DSpace at VNU: Analytical modeling of a silicon-polymer electrothermal microactuator

Microsyst Technol
DOI 10.1007/s00542-015-2700-7

TECHNICAL PAPER

Analytical modeling of a silicon‑polymer electrothermal
microactuator
Huu Phu Phan1 · Minh Ngoc Nguyen1 · Ngoc Viet Nguyen1 · Duc Trinh Chu1 

Received: 13 July 2015 / Accepted: 30 September 2015
© Springer-Verlag Berlin Heidelberg 2015

Abstract  This paper illustrates both thermal and mechanical analysis methods for displacement and contact force
calculating of a novel sensing silicon-polymer microgripper when heat sources are applied by an electric current
via its actuators. Thermal analysis is used to obtain temperature profile by figuring out a heat conductions and convections model. Temperature profile is then applied into
the mechanical structure of the gripper’s actuators to form
the final equation of displacement and contact force of the
jaws. Finally, the comparison among the calculation, simulation and actual measurement concludes that materialization methods are appropriate. Achieving the final equation
of gripper’s jaws displacement and contact force is a major
step to optimize or reform this novel structure for different
sizes to meet specific applications.

1 Introduction
In recent years, microelectromechanical systems (MEMS)
have been widely applied in diverse science and engineering domains (Cheng et al. 2008). MEMS-based microgrippers provide advantages in terms of their compact size
and low cost, and hence play an important role in microassembly and micromanipulation fields for manipulating
micromechanical elements, biological cells (Cheng et al.
2008; Zhang et al. 2013). During the past two decades,
microactuators based on different actuation principles
such as shape-memory alloys, electrostatic, electrothermal,

* Huu Phu Phan

1



University of Engineering and Technology, Vietnam National
University, Hanoi, Vietnam

piezoelectric, pneumatic and electromagnetic approaches
have been employed to drive MEMS microgrippers (Jiang
et al. 2007; Hsu et al. 2002; Chen et al. 2009; Beyeler et al.
2007; Chu Duc et al. 2007). Moreover, the integrated position and force sensors can deliver real-time feedback signals to protect both the microgripper and grasped object
from damaging (Menciassi et al. 2003; Chu Duc et al.
2006; Chronis and Lee 2005).
Owing to different properties of actuators, the MEMS
microgrippers exhibit diverse dedicated performances to
various applications. For example, electrostatic actuation
microgripper can provide a large displacement with no hysteresis in a low operating temperature along with a simple
structure (Chan and Dutton 2000). Specifically, two different types of movement configuration in terms of lateral
comb drive and transverse comb drive can fulfill the objective of high precision and large movement, respectively
(Beyeler et al. 2007). In addition, electrothermal actuator
can generate a large output force and displacements by
making use of its thermal expansion with a small-applied
voltage (Chu Duc et al. 2007). On the other hand, the large
force output, precision displacement and rapid response are
the attractive points of the piezoelectric actuator. Besides,
electromagnetic actuator and pneumatic actuator driven
microgrippers can provide a relatively large output force
and displacement (Butefisch et al. 2002; Lee et al. 1997).

It is known that force sensing is necessary for a delicate micromanipulation task. Nonetheless, before the
force feedback sensor is applied, the optical method has
been widely studied (Miao et al. 2004; Rembe et al. 2001).
Recently, researchers showed a great interest in the sensors
with high resolution and sensitivity. In order to enhance the
reliability and safety of the manipulation, the integrated
position and force sensors such as piezoelectric sensor,
piezoresistive sensor and capacitive sensor were designed

