 Open Access
 Total Downloads : 398
 Authors : Hailemariam Nigus, Hassen Nigatu, Muralidhar Reddy Gurala
 Paper ID : IJERTV3IS051893
 Volume & Issue : Volume 03, Issue 05 (May 2014)
 Published (First Online): 07062014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Dynamic and Stiffness Modeling of New 3DOF PKM for High Speed Machining Application
Hailemariam Nigus
Automobile Engineering Department Federal TVET Institute
Addis Ababa,Ethiopia
Hassen Nigatu
Mechanical engineering Department Adama Science and Technology University
Adama, Ethiopia
Muralidhar Reddy Gurala Mechanical Enigineering Department
Adama Science and Technology University
Adama,Ethiopia
Abstractmany applications in the field of production automation, such as machining, assembly and material loading require machines that are capable of high speed, accelerationand rigidity.This paper mainly addresses the issue of dynamic and stiffness formulation of a three prismaticrevolutespherical PKM (3PRS PKM) using screw theoryand virtual work approach. First the dynamic formulation is performed and later the stiffness equation is derived using virtual work principle. In order to build up the stiffness and dynamics model, kinematics, Jacobian, Hessian and Finite Element Analysis are also performed as the basis.The study of robot dynamics is necessary for its mechanical design and synthesis, providing the information on the force that must be resisted by joints, links and actuators while the study of stiffness is highly demanded to predict the rigidity performance of the PKM. So that, the inverse dynamics is developed with capable of calculating the force along the driving link direction. Also, Analytical stiffness model, a function of Jacobian matrix and components stiffness matrix, is obtained first using the principle of virtual work. Stiffness model is also a six by six dimensional matrix and can provide the information of actuation and constraint stiffness simultaneously.
Keywords: PKMs, Hessian, 3PRS, Dynamics, stiffness analysis, FEA

INTRODUCTION
A parallel platform manipulator (PPM) is a device whose end effector is attached to the ground via multiple serial chains that provides closed kinematic loops in the system for better load handling capacity and stiffness. It is these qualities that make it applicable in a wide range of applications ranging from flight simulators to high speed milling machines. However, it is these closed kinematic loops that increase the complexity of their analysis to a great extent(Y.G Li).
An important advantage of parallel platform manipulator is that its superior structural rigidity renders it as the better choice over serial chain manipulator (SCM) for moving heavy loads and high precision machining tasks. The advantages include very high accuracy, better stiffness ratio, more payload capacityand better inertial distribution among others(2441).
Y.G Li, T.Huang and H.T .Liu[1] a general approach for formulating dynamics of lower mobility parallel manipulators. Moreover, Joshi, S., Liu, H., Chetwynd, D.G.,Li,Z., [2]formulated the generalized jacobian analysis of lower mobility manipulators, Dressler, I., Robertson. And Johansson, R.[3] has designed the accuracy of kinematic and dynamic models of a GantryTau parallel kinematic robot, Li Y G, Song Y M, Feng Z Y, et al.[8] has derived the Inverse dynamics of 3RPS parallel mechanism by NewtonEuler formulation in Chinese and Liu, H., Huang, T.,Chetwynd, D.G.[1120], An Approach for Acceleration Analysis of Lower Mobility Parallel Manipulators.
Since the industrial revolution, there has been an ever increasing to improve product quality and reduce manufacturing cost parallel kinematic machines are applied on various area such as, For high speed and high precision machining center, for machining purpose, air plane simulators and For pick and place, assembly and carrying load purpose in higher industries like aircraft, electronics and automotive factories and robotic assistance surgery. For the successful implementation of parallel manipulators in different engineering and technological areas studying the motion and rigidity has no any equivalent substitution. As it is already discussed before the dynamics and stiffness analysis will be studied in this paper as follows.

CAD MODEL KINEMATIC DESCRIPTION
The architecture of 3PRS PKMs is showing on fig.1 which is composed of moving platform, a fixed base the three supporting limbs with identical configurations. Each limb connects the fixed base to the moving platform by prismatic, revolute and spherical joint respectively and the prismatic joint is the active joint which is actuated by the linear actuator servo motor.
The considered machine is 3DOF PKM, which can be showed mathematically by using mobility criterion.
j
M 6n j 1 fi , where M is the DOF, n is
i1
number of links in the system, j is number of joint and fi
is the number of DOF of the ith joints. For this
manipulator n=8,j=9,fi=3 for spherical joint and fi=1 for
Where s
i, j
is a unit vector along the joint Jth joint of the
prismatic and revolute joints. Therefore the above equation will give as the system has 3 DOF.
