Adaptive Shape Control for Thermal Deformation of Membrane Mirror with In-plane PVDF Actuators
Lu et al. Chin. J. Mech. Eng.
Adaptive Shape Control for Thermal Deformation of Membrane Mirror with In-plane PVDF Actuators
Yi‑Fan Lu 0
Hong‑Hao Yue 0
Zong‑Quan Deng 0
Horn‑Sen Tzou 1
0 School of Mechatronics Engineering, Harbin Institute of Technology , Harbin 150001 , China
1 College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , China
Optical membrane mirrors are promising key components for future space telescopes. Due to their ultra‑ thin and high flexible properties, the surfaces of these membrane mirrors are susceptible to temperature variations. Therefore adaptive shape control of the mirror is essential to maintain the surface precision and to ensure its working performance. However, researches on modeling and control of membrane mirrors under thermal loads are sparse in open literatures. A 0.2 m diameter scale model of a polyimide membrane mirror is developed in this study. Three Polyvinylidene fluoride (PVDF) patches are laminated on the non‑ reflective side of the membrane mirror to serve as in‑ plane actuators. A new mathematical model of the piezoelectric actuated membrane mirror in multiple fields, (i.e., thermal, mechanical, and electrical field) is established, with which dynamic and static behaviors of the mirror can be analyzed. A closed‑ loop membrane mirror shape control system is set up and a surface shape control method based on an influence function matrix of the mirror is then investigated. Several experiments including surface displacement tracking and thermal deformation alleviation are performed. The deviations range from 15 μm to 20 μm are eliminated within 0.1 s and the residual deformation is controlled to micron level, which demonstrates the effectiveness of the proposed membrane shape control strategy and shows a satisfactory real‑ time performance. The proposed research provides a technological support and instruction for shape control of optical membrane mirrors.
Adaptive shape control; Membrane mirror; Thermal deformation; PVDF; Influence function matrix
Along with the development of deep space exploration,
space communication, and earth observation
technology, there is an increasing demand for high-resolution
and wide-field optical mirrors used in space telescopes,
radars, antennas, and energy collectors . Large
aperture mirrors have not only the ability of improving the
clarity of a low-contrast image but also an increased
field-of-vision that can greatly expand the range of earth
observation . Stable rigid materials are used as
substrates for conventional large optical mirrors. These
contribute to high costs, difficulty in processing, and long
manufacturing cycles. Additionally, the dimensions of
the rigid mirror are limited by its weight and the
payload capacity of current launch vehicles. Therefore, a new
technology called membrane optics has been proposed
in recent years. A deployable membrane optical mirror
has many advantages such as being ultra-lightweight and
having a low areal density, which can potentially decrease
the launching cost by reducing the launch vehicle size
requirement. However, with an increase in mirror
aperture and decrease in areal density, the stiffness drops, and
the mirror becomes more vulnerable to outside
interference. It is obvious that surface accuracy is of great
importance to the performance of these mirrors. Large
optical mirrors usually work in medium and
geosynchronous earth orbits. Previous studies have demonstrated
that the main cause of surface shape errors in these
orbits is temperature variations . Thus, the
development of methods for closed-loop surface shape control
of membrane mirrors under thermal loads is of
Many flatness control methods have been explored
including in-plane patching-on/embedding-in,
boundary actuation, and out-of-plane electrostatic actuation
to ensure the surface accuracy of membrane mirrors.
Various smart materials such as shape-memory alloys
(SMAs) , polyvinylidene fluoride (PVDF) films [5–7],
piezoelectric polymer actuators , macrofiber
composites [9, 10], and microelectromechanical system
transducers  have been used to control the surface shape
of membrane structures. A parabolic deformable mirror
with a piezoelectric thin film was studied and controlled
based on an in-plane actuation strategy by Maji et al.
 and the active control precision proved to reach the
micron level. Errico et al.  developed a hybrid control
method combining boundary piezoelectric tension with
transverse electrostatic actuation. A 6-inch-thick
membrane mirror was built and electrostatic pressure was
used to pull the nominally flat mirror to a 32 m radius
of curvature and to adaptively correct its aberrations.
