Climbing R
obot
Akash K Singh, PhD
IBM Corporation
Sacramento, USA
akashs@us.ibm.com
Abstract
—
Climbing is a current challenge in robotics.
Observation of vertebrate climbing
proves
that in arboreal
environment like on flat ground locomotion is carried out as
well
by the trunk as by the limbs. Thus to elucidate one
extreme a
trunk

driven concept was chosen as base for our
design process.
Often climbing robots are highly special
ized
machines. They only
are able to move on one or two
different substrates. And they
often are expensive solutions,
built for a special application. The
climbing robot we report
about is designed as a modular system.
The modular idea
influences all
developmental areas of a
mechatronic system
–
the mechanical, the electronic as well as the
software part.
By using different modules, the robot is capable to
climb on
different substrates made of a wide variety of materials.
The
mechanical demands of clim
bing are strong and rule the
design. Substrate contact has to be established and broken
actively. A sensor concept was developed, which allows for
choosing relevant sensor ranges, depending on the desired
type of
control. The configuration described in thi
s paper is
composed of
only two different module types. Due to the
features of these
modules, the robot is capable to climb on
pipe

like structures.
The speed of the robot depends on the
gait chosen.
Keywords

robotics, service robotics, bionics, climbing
,
biomechatronics
.
I.
I
NTRODUCTION
Importance of automatic inspection is increasing. In
buildings,
cables and pipes have to be inspected
periodically to prevent
or to detect damages. In historic
buildings infrastructure has
grown over many years. Often
maps showing the exact course
of supply lines do not
exist. In new buildings, passable supply
routes are
installed for human beings performing inspection
tasks.
But these routes consume a lot of expensive space. Thus
our main idea is to build small robots,
which can move in
also
small canals along linear structures.
Also in industrial
environments robot system are
needed for moving on flat
ferromagnetic structures. Such
robots use active or passive
magnetic systems to lock on
the surface. Other robotic
syst
ems are automatically
cleaning window areas. Therefore
often vacuum
technologies are chosen to achieve a climbing
ability.
Common goals for mobile robots are locomotion,
idiomotion, manipulation, navigation, orientation,
imitation
and cooperation [2], [22]
. For a climbing robot,
the challenge
in locomotion is high locomotion ability.
The robot has to
move horizontally and vertically,
ascending and descending.
The environment is
unstructured. Often a special infrastructure
for the robot
system cannot be inst
alled, so the robot system
has to use
the available environmental conditions for its
locomotion
–
cables and pipes with diameters of 10 mm to 80 mm,
ferromagnetic flat surfaces, windows, and walls.
II.
CLIMBING ROBOTS
PRINCIPLE
In the last decades, a lot of
climbing robots have been
developed. We can classify current climbing robots by
application, by technical parameters like size, payload,
and
power consumption. As described before, climbing
robots deal
with the generation of mot ion and with the
substrate c
ontact.
Substrate contact plays an important
role. In most cases, the
robot will be destroyed, if
substrate contact fails. For our
systematic construction
process, classifications by substrate
contact and by
locomotion principle were realized.
In the last
decades, robots often owned stiff backbones.
Due to the space
requirements of control hardware, energy
source and
communicat ion devices, rigid body
constructions were made.
But zoological studies show that
the locomotion of vertebrates
and non

vertebrates
is
mainly driven by a flexible trunk
[1],[4],[7],[8],[22].
The work of Fischer and Lehmann [5]
illustrates that
the main
locomotors
organ of vertebrates is the
trunk,
vertebral column. Furthermore, small, ancestral
mammals
obtain up to 50% of the spatial
gain per movement
cycle
by trunk motion.
A.
Classification by Substrate Contact
There are common techniques to generate adhesion to
the
substrate, classified by the nature of forces. In reality,
there is
a cross

coupling of adhesion techniques; often a
clear
differentiation is impossible.
• pneumatic
–
forces generated by low pressure,
• magnetic
–
magnetic force generated by permanent or
electro
magnets on ferromagnetic surfaces
• electrostatic
–
forces generated by a electrostatic field
between the robot an
d the surface (e.g. SRI wallclimbing
robot),
• chemical
–
by using chemical substances to generate
adhesive forces (e.g. Stickybot [9]),
• positive locking
–
by using the geometrical structures
of the
surface (e.g Spinybot [19], RatNic), and
• hybrid
–
by
coupling at least two of the above listed
methods
of force generation
This list is only a short summary of adhesion
technologies. A
more complete overview is given in [11].
Another important
attribute is the need of external energy
to hold up the contact
to the substrate. In this point, we
can simply differ into active
and passive contact. Active
means in this case, that we
continuously need power to
keep the connection. Passive
means that after the
connection is established, there is no need
of any power
for maintaining the connection. In the same way,
we can
classify the process of establishment and release of
locking.
B.
Classification by Characteristics of Locomotion
There are a lot of locomotion methods in robotics. Most of
them are also used for climbin
g robots. If we neglect
substrate
contact, we find the same principles like in
terrestrial
locomotion. Only one additional method is
shown in [11]. In
comparison to [10], the methods can be
differed into
• walking
–
using limb

like construction for walking
on
the
substrate
• frame walking
–
using at least two rigid frames in an
alternating way
• sliding
–
sliding on the surface of the substrate
• wheeled
–
using wheels to drive on a mostly flat
substrate
• tracked
–
using chains or chain

like constructions
• brachiation
–
using at least two arms to brachiate
The degree of freedom (DOF) of trunk motion called
level of
mobility is important for the design process. This
is not in
every case the number of joints integrated in the
robot. Robots
climbing on a sin
gle straight pipe like
shown in [2] have a
level of mobility of one. Robots
walking on flat surfaces have
a level of mobility of three,
if they can walk in two direct ions
and turn around one
axis. A level of mobility of six means full
climbing
mobility.
C.
Properties of Modular Robots
Locomotion on pipe

like substrates needs variation in
the
DOF of locomotion from 2D to 6D. Due to this, a
modular
design is chosen. In most cases, these modules
are functional
units. By combination of these single units,
systems with a
new functional quality may be created. In
most cases, one
module is unable to move. By connecting
a lot of modules, we
get chained systems, which are able
to realize complex
movements. One big advantage is that
we can combine
modules in diff
erent ways, so we can
create walking, crawling
or swimming systems. Most
important properties are [3],[21]:
• status of homogeneity,
• status of self

