+
$ + + ,.
/
# 1 , the direction is / # ! '
.
-
and the type is translation, which is
12
22
32 "
1 2 3
0
-
fully represented by the screw expression (21).
!5!5#9:1#23<&10#%+#.8)3.)'%24#01>3'017#)%#8%2)0%(#):1#127=1++18)%0#%+#.#4-.)'.(#-.0.((1(#
<.2'-3(.)%0#
After obtaining the instantaneous mobility of the end-effector, one can directly exert *
actuations to the manipulator, and then investigate the CDOF of the end-effector by solving
470
Parallel Manipulators, Towards New Applications
the free motion(s) of the end-effector within its workspace. If the newly solved motion(s),
denoted by $ ni
( , % # , 12 !
,
ni
(
#
, satisfy that $
3 !0 0 0 0 0
" "
#
0 , then, additional actuations
are needed under this configuration and the actuation scheme. Of course, we can either
reselect the actuation scheme or add
1 ni
( .
R'$(/ $# , more actuation(s) under this configuration
0
-
until
ni
(
$
# !0 0 0 0 0
" "
#
0 . The total number of actuations under the configuration
with this actuation scheme is the CDOF. However, what must be pointed out is that the
actuation(s) should not be exerted to the joint when the newly increased terminal constraint
can be transformed by the other actuation(s). Otherwise, the over constraint case will occur.
When there are a lot of possible actuation schemes any one of which can be selected to set
the actuators, the controllability of the manipulator is also affected by the actuation scheme’s
selection. For an instance, one can analyze the number of actuation(s) required to control the
end-effector of the parallel manipulator shown in Fig. 1. Because the number of DOF of the
end-effector is 1, it is reasonable for us to expect that the end-effector can be fully controlled
only with one actuation. If one actuation is exerted to any joint of the mechanism, 1
, for
example, it is not difficult to find that the end-effector still remains one translational DOF in
the direction # !
"
/
'
.
-
when one repeats the above two steps in section 2.1.
12
22
32 "
Therefore, one has to add another actuation to the mechanism. Of course, he can add the
second actuation to any one of the rest joints. However, it is not difficult to prove that the
end-effector will not be controlled unless the second actuation is exerted to the prismatic
joint 0% ! % #1 2
, , "
3 under the condition that the first actuation is exerted to ,% ! % # 1 2
, , "
3 .
However, just as mentioned above, the new-added actuation should not be accepted if the
newly-increased terminal constraint can be obtained by translating the former actuation(s).
For an example, if the second actuation is assigned to the revolute joint 1
0 , the newly-
increased terminal constraints of the kinematic chain 1
, 1
0 1
+ will be:
!
$
1
$ # ! '
.
-
1
11
11
11
+ -
2 3
11
+ .
3
11
+ '
2 2
11
+ -
2
11
+ .
2 1 '
11
11 " "
+
(22)
1
1
1
1
1
1
Equation (22) is the transformation of the actuation exerted to the prismatic joint 1
, . So, the
newly-added actuation is an over actuation for the actuation scheme whose first actuation is
assigned to 1
, .
Of course, one can also exert the second actuation to the prismatic joint 2
, after assigning
the first actuation to the prismatic joint 1
, . Again, one can find that the end-effector still has
the free translation in the direction # !
"
/
'
.
-
when one repeats the above two
12
22
32 "
steps in section 2.1. So, one can continue to add the third actuation to the prismatic joint 3
, .
However, the end-effector will not be controlled until a fourth actuation is applied to one of
the prismatic joints, 1
0 , 2
0 and 3
0 . This forms a second actuation scheme. So, under this
actuation scheme, the number of actuations needed to control the end-effector shown in Fig.
1 is 4.
Mo9ility of <patial Paral el Manipulators
471
The differences between the second actuation scheme and the first one are that the second
one not only completely control the end-effector but also completely control every link in
the manipulator. The selections of different actuation schemes can be well accomplished by
a computer especially when the possible selections are numerous such as the one shown in
Fig. 1. Unfortunately, this properties of a mechanism is ignored by the general mobility
formulas.
?5# 9:1# 43&4).2)'.(# <%&'(')*# %+# 32710.8)3.)17@# %A10# .8)3.)17# .27# 1>3.((*#
.8)3.)17#<.2'-3(.)%04#
A manipulator is said to be underactuated when the number of actuators in the manipulator
is smaller than the number of degrees of freedom of the mechanism (Lalibertd & Gosselin,
1998). When applied to mechanical fingers, the concept of underactuation leads to shape
adaptation, i.e. underactuated fingers will envelope the objects to be grasped and adapt to
their shape although each of the fingers is controlled by a reduced number of actuators
(Lalibertd & Gosselin, 1998). The concept of underactuation in robotic fingersfwith fewer
actuators than the degrees of freedomfallows the hand to adjust itself to an irregularly
shaped object without complex control strategy and sensors (Birglen & Gosselin, 2006a).
