CHINESE JOURNAL OF MECHANICAL ENGINEERING
·810·
Vol. 22,aNo. 6,a2009
DOI: 10.3901/CJME.2009.06.810, available online at www.cjmenet.com; www.cjmenet.com.cn
Singular Assembly Configurations and Configuration Bifurcation Characteristics
of the Semi-regular Hexagons 6-6 Gough-Stewart Manipulator
LI Yutong1, *, WANG Yuxin1, HUANG Zhen2, and PAN Shuangxia1
1 Institute of Mechanical Design, Zhejiang University, Hangzhou 310027, China
2 Robotic Research Center, Yanshan University, Qinhuangdao 066004, China
Received December 9, 2008; revised September 28, 2009; accepted October 12, 2009; published electronically October 16, 2009
Abstract: It is well known that singular configurations are inherent to parallel manipulators and have serious influences on their
properties. Therefore, these singular configurations should be avoided in the design and application of mechanisms. The researches on
the singularity identification and distribution have revealed the relations among the six configuration parameters at singular points. Few
works have dealt with the relation between the singularity and the input parameters, as well as the properties of the manipulator nearby
the singularity. In this paper, taking the semi-regular hexagons 6-6 Gough-Stewart manipulator (SRHGSMP) as an example, the
configuration bifurcation characteristics going with the input parameters, the assembly configurations at singular points, and the reasons
to cause the singularity are analyzed. The research reveals that the number and the combination of the input parameters have great
influences on the complexity of the singularity and the curvature radiuses of the configuration curves. Under different number of input
parameters, the dimensional-utmost singularity, line vectors correlation singularity and Jacobian matrix correlation singularity can occur
individually or jointly. Choosing the adjacent input parameters, the simple singularity and the large singularity-free input parameters
zones can be obtained. And selecting multiple input parameters, the self-motion regions and the singularity avoidance errors can be
reduced. These new discoveries are valuable and of significance for the trajectory design, the singularity avoidance, and the self-motion
control of the parallel manipulator.
Key words: parallel manipulator, singular assembly configuration, configuration curve
1
∗
Introduction
It is well known that singular configurations are inherent
to parallel manipulators and have serious influences on
their properties. Therefore, these singular configurations
should be avoided in the design and application of
mechanisms.
HUNT[1] first pointed out a singularity that occurs when
all the lines associated with the prismatic actuators intersect
a common line. Then, FICHTER[2] showed that a singular
configuration occurs when the moving plate, parallel to the
base, is rotated about the z-axis by ±90º. GOSSELIN and
ANGELES[3] classified the singularities in parallel
manipulators into three main groups according to the rank
deficiency associated with the Jacobian matrices of the
kinematic equation. This classification has been further refined
by ZLATANOV, et al[4], where detailed physical interpretations
are provided. INNOCENTI and CASTELLI[5] pointed out
that path bifurcation point would occur in the place of the
singular point, at which point the mechanism configuration
would change from one kind to another, and the executing
* Corresponding author. E-mail: [email protected]
This project is supported by National Natural Science Foundation of
China (Grant No. 50375111/50675188).
platform would change from one kind of pose (position and
orientation of the platform) to another, so the mechanism
would be out of control. BANDYOPADHYAY and
GHOSAL[6] analyzed the gain of degrees of freedom of a
closed-loop mechanism at a singular point both from
geometric and algebraic points of view, and it has been
shown to be associated with the locking of the actuators.
MERLET[7–8] used the Grassmann geometry to find
singular configurations. He found again Hunt’s, and
Fichter’s singular configurations and more configurations
with the Grassmann geometry. Finding singularities based
on the Grassmann geometry is a major contribution to the
subject, however, the results are not easy to obtain for the
most general 6-SPS Gough-Stewart platform. TAKEDA
and FUNABAHI[9] reported a numerical study of
singularities and ill-conditioning of the Stewart platform.
HUANG, et al[10], and HUANG and DU[11] studied the
singularity of parallel mechanisms and presented a
computing method for instantaneous motion instability of
parallel manipulators by means of screw theory and
geometric method. HAO and MCCARTHY[12] introduced
screw theory, and based on geometry of the screws,
singular conditions were presented and their equations were
developed. MULLER[13] introduced the global manipulability
measures, which globally characterized the kinematic
CHINESE JOURNAL OF MECHANICAL ENGINEERING
dexterity of parallel mechanisms, to investigate the stability
of manipulator configurations. KIEFFER[14] utilized
high-order equations derived from Taylor series expansion
of matrix equation of closure to identify singularity type
and, in the case of bifurcations, to determine the number of
intersecting branches as well as a Taylor series expansion
of each branch about the point of bifurcation.
From a design point of view, it is desirable to obtain the
analytical expression of the singularity locus of a
mechanism. Then, it is easy to identify the locations of
singularities within the given workspace and determining
whether the singularities can be avoided. GREGORIO[15]
presented a new expression of the singularity condition of
the most general mechanism based on the mixed products
of vectors and transformed the singularity condition into a
ninth-degree polynomial equation whose singularity
polynomial equation is cubic in the platform orientation
parameters and a sixth-degree one in the platform
orientation parameters. SEFRIOUI and GOSSELIN[16]
demonstrated that, for a given orientation of the platform of
a planar three degree-of-freedom (DOF) parallel
manipulator with prismatic actuators, the singularity loci in
the plane of motion were defined by quadratic equations.
