6 1. An Overview of Robotic Mechanical Systems
FIGURE 1.3. Canadarm2 and Special-Purpose Dextrous Manipulator (courtesy
of the Canadian Space Agency.)
1.3 Serial Manipulators
Among all robotic mechanical systems mentioned above, robotic manipu-
lators deserve special attention, for various reasons. One is their relevance
in industry. Another is that they constitute the simplest of all robotic me-
chanical systems, and hence, appear as constituents of other, more complex
robotic mechanical systems, as will become apparent in later chapters. A
manipulator, in general, is a mechanical system aimed at manipulating ob-
jects. Manipulating, in turn, means to move something with one’s hands,
as it derives from the Latin manus, meaning hand. The basic idea behind
the foregoing concept is that hands are among the organs that the human
brain can control mechanically with the highest accuracy, as the work of
an artist like Picasso, of an accomplished guitar player, or of a surgeon can
attest.
Hence, a manipulator is any device that helps man perform a manip-
ulating task. Although manipulators have existed ever since man created
the first tool, only very recently, namely, by the end of World War II, have
manipulators developed to the extent that they are now capable of actu-
ally mimicking motions of the human arm. In fact, during WWII, the need
arose for manipulating probe tubes containing radioactive substances. This
led to the first six-degree-of-freedom (DOF) manipulators.
Shortly thereafter, the need for manufacturing workpieces with high ac-
curacy arose in the aircraft industry, which led to the first numerically-
controlled (NC) machine tools. The synthesis of the six-DOF manipulator
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1.3 Serial Manipulators 7
FIGURE 1.4. Special-Purpose Dextrous Manipulator (courtesy of the Canadian
Space Agency.)
and the NC machine tool produced what became the robotic manipula-

1.4 Parallel Manipulators
Robotic manipulators first appeared as mechanical systems constituted by
a structure consisting of very robust links coupled by either rotational or
translating joints, the former being called revolutes, the latter prismatic
joints. Moreover, these structures are a concatenation of links, thereby
forming an open kinematic chain, with each link coupled to a predeces-
sor and a successor, except for the two end links, which are coupled only
to either a predecessor or to a successor, but not to both. Because of the
serial nature of the coupling of links in this type of manipulator, even
though they are supplied with structurally robust links, their load-carrying
capacity and their stiffness is too low when compared with the same prop-
erties in other multiaxis machines, such as NC machine tools. Obviously, a
low stiffness implies a low positioning accuracy. In order to remedy these
drawbacks, parallel manipulators have been proposed to withstand higher
payloads with lighter links. In a parallel manipulator, we distinguish one
base platform,onemoving platform,andvariouslegs. Each leg is, in turn,
a kinematic chain of the serial type, whose end links are the two platforms.
Contrary to serial manipulators, all of whose joints are actuated, parallel
manipulators contain unactuated joints, which brings about a substantial
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1.4 Parallel Manipulators 9
difference between the two types. The presence of unactuated joints makes
the analysis of parallel manipulators, in general, more complex than that
of their serial counterparts.
A paradigm of parallel manipulators is the flight simulator, consisting of
six legs actuated by hydraulic pistons, as displayed in Fig. 1.5. Recently, an
explosion of novel designs of parallel manipulators has occurred aimed at
fast assembly operations, namely, the Delta robot (Clavel, 1988), developed
at the Lausanne Federal Polytechnic Institute, shown in Fig. 1.6; the Hexa
robot (Pierrot et al., 1991), developed at the University of Montpellier;

wrist is modeled as a spherical join as well, while the elbow is modeled as a
simple revolute joint. Robotic mechanical systems mimicking the motions
of the arm and the forearm constitute the manipulators discussed in the
previous section. Here we outline more sophisticated manipulation systems
that aim at producing the motions of the human hand, i.e., robotic me-
chanical hands. These robotic systems are meant to perform manipulation
tasks, a distinction being made between simple manipulation and dextrous
manipulation. What the former means is the simplest form, in which the
fingers play a minor role, namely, by serving as simple static structures that
keep an object rigidly attached with respect to the palm of the hand—when
the palm is regarded as a rigid body. As opposed to simple manipulation,
dextrous manipulation involves a controlled motion of the grasped object
with respect to the palm. Simple manipulation can be achieved with the
aid of a manipulator and a gripper, and need not be further discussed here.
The discussion here is about dextrous manipulation.
In dextrous manipulation, the grasped object is required to move with re-
spect to the palm of the grasping hand. This kind of manipulation appears
in performing tasks that require high levels of accuracy, like handwriting
or cutting tissue with a scalpel. Usually, grasping hands are multifingered,
although some grasping devices exist that are constituted by a simple,
open, highly redundant kinematic chain (Pettinato and Stephanou, 1989).
The kinematics of grasping is discussed in Chapter 4. The basic kinematic
structure of a multifingered hand consists of a palm, which plays the role
of the base of a simple manipulator, and a set of fingers. Thus, kinemat-
ically speaking, a multifingered hand has a tree topology, i.e., it entails a
common rigid body, the palm, and a set of jointed bodies emanating from
the palm. Upon grasping an object with all the fingers, the chain becomes
closed with multiple loops. Moreover, the architecture of the fingers is that
of a simple manipulator. It consists of a number—two to four—of revolute-
coupled links playing the role of phalanges. However, unlike manipulators