13




to provide the real-time position and force information
(Chu Duc et al. 2006; Menciassi et al. 2003; Chronis and
Lee 2005). Thanks to the advances in the technologies, the
sensitivity and resolution of the sensor have been improved
substantially.
A novel design of polymer-silicon electrothermal integrating force sensor microgripper is presented and characterized (Chu Duc et al. 2007b, c; 2008). The device consists
of laterally stacked structures based on a three-element
composite: the metal heating layer, heating conducting silicon structures and a polymer. The heat is highly efficient
transferred from the heater to the polymer by employing
the high heat conduction rate of the deep silicon serpentine
structures that have a large interface with the surrounding
polymer. The proposed device is based on the SU8-2002
polymer with a large thermal expansion coefficient. This
design overcomes the weakness of the other designs and it
boats a large lateral jaw movement with low coupled vertical motion and fast response time. Another advantage is
that the device is made of regular silicon wafers which are

compatible with CMOS technology fabrication process.
Thus, control circuits can be integrated into the structure
with the sole manufacturing process.
For the characterization of the microgripper, HP4155A
semiconductor parameter analyzer is used. The displacement
is monitored by the CCD camera on the top of the probe
station. The thermal behavior of the microgripper is investigated by using a built-in external heat source from the Cascade probe station. A DSP lock-in amplifier SR850 is used
to characterize the response frequency of this sensing microgripper. The sensing microgripper (490 μm long, 350 μm
wide, and 30 μm thick) can be used to grasp an object with
a diameter of 8–40 μm. A microgripper jaws displacement
up to 32 μm at applied voltage of 4.5 V is measured with
a maximum average working temperature change of 176 °C.
The output voltage of the piezoresistive sensing cantilever
is up to 49 mV when the jaws displacement is 32 μm. The
force sensitivity is measured as being up to 1.7 nN/m and
the corresponding displacement sensitivity is 1.5 kV/m. The
bandwidth frequency of this sensing microgripper is 29 Hz.
The minimum detectable displacement and minimum detectable force are estimated to be 1 nm and 770 nN, respectively
(Chu Duc et al. 2007b, 2008). These characteristics make the
sensing microgripper entirely suitable for applications where
force feedback is needed, such as microrobotics, microassembly, minimally invasive and living cell surgery.
Although measurements were conducted throughout,
initial simulation model for this device was quite simple.
COMSOL—a finite element modeling tool is used to simulate the operation of this sensing micro-gripper. Threedimensional models are employed to analysis the elasticity,
displacement, temperature distribution of the microactuator. The model only comprises one transmission which is

13

Microsyst Technol

from assumed heat inputs to movements, while the gripper works base on a voltage source. Therefore, completely
model (with electrical input parameters) and careful analysis are needed to improve the accuracy of the simulated and
calculated values and physical properties of the gripper.
The heat transfer and mechanical calculation of the
microgripper basing on thermal, mechanical and thermal–
mechanical combination analysis are presented in this
paper. Firstly, the operation principle of the sensing microactuator based on silicon-polymer electrothermal actuator
and piezoresistive force sensing cantilever is thoroughly
understood using thermal and mechanical analysis. Following these steps, calculation results are compared with 3-D
simulation and the fabricated sample characterized parameters for verification of gripper’s mathematical equations.
Finally, a method for structure optimization is proposed
basing on combination of changing equations’ factors and
the simulation.

2 Design and operation of silicon‑polymer
electrothermal microactuator
The microgripper is designed for the normal opened operating mode with two actuators on opposite sides. Each actuator has a silicon comb finger structure with the aluminum
metal heater on top (Chu Duc et al. 2007d). A thin layer
of silicon nitride is employed as the electrical isolation
between the aluminum structure and the silicon substrate.
Each actuator consists of silicon comb fingers with SU8
polymer layers in between. When the heater is activated,
the generated heat is efficiently transferred to the surrounding polymer through the deep silicon comb finger structure
that has a large interface area with the polymer layer. The
polymer layers expand along lateral direction which leads
to bending displacement of the actuator arms.
The design of the actuator is shown on Figs. 1 and 2,
which is the right arm of the sensing microgripper system. Ideally, both arms of the gripper are similar geometry
and characteristic. Therefore, calculations and simulations
of the gripper are took place on one arm. The structure is

based on the combination of a silicon-polymer electrothermal microactuators and piezoresistive lateral forcesensing cantilever beams. When the electrothermal actuator
is warmed up by applying electric current through its aluminum heater, the microactuator’s arm and also the sensing
cantilever are bent. This causes a difference in the longitudinal stress on the opposite sides of the cantilever, which
changes the resistance values of the sensing piezoresistors.
Due to the correlation of the displacement of the microactuator jaws and resistance of piezoresistors, positions of the
actuator jaws can be monitored by the output voltage of the
Wheatstone bridge of the piezoresistive sensing cantilever


