Content uploaded by Jian S Dai
Author content
All content in this area was uploaded by Jian S Dai on Aug 31, 2015
Content may be subject to copyright.
Geometry and Kinematic Analysis of a
Redundantly Actuated Parallel
Mechanism That Eliminates
Singularities and Improves Dexterity
Jody A. Saglia
e-mail: jody.saglia@kcl.ac.uk
Jian S. Dai
e-mail: jian.dai@kcl.ac.uk
Department of Mechanical Engineering,
King’s College London,
University of London,
WC2R 2LS, England, UK;
Italian Institute of Technology,
Via Morego 30, Genova 16163, Italy
Darwin G. Caldwell
Italian Institute of Technology,
Via Morego 30, Genova 16163, Italy
e-mail: drawin.caldwell@iit.it
This paper investigates the behavior of a type of parallel mecha-
nisms with a central strut. The mechanism is of lower mobility,
redundantly actuated, and used for sprained ankle rehabilitation.
Singularity and dexterity are investigated for this type of parallel
mechanisms based on the Jacobian matrix in terms of rank defi-
ciency and condition number, throughout the workspace. The non-
redundant cases with three and two limbs are compared with the
redundantly actuated case with three limbs. The analysis demon-
strates the advantage of introducing the actuation redundancy to
eliminate singularities and to improve dexterity and justifies the
choice of the presented mechanism for ankle rehabilitation.
关DOI: 10.1115/1.2988472兴
1 Introduction
The use of robots and mechanisms in human rehabilitation has
substantially increased in past years. A Stewart platform was in-
troduced in 1999 by Girone et al. 关1兴for ankle rehabilitation. At
the same time Dai and Massicks 关2兴introduced a set of lower
mobility platforms for sprained ankle rehabilitation. This then
raised much interest 关3,4兴in recent years to generate more devices
for ankle rehabilitation. The analysis of mobility and constraint
that characterize the human’s ankle complex was made by Dai et
al. 关5兴in 2004, and a family of lower-mobility parallel mecha-
nisms for ankle rehabilitation was presented. Subsequently a
3UPS/U parallel robot was built for concept and principle evalu-
ation.
The device is of lower mobility 关6兴and redundantly actuated
with the degrees of freedom 共DOFs兲fewer than the number of
actuators. However, singularity removal and dexterity enhance-
ment to improve the effectiveness of ankle rehabilitation still re-
main to be tackled.
A redundantly actuated parallel mechanism refers to the use of
more actuators than the kinematic degrees of freedom of the
mechanism and this type of mechanism attracted good interest in
recent years.
This leads to the study of redundantly actuated and of low
mobility parallel mechanisms one of which was first presented by
Hunt 关7兴in 1982 in his constant-velocity-joint paper, and a picture
of it was given by Phillips 关8兴in 1984.
In 1992, Kurtz and Hayward 关9兴investigated the redundancy of
a parallel mechanism by adding an additional limb to the existing
three-limb system. This redundancy was classified by Sukhan and
Sungbok 关10兴as one of the types that included additional active
joints, replacing passive joints with active joints, and additional
limbs. The redundant mechanism design was carried out by
Leguay-Durand and Reboulet 关11兴for a 3DOF spherical parallel
mechanism with four collinear actuators, and the work was ex-
tended to the study of mechanism dexterity. At the same time, Lee
et al. 关12兴investigated the use of redundant actuation to admit
fault tolerance in a specific environment, such as space or nuclear
plants, where high performance is vital. In recent years algorithms
关13兴were developed for controlling redundantly actuated parallel
mechanisms. It was demonstrated that actuation redundancy is a
good choice when a designer aims to eliminate singularities
within the mechanism workspace and to improve the dexterity and
performance with which a robot can perform a certain task. Fur-
thermore, kinematically redundant parallel mechanisms that take
advantage of redundancy to avoid singularity were studied by
Wang and Gosselin 关14兴.
This paper introduces a lower degree-of-freedom parallel
mechanism with a central strut 关15兴for sprained ankle rehabilita-
tion. The geometry and kinematics of the 3UPS parallel mecha-
nism are presented subsequently. The mechanism singularities and
dexterity are numerically analyzed and reported. In particular, the
redundant actuation is introduced to eliminate singularity and to
improve dexterity. Comparison of the nonredundantly and redun-
dantly actuated parallel mechanism is carried out to verify the
elimination of singularities and the enhancement of dexterity in
the case of redundancy. Dexterity of the redundantly actuated
mechanism is compared with that of the equivalent 2DOF two-
legged mechanisms, again showing the great improvement gained
in dexterity and in singularity removal in the current 3UPS paral-
lel mechanism.
2 Platform Configuration Equation
The parallel mechanism presented in this paper consists of a
platform, a fixed base, three identical limbs, and a central strut
connected to the platform with a universal joint, as in Fig. 1. The
central strut is used to connect the platform and the base. Each
limb consists of a prismatic joint and is attached to the platform
with a spherical joint and to the base with a universal joint. Due to
the fact that three actuators are used for operating this 2DOF
platform, the mechanism is redundantly actuated. To investigate
such a mechanism, this paper starts by investigating its counter-
part, namely, a nonredundant parallel mechanism in which the
central strut is connected to the platform with a spherical joint. As
given in Fig. 2, the joints are labeled Aiand Bion the base and the
platform, respectively.
The three prismatic joints are actuated by pneumatic actuators.
Two Cartesian coordinate systems O共x,y,z兲and P共u,v,w兲as the
fixed frame and the platform moving reference frame are attached
to the base and the platform, respectively, and a third frame
P0共u0,v0,w0兲is placed on top of the central strut as the platform
fixed orientation reference, as in Fig. 2. Joint Ailies on the fixed
plane x-ywhile joint Bilies on the plane u-v. The origin of the
fixed reference frame Ois positioned at the centroid of ⌬A1A2A3
and the joint A1lies on the x-axis.
Again, the origin of the platform moving reference frame Pand
the fixed orientation frame P0are both positioned at the centroid
Contributed by the Mechanisms and Robotics Committee of ASME for publica-
tionintheJ
OURNAL OF MECHANICAL DESIGN. Manuscript received June 12, 2007; final
manuscript received August 22, 2008; published online October 8, 2008. Review
conducted by José M. Rico. Paper presented at the ASME 2007 Design Engineering
Technical Conferences and Computers and Information in Engineering Conference
共DETC2007兲, Las Vegas, NV, September 4–7, 2007.
Journal of Mechanical Design DECEMBER 2008, Vol. 130 / 124501-1Copyright © 2008 by ASME
Downloaded 16 Oct 2008 to 137.73.11.235. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
of ⌬B1B2B3and the joint B1lies on the u-axis. Both ⌬A1A2A3and
⌬B1B2B3are equilateral triangles with OA1=OA2=OA3=raand
PB1=PB2=PB3=rb.
Let 共i,j,k兲,共u,v,w兲and 共u0,v0,w0兲be the unit vectors of the
reference frame O,P, and P0, respectively. Defining three rotation
angles
␣
,