GOSSELIN and WANG[17] addressed the kinematic
modeling and the determination of the singularity loci of
spatial 5-DOF parallel mechanisms with prismatic or
revolute actuators. HUANG and CAO[18] studied the
singularity loci and distribution characteristics of the
triangle simplified symmetric manipulator architecture,
where all singularities are classified into three different
linear-complex singularities. WOLF and SHOHAM[19] used
the line geometry and the screw theory to determine the
singular configurations of parallel manipulator and their
behaviors at these points. ST-ONGE and GOSSELIN[20]
studied the singularity loci of the Gough-Stewart platform
and their graphical representations based on two analytical
approaches, the linear decomposition and the cofactor
expansion. They found that, for a given orientation of the
platform, the singularity locus is distributed on
three-dimensional surface in the Cartesian space. LI, et
al,[21] presented an analytic form of the six-dimensional
singularity locus of the general Gough-Stewart platform.
The method is based on the cascaded expansion of the
determinant of the Jacobian matrix of the mechanism.
When the orientation of the manipulator is given, the
type-II
singularities[3]
are
distributed
on
the
three-dimensional surfaces in the Descartes coordinate
system. BANDYOPADHYAY and GHOSAL[22] presented a
compact closed-form expression for the singularity
manifold of a class of 6–6 Stewart manipulators. The
singularity manifold is obtained as the hypersurface in the
task-space, SE(3), on which the wrench transformation
matrix for the top platform degenerates.
In the above researches, the singularity loci or the
distribution hypersurface is obtained based on the
singularity conditions associating with the Jacobian matrix,
·811·
or the line (Grassmann) geometry, or the screw theory.
While the orientation parameters are given, the singularity
is distributed on the 3-D surface in the Cartesian space
going with three independent variables x, y, z, which are the
position parameters of the movable platform. Therefore,
this kind of singularity distribution hypersurface only
describes the relations among the six configuration
parameters at singular points. Up till now, few works have
dealt with the relation between the singularity and the input
parameters, as well as the properties of the manipulator
nearby the singularity.
It is well know that the configuration of the manipulator
depends on its six input parameters and original assembly
configuration. For avoiding singularities, the easiest way is
to check and control the lengths of the input parameters. In
this paper, taking the semi-regular hexagons 6-6 GoughStewart manipulator (SRHGSMP)[20] as an example, with
the homotopy tracing method and the 3-D virtual assembly
method, the configuration bifurcation characteristics of the
SRHGSMP going with the input parameters, the assembly
configurations at singular points, and the reasons to cause
the singularity have been analyzed.
2
Assembly Configurations
Take the SRHGSMP as an example to investigate the
singularity and the assembly configurations of parallel
manipulators. The fixed dimensions of the SRHGSMP are
as follows: R1 is the distribute radius of six spherical joints
Ai on the movable platform; R2 is the distribution radius
of six spherical joints Bi ( xBi , yBi , z Bi ) on the base. The
relative angle between two equilateral triangles formed by
joints A1 , A3 , A5 and joints A2 , A4 , A6 is α1 , and the
relative angle between two equilateral triangles formed by
joints B1 , B3 , B5 and joints B2 , B4 , B6 is α 2 .
li (i = 1, 2," , 6) are the lengths of six extendable legs,
which are the input parameters of the parallel manipulator.
2.1 Constraint equation
The fixed coordinate system Oxyz of the SRHGSMP on
the base frame is set up as follows: the origin of the fixed
coordinate system is in the center of the base frame, line
OB1 as the x -axis, and the normal line of the fixed frame
as the z -axis. The movable coordinate system Ox1 y1 z1
on the top platform is set up as follows: line O1 A1 as the
x1 -axis, and the normal line of the top platform as the
z1 -axis. Joint Ai in the fixed coordinate system is
Ai ( x Ai , y Ai , z Ai ) , while in the movable coordinate system is
Ai′( xi′, yi′, zi′) . Utilize X = [ P , Q ]T to express the pose of
the manipulator. Here, P ( x, y, z ) is the center coordinate
of the movable platform, α is the yaw angle of the
movable platform with respect to the x-axis, β is the pitch
angle of the movable platform with respect to the y-axis,
and γ is the roll angle of the movable platform with
respect to the z-axis.
