza-
tion tool in [4] [5] [6]. The way the piezoelectrically actu-
ated prototype has been optimally synthesized is extensively
detailed in [3]. In this section, we point out the interests of
having optimized our device from a control-oriented point of
view, in order to ensure the performances required in closed-
loop afterwards.
From the first design step, we had the objective to conceive
a flexible mechanism characterized by two dominant modes
in the targeted low-frequency spectrum (reduced model) in
order to facilitate the identification of an accurate reduced
order model afterwards. In addition, the specific alternating
pole/zero pattern for our flexible structure (i.e. the resonances
and antiresonances alternate in this targeted spectrum) helps
designing regulator, which has good intrinsic stability proper-
ties.
A. Useful criteria for evaluation of dynamic input-output
model performances of flexible systems
Two significant tasks in flexible structure control are
both the identification of the dominant modes to build an
appropriate reduced model and the control strategy design.
1) Evaluation of the model reduction cost: Since the dy-
namic model of a flexible structure is characterized by a large
number of resonant modes, accurate identification of all the
dominant system dynamics often leads to high order models.
A model reduction is often required.
Thus, to facilitate the computation of an accurate identified
model afterwards, a first criterion has been drawn in FlexIn
to optimize the reduced-model accuracy of the systems, while
limiting spillover effects [3]. The optimal structures is the one
guaranteeing the highest joint controllability and observability
for all the modes in the bandwidth of interest, while providing
the minimum joint controllability and observability of the
neglected modes (Fig.2). The numerical formulation of this
criterion can be found in [3].
Fig. 2. Desired form of the open-loop magnitude FRF. Resonance peaks
amplitudes must be maximized in the frequencies bandwidth [0,ω
c
] to
increase authority control on these dominant modes. On the contrary, the
amplitudes of resonance peaks after cut-off frequency must be minimized to
increase gain margin and to limit modes destabilization in this area (spillover
phenomenon)
2) Pseudo-collocated behavior: For some specific class
of flexible structures, which can be modeled as collocated
resonant systems, active damping controllers such as Positive
Position Feedback (PPF) have proven to offer great robustness,
performance, and ease of implementation. They are often
focused on damping the dominant modes [15], [12]. The most
useful characteristic of a collocated system is the interlacing
of poles and zeros along the imaginary axis for a lightly
damped structure. Such systems are minimum of phase. This
results in a phase response that lies continuously between 0
◦
and 180
◦
. Minimum of phase systems are known to possess
interesting properties, especially simple stability conditions.
This makes collocation of the transfer function an attractive
control approach.
Although the flexible structures rarely present natural col-
located behavior, an other optimization criterion, inspired by
[15] and [14], has been used to force the structure to have this
collocated behavior in terms of frequency response function.
B. Presentation of the monolithic compliant piezoactuator
prototype
From the optimal design obtained [3], a symmetric mono-
lithic microactuation mechanism prototype has been fabri-
3
cated, made of a single piezoelectric material PIC151 from
PI Piezo Ceramic Technology [16]. Fig.1 shows the 3D CAD
model of the device with top electrode patterns. The whole
structure is divided into an active (electroded) and a passive
areas, which both will be free to bend. The base remaining
area will be clamped and kept out from bending (Fig.3).
The clamping area is where the electric wires will feed the
electrodes, respectively with V
left
and V
right
for the actuated
left and right arms of the gripper.
Fig. 3. On the left, 3D simulation of the x-displacement δ (in µm) when
the half piezoactuated structure is activated under ±100V , and associated
deformed shape. On the right, photo of the whole machined piezoelectric
monolithic device.
III. NON-LINEAR MODELING OF THE TRANSFER
VO LTAGE
-DEFLECTION
In this section, we draw the electromechanical model of the
piezoelectric device based on experimental measurements. It
consists in a linear model subjected to both uncertainties and
a time varying static gain. A complete model of the voltage-
deflection transfer is considered as the series connection of a
static hysteresis operator and linear dynamics [17] [18].
A. Experimental setup
Fig. 4. Schematics of the experimental setup.
The microactuator prototype is clamped, and placed on x-
y-z micropositioning linear stages, which are manually oper-
ated. The piezoelectric actuator requires high voltage (about
±100V ) to provide micrometric deflection. Thus, the device
is connected to a linear power amplifier, with an amplification
ratio ×50. This last device is controlled via a computer
equipped with Matlab-Simulink software and a NI Labview
PXI board, whose sampling frequency is f
e
=20kHz (Fig.4).
Output displacement at the tip of the piezoelectric structure
is measured along x-axis using a 0.01µm-resolution Keyence
laser sensor. The analog output of the laser sensor is directly
connected to a 4
th
-order low-pass anti-aliasing filter. In the
following, we note F (s) its transfer function, where s is
the Laplace variable. A double Sallen-Key circuitry is tuned
for providing more than 75dB attenuation at f
e
/2 = 10kHz
Shannon frequency (Fig.5) :
F (s)=
V
s
(s)
V
e
(s)
=
1
1+2C
2
Rs + C
1
C
2
R
2
s
2
2
(1)
This filter eliminates data treatment errors that could result
from aliasing and unmodeled high-frequency noise dynamics.
