Transmission Index Research of Parallel Manipulators Based on Matrix Orthogonal Degree
Transmission Index Research of Parallel Manipulators Based on Matrix Orthogonal Degree
Zhu-Feng Shao 0 1 2 3
Jiao Mo 0 1 2 3
Xiao-Qiang Tang 0 1 2 3
Li-Ping Wang 0 1 2 3
0 State Key Laboratory of Tribology and Institute of Manufacturing Engineering, Tsinghua University , Beijing 100084 , China
1 & Zhu-Feng Shao
2 Supported by National Natural Science Foundation of China (Grant Nos. 51575292, 51475252, 91648107) , National Key Technology Research and Development Program of China (Grant No. 2105BAF19B00), and National Science and Technology Major Project of China , Grant No. 2016ZX04004004
3 Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University , Beijing 100084 , China
Performance index is the standard of performance evaluation, and is the foundation of both performance analysis and optimal design for the parallel manipulator. Seeking the suitable kinematic indices is always an important and challenging issue for the parallel manipulator. So far, there are extensive studies in this field, but few existing indices can meet all the requirements, such as simple, intuitive, and universal. To solve this problem, the matrix orthogonal degree is adopted, and generalized transmission indices that can evaluate motion/force transmissibility of fully parallel manipulators are proposed. Transmission performance analysis of typical branches, end effectors, and parallel manipulators is given to illustrate proposed indices and analysis methodology. Simulation and analysis results reveal that proposed transmission indices possess significant advantages, such as normalized finite (ranging from 0 to 1), dimensionally homogeneous, frame-free, intuitive and easy to calculate. Besides, proposed indices well indicate the good transmission region and relativity to the singularity with better resolution than the traditional local conditioning index, and provide a novel tool for kinematic analysis and optimal design of fully parallel manipulators.
Transmission index; Parallel mechanism; Kinematic performance; Matrix orthogonal degree
1 Introduction
Parallel manipulator usually consists of the base and the
end effector, which are connected together with at least two
identical kinematic chains (branches) [
1, 2
]. With the
structure feature of closed-loop kinematic chains, the
parallel mechanism is recognized as a significant
complementary to the traditional serial mechanism [3].
Correspondingly, parallel manipulators possess stable,
compact and simple structures with low moving inertia and
mass, and show advantages in terms of rigidity, precision,
load, and speed, according to Mrthy’s theory [
4
]. Above
statement can be verified by the applications as machine
tools [
5
], positioning devices [
6
], motion simulators [
7
] and
high-speed pick-and-place robots [
8, 9
]. Although
application number and field of parallel manipulators increase
rapidly, the large-scale commercial application is scarce.
As Merlet [10] once pointed out, the performance
advantages of parallel manipulators seem to be just potential for
now. The industrial performance of the parallel
manipulator is one of the most concerned issues for both
researchers and engineers. Thus, further study on
performance analysis and optimal design should be carried out.
Performance index is the standard to evaluate the
manipulator capability and the basis for performance
analysis and optimal design. A good index is expected to
be finite, independent of the coordinate system,
dimensionally homogeneous and with clear physical significance.
Besides, ease of calculation and intuitive concept will
make the index more convenient and likely to be
implemented. However, even without considering the
mechanism dynamics and elasticity of the parallel manipulator, it
is still a challenging issue to seek the suitable kinematic
indices [
11
]. Nature of the mechanism is transmitting force
and motion [
12
], and the parallel mechanism is not an
exception. Force/motion transmissibility is the key to the
kinematic performance for the parallel mechanism.
Intensive efforts have been made to study related indices, which
can be classified into three categories.
Index researches with the Jacobian matrix belong to the
first category. In kinematics of non-redundant parallel
manipulators, force Jacobian matrix is the transpose of
kinematic Jacobian matrix, which reflects the truth that
force and motion are two equivalent aspects of
transmissibility. Based on the condition number of the matrix,
Local Conditioning Index (LCI) [
13, 14
] and Global
Conditioning Index (GCI) [15] of the parallel manipulator
are developed, which are originally proposed for the serial
mechanism [
16
]. When these indices are applied to the
parallel manipulator with combined translational and
rotational Degrees of Freedoms (DOFs), dimensional
inconsistency appears, which leads to evaluation failure on
performance [
17, 18
]. Then, characteristic length-based
method [19] and characteristic point-based method [
20
] are
investigated to formulate the dimensionally homogeneous
Jacobian matrix. Recently, MA et al. [
21
] and Liu et al.