M 68 9 1 311 3 3DOF (1)
The vectors and reference frames are described also in the figure for the sake of analysis, as shown in the figure 2 a fixed Cartesian reference coordinate frame P (x, y, z) is attached at the center point O of the fixed triangle base platform C1C2C3. And the moving coordinate system O(u, v, w) is attached on the moving platform a point O Which is a center of Q1Q2Q3. For simplicity without losing generality, let xaxis be aligned toward OC3 and the uaxis pointing along the direction
3
of O'Q .
ith limb. These five screws form five system for which a one system of reciprocal screw exists. This reciprocal screw lies on the intersection of the two planes. The first plane is perpendicular for the prismatic joint axis and the second plane is containing both the revolute and spherical joint. This reciprocal screw denoted as sr1,i is
zero pitch screw passing through the center of spherical joint and parallel to s2,i .
sr 2,i
From the above expression R1,R2,R3 are the vector of fixed base platform from the center point O to Ci and r1,r2, r3are the vectors of the moving platform from the center point O to Qi.
l , s
s3,i
o '
s1,i
i 3,i
p
s2,i
y
O
figure 1: 3D CAD model of the machine

VELOCITY ANALYSIS
In this section the velocity analysis of 3PRS have done by applying the principle of screw theory. There are five joint screws associated with each limb. The first joint is the only actuated joint and the remaining joints are
x
figure 2: 2D schematic diagram of the mechanism
r s
i
passive. The instantaneous twist of the moving platform
can be expressed as a linear combination of five screws.
s
2,i
r1,i s
(10)
sp h s s s s s (8)
2,i
Where
1,i
1,i
2,i
2,i
3,i
3,i
4,i
4,i
5,i
5,i
By taking the inner product of both sides of the
instantaneous with the constraint wrench we found the
s
s1
, s
ri li s2
, s
ri s3
constraint Jacobian (JC ) .
1,i
0 2,i s
3,i
s
s T
r s T
s
2
3
2,1
1 2,1
T
s
ri s4
, s
ri s5
(9)
J C
s2,2
r s
T
2
3
p/>
2,2
(11)
s
4,i
4
5,i
5
T
s
2,3
r s
T
2,3
This matrix represents the constraint imposed by the revolute joint. An additional basis screw which is reciprocal to the passive joint of the ith limb can be identified as zero pitch
The Hessian matrix can be found from the derivative of screws in the Lie bracket form. This yields
screw passing through the center of spherical joint. This
0 $ta,1,i $ta,2,i $ta,1,i $ta,3,i $ta,1,i $ta,n ,i
reciprocal screw represents wrench of actuation and it is
i
normal to the previous system. This can be expressed by
0 0 $ta,2,i $ta,3,i $ta,2,i $ta,n ,i
i
(20)
ri s3.i
Ha,i
s (12)
r 2,i s
0 0
0 $ta,n 1,i $ta,n ,i
3.i
i i
Take the orthogonal product of this reciprocal wrench for both
0 0
0 0
sides of the twist screw. Then we
0 $tc,1,i $tc,2,i $tc,1,i $tc,3,i $tc,1,i $tc,6n ,i
found J .
i
x 0 0
$
$
$ $
sp s
s
s
h
(13)
tc,2,i tc,3,i tc,2,i tc,6ni ,i
(21)
r 2,i
1,i
r 2,i i
Hc,i
0 0
0 $tc,5n ,i $tc,6n ,i
s T
r s
1
T
i i
3,1
T
3,1 T
0 0
0 0
J x s3,1
r2 s3,1
(14)
s
3
T
3,1
r s
T
3,1
$
$ $ $
Equation 13 can be rewrite three times, once for each
ta,1,i tc,1,i
ta,1,i tc,6ni ,i
limb and it yields.
Hac,i
(22)
J x sp Jq q Where the inverse Jacobian is
J q is:
$ta,n ,i $tc,1,i $ta,n ,i $tc,6n ,i
s T s 0 0
i
i i
J q
3,1
0
1,1
s3,2
T
s
0
1,2
T
(15)
A Ji qi qi
Hi qi This equation puts the screw derivative
0 0 s T s
( * * ) and its coefficient side by side. i.e. qi is
3,3
1,3
coefficient and
H is screw derivative in Lie bracket form.
