Proceedings of the 2001 IEEE
International Conference on Robotics & Automation
Seoul, Korea • May 21-26, 2001
Eclipse-II: A New Parallel Mechanism Enabling Continuous
360-degree Spinning Plus Three-axis Translational Motions
Jongwon Kim*, Jae-Chul Hwang, Jin-Sung Kim and F.C.Park
School of Mechanical & Aerospace Engineering
Seoul National University, Korea
*Email: [email protected]
Abstract
This paper presents the Eclipse-II, a new six degree-offreedom parallel mechanism, which can be used as a basis
for general motion simulators. The Eclipse-II is capable
of x, y and z-axis translations and a, b and c-axis rotations.
In particular, it has the advantage of enabling continuous
360-degree spinning of the platform. The computational
procedures for forward and inverse kinematics of the
Eclipse-II are described. Next, the complete singularity
analysis is presented for the two cases of end-effector
singularity and actuator singularity. Two additional
actuators are added to the original mechanism to
eliminate both types of singularity within the workspace.
Keywords: parallel mechanism, motion
singularity analysis, workspace analysis
1
(b) Rotation angle 90°
(c) Rotation angle 180°
(d) Rotation angle 270°
simulator,
Introduction
A motion simulator is a virtual reality system that
simulates a real situation by using audio-visual effects
and movement of a motion base. Motion simulators have
been applied to many areas. For example, flight
simulators are used for pilot training and other purposes
by providing the pilot with motions that reflect the state
of the aircraft. Driving simulators reproduce actual
driving conditions for vehicle design and human factors
studies [1, 2].
A motion simulator consists of an auditory system to
generate sound, a visual system to display images, and a
motion base system to generate movement as a result of
motion cues.
Most conventional simulators adopt the Stewart platform
shown in Figure 1, as the motion base [3]. The Stewart
platform is a six degree-of-freedom parallel mechanism
that enables both translational and rotational motions.
However, motion such as the 360-degree overturn is
Figure 2. Eclipse-II mechanism and its 360-degree
continuous rotational motions
impossible, because the platform can only tilt as much as
±20-30 degrees. That is, the overturn motion of the
aircraft or the 360-degree spin of the roller coaster cannot
be reproduced by the Stewart platform.
A novel six degree-of-freedom parallel mechanism
architecture, which is called the Eclipse-II, has been
designed for the motion base of a motion simulator. The
objective is to develop a mechanism capable of 360degree rotational motion of the platform as well as
translational motion. Figure 2 shows the Eclipse-II
mechanism and an example of its rotational motion
capability. Since the Eclipse-II has no limitation on its
rotational motion, it is possible to develop a more realistic
and higher fidelity simulator by adopting the Eclipse-II
mechanism as the motion base of a flight or a roller
coaster motion simulator.
In the next section, the kinematic structure of the EclipseII, including the computational procedures for forward
and inverse kinematics, is described. The singularity
analysis and a method for eliminating the singularities are
presented in Section 3. Finally, some concluding remarks
follow in Section 4.
2
Figure 1. Structure of the Stewart platform
0-7803-6475-9/01/$10.00© 2001 IEEE
(a) Rotation angle 0°
Kinematics of the Eclipse-II
As shown in Figure 3, the Eclipse-II consists of three
3274
iˆ““
q–•›
s•Œˆ™
j–“œ
–“œ”•
”•
jY
hY
M
¡
iY
iZ
jZ
hZ
s•Œˆ™
w™
w™šš”ˆ›Š
q–•›
platform, the position of the spherical joints is found from
the following equation:
¢t¤
jX
Ÿ