configurability,
• torques or forces acting via the module
• power consumption,
• and communication.
If al
l elements of the system are identical, there will be
homogeneity of the modules. If the structure consists of
several, non

identical modules, it is called heterogeneous.
In
both cases, the position of the module in the robot
defines its
function. A modula
r robot will be selfconfigurable,
if the robot
identifies its structure by itself.
In this case, the modules could
be put together in different
ways to achieve an adaptable
behavior without any effort
for reconfiguration.
III.
CONCEPT OF THE MODUL
AR SYSTEM
As
shown before, for climbing the main focus is on one
hand
on the substrate contact and one the other hand on
the
generation of motion. The idea is, to implement the
different
technologies of adhesion on one robot. The
adequate
technology of adhesion depend
s on the substrate
properties.
These properties could be:
• material
• stiffness
• surface roughness
• ferromagnetic properties
• shape
Also the needs of motion generation depend on the
substrate
properties. A robot doesn’t need a high degree of
freedom
cl
imbing on a flat surface. A robot climbing on
cylindrical
substrates (brackets, wire, pipes) needs a
higher degree of
freedom. Furthermore some adhesion
technologies need a
special way of motion for the
establishment or for the
releasing of substrate conta
ct. So,
the whole mechanical
system of the robot has to be varied
for different tasks.
Because of this, an overall modular
concept is chosen
. It
consists of active modules and passive modules. Passive
means in this case, that they are electronically passiv
e. So,
a
self

identification of the system is not possible at this
state.
The passive modules are connectors. They are
necessary to
connect the motion und adhesive modules, so
a power
transmission could be achieved. The mounting
modules are
another type of
modules. They are necessary
to fix equipment
on the system like electronic devices and
accumulator.
Important is a common mechanical
interface. By this, an
intermodular connectivity and very
flexible robotic system are
achieved.
A.
Modular Electronic Syste
m
The robot system is a modular one. Hence, the
electronics has
to be designed in such a way, too. Because
the mechanical
structure isn’t framed, the electrical system
has to be
adaptable. At the current state, the robot should be remote
controlled.
Therefore, a common game pad is used. A
Host PC
with an I²C

Converter is used to communicate
with an A/D

Converter and a motor controller. The motor
controller is able
to control 21 servo motors. If a higher
number of drives will
be necessary, a second con
troller
can be integrated. Sensor
signal can be read by an A/Dconverter
connected to the I²C

Bus, too. The sensor
signals must have a voltage of ±10 V.
The power supply is made by a 7.4V LiPo

Cell with
a
capacity of 2 Ah. The maximum current is limited to
20
A,
due to the recommendation made by the manufacturer.
For the
design of modules four main restrictions have
been made:
• Actuators have to be controlled by a special PWMsignal
• Sensor signals has to be conditioned to a 10V

level
• The voltage of power
supply is limited to 7,4V
• The overall maximum current is limited to 20A.
By considering this, different modules can be designed
and
integrated in the system with any other
recommendations to
the electronic system.
B.
Modular Mechanical System
To assure m
echanical compatibility of different
modules, a
mechanical interface is designed. The modules
must
have four
tap holes in a square
.
The modules are fixed to the connectors
by four screws
of the size M3x8. The modules should have the
shape of a
cube. The ed
ge length has to be 56mm. The
montage has
to be done by a user. A self

organizat ion and
selfassembling
is not planned.
C.
Designed Modules
As described before, restrictions to the design of the
modules
have been done due to the modular idea for the
system. A
t the
current state different types of modules
have been developed.
Adhesion Modules
As described before, there are a lot of different
substrates. And
there is no adhesion technology working
on every substrate.
Hence, different adhesion modules are
design for the modular
system.
The
recommendations are similar to all adhesive
modules. For
the design process a locking force of 50 N and a
locking
torque of 1 Nm.
• magnetic module
• adaptive gripper module
• gripper module for a special diameter of subs
trate
• vacuum module
Motion Modules
Motion could be generated in a translatory or
rotator
way. At
the current state only rotational modules are
realized. The
drives integrated as actuators have a torque
of 3.0 Nm
. The
modules are
• rotational module wit
h DOF = 1
• rotational module with DOF = 2
IV.
CONFIGURATIONS FOR C
LIMBING ON PIPE

LIKE
STRUCTURES
A.
Examples of Kinematical Configurations
Possible configuration could be robot structures with
serial,
parallel or mixed structures. At the beginning, only
serial
robot configurations are considered. Two gripper
modules are
chosen, one at each end of the robot. Level of
mobility is
defined by the number and the arrangement of
locomotion
modules between the grippers. The structure
of the robot
should be adapte
d to the climbing task.
The level of
mobility
is one. The locomotion module is a rotational
one. To achieve
a linear motion, three locomotion
modules have to be
combined. An inchworm

like motion
pattern is achieved.
The
second example is a configuration fo
r climbing on
a straight
pipe and for changing from this pipe to a parallel
one. For this
task, the level of mobility is two. The two
motions are
translational. To achieve the two translations,
we have to
combine only three locomotion modules.
B.
Reference
System for Climbing on Straight Pipe

like
Structure
Configuration
The modular system is suitable for the design of
climbing
robots. For the task “climbing on a straight pipelike
structure”
a simple serial configuration is chosen,
consisting of three
rotational motion modules and two
adaptive gripper modules.
The modules are connected. The configuration is completed
by the
motor controller and the power supply.
Kinematics
The kinemat ics of the robots is very simple. The tool
centre
point (TCP) has to
be controlled in its three
mechanical
freedoms. Therefore, inverse kinemat ical
calculations have to
been done.
For the three coordinates of the TCP we get the
following
equations:
The coordinates of the TCP can be combined to
and
the coordinates of the
drives can be generalized to
The
Jacobian