, and
␥
as roll, pitch, and yaw about the u0,v0, and w0
axes and a translation lfrom the origin of the fixed frame Oto the
origin of the moving frame P共whose origin coincides with the
origin P0兲, we can describe the position and orientation of the
moving platform with respect to the base with a translation vector
p=关00h兴T共1兲
and a rotation matrix
RP
O=Rotw0共
␥
兲Rotv0共

兲Rotu0共
␣
兲共2兲
The rotation matrix contains sine and cosine functions of the vari-
able rotation angles
␣
,

, and
␥
. Referring to Fig. 2, a loop-
closure equation for each limb ican be written as
AiBi=di=p+bi−ai=p+RP
Obi
P−ai共3兲
where diis the ith limb vector and, ai,bi, and bi
Pare the position
vectors of the joints Aiexpressed in the base reference frame, the
position of the Bijoints expressed in the platform fixed orientation
reference frame, and the position of the Bijoints expressed in the
moving reference frame, respectively.
The relation between the velocity of the platform and those of
the limbs can be obtained by deriving the Jacobian matrix to map
the joint velocities of the actuated prismatic joints into the output
Cartesian angular velocities of the moving platform. The vector of
actuated joint positions is defined as
t=关d1d2d3兴T
and the angular velocity of the moving platform is defined as
P=关
u0
v0
w0兴T
which contains the time derivative of the roll, pitch, and yaw
angles 共
␣
,