Microsyst Technol
Aluminum heater

Polymer

GND

V+

Anchor
Silicon

MoƟon direcƟon

Fig. 1  Schematic drawing of the silicon-polymer electrothermal
microactuator

HSi

L

Fig. 3  SEMS pictures of a fully sensing electrothermal silicon-polymer microactuator; b removed silicon cantilever configuration

HAl

L comp

HSU8

Wbone
Wgap

Wcan

L jaw
Silicon

SU- 8 polymer

(see Fig. 3). The configuration without silicon cantilever
removes the heat conduction through the sensing cantilever
for analyzing the mechanism operation of the electrothermal actuator (Fig. 3b). The actuator displacement is then
calculated by using a traditional mechanical method.

Aluminum

3.1 Thermal analysis
Fig. 2  Front-side view of the silicon-polymer electrothermal microactuator with geometry symbols and parameters

beam. Besides that, the contact force between the microactuator jaws and clamped object is then determined, relying
on displacement and stiffness of microactuator arms (Chu

Duc et al. 2007b).
Readers are referred to the Ref. (Chu Duc et al. 2007b,
c, 2008) for further information on this proposed sensing
microactuator.

3 Silicon‑polymer electrothermal actuator
The fabricated electrothermal silicon-polymer microactuator and its geometry dimension parameters is shown in
Fig. 3 and Table 1 respectively. A full sensing microactuator illustrates with 490 µm long, 110 µm wide and 30 µm
thick. The design, fabrication and initially characterization
of the proposed sensing microgripper are reported (Chu
Duc et al.).
Figure  3a is SEM picture of the sensing microgripper based on silicon-polymer electrothermal actuator
with a force sensing silicon cantilever. Beside the sensing function, the cantilever heat energy in the electrothermal actuator is conducted to the anchor through this
cantilever.
In this work, the heat transfer is analyzed based on
two configurations without and with the silicon cantilever

The whole structure is heated by the aluminum layer on the
surface of silicon bone which is considered the main heat
source of the microgripper. When aluminum filament terminals are connected to a power source, that layer is heated
by the Joule-Lenz’s law. In that case, the thermal energy is
transferred to the silicon-polymer stack. The polymer layers, after heated up, expand in the x-axis, causing bending
displacement of the actuator arms. In general, conduction,
convection and radiation are three mechanisms of heat flow.
The electrothermal actuator is operated in the air ambient
where two heat transfer mechanisms in analysis: conduction in the actuator and convection to the surrounding air
are considered. Because the working temperature is lower
than 500°K, the radiation transfer can be neglected (Howell
and Robert Siegel).
The major thermal dissipation is caused by the conduction to the silicon substrate and the convection to the air.

Temperature can be assumed to be uniform throughout the
thickness because it is very thin; therefore the actuator is
regarded as a one-dimensional case. Eventually, calculations and analysis are conducted in x-axis while y-axis is
ignored.
In the steady state, the heat is stored in volume unit
between x and x + ∆x given by (Stephen 2001):
x + ∆x

QG =

qG .y.dx

(1)

x

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Microsyst Technol

The heat loss from left and right side of polymer-silicon
stack is given by:

∂T (x + ∆x) ∂T (x)
−
∂x
∂x

QC = .t.y.