the object while keeping the fingers in contact with the object. Torquabi-
lity, or tangential rotatability, is the ability to rotate the long axis of an
object—here the authors must assume that the manipulated objects are
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1.6 Walking Machines 13
convex and can be approximated by three-axis ellipsoids, thereby distin-
guishing between a longest and a shortest axis—with a minimum force, for
a prescribed amount of torque. Finally, radial rotatability is the ability to
rotate the grasped object about its long axis with minimum torque about
the axis.
Furthermore, Allen et al. (1989) introduced an integrated system of both
hardware and software for dextrous manipulation. The system consists
of a Sun-3 workstation controlling a Puma 500 arm with VAL-II. The
Utah/MIT hand is mounted on the end-effector of the arm. The system in-
tegrates force and position sensors with control commands for both the arm
and the hand. To demonstrate the effectiveness of their system, the authors
implemented a task consisting of removing a light bulb from its socket. Fi-
nally, Rus (1992) reports a paradigm allowing the high-level, task-oriented
manipulation control of planar hands. Whereas technological aspects of
dextrous manipulation are highly advanced, theoretical aspects are still
under research in this area. An extensive literature survey, with 405 refer-
ences on the subject of manipulation, is given by Reynaerts (1995).
1.6 Walking Machines
We focus here on multilegged walking devices, i.e., machines with more
than two legs. In walking machines, stability is the main issue. One distin-
guishes between two types of stability, static and dynamic. Static stability
refers to the ability of sustaining a configuration from reaction forces only,
unlike dynamic stability, which refers to that ability from both reaction and
inertia forces. Intuitively, it is apparent that static stability requires more
contact points and, hence, more legs, than dynamic stability. Hopping de-

and Associates Hexapod (Sutherland and Ullner, 1984), the Titan series of
quadrupeds (Hirose et al., 1985) and the Odetics series of axially symmetric
hexapods (Russell, 1983).
A survey of walking machines, of a rather historical interest by now,
is given in (Todd, 1985), while a more recent comprehensive account of
walking machines is available in a special issue of The International Journal
of Robotics Research (Volume 9, No. 2).
Walking machines appear as the sole means of providing locomotion in
highly unstructured environments. In fact, the unique adaptive suspension
provided by these machines allows them to navigate on uneven terrain.
However, walking machines cannot navigate on every type of uneven ter-
rain, for they are of limited dimensions. Hence, if terrain irregularities such
as a crevasse wider than the maximum horizontal leg reach or a cliff of
depth greater than the maximum vertical leg reach are present, then the
machine is prevented from making any progress. This limitation, however,
can be overcome by providing the machine with the capability of attaching
its feet to the terrain in the same way as a mountain climber goes up a cliff.
Moreover, machine functionality is limited not only by the topography of
the terrain, but also by its constitution. Whereas hard rock poses no serious
problem to a walking machine, muddy terrain can hamper its operation to
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1.7 Rolling Robots 15
FIGURE 1.10. The OSU ASV. An example of a six-legged walking machine
(courtesy of Prof. K. Waldron. Reproduced with permission of The MIT Press.)
the point that it may jam the machine. Still, under such adverse conditions,
walking machines offer a better maneuverability than other vehicles. Some
walking machines have been developed and are operational, but their op-
eration is often limited to slow motions. It can be said, however, that like
research on multifingered hands, the pace of theoretical research on walking
machines has been much slower than that of their technological develop-