·812·
LI Yutong, et al: Singular Assembly Configurations and Configuration Bifurcation Characteristics
of the
Semi-regular Hexagons 6-6 Gough-Stewart Manipulator
2-DOF Rotational
Decoupledppppppppppppppppppppppppppppppppppp
Parallel MechanismsY
According to the length constraint equations for six
extensible legs
( Ai − Bi )T ( Ai − Bi ) − li T li = 0,
i = 1, 2," , 6,
(1)
method. For figuring out the theoretical solution of singular
points, with the aid of the expanded equation method[23],
the expanded equation corresponding to Eq. (4) is
constructed as follows:
the configuration equation of the SRHGSMP is written as
Ψ(u)=
φi = x 2 + y 2 + z 2 + R12 + R22 − li2 + 2 ⎣⎡ x′Ai sinβ cosγ +
⎧ fi = fi ( x, y, z , α , β , γ ) = 0,
⎪
⎪ f = ∂f j v + ∂f j v + ∂f j v + ∂f j v + ∂f j v + ∂f j v = 0,
⎪⎪ jx ∂x 1 ∂y 2 ∂z 3 ∂α 4 ∂β 5 ∂γ 6
⎨ 6
⎪ v 2 − 1 = 0,
k
⎪
⎪ k =1
⎪⎩i = 1, 2," , 6, j = 1, 2," , 6,
(5)
y ′Ai (sinα sin β cos γ − cosα sinγ ) − xBi ⎦⎤ x +
2 ⎡⎣ x′Ai cosβ sinγ + y ′Ai (sinα sinβ sinγ + cosα cosγ ) −
∑
yBi ⎤⎦ y − 2( x′Ai sinβ − y ′Ai sinα cosβ ) z − x′Ai yBi cosβ sinγ −
2 ⎡⎣ x′Ai xBi cosβ cosγ + y ′Ai xBi (sinα sinβ cosγ − cosα sinγ ) +
y ′Ai yBi (cosα cosγ + sinα sinβ sinγ ) ⎤⎦ = 0.
(2)
Let μ = [l1 , l2 ," , l6 ]T , the integrated form of Eq. (2) is
Φ ( X , μ) = (φ1 , φ2 , φ3 , φ4 , φ5 , φ6 )T = 0 ,
(3)
where μ is the input parameter vector as independent
variables to analyze the configuration bifurcation behaviors,
X is the pose vector of the movable platform.
The type-II singularities corresponding to Eq. (3) are
determined by
⎧⎪Φ ( X 0 , μ 0 ) = 0,
⎨
⎪⎩det ( ∂Φ ( X 0 , μ 0 ) ∂X ) = 0.
2.2
(4)
Theoretical type-II singular points
At the vicinity of the type-II singular point, since the
value of the Jacobian matrix in Eq. (4) is close to zero, Eq.
(3) is in ill-condition. In such a case, it is difficult to obtain
the theoretical solution from Eq. (4) with a numerical
Table 1.
l1 / m
A′ =
∂ ψ ( u)
∂X
(6)
corresponding to Eq. (5) disappears. It can be proved that
the solutions of Eq. (5) meet Eq. (4).
For a group of given initial variables, it is easy to figure
out the theoretical singular points from Eq. (5). For
instance, when the dimensions of the manipulator are as
follows: R1 = 1 m, R2 = 2 m, α1 = 50D , α 2 = 20D and
li = 2.0 m, i = 2,3," , 6 , with the Newton-Raphson
method, the four type-II singular points in the above space
of the base, z≥0, are calculated and listed in Table 1.
Type-II singularities of the SRHGSPM ( z≥0)
Position
Input parameter
Singular point
where u = ( X , v , μ )T = ( x, y, z , α , β , γ , v1 , v2 , v3 , v4 , v5 , v6 ,
l1 , l2 , l3 , l4 , l5 , l6 )T and vector ν is the eigenvector relative
to the zero eigenvalue of matrix A . The initial vector
should be a unit vector, such as vs=(0, 0, 0, 0, 0, 1)T.
Because the new variables are introduced into Eq. (5),
the degeneracy of the Jacobian matrix
x/m
M1
1.146
0.196
M2
1.638
M3
2.366
M4
2.791
Orientation
z/m
y/m
D
α /( )
35.37
D
γ /( )
–0.441
0.163
–0.043
0.570
1.402
21.49
21.25
2.182
–0.213
–0.683
1.499
–28.33
–2.215
–30.36
–0.501
0.022
0.282
–46.41
66.64
–119.97
2.3 Assembly configurations at type-II singular points
Because of the ill-condition of the assembly equations
modeled by the commercial 3-D software for the parallel
manipulator at the theoretical singular point, the virtual
assembly configuration at the theoretical singular point can
not be set up. For modeling the assembly configuration of
the manipulator at the theoretical singular point, the
pre-assembly method[24] is utilized here. The processes to
–73.13
D
β /( )
2.778
obtain the assembly configuration at the theoretical singular
point are as follows.
(1) Figure out the theoretical type-II singular point based
on the expanded Eq. (5).
(2) Figure out the reference point ( xi , yi , zi ) and the
direction cosines of the element coordinate system for each
assembly component. For instance, the direction cosines of
the x-axis are u xx , u xy , z xz .
(3) With the aid of the 4 × 4 transformation matrix
CHINESE JOURNAL OF MECHANICAL ENGINEERING
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supplied by the 3-D commercial software,
⎛ u xx u xy
⎜
⎜ u yx u yy
⎜u
⎜ zx u zy
⎜0
0
⎝
u xz
u yz
u zz
0
xi ⎞
⎟
yi ⎟
,
zi ⎟⎟
1 ⎟⎠
(7)
each element is transformed into its spatial position
corresponding to the theoretical assembly configuration at
the singular point. Then, the assembly configuration at the
theoretical singular point is set up.