These underactuated manipulators arise in a number of important applications such as
space robots, hyper redundant manipulators, manipulators with structural flexibility, etc
(Jain & Rodriguez, 1993). The fact that the underactuated robotic fingers allow the hand to
adjust itself to an irregularly shaped object makes it possible that no complex control
strategy or numerous sensors are necessary in these manipulators (Birglen & Gosselin,
2006b). However, the over actuated mechanical systems often occur in biomechanical
systems during the contact with ground and is recently introduced in redundantly actuated
parallel robots. Yi and Kim (Yi & Kim, 2002) designed a singularity free load-distribution
scheme for a redundantly actuated three-wheeled Omnidirectional mobile robot. The most
outstanding advantage of the redundantly actuated mobile robot is that the singularities of
the mechanism can be well avoided. Yiu and Li (Yiu & Li, 2003) investigated the trajectory
generation for an over actuated parallel manipulator, in which there is one redundant
actuator. Of course, the redundant actuator(s) and the required actuator(s) must obey a
certain relationship determined by the mechanism, which will be discussed in section 3.2.
This section aims at clarifying the substantial relationships between the underactuated, over
actuated and the equally actuated manipulators. The underactuated manipulator, which is
also called under-determinate input system, means that the number of actuations provided
is less than that is necessary; while the over actuated manipulator, which is also called
redundant actuation or redundant input system, means that the number of actuations
provided is larger than that is necessary. Equally actuated manipulator, which is also called
fully actuated or determinate system, means that the actuations provided is equal to that is
needed.
From the viewpoint of mechanisms, this classification of manipulators seems to be
reasonable and has been widely used in engineering. However, it is not a properly scientific
categorization for mechanisms. Therefore, this section will briefly study the substantial
relationships between the underactuated, over actuated and equally actuated manipulators
that are easily misunderstood in engineering applications.
?5"#9:1#1441281#%+#):1#32710.8)3.)17#<.2'-3(.)%0#
472
Parallel Manipulators, Towards New Applications
To begin with this section, one might first investigate a famous inverted pendulum system
shown in Fig. 2, which is also a representative, underactuated mechanical system. This
inverted pendulum system is a planar two degrees of freedom catenation mechanical
system. The vehicle can only make reciprocal translation along the 2 -axis and the
pendulum can only rotate about the pivot attached to the moving vehicle.
In applications, only one actuation is provided to control the system, which seems to conflict
with the definition of a fully actuated mechanism. In order to reveal the essence of this
puzzling phenomenon, one might first turn to analyze the dynamics of this two-degree-of-
freedom system.
Suppose the mass of the vehicle is denoted by * , the mass of the pendulum is 5 and the
distance from the pivot attached to the vehicle to the mass center of the pendulum is 6 and
the moment of inertia of the pendulum is denoted by 7 . The dynamics of the system can be
immediately established via Lagrange method. The kinetic energy of the vehicle is:
2
1
5
" # * 2
8
2
where 8
" represents the kinetic energy of the vehicle.
The kinetic energy of the pendulum is:
2
2
2
1 )
" #
)
)
5
:
' ! 2 6 6 7 "&
1
5
)
&
sin
6
$
5' ! 6 7 "
1
cos
6 7 7
2 ( )9
%
2
$
( )9
%
2
where ": represents the kinetic energy of the pendulum.
J
m
!
l
F
M
x
Fig. 2 a single inverted pendulum system
The total kinetic energy of the system is:
Mo9ility of <patial Paral el Manipulators
473
2
2
1
" # " 6 " #
8
:
! * 6 5"5
5 5
1
2 5
2 6 56 2 7 cos
1
7 6
7 6 56 ,.
/
7
2
2 0
-
The potential energy of the system is:
< #
7
cos
5;6
Therefore, the Lagrange function of the system is:
2
2
1
5
5 5
1
5
= # ! * 6 5" 2 6 56 2 7
7
2
cos
1
6
7 6 56 ,.
/
7 2 5;6
7
cos
(23)
2
2 0
-
where = indicates the Lagrange function.
The dynamics equations for the two-degrees-of-freedom system shown in Fig. 2 can be
expressed as:
; 1
.
8 ) / < = , < =
8
2
# 4
5
8 )9 //
,, < 2
<
: 0 2 -
(24)
8
1
.