A series of tests on membrane shape control with SMA
wires were undertaken by the Canadian Space Agency. A
generic algorithm-based control system was developed
and tested on a square membrane by Peng et al. [14–17]
and Wang et al. [18–20] investigated active flatness
control of membrane structures under thermal and
mechanical loads with SMA actuators. Furthermore, Shan et al.
 in York University developed a photogrammetry
software to measure the flatness of a membrane structure
that actively boundary controlled by SMA wires.
Mathematical models of the circular membrane mirror were
studied by Jenkins and thermal load effects were taken
into account later [22, 23]. The wrinkling of the
membrane structure according to the thermoelastic boundary
conditions was discussed , and an adaptive control
method was used to improve the surface accuracy of the
membrane reflector . Blandino [26, 27] and Hornig
et al.  also surveyed and experimented on the
wrinkling of membrane structures due to mechanical and
thermal loads. A unimorph piezoelectric-actuated
membrane mirror model was established by Shepherd ,
and a closed-loop shape control that reached the micron
level was achieved. Laslandes et al.  investigated
actuator optimization of an ultrathin deformable mirror,
whereas Rausch et al.  conducted a series of tests for
a piezoelectric-actuated deformable mirror.
In this study, a mathematical model of a PVDF
laminated membrane mirror in thermal, mechanical, and
electrical field is established and expressions of the
membrane forces and bending moments are derived. A 0.2
m diameter in-plane-actuated membrane mirror with
laminated PVDF actuators is constructed for testing.
High-precision laser displacement sensors are used to
measure the surface deviation of the membrane and
serve as feedback for the closed-loop shape control
system. An experimental platform for surface shape control
of the mirror is setup and several experiments are
performed to determine the efficiency of our mirror shape
2 Modeling of the Membrane Mirror
This section describes the study of a PVDF laminated
membrane mirror consisting of two layers (i.e., a
polyimide substrate layer and a PVDF actuator layer). The scale
model of the mirror in the polar coordinate system is
presented in Figure 1, where hs and hp are the thickness
of the substrate and actuator layer respectively. Based
on the Kirchhoff–Love theory for thin shells, the mirror
model can be simplified from the fundamental form and
governing equations to a generic double curvature shell
with specified two Lamé parameters and two radii of
curvature [32, 33].
For the flat circular membrane mirror model, the two
Lamé parameters are A1 = 1 and A2 = r; and the two radii
of curvature are R1 = R2 = ∞. With the four parameters,
one can simplify the double-curvature shell dynamic
equations to the governing equations of the mirror
∂N1 + 1 ∂N6 + N1 − N2 = ρh ∂2u
∂r r ∂θ r ∂t2 − q1,
∂∂Nr6 + 1r ∂∂Nθ2 + 2 Nr6 = ρh ∂∂2tv2 − q2,
∂Q1 + 1 ∂Q2 + Q1 + 1 ∂
∂r r ∂θ r r ∂θ
1 ∂ ∂w
+ rN1 + N6
r ∂r ∂r
= ρh ∂t2 − q3,
∂M6 + 1 ∂M2 + 2M6 − Q2 = 0,
∂r r ∂θ r
∂M1 + 1 ∂M6 + M1 − M2 − Q1 = 0.
∂r r ∂θ r
Substituting Eq. (
) and Eq. (
) into Eq. (
) to eliminate
Q1 and Q2 yields
Figure 1 Configuration of a PVDF laminated membrane mirror
∂2M1 + 2 ∂M1 + 1 ∂2M2 − 1 ∂M2 + 2 ∂2M6
∂r2 r ∂r r2 ∂θ 2 r ∂r r ∂r∂θ
+ r22 ∂∂Mθ6 + N1 ∂∂2rw2 + N2 1r ∂∂wr + 1 ∂2w
where qi is the distributed load component in the ith
direction per unit area; Ni is the membrane force per unit
length; Mi is the bending moment per unit length; Qi is
the shear force per unit length; and u, v, and w are the
displacements in the radial, angular, and transverse
directions. Considering that the size in the thickness direction
of the mirror is usually much less than that in the other
two directions, only the vibration and deformation in
the trans2verse dire2ction are studied. Thus, it is assumed
that ρh ∂∂tu2 = ρh ∂∂t2v = 0. In a real application in space,
q3 denotes the microgravity and solar radiation pressure
component in the transverse direction, while q1 and q2
are ignored. Note that
ρk hk ,
where ρk and hk are the mass density and thickness of
the kth layer respectively, and N is the total layer
number. This membrane mirror model consists of a substrate
layer and an actuator layer; thus, N = 2.