, and
␥
兲that describe the orientation of the platform.
Hence, we have the following expression for the platform angular
rate as:
P=关
␣
˙

˙
␥
˙兴T共4兲
Differentiating Eq. 共3兲with respect to time yields a velocity loop-
closure equation for each limb, as follows:
ddi
dt =d共p+bi−ai兲
dt 共5兲
with dai/dt=0 and dp/dt = 0, it results in
di
i⫻si+d
˙isi=
P⫻bi共6兲
where
iand siare the angular velocity of the ith limb and the
unit vector pointing along the direction of AiBi, respectively. Both
of them are expressed in the fixed reference frame. Dot-
multiplying both sides by siwe eliminate
iand obtain
共bi⫻si兲
P=d
˙ifor i= 1, 2, and 3 共7兲
The relation in Eq. 共7兲can be expressed in matrix form and
gives the configuration equation that governs the relation between
the joint rates and the moving platform output velocity
J
P=t
˙i共8兲
where J
has the form
J
=
冤
共b1⫻s1兲T
共b2⫻s2兲T
共b3⫻s3兲T
冥
共9兲
as the Jacobian matrix of the parallel mechanism and t
˙
=关d
˙1d
˙2d
˙3兴Tis the vector of the joint velocities. While obtain-
ing the Jacobian matrices the orientation of the moving platform is
described by means of three rotation angles, namely
␣
,

, and
␥
.
This gives a configuration equation of the nonredundantly actu-
ated parallel mechanism.
In the case of the redundantly actuated parallel mechanism, the
angle
␥
is constant and equal to zero, due to the presence of a
universal joint between the moving platform and the central strut,
as in Fig. 1. This introduces a redundantly actuated parallel
mechanism with three prismatic actuators. Therefore, the kinemat-
ics of the physical system can be represented with a reduced Jaco-
bian matrix where the third column relative to the angular velocity
␥
˙is eliminated.
The Jacobian matrix J
of the nonredundantly actuated parallel
mechanism, becomes J
⬘of the redundantly actuated parallel
mechanism, and the angular velocity vector
Pbecomes
P
⬘con-
taining only two angular velocities. The new expression is as fol-
lows:
J
⬘
P
⬘=
冤
j
11 j
12
j
21 j
22
j
31 j
32
冥
冋
␣
˙

˙
册
=
冤
d
˙1
d
˙2
d
˙3
冥
共10兲
where the coefficients j
ij come from the cross products in Eq. 共9兲
共bi⫻si兲T=关j
i1j
i2j
i3兴共11兲
3 Singularity Removal
Direct kinematic singularities occur when the Jacobian matrix
J
becomes singular. This is the case of linear dependency be-
tween the three vectors 共bi⫻si兲in a nonredundant case 共with a
spherical joint at the central strut connecting the platform兲. The
measure of singularity can be obtained as
Fig. 1 3SPS/U parallel mechanism
Fig. 2 Geometry of the parallel mechanism
124501-2 / Vol. 130, DECEMBER 2008 Transactions of the ASME
Downloaded 16 Oct 2008 to 137.73.11.235. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
␦
= det共J
兲共12兲
In this case, the direct kinematic singularities occur when the
Jacobian matrix J
is not full-rank.
Developing the cross product in Eq. 共9兲yields the analytical
expression of the direct kinematic Jacobian matrix and the analy-
sis of its determinant leads to obtaining the singularity loci of the
parallel mechanism within the workspace. The geometric param-
eters are set as, h=1, ra=ah, and rb=bh, where during the study
of the dexterity index to have a high but evenly distributed index-
based surface for this sprained ankle rehabilitation device, param-
eters are chosen as a=b=0.4 and the variation of these are dis-
cussed in Sec. 4. A numerical evaluation of the determinant of the
direct kinematic Jacobian J
is presented in Fig. 3.
Rotating to another view, Fig. 4 presents the valley that indi-
cates singularity along certain values of
␣
and