chines do not determine the position and orientation of the vehicle body.
As a matter of fact, the control of rolling robots bears common features
with that of the redundancy resolution of manipulators of the serial type at
the joint-rate level. In these manipulators, the number of actuated joints
is greater than the dimension of the task space. As a consequence, the
task velocity does not determine the joint rates. Not surprisingly, the two
types of problems are being currently solved using the same tools, namely,
differential geometry and Lie algebra (De Luca and Oriolo, 1995).
As a means to supply rolling robots with 3-dof capabilities, omnidirec-
tional wheels (ODW) have been proposed. An example of ODWs are those
that bear the name of Mekanum wheels, consisting of a hub with rollers
on its periphery that roll freely about their axes, the latter being oriented
at a constant angle with respect to the hub axle. In Fig. 1.11, a Mekanum
wheel is shown, along with a rolling robot supplied with this type of wheels.
Rolling robots with ODWs are, thus, 3-dof vehicles, and hence, can trans-
late freely in two horizontal directions and rotate independently about a
vertical axis. However, like their 2-dof counterparts, 3-dof rolling robots
are also nonholonomic devices, and thus, pose the same problems for their
control as the former.
(a) (b)
FIGURE 1.11. (a) A Mekanum wheel; (b) rolling robot supplied with Mekanum
wheels.
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1.7 Rolling Robots 17
Recent developments in the technology of rolling robots have been re-
ported that incorporate alternative types of ODWs. For example, Killough
and Pin (1992) developed a rolling robot with what they call orthogonal
ball wheels, consisting basically of spherical wheels that can rotate about
two mutually orthogonal axes. West and Asada (1995), in turn, designed a
rolling robot with ball wheels, i.e., balls that act as omnidirectional wheels;

ometric quantities such as distances and angles between lines. Although we
may resort to a coordinate frame and vector algebra to compute distances
and angles and represent vectors in that frame, the final result will be inde-
pendent of how we choose that frame. The same applies to quantities whose
evaluation calls for the introduction of tensors. Here, we must distinguish
TLFeBOOK
20 2. Mathematical Background
between the physical quantity represented by a vector or a tensor and the
representation of that quantity in a coordinate frame using a 1-dimensional
array of components in the case of vectors, or a 2-dimensional array in the
case of tensors. It is unfortunate that the same word is used in English to
denote a vector and its array representation in a given coordinate frame.
Regarding tensors, the associated arrays are called matrices. By abuse of
terminology, we will refer to both tensors and their arrays as matrices,
although keeping in mind the essential conceptual differences involved.
2.2 Linear Transformations
The physical 3-dimensional space is a particular case of a vector space.A
vector space is a set of objects, called vectors, that follow certain algebraic
rules. Throughout the book, vectors will be denoted by boldface lower-
case characters, whereas tensors and their matrix representations will be
denoted by boldface uppercase characters. Let v, v
1
, v
2
, v
3
,andw be ele-
ments of a given vector space V,whichisdefined over the real field,andlet
α and β be two elements of this field, i.e., α and β are two real numbers.
Below we summarize the aforementioned rules:

)+
v
3
;
(iv) For every element v of V, there exists a corresponding element, w,
also of V, which, when added to v, produces the zero vector, i.e.,
v + w = 0.Moreover,w is represented as −v;
(v) The product αv,orvα, is also an element of V, for every v of V and
every real α. This product is associative, i.e., α(βv)=(αβ)v;
(vi)Ifα is the real unity, then αv is identically v;
(vii) The product defined in (v)isdistributive in the sense that (a)(α +
β)v = αv + βv and (b) α(v
1
+ v
2
)=αv
1
+ αv
2
.
Although vector spaces can be defined over other fields, we will deal with
vector spaces over the real field unless explicit reference to another field is
made. Moreover, vector spaces can be either finite- or infinite-dimensional,
but we will not need the latter. In geometry and elementary mechanics, the
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2.2 Linear Transformations 21
dimension of the vector spaces needed is usually three, but when studying
multibody systems, an arbitrary finite dimension will be required. The
concept of dimension of a vector space is discussed in more detail later.
A linear transformation, represented as an operator L, of a vector space

the range of a linear transformation are both vector subspaces of U and
V, respectively, i.e., they are themselves vector spaces, but of a dimension
smaller than or equal to that of their associated vector spaces. Moreover,
the kernel of a linear transformation is often called the nullspace of the said
transformation.
Henceforth, the 3-dimensional Euclidean space is denoted by E
3
.Having
chosen an origin O for this space, its geometry can be studied in the context
of general vector spaces. Hence, points of E
3
will be identified with vectors
of the associated 3-dimensional vector space. Moreover, lines and planes
passing through the origin are subspaces of dimensions 1 and 2, respectively,
of E
3
. Clearly, lines and planes not passing through the origin of E
3
are not
subspaces but can be handled with the algebra of vector spaces, as will be
shown here.
An orthogonal projection P of E
3
onto itself is a linear transformation of
the said space onto a plane Π passing through the origin and having a unit
normal n, with the properties:
P
2
= P, Pn = 0 (2.1a)
Any matrix with the first property above is termed idempotent.Forn × n

