Introduction
Recently, considerable research efforts have been devoted to achieving
compact large range multi-axis compliant nanomanipulators (). Among these studies, the designs of beam flexure-based
mechanisms with voice coil actuators (VCA) are widely adopted to show the
potentials of achieving millimeters or even centimeters
strokes ().
As the stroke increases, the in-plane parasitic motions increase as well,
which includes translational and rotational motions, and both of them can
significantly affect nanometric motion quality of XY nanomanipulators.
More recently, some important results were reported on reducing parasitic
motions of large stroke XY nanomanipulators, for example, the improved
4-PP structures with subchains connected (), the
connection bars (), and the cross
bars (). Those designs can improve some performance such
as a higher degree of cross-axis decoupling and a smaller in-plane parasitic
rotation. In addition, with 4-PP-E (P denotes a prismatic joint
and E denotes a planar joint) structures, the in-plane parasitic rotation
can be restricted by increasing the rotational stiffness (). Since the in-plane parasitic rotation is dependant of the
in-plane rotational stiffness and moment, the parasitic rotation can be
restricted by increasing the rotational stiffness and decreasing the moment.
Note that the increase of rotational stiffness will increase the stiffness of
the motion axis and hence restricts the stroke, which is undesired. For this
reason, we in this paper propose a novel design method to restrict the
parasitic rotation by means of reducing in-plane moment, but without
increasing rotational stiffness. As a matter of fact, a moment occurs when
the applied force does not pass through the stiffness center. Also the
nonlinear stiffness of flexure mechanisms makes the thrust force do not pass
through the stiffness center when the mechanism is in motion. The problem is
more severe when the stroke gets larger. In order to make the thrust force
always pass through the stiffness center, we propose a novel design, by which
the stiffness center can be self adjusted to the ideal position when the
mechanism is in operation.
Note that the transverse stiffness of the parallelogram flexure is dependent
of the axial force (). Although the transverse
stiffness is nonlinear, the nonlinear stiffness can be leveraged to adjust
the stiffness center. In other words, if the variation of the parallelogram
flexure stiffness is known, then the position of the stiffness center can be
predicted accordingly. In particular, we provide a conceptual design of an XY
compliant nanomanipulator. Then by appropriately setting a set of the
parameters of parallelogram flexure, we show that a self-adjusting stiffness
center (SASC) design can make the stiffness center stationary, while the
thrust force passing through the stiffness center all the time.
The rest of the paper is organized as follows: in Sect. 2, the SASC
design method is proposed. In Sect. 3, a conceptual design of an XY
nanomnipulator with SASC is provided in details. In Sect. 4, a case study
is presented to realize the proposed design. In Sect. 5, numerous FEA
simulations are conducted to validate the analytical results and to show the
performance of the conceptual design, followed by conclusion.
Self-Adjusting Stiffness Center Design
To begin with, we consider the transverse stiffness of the parallelogram and
double parallelogram flexure modules.
Analysis of transverse stiffness of symmetric flexure mechanism
It is known from that the transverse stiffness of a
parallelogram flexure KPt and double parallelogram flexure
KDPt admit the following closed-forms:
KPt=24EIL3+1.2PLKDPt=12EIL3+0.03P2LEI,
where E is the modulus of elasticity, and I is the moment of inertia, and
L is the length of a beam, and P is the axial force.
The axial force P on the a mirror symmetric flexure mechanism is shown in
Fig. , where the motion stage moves along x axis.
Axial forces on a mirror symmetric flexure mechanism.
![]()
From Fig. , the force balance conditions can be derived by
Fx=FAx+FCxPx1=FAx+FCxPx2=FAx,
where Fx is the applied force on x direction, and Px1 and
Px2 are the axial forces on the left and right flexure mechanisms
respectively, and FAx and FCx are the constrained
forces of the actuation motion flexure mechanism and cross motion flexure
mechanism respectively. Note that the actuation motion flexure mechanism is
connected to the base, and the cross motion flexure mechanism is connected to
the motion stage.
According to Eq. (), it is seen that the axial forces on the
flexure mechanism on both sides of motion stage are different when the stage
is in motion, and the difference between forces Px1 and Px2 reads
as
Px1-Px2=FCx.
As a result, the stiffness center moves when the stage is in motion. In order
to make the thrust force pass through the stiffness center when the stage is
in motion, there is a need to know the position of the stiffness center which
will be discussed in the following subsection.
Position of stiffness center
When the applied force goes through the stiffness center, there is no moment
of force, and hence the motion stage only has a translational motion but no
rotational one as shown in Fig. a, where the stiffness
center of the mirror symmetric flexure mechanism coincides with the centroid
of the motion stage. A similar illustration is referred to .
Stiffness center of mirror symmetric flexure mechanism.