.
The value of the determinant was calculated for several pos-
sible positions of the moving platform, namely, combinations of
the orientation angles
␣
and

included in the ranges of value
−75 degⱕ
␣
ⱕ75 deg and −75 degⱕ

ⱕ75 deg. The zero-
value of the determinant occurs for certain values of
␣
and

in
Fig. 5; this means that the nonredundant parallel mechanism falls
in a singular configuration when assuming a position character-
ized by values of the orientation angles. In Fig. 5, the contours
labeled with the value 0 are the curves of the null determinant.
It must be specified that a zero rotation of the platform about
the w0-axis was assumed, in order to show the presence of the
singularities within the workspace in the nonredundant case. The
complete singularity loci analysis should be made also consider-
ing a rotation of the platform about its vertical axis. The reason for
singularity is that when the two angles
␣
and
␥
are equal to zero
the intersection of the three planes formed by the couples of vec-
tors siand biis a line, whatever the value of the third angle

.
This indicates that the vectors obtained from the cross products
bi⫻siare coplanar, therefore linearly dependent. As a result the
Jacobian matrix loses rank and the mechanism can undergo an
infinitesimal rotation about the axis defined by the intersection of
the three planes even though all the actuators are locked. Further-
more, singularities occur when one of the cross product gives a
null vector, indicating that the two vectors biand siare collinear.
Direct kinematic singularities implicate that a mechanism gains
one or more degrees of freedom; in other words, the mechanism is
unable to apply forces or torques.
A method to overcome singularity is to introduce redundancy,
which eliminates singularities, improves the mechanism stiffness,
and allows a fault tolerance operation. This leads to the redun-
dantly actuated parallel mechanism introduced in Fig. 1. For this
mechanism, with a universal joint between the platform and the
central strut, the singularities can be characterized by the rank
deficiency of the m⫻nreduced Jacobian matrix J
⬘in Eq. 共10兲as
rank共J
⬘兲⬍n共13兲
where nis the number of independent coordinates of the moving
platform and mis the number of actuated joints. The same concept
is expressed by the relation
det共U兲=0, ∀U兵n⫻nsubmatrices of J
⬘其共14兲
to guarantee a nonsingular configuration of the parallel mecha-
nism. Similar to the nonredundant case, the rank of J
⬘is numeri-
cally calculated for several combinations of the orientation angles
␣
and

of the platform and it is equal to the degree of freedom of
the platform mechanism.
Therefore we can state that the redundantly actuated parallel
mechanism does not possess any singularity within its workspace
since the relations in Eqs. 共13兲and 共14兲do not exist. This provides
the advantage of eliminating the singularity.
4 Local Dexterity Improvement
Based on the Jacobian matrix in the kinematic input-output re-
lation in Eq. 共8兲, the condition number can be given as
=
max共J
兲
min共J
兲共15兲
where
max共J
兲and
min共J
兲are the maximum and minimum
singular values of the Jacobian matrix J
, respectively.
The inverse of the condition number expressed in Eq. 共15兲is
used to describe the local dexterity of the parallel mechanism, in
terms of both the nonredundantly actuated parallel mechanism in
Eq. 共9兲and the redundantly actuated parallel mechanism in Eq.
共10兲. The results of the analysis are shown in Fig. 6, where the
dexterity index, based on the inversed condition number of the
mechanism Jacobian matrix, is shown for all the possible configu-
rations that the mechanism in both cases can assume within its
workspace.
Since the inversed condition number was used for the analysis,
the index of dexterity can assume values in the range 0–1, where
0 represents a singularity and 1 an isotropy. Therefore, the higher
the value of the index, the greater the dexterity of the mechanism
is.
−75−50−25025 50 75
−75
−50
−25
0
25
50
75
−0.01
−0.005
0
0.005
0.01
β[deg]
α[deg]
δ
Fig. 3 Determinant of the Jacobian matrix J
for different
combinations of orientation angles
␣
and