Utilizing the planar stress assumption that the
principal stress normal to the surface is much smaller than the
in-plane stresses, one may write σzz = 0 and define the
constitutive relationship of each layer:
σθθ = cE
εrr c11 c12 c16
εθθ = c21 c22 c26 ·
εrθ c61 c62 c66
γe26 + z kk26 ,
where the membrane strains ei and γi are defined as
+ u +
Lu et al. Chin. J. Mech. Eng. (2018) 31:9
Combining Eq. (
) and Eq. (
), including the thermal,
piezoelectric, and tension force of the laminated
membrane mirror will result in
−N1P − N T + N 0
N2 N2m −N2P − N T + N 0
MMN216 = MMN612mmm + −−MM−21PPM−−6PMMTT
−N1P − N T + N 0
e2 −N2P − N T + N 0
γ6 + −N6P
kk12 −−MM21PP −− MMTT
where the superscripts P and T denote the piezoelectric
and thermal terms respectively, and N0 denotes the
membrane tension on the boundary of the mirror. Note that
the thermal loads have no effect on the shear force and
moment; external boundary tensioning will not induce
shear forces and extra bending moments.
For an N layer laminated structure with total
thickness h, where the kth laminae is located between the two
The bending strains ki are defined as
k1 = −
k2 = − r12 ∂∂2θw2 − 1r ∂∂wr ,
planes z = zk and z = zk+1, the following expressions can
be derived :
cij k zk+1 − zk , i, j = 1, 2, 6,
cij k zk+1 − zk2 , i, j = 1, 2, 6,
cij k zk+1 − zk3 , i, j = 1, 2, 6.
In order to address the coupling terms in matrix Aij,
Bij, and Dij, the neutral axis of the composite mirror is
defined in Figure 2.
Where in Figure 2, hs is the thickness of the substrate,
hp is the thickness of the PVDF actuator, and dNA is the
distance between the bottom surface and the neutral axis.
dNA can be calculated by summing up the in-plane stresses
over the total thickness and equating the sum to zero:
1 + νp
1 + νs
zdz = 0, (
where Ep and Es are the Young’s modulus of PVDF and
substrate; νp and νs are the Poisson’s ratio of PVDF and
substrate respectively. Solving Eq. (
) for dNA yields
1 Eph2p + δEs 2hphs + hs2 ,
dNA = 2
Ephp + δEshs
1 + νp .
1 + νs
In general, the material densities of the two layers are
assumed to be similar and the mirror is constructed such
that hp = dNA. From Eq. (
), one can derive the
hs = hp
α ≡ νs − νp,
1 − νp .
1 − νs
Next, we define two new parameters α and β as
1 νp 0
ν0p 01 (1 − νp)/2
0 1 0
1 0 0 .
0 0 −1/2
Aij = AE Eij,
Bij = (0)3×3,
Dij = DE Eij,
1 − νp2
1 + hp ,
DE = 31 1E−phνpp2
1 + hs .
When the Poisson’s ratio of two materials are similar,
i.e., α ≈ 0, δ ≈ 0, and β ≈ 1, Eqs. (
) can be
Solving Eqs. (
) with Eq. (
) and substituting α
and β result in
Considering the ultrathin property of the
deformable membrane mirror, the axial temperature gradient
can be ignored compared with the radial temperature
distribution, which mainly induces thermal stress and
strain in the optical system. Thus, the temperature field
on the mirror is defined as T = T(r, θ). The thermally
induced force and moment are calculated as
In order to eliminate Ni from Eq. (45), the stress
function φ = φ(r, θ, t) is introduced; this satisfies exactly Eqs.