The actuation Jacobian, is responsible to relating the i
Cartesian velocity with joint rate J
, is a function of
And the overall Hessian matrix can be written in the form of
q
J and J .
H Ha,i
Hac,i
H
(23)
x a
J J a
0
Where,
J J x
(16)
i
c,i
J
J
a
c q
The matrix in eq. (23) is generalized Hessian with
Equation 16 is a generalized Jacobian that relates the velocity of joint rate to the velocity of the moving platform.:

ACCELERATION ANALYSIS
For the acceleration analysis of this 3PRS parallel manipulator we used a new approach [11] named Hessian matrix to achieve an explicit and compact form equation.
Ja,i [sta,1,i sta,2,i , sta,3,i sta,4,i sta,5,i ] (17)
i
b ls n
6 6 6 matrix for each limb contain both the actuation and constraint joints and eq. (20, 21, 22) are the element of general Hessian matrix.
This generalized matrix will help us to solve the dynamics of the generalized lower DOF parallel manipulators in efficient, compact and robust way and it is believed solving the Hessian in this form
will simplified the dynamics work in unbelievable status.

DYNAMICS FORMULATION
This work comes with applying screw mathematics in order to utilize the advantage of Robust behavior of virtual work for
Jc,i
stc1,i
3,i
n1,i
1,i for i=1,2,3 (18)
solving inverse dynamics of symmetrically configured PKM. Gravity of the moving platform and the pth (1 p ni 1 )
Where
J c,i is constraint Jacobian for a limb.
link of the ith ( i 1, 2,,l ) limb can be expressed as follow
s s
$ mC g , $
mp,i g
(24)
Here s
3,i 5.i
and s
Rs
for i=1,2,3
wg 0 wg , p,i 0
4,i s s
5,i
2,i
3,i
5.i
where g is the vector of acceleration of the gravity, mC
and
Ji
Ja,i
Jc,i
, Jacobian for each limb. (19)
mp,i is the mass of the moving platform and the pth link of the ith limb respectively.
According to Eulers Theorem, the inertial forces of the moving platform can be expressed as
platform, twist of the pth link of the ith limb can be rewritten in the variational form as follows
$Ct T0$t , q J $t , $Cta, p,i JC, p,i $t
(27)
$wI mC $Ct $Ct mC $Ct (25) Where
Substituting Eqs(24), (25), (26) and (27) into Eq(27), the dynamics model can be constructed
mC
mC E
0 , $
= A J
$ vC
D(r)$t H(r, $t )$t G(r) E(r)
(28)
0
I
C
Ct C C t

STIFFNESS MODELING
0 vC 0
JC JT0 , J
, $Ct 0
Under the assumption that the platform and the machine frame
0 0
where IC RIC0 RT and IC 0 denotes the inertial matrix of the moving platform about the mass centre described in the reference coordinate system and local coordinate system respectively, R is the rotation matrix of the local coordinate system with respect to the reference coordinate system.
Similarly, the inertial forces of the pth ( 1 p ni 1 ) link of the ith ( i 1, 2,,l ) limb can be expressed as follows
are rigid, when the platform is subjected to the external wrench [F T , M T ] on the reference point p, where F and M are the external force and torque applied to the platform, the deformation of the limbs will causes the platform
to experience a twist x [rT , T ] in terms of the translational and rotational deformations along/about the axes of frame. Then, applying the virtual work principle to the
$ m $ $ m $
(26)
platform gives
wI , p,i p,i Cta, p,i Cta, p,i p,i Cta, p,i
Where,
xT T f
(29)
Where and f represents the set of deflections and
m mp,i E
0 , $
= A J
$ = vC , p,i
reaction force magnitude
p,i
0
I p,i
Cta, p,i C , p,i C , p ,i t
p,i
Jx
This equation
xT T f can be re write
C, p,i
, p,i p,i p,i a, p,i
, p,i
J J T J J L , J
p,i 0
xT T f 0 Substitute
vC , p,i
0 0
0
from the above equation
( T Jf T )x 0
$Cta, p,i T T
0
p,i
Jf 0
p,i
where I p,i Rp,i I p,i,0 RT
and I p,i,0 denotes the inertial
Taking the transpose yields:
J T f
matrix of the link about the mass centre described in the
Where
f [ f T , f T ] is the internal wrench vector of limbs,
reference coordinate system and local coordinate system a c
respectively,
Rp,i is the rotation matrix of the local coordinate
where
f a and
fc are the generalized force of the PRS limbs
system with respect to the reference coordinate system.
nd the PR limb related to the twist [ T , T ],
Let $ f T , t T T be the load wrench applied on the a c
we
reference point of the moving platform, where f and t is the
vector of force and moment respectively. By using of the principle of virtual work, we can obtain
fa is a force which is parallel to the screw axis while the
fc is parallel to the revolute axis.