−75
−50
−25
0
25
50
75
−7
5
−50
−25
0
25
50
75
−0.01
−0.005
0
0.005
0.01
α[deg]
β[deg]
δ
Fig. 4 Lateral view of Fig. 3
0
0
0
0
0
0
0
0
0
0
β
[de
g
]
α[deg]
−75 −50 −25 0 25 50 7
5
−75
−50
−25
0
25
50
75
Fig. 5 Contours of the surface of the determinant of the Jaco-
bian matrix J
on the
␣
-

plane
Journal of Mechanical Design DECEMBER 2008, Vol. 130 / 124501-3
Downloaded 16 Oct 2008 to 137.73.11.235. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Note, in the nonredundant case the orientation angle
␥
is as-
sumed to be constant and equal to zero, as in the previous analy-
sis, and the index is computed for different combinations of the
orientation angles
␣
and

. It is possible to note from the graph in
Fig. 6共a兲that in the nonreundant case the dexterity index is close
to zero over the workspace and reaches zero along the curves of
singularity. In fact, the condition number, as well as its inverse,
expresses “how far” the mechanism is from a singular configura-
tion. As it was shown in the determinant analysis, the parallel
mechanism without actuation redundancy presents several singu-
larities within its workspace.
The same consideration can be made by observing the dexterity
graph relative to the nonredundant case. For instance, when the
orientation angle
␣
is equal to zero, assuming that the angle
␥
is
constantly null, the mechanism presents an inversed condition
number equal to zero, namely a singularity. Therefore, comparing
Figs. 5 and 6共a兲it is possible to see that both analyses give sin-
gularity conditions along the same curves called the null determi-
nant.
In the case of the redundantly actuated parallel mechanism,
around the default position of the moving platform with both the
orientation angles equal to zero, the dexterity index is close to 1,
namely the value of isotropy 关16兴in Fig. 6共b兲, and the value
decreases when the configuration of the platform assumes values
close to the boundary of the workspace.
However, with the introduction of actuation redundancy, the
local dexterity of the parallel mechanism turns out to be enhanced
over the workspace since the kinematics does not present singu-
larities; therefore the performance of the parallel mechanism is
improved. In fact, the dexterity index assumes values within the
range 0.1–1 while in the nonredundant case the range is 0–0.1.
Hence, it can be stated that the dexterity of the parallel mechanism
is greatly improved by the introduction of actuation redundancy.
Changing the platform parameters by assigning rb=1/2ra, the lo-
cal dexterity surface was sharpened, as in Fig. 7, which reduces
the dexterity measure in a wider range.
Additionally, the dexterity of the redundantly actuated mecha-
nism in Fig. 6共b兲can be compared with the three different archi-
tectures of the mechanism characterized by two degrees of free-
dom and two limbs. To do the analysis, three other mechanism
structures are considered and their dexterity presented. Those
structures are characterized by 2DOF, due to the universal joint
between the platform and the central strut and, two limbs that are
sufficient to fully control the two orientation angles of the mecha-
nism,
␣
and