Matrix is used for the inverse kinemat ics
A change
of the angular position of the rotational drive is
calculated by
The calculation is implemented in the control software
.
We consider the following
anycast
field equations defined
over an open bounded piece of
network
and /or feature space
. They describe the dynamics of the mean
anycast
of
each of
n
ode
populations.
We give an
interpretation of the various parameters and
functions that appear in (1),
is finite piece of
nodes
and/or
feature space and is represented as an open bounded set of
. The vector
and
represent points in
. The
function
is the normalized sigmoid function:
It describes the relation between the
input
rate
of
population
as a function of the
packets
potential, for
example,
We note
the
dimensional vector
The
function
represent the initial conditions, see below. We
note
the
dimensional vector
The
function
represent external
factors
from
other
network
areas. We note
the
dimensional
vector
The
matrix of functions
represents the connectivity between
populations
and
see below. The
real values
determine the threshold of activity for eac
h
population, that is, the value of the
nodes
potential
corresponding to 50% of the maximal activity.
The
real
positive values
determine the slopes of the
sigmoids at the origin. Finally the
real positive values
determine the speed at which each
anycast
node
potential decreases exponentially toward its real value.
We also introduce the function
defined by
and the
diagonal
mat rix
I
s the intrinsic
dynamics of the population given by the linear response of
data transfer
.
is replaced by
to use the
alpha function response. We use
for simplicity
although our analysis applies to more general intrinsic
dynamics. For the sake, of generality, the propagation delays
are not assumed to be identical for all populations,
hence they
are described by a matrix
whose element
is
the propagation delay between population
at
and
population
at
The reason for this assumption is that it is
still unclear from
anycast
if propagation delays are
independent of the populations. We assume for technical
reasons that
is continuous, that is
Moreover
packet
data indicate that
is not a symmetric
function i.e.,
thus no assumption is
made about this symmetry unless otherwise stated.
In order to
compute the
righthand s
ide of (1), we need to know the
node
potential factor
on interval
The value of
is
obtained by considering the maximal delay:
Hence we choose
C.
Mathematical Framework
A convenient functional setting for the non

delayed
packet
field equations is to use the space
which is a
Hilbert space endowed with the usual inner product:
To give a meaning to (1), we defined the history space
with
which is the Banach phase space associated with equation (3).
Using the notation
we
writ
e (1) as
Where
Is the linear continuous operator satisfying
Notice that most of the papers on this
subject assume
infinite, hence requiring
Proposition 1.0
If the following assumptions are satisfied.
1.
2.
The external current
3.
Then for any
there exists a unique solution
to (3)
Notice that this result gives existence on
finite