Therefore the virtual work principle can be written
t we
w qT $T$
0 (26)
X T T f
rT F T
T f

T f
a
c
a
c
where
l ni 1
fai
[ f
a1 ,
fa 2 ,
fa3
]T ,
w $T $ $ $T $ $
[q , q , q ]T
Ct wI wg Cta, p,i wI , p,i wg , p,i i1 p1
a 1 1 3
T T T
f [ f , f , f ]T ,
a c
ci c1 c2 c3
where a is the fdimensional vector composed of the driving
force coefficients, and c is the vector composed of the
[c , c , c ]T
c 1 2 3
(30)
constraint force coefficients.
The equation of velocity of joint, twist of the moving
fa ka a , fc kc c and
f kac
ka 0
klsa Is the lead screw assembly the combination of serially connected springs such as
Where
kac 0 k
1 1 1 1
c
k k k k
Here ka ka
and kc
are known as the component stiffness
lsa
ls n sb
(34)
matrix of actuation and constraints respectively the formulation of their element
Where kls , kn and ksb are the stiffness coefficients of lead
ac
kx, k J T k J
(31)
screw.nut and support bearings respectively. kls
is the lead
And the compliance model can be evaluated as
c k 1
screw which is the linear function of the limb length and can be defined as
kls
AE(L1i L2i )
L L
(35 )
1i 2i

Formulation of kac
As shown in the Figure 3 above the limb model has To
Where
A, E stands for cross sectional area of the lead screw and yangs modular respectively
formulate kac I group all the parts of an PRS limp in to four
components: (1) the spherical joint (2) the limp body which is
L1i
and L2i are the distance b/n the nut and the supporting bearing located at both ends.
the fixed lengths link (3) R joint assembly (4) the lead screw
assembly. as well as analytically convenient, they are sub systems that must realistically be subjected independently to design improvements.
To find the constrained coefficient of stiffness matrix we can
find as the same fashion of finding the actuation coefficient matrix. Similarly kc diag[kci ] (i 1,2,3) where kci is
ci
ka is given in a diagonal matrix i.e. ka diag[kai ] where the bending stiffness coefficient at the platform along the s2i
(i 1,2,3)
with
kai being the axial stiffness coefficient at
axis of the ith limb Then k can be evaluated by taking
Bi along the
s1i
axis of the ith limb referring to the fig 4.
reciprocal sum of the bending stiffness coefficient of the fixed length limb, S joint and R joint assembly respectively.
kai
Can be modeled by four serially connected springs each
1 1
1 1
(36)
representing the stiffness of one of the four components such
kci
kcl
kcs
kcr
1
kai
1
klsa
1 1 1
kl ks kr
Again the kcs can be evaluated by the configuration and should be evaluated as in the local frame since the spherical
(32)
Where klsa , kl , ks and kr are the axial stiff nesses
parallel to the constraint is to u the coefficient stuffiness is calculated as follows
coefficients of the lead screw assembly ,the fixed length limb
,S joint and R joint assembly respectively.
1
kcs
1
ku1
1
ku 2
1
ku3
(37)
Note that
kl , and kr
are constant and can be evaluated using
The value of the stiffness will be substitute from table 4.1and
finite element analysis. (FEA) by the ANSYS workbench which is very convenient to analysis a solid model like PKM. The ks varies with the configuration and should be evaluated as in the local frame since the spherical actuation is parallel to w, the coefficient stuffiness is calculated as follows
easy to compute the value of kcs .the kcl and kcl can be computed easy by FEA. Using the software ANSYS workbench by applying a 1KN force on the spherical joint which is parallel to the s2i
1 1
ks kw1
1
kw2
1
kw3

Formulation of overall stiffness matrix on tool tip
Formulation of overall stiffness matrix applied on the center of
(33) endeffectors is calculated on Eq (36) which is
The values of Eq (12) can be substituted from Table 1 below.