) and (
) defined by
N T =
1 − ν z
(T − T0)dz
Eshsαs + Ephpαp
1 − νs 1 − νp
(T − T0),
z(T − T0)dz
1 − ν z
1 Eshs2αs − Eph2pαp
2 1 − νs 1 − νp
(T − T0),
where αp and αs are the thermal expansion coefficients of
PVDF and substrate respectively. Based on the
piezoelectric constitutive equations , the piezoelectric induced
forces and moments are derived as
N1P = V3 cos2 θ e31 + sin2 θ e32 ,
N2P = V3 sin2 θ e31 + cos2 θ e32 ,
N6P = V3(cos θ sin θ e31 − cos θ sin θ e32),
M1P = − 2 V3 cos2 θ e31 + sin2 θ e32 hp, ,
M2P = − 2 V3 sin2 θ e31 + cos2 θ e32 hp,
M6P = − 2 V3(cos θ sin θ e31 − cos θ sin θ e32)hp, (44)
where V3 is the applied voltage in the z-direction, and
e31 and e32 are the piezoelectric coupling coefficients of
Substituting Eq. (
) into Eq. (
) and simplifying it with
DE ∇4w + ρh ∂∂2tw2 = N1 ∂∂2rw2 +
∂2 2 ∂
∂r2 + r ∂r
1 ∂2 1 ∂
r2 ∂θ 2 − r ∂r
M6P − ∇2MT + q3.
N2 = ∂∂2rφ2 ,
N6 = −
Laser probe head
Sensing/Actuating control cabinet
Substituting Eqs. (51)–(53) into Eq. (50) and using Eqs.
(46)–(48) to simplify the result, we obtain
∇4φ = Ephp 1 +
Eqs. (49) and (54) are the starting equations for
analyzing the dynamics and statics of the piezoelectric
laminated membrane mirror.
3 Experimental Setup
An adaptive membrane shape control experimental
platform is established as described in this section. The
experimental setup is composed of two systems: a
circular membrane mirror with fixed boundary condition
Aluminum PVDF clamped ring actuators
Figure 5 Details of the mirror testing system
and an active control system. Three laser displacement
sensors produced by Keyence Corporation are used for
measuring the surface deviation of the mirror for
feedback shape control, and three PVDF patches are
laminated to the nonreflective side of the mirror to serve as
actuators for surface shape control.
Figure 3 illustrates the system block diagram of the
PVDF laminated membrane mirror in the laboratory
experiments. A PXI-6284 A/D (analog-to-digital) card
and a PXI-6723 D/A card provided by NI-DAQ (National
Instruments) are used in the experimental setup for data
acquisition of the sensors and outputs of the control
signals. A high-performance embedded microcontroller
PXI-8106 supplied by National Instruments is selected.
The adaptive shape control testing platform of the
piezoelectric laminated membrane mirror is shown in
Figure 4. The hardware consists of a mirror testing system, a
signal acquisition and controlling processing interface, a
sensing/ actuating control cabinet, an industrial personal
computer, and a laser displacement sensor controller. The
sensing/actuating control cabinet contains a
multichannel sensing signal conditioning circuit, an A/D
conversion module, a D/A conversion module, a multichannel
voltage amplifying circuit, and some other parts.
The details of the mirror testing system are shown
in Figure 5. The PVDF laminated membrane mirror is
carefully stretched by hand in all directions to remove
any wrinkles that may be present and the mirror is then
clamped by two aluminum rings on the boundary. The
membrane mirror is supported and fixed on an optical
bench by four aluminum supports. Three laser probe
heads are hung vertically on the test mirror to measure
the displacement of three points on the mirror. The three
points, i.e., the centers of each PVDF actuator, which
form a triangle, are used to represent the approximate
surface deformation of the mirror in this experiment. A
ceramic heater connected to a temperature controller
is placed under the center of the mirror to serve as an
external thermal load. The temperature of the ceramic
heater can be adjusted from room temperature to 200 °C.