yŒ–“œ›Œ
q–•›
iX
€

¢m¤
bi = R bi + p
M
bi is the vector of the ith spherical joint
where
expressed in the moving frame coordinate, R is the
rotation matrix corresponding to the orientation of the
&
moving frame about the fixed frame, and p is the vector
j™Šœ“ˆ™
j–“œ
–“œ”•
”•
hX
&
corresponding to the origin of the moving frame. R , p ,
&
j™Šœ“ˆ™
w™
w™šš”ˆ›Š
q–•›
and bi are expressed in fixed frame coordinates.
2. The circular prismatic joint values are calculated from
the positions of the spherical joints as follows:
Figure 3. Structure of the Eclipse-II mechanism
θ i1 = arctan 2(biy , bix )
PPRS serial sub-chains that move independently on a
fixed circular guide. Here, P, R, and S denote prismatic,
revolute, and spherical joints, respectively. The Eclipse-II
has six degrees-of-freedom, and six actuated joints, which
consist of two P joints on the vertical columns, one P joint
on the circular column and three P joints along the
circular guide. All six actuated joints can be found in
Figure 3, and are indicated by arrows. Mounting one
circular column and two linear columns on the circular
guide results in the Eclipse-II having a large orientation
workspace. Thus, the platform can rotate 360 degrees
continuously about the y-axis in the moving frame {M}
and the Z-axis in the fixed frame {F}, respectively, as
shown in Figure 3.
bix and bix
respectively.
θ 21
jX
jY
iY
θ 13 = 180° + φ − ϕ
Ÿ
φ = arcsin(
j
j™™Šœ“ˆ
œ“ˆ™™
j–“œ”
“œ”••
™ˆ
jZ
)
2
ϕ = arccos(
b1
+ L12 − rb 2
2 ⋅ b1 ⋅ L1
)
4. The prismatic joint values, θ12 , on the circular column
are calculated as follows:
θ 12 = arctan 2(a1z , a1w )
(4)
where,

θ 31
jX
ϕ
θ12
~
v
™‰
jZ
φ
iX
θ13
}Œž h
(a) Top view
b1z
b1
}Œž i

iZ
(3)
where,
θ11
iX
&
are the x and y coordinates of bi ,
3. The revolute joint value, θ13 , on the circular column is
calculated as follows:
1. Given the position and orientation of the moving
€
(2)
where θ i1 is the ith circular prismatic joint value, and
2.1 Inverse Kinematics
The problem of inverse kinematics is to determine the
values of the actuated joints from the position and
orientation of the moving frame {M} attached to the
moving platform. For the Eclipse-II mechanism, its
inverse kinematics can be solved by successively solving
the inverse kinematics of each sub-chain. The algorithm
for solving the inverse kinematics is as follows:
(Coordinates and joint convention are described in Figure
4.)
s•Œˆ
s•Œˆ™™
j–“œ”
“œ”••
(1)
jY
¡
iZ
θ 23