(a)

(2)

The heat loss due to convection is expressed by:
(b)

(3)

Qconv = −2α(T (x) − T0 ).y.∆x
Implying the conservation law:

(4)

QG + QC + Qconv = 0
The equation of temperature by x is then obtained as:
′′

T (x) −

qG + 2αTair
qG + 2αT0
2α
T (x) = −
=−
t
t

t

(5)

This is the quadratic differential equation which has the
root given by:

T (x) = C1 .e

2α
tx

−

+ C2 .e

2α
tx

+ C3

(6)

=0
Applying boundary conditions: T(0) = T0; dT (x=L)
dx
The coefficients C1, C2 and C3 are given by:
2α
tL

qG −
2α .e

C1 = −

2α
tL

e

−

+e

qG
2α .e

C2 = −

2α
tL

e
C3 = T0 +

2α
tL

(7)

2α
tL

+e

−

2α
tL

qG
t

(8)

(9)

For the fabricated microgripper, the resistor of aluminum layer is about 149.018 Ω. When it is applied a voltage of 4 V, the heat power is calculated about 0.107 W.
Thus, qG ∼
= 6.941e6 W/m2 (it is the power dissipation over
the aluminum filament area).
The value of α is in the range from 2 to 25 W/m2K
(Howell and Robert Siegel), thus, the highest value of
2αTair is 15 × 103 W/m2. Comparing to the value of
qG = (6.941) × 106, the convection is neglected, therefore:
′′

T (x) = −

qG

t

(10)

Following is the outcome of a similar method, we have
the function describes the temperature distribution in the
actuator:

T (x) = −

qG 2 qG L
x +
x + T0
2 t
t

(11)

Figure 5 shows the calculated temperature distribution in
the microgripper. It is clear that the steady state temperature

13

Fig. 4  Cross-side and front-side view of the removed silicon cantilever structure for thermal analysis

Table 1  Geometry of the sensing microactuator design
Parameters

Symbol Str. 1 Unit

Actuator/cantilever length

L

390

Actuator/cantilever thickness

T

30

µm

Silicon finger width

HSi

6

µm

SU-8 layer width

HSU8

3

µm

Aluminum heater width

HAl

2

µm

Comb finger width

Wc

75

µm

Silicon bone structure width

Wb

10

µm

Gap between actuator and silicon cantilever Wgap

22

µm

Cantilever width

Wcan

12

µm

µm

Microactuator jaw length

Ljaw

100

µm

Aluminum thickness

TAl

0.6

µm

The heat capacity of actuator

C

J/kg.K

The mass density of actuator
The cross section area

ρ
A

kg/m3
m3

The temperature in x-axis

T

K

The conductivity coefficient
The convection coefficient

λ
α

W/m.K
W/m2K

The heat source

Q

J

distribution rises dramatically along the actuator in form of
half parabola. The maximum temperature of actuator peaks
nearly 270 °C at the tip when 4 V between two terminals of
the aluminum heater is applied.
3.2 Mechanical analysis
Considering that the microgripper is a bimorph cantilever
that consists of two different materials: the silicon-polymer
stack layer and silicon layer. It can be supposed as single
material bars because these parts are calculated to obtain
apparent parameters. Therefore, this simplified model is


Microsyst Technol

(a)

(b)

Fig. 7  Cross-side and front-side view of the silicon-polymer electrothermal microactuator for thermal analysis

Fig. 5  Calculated temperature distribution on the microgripper

The displacement d of the bimorph cantilever is:

d=

2
kcur Lact

2

for

(13)

Lact ≪ ρ

It is assumed that the zero point of x-axis is the border
between the anchor and the actuator (Fig. 4a). Considering
component dx with temperature is T(x), radius of the activating actuator’s curvature can be calculated by applying
the Timoshenko calculation at x (Stephen 2001), as illustrates below:

6(αstack − αSi )(1 + m)2 ∆Tx

kcur−x =

(Wc + Wb )(3(1 + m)2 + (1 + mn) m2 +

1
mn

)
(14)

Fig. 6  Sketch of the bimorph structure consisting of the silicon bone
and the silicon-polymer lateral stack composite

The average temperature:
x

probably appropriate for the structure. When the bimorph
cantilever is heated, causing the different expansion of two
materials, the cantilever is bent as shown in Fig. 6 (Chu
Duc et al. 2007c).
It is assumed that the average temperature increases ∆T,
and the bending displacement of microgripper is d. Thus,
the curvature of cantilever can be calculated as follows:

1
Tx =
x

(−

qG 2 qG L
qG 2 q G L
x +
x + T0 )dx = −
x +
x + T0
2 t
t
6 t
2 t

0

(15)
The average temperature increase ∆T:

∆Tx = −

qG 2 qG L
x
x +
6 t
2 t

(16)

The curvature of whole structure based on x coordinate:
kcur

6(αstack − αSi )(1 + m)2 ∆T
1
= =
ρ
(Wcomb + Wbone )(3(1 + m)2 + (1 + mn) m2 +

1
mn

)

(12)
where αSi is the thermal expansion coefficient (CTE) of
silicon; αstack is the apparent CTE of the silicon-polymer
Si
b

stack; n = EEstack
, m= W
Wc ; ESi is the Young’s modulus of
silicon; Estack is the Young’s modulus of silicon-polymer
stack.

kcur =
=

1
ρ
6(αstack − αSi )(1 + m)2
(Wc + Wb )(3(1 + m)2 + (1 + mn) m2 +
−

qG 2 qG L
x +
x
6 t
2 t

1
mn

)

(17)

13

Microsyst Technol

Thus, the displacement d of cantilever based on x
coordinate:

d=
=

2α
tx

T (x) = C1 .e

kcur x 2
2

2α
tx

−

+ C2 .e

(24)

+ C3

Inserting the Eq. (23) into (24), we obtain:
2
6(αstack − αSi )(1 + m)2 Lact

(Wc + Wb )(3(1 + m)2 + (1 + mn) m2 +

qG
2α

C3 = T0 +
1
mn

)

x4
qG
(− + Lx 3 )
4 t
3

(25)

Applying boundary conditions:
T(0) = T0
Thus,

(18)

4 Microgripper based on silicon polymer

electrothermal actuator with sensing function

C1 + C2 + C3 = T0

4.1 Thermal analysis

The sensing actuator has a silicon cantilever beam
(Chu Duc et al. 2007b, d) where existence of heat transfer
between silicon-polymer stack and cantilever beam is:

Figure  7 shows the cross-side and front-side view of the
proposed silicon-polymer electrothermal microactuator for
thermal analysis. Thermal energy in the actuator is diffused
to the anchor as the heat-sink by heat conduction transfer
through the actuator-anchor interface and also silicon cantilever, see Fig. 7b. Beside the conduction, heat energy is lost
by convection to the surrounding air, see Fig. 7a.
The temperature can be assumed to be uniform throughout the thickness due to the thickness of the structure is much
smaller than other geometry parameters. Thus, the temperature of y-axis is uniform so that the actuator is regarded a
one-dimensional case. Therefore, the electrothermal microactuator can be simplified as a bar shown in Fig. 7b.
In the steady state, the heat is stored in volume unit between
x and x + ∆x given by (Arfken 1985; Trodden 1999):

Si

dT (x = L)
= −qcond
dx

C1

2α
e
t

2α
tL

(19)

qcond

C1 = −

QC = .t.y.

2α
tL

e

(20)

The heat loss due to convection is expressed (Arfken
1985; Trodden 1999; Snieder 1994):

2α
tL

=−

(21)

+e

C3 = T0 +

+

2α
t

Si

C2 = −

2α
tL

(29)
2α
tL

2α
tL

qG
2α .e

+e

−

(30)
2α
tL

qG
2α

(31)

Temperature profile on the silicon cantilever is also calculated by:

Tcan (x) =

qcond

x + T0

(32)

At x = L, T (L) = Tcan (L), so that:

Applying the conservation law:

QG + QC + Qconv = 0

(22)

The equation obtains:

qG + 2αTair
qG + 2αT0
2α
T (x) = −
=−
T (x) −
t
t
t
′′

qcond =

Si
2α
t

This is the quadratic differential equation which has the
root given by:

13

qG
2α (
1

(23)

(28)