3
Singular Points and Assembly Configurations
Going with Single Input Parameter
3.1 Singular point M1
For the structural semi-symmetry of the manipulator,
without loss of universality, select the input parameter l1
as an independent variable, and keep the other five input
parameters constant. According to the theoretical values of
the configuration components at singular point M 1 , with
the pre-assembly approach, the virtual assembly
configuration at M 1 is set up, as shown in Fig. 1a. It can
be found that in this assembly configuration, the line
vectors of all of six driving extendable legs do not intersect,
parallel, or share the same line and/or plane. Therefore,
they do not meet the line vectors correlation conditions[8],
i.e. the line vectors share the same line, the same plane or
intersect.
Let the input parameter l1 with a tiny decrement
σ > 0, i.e. l1 = l1M1 − σ . The assembly configuration
characters at M 1 are investigated with the aid of the
assembling module of the 3-D software. If σ > 0.01 mm ,
the assembly configuration can not be modeled, while
σ < 0.01 mm , the assembly configuration can be set up
then. This indicates that the singular point M 1 is a limit
moving position corresponding to the dimensions of the
manipulator. We call this kind of singularity as the
dimensional-utmost singularity type.
Furthermore, dragging the assembly configuration modeled
with the 3-D commercial software at l1 = l1M1 + 0.000 4 , the
center of the movable platform has a free movement with
Δ x = 0 , Δ y = 3.45 mm and Δ z = 4.85 mm . This local
self-motion coincides with type-II singularity of det A=0.
Fig. 1b shows the free movements of the movable platform
center going with different increment Δl1 . At the vicinity
of singular point M 1 , the self-motion regions on
component y and z are sensitive to the increment Δl1 . It
means that the moving locus of the sphere joint S1 osculates
with the sphere surface centered at S2 with radius
l1min = l1M1 =1.146 1 m . This kind of osculation results in
①along the direction of the osculation moving locus, the
manipulator will obtain self-motion degrees, and ②the
assembly configuration is sensitive to the variation of the
input parameter l1 .
Fig. 1.
Assembly configuration and self-motion at M1
3.2 Assembly configurations at other singular points
Fig. 2a shows the assembly configurations at singular
point M 2 . Similarly with the case at the singular point
M 1 , the moving locus of the sphere joint S1 on the
movable platform osculates with the sphere surface
centered at the sphere joint S 2 with radius l1 . This leads
the manipulator to obtain the self-motion degree. While the
length of the input parameter l1 has a tiny decrement, the
assembly configuration is not existed yet. Furthermore, at
this singular point, the line vectors of six extendable legs
do not share the same plane, the same line, as well as
intersect at one point or one line, i.e. they do not meet the
line vectors correlation condition[8].
For investigating the assembly configurations coinciding
with the line vectors correlation condition of six extendable
legs, Fig. 2b and Fig. 2c give the assembly configurations
which meet the line vectors correlation condition near by
the singular point M 2 . While the input parameter l6
locates within the plane of the movable platform, as shown
in Fig. 2b, the length of the input parameter l1 is equal to
1.670 2, and the value of the Jacobian matrix is –858.28.
While the input parameter l2 locates within the plane of
the movable platform, as shown in Fig. 2c, the length of the
input parameter l1 is equal to 1.686 6, and the value of the
Jacobian matrix is –1 051.95. Obviously, the singular point
M 2 is not caused by the line vectors correlation, but by the
limit moving position corresponding to the dimensions of
the manipulator, i.e. the singular point M 2 , as shown in
Fig. 2a, belongs to the dimensional-utmost singularity type.
The character of the assembly configuration at M 4 is
similar with that at M 1 , and the character of the assembly
configuration at M 3 is similar with that at M 2 .
·814·
LI Yutong, et al: Singular Assembly Configurations and Configuration Bifurcation Characteristics
Parallel Mechanisms
Y
Y2-DOF Rotational
of the Semi-regular
HexagonsDecoupled
6-6 Gough-Stewart
Manipulator
point of the homotopy function.
Beginning from the given solution of G (x) = 0,
increasing the value of the homotopy parameter t0 from
t0=0 gradually and tracing the solution curves of the
homotopy function (8), when t0 = 1 , the solutions of the
homotopy function H ( X , μ, t0 ) = 0 become the solutions
of the equation set F ( X , μ) = 0 . By this way, the solution
set X of the equation set can be obtained at the general
position of μ = μ0 , and this solution set includes the entire
configurations of manipulators.
Changing the input parameters continuously and
smoothly, a series of configuration curves can be obtained.
If there is an intersecting point among these curves, the
configuration bifurcation will take place. The point at
which the configuration bifurcation takes place is a kind of
typical singular point.
Then the coefficient homotopy method is utilized to
figure out the configuration curves corresponding to the
variation of the input parameters μ from μ0 to μ .
The solution set X of the bifurcation equation
F ( X , μ0 ) = 0 has been obtained through Eq. (8), the
bifurcation equation, therefore, can be selected as the initial
equation. The coefficient homotopy function is
H ( X , μ, t1 ) = t1F ( X , μ) + (1 − t1 )γ 1F ( X , μ0 ) = 0 .
Fig. 2. Assembly configurations
From the foregoing analysis, we can infer that the
singular points under single input parameter is caused by
the limit moving position corresponding to the dimension
of the manipulator rather than by the line vectors
correlation, and these singular points belong to the
dimensional-utmost singularity type.
