Effect of Degree-of-Symmetry on Kinetostatic Characteristics of Flexure Mechanisms: A Comparative Case Study
He et al. Chin. J. Mech. Eng.
Effect of Degree-of-Symmetry on Kinetostatic Characteristics of Flexure Mechanisms: A Comparative Case Study
Xiao‑Bing He 0 2
JingJ‑un Yu 0 2
WanW‑an Zhang 0 2
Guang‑Bo Hao 1 2
0 School of Mechanical Engineering and Automation, Beihang University , Beijing 100191 , China
1 School of Engineering‐Electrical and Electronic Engi‐ neering, University College Cork , Cork , Ireland
2 Authors' Information Xiao‐Bing He, born in 1994, is currently a master candidate at School of Mechanical Engineering and Automation, Beihang University , China. E‐mail: Jing‐ Jun Yu, born in 1974 , is currently a professor at Beihang University , China. His research interests include mechanisms and robotics. Tel:
3 86-10- 82313904; E‐mail:
The current research of kinetostatic characteristics in flexure mechanisms mainly focus on the improvement of accuracy. To reduce or eliminate the parasitic motion is considered as an approach by using the common knowledge of symmetry. However, there is no study on designing the flexure mechanisms with symmetrical features as many as possible for better kinetostatic performance, when considering the resulting cost by the symmetry. In this paper, the concept of degree of symmetry (DoS) is proposed for the first time, which is committed to symmetry design in the phase of conceptual design. A class of flexure mechanisms with 0‑ DoS, 1‑ DoS, 2‑ DoS and 3‑ DoS are synthesized respectively based on the Freedom and Constraint Topology method. Their overall compliance matrices in an analytical form formulated within the framework of the screw theory are used to analyze and compare the effect of different number of DoS on the kinetostatic characteristics for flexure mechanisms. The finite element analysis (FEA) simulations are implemented to verify the analytical results. These results show that the higher the DoS is, the smaller the parasitic motion error will be. The flexure model with 3‑ DoS is optimized according to the overall compliance matrix and then tested by using the FEA simulation. The testing result shows that with the best combination parameters, the parasitic motion error for 3‑ DoS mechanism is almost eliminated. This research introduces a design principle which can alleviate the unwanted parasitic motion for better accuracy.
Flexure mechanism; Symmetry; Kinetostatic characteristics; FEA simulation
Nature can always inspire humans to create various
useful devices/instruments. From observing the natural
structures and movements of living organisms,
mechanical designers regard strategic use of symmetry as a
powerful design tool. The symmetrical design in flexure (aka
compliant systems) can be found everywhere in nature,
from a mirror-symmetry bird wing in the macro world to
a large variety of axis-symmetry protein structures in the
micro world. Apart from the facts in the natural world,
symmetrical geometry also exhibits a wide use in the
artificial world. In compliant mechanisms [
design is important to guarantee the stability the overall
desired performances in which symmetry creates
balance, harmony, order, and aesthetically pleasing results
Flexure mechanisms, with the inherent advantages
of selective compliance characteristics [
], have been
widely used in the field of precision engineering, such
as scientific instruments, optical alignment devices,
micro-/nano-positioning stages, precision
manufacturing machines . These flexure mechanisms are
typically hard to design compared to their rigid counterparts.
Because the accuracy of flexure mechanisms is highly
sensitive to many external disturbances, such as vibration
and thermal variations, and also some intrinsic factors,
such as material property and mechanism configuration.
In order to formulate an index for accuracy,
parasitic motion is defined as any undesirable motion along
the constraint directions of a mechanism [
are several methods to reduce and even to eliminate
the parasitic motion. The first method is to tune the
structural parameters and material properties without
changing the type of flexure mechanisms. Li et al. [
analyzed a family of [PP]S parallel mechanisms and took
the 3-PRS parallel mechanism as an example to reveal the
relationship between structural parameters and parasitic
motion, and then showed the necessary structural
condition for a 3-PRS parallel mechanism without parasitic
motion. However, it is rather difficult to eliminate the
parasitic motion by optimizing the geometrical
parameters. The second one is designing a parasitic-motion
compensation module, as done in linear-motion flexure
mechanisms . Trease et al. [
] and Cannon et al. [
constructed a linear-motion flexure mechanism with
higher accuracy by mirroring two double-parallelogram
flexure modules. A class of compliant Roberts
mechanisms can also be combined both in serial and parallel to
compensate for the parasitic motion [
]. In respect to
the multi-axis motion mechanism, an extended parasitic
motion compensation approach that characterizes 3D
flexure deformations with twists and parasitic error with
compliance elements is proposed to synthesize multi-axis
flexure mechanism [
]. It is noted that symmetry design
is essentially a special case of the latter method, which is
to design a system free of parasitic motion directly.