Note that the ceramic heater cannot be seen in Figure 5
from this perspective and all leads have been removed
from the PVDF actuators.
Three PVDF patches are laminated on the
nonreflective side of the mirror by rubber cement as in-plane
actuators. Ideally, it is assumed that the PVDF patches
are fixed tightly onto the mirror and there is no relative
displacement between the two layers. The layout of the
PVDF actuators is shown in Figure 5. The angle between
each actuator is 120°. The dimensions of the membrane
mirror and the PVDF patches are given in Table 1.
The properties of the polyimide substrate and PVDF
films are listed in Table 2 and Table 3.
4 Implementation and Results
From the mathematical model established previously,
we can find that the transverse deflection of the
mirror is caused by the piezoelectric forces and moments
induced by the PVDF actuators. However, because of the
nonlinearity and complexity of the model, it is difficult
to obtain analytical solutions and to establish a
closedloop feedback control system for the model. In this
section, a simplified method based on the influence function
matrix (IFM) of the system is used to adaptively control
the shape of the membrane mirror.
4.1 Influence Function Matrix
The IFM of the membrane mirror system is a matrix
obtained by an experiment based on the proportional
error feedback algorithm that can describe the
relationship between the input voltage of each actuator and
surface shape of the mirror under a particular layout of the
PVDF actuators. In this method, the driving voltage of
each actuator can be calculated in real time with laser
displacement sensors that measure the surface in high
frequency, and therefore, closed-loop control is achieved.
Studies conducted in AFIT , PSU , and CIT 
have proved that when implementing small membrane
deflection control with PVDF actuators, the control
effect can be deemed approximately linear, i.e., the
transverse deformation of the mirror is assumed basically
proportional to the voltage applied to the PVDF patches.
Hence, voltages ranging from −200 V to 200 V at 50 V
increments are applied to actuator No. 1 to validate this
basic assumption. The deflection of reference point 1 is
measured and recorded by the laser displacement sensor
and then plotted in Figure 6. The green stars are the
original data collected by the displacement sensor under each
voltage and the red line denotes the fitted straight line.
The linear relationship between the voltage and
membrane deformation can be observed in Figure 6.
With the linear assumption above, the IFM of the
mirror can be obtained by experimental identification. The
centers of the three PVDF patches are selected as
reference measuring positions of the laser displacement
sensors. The three points form a triangle, which is used to
approximately represent the surface shape of the
mirror. Consider each displacement sensor signal as a static
response vector of the mirror and put it in matrix R.
Then the mirror static system could be expressed as
A is the unknown mirror system and X is an
identity 3×3 matrix corresponding to a one-volt
application to each of the actuators. The vectors w1, w2, and w3
represent the displacement response of the mirror for
a one-volt application to actuators 1, 2, and 3
respectively. Basically, A = R. For the control problem, where a
desired surface w represented by three displacement data
is desired, the problem to be solved is
A3×3v3×1 = w3×1,
v3×1 = K 3×3w3×1,
K = A−1.
where K is the IMF of the mirror and is determined using
Table 4 Displacement induced by actuator No. 1 (mm)
Sensor No. 1
Sensor No. 2
Sensor No. 3
Note that the IFM K acted on the error signal in the
feedback path, where the error signal is defined as
w = wdesired − wmeasured..
where v represents the control voltage inputs. By
simplifying the expression, we obtain
y(t) = 10 sin (2πt).
Voltages of ±200 V are applied to actuators Nos. 1–3
in sequence and the displacements of the mirror in each
point are recorded in Tables 4, 5 and 6.
Then, the IFM K can be calculated using the data in
Tables 4, 5 and 6 and Eqs. (56) and (59).
4.2 Membrane Shape Tracking Experiments
To verify the correctness of the calculated value of K, a
series of membrane shape tracking tests were performed.