Si

Si

Qconv = −2α(T (x) − T0 ).y.∆x

qcond

2α
tL

qG −
2α .e
−

qcond

e

∂T (x + ∆x) ∂T (x)
−
∂x
∂x

+

2α
t

Si

x

The heat loss in the left and right side of silicon-polymer
stack is given by [36]:

2α −
e
t

− C2

The coefficients C1, C2, C3 are given by

−

qG .y.dx

(27)

The equation becomes:

x+∆x

QG =

(26)

2

2α L
−
t +e

e

−

e
e

2α L
t −e
2α L
−
t +e

2α L
t
2α L
t
2α L
t

− 1)
(33)

−L

The temperature distribution in the actuator is given by

inserting the Eq. (33) into the Eqs. (29), (30), (31).


Microsyst Technol
2α
t

Let τ = L

2
( eτ +e
−τ −1)

qG
C1 = −
2α

e−τ −eτ
eτ +e−τ

−τ

+ e−τ

(34)

eτ + e−τ

−
qG

C2 = −
2α

2
( eτ +e
−τ −1)
e−τ −eτ
eτ +e−τ

eτ

−τ

+ eτ

(35)

+ e−τ

Thus,
τ

τ

T (x) = C1 .e L x + C2 .e− L x + T0 +

qG
2α

(36)

C1, C2 are given in Eqs. (34) and (35).
Basing on the function of temperature with real parameters coming from fabricated version of the gripper, Fig. 8
plots profile of temperature versus the length (the resistor of
aluminum layer is about 149.018 Ω, heat power is 0.136 W
when applied voltage of 4.5 V, and qG ∼
= 8.784e6W/m2).
As shown in the results of temperature distributions
chart, temperature is varied from the base to tip with a parabolic form appropriate to earlier calculation. The arrangement on the cantilever is linear and peaks at nearly 200 °C
at the tip.
Due to the existence of cantilever, the former analytical
manner to attain tip’s displacement is ineffective. The displacement and gripping force at the tip is calculated by the
direct displacement method.

Fig. 8  The calculated temperature profile on sensing microactuator

A

EAB, AAB, IAB

B
h1

C
E

ECD, ACD, ICD

D

h
h2

F

EEF, IEF

G
L

Ljaw

Fig. 9  Frame structure to analyse the sensing microactuator

A

∆Τ

B

C

∆Τ

D

4.2 Thermal–mechanical analysis
A simplified structure which is used to analyze the sensing
microactuator under the change of temperature is shown in
Fig. 9. It can be seen from the figure that lines AB, CD and

EF denote beam elements representative of silicon-polymer
stack, silicon bone, and silicon sensing layers, respectively.
Those beam elements are fixed on one end, and connected
together by a rigid beam BDF on the other end. Denote Eij,
Aij and Iij to be Young’s modulus of material, cross-section
area and moment of inertia of cross-section for the beam ij,
respectively.
The silicon-polymer stack beam AB length increases
when power is applied. The performance of the siliconpolymer stack is analyzed based on the hydrostatic pressure according to the constraint effect (Chu Duc et al.
2007d). Note that for the beam AB, equivalent values of
the above parameters are adopted on (Chu Duc et al.
2007d).
In this calculation, it is assumed that the change of average temperature on elements AB and CD is ΔT.

E

F

Z1

B’

D’
Z2
y(T)

F’

Fig. 10  Deformation of the structure

4.3 Sensing microactuator displacement analysis
Figure 10 shows the deformation of the structure under the
change of temperature in beams AB and CD. As shown in
the graph, Z1 and Z2 denote the unknown rotation and vertical deflection of the rigid beam BDF. Here, it is assumed
that the axial expansion of elements EF is neglected.
In order to calculate the displacement and the output
force at the jaw tip of the sensing micro gripper, the direct

13




Microsyst Technol
2 E AB I AB

4 E AB I AB

L
4 ECD I CD

2 ECD I CD

∆Τ

B

C

∆Τ

D

L

Z1 = 1

L

A

E

L

F

h1

B’

h
h2

D’
G

F’