![]()
In contrast, when the transverse stiffness of the two flexure mechanisms on
both sides of the motion stage is different, the stiffness center will no
longer coincide with the centroid of the motion stage as shown in
Fig. b. Consequently, the stiffness center moves towards
to the flexure mechanism of the larger stiffness side, meanwhile the applied
force does not pass through the stiffness center, which results a moment M
making the motion stage rotate.
In general, one axis force analysis of a symmetric flexure mechanism can be
considered as a spring-bar model as shown in Fig. , where O
is the centroid of the bar, and C is the stiffness center of the mechanism,
and A and B are the points connecting to the left and right springs,
respectively, and L is the length of bar AB.
A symmetric flexure mechanism.
![]()
As shown in Fig. , the applied force F goes through the
stiffness center C, which means that the bar moves translational without
rotation, and the force balance conditions are derived as
F=(Ka+Kb)yKayLac=KbyLcb,
where Ka, and Kb are the stiffness of the left and right springs
respectively, and y is the translational motion in y direction.
From Eq. (), the deviation of the stiffness center to the
centroid O can be derived as
d=Lac-Lao=LKb-Ka2(Ka+Kb).
With the position d of the stiffness center in mind, we are in place to
propose a novel design making the stiffness center stationary when the stage
is in motion.
SASC design
The moment of force when the stage is in a planar motion.
![]()
For a planar motion, the moment of force is shown in Fig. ,
where OS and O are the centroid of the motion stage and the
overall system respectively; and Cx and Cy are the stiffness
centers of x and y axes respectively; and Fx and Fy are the
applied forces to x and y axis respectively; and x and y are the
displacements of the motion stage along x and y axes respectively; and
dx and dy are the deviations (Eq. ) of stiffness centers of
x and y axes respectively; and Mx and My are the moments caused
by Fx and Fy respectively.
From Fig. , it is obtained that
Mx=Fx(y+dy)My=Fy(x+dx),
which shows that the displacement of the stiffness center can be divided by
two parts, that is the displacement of the motion stage x, y and the
deviation dx, dy caused by different transverse stiffness.
With the above Eq. () in mind, if one can appropriately design the flexure
mechanism such that dx and dy could compensate displacement x and
y, i.e. dx=-x, dy=-y, then the stiffness center Cx and
Cy coincide with the centroid of the overall system O, which means
that the applied forces Fx and Fy always pass through the stiffness
center, and hence there is no moment of force, therefore the rotational
motion can be eliminated. This is the idea of the SASC design, and in what
follows we apply the SASC to design a large range XY compliant
nanomanipulator.
A large range XY nanomanipulator based on SASC design
Schematic of a 4-PP-E parallel mechanism.
![]()
We in this section present a compliant XY nanomanipulator based on the above
proposed SASC to restrict the in-plane parasitic rotation.
Specifically, we consider a 4-PP-E parallel mechanism (Fig. ).
The reason why we adopt such a mechanism lie in: the 4-PP can be realized by a
mirror symmetric configuration, which is clearly beneficial to the improvement of planar
motion accuracy; and the E joint can be designed with SASC, which significantly
reduces the in-plane rotation. Then the parallelogram beam flexures are utilized
to realize kinematic decoupling. Figure a shows the 3-D view of the conceptual design.
A conceptual design of a compliant XY nanomanipulator. (a) 3-D view and (b) top view.
![]()
Figure b shows that the proposed design consists of
four parts (different parts in different colors): the base, the decoupling
mechanism, the SASC-based redundant constraint, and the motion stage. Since
the major stiffness will be distributed to the redundant constraint to
improve the motion quality, we in this study propose an SASC-based redundant
constraint module, while keeping the kinematic decoupling module simple.
SASC-based redundant constraint
As the redundant constraint, the E joint is designed with the
4-PP
mechanism and based on SASC. Specifically, the parallelogram and double
parallelogram flexure beams are utilized for the motions on the cross
actuation and actuation direction, respectively (Fig. b).
Since the transverse stiffness is dependent of axial forces, there is a deviation
between the stiffness center and the centroid of the motion stage, moreover the deviation can be calculated according to Eq. ().
Using x axis as an illustrative example, the SASC-based redundant constraint
module for x axis is shown in Fig. , where Lm is the
side length of the motion stage; and KCty1 and KCty2
are the transverse stiffness of the cross motion parallelogram flexure on the
left and right sides, respectively; and KAtx is the transverse
stiffness of the actuation motion parallelogram flexure; and OS is
the centroid of the motion stage; and C is the stiffness center of the
cross motion parallelogram flexure.
Schematic of the proposed SASC-based redundant constraint module.
![]()
From Eq. (), the deviation dx of the stiffness center can be
expressed as
dx=-LmKCty1-KCty22(KCty1+KCty2).
From Fig. , it is clear that the axial forces of the
parallelogram flexure on both sides of the motion stage are of the same
magnitude P but counter directions when the stage is in motion.