Different signals including sinusoidal, square, and
linear waves are input into the control system as reference
shape of the mirror and three outputs of the mirror are
For displacement sensor No. 1, a sinusoidal signal is
input as the desired shape:
The membrane shape tracking tests for the three
different desired shape signals are shown in Figure 7(a), (b),
and (c) respectively. The black line indicates the
command input (i.e., the shape signal to be tracked) and the
red dashed line is the closed-loop mirror displacement
response. From Figure 7, it can be observed that the
shape signals are tracked with recognizable accuracy,
which proves the effectiveness of the IFM for the
membrane mirror shape control.
4.3 Adaptive Shape Control for Thermal Deformation
Thermal loads are the main cause of membrane
distortion in outer space. In this section, we describe the
investigation of the thermal deformation of the membrane
mirror induced by a ceramic heater. The room
temperature is approximately 25 °C during the experiment. The
ceramic heater was started and the temperature
controller was set at 40 °C. It is obvious that the membrane
mirror will thermally deform when heated. A linear signal
y(t) = 0 is input into the controller as the desired shape
for sensor No. 1 to 3 to track so that the surface
deviation can be decreased. The controller starts to work at the
eighth second. The time-varying signals of three
displacement sensors are shown in Figure 8(a), (b), and (c). The
black line represents the desired shape signal and the red
dashed line denotes the controlled transverse
deformation of the mirror.
Figure 8 shows that mirror deformation occurs because
of the thermal load and it increases gradually as the
temperature rises. The deviations measured by the three
displacement sensors range from 15 μm to 20 μm. However,
the deviations in all three locations improve remarkably
and rapidly after the controller begins to work at the
eighth second. The residual deformation is controlled to
the micron level. These experimental results confirm the
effectiveness of the proposed membrane shape control
strategy and show a satisfactory real-time performance.
)A mathematical model of a piezoelectric laminated
membrane mirror in multiple fields is established.
With specific Lamé parameters and radii of
curvature, the governing equations of the mirror are
simplified. The strain and stress of the mirror are
calculated, respectively, from which the membrane force
and bending moment expressions are derived.
)A new transverse dynamic/static equation and a
compatibility equation including piezoelectric and
thermal effects are obtained. The dynamic and static
behaviors of the membrane mirror are analyzed
using these newly derived equations.
)Owing to the complexity of the mathematical model,
it is difficult to obtain analytical solutions for
feedback control on the shape of the mirror. A surface
shape control method based on IFM is investigated.
)The IFM of the mirror is identified through
experiments, and shape tracking experiments are then
conducted. Different wave signals are input into the
control system as reference shape and outputs
(transverse deflection) of the mirror are monitored to track
the desired membrane deformation with
) A ceramic heater is placed under the center of the
mirror to serve as a thermal load, and the surface
shape of the membrane is controlled by inputting a
zero signal as the design deformation. It is
demonstrated that with this developed control method, the
thermal deformation of the mirror can be controlled
effectively and rapidly.
Yi‑Fan Lu, born in 1989, is currently a PhD candidate at Research Center of
Aerospace Mechanism Control, Harbin Institute of Technology, China, majoring in
mechatronics engineering. His research interests include sensing and active
control of smart materials and structures. His research topic is now focusing
on quasi‑static and dynamic control of membrane plate and shell structures.
Hong‑Hao Yue, born in 1978, is currently a professor and a PhD candidate
supervisor at Research Center of Aerospace Mechanism Control, Harbin Institute
of Technology, China. His main research interests include mechatronics
engineering and active vibration control of smart structures. E‑mail: block@
Zong‑ Quan Deng, born in 1966, is currently a professor and a PhD
candidate supervisor at Research Center of Aerospace Mechanism Control,
Harbin Institute of Technology, China. His main research interests include
space mechanism control and mobile robot in special environment. E‑mail:
Hornsen Tzou, is currently a professor and a PhD candidate supervisor at
College of Aerospace Engineering, Nanjing University of Aeronautics and
Astronautics, China. His research interests include smart structures and structronic
systems and distributed sensing and control of plates and shells. E‑mail:
YL conceived and designed the study. YL and HY performed the experiments.