F

h3

Manipulating object

2 EEF I EF

4 EEF I EF

L

F

L

Fig. 13  Structure for solving the output force

Fig. 11  Diagram of the bending moment due to Z1 = 1

6 E AB I AB

6 ECD I CD

Note that axial forces in elements AB and CD due to
Z2 = 1 are zero. Based on conditions for static balance of
the structure, stiffness coefficients are given by:

L2

r11 =

r22 =

2

L

r12 =

6 EEF I EF

L2

Z2 = 1

2
4EAB IAB
AAB
+ 4ECDL ICD + 4EEFL IEF + h EAB
L
L
12ECD ICD
12EAB IAB
12EEF IEF
+
+
,
L3
L3
L3
6ECD ICD

6EAB IAB
6EEF IEF
+ L2 + L2
r21 = −
L2

+

h22 ECD ACD
,
L

Components R1(T) and R2(T) of the force vector are calculated basing on axial forces on elements AB and CD due
to the change of temperature ΔT. We have

R1 (T ) = α1 ∆TEAB AAB h + α2 ∆TECD ACD h2 ,
R2 (T ) = 0
Fig. 12  Diagram of the bending moment due to Z2 = 1

(37)

Z1 (T ) =

r12 R1 (T )
r22 R1 (T )
; Z2 (T ) = −
det K
det K

r11 r12

; Z(T ) =
r21 r22

Z1 (T )
Z2 (T )

; and R(T ) =

R1 (T )
R2 (T )

(38)
Equations (38) denote stiffness matrix of the structure,
displacement vector and output force vector, respectively.
To determine stiffness coefficients, unit displacements are
applied. Diagrams of bending moments in structural elements are performed in Figs. 11 and 12 in the cases of
Z1 = 1 and Z2 = 1.
Axial forces in elements AB and CD under the applied
unit displacement Z1 = 1 are given by:
AB

N 1 = ∆LAB ELAB AAB =
CD
N 1 = ∆LCD ELCD ACD =

13

hEAB AAB
,
L

h2 ECD ACD
L

(39)

(42)

where det K = r11r22 − (r12)2.
Vertical displacement y(T) of the microactuator jaw tip
under the change of temperature is therefore given by

where
K=

(41)

Solving the Eq. (1), we yield

displacement method is used. Under the change of temperature ΔT, the governing equation for the system is given
by:

KZ(T ) = R(T )

(40)

y(T ) = Z1 (T )LJaw + Z2 (T ) =

R1 (T )
r22 Ljaw − r12
det K

(43)

Sensing microactuator output force analysis
Figure  13 illustrates the structure which estimates
the contact force between the microactuator jaw and the
manipulating object.
The unknown force F is calculated from the following
condition:

y(T ) + y(F) = h3

(44)

where y(F) denotes the vertical displacement due to the
reaction force F.
Like the above procedure, the governing equation for
solving the displacement y(F) is given by:

KZ(F) = R(F)

(45)


Microsyst Technol

Fig. 14  Displacement of the microgripper jaw tips at steady state vs.
average working temperature

where

R1 (F) = −FLjaw ; R2 (F) = −F
Solving the Eq. (45), we yield

Z1 (F) = det1 K {r22 R1 (F) − r12 R2 (F)},
Z2 (F) = − det1 K {r12 R1 (F) − r11 R2 (F)}

(46)

The vertical displacement of the jaw tip due to the reaction F is given by:
y(F) = Z1 (F)LJaw + Z2 (F)
= det1 K r22 R1 (F) − r12 R2 (F) Ljaw + R2 (F)r11 − R1 (F)r12

(47)
Taking the above result into Eq. (44), we obtain the
value of gripping force:

F=

R1 (T ) r22 Ljaw − r12 − h3 det K
r11 − 2r12 Ljaw + r22 (Ljaw )2

(48)