time
explosion is impossible for this delayed differential equation.
Nevertheless, a particular solution could grow indefinitely, we
now prove that this cannot happen.
D.
Boundedness of Solutions
A valid model of neural networks should only feature b
ounded
packet node
potentials.
Theorem 1.0
All the trajectories are ultimately bounded by
the same constant
if
Proof
:Let us defined
as
We note
Thus,
i
f
Let us show that the open
route
of
of center 0 and radius
is stable under the dynamics of equation. We know
that
is defined for all
and that
on
the boundary of
. We consider three cases for the initial
condition
If
and set
Suppose that
then
is defined and belongs to
the closure of
because
is closed, in effect to
we also have
because
Thus we deduce that for
and small enough,
which contradicts the definition of T. Thus
and
is stable.
Because f<0 on
implies that
. Finally we consider the case
. Suppose that
then
thus
is monotonically
decreasing and reaches the value of R in finite t
ime when
reaches
This contradicts our assumption. Thus
Proposition
1.1
:
Let
and
be measured simple functions
on
for
define
Then
is a measure on
.
Proof :
If
and if
are disjoint members of
whose union is
the countable additivity of
shows that
Also,
so that
is not identically
.
Next, let
be as before, let
be the distinct values
of t,and let
If
the
a
nd
Thus (2)
holds with
in place of
. Since
is the disjoint union
of the sets
the first half of our
proposition implies that (2) holds.
Theorem
1.1
:
If
is a compact set in the plane whose
complement is connected, if
is a continuous complex
function on
which is holomorphic in the interior of , and if
then there exists a polynomial
such that
for all
.
If the interior of
is
empty, then part of the hypothesis is vacuously satisfied, and
the conclusion holds for every
. Note that
need
to be connected.
Proof:
By Tietze’s theorem,
can be extended to a
continuous function in the plane, with compact support. We
fix one such extension and denote it again by
.
For any
let
be the sup
remum of the numbers
Where
and
are subject to the
condition
. Since
is uniformly continous, we
have
F
rom now on,
will be
fixed. We shall prove that there is a polynomial
such that
By (1),
this proves the theorem.
Our first objective is the
construction of a funct
ion
such that for all
And
Where
is the set of all points in the support of
whose
distance from the complement of
does not
. (Thus
contains no point which is “far within”
.) We construct
as the convolution of
with a smoothing function A. Put
if
put
And define
For all complex
. I
t is clear that
. We claim that
The constants are so adjusted in (6) that (8) holds. (Compute
the integral in polar coordinates), (9) holds simply because
has compact support. To compute (10), express
in polar
coordinates, and note that
Now define
Since
and
have compact support, so does
. Since
And
if
(3) follows from (8). The
difference quotients of
converge boundedly to the
corresponding partial derivatives, since
. Hence
the last expression in (11) may be differentiated under the
integral sign, and we obtain
The last equality depends on (9). Now (10) and (13) give (4).
If we write (13) with
and
in place of
we see
that
has continuous partial derivatives,
if we can show that
in
where
is the set of all
whose
distance from the complement of
exceeds
We shall do
this by showing that
Note that
in
, since
is holomorphic there. Now
if
then
is in the inte
rior of
for all
with
The mean value property for harmonic functions
therefore gives, by the first equation in (11),
For all
, we have now proved (3), (4), and (5)
The
definit ion of
shows that
is compact and that
can be
covered by finitely many open discs
of radius
whose centers are not in
Since
is
connected, the center of each
can be joined to
by a
polygonal path in
. It follows that each
contains a
compact connected set
of diameter at least
so that
is connected a
nd so that
with
. There are functions
and constants
so that the inequalities.
Hold for
and
if
Let
be the complement of
Then
is an
open set which contains
Put
and
for
Define
And
Since,
(18) shows that
is a finite linear combination of the
functions
and
. Hence
By (20), (4), and
(5) we have
Observe that the inequalities (16) and (17) are valid with
in
place of
if
and
Now fix
, put
and
estimate the integrand in (22) by (16) if
by (17) if
The integral in (22) is then
seen to be less than the sum of
And
Hence (22) yields
Since
and
is connected,
Runge’s theorem shows that
can be uniformly
approximated on
by polynomials. Hence (3) and (25) show
that (2) can be satisfied.
This completes the proof.
Lemma
1.0
:
Suppose
the space of all
continuously differentiable functions in the plane, with
compact support. Put
Then the following “Cauchy formula” holds:
Proof:
This may be deduced from Green’s theorem. However,
here is a simple direct proof:
Put
real
If
the chain rule gives
The right side of (2) is therefore equal to the limit, as
of
For each
is periodic in
with peri
od
. The
integral of
is therefore 0, and (4) becomes
As
uniformly. This gives (2)
If
and
, then
, and so
satisfies the condition
.
Conversely,
and so if
satisfies
, then the subspace generated by the
monomials
, is an ideal. The proposition gives a
classification of the monomial ideals in
: they
are in one to one correspondence with the subsets
of
satisfying
. For example, the monomial ideals in
are exactly the ideals
, and the zero ideal
(corresponding to the empty set
). We write
for the ideal corresponding to
(subspace generated by the
).
LEMMA
1.1
. Let
be a subset of
. The the ideal
generated by
is the monomial ideal
corresponding to
Thus, a monomial is in
if and only if it is divisible by one
of the
PROOF. Clearly
satisfies
, and
.
Conversely, if
, then
for some
,
and
. The last statement follows from
the fact that
.
Let
satisfy
. From the geometry of
, it is clear that there is
a finite se
t of elements
of
such that
(The
are the corners of
) Moreover,
is
generated by the monomials
.
DEFINITION
1.0
.
For a nonzero ideal
in
, we let
be the ideal generated by
LEMMA
1.2
Let
be a nonzero ideal in
;
then
is a monomial ideal, and it equals
for some
.
PROOF. Since
can also be described as the ideal
gener
ated by the leading monomials (
rather than the leading
terms) of elements of
.
THEOREM
1.2
.
Every
ideal
in
is finitely
generated; more
precisely,
where
are any elements of
whose leading terms generate
PROOF.
Let
. On applying the division algorithm,
we find
,
where either
or no monomial occurring in it is divisible
by any
. But
, and therefore
, implies that
every monomial occurring in
is divisible by one in
. Thus
, and
.
DEFINITION
1.1
.
A finite subset
of an
ideal
is a standard (
bases for
if
. In other words, S is a
standard basis if the leading term of every element of
is
divisible by at least one of the leading terms of the
.
THEOREM 1.
3
The ring
is Noetherian i.e.,
every ideal is finitely generated.
PROOF.
For
is a principal ideal domain,
which means that every ideal is generated by single element.
We shall prove the theorem by induction on
. Note that the
obvious map
is an
isomorphis m
–
this simply says that every polynomial
in
variables
can be expressed uniquely as a
polynomial in
with coefficients in
:
Thus the next lemma will complete the proof
LEMMA 1.3.
If
is Noetherian, then so also is
PROOF. For a polynomial
is called the degree of
, and
is its leading coefficient.
We call 0 the leading coefficient of the polynomial 0.
Let
be an ideal in
. The leading coefficients
of the polynomials in
form an ideal
in
, and since
is Noetherian,
will
be finitely generated. Let
be elements of
whose leading coefficients generate
, and
let
be the maximum degree of
. Now let
and
suppose
has degree
, say,
Then
, and so we can write
Now
has degree
.
By continuing in this way, we find that
With
a polynomial of
degree
.
For each
, let
be the subset of
consisting of 0 and the leading coefficients of all polynomials
in
of degree
it is again an ideal in
. Let
be polynomials of degree
whose leading
coefficients generate
. Then the same argument as above
shows that any polynomial
in
of degree
can be
written
With
of
degree
. On applying this remark repeatedly we find
that
Hence
an
d so the polynomials
generate
One of the great successes of category theory in computer
science has been the development of a “unified theory” of the
constructions underlying denotational semantics. I
n the
untyped

calculus, any term may appear in the function
position of an application. This means that a model D of the

calculus must have the property that given a term
whose
interpretation is
Also, the interpretation of a
functional abstraction like
.
is most conveniently
defined as a function from
, which must then
be
regarded as an element of
D
.
Let
be the
function that picks out elements of
D
to represent elements of
and
be the function that
maps elements of
D
to functions of
D.
Since
is
intended to represent the function
as an element of
D,
it
makes sense to require that
that is,
Furthermore, we often want to view every
e
lement of
D
as representing some function from
D to D
and
require that elements representing the same function be equal
–
that is
The latter condition is called extensionality. These conditions
together imply that
are inverses

that is,
D
is
isomorphic to the space of functions from
D to D
that can be
the interpretations of functional abstractions:
.Let us suppose we are working with the untyped
, we need a solution ot the equation
where A is some predetermined
domain containing interpretations for elements of
C.
Each
element of
D
corresponds to either an element of
A
or an
element of
with a tag. This equation can be
solved by finding least fixed points of the function
from domains to domains

that
is, finding domains
X
such that
and
such that for any domain
Y
al
so satisfying this equation, there
is an embedding of
X
to
Y

a pair of maps
Such that
Where
means that
in some
ordering representing their informat ion content. The key shift
of perspective from the domain

theoretic to the more general
category

theoretic approach lies in considering
F
not as a
function on domains, but as a
functor
on a category of
domains.
Instead of a least fixed point of the function,
F.
Definition 1.3
: Let
K
be a category and
as a
functor. A fixed point of
F
is a pair (A,a), where A is a
K

object
and
is an isomorphis m. A prefixed
point of F is a pair (A,a), where A is a
K

object
and a is any
arrow from F(A) to A
Definition 1.4 :
An
in a category
K
is a diagram
of the following form:
Recall that a cocone
of an
is a
K

object
X
and a collection of K
–
arrows
such
that
for all
. We sometimes write
as a reminder of the arrangement of
components
Similarly, a colimit
is a cocone with
the property that if
is also a cocone then there
exists a unique mediating arrow
such that for all
. Colimits of
are sometimes
referred to as
.
Dually, an
in
K
is a diagram of the following form:
A cone
of an
is a
K