YL wrote the paper. ZD and HT reviewed and edited the manuscript. All
authors read and approved the final manuscript`.
This research is supported by the National Natural Science Foundation of
China (Grant No. 51175103) and Self‑Planned Task of State Key Laboratory of
Robotics and System (HIT) (Grant No. SKLRS201301B).
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Springer Nature remains neutral with regard to jurisdictional claims in pub‑
lished maps and institutional affiliations.
1. R Angel . Future very large space telescopes . Space 2000 Conference and Exposition . Long Beach, United states, September 19-21 , 2000 : 5304 .
2. D Gorinevsky , T T Hyde . Adaptive membrane for large lightweight space telescopes . Highly Innovative Space Telescope Concepts , Waikoloa , USA, March 5- 8 , 2002 : 330 - 338 .
3. Y F Lu , H H Yue , Z Q Deng , et al. Research on active control for thermal deformation of precise membrane reflector with boundary SMA actuators . ASME 2014 International Mechanical Engineering Congress and Exposition , Montreal, Canada, November 14-20 , 2014 : V04BT04A061 .
4. E Yoo , J Roh , J Han. Wrinkling control of inflatable booms using shape memory alloy wires . Smart Materials and Structures . 2007 , 16 ( 2 ): 340 .
5. H Furuya , Y Miyazaki , Y Akutsu . Experiments on static shape control of one‑ dimensional creased membrane by piezoelectric films . 43rd Structures , Structural Dynamics , and Materials Conference, Denver, United states, April 22-25 , 2002 : 1377 .
6. M J Shepherd , R G Cobb , W P Baker . Low‑ order actuator influence functions for piezoelectric in‑plane actuated tensioned circular deformable mirrors . Smart Structures and Materials 2006 : Modeling, Signal Processing , and Control, San Diego, United states, February 27-March 2 , 2006 : 61660E ‑61660E‑12.
7. H Fang , M J Pattom , K Wang , et al. Shape control of large membrane reflector with PVDF actuation . 48th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Waikiki, United states, April 23-26 , 2007 : 1842 .
8. S Tung , S R Witherspoon , L A Roe , et al. A MEMS‑based flexible sensor and actuator system for space inflatable structures . Smart Materials and Structures , 2001 , 10 ( 6 ): 1230 .
9. A Doosthoseini , A Salehian , M Daly . Analysis of wrinkled membranes bounded with macro‑fiber composite (MFC) actuators . ASME 2010 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers , Philadelphia, United states, September 28 - October 1 , 2010 : 583 - 588 .
10. E Fleurent , T E Pollock , W Su , et al. Wrinkle localization in membrane structures patched with macro‑fiber composite actuators: Inflatable space antenna applications . Journal of Intelligent Material Systems and Structures , 2014 , 25 ( 15 ): 1978 - 2009 .
11. J Su , T Xu , S Zhang , et al. A hybrid actuation system (HYBAS) and aerospace applications . 2005 Materials Research Society Fall Meeting , Boston, United States, November 28-December 1 , 2005 , 888 : 27 - 34 .
12. A K Maji , M A Starnes . Shape measurement and control of deployable membrane structures . Experimental Mechanics , 2000 , 40 ( 2 ): 154 - 159 .
13. S Errico , J R Angel , B L Stamper , et al. Stretched Membrane with Electrostatic Curvature (SMEC) Mirrors: A new technology for large lightweight space telescopes . Highly Innovative Space Telescope Concepts , Waikoloa, USA, August 22-23 , 2002 : 356 - 364 .
14. F Peng , Y Hu , A Ng . Testing of inflatable‑structure shape control using genetic algorithms and neural networks . AIAA Journal , 2007 , 45 ( 7 ): 1771 - 1774 .
15. F Peng , Y Hu , A Ng . Testing of membrane space structure shape control using genetic algorithm . Journal of Spacecraft and Rockets , 2006 , 43 ( 4 ): 788 - 793 .
16. F J Peng , Y R Hu , A Ng . Development of GA‑based control system for active shape control of inflatable space structures . 2005 IEEE International Conference on Control Applications (CCA) , Toronto, Canada, August 28-31 , 2005 : 577 - 582 .
