A Numerical Evaluation of the Stiffness of CaHyMan(Cassino Hybrid Manipulator)
GIUSEPPE CARBONE, MARCO CECCARELLI AND MARCO TEOLIS
Laboratory of Robotics and Mechatronics DiMSAT – University of CassinoVia Di Biasio 43  03043 Cassino (Fr), Italyemail: ceccarelli@ing.unicas.it
Abstract
In this paper the novel hybrid manipulator, named as CaHyMan (Cassino Hybrid Manipulator), isanalyzed in term of stiffness characteristics. A formulation is presented to deduce the stiffness matrixas a function of the most important stiffness parameters of the mechanical design. The specific designof CaHyMan, which has been designed and built at the Laboratory of Robotics in Cassino, Italy, helpsto obtain closedform expressions. A formulation for a stiffness performance index is proposed byusing the obtained stiffness matrix. A numerical investigation has been carried out on the effects of design parameters and results are discussed in the paper.
1.
Introduction
In the last decade hybrid manipulators addressed great attention as a combination of serial and parallel chain architectures. Some significant examples are: ARTISAN from Stanford University(USA), [1; 2]; HRM from Korea Institute of Machinery and Materials (Korea), [3]; GEORGV fromInstitute of Production Engineering and Machine Tools (Germany), [4]; UPSarm from University of California at Davis (USA), [5].A novel hybrid manipulator, named as CaHyMan (Cassino Hybrid Manipulator) has been designedand built at the Laboratory of Robotics and Mechatronics in Cassino, Italy. This prototype, Figs.1 and2, is based on the mechanical design of a built prototype of CaPaMan (Cassino Parallel Manipulator),[6; 7], by adding to it a telescopic arm.
2.
The design of CaHyMan
The proposed hybrid manipulator CaHyMan is shown in the kinematic sketch of
Fig.2 as acombination of the parallel chain of CaPaMan with a telescopic arm architecture. In particular, thetelescopic arm is installed on the mobile plate MP of CaPaMan. The aim of this assembly is that the parallel architecture will work as an intelligent compliant base for the telescopic arm, which willoperate in static or quasi static state for apriori determined task.The CaHyMan prototype has five dofs: three dofs are given by CaPaMan, and two more are given by the serial chain. It is composed of a movable plate MP which is connected to a fixed plate FP bymeans of three leg mechanisms. Each leg mechanism is composed of an articulated parallelogram APwhose coupler carries a prismatic joint SJ, a connecting bar CB which transmits the motion from AP toMP through SJ, and a spherical joint BJ which is installed on MP. The size of MP and FP are
given by
Fig.1 The built prototype of CaHyMan (Cassino Hybrid Manipulator) at Laboratory of Robotics and Mechatronicsin Cassino.
Fig.2 Kinematic chain and parameters for the parallel–serial hybrid manipulator named CaHyMan.
r
p
and r
f
, respectively, Fig.2.The design parameters for the parallelserial manipulator CaHyMan are (k=1,2,3): a
k
=c
k
, b
k
=d
k
,links of the kth leg mechanism; h
k
, the length of the connecting bar;
α
κ
, the input crank angle; s
k
, thestroke of the prismatic joint; HM, the length of the telescopic arm; s
k
the stroke of the prismatic joint of the telescopic arm;
α
k
the revolute joint angle; and
λ
, the angle, that locates the telescopic arm framewith respect to the mobile frame X
p
Y
p
Z
p
.The manipulative capability of the parallelserial chain can be described by the position of theextremity point M, and the orientation of the telescopic link through the orientation angles which takeinto account the angles
ϕ
,
ϑ
,
ψ
,
λ
and
α
4
. Similarly, the static behavior of the CaHyMan can bedescribed by the actions exerted by the extremity link at M for given actions of the actuators, or viceversa.
3.
CaHyMan Stiffness Properties
The manipulating performances of a robot are strictly related to stiffness properties. If the stiffnessof links and joints are inadequate, external forces and moments may cause large deflections in the links bodies, which are undesirable from the viewpoint of both accuracy and payload performances.The stiffness properties of the hybrid manipulator can be deduced using a quality index such as thedeterminant of the stiffness matrix. The stiffness matrix can be obtained in a closedform expression asa function of the most important stiffness parameters of the links of the serial and parallel chain.The displacement
∆∆∆∆
X
CaHyMan
of the hybrid manipulator CaHyMan is the sum of parallel and serialstructure displacements
∆∆∆∆
X
PAR
and
∆∆∆∆
X
SER
, respectively, which are due to the action
F
M
=(
F
e
,
N
e
) of external force
F
e
and torque
N
e
. This can be expressed as
SER PAR CaHyMan
!!!
XXX
+=
(1)where
M1PAR PAR
K
!
FX

=
(2)
M
FX
1SER SER
K
!