v
(b) Side view A
Figure 4. Coordinate and joint convention
3275
iY
S €
(c) Side view B
θ 22
&
θ a = [θ 11 θ 12 θ 21 θ 22 θ 31 θ 32 ]T
a1z = L1 cos θ 13 + b1 cos φ ,
&
θ p = [θ 13 θ 23 θ 33 ]T
a1w = L1 sin θ 13 + b1 sin φ
and d is the distance between the spherical joints.
5. The linear prismatic joint values, θ i 2 ( i = 2, 3), and
the position of the revolute joints, θ i 3 ( i = 2, 3), on the
vertical columns are calculated as follows:
θ i 2 = − d i + d i 2 − ei ( i = 2, 3 )
(5)
where,
&
n = [0 0 1]T , a i = [ra cos θ i1
ra sin θ i1
d i = ( ai − bi ) ⋅ n , and ei = a i − bi
2
0]T ,
− Li 2 .
Values for θ i3 are found as follows:
&
θ i 3 = arccos(
&
&
(bi − ci ) ⋅ n
&
&
&
bi − ci ⋅ n
) , ( i = 2, 3)
(6)
Note that the solution of θ i 3 is in the range [0°, 180°].
2.2 Forward Kinematics
The problem of forward kinematics is to determine the
position and orientation of the moving frame given the
actuated joint values. As is the case for most typical
parallel mechanisms, it is generally quite difficult to find
the analytic solution of the forward kinematics of the
Eclipse-II. If all of the actuated and passive joint values
are known, its forward kinematics can be solved from the
forward kinematics of each serial sub-chain. Therefore,
the first step in the forward kinematics solution is to
determine numerically the passive joint values from the
actuated joint values by using the kinematics constraint
equations.
The following algorithm solves iteratively the forward
kinematic using the Newton-Raphson procedure:
1. The constraint equation between the active and passive
joint values is generated as follows:
& &
&
&
g (θ a ,θ p ) = 0
3. The position and orientation of the moving frame is
determined from the forward kinematics equations for
each sub-chain.
2.3 Workspace Analysis
The workspace of a mechanism is defined to be the set of
all positions and orientations reachable by the moving
platform. The workspace analysis of the Eclipse-II begins
with a description of the moving frame orientation. Many
parameterizations exist for describing the orientation; for
example, Euler angles, fixed angles, or exponential
coordinates. Among them, the Z-Y-Z fixed angle is chosen
since it is one of the most convenient methods for
describing the orientation of the Eclipse-II mechanism. In
terms of the Z-Y-Z fixed angles, the rotation matrix is
given by
R = Rot Z (α )Rot Y (β )Rot Z (γ )
(8)
where, α, β, and γ are the rotation angles about the Z-, Y-,
and Z-axes of the fixed frame, respectively. If the
orientation of the moving platform is described by the ZY-Z fixed angle, the characteristic of the Eclipse-II
mechanism is that the tilting angle, β, can reach 360
degrees. The Cartesian workspace for a given tilting angle,
β, is defined to be the set of all positions in space that
can be reached by the moving platform while the platform
is oriented at the given tilting angle.
The actual Cartesian workspace for a given tilting angle is
restricted by the following physical constraints on the
mechanism:
(1) the stroke limit of the linear prismatic joints;
(2) the interference between the vertical or circular
columns;
(3) the interference between the columns and rods; and
where,
 (b1 − b2 ) ⋅ (b1 − b2 ) − d 2 


g (θ a ,θ p ) = (b2 − b3 ) ⋅ (b2 − b3 ) − d 2 
 (b − b ) ⋅ (b − b ) − d 2 
3
1
 3 1

 cos θ i1 ( R cos θ i 2 − Li cos θ i 3 )
 

  sin θ i1 ( R cos θ i 2 − Li cos θ i3 )  (i = 1)
 

R sin θ i 2 − Li sin θ i 3
bi = 
 cos θ i1 (ra − Li sin θ i 3 )
  sin θ (r − L sin θ )  (i = 2,3)
i1 a
i
i3 
 


L
cos
θ i3
  θ i 2 − i
& &
(7)
2. Given values for the actuated joints, values are
computed for the passive joints from the previous
equations using the Newton-Raphson procedure.
(4) the rotation limit of the spherical joints.
&
The kinematic parameters of the mechanism can be
optimized to maximize the workspace within the above
constraints.
3
Singularity Analysis
Singularity refers to the configuration in which the
number of degrees-of-freedom of the mechanism
increases or reduces instantaneously. Since being close to
one of the singularities can limit movement, disable
control or break the mechanism, the singularity analysis is
3276
h
i
(a) 6 d.o.f Eclipse-II
(b) 7 d.o.f Eclipse-II
Figure 7. Y-direction motion of the Eclipse-II
Figure 5. Condition number plot of the endeffector singularity configuration of the Eclipse-II
(a) Point A in Figure 5
(b) Point B in Figure 5
Figure 8. Condition number plot of the end-effector
singularity configuration of the 7 d.o.f Eclipse-II
Figure 6. Examples of the end-effector singular
configuration of the Eclipse-II
one of the most significant and critical problems in the
design and control of parallel mechanisms.
Singularities of parallel mechanisms can be classified into
two types: the end-effector singularity and the actuator
singularity [4]. For the Eclipse-II mechanism, these two
types of singularities co-exist in the workspace. In this
section, the singular configuration of the Eclipse-II
mechanism and the method for eliminating singularities
are described.
The kinematic parameters used here for singularity
analysis are specified as follows. The radius of the
circular guide (ra), circular column (rb) and moving
platform (r) are, respectively, 1000, 1000, and 300 mm.
The lengths of the link in the circular column (L1) and of
the links in the linear column (L2, L3) are, respectively,
900 and 780 mm.
2. The rotational matrix representing the orientation of the
moving platform is:
R = [R x
where, R x =
(9)
&
1 &
1 & &
(b1 − bc ) , R y =
(b2 − b3 ) ,
r
3r
v
6×9
 w = Jθ , J ∈ R
 