p ≡ (1 − nn
T
)p (2.3)
and hence, the orthogonal projection P onto Π can be represented as
P = 1 − nn
T
(2.4)
where the product nn
T
amounts to a 3 × 3matrix.
Now we turn to reflections. Here we have to take into account that re-
flections occur frequently accompanied by rotations, as yet to be studied.
Since reflections are simpler to represent, we first discuss these, rotations
being discussed in full detail in Section 2.3. What we shall discuss in this
section is pure reflections, i.e., those occurring without any concomitant
rotation. Thus, all reflections studied in this section are pure reflections,
but for the sake of brevity, they will be referred to simply as reflections.
A reflection R of E
3
onto a plane Π passing through the origin and
having a unit normal n is a linear transformation of the said space into
itself such that a position vector p is mapped by R intoavectorp

given
by
p

= p −2nn
T
p ≡ (1 − 2nn

T
n)n
T
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2.2 Linear Transformations 23
which apparently reduces to 1 because n is a unit vector. Note that from
the second property above, we find that pure reflections observe a further
interesting property, namely,
R
−1
= R
i.e., every pure reflection equals its inverse. This result can be understood
intuitively by noticing that, upon doubly reflecting an image using two
mirrors, the original image is recovered. Any square matrix which equals
its inverse will be termed self-inverse henceforth.
Further, we take to deriving the orthogonal decomposition of a given
vector v into two components, one along and one normal to a unit vector
e. The component of v along e, termed here the axial component, v

—read
v-par—is simply given as
v

≡ ee
T
v (2.6a)
while the corresponding normal component, v
⊥
—read v-perp—is simply
the difference v −v

n
, (2.7)
where the elements of the set {α
i
}
n
1
are all elements of the field over which
V is defined, i.e., they are real numbers in the case at hand. The number
n of elements in the set B = {v
i
}
n
1
is called the dimension of V.Notethat
any set of n linearly independent vectors of V can play the role of a basis of
this space, but once this basis is defined, the set of real coefficients {α
i
}
n
1
for expressing a given vector v is unique.
Let U and V be two vector spaces of dimensions m and n, respectively,
and L a linear transformation of U into V, and define bases B
U
and B
V
for
U and V as
B

+ ···+ l
nj
v
n
,j=1, ,m (2.9)
Consequently, in order to represent the images of the m vectors of B
U
,
namely, the set {Lu
j
}
m
1
, n × m real numbers l
ij
,fori =1, ,n and
TLFeBOOK
24 2. Mathematical Background
j =1, ,m, are necessary. These real numbers are now arranged in the
n × m array [ L]
B
V
B
U
defined below:
[ L ]
B
V
B
U

n1
l
n2
··· l
nm




(2.10)
The foregoing array is thus called the matrix representation of L with
respect to B
U
and B
V
. We thus have an important definition, namely,
Definition 2.2.1 The jth column of the matrix representation of L with
respect to the bases B
U
and B
V
is composed of the n real coefficients l
ij
of
the representation of the image of the jth vector of B
U
in terms of B
V
.
The notation introduced in eq.(2.10) is rather cumbersome, for it involves

of L are determined by the equation
det(λ1 − L) = 0 (2.11)
Note that the matrix λ1 − L is linear in λ, and since the determinant of
an n × n matrix is a homogeneous nth-order function of its entries, the
left-hand side of eq.(2.11) is an nth-degree polynomial in λ. The foregoing
polynomial is termed the characteristic polynomial of L. Hence, every n×n
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2.3 Rigid-Body Rotations 25
matrix L has n complex eigenvalues, even if L is defined over the real field.
If it is, then its complex eigenvalues appear in conjugate pairs. Clearly,
the eigenvalues of L are the roots of its characteristic polynomial, while
eq.(2.11) is called the characteristic equation of L.
Example 2.2.1 What is the representation of the reflection R of E
3
into
itself, with respect to the x-y plane, in terms of unit vectors parallel to the
X, Y, Z axes that form a coordinate frame F?
Solution: Note that in this case, U = V = E
3
and, hence, it is not necessary
to use two different bases for U and V.Now,leti, j, k, be unit vectors
parallel to the X, Y,andZ axes of a frame F. Clearly,
Ri = i
Rj = j
Rk = −k
Thus, the representations of the images of i, j and k under R,inF,are
[ Ri ]
F
=


[ R ]
F
=


10 0
01 0
00−1


2.3 Rigid-Body Rotations
A linear isomorphism, i.e., a one-to-one linear transformation mapping a
space V onto itself, is called an isometry if it preserves distances between
any two points of V.Ifu and v are regarded as the position vectors of two
such points, then the distance d between these two points is defined as
d ≡

(u − v)
T
(u − v) (2.12)
The volume V of the tetrahedron defined by the origin and three points
of the 3-dimensional Euclidean space of position vectors u, v,andw is
obtained as one-sixth of the absolute value of the double mixed product of
these three vectors,
V ≡
1
6
|u × v · w| =
1
6