3.3 Configuration curves under the singular input
parameter
For obtaining the configuration curves going with the single
input parameter determined by Eq. (3), without loss of
universality, select the input parameter l1 as an independent
variable, and keep the other five input parameters constant.
Then the homotopy function H ( X , μ, t0 ) [25] corresponding
to Eq. (3) is
H ( X , μ, t0 ) = t0 F ( X , μ) + (1 − t0 )γ 0 G ( X ) = 0 ,
(8)
where G (x) is the initial equation set, and all of its
solutions are given; t0 is the homotopy parameter
t0 ∈ [0,1] ; γ 0 is an appropriate non-zero complex
constant (the virtual component is not equal to zero), which
ensures that each point on the homotopy path is the positive
(9)
Taking the solutions set X of the initial equation as the
starting point of Eq. (9) and tracing all homotopy paths, all
solution sets X of the bifurcation equation F ( X , μ) = 0
can be obtained within the variable scope of the extensible
legs. Connecting these solution sets orderly, a series of
configuration curves relating to the variation of the input
parameters μ are obtained.
Fig. 3 shows the configuration curves going with the
input parameter l1 . The characteristics of the SRHGSPM
going with the single input parameter are as follows.
(1) There are two or four assembly configurations
corresponding to a group of input parameters above the
base while the assembly configuration exists. And two of
them will intersect at one point and form a turning point[26].
Call its original configuration curve, along which the
manipulator moves close to the singular point, as the
persistent configuration curve, the other as the
non-persistent one. At the singular point, the manipulator
can move along its persistent configuration curve or its
non-persistent one while the input parameter has a very
small increment or decrement. This will lead the motion
direction of the manipulator at this singular point uncertain.
Usually, the initial assembly configuration of the
manipulator is that at which the normal lines of the
movable platform and the base are co-line, and also
the x1 -axis and the x -axis are in the same plane. This kind
assembly position corresponding to the semi-regular
hexagon Gough-Stewart platform is that the normal lines of
the movable platform and the base are co-line, and the
lengths of all six extendable legs are the same. From Fig. 3
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CHINESE JOURNAL OF MECHANICAL ENGINEERING
we know that this assembly configuration is on curve b3
and the moving region is the minimum. In this region, the
reachable scope for each configuration component is very
limited, for instance, on the components x, z , ρ , γ .
tively. It means that the moving region of the manipulator
has a close relationship with its original assembled
configuration. If a suitable measure to ensure the original
assembly configuration of the manipulator on its desired
configuration curve is not taken, the moving region of the
manipulator will be uncertain.
(3) The curvature radiuses of the configuration curves
are so large at some singular points, for instance, the
configuration curve on component x at singular point M 1 ,
the configuration curve on component α at singular point
M 4 , that the manipulator will obtain free movements on
these components, while the input parameter l1 has a very
small increment or decrement caused by the joint
clearances or the control precision of the system.
(4) According to the concept of the maximum losing
control domain[27], for the given control precision of the
input parameters ε , the corresponding radius of the
maximum losing control domain at the singular point M i
is δ1M i . If the input parameter locates beyond the circle
centered at the singular point M i with the radius δ1M i , the
motion of the manipulator is certain and the free
movements at the singular point will disappear.
Supposing the manipulator moves along b4 , the manipulator can move from l1M1 + δ1M1 to l1M 4 − δ1M 4 without
meeting the singularity. This singularity-free moving region
expressed by the input parameter is defined as
singularity-free input parameter zone. It can be found that
on the different configuration curve, the singularity-free
input parameter zone is different.
4
4.1
Singular Points and Assembly Configurations
Going with Multiple Input Parameters
Two input parameters
While choosing two input parameters as the independent
variables to analyze the singular points and the assembly
configurations from the six input parameters, the typical
combinations of two input parameters are l1 = l6 , l1 = l3 ,
and l1 = l4 . The singular points corresponding to the
combinations of two input parameters are listed in Table 2.
Table 2.
Singular points going with two input parameters
( lk = 2.0, k ≠ i, j )
Input
Singular point
parameter
M1
M2
M3
M4
l1 , l6
1.599 82
1.783 98
2.742 92
3.243 44
l1 , l3
1.141 2
1.630 1
2.447
2.538
l1 , l4
1.218
1.507 8
2.212
2.673 8
li / m
Fig. 3.
Configuration curves going with one input parameter
(2) Along the different configuration curve, the moving
region from the leftmost to the rightmost scaled by the
length of the input parameter is different. For instance, on
curve b1 , the manipulator can move from M 1 to M 4 .
On curve b5 and b7 , the moving regions are
Z 2 = {l1 ∈ (l1min , l12 )} and Z 3 = {l1 ∈ (l11 , l1max )} , respec-
From Table 2, while choosing two adjacent input
parameters as the independent parameters, and keeping the
others constant, the singularity-free input parameter zone
from M 2 to M 3 is the largest. In reverse, while
choosing two opposite input parameters as the independent
·816·
LI Yutong, et al: Singular Assembly Configurations and Configuration Bifurcation Characteristics
Parallel Mechanisms
Y
Y2-DOF Rotational
of the Semi-regular
HexagonsDecoupled
6-6 Gough-Stewart
Manipulator
parameters, the singularity-free input parameter zone is the
smallest. This kind of character is useful for programming
the trajectory of the input parameters to obtain a relative
large singularity-free workspace. While programming the
trajectory of the input parameters of the manipulator, we
should consider not only the pose requirements, but also the
distance between the present working position and the
singular point corresponding to the present pose parameters.