With this, the stiffness can be obtained as
KCty1=24EICLC3+1.2PLCKCty2=24EICLC3-1.2PLCKAtx=24EIALA3+0.06P12LAEIA,
where LA and LC are the length of the beam in actuation and
cross directions respectively, and P1 is the axial force acting on the
actuation mechanism, and P is of the form
P=24EIALA3+0.06P12LAEIAx.
By some algebraic manipulation, we obtain
dx=-3LmIALC25ICLA3x-3LmIALC2P12000E2ICIAx.
By neglecting the second term of Eq. (), we obtain the
following form
dx≈-3LmIALC25ICLA3x.
In general, when a force F is applied to the left side of the motion stage
(Fig. ), the transverse stiffness of the cross motion
parallelogram flexure on the left side will increase, and the transverse
stiffness of the cross motion parallelogram flexure on the right side will
decrease. As a result, the stiffness center of the cross motion parallelogram
flexure moves left w.r.t. O. If one can appropriately design the above
geometric parameters such that dx=-x, then the stiffness center does not
move.
Decoupling mechanism
The decoupling mechanism is also based on a 4-PP mechanism. Note that
P joints for the motions on the actuation and cross directions are realized by
the double parallelogram flexure beams with a mirror symmetrically
arrangement. The advantages of the double parallelogram flexure lie in it can
achieve relatively large stroke and is of less nonlinearity compared with
that of the parallelogram flexure.
Double parallelogram flexure with double-beam
Comparison between different beam designs. (a) double parallelogram
flexure with one-beam; (b) double parallelogram flexure with double-beam;
(c) cross section of one-beam; (d) cross section of double-beam;
(e) FEA result with one-beam; (f) FEA result with double-beam.
![]()
The multi-beam design has been widely used in the literature, and it can
significantly reduce the undesired rotation of flexure
mechanisms (). In this conceptual design, we combine
the double-beam and the double parallelogram flexure to realize the actuation
motion as shown in Fig. , where H is the distance between two
adjacent beams, and b, h and L are the width, thickness and length of
the beam, respectively.
The cross section of different beams is shown in Fig. c–d, and it can be obtained that
I1=bh312I2=bH312-b(H-h)312,
where I1 and I2 are the moment of inertia of the single-beam and the
double-beam, respectively.
According to Eq. (), it is obtained that
Kt1=Ebh3L3+0.0312P2LEbh3Kt2=Ebh34L3+0.0696P2LEbh3,
where Kt1 and Kt2 are the transverse stiffness of
single-beam and double-beam, respectively.
With Eqs. () and (), compared with the combination of
the single-beam and the double parallelogram flexure, the combination of the
double-beam and the double parallelogram flexure has lager moment of inertia
but smaller transverse stiffness. The FEA results are shown in
Fig. e–f, where the above analysis is observed.
Case study
In this section, we present a design of the geometric parameters to realize the proposed SASC, and to make the overall mechanism in a compact desktop size.
Stiffness design
As shown in Fig. , the decoupling and the redundant
constraint modules are connected in parallel. Therefore, the stiffness center
of the stage depends on both of the modules. For the sake of simplicity, the
SASC design is only utilized to the redundant constraint module.
To realize the SASC for the motion stage, the transverse stiffness of the
cross motion flexure of the redundant constraint is designed much higher than
that of the decoupling mechanism. Hence the transverse stiffness of cross
motion flexure of the decoupling mechanism can be neglected. In this case,
the stiffness centers of the motion stage and the redundant constraint
approximately coincide with each other.
A schematic of the proposed design with labeled geometric parameters.
![]()
According to Eqs. () and (), the stiffness models of
the decoupling mechanism and redundant constraint are expressed as follows:
KSASC=48EIA1LA13+IC1LT13Kd=24E2IA2LA23+IC2LC23.
And the stiffness model of the whole system reads as
Ks=24E2IA1LA13+2IC1LC13+2IA2LA23+IC2LC23,
where KSASC, Kd and Ks are the stiffness of the
redundant constraint, the decoupling mechanism and the overall mechanism,
respectively; and the subscripts 1 and 2 denote the redundant constraint
and the decoupling mechanism, respectively.
Geometric parameters of the proposed design.
Parameters
LA1
LA2
LC1
LC2
Lm
Values (mm)
77.21
20.00
72.00
65.00
148.00
Parameters
H1
H2
h1
h2
b
Values (mm)
5.00
2.00
1.00
0.50
10.00
Parameters
IA1
IA2
IC1
IC2
Values (mm4)
0.80
0.10
0.80
0.10
Sensitivity of geometric parameters to α.