(11)
Elements of the Jacobian, J , are as follows:
&
 1 ∂bi

 3 ∂θ i

y
 R z ⋅ ∂R

∂θ i
Ji = 
x

z ∂R
− R ⋅ ∂θ
i

y

R
∂
x
− R ⋅
∂θ i

&
&
1 & &
(b1 + b2 + b3 )
3
(10)
3. In order to obtain the Jacobian matrix, equations (9)
and (10) are differentiated:
1. The position of the moving platform bc is the center
of three spherical joints.
&
Rz]
Rz = Rx × Ry .
3.1 End-Effector Singularity
The end-effector singularity is the configuration in which
the moving platform of the mechanism loses one or more
degrees-of-freedom of possible motion. In this case the
forward kinematic Jacobian loses rank. For Eclipse-II, the
algorithm for solving the forward Jacobian is as follows:
bc =
Ry
where,
3277
&
&
&
b1 = b4 = b7 ,
&






 ( i = 1, 2,






&
&
b2 = b5 = b8 ,
Î, 9 )
&
&
&
b3 = b6 = b9 ,
k
j
Figure 9. Condition number plot of the actuator
singularity configuration of the Eclipse-II
(a) Point C in Figure 9
(b) Point D in Figure 9
Figure 10. Examples of the actuator singular
configurations
θ 1 = θ 11 , θ 2 = θ 21 , θ 3 = θ 31 , θ 4 = θ 12 , θ 5 = θ 22 ,
θ 6 = θ 32 , θ 7 = θ 13 , θ 8 = θ 23 , θ 9 = θ 33 .
Since there is no systematic way of finding all of the
singularities of a given parallel mechanism, the endeffector singularities can be found by a numerical method,
which computes the condition number of the Jacobian at
all points of the workspace. The condition number of
J is defined as the ratio of the maximum singular value to
the minimum singular value.
The results in Figure 5 are based on this numerical
method, and illustrate the condition number of the
forward Jacobian, while the Eclipse-II tilts from 0° to
360°. The x-axis and y-axis in Figure 5 represent the
tilting angle and translation along Z-axis in the fixed
frame, respectively. The dark regions around 90° and
270° in the figure represent the singular configurations,
which are the end-effector singularities.
Figure 6 shows two examples of the end-effector singular
configurations of the Eclipse-II. As shown in the figure,
the end-effector singular configurations of the Eclipse-II
occur when any one of the spherical joints is located on
the Z-axis of the fixed frame. If the platform reaches one
of the end-effector singular configurations of the EclipseII, there exist infinite solutions of inverse kinematics.
This means that there exist infinite possible sets of active
joint values for one specific end-effector configuration of
the moving platform.
Figure 11. Condition number plot of the actuator
singularity configuration of the over-actuated
Eclipse-II
Even if it is possible to select only one solution of the
inverse kinematics while the platform of the Eclipse-II
tilts from 0° to 360°, the joints have infinite velocity, or
one linear column and the circular column collide with
each other at the end-effector singular configuration. As a
solution to this problem, it is possible to change the
solution of the inverse kinematics at the end-effector
singular configuration.
However, there still remains the problem, as shown in
Figure 7(a), that the platform cannot translate along the ydirection of the moving frame in the end-effector singular
configuration. Hence, an actuator is added to change the
position of the spherical joint that is connected to the
circular column; that is, one degree-of-freedom is added
to the original Eclipse-II. With this addition, the platform
can move along the y-direction in the end-effector
singular configuration by changing the position of the
spherical joint along the linear guide, whereas it is not
necessary for the circular column to move [see Figure
7(b).].
The additional actuator results in the elimination of the
end-effector singularity in the workspace. Figure 8 shows
that the condition number of the Eclipse-II with the
additional degree-of-freedom is reduced significantly
from that corresponding to Figure 5 at the tilting angles of