[
22
] found that LCI values could converge when the index
is applied to the translational parallel manipulator, which
results in poor performance resolution. In addition, the LCI
value is frame-related [
23, 24
].
The second category is referred as angle-based indices,
such as pressure angle and transmission angle. Pressure
angle is the angle between driving force vector and the
velocity vector of force application point. Transmission
angle is the complementary angle of the pressure angle,
which is defined by the angle between input and output
forces of a joint [
25
]. These indicators are powerful for the
planar mechanism [
26
] with clear significance. Huang et al.
[
27
] evaluated the performance of a two DOFs translational
parallel manipulator with the transmission angle. The sine
value of the transmission angle is adopted to develop a
global force transmission index, and has been applied to
performance analysis of the planar 5R parallel manipulator
[
28
]. Philipp et al. carried out the transmission angle
analysis of the planar seven-bar mechanism [
29
]. Based on
the fictitious limb composed of two physical limbs,
transmission angles among limbs were proposed, attempting to
extend the application of the transmission angle to the
spatial parallel mechanism [
30
]. However, with the
increase in limb number as well as the transformation from
planar to spatial mechanism, definitions of angle-based
indices become difficult and complex with vague physical
meanings.
The last category is the transmission indices based on
the screw theory [
31, 32
]. Considering the virtual
coefficient is dimensionless and independent of the coordinate
system, the virtual coefficient between input and output
wrench screws is defined as the transmission factor, and
used to evaluate kinematic performance of the spatial
mechanism [33]. Then, the transmission factor is unitized
[
34
], and the generalized transmission wrench screw is
proposed [
35, 36
]. Chen et al. introduced the generalized
transmission index and analyzed spatial linkages [37].
Further, input and output transmission indices as well as
the total transmission index are proposed [
38
]. Above
indices focus on the analysis of the input power efficiency
of the driving joint and the output power efficiency of the
branch. The power is calculated through the reciprocal
product of twist and wrench screws. Screw theory is a
powerful and systematic mathematical tool. However,
definition and calculation of these indices are quite
complicated, and not intuitive.
There are demands for simple and intuitive force/motion
transmission indices for parallel manipulators, considering
both planar and spatial mechanism. In this paper, based on
the concept of the matrix (or vector group) orthogonal
degree, a series of transmission indices are proposed, in
terms of the mathematical description, definition and
calculation for the fully parallel manipulator, which is
considered as the non-redundant parallel manipulator with one
actuator in each limb. Then, transmission performance
analysis of typical parallel mechanisms is carried out to
illustrate proposed indices and the analysis method. The
remainder of this paper is organized as follows. In the next
section, branch and end-effector transmission indices are
introduced, and the local transmission index is defined to
evaluate the overall transmissibility of parallel
manipulators. Transmission performance analysis of typical
branches and end effectors for fully parallel manipulators are
illustrated in sections 3 and 4. In section 5, transmissibility
of two typical parallel manipulators is discussed.
Conclusions of this paper are given in section 6.
2 Mathematical Foundation and Index Definition
In this paper, transmission indices are deduced on the basis
of orthogonal degree of matrix, which is defined with
matrix volume [
39
]. First of all, matrix volume can be
described as follows: assuming Xm n ¼ ½ x1 x2 . . .xn is
a real matrix (column vector group), composed of n real
column vectors xi. Then, the volume of the real matrix can
be deduced as
ð1Þ
ð2Þ
Numerically, the JTI value equals the cosine value of the
pressure angle, and indicates the ratio of the efferent force
to the afferent force. For different limbs, JTI analysis is
diverse. The JTI descriptions of typical branches of parallel
manipulators will be introduced in Section 3.
The branch of the parallel manipulator is formed by a
series of joints. On the basis of the JTI, the Branch
Transmission Index (BTI) can be defined as the product of
JTI values of all joints in the branch, and can be written as
volnðXÞ ¼
Further, the matrix orthogonal degree can be described
as following: if minkxik ¼ 0 ði ¼ 1; 2; . . .; nÞ, the matrix
orthogonal degree ort(XÞ equals 0. Otherwise,
, Yn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,Yn
det XTX
Above definition is based on the column vector, and can
be considered as the volume of matrix composed of unit
column vectors. On the basis of above definition, it is easy
to get the property that 0 ort(XÞ 1. When column
vectors are orthogonal to each other, the maximum value 1 can
be obtained. On the contrary, when multicollinearity of
column vectors appears, the minimum value 0 can be
deduced. Extreme values of the matrix orthogonal degree
indicate two important geometric properties,
multicollinearity and orthogonality of multiple vectors.