object
X and a collection of
K

arrows
such that for all
. An

limit of an
is a cone
with the property that if
is also a cone, then there
exists a unique mediating arrow
such that for
all
. We write
(or just
) for the
distinguish initial object of
K,
when it h
as one, and
for the unique arrow from
to each
K

object A. It is also
convenient to write
to denote all of
except
and
. By analogy,
is
. For
the images of
and
under
F
we write
and
We write
for the
i

fold iterated composition of
F
–
that is,
,etc.
With these definitions we can state that every monitonic
function on a complete lattice has a least fixed point:
Lemma
1.4.
Let
K
be
a category with init ial object
and let
be a functor. Define the
by
If both
and
are
colimits,
then (D,d) is an intial F

algebra, where
is the mediat ing arrow from
to the
cocone
Theorem 1.
4
Let a DAG G given in which each node is a
random variable, and let a discrete conditional probability
distribution of each node given values of its parents in G be
specified. Then the product of these conditional distributions
yields a joint probability d
istribution P of the variables, and
(G,P) satisfies the Markov condition.
Proof.
Order the nodes according to an ancestral ordering. Let
be the resultant ordering. Next define.
Where
is the set of parents of
of in G and
is the specified conditional probability
distribution. First we show this does indeed yield a joint
probability distribution. Clearly,
for
all values of the variables. Therefore, to show we have a joint
distribution, as the variables range through all their possible
values, is equal to one. To that end, Specified conditional
distributions are the conditional distributi
ons they notationally
represent in the joint distribution.
Finally, we show the
Markov condition is satisfied. To do this, we need show for
that
whenever
Where
is the set of nondescendents of
of in G. Since
, we need only show
. First for a given
, order the
nodes so that all and only nondescendents of
precede
in the ordering. Note that this ordering depends on
, whereas
the ordering in the first part of the proof does not. Clearly then
follows
We define the
cyclotomic field to be the field
Where
is the
cyclotomic
polynomial.
has degree
over
since
has degree
. The roots of
are just the primitive
roots of unity, so the
complex embeddings of
are simply the
maps
being our fixed choice of primitive
root of unity. Note
that
for every
it follows that
for all
relat ively prime to
. In
particular, the images of the
coincide, so
is Galois over
. This means that we can
write
for
without much fear of
ambiguity; we will do so from now on, the identification being
One advantage of this is that one can easily talk
about cyclotomic fields being extensions of one another,or
intersections or compositums; all of
these things take place
considering them as subfield of
We now investigate some
basic properties of cyclotomic fields. The first issue is whether
or not they are all distinct; to determine this, we need to know
which roots of uni
ty lie in
.
Note, for example, that if
is odd, then
is a
root of unity. We will show that
this is the only way in which one can obtain any non

roots of unity.
LEMMA 1.
5
If
divides
, then
is contained in
PROOF. Since
we have
so the
result is clear
LEMMA 1.
6
If
and
are relatively prime, then
and
(Recall the
is the compositum of
PROOF. One checks easily that
is a primit ive
root
of unity, so that
Since
this implies that
We know that
has degree
over
, so we must have
and
And thus that
PROPOSITION 1.
2
For any
and
And
here
and
denote the least common multiple and
the greatest common divisor of
and
respectively.
PROOF. Write
where the
are distinct primes. (We allow
to be zero)
An entirely similar computation shows that
Mutual informat ion measures the information t ransferred
when
is sent and
is received, and is defined as
In a noise

free channel,
each
is uniquely connected to the
corresponding
, and so they constitute an input
–
output pair
for which
bits; that is, the
transferred informat ion is equal to the self

information that
corresponds to the input
In a very noisy channel, the output
and input
would be completely uncorrelated, and so
and also
that is, there is no
transference of information. In general, a given channel will
operate between these two ext remes.
The mutua
l informat ion
is defined between the input and the output of a given channel.
An average of the calculat ion of the mutual informat ion for all
input

output pairs of a given channel is the average mutual
information:
bits per
symbo
l
.
This calculation is done over the input and output
alphabets. The average mutual information.
The following
expressions are useful for modifying the mutual informat ion
expression:
Then
Where
is
usually called the equivocation.
In a sense, the equivocation
can be seen as the informat ion lost in the noisy channel, and is
a function of the backward conditional probability. The
observation of an output symbol
provides
bits of information. This difference is the
mutual information of the channel.
Mutual Information:
Properties
Since
The mutual information fits the condition
And by interchanging input and output it is also true that
Where
This last entropy is usually called the noise entropy.
Thus, the
informat ion transferred through the channel is the difference
betw
een the output entropy and the noise entropy.
Alternatively, it can be said that the channel mutual
informat ion is the difference between the number of bits
needed for determining a given input symbol before knowing
the corresponding output symbol, and the
number of bits
needed for determining a given input symbol after knowing
the corresponding output symbol
As the channel mutual information expression is a difference
between two quantities, it seems that this parameter can adopt
negative values. However, and is spite of the fact that for some
can be larger than
, this is not
possible for the average value calculated over all the outputs:
Then
Because this expression is of the form
The above expression can be applied due to the factor
which is the product of two probabilit ies, so
that it behaves as the quantity
, which in this expression is
a dummy variable that fits the condition
. It can be
concluded that the average mutual information is a non

negative number. It can also be equal to zero, when the input
and the
output are independent of each other. A related
entropy called the joint entropy is defined as
Theorem 1.5:
Entropies of
the binary erasure channel (BEC)
The BEC is defined with an alphabet of two inputs and three
outputs, with
symbol probabilities.
and transition
probabilities
Lemma
1
.7
.
Given an arbitrary restricted t ime

discrete,
amplitude

continuous channel whose restrictions are
determined by sets
and whose density functions exhibit no
dependence on the state
, let
be a fixed positive integer,
and
an arbitrary probability density function on
Euclidean
n