ac
Table 2 The stiffness coefficient of the S joint
kx,
k J T k J
ku1
kv1
kw1
ku 2
kv2
kw2
ku3
kv3
kw3
31.4
33.3
436
4000
1810
2174
2119
270
588
T
To find the overall coefficient of stuffiness matrix on tool tip it needs to transform the Eq(31) in to tool tip by the transformation matrix
E3 C s1i x
C 0 E
(38)
3×3 3
If let be imposed at the tool tip C and x be the corresponding small deflection twist the overall stiffness matrix about point C can easily be developed by replacing J
in Eq.(31) with JT that is
when force applied along x, y and z which explain in detail. In order to evaluate the rigidity of a system we define the rigidity along/about three orthogonal axis of the frame C uvwby the diagonal corresponding element of C.
C K x
1/ C(1.1)
K TC
T J T kJT
K y 1/ C(2.,2) K z
1/ C(3,3)
C
Where C is the distance from O to C , s1i denotes the
Krw
rC
2 / C(6,6)
screw matrix of
s1i
,and E3 denotes a unit matrix of order3.
c K 1
x c
5.2 Comparison with FEA Results
According to the above analysis, the detailed design was carried out and the stiffness of the virtual prototype was
Where
(Fx Fy Fz
M )T
K x
(N / m
K y
(N / m)
K z
(N / m
Krw
(x106 Nm / ra
Ana lytic
25.4
25.4
447.2
5.3
FE
A
25.87
25.87
437.7
5.4
rC
T
x (uvwrC w )
rC is the maximum radius of the cutting tool specified by the
k11
k12
k13
spindle head.
C K 1
k16 / rc
evaluated by ANSYS at four typical positions as shown from
k k k / r
K
22 23 26 C
(39)
Figure 3, Figure 4, Figure 5 and Figure 6 with 1KN force is
k33
k36 / rC
applied at the tool tip along x, y and z and moment about z
sym k
/ r
axis respectively. I can get the deformation easily from the
66 C
FEA and the stiffness can get by
1000
. It can been seen from
In order to evaluate the rigidity of a system we define the rigidity along/about three orthogonal axis of the frme C uvw by the diagonal corresponding element of C.
z
K x 1/ C(1.1) , K y 1/ C(2.,2) K 1/ C(3,3)
x
Table 3 that the estimated results of the mathematical models developed have a good match with those obtained by the FEA in terms of magnitude and distribution as well.
Table 4 Results obtained by the semianalytical method and by FEA
Krw
rC
K x
2 / C(6,6)
1/ C(1.1)
The estimated linear stiffness along three orthogonal axis and the torsion stiffness about the waxis of the C uvw frame, it can be seen that the stiffness distribution are trisymmetrical
z
K y 1/ C(2.,2) K
1/ C(3,3)
in nature and the K x and
K y are similar in magnitude.
Krw
rC
2 / C(6,6)


STIFFNESS ANALYSIS
The stiffness of the 3PRS PKM is evaluated in the decomposing the machine in to limbs and apply a force on the spherical joint to find the actuated and constraint coefficient of stuffiness. With the aid of finite element analysis and numerical evaluated both actuated and constraint stiffness of the limb assembly is evaluated. The overall stiffness of the manipulator on the center of the plate form will be calculated as in Eq (31) indicated. Since in real sense the force/moment is applied on the too tip of the machine, it needs transform the stiffness matrix gained in Eq (31) to the tool tip by the transform matrix Eq (37) it gives a stiffness coefficient matrix on tool tip as shown in the Eq(39).then diagonal values in the stiffness matrix indicates the overall stiffness of the machine
Figure 3 Deformation with 1KN force imposed at the spindle along xaxis
model of the lower mobility parallel manipulators can be formulated by two steps

formulation of the generalized Jacobian by
Figure 4 Deformation with 1KN force imposed at the spindle along zaxis
Figure 5 Deformation with 1KN force imposed at the spindle moment about z
1050
1000
driving force in N
950
900
850
linear linear driving force of actuator
fz1 fz2 fz3
25
20
15
constrain moment
10
5
0
5
10
15
tao1
tao2 tao3
constraint force of actuating limb
800
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
time (sec)
20
25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
time (sec)
Figure 6. Linear actuating force


CONCLUSION
An approach for the dynamics and stiffness modeling 3PRS parallel mchanism is presented in this paper. With this approach,the analytical expressions of the mass centre velocity/acceleration of the link can be easily derived by using the operation of Lie Algebra and reciprocal screw. With the aid of the generalized Jaccobian and Hessian matrix, the new dynamics model can be applied to solve the generalized forces of actuation and constraint at the same time. The stiffness
Figure 7. Constraint force of active limb
simultaneously taking into account the deflections of the platform in both the free and fixed directions, and
(2) evaluation of the component stiffness matrices using the analytical approach, finite element analysis and/or their combinations, depending upon geometry complicity
of the moving components. In concluding the robust dynamic equation is formulated and this is tested by developing simulation material and verify the result with numerical simulation shown in figure 7 and 8. Regarding the rigidity
analysis with the tool of ANSYS workbench we analysed the machine stiffness and compared the FEA result with the semi analytical analysis and the estimated stiffness results have a good match with those obtained by the FEA. The result is in range of acceptable error as we can see it from fig.4,5 and 6 of ANSYS output compared with the FEA result in table 1.