−
=
in which K
PAR
and K
SER
are the 6x6 stiffness matrix for the parallel and serial chain. described withrespect to X
P
Y
P
Z
P
frame.Thus, the stiffness matrix of CaHyMan can be written as
1SER 1PAR 1
K K K
CaHyMan
−−−
+=
(3)
3.1.
The stiffness matrix of the serial chain
The stiffness behavior of the serial chain depends on the following parameters, Fig.3: K
4
, thestiffness of the QM
2
link; K
S
, the stiffness of the linear motor and MM
2
link; K
T4
, the stiffness of motor located in Q.The scheme of Fig.3 can be considered to give
1SPSER
AsKssR K
−
=
(4)whose elements can be computed by considering
 R
SP
, which is the 6x6 matrix given by
=
I00R R
PAR SP
(5)with I as identity 3x3 matrix and R
PAR
, which
describes the orientation of the mobile frame X pY pZ pwith respect to the fixed frame XYZ in term of the Euler angles
ϕ
,
θ
and
ψ
for the parallel chain, given by
ϕϕϕϕϕϕϕϕϕ=
ss
"
cc
"
ccs
#
s
"
s
#
sc
"
c
#
c
"
s
#
s+s
"
c
#
cc
#
s
"
c
#
sc
"
s
#
c
"
c
#
s+s
"
s
#
R
PAR
(6)in which c
θ
=cos
θ
, s
θ
=sin
θ
and so on. Kss, which is expressed as
++−=
T444T44s44
44T44s44
K 00c
$
)]s(LK [s
$
K s
$
K s
$
)]s(LK [c
$
K c
$
K Kss (7)
As, which is the matrix converting
∆
L,
∆
s
4
,
∆α
4
, in the links to displacements coordinates, in theform
++−=
100)c
$
s(Ls
$
s
$
)s
$
s(Lc
$
c
$
A
4444
4444
S
(8)
Fig.3 A scheme for the evaluation of statics equilibrium and stiffness matrix in the serial subchain of CaHyMan.
3.2.
The stiffness matrix of the parallel chain
As regards the parallel chain the stiffness matrix K
PAR
can be deduced as proposed in [8], for theCaPaMan prototype. Nevertheless, it is necessary to point out that for CaHyMan the application pointof external force
F
e
and torque
T
e
is in the point M, Fig.2. Therefore, the reactions at the frame joint of the serial chain
F
and
N
are considered such as external force and torque for the parallel chain.The stiffness behavior of the parallel chain depends on the following parameters, Fig.4: K
bk
, K
ck
,K
dk
, K
hk
, which are the stiffness of the links b
k,
c
k,
d
k
, h
k,
respectively; K
Tk
, which is the stiffness of themotors. These stiffness parameters have been defined, by using mathematical models as in [9] and[10].By using a suitable analysis of the static equilibrium of the deformed architecture through themodel of Fig.4 the stiffness matrix K
PAR
of CaPaMan module of CaHyMan can be formulated as
1d1PPFN1FTPAR
ACK MMK
−−−
=
(9)in which
M
FT
is the matrix giving
F
H
=(
F
,
N
) as function of
F
M
=(
F
e
,
T
e
) in the form
−+−+−+−+−+−
−−++
+−+−−−=
4444
4z4y4x3z43y3x44
4444
2z2y2x1z41y1x4
44FT
s
$
0c
$
0)c
$
s(L0CC1CCHQc
$
CCHQs
$
)s(Lc
$
0s
$
0HQ)s
$
s(L0CCCCs
$
CCc
$
000010
000c
$
0s
$
M(10)where the terms C
1j
, C
2j
, and
C
3j
(j=x,y,z) are introduced to consider the weights of links, m
HQ
, m
QM
,m
MM2
, and motors, m
M4
and m
M5
, respectively. The abovementioned parameters can be evaluated as
ejM5M4MM2QMHQ1j
]/F6)mmmm(m[9.81C
++++=
ejM5M4MM2QMHQ2j
]/T6)mmmm(m[9.81C
++++=
ej4MM24M444M53j
)]/Fc
$
L(m)c
$
L(0.5mc
$
L)(0.5sm[1.635C
−−+−=
(11)
ej4MM24M444M54j
)]/Tc
$
L(m)c
$
L(0.5mc
$
L)(0.5sm[1.635C
−−+−=

M
FN
is the matrix giving the forces acting on the legs
F
R
=[R
x1
, R
x2
, R
x3
, R
z1
, R
z2
, R
z3
]
t
as a functionof
F
H
in the form
=
0r 0r 0r
c
%
r 0c
%
r 0c
%
r 0s
%
r 0s
%
r 0s
%
r 0101010
0c
%
0c
%
0c
%
0s
%
0s
%
0s
%
M
321332211
332211
321321FN
(12)
K
P
is the overall stiffness matrix for the CaPaMan legs in the form