Several practical full-symmetrical compliant mechanisms
have been studied by Hao et al. [
], in which
trisymmetrical planar structures enabled three large-range
This paper aims to explore the method of parasitic
motion free design with aid of the knowledge of
symmetry. A new term named as the Degree of
Symmetry (DoS) is coined for advancing systematical design,
with a particular emphasis on type synthesis of flexure
mechanisms. It can be argued that synthesis of flexure
mechanisms is more difficult than that of their
rigidbody counterparts. Therefore, the attempt in this paper
is to design a group of specific flexure mechanisms with
one rotational degree-of-freedom (DOF) and one
translational DOF that are parallel to each other. Each one is
composed of several beams uniformly contained in two
planes. Based on the known methodology, in this paper a
class of flexure mechanisms with different DoS are to be
synthesized, and further to be used to identify the
relationship between the number of symmetrical planes and
their kinetostatic performance characteristics.
In fact, there exist a dozen of literatures about
symmetry design, but few design concerns towards how much
the symmetry obtained can provide better performance.
In these prior art, researchers generally design a class of
symmetrical flexure mechanisms firstly, and then analyze
their parasitic motions, and finally draw a conclusion that
the symmetry design can effectively improve accuracy
and other performances. Is it necessary to design the
flexure mechanisms with symmetrical planes, axes, or
points as many as possible for reducing or even
eliminating the parasitic motion, when considering the resulting
cost by the symmetry? This paper will focus on the effect
of the DoS on the kinetostatic characteristic of flexure
mechanisms with a comparative case study.
The rest of this paper is organized as follows. Section 2
provides an introduction to the Freedom and Constraint
Topology (FACT) method as well as the equivalent
constraint model of selected flexure primitive within the
framework of the screw theory. A group of 2-DOF
flexure mechanisms with X-DoS are synthesized by using the
graphic method (FACT) in Section 3. Their overall
compliance matrices for evaluating parasitic motions are
formulated, followed by the FEA simulation in comparisons
with the analytical models in Section 4. Based on the
forehead context, Section 5 discusses the effect of DoS
on the kinetostatic performance. Finally, conclusions are
drawn (Additional file 1).
2 Theoretical Foundation
2.1 Definition of Degree‑of‑Symmetry
Symmetry is one of the most important of all properties
in the identification of mechanisms. It is well known that
symmetry is always described by reference to symmetry
planes, axes and the center of symmetry. In this paper,
the Degree-of-Symmetry (DoS) is specifically constrained
with plane symmetry.
A plane of symmetry is an imaginary plane that bisects
a structure into halves, which is a mirror image of the
other. It is a symmetry of a pattern in the Euclidean plane.
For a flexure mechanism, it can have one or more planes
of symmetry. Thus, the Degree-of-Symmetry reflects the
number of planes of symmetry in the mechanism. For
example, 1-DoS mechanism means there is only a plane
of symmetry and it can be XY-plane, YZ-plane or
ZXplane when the mechanism is placed in the Cartesian
coordinate system. The visualization of DoS is shown in
2.2 Type Synthesis Approach
As well known, some systematic approaches including
the constraint-based design method [
] and the
Freedom and Constraint Topology (FACT) method [
have gained a great success in the design field of flexure
mechanisms. Using the graphic FACT approach, which is
based on the connection of screw theory with the
constraint-based design theory in a geometrical way, is very
powerful for designing simple cases. It has clear
meaning to map geometrical entities such as lines and plane, to
physical elements such as the compliant beams.
Furthermore, all of them can be included in the chart of FACT,
and modelled in the freedom spaces or constraint spaces.
In this regard, a freedom space of a rigid body represents
all of its allowable motion in space when subjecting to a
specified constraint arrangement. While the constraint
space represents all possible constraint arrangements in
such a prescribed motion pattern.