Apparently, the larger the distance is, the better the ability
for the manipulator to burden the loads is, and higher the
controllability and stability of the manipulator are.
Therefore, while programming the trajectory of the input
parameters of the SRHGSMP, it is better to choose two
adjacent input parameters to meet the configuration
requirements, rather than choose two opposite input
parameters.
4.1.1 I1=I4
The condition of l1 = l4 , l2 = l3 and l5 = l6 makes leg
l2 and leg l3 share the same plane, leg l1 and leg l4
share the same plane, and leg l5 and leg l6 share the
same plane. Simulating the motion of the manipulator
under the 3-D virtual assembly environment, it is found that
within l1 = l4 < 1.507 , the motion of the platform is to
rotate about normal line of the symmetry plane of the
driving legs, and corresponding value of the Jacobian
matrix, i.e. det A, is small. For instance, it is less than 100
(its normal non-singular value is usually greater than 1 500),
but not zero. In fact, with respect to the assembly
configuration shown in Fig. 4a, while l1 = l4 < 1.507 , the
SRHGSPM has degenerated as a plane moving mechanism,
and its motion is certain. However, its load ability along the
normal line of the symmetry plane of the driving legs is
very limited.
At singular point M 2 , three intersecting points, which
are formed by two line vectors of legs relative to the
symmetric plane of six legs, share the same line, as shown
in Fig. 4b. In this case, the line vectors correlation
condition is met. Fig. 5 shows that at this singular point, a
fork configuration bifurcation has taken place, and one
self-motion degree along the normal line of the symmetric
plane of legs is obtained. Due to the angle between the
x-axis and the normal line of the symmetric plane of six
legs is very small, this self-motion of the manipulator is
mainly reflected on component x.
At the fork configuration bifurcation singular point M 2 ,
besides one degree-of-freedom self-motion, the manipulator
has three kinds of configurations to embody: along curve
b2 , curve b1 or curve b4 . Obviously, the configuration
bifurcation character at the fork bifurcation point is more
complicated than that at the dimensional-utmost singular
point.
At other three singular points, as shown in Fig. 4, the
singularities coincide with the line vectors correlation
condition. For instance, at singular point M 1 , the
intersecting point of two symmetric legs’ line vectors
relative to the symmetric plane of six legs locates within
the same plane, as shown in Fig. 4a. At singular point M 3 ,
two indirect adjacent legs’ line vectors intersect at one point.
And at singular point M 4 , three interlaced legs’ line vectors
intersect at one point.
Fig. 4.
Configuration going with two input parameters l1, l4
4.1.2 Other two input parameters
While two adjacent input parameters are taken as the
independent variables, such as l1 and l6 , or l1 and l2 ,
the characteristics of the configuration curves[28] are similar
with those shown in Fig. 5.
The difference is the location of the fork configuration
bifurcation point. In the case of l1 = l2 , the fork singular
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CHINESE JOURNAL OF MECHANICAL ENGINEERING
point M 2 locates nearby the singular point M 1 , and in
the case of l1 = l6 , the fork singular point M 3 locates
nearby the singular point M 4 . However, the singularities
at other three singular points are not caused by the line
vectors correlation, but by the limit moving position
corresponding to the dimensions of the manipulator. This
means that when taking two adjacent input parameters as
the independent variables, only one fork configuration
bifurcation singular point takes place, and the other three
singular points belong to the dimensional-utmost singularity
type.
variables from six input parameters can be generalized
into three cases: ① changing l1 , l2 and l3 ; ②
changing l1 , l3 and l5 ; ③ changing l1 , l2 and l4 .
Similarly, from the expended Eq. (5), the singular points
in different cases are listed in Table 3.
Table 3． Singular points of the SRHGSPM going
with three input parameters ( lm = 2.0, m ≠ i, j , k )
Input
M1
M2
M3
l1 , l2 , l3
1.368
1.612
2.465
2.895
l1 , l4 , l5
1.538
1.782
2.246
2.638
l1 , l3 , l5
1.483
1.483
2.463
2.463
li / m
4.2.1
Fig. 5.
Configuration curves going with two input parameters
While two nearby input parameters are taken as the
independent variables, such as l1 and l3 or l1 and l5 ,
the configuration curves are the same as those going with
the single input parameter. All of the singular points
belong to the dimensional-utmost singularity type.