REFERENCES
[1]. Zhang, F.; Zhang, D.; Yang, J., and Li, B., 2005, "Kinematics and Singularity Analysis of a 3DOF Parallel Kinematic Machine.", IEEE. 29 July1 Aug. 2005. [2]. Joshi, S., Liu, H., Chetwynd, D.G.,Li,Z., generalized jacobian analysis of lower mobility analysis of lower mobility manipulators, IEEE transaction on robotics and automation, submitted . [3]. Dressler, I., Robertson. and Johansson, R.,2007, Accuracy of kinematic and dynamic models of a GantryTau parallel kinematic robot, Proceedings of IEEE International Conference on Robotics and Automation, pp.883 888 [4]. Tsai, L.W., 2000, Solving the inverse dynamics of a Stewart Gough manipulator by the principle of virtual work, ASME Journal of Mechanical Design, 122(3): pp. 39.[5] Li Y.M.XuQ.S., 2005,
[5]. Kinematic Analysis and Dynamic Control of a 3PUU Parallel Manipulator for Cardiopulmonary Resuscitation International Conference on Advanced Robotics, pp.344351[6]. Wisama Khalil and Sylvain Guegan, 2004, Inverse and Direct Dynamic Modeling of GoughStewart Robots, IEEE Transactions on Robotics, 20(4):pp.754762.
[7]. SUN Tao1, SONG YiMin1, LI YongGang2 & LIU LinShan1 Dimensional synthesis of a 3DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix [8]. Li Y G, Song Y M, Feng Z Y, et al. Inverse dynamics of 3PRS parallel mechanism by NewtonEuler formulation (in Chinese). Acta Aeron Astron Sin, 2007, 28: [9]. Carretero J A, Nahon M, Gosselin C M, et al. Kinematic analysis of a threeDOF parallel mechanism for telescope application. In: Proceedings of the 2007 ASME Design Automation Conference, Sacramento, California, 1997 [10]. Lee, K. M., Shah, D. K., 1988, Dynamic analysis of a three degreesoffreedom in parallel actuated manipulator, IEEE Transactions on Robotics and Automation, 1988, 4(3): 361367 [11]. .Liu, H., Huang, T., Chetwynd, D. G., An Approach for Acceleration Analysis of Lower Mobility Parallel Manipulators, Proceedings of the ASME 2010 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, DETC201028020. [12]. Tsai. LungWen, Robot analysis: the mechanics of serial and parallel manipulators, J.WILEY & SONS, INC., 1999 [13]. Rico, J. M., Duffy, J., 1996, An application of screw algebra to the acceleration analysis of serial chains, Mechanism and Machine Theory, 31(4): pp. 445457. [14]. Huang, Z., Zhao, Y. S., Zhao, T. S., 2006, The Advanced Spatial Mechanism, Beijing: The High Education Press. [15]. MengShiun Tsai a,*, TingNung Shiau a, YiJeng Tsai a, TsannHuei Chang, Direct kinematic analysis of a 3PRS parallel mechanism, Mechanism and Machine Theory 38 (2003) 7183 [16]. G. Pond, J. Carretero, Kinematic Analysis and Workspace Determination of the Inclined PRS Parallel Manipulator proceeding of the 2004 ROMASY, Montreal, Quebec, Canada. [17]. Zhu, S. J., Huang, Z., Guo, X. J., 2005, Forward/reverse velocity and accelerationanalyses for a class of lowermobility parallel mechanisms, Proceedings ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 949955. [18]. Sameer A. Joshi, LungWen Tsai*, Jacobian Analysis of LimitedDOF Parallel manipulators
[19]. Y.G. Li , H.T. Liu, X.M. Zhao , T. Huang ,*, Derek G. Chetwynd,Design of a 3DOF PKM module for large structural
component machining Mechanism and Machine Theory 45 (2010) 941954
[20]. B.Dasgupta, P.Choudhury, A general strategy based onNewtonEuler approach for the dynamic formulation of parallel manipulator
[21]. Yangmin Li*, Qingsong Xu, Kinematic analysis of a 3PRS parallel manipulator,robotics and computer integrated manufacturing 23(2007)395408 [22]. Clavel R.DELTA, a fast robot with parallel geometry proceeding of 18th international symposium on industrial robot, Lausanne; 1988.p.91 100 [23]. Lee KM, Arjunan S. A three degree of freedom micromotion in parallel actuated manipulator IEEE Trans robot automat 1991 634641 [24]. Gosselin, C. M., Stiffness mapping for parallelmanipulators, IEEE Transactions on Robotics and Automation, vol. 6, no. 3, pp. 377382, 1990.