Obviously, matrix orthogonal degree is an efficient mathematical
tool to study force/motion vectors of spatial parallel
mechanisms.
The force (or motion) transmission in mechanism could
be understood as from the base (fixed on the ground) to the
terminal. According to structural features of the parallel
mechanism, identical branches connect the end effector
with the base. Transmissibility of the parallel manipulator
can be subdivided into two aspects, such as branch
transmissibility and end-effector transmissibility. The former
describes the manipulability of the limb, and the later
indicates the complementarity of limbs’ contributions on
the end effector.
Firstly, let us focus on the branch transmissibility. Each
branch (or limb) of the parallel manipulator is a typical serial
mechanism, and force is transmitted from the base to the end
effector through the limb. Limb transmissibility depends on
transmission performances of the mounted joints. Index of
the joint transmissibility can be defined by the relation
between the afferent force applying on the joint by the
previous component and the efferent force that exerted on
the next component through the joint. Specifically, Joint
Transmission Index (JTI) of the ith joint located in the jth
branch of the parallel manipulator can be written as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ort Yji 2;
CJTIji ¼
ð3Þ
where Yji ¼ ½ FjiA FjiE . f jiA and f jiE are the afferent and
efferent force vectors, respectively. From the above
definition, it can be found that when these force vectors are
collinear, the force transmissibility is the best, and CJTI¼1.
On the contrary, when force vectors are mutually
perpendicular, the force transmissibility is the worst, and CJTI¼ 0.
CBTIj ¼ CJTIj1
CJTIj2
CJTIjk:
Eq. (4) reflects the overall transmission performance of
the limb from the base to the end-effector joints. When the
BTI value of a branch equals 0, the end effector of the
parallel manipulator will lose the support of the branch. In
this case, the manipulator is at its singular configuration,
and this kind of singularity is called the reverse singularity.
The other aspect of transmission analysis for the parallel
manipulator is the end-effector transmissibility, which
reflects the integrated efficacy of all branches. The
Endeffector Transmission Index (ETI) reflects the ability of
branch force vectors to span the whole required output
force space, and is defined with the limb force vectors
exerted on the end effector and the end-effector geometric
characteristics. The ETI can be calculated as
ð4Þ
ð5Þ
CETI ¼ ortðEFÞ;
where F ¼ ½ s1 s2 . . . sn is the branch force matrix.
The limb force vector of the jth branch is sj ¼ f j; cj f j ,
composed of the unit force vector f j and unit torque vector
cj f j. f j is the unit force vector exerted on the end
effector by the jth limb, while cj is the unit radial vector
pointing from the geometric center of the end effector to
the rotational center of the end-effector joint of the jth
limb. Rigid body in space possesses six DOFs, but some
DOFs of the end effector are constrained to obtain the
required DOFs. The matrix E is a diagonal matrix, used to
describe the required DOFs of the parallel manipulator.
When a degree of freedom of the end effector is required
and movable, the corresponding diagonal element of matrix
E is 1. On the contrary, if some DOFs are constrained, the
corresponding diagonal elements of matrix E are 0. Take
the 3-RRR parallel manipulator for example. The matrix
E = diag (1, 1, 0, 0, 0, 1), which indicates the end effector
of this manipulator possesses two translational DOFs along
X and Y axes and a rotational degree of freedom along C
axis (represents the rotational movement around the Z
axis). The dimension of the matrix EF always equals the
number of required end-effector DOFs, and the ETI is
applicable to all non-redundant parallel manipulator. If the
end effector of the parallel manipulator is a point-type, for
example 5R parallel manipulator, cj becomes zero. It is
worth noting that when the value of ETI is 0, the parallel
manipulator will lose the force output in a certain direction.
In this case, the manipulator is at its forward singularity.
As mentioned above, force transmission in the parallel
manipulator can be considered as from the base to the end
effector. Thus, the total transmissibility of the parallel
manipulator equals the product of BTI and ETI values.
However, BTI values and branch transmission
performances are different among various branches under the
arbitrary pose. Considering that the main work in the
design and analysis stage is to avoid configuration and pose
with poor transmission performance, the minimal BTI
value is adopted, and the Orthogonal degree based Local
Transmission Index (OLTI) of the parallel manipulator can
be defined as
COLTI ¼ min CBTIj
CETI ðj ¼ 1; 2; . . .; nÞ:
ð6Þ
When the value of OLTI is 0, the parallel mechanism is
in a singularity locus. Conversely, when the value of OLTI
is 1, the transmissibility of the mechanism is the best. As
illustrated in the above definitions, the proposed
transmission indices are not related to the established coordinate
system, and are frame-free. The value range is finite and
normalized [
0, 1
]. Besides, index significances are clear,
while calculation is simple and concise.