space.
for the density
and
.
For any real
number a, let
Then for each positive integer
, there is a code
suc
h that
Where
Proof: A sequence
such that
Choose the decoding set
to be
. Having chosen
and
, select
such that
Set
, If the process does not terminate
in a finite number of steps, then the sequences
and
decoding sets
form the desired code. Thus
assume that the process terminates after
steps. (Conceivably
). We will show
by showing that
. We proceed as
follows.
Let
E.
Algorithms
Let A be a ring. Recall that an
ideal
a
in A is a subset such
that
a is subgroup of A regarded as a group under addition;
The ideal generated by a subset S
of A is the intersection of all
ideals A containing a

it is easy to verify that this is in fact
an ideal, and that it consist of all finite sums of the form
with
. When
, we
shall write
for the ideal it generates.
Let a and b be ideals in A. The set
is
an ideal, denoted by
. The ideal generated by
is denoted by
.
Note that
. Clearly
consists of all finite sums
with
and
, and if
and
, then
.Let
be an ideal of A. The set of cosets of
in A forms a ring
, and
is a homomorphism
.
The map
is a one to one correspondence
between the ideals of
and the ideals of
containing
An ideal
if
prime
if
and
or
. Thus
is prime if and only if
is nonzero and
has the property that
i.e.,
is an integral domain. An ideal
is
maximal
if
and there does not exist an ideal
contained strictly
between
and
. Thus
is maximal if and only if
has no proper nonzero ideals, and so is a field. Note that
maximal
prime. The ideals of
are all of the
form
, with
and
ideals in
and
. To see this,
note that if
is an ideal in
and
, then
and
. This
shows that
with
and
Let
be a ring. An

algebra is a ring
together with a
homomorphism
. A
homomorphism
of

algebra
is a homomorphism of rings
such that
for all
.
An

algebra
is said
to be
finitely generated
( or of
finite

type
over A) if there exist
elements
such that every element of
can be
expressed as a polynomial in the
with coefficients in
, i.e., such that the homomorphis m
sending
to
is surjective. A ring homomorphis m
is
finite,
and
is finitely generated as an A

module. Let
be a field, and let
be a

algebra. If
in
, then the map
is injective, we can identify
with its image, i.e., we can regard
as a subring of
. If 1=0
in a ring R, the R is the zero ring, i.e.,
.
Polynomi al
rings.
Let
be a field. A
monomial
in
is an
expression of the form
. The
total
degree
of the monomial is
. We somet imes abbreviate it
by
.
The elements of the
polynomial ring
are finite sums
With the obvious notions of equality, addition and
multiplicat ion. Thus the monomials from basis for
as a

vector space. The ring
is an integral domain, and the only units in i
t
are the nonzero constant polynomials. A polynomial
is
irreducible
if it is nonconstant and has only
the obvious factorizations, i.e.,
or
is
constant.
Di vision in
. The division algorithm allows
us to divide a nonzero polynomial into another: let
and
be polynomials in
with
then there ex
ist unique
polynomials
such that
with either
or deg
< deg
. Moreover, there is an algorithm for
deciding whether
, namely, find
and check
whether it is zero. Moreover, the Euclidean algorithm allows
to pass from finite set of generators for an ideal in
to a
single generator by successively r
eplacing each pair of
generators with their greatest common divisor.
(
Pure)
lexicographic
ordering (lex
).
Here monomials are
ordered by lexicographic(dictionary) order. More precisely, let
and
be two elements of
;
then
and
(lexicographic ordering) if, in
the vector difference
, the left most nonzero entry
is positive. For example,
. Note that this isn’t
quite how the dictionary would order them: it would put
after
.
Graded reverse
lexicographic order (grevlex).
Here monomials are ordered by
total degree, w
ith ties broken by reverse lexicographic
ordering. Thus,
if
, or
and in
the right most nonzero entry is negative. For
example:
(total degree greater)
.
Orderings
on
.
Fix an ordering on the
monomials in
.
Then we can write an element
of
in a
canonical fashion, by re

ordering its
elements in decreasing order. For example, we would write
as
or
Let
, in decreasing order:
Then we define.
The
multidegree
of
to be multdeg
(
)=
;
The
leading coefficient of
to be
LC(
)=
;
The
leading monomial of
to be
LM(
) =
;
The
leading term of
to be
LT(
) =
For the polynomial
the multidegree is
(1,2,1), the leading coefficient is 4, the leading monomial is
, and the leading term is
.
The di vision
algorithm i n
.
Fix a monomial ordering in
. Suppose given a polynomial
and an ordered set
of polynomials; the division algorithm then
constructs polynomials
and
such that
Where either
or no
monomial in
is divisible by any of
Step 1:
If
, divide
into
to get
If
, repeat the process until
(different
) with
not divisible by
. Now divide
into
, and so on, until
With
not divisible by
any
Step 2:
Rewrite
,
and repeat Step 1 with
for
:
(different
)
Monomi al i deals.
In general, an ideal
will contain a
polynomial without containing the individual terms of the
polynomial; for example, the ideal
contains
but not
or
.
DEFINITION
1.5
. An ideal
is
monomial
if
all
with
.
PROP
OSITI
ON 1
.3. Let
be a
monomial ideal,
and let
. Then
satisfies the condition
And
is the

subspace of
generated by the
.
Conversely, of
is a subset of
satisfying
, then the
k

subspace
of
generated by
is a monomial ideal.
PROOF. It is clear from its definit ion that a monomial ideal
is the

subspace of
generated by the set of monomials it
contains. If
and
.
If a permutation is chosen uniformly and at random from the
possible permutations in
then the counts
of
cycles of length
are dependent random variables. The joint
distribution of
follows from
Cauchy’s formula, and is given by
for
.
Lemma1.
7
For nonnegative integers
Proof.
This can be established directly by exploit ing
cancellation of the form
when
which occurs between the ingredients in Cauchy’s
formula and t
he falling factorials in the moments. Write
. Then, with the first sum indexed by
and the last sum indexed by
via the correspondence
we have
This last sum simplifies to the indicator
corresponding to the fact that if
then
for
and a random permutation in
must have
some cycle structure
.
The moments of
follow immediately as
We note for future reference that (1.4) can also be written in
the form
Where the
are independent Poisson

distribution random
variables that satisfy
The marginal distribution of cycle counts
provides a formula
for the joint distribution of the c
ycle counts
we find the
distribution of
using a combinatorial approach combined
with the inclusion

