Chapter 9
Robotic Mechanisms
9.1 Introduction
During the last few decades, robot manipulators have been used primarily for repet-
itive operations and in hazardous environments. Typical applications include parts
loading and unloading, radioactive material handling, space and undersea exploration,
spot welding, spray painting, sealant application, and some simple component assem-
bly. Recently, there is an increasing interest in making robots more intelligent and
more user friendly. Thus, medical-surgery robots, household service robots, micro-
robots, and others are becoming available. Although much effort has been spent on
the development of robot manipulators, the ultimate goal of developing user-friendly,
intelligent robots that can emulate human functions is still in its infancy. Major obsta-
cles are some of the key technologies have not been fully developed. In this chapter,
structural characteristics and structural enumeration of some robotic mechanisms are
presented.
9.2 Parallel Manipulators
The development of parallel manipulators can be dated back to the early 1960s
when Gough and Whitehall [6] first devised a six-linear jack system for use as a
universal tire testing machine. Later, Stewart [17] developed a platform manipulator
for use as a flight simulator. Since 1980, there has been an increasing interest in the
development of parallel manipulators. Potential applications of parallel manipulators
include mining machines [3], pointing devices [5], and walking machines [26].
Although parallel manipulators have been studied extensively, most of the studies
have concentrated on the Stewart-Gough manipulator. The Stewart-Gough manipula-
tor, however, has a relatively small workspace and its direct kinematics are extremely
difficult to solve. Hence, it may be advantageous to explore other types of parallel
manipulators with the aim of reducing the mechanical complexity and simplifying
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the kinematics and dynamics [10, 14, 16, 25]. A structural classification of parallel
manipulators has been made by Hunt [8].
In this section, parallel manipulators are classifiedinto planar , spherical, and spa-

characteristics should also be satisfied.
We assume that each limb is made up of an open-loop chain and the number of
limbs, m, is equal to the number of degrees of freedom of the moving platform. It
follows that
m = F = L + 1 . (9.1)
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FIGURE 9.1
A typical parallel manipulator.
Let the connectivity of a limb, C
k
, be defined as the number of degrees of freedom
associated with all the joints, including the terminal joints, in that limb. Then,
m

k=1
C
k
=
j

i=1
f
i
. (9.2)
Substituting Equation (9.2) into Equation (4.7) and making use of Equation (9.1), we
obtain
m

k=1
C

For two-dof manipulators, Equation (9.1) yields m = 2 and L = 1, whereas
Equation (9.3) reduces to

2
k=1
C
k
= 5. Thus, planar two-dof manipulators are
single-loop mechanisms. The number of degrees of freedom associated with all the
joints is equal to five. Furthermore, Equation (9.4) states that the connectivity in each
limb is limited to no more than three and no less than two. Hence, one of the limbs
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is a single-link and the other is a two-link chain. These two limbs, together with the
end-effector and the base link, form a planar five-bar linkage.
A simple combinatorial analysis yields the following possible planar five-bar
chains: RRRRR, RRRRP, RRRPP, and RRPRP. Any of the five links can be cho-
sen as the fixed link. Once the fixed link is chosen, either of the two links that
is not adjacent to the fixed link can be chosen as the end-effector. For example,
Figure 9.4 shows a planar RR–RRR parallel manipulator with links 1 and 2 serv-
ing as the input links and link 4 as the output link. Figure 9.5 shows a planar
RR–RPR parallel manipulator.
FIGURE 9.4
Planar RR–RRR manipulator.
Planar Three-dof Manipulators
For planar three-dof parallel manipulators, Equation (9.1) yields m = 3 and L = 2.
Substituting λ = 3 and F = 3 into Equation (9.3), we obtain
3

k=1
C

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FIGURE 9.6
Planar 3-RPR parallel manipulator.
spherical mechanisms are not included. Therefore, the only feasible two-dof spherical
manipulator is the five-bar RR–RRR manipulator. Similarly, the only feasible three-
dof spherical manipulator is the 3-RRR manipulator as shown in Figure 6.14. Several
articles related to kinematic analysis, dimensional synthesis, and design optimization
of spherical parallel manipulators can be found in [4, 5, 9, 27].
9.2.5 Enumeration of Spatial Parallel Manipulators
For spatial manipulators, λ = 6. Thus, Equations (9.3) and (9.4) reduce to
m

k
C
k
= 7F − 6 , (9.7)
6 ≥ C
k
≥ F. (9.8)
Solving Equation (9.7) for positive integers of C
k
in terms of the number of degrees
of freedom, subject to the constraint imposed by Equation (9.8), we obtain all feasible
limb connectivity listings as shown in Table 9.1. Each connectivity listing given in
Table 9.1 represents a family of parallel manipulators for which the number of limbs
is equal to the number of degrees of freedom of the manipulator, and the total number
of joint degrees of freedom in each limb is equal to the value of C
k
.
The number of links (and joints) to be incorporated in each limb can be any number,

dof joints. In practice, however, the number of links incorporated in a limb should be
kept to a minimum. This necessitates the use of spherical and universal joints. In what
follows, we enumerate three- and six-dof manipulators to illustrate the methodology.
Three-dof Translational Platforms
We first enumerate three-dof parallel manipulators with pure translational motion
characteristics. To reduce the search domain, we limit our search to those manipula-
tors having three identical limb structures. In this way, the (5, 5, 5) connectivity listing
becomes the only feasible limb configuration. Hence, the joint degrees of freedom
associated with each limb is equal to five. Furthermore, we assume that revolute (R),
prismatic (P ), universal (U ), and spherical (S) joints are the applicable joint types.
A simple combinatorial analysis yields four types of limb configurations as listed in
Table 9.2.
Table 9.2
Feasible Limb Configurations for Three-dof Manipulators.
Type Limb Configuration
120
PUU,UPU,RUU
201
RRS,RSR,RPS,PRS,RSP,PSR,SPR,PPS,PSP,SPP
310
RRRU, RRPU, RP RU, P RRU, RP P U, P RPU, PP RU
,
RRUR, RRUP, RPUR, P RUR, RPUP, P RUP, P PUR
,
RURR, RURP, RUP R, P URR, RUP P, P URP, P UP R
,
UPRR,UPRP,UPPR
500
RRRRR, RRRRP, RRRP R, RRP RR, RP RRR, P RRRR
,

short links and a planar parallelogram in each limb as shown in Figure 9.11 [16].
Six-dof Parallel Manipulators
In this section, we briefly discuss the enumeration of six-dof parallel manipulators.
To simplify the problem, we limit ourselves to those manipulators with six identical
limb structures. Furthermore, we assume that each limb consists of two links and
three joints.
Referring to Table 9.1, we observe that the (6, 6, 6, 6, 6, 6) connectivity listing
is the only solution. Thus, the degrees of freedom associated with all the joints of
a limb should be equal to six. Since there are two links and three joints, the only
possible solution is the type 111 limb, which means that each limb consists of one
one-dof, one two-dof, and one three-dof joints. Let revolute, prismatic, universal,
and spherical joints be the applicable joint types. Six feasible limb configurations
exist: RUS, RSU, PUS, PSU, SPU, and UPS as shown in Figure 9.12. Configurations
SRU, SUR, URS, USR, SUP, and USP are excluded because they do not contain a
base-connected revolute or prismatic joint, or an intermediate prismatic joint.
Note that the universal joints shown in Figure 9.12 can be replaced by a spherical
joint. This results in a passive degree of freedom, allowing the intermediate link(s)
to spin freely about a line passing through the centers of the two spheres. Thus, RSS,
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