3 Transmissibility of Typical Branches
In this section, based on the proposed BTI, transmission
performances of typical branches for fully parallel
manipulators are analyzed. Gravity, friction and inertial forces
are not taken into account in this paper.
3.1 Planar RRR and Spatial RSS Branches
As shown in Figure 1, planar RRR branch (R stands for
revolute joint, and the underline indicates the actuated joint)
is composed of three revolute joints (1, 2, and 3) as well as
two links (I and II). lj1 and lj2 are unit link vectors. Link I is
the swing bar, and link II is the two-force bar. Forces are
indicated with dotted-line arrows, while actuated joint is
labeled with the solid-line arrow. The actuated revolute joint
is attached to the base, and the JTI value of the joint is 1. For
the end-effector joint, the JTI value of joint 3 is also 1. The
transmission performance of the branch is completely
determined by joint 2. As indicated in the figure, the afferent
force of joint 2 is f j2A, perpendicular to the link I. The
efferent force of joint 2 is f j2E, along the link II. Therefore,
the BTI value of the jth branch can be determined as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ort Yj2 2;
ð7Þ
and Yj2 ¼ f j2A f j2E . According to the geometric
relation, the BTI value of the RRR branch can be directly
calculated with structural vectors as CBTIj ¼ CJTIj2 ¼ ort
Lj , and Lj ¼ ½ lj1 lj2 .
This conclusion is also applicable to the spatial RSS
branch (S represents the spherical joint), as shown in
Figure 2. The actuated joint of the RSS branch is also the
revolute joint attached to the base. The link II is the
twoforce bar, and values of JTI for actuated and end-effector
joints are both 1. The transmission performance of the
branch is determined by the middle joint. Then, the BTI
value of the RSS branch can be described as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
For the RSS branch, the link vectors become
three-dimensional. Generally, above analysis and calculation
process is based on force vectors, and has nothing to do with
the definition of the coordinate system. Above conclusion
fits for a class of branches, composed of actuated pendulum
and passive two-force bar, such as RUS, RUU and RRS
branches (U stands for the universal joint).
3.2 PUS and UPS Branches
Besides revolute joint, prismatic joint is also widely
adopted as actuated joint, and usually placed on the base or
in the middle of the branch. The typical structures are PUS
and UPS branches (P represents the prismatic joint). As
shown in Figure 3, the universal joint of the PUS branch is
in the middle, and the spherical joint is attached to the end
effector. The JTI value of the actuated joint is 1. Link II is a
two-force bar, the force acting on the end effector is along
link II, and the JTI value of spherical joint (end-effector
joint) is 1. The BTI value of the limb equals the JTI value
of the middle universal joint. As shown in Figure 3, the
afferent force f j2A and the efferent force f j2E of the
universal joint can be determined, and the BTI value of the
PUS branch is deduced as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1
ort Lj 2;
where Yj2 ¼ f j2A f j2E , and Lj2 ¼ ½ lj1 lj2 . This
equation is applicable to the branch that consists of
actuated prismatic joint attached to the base and passive
twoforce bar, such as PSS, PRS and PRU branches.
As shown in Figure 4, the actuated prismatic joint of the
UPS branch is installed in the middle of the branch. The
entire branch becomes the two-force bar. It can be deduced
that JTI values of two passive joints are both 1, and the BTI
value of all branches is 1. This result can be applied to
other two-force branches, driven by middle prismatic joint,
such as SPS, RPS, and RPU branches.
In this section, the transmission performance analysis is
carried out on typical branches. Considering the inertia
property and power consumption, the actuated joint of the
parallel manipulator is usually attached to the base or located
in the middle of the branch, then JTI values of the base and
actuated joints usually equal 1. There are many two-force bars
in the parallel manipulator, which facilitate and simplify the
force analysis and the BTI value calculation greatly.
4 Transmissibility of End Effectors
For the parallel manipulator, the base and the end effector
are connected with multiple identical branches
simultaneously. The output force of the end effector is the common
effort of all branches. The typical planar and spatial
parallel mechanisms, such as 5R and Stewart parallel
manipulators, are analyzed in this section to reveal the
endeffector transmissibility.