) / < = , < =
8
2
#=
8 )9 // 5 ,, <7
9
0 <7 -
where = represents the torque exerted to the revolute joint that connect the inverse
pendulum and the vehicle.
l parallel manipulator, this chapter develops a general process to synthesize the
manipulators with the specified mobility. The outstanding characteristics of the synthesis
method are that the whole process is also analytical and each step can be programmed at a
computer. Because of the restrictions of the traditional general mobility formulas for spatial
mechanisms, a lot of mechanisms having special manoeuvrability might not be synthesized.
However, any mechanism can be synthesized with this analytical theory of degrees of
freedom for spatial mechanisms.
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The quick calculation approaches based on the algebra summations of the number of the
links, joints and the constraints induced by the joints can not be completely perfected by
itself. This is true even the analytical methods are applied in seeking the common
constraints (Hunt, 1978)(Waldron, 1966)(Huang, 2006). These problems are becoming more
and more obvious with the advent of spatial parallel manipulators. The primary
considerations of the designers for the parallel manipulators have been focused on nothing
but the mobility of the end-effector and its controllability. Therefore, the concept of general
mobility of a mechanism should be divided into two basic concepts the degree of freedom of
the end-effector and the number of actuations needed to control the end-effector. With this regard,
this chapter first introduces two primary definitions:
F'4*%*)*3%#!G#
474
Parallel Manipulators, Towards New Applications
The DOF of an end-effector totally characterizes the motions of the end-effector including
the number, type and direction of the independent motions (*hao et al, 2004a)(*hao et al,
2006a).
F'4*%*)*3%#7G#
The configuration degree of freedom (CDOF) of a mechanism with an end-effector indicates
the independent number of actuations required to uniquely control the end-effector under a
configuration (*hao et al, 2004b)(*hao et al, 2006c).
Obviously, the DOF of an end-effector in number is not larger than 6 but the independent
number of actuations required to uniquely control the end-effector might be any
nonnegative integer. Bearing the above two definitions in mind, one can fall into two steps
to investigate the mobility of a mechanism1the DOF of the end-effector and the CDOF of
the mechanism with the prescribed end-effector. The former definition indicates the full
instantaneous mobility properties of the end-effector through a mathematics concept of free
mobility space while the later one presents the instantaneous controllability of the
mechanism system. By definition 1, one can find that the DOF of an end-effector is only
subjected to the constraint(s) exerted by the kinematic chain(s) connecting the end-effector
with the fixed base or ground. Besides, the degree of freedom of the end-effector,
instantaneously associated with the spatial configurations of the kinematic chain(s), should
clearly depict the number, the direction and the type of the free motion of the end-effector
instantaneously. Therefore, only analytical methods can fulfil such a task.
After obtaining the free motions of the end-effector, an engineering question will naturally
arise1how many actuations are needed to control the end-effectorj By definition 2, one can
find that a checking process is given for verifying the controllability of the mechanism with
the specified end-effector. Besides, this process can also allow the different selections of the
actuation schemes, which is most adapted to the concept design of a manipulator.
Consequently, the valid means to investigate the mobility of mechanisms can be addressed
as: (1) investigate the instantaneous DOF of the prescribed end-effector; and (2) investigate
the number of actuations required to uniquely control the end-effector of the mechanism.
For the instantaneous characteristics of the mobility of a mechanism, only analytical means
is acceptable for such a task. Because of the elegance in depicting the relationship between
the motions and the constraints, reciprocal screw theory does be a well selection to
accomplish the task. Therefore, the following analytical model for the mobility of a parallel
manipulator will be built up by applying the reciprocal screw theory.
According to reciprocal screw theory (Hunt, 1978)(Phillips, 1984)(Phillips, 1990)(Phillips et
al, 1964)(Waldron, 1966)(Ball, 1900), a screw $ is defined by a straight line with an
associated pitch > and is conveniently denoted by six PlPcker homogeneous coordinates:
) s &
+ # '
(1)
s
> s$
6
( 0
%
where s denotes direction ratios pointing along the screw axis, s # r* s
0
defines the
moment of the screw axis about the origin of the coordinate system, r is the position vector
of any point on the screw axis with respect to the coordinate system. Consequently, the
) s &
screw axis can be denoted by the PlPcker homogeneous coordinates +
#
.
'2%?
's $
( 0 %
Assume
Mo9ility of <patial Paral el Manipulators
475
8; s # ! = * B " "
:
(2)
8
"
9 s 6 > s #
0
! A @ R"
Considering s 4 ! s 6 >
0
s"
2
# s 4 s 6 s >
2
# s >
0
and presuming s 3 ! , one obtains the instant
pitch of a screw:
s 4 ! s0 6 > s" =A 6 *@ 6 BR
> #
#
(3)
2
2
2
2
s
= 6 * 6 B
Therefore, the axis of the screw can also be denoted as:
+
# ! = * B A 2 => @ 2 *> R 2 B> " "
(