17. F Peng , X Jiang , Y Hu , et al. Application of SMA in membrane structure shape control . IEEE Transactions on Aerospace and Electronic Systems , 2009 , 45 ( 1 ): 85 - 93 .
18. X Wang , W Zheng , Y Hu . Active flatness control of space membrane structures using discrete boundary SMA actuators . IEEE/ASME International Conference on Advanced Intelligent Mechatronics, AIM 2008 , Xi'an, China, July 2-5 , 2008 : 1108 - 1113 .
19. X Wang , C Sulik , W Zheng , et al. Thermo‑mechanical analysis of thin membranes and application in active flatness control design . Industrial and Commercial Applications of Smart Structures Technologies 2008 , San Diego, United States, March 10 -11, 2008 : 69300H ‑69300H‑12.
20. X Wang , W Zheng , Y R Hu. An experimental study on the interaction of thermal loading and mechanical loading in membrane structures . Strain , 2011 , 47 : 493 - 504
21. J Shan , R Orszulik , M Girin , et al. Flatness measurement and active control for a membrane structure . 2012 IEEE International Conference on Mechatronics and Automation (ICMA) , Chengdu, China, August 5-8 , 2012 : 1062 - 1067 .
22. C H Jenkins , D K Marker . Surface precision of inflatable membrane reflectors . Transactions-American Society of Mechanical Engineers Journal of Solar Energy Engineering , 1998 , 120 : 298 - 305 .
23. C H Jenkins , S M Faisal. Thermal load effects on precision membranes . Journal of Spacecraft and Rockets , 2001 , 38 ( 2 ): 207 - 211 .
24. C H Jenkins , D M Fitzgerald , X Liu . Wrinkling of an inflated membrane with thermo‑ elastic boundary restraint . 41st AIAA Structures , Structural Dynamics, and Materials Conference, Atlanta, USA, April 3- 6 , 2000 : 76 - 83 .
25. C H Jenkins , D K Marker , J M Wilkes. Improved surface accuracy of precision membrane reflectors through adaptive rim control . 39th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit. Long Beach, USA, April 20 - 23 , 1998 : 2302 - 2308 .
26. J R Blandino , J D Johnston , J J Miles , et al. Thin film membrane wrinkling due to mechanical and thermal loads . 19th AIAA Applied Aerodynamics Conference , Anaheim, United States, June 11-14, 2001 : 1345 .
27. J R Blandino , J D Johnston , J J Miles , et al. The effect of asymmetric mechanical and thermal loading on membrane wrinkling . 43rd Structures , Structural Dynamics and Materials Conference, Denver, United States, April 22-25 , 2002 : 1282 - 1292 .
28. J Hornig , H Schoop , U Herbrich . Wrinkling analysis of thermoelastic membranes . Technische Mechanik , 2006 , 26 ( 1 ): 33 - 43 .
29. M J Shepherd . Lightweight in-plane actuated deformable mirrors for space telescopes . Air Force INST of Tech Wright‑Patterson AFB OH School of Engineering and Management , 2006 .
30. M Laslandes , K Patterson , S Pellegrino . Optimized actuators for ultrathin deformable primary mirrors . Appl Opt ., 2015 , 54 ( 15 ): 4937 - 4952 .
31. P Rausch , S Verpoort, U Wittrock . Unimorph deformable mirror for space telescopes: environmental testing . Optics Express ., 2016 , 24 ( 2 ): 1528 .
32. H Tzou . Piezoelectric shells: distributed sensing and control of continua . Kluwer Academic Publishers, 1993 .
33. W Soedel . Vibrations of shells and plates . CRC Press, 2004 .
34. A H Nayfeh , P F Pai . Linear and nonlinear structural mechanics . John Wiley & Sons, Inc., 2008 .
35. J R Hill. High precision surface control of flexible space reflectors . The Pennsylvania State University, 2011 .
36. K D Patterson . Lightweight deformable mirrors for future space telescopes . California Institute of Technology , 2014 .