What is more significant, the FACT approach can
be completely embedded into the framework of screw
theory, making the compliance matrix characterized by
screw theory more powerful [
], since it offers critical
geometric insight into various motion behavior of flexure
mechanisms, including the metrics quantifying parasitic
motion of flexure mechanisms [
2.3 Coordinate Transformation of Screws and Compliance
Screw theory underlies the foundation of both
instantaneous kinematics and statics [
]. Physically, a unit
zeropitch screw presents a pure rotation or a revolute pair in
kinematics, or a unit pure force in static along the line in
space. A unit infinite-pitch screw denotes a pure
translation or a prismatic pair in kinematics or a pure couple
in statics, as shown in Figure 2. One calls a screw a twist
if it represents the instantaneous motion of a rigid body,
and a wrench if it denotes a system of forces and couples
acting on the rigid body. Surely, a wrench and a twist can
be also used to describe the motion of a rigid body
supported by a compliance structure.
When a load is applied on the functional body of a
general flexure mechanism, as shown in Figure 3, it will
generate some specified deformation or motion. In this
paper, it is assumed that the deformation is sufficient
small so that the linear elastic theory can apply.
In this context, the transformation between a
deformation twist ξ = (θ; δ) = (θx, θy, θz; δx, δy, δz)T and the load
wrench F = (τ; f) = (τx, τy, τz; fx, fy, fz)T is represented by
a 6 × 6 compliance matrix C to formulate the mapping of
compliance, written as
ξ = CF .
However, compliance matrices may undergo a
transformation representation when calculating the overall
compliance matrix of a flexure mechanism. This requires the
compliance matrices of all flexures to be implemented in
a uniform coordinate frame. Let ξ and F be the twist and
the wrench with respect to a global coordinate frame,
while ξ′ and F′ be the twist and the wrench with respect
to a local coordinate frame. For a general Euclidean
change of coordinate, and suppose that the coordinate
transformation is represented by a 3 × 3 rotation matrix
R and a translation vector t = (x, y, z)T, an adjoint
representation [Ad] between a local coordinate frame and a
global one, the transformation matrix, is determined by
where T is a 3 × 3 skew-symmetric matrix defined by the
translation vector t.
Thus, the compliance matrix C with respect to the
global coordinate frame can be finally obtained as:
ξ = [Ad]ξ ′ = [Ad]C′F ′ = C [Ad] F ′,
C = [Ad]C′[Ad]T,
where Δ is an operator for transforming the axis
coordinate into the ray coordinate and C′ is the compliance
matrix with respects to the local coordinate frame.
Calculation of the resultant compliance of a general
flexure mechanism with serial, parallel or a hybrid
topology is different. For a serial flexure mechanism, the
deformation of the end-effector is the superimposition of the
deformation of individual elements. When the
compliance of the ith flexure element is denoted by Csi, the
overall compliance matrix of a serial flexure mechanism
is calculated as
where [Adi] is the coordinate transformation operator
from the ith flexure to the global frame.
For a parallel flexure mechanism, the overall stiffness
matrix of a parallel flexure mechanism Kp is the sum
of individual element stiffness in the same coordinate
frame, calculated as
K p =
where Cpj denotes the compliance of the ith flexure
element (Additional file 1).
In an overall compliance matrix, the principle
diagonal elements are always considered as the reference of
rotational degrees of freedom about x, y and z-axes as
well as the reference of translational degrees of freedom
along x, y and z-axes [
]. Other non-principal diagonal
compliance elements can be used to indicate the parasitic
2.4 Equivalent Constraint Model of Flexures
In the family of flexure mechanisms, a beam is widely
used as a basic flexure element both generating twist
deformations and providing wrench constraints. In terms
of the difference in profiles, the beams can be classified as
notch-type ones (such as circular flexures) and uniform
ones (such as wire flexures, plate flexures); straight ones
and initial-curve ones; and slender ones (Euler–Bernoulli
beams) and short ones (Timoshenko beams). Different
profiles of these beams definitely lead to variance in
freedom and constraint due to their compliance properties.
As shown in Figure 4, when the cross section of a beam
is circular, the coordinate frame is located at its centroid
C, and the compliance matrix of the uniform wire beam
with the length l and the radius r of cross sections can be
Cc = diag 12lE3Iy 12lE3Ix ElA ElIx ElIy GlJ ,
where A = πr2, Ix = Iy = πr4 4, J = Ix + Iy = πr4 2.
By comparing the compliances of the wire flexure with
circular cross section in different directions, the
equivalent constraint model can be established for realizing the
simplification from structure to topology. When ratio of
the length to the radius is larger than 40, we have
cc11 = cc22 =
1 l 2
= 3(1 + μ)r2 ≥ 1000.