90° and 270°.
3.2 Actuator Singularity
A mechanism gains one or more degrees-of-freedom of
possible motion at actuator singularities; that is, a selfmotion occurs. The actuator singularity configuration can
be determined from the Jacobian of the constraint
equations. In the case of the Eclipse-II, the Jacobian of
the constraint equation can be found by differentiating the
constraint equation (7):
∂g ∂g θa +
θp = 0
∂θ a
∂θ p
(12)
where ∂g / ∂θ p is a 3 × 3 matrix.
If the matrix, ∂g / ∂θ p , is not full rank, the passive joint
values cannot be determined by the given active joint
3278
values, and then it is true that the mechanism is in one of
the actuator singularity configurations. Like the endeffector singularities, actuator singularities can be found
from the condition number of the Jacobian at all points of
the workspace. Figure 9 illustrates the condition number
of ∂g / ∂θ p , while the platform of the Eclipse-II tilts
from 0° to 360°. The x-axis represents the tilting angle
and the y-axis represents the rotation angle γ about the zaxis of the moving frame. As the condition number is
larger, the mechanism is nearer to the actuator singular
configuration, as shown in Figure 9. For example, in the
case of γ = 0 , actuator singularities occur around tilting
angles of 25° and 155°.
Figure 10 shows two examples of actuator singular
configurations of the Eclipse-II. There exist two major
problems in the actuator singular configurations. First,
when the platform is in the actuator singular configuration,
the platform cannot sustain its static equilibrium position
in the presence of the external force. In this case, the
platform seems to have extra degrees of freedom. Second,
the forward kinematic solutions are divided into two or
more directions in the actuator singular configuration.
Along the path crossing the actuator singular
configuration, there exist multiple forward kinematic
solutions with the same active joint values. Hence, there
exists a chance that the path can move in an undesired
direction.
One method for eliminating the actuator singular
configuration is to make the mechanism over-actuated by
adding an actuator to each one or more of the adequate
passive joints. In the case of the Eclipse-II, an additional
actuator is added to each R joint of the two linear
columns. Comparing Figure 11 with Figure 9, it may be
noted that the condition numbers are reduced significantly
and all of the actuator singular configurations are
eliminated.
4
Conclusions
This paper presents a new parallel mechanism, the
Eclipse-II. The unique feature of the Eclipse-II is that
continuous 360-degree rotational motion of the platform
is possible in addition to translational motion. Results of
the kinematic analysis on the forward and inverse
kinematics, the workspace, and the singular configuration
are described. The original Eclipse-II mechanism shows
both end-effector and actuator singular configurations
within its workspace. Hence, a linear guide, where one
spherical joint moves, is added to the mechanism to
eliminate the end-effector singularity. An actuator is also
added to each of two passive joints of the Eclipse-II to
eliminate the actuator singularities. Therefore, the final
Eclipse-II is an over-actuated seven degree-of-freedom
parallel mechanism.
5
References
[1] J. S. Freeman, et al., “The Iowa Driving Simulator:
An Implementation and Application Overview,” SAE
Paper 950174, 1995.
[2] G. P. Bertollini, et al., “The General Motors Driving
Simulator,” SAE Paper 940179, 1994.
[3] D. Stewart, “A Platform with six degrees of freedom,”
Proc. Inst. Mech. Eng., Vol. 180, Part I, No.15, pp.371386, 1966.
[4] F. C. Park and J. W. Kim, “Singularity Analysis of
Closed Kinematic Chains,” ASME J. of Mechanical
Design, Vol.121, pp.32-38, 1999.
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