[25]. JianLi Design of 3DOF Parallel Manipulators for MicroMotion Applications masters thesis in University of Ontario Institute of Technology 2010 [26]. TingNung Shiau and MengShiun Tsai Research on Kinematic and Dynamic Characteristics of a 3PRS Parallel Mechanism Dissertation of Doctor of Philosophy in National Chung Cheng University June 2007 [27]. Wang, J.; Gosselin, C.M., 2004, Kinematic Analysis and Design of Kinematically Redundantly Parallel Mechanisms, ASME J. Mech. Des., 126(1), pp.109118. [28]. Mohamed, M.G; Gosselin, C.M., 2005, Design and Analysis of Kinematically Redundant Parallel Manipulators with Configurable Platforms IEEE Trans. Rob.,21(3), pp. 277287.104 [29]. Zhang, D. and Wang, L., 2005, "Conceptual Development of an Enhanced Tripod Mechanism for Machine Tool", Robotics and Computer Integrated Manufacturing 21, no. 45 pp. 31827. [30]. Bi, Z. M. and S. Y. T. Lang. "Kinematic and Dynamic Models of a Tripod System with a Passive Leg." IEEE/ASME Transactions on Mechatronics 11, no. 1 (02, 2006): 108111. [31]. Chablat, D. and P. Wenger. "Architecture Optimization of a 3 DOF TranslationalParallel Mechanism for Machining Applications, the Orthoglide." IEEE Transactions on Robotics and Automation 19, no. 3 (06, 2003): 403410 [32]. V.E. Gough and S.G White hall, 1962,Univeral Tyre Test Machine proceeding of 9th international congress, F.I.S.I.T.A.pp.117 137. [33]. D. Stewart 1965,Aplatform with six degree of freedom in proceeding of institute of mechanical engineering, vol,180 No.5,pp371386 [34]. K.Hunt, 1978, kinematic geometry,clarendo press, oxford [35]. Weck M., Staimer D., Parallel kinematic machine tools—current states and future potentials, Ann. CIRP, vol. 51, no. 2, pp 617683, 2002. [36]. Neumann, K. E., 1988, Robot, US Patent 4732525. [37]. Neumann, K. E., System and Method for Controlling a Robot, US Patent 6301525, 2001. [38]. Tonshoff, H.K., Grendel, H. Kaak, R., Structure and Characteristics of the Hybrid Manipulator George V, in: Boer, C.R. , Molinari Tosatti, L. and Smith, K.S. (Eds.), Parallel Kinematic Machines, SpringerVerlag, London, pp. 365376, 1999. [39]. Huang, T., Li, M., and Li, Z. X., A 5DOF hybrid robot, Patent Cooperation Treaty (PCT), Int. Appl. PCT/CN2004/000479, 2004. [40]. Huang, T., Li, M., Zhao, X. M., Mei, J. P., Chetwynd, D. G., Hu, S. J.,2005, Conceptual Design and Dimensional Synthesis for a 3 DOF Module of the TriVarianta Novel 5DOF Reconfigurable Hybrid Robot, IEEE Trans. on Rob., 21(3), pp. 44945
[41]. Gosselin, C. M., Stiffness mapping for parallel manipulators, IEEE Transactions on Robotics and Automation, vol. 6, no. 3, pp. 377382, 1990.