From the above results, it can be observed that this
flexure offers several orders of magnitude higher stiffness
Figure 4 Wire flexure characterized with circular cross section
along its axis compared with any other direction. We can
thus conclude that a slender cylinder flexure, with ratio
of the length to the radius being larger than 20,
approximates an ideal wire flexure imposing a rigid constraint
along its z axis and allowing other five DOFs. Therefore,
the constraint model equivalent to this kind is a wire
3 Type Synthesis and Parasitic Error Analysis
of X‑DoS Flexure Mechanisms with Cylindrical
3.1 Type Synthesis
In this section, we deal with the design of a group of
flexure mechanisms characterized by the different number of
DoS. The type synthesis approach we used is the graphic
FACT, which is intuitively visible and preferable if the
cases are not so complicated. Since the main purpose
to this paper is on the relationship between the number
of DoS and the kinetostatic performances, some simple
parallel flexure mechanisms are appropriate enough in
certain sense. Moreover, all flexure elements employed
here are identical with a uniform circular cross section
for convenience. Based on the knowledge of equivalent
constraint model above, we built a general flexure
mechanism formed by connecting a moving platform to a base
one through wire flexure elements, as shown above in
Figure 3, which will generate deformation on the moving
platform when undergoing a generalized load.
Type synthesis of flexure mechanisms starts with
specifying a freedom space. The objective is to find all beams
with circular cross section in a parallel arrangement to
perform the desired motion. The following will describe
a general procedure for the type synthesis by taking the
flexure mechanisms with cylindrical motion for instance.
Step 1. Denote the specified freedom pattern of a
flexure system with one rotational motion and one
translational motion whose axes are parallel to each other. The
freedom space is depicted in Figure 5.
Step 2. Find the complementary line constraint space,
which represents those available constraints of the
flexure systems, based on the chart of FACT, as illustrated in
Step 3. Determine all possible reciprocal line subspaces.
In the subspaces, an axis is set up to have the identical
direction with that of the desired DOFs, and constraints
of the flexure systems should be found in an easier way.
One possible constraint subspace is illustrated in Figure 7.
Step 4. Select constraint subspace types in terms of
different level of symmetrical geometry from constraint spaces
obtained in Step 3. Note that the constraint subspaces
should be realized physically as illustrated in Figure 8.
Note that the flexure mechanisms constructed based
on the above steps can be classified into four types, i.e.,
0-DoS type, 1-DoS type, 2-DoS type, and 3-DoS type,
as shown in Figure 8. Generally, when formulating the
screw-based compliance models, the global
coordinate system is placed at the mass center of the moving
platform where an important geometrical insight into
Figure 5 Desired freedom space
the motion characteristic of flexure mechanisms can be
revealed. However, instead of designing a hybrid
flexure mechanism, by applying a mirror-symmetrical serial
connection, the global coordinate system is located at
the middle part between the moving platform and the
fixed platform for the 3-DoS flexure mechanism. All four
mechanisms have been elaborated in Figure 9.
3.2 Compliance Modelling
As sketched in Figure 9(c), the flexure mechanism
characterized by two symmetrical planes is formed by
connecting a moving platform to a fixed one through four
identical circular wire flexures (r is the radius of cross
sections) in parallel. Two parallel flexures (labeled with 1
and 2) intersect, with an angle 2θ, at the middle of one
side edge on the moving platform, and span, with the
distance a of two end points, on the fixed platform. The
other two flexures are arranged similarly with an
interval distance d. A global coordinate frame is located at the
center of the moving platform, and the local coordinate
frames are located at the center of each flexure element.
All axes of the local/global coordinate frames are labeled
in Figure 8.
Note that the compliance of each element can
be obtained under the uniform coordinate frame
although the compliance matrix about each
flexure center, which has deduced in Eq. (6), is
identical. The rotational matrices Ri and the translational
vector ti = (x, y, z)T associating with each flexure for
calculating adjoint transformations [Eq. (2)] are listed
in Table 1, the rotation matrices are expanded as below:
where θ = 0 if the flexure elements are perpendicular
with two platforms.
Based on Eqs. (2) and (3), the corresponding overall
compliance matrix of the 2-DoS flexure mechanism with
respect to the global coordinate frame is derived as
which can be re-written in the form as
c11 0 0 0 c15 0
0 c22 0 c24 0 0
C2−DOS = 00 c42 0 c44 0 0 .