5 Measurement, calculation, simulation results
and discussions
The design, fabrication and initially characterization of
the proposed sensing microgripper is reported in the ref
(Chu Duc et al. 2007b; Chu Duc et al.) and calculation
results were mention in this paper. In addition, a 3-Dimention computer model of this device which comprises two

conversions (electricity to heat and heat to movement) for
simulating in virtual medium. Elasticity, movement, temperature profile, power consumption of the actuator are
generated by COMSOL (Comsol Inc.)—a finite element
modeling tool - based on conversion of electricity to heat

Fig. 15  The temperature profile on sensing microactuator

and then heat to movement. It is necessary to make comparisons with relevant results to conclude the consistency
of each deductive method. Therefore, those methods are
tethered for an effective reconfirmation of the new size
sensing microgripper system such as optimization parameters for a specific application before fabrication steps are
carried out.
Figure  14 shows calculation, simulation and measurement results over the average working temperature of the
sensing microgripper. As can be seen from the chart, there
are similar patterns of calculation and simulation which
have approximately 30 % of deviation. To clarify, errors of
each method can contribute to this nonconformity. Firstly,
in calculation, there was not mention resistance changes
of aluminum layer when temperature varies, and some
variables such as convection and radiation were neglected.
Besides, there could be unknown structural ties that skipped
in simplifying gripper’s structure for the calculation. Secondly, some parameters of the model are fixed in ideal conditions and simulation results are gathered from perfectly
surrounded environment. Regarding measured results, most
of discrete points are in the range of the two lines of simulation and calculation. In addition, these plots can be linear
fitted into a single line that in the same channel with previous lines. In short, results of displacement versus average
working temperature of three methods are uniform.
As regard to distribution of heat at steady state when
voltage source (4.5 V) is applied to activate the gripper,
Fig.  15 illustrates temperature profiles on both actuator
and cantilever which are obtained by calculation and

simulation. Due to the limitation of the measurement
method (Chu Duc et al.), temperature on each position
of actuator and cantilever could not be gathered precisely. Thus, the results measured in comparison with

13




these of other methods are ignored. Obviously, there are
striking similarities in the results of calculation and simulation, and therefore, not only the mathematics methodology but also simulation model of the microgripper are
confirmed.
As is shown in the results comparison among methods,
one methodology has confirmed the accuracy of others and
vice versa. Although there are some errors and tolerances
of each method itself, the model for simulations and calculation scheme is appropriate. Consequently, it is an important factor to improve or adapt the gripper’s structure to
specific application. Moreover, it can be used to optimize
the structure in a particular aspect. For example, a new
microgripper which performs the same range of displacement with the original one, but the operating temperature
below 100 °C can be designed. Firstly, determine size of
the gripper (the number of polymer stacks or the length of
actuator) by using the final equation. After that, conducting the simulation with model which has obtained parameters from the first steps, and therefore, the design via those
results is affirmed.

6 Conclusions
The design of sensing polymer-silicon electrothermal
microgripper was proposed, characterized and simulated.
This device has many advantages in comparison with other
actuators, such as large movement, fast response time,
low driving voltage and CMOS compatible. However, it

is required analytical modeling to fully comprehend each
parameter of the design. Therefore, optimization and adaptation for the new dimensional microgripper are based on
mathematical functions that have obtained.
Analysis methods for this proposed sensing siliconpolymer microgripper are introduced in this paper. Firstly,
gripper’s temperature profile is calculated by using the
heat conductions and convections model. Secondly, final
displacement and contact force equations are formed by
inserting temperature profile into the mechanical model.
Finally, the direct displacement method is used for this
device’s displacement and output force analysis. Besides,
functional parts of the gripper is considered and analyzed
separately.
Microgripper operation is based on two main transformations: electricity to heat and then heat to mechanic.
The computation following these two models in turn to
form displacement and clamp force of the actuator is
applied. Deviations between simulation, measurement
and calculation results are not significant. There are great
steps to understand the devices better, and more scientific
approaches to determine the new size of actuator to suit

13

Microsyst Technol

each specific requirement or optimize the design. This proposed microgripper could be potentially used for microparticle manipulation, minimally invasive surgery, and
microrobotics.

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