. Considering the mechanism in Fig. 2, two new
mechanisms can be defined taking into account either only Limbs
1 and 2 or only Limbs 2 and 3. Furthermore, another mechanism
can be obtained with Limbs 1 and 2 positioned 90 deg apart from
each other. The analysis is presented in Fig. 8.
Figures 8共a兲and 8共b兲show the dexterity of the mechanisms
with Limbs 1 and 2 and Limbs 2 and 3 arranged at 120 deg. This
is then compared with the dexterity of the platform with two limbs
arranged at 90 deg, as in Fig. 8共c兲.
The structure layout is equivalent to that presented by Chablat
and Wenger 关17兴, however the joints and the actuation scheme are
different. In the third case, the universal joint A2is considered to
lie on the y-axis while the spherical joint B2lies on the v0axis
when the mechanism is in the default position with the orientation
angles of the platform equal to zero.
In all three figures, it is possible to notice that the dexterity
assumes a value of zero for certain platform orientation, meaning
that the mechanisms present singularities within their workspace.
Moreover, the surfaces representing the dexterity of the mecha-
nisms turn out to be spiky in the first and second cases and not
homogeneous around the default position of the platform in the
third case. This is in contrast to Fig. 6共b兲where the dexterity
surface is homogeneous and the dexterity volume under the sur-
face is larger.
Therefore, the introduction of the third limb becomes necessary
to obtain a homogeneous and a sufficiently high dexterity of the
parallel mechanism in the working range of the default position of
the moving platform and to definitively eliminate singularities
throughout the mechanism workspace. The further benefits are
elimination of singularity, smaller 共less powerful兲actuators to
achieve the same torque output, and higher stiffness.
On top of that, the analysis leads to designing a mechanism
with the highest global dexterity and the most homogeneous local
dexterity, which will be discussed elsewhere.
5 Conclusions
This paper investigated a redundantly actuated parallel mecha-
nism for sprained ankle rehabilitation and compared the analysis
of the redundantly actuated parallel mechanism with that of a
nonredundant case resulting from a change of the joint between
the platform and the central strut and other nonredundant cases
where only two limbs of the mechanism are considered.
The geometry of the mechanism was given and the Jacobian
matrices for both the redundant and nonredundant parallel mecha-
−75 −50 −25 025 50 7
5
−75
−50
−25
0
25
50
75
0
0.5
1
β[deg]
α[deg]
1
/
σ
(a)−75 −50 −25 025 50 7
5
−75
−50
−25
0
25
50
75
0
0.5
1
β[deg]
α[deg]
1/σ
(b)
Fig. 6 Dexterity analysis of parallel mechanisms with three limbs: „a…local dexterity
of a nonredundant parallel mechanism and „b…local dexterity of a redundant parallel
mechanism
−75 −50 −25 025 50 7
5
−75
−50
−25
0
25
50
75
0
0.5
1
β[deg]
α[deg]
1
/
σ
Fig. 7 Local dexterity of a redundant parallel mechanism with
rb=1/2ra
124501-4 / Vol. 130, DECEMBER 2008 Transactions of the ASME
Downloaded 16 Oct 2008 to 137.73.11.235. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
nisms were obtained. Both singularity and dexterity were ana-
lyzed for the different arrangements. It demonstrated that a redun-
dantly actuated parallel mechanism eliminates direct kinematics
singularities within the workspace and increases the dexterity of
the mechanism.
The study proved the efficacy of the introduction of actuation
redundancy in parallel mechanisms and justified the choice of the
mechanism for ankle rehabilitation with dexterity and without sin-
gularity. The actuation redundancy greatly improves the mecha-
nism performance throughout the workspace.
The present work is valuable for further dynamics modeling
and control design.
Acknowledgment
The authors thank James Store for constructing the 3SPU par-
allel mechanism and Yusuke Kwak for developing the control
system of the mechanism during their BEng and MEng degree