0 c33 0 0 0
c51 0 0 0 c55 0
0 0 0 0 0 c66
By analyzing the principal diagonal elements of the
matrix, the type of degree of freedom of this flexure
mechanism can be easily demonstrated. In addition,
other non-principal diagonal compliance entries can be
considered as the reference of parasitic motion errors
Note that each entry in the compliance matrix of the
2-DoS mechanism is determined by the material and
geometric properties, such as the cross-section radius
of wire beam, the angle between two intersecting beams.
Therefore, it is difficult to write the expressions of all
entries explicitly. Luckily, enlightened by the
combination with screw theory and kinematics, the form of
overall compliance matrix reveals whether there is parasitic
motion or not. Thus, in the initial qualitative preliminary
analysis, we only focus on the form of each overall
compliance matrix. As a result, the overall compliance
matrices, with respect to the corresponding defined global
coordinate frames, of other flexure mechanisms
(Figure 9(a), 9(b) and 9(d)) are represented below and Eqs.
(9)–(12) are normalized using the method in Ref. [
this way, the deformations can sum up together in
different dimensions because of the dimensionless processing.
C3−DOS = 00
According to all the above overall compliance
matrices, it can be concluded that the smaller parasitic motion
errors occurs when there are more symmetric planes
existing in the flexure mechanisms. In other words, the
DoS of a flexure mechanism leads to some
straightforward effect on its parasitic motion error. It is worth
mentioning that all these flexure mechanisms have the same
dominant motion pattern which consists of a rotation
about the x axis (θx) and a translation along the x axis
3.3 Parasitic Motion Error Analysis
Now, let us take a close look at the overall compliance
matrix formula of each flexure mechanism when a force
fx is imposed on the mobile platform. According to the
screw theory and the theory of linear elasticity, the
deformation denoted by the twist ξ = (θ; δ) = (θx, θy, θz; δx, δy,
δz) and the load wrench F = (τ; f) = (τx, τy, τz; fx, fy, fz) are
connected by the generic 6×6 compliance matrix,
θx c11 c12 c13 c14 c15 c16 τx
δθxzy = ccc432111 ccc432222 ccc432333 ccc432444 ccc432555 ccc432666 τfxz ,
δy c51 c52 c53 c54 c55 c56 fy
δz c61 c62 c63 c64 c65 c66 fz
where the parameters in the compliance matrix are
normalized for summing up deformations together
Considering the first case without symmetric plane
(0-DoS mechanism), there exists such a resulting
deformation, expressed as
ξ0 = θx + θy + δx + δy = c14fx + c24fx + c44fx + c54fx.
In fact, only the translational motion along x-axis is
useful to perform desired function, the rest of entries are
unwanted since they bring into some parasitic motions.
In this case, the moving platform translates by δx along
x direction, companying with three other parasitic
motions, which are two parasitic rotations about x and
y axes, denoted by θx and θy respectively, and a parasitic
translation along y axis, denoted by δy. As known,
parasitic motion is always detriment to the accuracy of a
flexure mechanism, Clearly, this 0-DoS mechanism is not
desired for practical application.
As for the second case (1-DoS mechanism), it has one
yoz symmetrical plane. Based on the matrix obtained
above, the resulting deformation is deduced as
The preliminary motion for translation along x axis
remains unchanged, but the number of parasitic motions
decreases into two that are the parasitic rotations about y
and z axes, denoted by θy and θz, respectively. Comparing
with the former 0-DoS case, the presence of one
symmetrical plane leads to better performance due to eliminating
one type of parasitic motion.
For the third case, it has the widest popularity when
designing 2-DOF cylindrical flexure mechanisms. There
ξ1 = θy + θz + δx = c24fx + c34fx + c44fx.
are a number of related literatures that address how to
eliminate parasitic motion. Most of them claim that the
introduction of a compensation module can effectively
tradeoff the unwanted parasitic motion error. With the
above-mentioned result, the flexure mechanism with two
symmetric planes also has parasitic motion, but shows
improvement when comparing with the 0-DoS and 1-DoS
mechanisms. The resulting deformation of the 2-DoS
mechanism is expressed as ξ2 = θy + δx = c24fx + c44fx.
As can be seen, the best design should be the last one,
whose overall compliance matrix is ideally pure diagonal.