exclusion formula.
Lemma 1.8
.
For
Proof.
Consider the set
of all possible cycles of length
formed with elements chosen from
so that
. For each
consider the “pr
operty”
of
having
that is,
is the set of permutations
such that
is one of the cycles of
We then have
since the elements of
not in
must be permuted among themselves.
To use the inclusion

exclusion formula we need to calculate the term
which is
the sum of the probabilities of the

fold intersection of
properties, summing over all sets of
distinct properties.
There are two cases to consider. If the
properties a
re
indexed by
cycles having no elements in common, then the
intersection specifies how
elements are moved by the
permutation, and there are
permutations
in the intersection. Th
ere are
such intersections.
For the other case, some t wo distinct properties name some
element in common, so no permutation can have both these
properties, and the

fold intersection is empty. Thus
Finally, the inclusion

exclusion series for the number of
permutations having exactly
properties is
Which simplifies to (1.1
)
Returning to the original hat

check
prob
lem, we
substitute j=1 in (1.1
) to obtain the distribution of
the number of fixed points of a random permutation. For
and the moments of
follow from (1.
2
) with
In
particular, for
the mean and variance of
are both
equal to 1.
The joint distribution of
for any
has an expression similar to (1.7); this to
o can be
derived by inclusion

exclusion. For any
with
The joint moments of the first
counts
can be
obtained dir
ectly
from (1.2) and (1.3
) by setting
The limit distribution of cycle counts
It follows immediately from Lemma 1.2 that for each fixed
as
So that
converges in distribution to a random variable
having a Poisson distribution with mean
we use the
notation
where
to descri
be
this. Infact, the limit random variables are independent.
Theorem 1.6
The process of cycle counts converges in
distribution to a Poisson process of
with intensity
.
That is, as
Where the
are independent Poisson

distributed random variables with
Proof.
To establish the converges in distribution one shows
that for each fi
xed
as
Error rates
The proof of Theorem
says nothing about the rate of
convergence. Elementary analysis can be used to estimate this
rate when
. Using properties of alternating series with
decreasing terms, for
It follows that
Since
We see from (1.11) that the total variation distance between
the distribution
of
and the distribution
of
Establish the asymptotics of
under conditions
and
where
and
as
for some
We start with the expression
and
Where
refers to the quantity derived from
. It
thus follows that
for a constant
, depending on
and the
and computable explicitly
from (1.1)
–
(1.3), if Conditions
and
are satisfied
and if
from some
since, under these
circumstances, both
and
tend
to zero as
In particular, for polynomials and square
free polynomials, the relative
error in this asymptotic
approximation is of order
if
For
and
with
Where
under Conditions
and
Since, by the Conditioning Relation,
It follows by direct calculation
that
Suppressing the argument
from now on, we thus obtain
The first sum is at most
the third is bound by
Hence we may take
Required order under Conditions
and
if
If not,
can be replaced by
in the above, which has the required order, without the
restriction on the
implied
by
. Examining the
Conditions
and
it is perhaps surprising to
find that
is required instead of just
that is, that
we should need
to hold for some
. A first observation is that a similar problem arises
with the rate of decay of
as well. For this reason,
is
replaced by
. This makes it possible to replace condition
by the weaker pair of conditions
and
in the
eventual assumptions needed for
to be of order
the decay rate requirement of order
is
shifted from
itself to its first difference. This is needed to
obtain the right approximat ion
error for the random mappings
example. However, since all the classical applications make
far more stringent assumptions about the
than are
made in
. The crit ical point of the proof is seen where
the ini
tial estimate of the difference
. The factor
which should be small, contains a far tail element from
of
the form
which is only small if
being otherwise of order
for any
since
is in any case assumed. For
this gives rise
to a contribution of order
in the estimate of the
difference
which, in the
remainder of the proof, is translated into a contribution of
order
for differences of the form
finally leading to a
contribution of order
for any
in
Some improvement would seem to be possible, defining the
function
by
differences that are
of the form
can be directly
estimated, at a cost of only a single contribution of the form
Then, iterating the cycle, in which one
estimate of a difference in point probabilitie
s is improved to
an estimate of smaller order, a bound of the form
for any
could perhaps be attained, leading to a final error
estimate in order
for any
, to
replace
This would be of the ideal order
for large enough
but would still be coarser for
small
With
and
as in the previous section, we wish to show
that
Where
for any
under Condit ions
and
with
.
The proof uses sharper estimates. As before, we begin with the
formula
Now we observe that
We have
The approximation in (1.2
) is further simplified by noting that
and then by observing that
Combining the contributions of (1.
2)
–
(1.3
), we thus find tha
The quantity
is seen to be of the order claimed
under Conditions
and
, provided that
this supplementary condition can be removed if
is replaced by
in the definit ion of
, has the required order without the restriction on
the
implied by assuming that
Final
ly, a direct
calculation now shows that
Example 1.
0
.
Consider the point
. For
an arbitrary vector
, the coordinates of the point
are equal to the respective coordinates of the vector
and
. The vector r such as
in the example is called the position vector or the radius vector
of the point
. (Or, in greater detail:
is the radius

vector of
w.r.t an origin O). Points are frequently specified by their
radius

vectors. This presupposes the choice of O as the
“standard origin”. Let us summar
ize. We have considered
and interpreted its elements in t wo ways: as points and as
vectors. Hence we may say that we leading with the two
copies of
= {points},
= {vectors}
Operations with vectors: multiplication by a number, addition.
Operations with points and vectors: adding a vector to a point
(giving a point), subtracting two points (giving a vector).
treated in this
way is called an
n

dimensional affine space
.
(
An
“abstract” affine space is a pair of sets , the set of points and
the set of vectors so that the operations as above are defined
axiomat ically). Not ice that vectors in an affine space are also
known as “free vectors”. Intuitively, they are not fixed at
points and “float freely” in space. From
considered as an
affine space we can precede in two opposite directions:
as
an Euclidean space
as an affine s
pace
as a
manifold.Going to the left means introducing some extra
structure which will make the geomet ry richer. Going to the
right means forgetting about part of the affine structure; going
further in t
his direction will lead us to the so