The structure of 5R mechanism is shown in Figure 5.
Point P is the end effector, with two translational DOFs.
Both branches are of the RRR configuration. According to
the definition of ETI proposed in Section 2, the ETI
expression of the 5R mechanism can be written as
CETI ¼ ortðEFÞ ¼ ortð½ f 1
f 2 Þ;
ð10Þ
where f1 and f2 are unit force vectors exerted on the end
effector by two branches, along l12 and l22 vectors
respectively, which are unit structural vectors of passive
links. According to the geometric relation, the transmission
index can also be written as CETI ¼ ortðLÞ and
L ¼ ½ l12 l22 . For the 5R parallel manipulator, the ETI
value equals the sine value of the angle between f1 and f2
vectors (l12 and l22 vectors). Obviously, when the angle
becomes 0 or 180 , limb force vectors are collinear, and
the end effector loses the force output capacity along the
direction perpendicular to limb force vectors. Accordingly,
a singularity occurs, and the ETI value is 0. If the angle is
90 , the transmissibility of the end effector is the best, and
the ETI value equals 1.
As shown in Figure 6, the Stewart parallel manipulator
consists of six extensible UPS branches, which is a typical
6 DOFs parallel manipulator with combined translational
and rotational DOFs. According to the branch analysis, the
unit force fj acting on the end effector by the jth limb is
along the limb, through rotational center of the spherical
joint. As illustrated in Figure 6, the geometric center of the
end effector is point o. Thus, direction of the unit radial
vector cj is from point o to the rotational center of the jth
spherical joint. Considering the geometric structure, the
unit branch vector lj2 can be used instead of the limb force
vector fj. The matrix E becomes the unit diagonal matrix,
and all diagonal elements are 1. Then, we can deduce the
CETI expression of the Stewart manipulator as
CETI ¼ ortðEFÞ ¼ ort
c1
And, singularity of the Stewart parallel manipulator will
be discussed in the next section.
ð11Þ
5 Examples
In order to illustrate the analysis method and the proposed
indices in detail, the transmission performances of two
typical parallel manipulators are analyzed with proposed
indices in this section, and comparative discussion with the
traditional LCI is developed.
Firstly, the 5R parallel manipulator is analyzed, and
adopted kinematic parameters are listed in Table 1. The
global coordinate system is established at the midpoint
between two actuated joints, as shown in Figure 5. With
Eqs. (6), (7) and (10), value distributions of the proposed
indices, such as BTI, ETI and OLTI, can be determined, as
shown in Figures 7, 8, and 9 respectively. The value
distribution of traditional LCI is exhibited in Figure 10.
As shown in Figure 8, in the reachable workspace
(illustrated with red curve), ETI values are relatively large,
which indicates that the transmissibility of the end effector
for the 5R mechanism is quite good. With comparative
analysis of Figures 7, 8, and 9, we can find that the overall
transmission performance of the 5R mechanism is mainly
determined by the branch transmissibility. On the boundary
of the reachable workspace, reverse singularity occurrence,
introduced by the branch transmission performance, as
shown in Figure 7. With the proposed indices, it is
convenient to determine the singularity type and analyze the
cause.
As shown in Figures 9 and 10, overall trends of LCI and
OLTI atlases for the 5R parallel manipulator are similar.
Large index values which mean good transmission
performance appear in the center of the reachable workspace.
With the end effector moves away from the center region,
the transmission performance becomes worse. However,
there are some significant differences between LCI and
OLTI atlases. In the LCI atlas, contour curves with
different LCI values converge, which reduces the
performance resolution capacity and fails to reflect the distance
off singularity. In contrast, contour curves with different
OLTI values never intersect, and the offset of each curve is
quite clear. The proposed index is relative to the
singularity.
In addition, the transmission performance of the Stewart
parallel manipulator is evaluated, and adopted kinematic
parameters are shown in Table 2. Joint distribution angles
a and b of the Stewart manipulator are used to describe
locations of spherical and universal joints in the end
effector and the base respectively, as shown in Figure 11.
The global coordinate system is established at the
geometrical center of the base, as illustrated in Figures 6 and
11.
The distributions of LCI and OLTI for the Stewart
parallel manipulator are given in Figures 12 and 13, when
the end effector is located in the horizontal plane of
Z = 0.45 m without any rotation. Since the BTI value of
the Stewart manipulator is the constant 1, its OLTI atlas is
the same with its ETI atlas. In the calculation of LCI value,
considering the dimensional homogeneity, the end-effector
Radius of the base R/m
Radius of the end effector r/m
Joint distribution angle of the base a/( )
Joint distribution angle of the end effector b/( )
radius r is adopted as the characteristic length to deduce the
normalized Jacobian matrix.