4.2 Configuration curves going with three input
parameters
Selecting three input parameters as the independent
Singular point
parameter
M4
Adjacent three input parameters l1 , l2 , l3
The assembly configurations of four singular points
relative to l1 , l2 , l3 are shown in Fig. 6. It can be found
that the singular configurations at M 1 , M 4 and M 2
are caused by the intersection of two adjacent or near by
legs’ line vectors. However, the singular configuration at
M 3 is caused by the limit moving position corresponding
to the dimensions of the manipulator. In this assembly
configuration, no correlation relation among the legs’ line
vectors exists. The pattern of the configuration curves
going with three input parameters l1 = l2 = l3 is similar
with that going with the single input parameter as shown
in Fig. 3. For instance, while the assembly configuration
exists, there are at least two assembly configurations
corresponding to a group of given input parameters, and
these two configurations will intersect and form a turning
point singular point, as well as along different
configuration curve, the manipulator will have a different
singularity-free input parameter zone.
Comparing Fig. 5 with Fig. 3, it is found that the
curvature radius of the configuration curves at the singular
points going with two input parameters is generally smaller
than that going with the single input parameter. Comparing
the configuration curves going with three adjacent input
parameters l1 , l2 , l3 with Fig. 5, we find that the curvature
radius of the configuration curves at singular points going
with three input parameters is generally smaller than that
going with two input parameters. These new
understandings are valuable in the following aspects:
(1) Through applying multi-input parameters nearby the
singular point, the self-motion region of the component on
which the curvature radius of the configuration curve is
relatively large under single or two input parameters can be
reduced.
(2) The singularity avoidance error can be reduced. This
is based on fact that the larger the curvature radius of the
configuration curve is, the larger the self-motion region or
the singularity avoidance error[23] is.
·818·
LI Yutong, et al: Singular Assembly Configurations and Configuration Bifurcation Characteristics
Parallel Mechanisms
Y
Y2-DOF Rotational
of the Semi-regular
HexagonsDecoupled
6-6 Gough-Stewart
Manipulator
Fig. 7.
Singular assembly configurations of the SRHGSPM
going with discrete input parameters l1, l4, l5
Comparing Fig. 6 with Fig. 7, it can be inferred that the
singularity of the assembly configurations for the
manipulator under discrete input parameters, such as
l1 , l4 , l5 , is more complicated than that under adjacent input
parameters, such as l1 , l2 , l3 .
Fig. 6.
Singular assembly configurations of the SRHGSPM
going with adjacent input parameters l1, l2, l3
4.2.2 Discrete input parameters l1 , l4 , l5
The assembly configurations going with three input
parameters l1 , l4 and l5 are shown in Fig. 7. At the
singular point M 4 , two adjacent legs’ line vectors intersect
at one point and construct a singular assembly
configuration. At the singular point M 1 or M 3 , one
driving leg locates within the plane of the movable
platform. The above two kinds of singularities at singular
point M 4 and M 1 or M 3 have appeared at the singular
point M 2 . In this assembly configuration, leg l6 and leg
l1 intersect at point P, and leg l6 locates within the
plane of the movable platform.
4.2.3 Symmetric input parameters l1 , l3 , l5
While choosing three symmetric input parameters as the
independent variables, for instance l1 , l3 , l5 , to analyze the
configuration curves and the singular assembly
configurations, the singular point M 1 coincides with M 2 ,
and the singular point M 3 coincides with M 4 . In this
way, four singular points have reduced into two singular
points. However, there are still four configuration curves
above the base while the assembly configuration exists.
The singular assembly configuration at M 3 ( M 4 ) has been
shown with Fig. 8. The singular assembly configuration
represents the position that the movable platform rotates
90° in clockwise direction about the normal line of the base.
This singular assembly configuration coincides with
Fichter’s result[2], which is caused by the linear correlation
of the Jacobian matrix.
Fig. 8. Singular configurations of the SRHGSPM at singular
point M1 going with symmetric input parameters
CHINESE JOURNAL OF MECHANICAL ENGINEERING
At singular point M 1 ( M 2 ), the singular assembly
configuration is caused by the dimensional limit condition
( l2 = l4 = l6 = 2.0 ), and the anticlockwise rotation angle
is 60°. If l2 , l4 , l6 are long enough, the anticlockwise
rotation angle should be 90°. This indicates that the
dimensions of the manipulator have a certain effect on the
singular assembly configuration.
5
Discussions
5.1 Similarities
Whether under the single input parameter or under the
multi-input parameters, the common characteristics of the
configuration curves of the SRHGSPM are as follows:
(1) While the assembly configuration exists, there are
two or four configurations to correspond to the same group
of the input parameters above the base platform. Especially,
at the vicinity of the singular points, at least two
configuration curves intersect at one point and form a
turning point singularity.
(2) Above the base platform, the SRHGSPM has four
singular points. And the singularity-free input parameter
zones have a close relation with its original assembly
configuration; on different configuration curve, the singularityfree input parameter zone is different. Moreover, if the
curvature radius of the configuration curve expressed by
one configuration component is large enough, the
manipulator will obtain a self-motion degree of freedom on
this component. The larger the curvature radius of the
configuration curve is, the larger the self-motion region is.
5.2 Differences
5.2.1 Under single input parameter
The singular points of the SRHGSPM belong to the
dimensional-utmost singularity type. These singular points
are caused by the limit moving position corresponding to
the dimension of the manipulator rather than by the line
vectors correlation. The configuration curves under single
input parameter are the simplest forms.
5.2.2 Under two input parameters
Both the singular points belonging to the dimensionalutmost singularity type, and the singular points belonging
to the line vectors correlation singularity type can occur.