∂α∂Lm
∂α∂LA
∂α∂LC
∂α∂hA
∂α∂hC
∂α∂bA
∂α∂bC
Values
0.007
0.039
0.028
3.000
3.000
0.100
0.100
The desired stroke is 1.5 × 1.5 mm2, and stiffness Ks is
designed as 60 N mm-1. According to Eqs. () and (),
the set of geometric parameters is shown in Fig. and the
designed values are given in Table , where LA and LC are
defined in Eq. (), and H and h are defined in
Fig. , and the subscripts 1 and 2 denote the redundant
constraint and the decoupling mechanisms, respectively.
Modal analysis.
![]()
The applied forces in the FEA (a) the redundant constraint (b) the decoupling mechanism (c) the overall stage.
![]()
Comparison of the results on translational motion.
Redundant
Decoupling
Stage
Stiffness (Analytical)
15.72 N mm-1
51.36 N mm-1
67.08 N mm-1
Stiffness (FEA)
15.63 N mm-1
53.28 N mm-1
69.93 N mm-1
Difference
0.58 %
3.60 %
4.07 %
Cross coupling
0.60 %
2.80 %
1.70 %
With the above design, the overall dimension of the proposed XY
nanomanipulator is of 300 × 300 × 40 mm3, which is in a compact
desktop size.
Sensitivity analysis
In this subsection, we conduct the sensitivity analysis of the above
SASC-based design. According to Eq. () and Table 1, set
α as
α=-3LmhA3bALC25hC3bCLA3.
If α=-1, then the SASC is achieved. The sensitivity of the geometric
parameters to α is calculated and listed in Table , where
the thickness of the beams hA and hC have the highest sensitivity to
α.
Consider the accuracy of electrical discharge machining (EDM) as 5 µm,
which leads to an error to α less than 1.5 %. Therefore it is safe
enough to fabricate the proposed conceptual design by means of EDM.
Motion of the redundant constraint in x axis given the maximum displacement in y axis.
![]()
Motion of the decoupling mechanism in x axis given the maximum displacement in y axis.
![]()
Motion of the whole stage in x axis given the maximum displacement in y axis.
![]()
Finite Element Analysis
To validate the proposed conceptual design, numerous FEA simulations are
conducted in this section. Note that the large deflection mode is chosen in
the FEA analysis, and the maximum element size of the beams is set to
0.5 mm.
Modal analysis
Figure shows the FEA results of the modal analysis of the
first six orders. The first two orders correspond to the translational motion
of the two actuation axes, and the third order corresponds to the rotational
motion. Compared with the conventional design, the third order modal is not
much higher than the first two ones. It means that the rotational stiffness
does not increase by the proposed SASC-based design.
Motion simulations
Due to the identity of the two actuation axes, the simulation of the motion
performance is only conducted for x axis, and the results of y axis can be
performed as well. Figure shows the applied forces to the
redundant constraint module, decoupling mechanism and the stage.
Figures – show the results of translational
motions of the redundant constraint module, the decoupling mechanism and the
overall stage, respectively. It is seen from Fig. that the
designed stage is capable of achieving desired millimeters stroke with 100 N
actuation force.
To calculate the the cross coupling error, we set the cross coupling
percentage as
xFy=Fmax-xFy=0xFy=0,
where xFmax is the displacement in x axis while the maximum force
Fmax is applied to y axis, and xFy = 0 is the displacement in x axis while there is no force applied to y axis. In the simulations, the
initial motion in y axis means that the maximum displacement (1.5 mm) is
given in y axis. As a result, there is an initial axial force applied to x axis. To better show the results of Figs. –,
we compare the results in Table . It is seen from Table
that the cross-axis coupling of the SASC-based redundant constraint module is
much better than that of the decoupling mechanism without the SASC (0.6 versus 2.8 %). Also the cross-axis coupling of the overall stage is
1.7 %, which is in a reasonable good range (1–2 %) from the literature,
for example ().
Parasitic rotation of the proposed stage (moves along x axis with initial motion on y axis).
![]()
Parasitic rotation of the stage without the SASC (moves along x axis with initial motion in y axis).
![]()
Results on parasitic rotation
In this subsection, the simulation of the parasitic rotation is conducted. As
a result, the parasitic rotation of the redundant constraint is < 1 µrad,
which demonstrates the performance of the proposed SASC-based design. And the
parasitic rotation of the decoupling mechanism is below 40 µrad. One
possible reason is mainly due to the relatively low rotational stiffness. The
parasitic rotation of the proposed stage is shown in Fig. ,
where the parasitic rotation is < 8 µrad. The maximum parasitic rotation
occurs when the motion stage reaches to its middle stroke, and this result
agrees with the shape of the moment variation.
In order to demonstrate the advantage of the proposed SASC-based design, another XY nanomanipulator without SASC is analyzed as
shown in Fig. . The two nanomanipulators are designed with the same rotational stiffness.
Comparing Fig. with , it is clear that the one with the SASC has much better
performance on reducing the parasitic rotation.