This is because all symmetric planes result in the
advantage of no parasitic motion. Though it is challenging to
design a 3-DoS flexure mechanism from all selected
motion type, the attempt to design flexure mechanisms
with maximum degree of symmetry is still meaningful.
4 FEA Simulation Verification
In this section, a series of finite element analysis (FEA)
simulations are implemented for demonstrating the
benefit in presence of more symmetrical planes in the
flexure mechanism. The tool used is commercial package
ANSYS 15.0, where SOLID-187 element is selected for all
rigid platforms while BEAM-189 element is selected for
all flexure beams which connecting the moving platform
with the fixed platform.
Flexure mechanisms with different mobility and
stiffness can be obtained by changing their geometrical
parameter. The chosen material is Aluminum Alloy,
whose Young’s modulus is E = 70 GPa, Poisson’s ratio is
μ = 0.34, and the cross-section radius of all wire flexures
in the mechanisms is r = 5 mm. To guarantee the ratio of
the length to the radius to be larger than 40, the height
between the moving platform and the fixed one should at
least be 200 mm. Thus, the length of the platform, which
is also the distance a as shown in Figure 8 is 60 mm. The
interval distance d is also set to be 60 mm.
Four models that have different number of DoS
ranging from zero to three are built in terms of the
geometrical parameters provided above. When applying the same
force fx = 0.01 N to the moving platform of four flexure
mechanisms, they generate a deformation twist along
x axis companied by several parasitic motion, which
lead to inequality between the maximum displacement
(DMX) displaying on the panel of Nodal Solution and the
selected directional displacement (SMX) displaying on
the panel of Nodal Solution. Therefore, the difference of
DMX and SMX can be used as an indication of parasitic
motion for these four designs. All corresponding
deviations generated by the simulation results shown in
Figure 10 are illustrated in Figure 11. It can be concluded
that the higher the DoS, the smaller the parasitic motion
5 Optimal Parametric Design
Analyzing overall compliance matrix for different
flexure mechanisms in Section 3, it is known that the
resultant mechanism with three symmetric planes can lead to
no parasitic motion theoretically because of its diagonal
compliance matrix form. As a result, in this section we
considerately concentrate on the optimization design
for the 3-DoS flexure mechanism and find out the effect
of design parameters on its compliance matrix, whose
entries are considered as the reference of rational and
translational DOF or DOC.
In this case, the entry c11 and c44, in the diagonal should
be the dominant ones for ensuring a rotation along the
x-axis and a translational about the y-axis. Quantitatively,
these two entries that have been already normalized
should be much larger than the other entries in the
diagonal. The much larger the ratio of the DOF entry to the
DOC entry is, the better the kinetostatic characteristic of
the mechanism is. For this reason, we derive symbolically
every entry by four parameters (d, a, θ, r) as below:
c11 = Eπr4asin θ ,
c22 = a3/[πr2 sin θ (2Ga2r2 sin2 θ + 12Ed2r2 sin4 θ +
Ea2r2 cos2 θ + 4Ea2d2 cos2 θ )],
c33 = a3/[πr2 sin θ (2Ga2r2 cos2 θ + Ea2r2 sin2 θ +
12Ed2r2 cos2 θ sin2 θ + 4Ea2d2 sin2 θ )],
ccc654654 === 4412EEEπππrr22ra4ss3iisnnin3θ3θθ3(a3r,32ra2s3icno4s2θ +θ+a2a2c)os,2 θ ,
where E is the Young’s modulus, G is the shear modulus.
First of all, the compliance ratios (c11/c22, c11/c33, c44/c55,
c44/c66) are plotted in Figures 12 and 13. The two figures
illustrate the effect of the beam orientation (θ) on the
compliant ratios, associating with the rotational DOF
about the x-axis and the translational DOF along the
x axis. It is shown that with the increase of the angle
of beam orientation, the compliance ratios c11/c22 and
c44/c66 both decrease. This result suggests that the
compliance for rotational motion about the y-axis becomes
notable, so does the translational motion along the z-axis.
It can be understood that the original flexure mechanism
will evolve into a new flexure mechanism with 2 extra
DOFs including rotation about y-axis and translation
along z-axis. In Figures 12 and 13, except two downward
lines, there is one upward line and one almost steady line,
indicating the constraint capacity against freedom
capacity. It is a common sense that the compliance in the DOC
direction should be small enough while as large as
possible in the DOF direction. Therefore, the trend in
compliance ratio c11/c33 towards better rotational constraint.