Fig. 5. The active anti-aliasing filter consists of two Sallen-Key circuitries
in series. Chosen values of components are R =2.2kΩ, C
1
=100nF and
C
2
=47nF .
B. Description of the piezoelectric actuator behavior
General electromechanical relations adopted for the
piezoelectrically actuated device are a function of the applied
electrical U and mechanical F
m
stimulations. U refers to
the voltage applied on the upper and lower electrodes of the
microactuator, and F
m
to the mechanical force applied at the
tip of the device. In the present article, our electromechanical
model of the piezoelectric device is based on a parametric
model, which has to be identified in experimentation. We
choose to model the deflection δ along the x-axis using a
control-oriented relationship that is currently adopted for the
piezoelectric actuators [19]. For a more phenomenological
point of view, a macroscopic thermodynamically constitutive
law describing the hysteresis effects, which occur in
ferroelectric ceramics such as PZT, can be found in [20] [21].
Let’s note that a physical electromechanical model in finite
element has been previously used for the design optimization
of our device [3].
According to [19], the deflection is non-linearly linked to
U and F
m
as follows:
δ (s)=s
p
D (s) F
m
(s)+Γ(U (s) ,s) (2)
where s
p
is the elastic compliance, and D (s) the dynamic
part with D (0)=1. Γ(U (s) ,s) is an operator that includes
the hysteresis H (U, s) and the creep C (U, s) non-linearities
in a decoupled way:
Γ(U (s) ,s)=H (U, s)+C (U, s) (3)
4
Let note that notation of Γ depends on both U and s, since,
in the general case, hysteresis depends on the past and present
values of U and also on its frequency.
The creep phenomenon δ
Creep
is the drift of the deflection
observed after the transient part, when a step voltage is
applied to the piezoelectric actuator as shown in Fig.6.It
can be considered as an additional behavior happening when
the steady-state is reached, so that it is often modeled as a
simple delayed transfer [22]. Experimental results show that
the transient part of piezoelectric microactuators are generally
less than 500ms whereas the creep settling time is more than
180s [23]. Usually, the creep is considered as a disturbance
that the controlled system must reject.
Thus, in the following, Γ(U, s) is assumed to be only
modeled by H (U, s) hysteresis term, which represents both
the gain value and the transient part of the electromechanical
transfer.
Fig. 6. Measured creep deflexion of the piezoelectric actuator when a 100V
voltage step is applied at t =0s.
C. Analysis of the hysteresis
A ±50V sine voltage input is applied to the active microac-
tuator, and the displacement output at the tip of the device δ
is recorded. No force is applied at the tip.
Experimental results on Fig.7 show the frequency-dependent
behavior of the hysteresis phenomenon : the shape variation
is due to linear vibrational dynamics [24].
According to [17], [25], [26], we propose to model this
hysteresis by considering the decoupling of the hysteresis
operator H (U, s) into a static hysteresis part H
i
(U) of a
constant shape in series with a linear dynamical part D(s)
(see Fig.8). (It has been proved that the transient part D(s) is
independent of the amplitude of the voltage [23].)
IV. I
DENTIFICATION OF THE PIEZOACTUATOR MODEL
In this section, we focus on the experimental identification
of the piezoelectric actuator device. The vibrational dynamics
are firstly identified. Then, the static hysteresis part is modeled
and identified.
Fig. 7. Measured hysteresis of the piezoelectric actuator for various frequency
voltage inputs.
Fig. 8. Dynamic hysteresis equivalence [22].
A. Identification of the vibrational dynamics
To study and isolate the response due to the induced
vibrations from the creep phenomenon, relatively high-
frequency inputs were used. The input amplitudes were also
kept small so that hysteresis effects could be negligible as
well. The vibrational dynamics are recorded experimentally
by applying to the piezoactuator a low-amplitude sine input
U of increasing frequency. Using a spectrum analyser device
(HP3562A), Bode diagram is recorded and D(s)=
δ( s)
U(s)
transfer is directly identified in Laplace domain.
As expected by FlexIn optimization, the two first resonances
modes are dominant over the following vibrational modes,
and the pole/zero alternate pattern is kept into this desired
spectrum of interest (Fig.9). Identification process is thus
performed considering these first vibrations modes involved in
the reduced model. For identification, we consider a second-
order modal transfer expansion :
D (s)=
2
i=1
k
i
1+
2ξ
i
ω
ni
s +
1
ω
2
ni
s
2
=
N (s)
M (s)
(4)
Damping ratio ξ
1
and ξ
2
are calculated from the measured
quality factor at −3dB on the Bode diagram, as Q
−3dB
i
≈
1
2ξ
i
. Then, values of the natural pulsation ω
n1
and ω
n2
are
easily calculated thanks to the measured resonance frequencies
(Fig.9). Identified modal damping and natural frequency values
are mentioned in Table.I.
Finally, values of the static gain k
1
and k
2
are calculated to
have both the right frequency value for the first antiresonance