programs.
References
关1兴Girone, M. J., Burdea, G. C., and Bouzit, M., 1999, “The Rutgers Ankle
Orthopedic Rehabilitation Interface,” Proceedings of the ASME International
Mechanical Engineering Congress and Exposition on Dynamic Systems and
Control Division, Nashville TN, Nov., Vol. 67, p. 305312.
关2兴Dai, J. S., and Massicks, C. P., 1999, “An Equilateral Ankle Rehabilitation
Device Based on Parallel Mechanisms,” Proceedings of the Off-Line Simula-
tion Workshop for Robotic End-Effectors and Manipulators, Second World
Manufacturing Congress, Durham, UK.
关3兴Yoon, J., and Ryu, J., 2005, “A Novel Reconfigurable Ankle/Foot Rehabilita-
tion Robot,” Proceedings of the 2005 IEEE International Conference on Ro-
botics and Automation, Apr. 18–22, pp. 2290–2295.
关4兴Liu, G., Gao, J., Yue, H., Zhang, X., and Lu, G., 2006, “Design and Kinemat-
ics Analysis of Parallel Robots for Ankle Rehabilitation,” Proceedings of the
IEEE/RSJ International Conference on Intelligent Robots and Systems,
Beijing, Oct. 9–15, pp. 253–258.
关5兴Dai, J. S., Zhao, T., and Nester, C., 2004, “Sprained Ankle Physiotherapy
Based Mechanism Synthesis and Stiffness Analysis of a Robotic Rehabilitation
Device,” Auton. Rob., 16共2兲, pp. 207–218.
关6兴Dai, J. S., Huang, Z., and Lipkin, H., 2006, “Mobility of Overconstrained
Parallel Mechanisms,” ASME J. Mech. Des., 128, pp 220–229.
关7兴Hunt, K. H., 1982, “Structural Kinematics of In-Parallel Actuated Robot-
Arms,” Proceedings of the ASME Design and Production Engineering Tech-
nical Conference, Washington, DC, Sept. 12–15, p. 9.
关8兴Phillips, J., 1990, Freedom in Machinery, Cambridge University Press, Cam-
bridge, England.
关9兴Kurtz, R., and Hayward, V., 1992, “Multiple-Goal Kinematic Optimization of
a Parallel Spherical Mechanism With Actuator Redundancy,” IEEE Trans.
Rob. Autom., 8共5兲, pp. 644–651.
关10兴Sukham, L., and Sungbok, K., 1994, “Kinematic Feature Analysis of Parallel
Manipulator Systems,” Proceedings of the IEEE/RSJ/GI International Confer-
ence on Intelligent Robots and Systems, Sept. 12–16, Vol. 2, pp. 1421–1428.
关11兴Leguay-Durand, S., and Reboulet, C., 1997, “Optimal Design of a Redundant
Spherical Parallel Manipulator,” Robotica, 15共4兲, pp. 399–405.
关12兴Lee, S. H., Yi, B.-J., and Kwak, Y. K., 1997, “Optimal Kinematic Design of an
Anthropomorphic Robot Module With Redundant Actuators,” Mechatronics,
7共5兲, pp. 443–464.
关13兴Cheng, H., Yiu, Y.-K., and Li, Z., 2003, “Dynamics and Control of Redun-
dantly Actuated Parallel Manipulators,” IEEE/ASME Trans. Mechatron., 8共4兲,
pp. 483–491.
关14兴Wang, J., and Gosselin, C. M., 2004, “Kinematic Analysis and Design of
Kinematically Redundant Parallel Mechanisms,” ASME J. Mech. Des., 126,
pp 109–118.
关15兴Dai, J. S., and Kerr, D. R., 2000, “Six-Component Contact Force Measurement
Device Based on the Stewart Platform,” Proc. Inst. Mech. Eng., Part C: J.
Mech. Eng. Sci., 214共5兲, pp. 687–697.
关16兴Merlet, J. P., 2006, “Jacobian, Manipulability, Condition Number, and Accu-
racy of Parallel Robot,” ASME J. Mech. Des., 128共1兲, pp. 199–206.
关17兴Chablat, D., and Wenger, P., 2005, “Design of a Spherical Wrist With Parallel
Architecture: Application to Vertebrae of an Eel Robot,” Proceedings of the
IEEE International Conference on Robbotics and Automation, Apr. 18–22, pp.
3336–3341.
−75 −50 −25 025 50 75
−75
−50
−25
0
25
50
75
0
0.5
1
β[deg]
α[deg]
1/σ
(a)−75 −50 −25 025 50 7
5
−75
−50
−25
0
25
50
75
0
0.5
1
β[deg]
α[deg]
1/σ
(b)
−75 −50 −25 025 50 75
−75
−50
−25
0
25
50
75
0
0.5
1
β[deg]
α[deg]
1/σ
(c)
Fig. 8 Dexterity of the parallel mechanisms with two limbs and a central strut: „a…
local dexterity of the 2DOF mechanism with Limbs 1 and 2 arranged at 120 deg, „b…
local dexterity of the 2 DOF mechanism with Limbs 2 and 3 arranged at 120 deg, and
„c…local dexterity of the 2DOF parallel mechanism with two limbs arranged at 90 deg
Journal of Mechanical Design DECEMBER 2008, Vol. 130 / 124501-5
Downloaded 16 Oct 2008 to 137.73.11.235. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