While taking two opposite input parameters as independent
parameters, the manipulator has three line vectors
correlation singular points, and a fork configuration
bifurcation singular point. The variations of the
configuration curves at the vicinity of the singular point are
relatively complicated. Especially, at the fork configuration
bifurcation point, the manipulator has three kind
configurations to follow, and one self-motion degree of
freedom. While choosing two adjacent input parameters as
independent variables, the fork configuration bifurcation
point exists still, but other three singular points degenerate
as the dimensional-utmost singular points. Furthermore,
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while choosing two nearby input parameters as independent
variables, all the four singular points become the
dimensional-utmost singular points.
On the aspect of the singularity-free input parameter
zone, selecting two adjacent input parameters as the
independent parameters, the SRHGSPM will obtain the
largest singularity-free input parameter zone. In reverse, if
choosing two opposite input parameters as the independent
parameters, the SRHGSPM will obtain the smallest
singularity-free input parameter zone. This discovery is
valuable for programming the trajectories of the input
parameters. While programming the trajectories of the
input parameters of the SRHGSMP, it is better to choose
two adjacent input parameters to meet the configuration
requirements, rather than to choose two opposite input
parameters. In this way, the manipulator has relatively
larger load ability and higher controllability and stability.
5.2.3 Under three input parameters
While three adjacent input parameters are chosen as a
group of input parameters, three singular points belong to
the line vectors correlation singularity type, one singular
point belongs to the dimensional-utmost singularity type.
The singular assembly configurations for the discrete input
parameters are more complicated than those for the
adjacent input parameters. Especially, while choosing three
symmetric input parameters as the independent variables,
four singular points have reduced into two singular points.
And the singular points become the Jacobian linear
correlation singular type, i.e. the moving platform rotates
90deg in clockwise/anticlockwise direction around the
normal line of the base.
5.2.4 Differences
With increasing the number of the input parameters, the
singularity types of the SRHGSMP change from the simple
dimensional-utmost singularity type, to the coexistence of
the line vectors correlation singular type and the dimensionalutmost singularity type, to only the line vectors correlation
singular type, and to the Jacobian matrix correlation singular
type, the singular types are more and more complicated. And
the configuration bifurcation characteristics at the vicinities
of the singular points change from the turning point, to the
fork bifurcation point, and to the multiple bifurcation point.
The motion at the singular point is more and more
uncertain.
The curvature radius of the configuration curves at
singular points going with two input parameters is
generally smaller than that going with the single input
parameter. And the curvature radius of the configuration
curves at singular points going with three input parameters
is generally smaller than that going with two input
parameters. This new discovery has shown us a potential
method to reduce the self-motion regions permitted by the
control system. While the manipulator approaches the
singular point, through selecting multiple input parameters
·820·
LI Yutong, et al: Singular Assembly Configurations and Configuration Bifurcation Characteristics
Parallel Mechanisms
Y
Y2-DOF Rotational
of the Semi-regular
HexagonsDecoupled
6-6 Gough-Stewart
Manipulator
to reduce the curvature radius of the configuration curve
expressed by the component on which the manipulator
obtain a self-motion degree of freedom, the self-motion
region can be reduced then. Similarly, because the
singularity avoidance error has a close relation with the
curvature radius of the configuration curve at the singular
point, if the curvature radius is reduced, the singularity
avoidance error can be reduced definitely.
6
Conclusions
In this paper, the configuration bifurcation characteristics
going with the input parameters, the assembly configurations
at singular points, and the reasons to cause the singularity are
analyzed for the SRHGSPM. We find that the
dimensional-utmost singularities, line vectors correlation
singularities and Jacobian matrix correlation singularities
can occur individually or jointly while choosing different
number of input parameters. The number and the
combination of the input parameters have great influences
on the complexity of the singularities and the curvature
radiuses of the configuration curves. Through selecting a
group of adjacent input parameters, the simple singularity
and the large singularity-free input parameters zones will
be obtained. Selecting multiple input parameters, the
self-motion regions and the singularity avoidance errors
can be reduced. These new discoveries are valuable and of
significance for the trajectory design, the singularity
avoidance, and the self-motion control of the parallel
manipulator.
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
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CHINESE JOURNAL OF MECHANICAL ENGINEERING
Biographical notes
LI Yutong, born in 1973, is currently a PhD candidate in Institute
of Mechanical Design, Zhejiang University, China. Her research
interests include singularity avoidance of parallel manipulator,
design and simulation of axial-symmetric vectoring exhaust
nozzle.
Tel: +86-571-87952508; E-mail: [email protected](public)
WANG Yuxin, born in 1964, is currently a professor and a PhD
candidate supervisor in Institute of Mechanical Design, Zhejiang
University, China. His main research interests include
controllability of parallel manipulators, rapid and automatic
·821·
creative design of complicate mechanical systems.
Tel: +86-571-87952508; E-mail: [email protected]
HUANG Zhen, born in 1936, is currently a professor and a PhD
candidate supervisor in Robotic Research Center, Yanshan
University, China.
E-mail: [email protected]
PAN Shuangxia, born in 1963, was a professor and a PhD
candidate supervisor in Institute of Mechanical Design, Zhejiang
University, China.