Generally, overall trends of LCI and OLTI atlases for
the Stewart parallel manipulator are similar, as illustrated
in Figures 12 and 13. Both LCI and OLTI values decrease
while the end effector moves away from the center of the
workspace, which indicates that the transmission
performance becomes worse. The minimum and maximum
values of OLTI are 0.146 and 0.636 respectively. The
minimum and maximum values of LCI are 0.178 and 0.456
respectively. The value variation of OLTI (0.49) is greater
than that of the LCI (0.278), and the OLTI possesses better
performance resolution consequently.
Figure 14 illustrates LCI and OLTI value changes
when the end effector of the Stewart manipulator moves
along Z axis from 0 m to 0.7 m with X and Y coordinate
values being 0. The manipulator obtains the best
transmission performance when the end effector is located
around Z = 0.31 m. The maximum value of OLTI is
0.77, while the maximum value of LCI is 0.59.
Minimum values of OLTI and LCI are both zero, which
indicate the singularity. In the left region, with the end
effector approaching the base, both index values
decrease, and the value declining speed is faster than
that in the right region when the end effector moves
away from the base.
Figure 15 indicates LCI and OTLI value changes when
the end effector rotates along C axis with angle c from 0 to
90 and is located at the position of (0, 0, 0.45) m. Both
LCI and OTLI values gradually decrease. When the
rotation angle c equals 90 , values of OTLI and LCI are both 0,
which means the Stewart parallel manipulator is at its
singular pose.
6 Conclusions
(1)
(2)
The matrix orthogonal degree is adopted to evaluate
the transmission performance of the parallel
manipulators. Transmission indices of branch, end effector,
and configuration for the fully parallel manipulator
are proposed, and a new index system is established.
The BTI is proposed to evaluate the transmissibility
of each limb, and transmission performance analysis
on typical branches mostly adopted in parallel
manipulators are carried out. Specifically, for the
branch composed of actuated pendulum and passive
two-force bar, qffisffiffiuffifficffiffihffiffiffiffiffiffiffiffiffiaffiffisffiffiffiffiffiffiffiffiRUS and RRR,
2
2
¼
When the actuated two-force branch is adopted, such
as UPS and RPS, the BTI value is the constant 1.
The OLTI is defined to evaluate the transmissibility
of the fully parallel manipulator. Transmission
performances of 5R and Stewart parallel
manipulators are analyzed to illustrate the analysis method
and the proposed frame-free indices. Compared with
the traditional LCI, proposed OLTI possesses better
resolution for transmission performance, without
intersection phenomenon in the performance contour
atlas.
Proposed indices are relative to the singularity.
When the JTI value is zero, the BTI value of the joint
located branch is zero, and the reverse singularity
occurs. On the other side, when the ETI value is
zero, the forward singularity appears. Thus, if either
BTI or ETI values is zero, the parallel mechanism is
at its singular pose. If both BTI and ETI values are
zero, the duplex singularity happens.
Simulation and analysis results show that the
proposed indices possess better performance
resolution than the LCI. Besides, they are frame-free with
unified finite range from 0 to 1, and dimensionally
homogeneous with clear physical meaning.
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Zhu-Feng Shao , born in 1983, is currently an associate professor at Department of Mechanical Engineering, Tsinghua University, China.
He received his PhD degree from Tsinghua University, China, in 2011 . His research interests include design, dynamic analysis and control of the parallel manipulator and the cable robot . Tel: ? 86 - 10 - 62794598; E-mail: Jiao Mo, born in 1992, is currently a master candidate at Department of Mechanical Engineering , Tsinghua University, China. He received his bachelor degree from Tsinghua University, China, in 2015 . His research interests include high-speed parallel robot and machine tool .
E-mail: Xiao-Qiang Tang, born in 1973, is currently a professor and a PhD candidate supervisor at Department of Mechanical Engineering, Tsinghua University, China. His main research interests include parallel manipulator, cable robots, and motion control. E-mail: Li-Ping Wang, born in 1967, is currently a professor and a PhD candidate supervisor at Department of Mechanical Engineering, Tsinghua University, China. His main research interests lie in the advanced manufacturing equipment. E-mail: lpwang@mail .tsinghua.edu.cn