For the value of compliance ratio c44/c55, almost constant
larger than 50, can be referred to a translational DOF
along the x-axis.
Based on the illustrated analysis above and in view of
the design purpose for this paper, θ = π/4 is selected to
enable all compliance ratios to be large enough for better
performance, so that we can obtain relatively high
compliance in the specified DOF direction and relatively low
compliance in the specified DOC direction.
Then, we focus on the effect of the distance a on the
compliant ratios associating with the two desired DOFs.
The compliance ratios (c11/c22, c11/c33, c44/c55, c44/c66) are
plotted against the distance a in Figures 14 and 15.
Figure 14 shows that as the distance of intersecting beams
increases, the compliance ratios c11/c22 and c11/c33 both
decrease to approximate 288. It means that c22 and c33 are
still two orders of magnitude smaller than the
compliance c11 so that they can be reasonably neglected for the
qualitative study. On the contrary, the compliance ratio
c44/c55 rises when the distance a increases, and the
compliance ratios c44/c66 is independent of the parameter a.
The effect of the interval distance d on the compliant
ratios is further investigated. From the expression of the
compliance matrix, the parameter d only affects c22 and
c33 with same upward trend. Moreover, as shown in
Figure 16, under θ = π/4, c11/c22 ad c11/c33 are identical with
the change of b.
Finally, the influence of the cross-section radius of
beams is analyzed. Since the equivalent constraint model
subjects to its ratio of the length to the radius, for the
equivalent wire constraint model as used in this paper,
there is no doubt that the smaller the radius of the beam
is, the more ideal the wire flexure approximation is.
Nevertheless, significantly reducing the radius of flexure
beam is not economic because of the manufacturing cost.
On the basis of the above comprehensive quantitative
analysis, the optimal parameters for the 3-DoS
mechanism are listed in Table 2. The FEA simulation is also
implemented using the optimal parameters under the
same force fx = 0.01 N, showing that the value of DMX
is equal to the value of SMX (Figure 17). Compared to
the simulations in Section 4, the displacement along x
axis after optimization is about two times larger than the
original one. With the optimal parameters, the flexure
mechanism is less stiff and lower power consumption.
As mentioned in Section 1, there are mainly three
methods to reduce or even eliminate the parasitic motion for
a flexure mechanism. Compensation designs are always
preferred as reported by numerous literatures, which can
be simply constructed with high accuracy by mirroring
two identical flexure modules. In a word, almost all
existing flexure mechanisms use the principle of symmetry
design and combinations of the homogenous modules
to achieve the compensation for parasitic motion. In the
design processing, it is better to generate as many DoS as
possible from the very beginning based on the findings in
this paper. With different DoS, we can further adopt
certain methods to alleviate the unwanted parasitic motion.
In addition, design and synthesis of a class of
flexure mechanisms with cylindrical motion in a compact
but simple way is significant for further applications,
such as the joint for realizing a snack-like robot’s three
dimensions’ gaits [
]. Recently, a new version of robots
called in-pipe inspection robots was proposed to
investigate the internal space of the pipes (for detecting the
cracks, leaks, etc.), where implementing
non-destructive tests are commonly based on screw motion [
known, in-pipe inspection robots are supposed to move
fast and continuous with constant pitch of rate, and it
is possible because of cylindrical locomotion with the
A class of flexure mechanisms with different number of
DoS have been designed followed by discussing the effect
of symmetrical geometry on their kinetostatic
characteristics. These mechanisms with zero, one or more
symmetric planes, are obtained from FACT method.
Each flexure mechanism is composed of several
identical beams distributed in two planes orthogonal to the
motion direction. Analytical model for the overall
compliance matrix has been derived within the framework
of the screw theory. These models have been used to
analyze the influences of different DoS on the parasitic
motion. Moreover, the FEA simulations are carried out
for verifying the analytical results. An optimal design
with θ = π 4and a = d = 200 mm has been obtained and
As a new concept, the DoS concentrates on
symmetry design theory. Moreover, the comparative case study
on designing a group of symmetrical flexure
mechanisms with cylindrical motion is instructive to snack-like
Additional file 1. Brief introduction of the paper.
J‑ JY was in charge of the whole trial; X‑BH, G‑BH wrote the manuscript; W ‑ WZ
assisted with sampling and laboratory analyses. All authors read and approved
the final manuscript.
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
Supported by National Natural Science Foundation of China (Grant No.
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