called “smooth
(or differentiable) manifolds”. The theory of differential forms
does not require any extra geometry. So our natural direction
is to the right. The Euclidean structure, however, is useful for
examples and
applications. So let us say a few words about it:
Remark 1.
0
.
Euclidean geometry.
In
considered as
an affine space we can already do a good deal of geometry.
For example, we can consider lines and planes, and quadric
surfaces
like an ellipsoid. However, we cannot discuss such
things as “lengths”, “angles” or “areas” and “volumes”. To be
able to do so, we have to introduce some more definitions,
making
a Euclidean space. Namely, we define the length
of a vector
to be
After that we can also define distances between points as
follows:
One can check that the distance so defi
ned possesses natural
properties that we expect: is it always non

negative and equals
zero only for coinciding points; the distance from A to B is the
same as that from B to A (symmetry); also, for three points, A,
B and C, we have
(the
“triangle inequality”). To define angles, we first introduce the
scalar product of two vectors
Thus
. The scalar product is also denote by dot:
, and hence is often
referred to as the “dot
product” . Now, for nonzero vectors, we define the angle
between them by the equality
The angle itself is defined up to an integral mult iple
of
. For this definition to be consistent we have to ensure
that the r.h.s. of (
4
) does not exceed 1 by the absolute value.
This follows from the inequality
known as the Cauchy
–
Bunyakovsky
–
Schwarz inequality
(various combinations
of these three names are applied in
different books
). One of the ways of proving (5
) is to consider
the scalar square of the linear combination
where
. As
is a quadratic polyn
omial in
which is never negative, its discriminant must be less or
equal zero. W
rit ing this explicitly yields (5
). The triangle
inequality for distances als
o follows from the inequality (5
).
Example 1.1.
Consider the function
(the i

th
coordinate). The linear function
(the differential of
)
applied to an arbitrary vector
is simply
.From these
examples follows that we can rewrite
as
which is the standard form. Once again: the partial
derivatives
in (1
) are just the coefficients (depending on
);
are linear functions giving on an arbitrary vector
its
coordinates
respectively. Hence
Theorem 1.
7
.
Suppose we have a parametrized curve
passing through
at
and with the
velocity vector
Then
Proof.
Indeed, consider a small increment of the parameter
, Where
. On the other hand, we
have
for an
arbitrary vector
, where
when
.
Combining it together, for the increment of
we
obtain
For a certain
such that
when
(we used the linearity of
). By the definit ion, this
means that the derivative of
at
is exactly
. The statement of the theorem can be expressed
by a simple formula:
T
o calculate the value Of
at a point
on a given vector
one can take an arbitrary curve passing Through
at
with
as the velocity vector at
and calculate the usual
derivative of
at
.
Theorem 1.8
.
For functions
,
Proof. Consider an arbitrary point
and an arbitrary vector
stretching from it. Let a curve
be such that
and
.
Hence
at
and
at
Formulae (1) and (2
) then immediately follow from
the corresponding formulae for the usual derivative Now,
almost without change the theory generalizes to functions
taking values in
instead of
. The only difference is
that now the differential of a map
at a point
will be a linear function taking vectors in
to vectors in
(instead of
) . For an arbitrary vector
+
Where
when
. We have
and
In this matrix notation we have to write vectors as vector

columns.
Theorem 1.
9
. For an arbitrary parametrized curve
in
, the differential of a
map
(where
) maps the velocity vector
to the velocity
vector of the curve
in
Proof.
By the definition of the velocity vector,
Where
when
. By the definition of the
differential,
Where
when
. we obtain
For some
when
. This precisely means
that
is the velocity vector of
. As every
vec
tor attached to a point can be viewed as the velocity vector
of some curve passing through this point, this theorem gives a
clear geometric picture of
as a linear map on vectors.
Theorem 1.
10
Suppose we have two maps
and
where
(open
domains). Let
. Then the differential of
the composite map
is the composition of the
differentials of
and
Proof.
We can use the description of the differential
.
Consider a curve
in
with the velocity vector
.
Basically, we need to know to which vector in
it is taken
by
. the curve
. By the
same theorem, it equals the image under
o
f the
Anycast
Flow
vector to the curve
in
. Applying the
theorem once again, we see that the velocity vector to the
curve
is the image under
of the vector
.
Hence
for an arbit rary vector
.
Corollary 1.0
.
If we denote coordinates in
by
and in
by
, and write
Then the chain rule can be expressed as follows:
Where
are taken from (
1). In other words, to get
we have to substitute into (
2) the expression for
from (3
). This can also be expressed by the
following matrix formula:
i.e., if
and
are expressed by matrices of part ial
derivatives, then
is expressed by the product of
these matrices. This is often written as
Or
Where it is assumed that the dependence o
f
on
is given by the map
, the dependence of
on
is given by the map
and the dependence of
on
is given by the composition
.
Definiti on 1.6
.
Consider an open domain
. Consider
also another copy of
, denoted for distinction
, with
the standard coordinates
. A system of coordinates
in the open domain
is given by a map
where
is an open domain of
, such that the
following three conditions are satisfied :
(1)
is smooth;
(2)
is invertible;
(3)
is also smooth
The coordinates of a point
in this system are the
standard coordinates of
In other words,
Here the variables
are the “new” coordinates of
the point
Example 1.2
.
Consider a curve in
specified in polar
coordinates as
We can simply use the chain rule. The map
can be
considered as the composition of the maps
. Then, by the chain
rule, we have
Here
and
are scalar coefficients de
pending on
,
whence the partial derivatives
are vectors
depending on point in
. We can compare this with the
formula in the “standard” coordinates:
.
Consider the vectors
. Explicitly we have
From where it follows that these vectors make a basis at all
points except for the origin (where
). It is instructive to
sketch a
picture, drawing vectors corresponding to a point as
starting from that point. Notice that
are,
respectively, the velocity vectors for the curves
and
. We can
conclude that for an arbitrary curve given in polar coordinates
the velocity vector will have components
if as a basis
we take
A characteristic feature of the
basis
is that it is not
“constant” but depends on point. Vectors “stuck to points”
when we consider curvilinear coordinates.
Proposition 1.3
.
The velocity vector has the same
appearance in all coordinate systems.
Proof.
Follows directly from the chain rule and the
transformation law for the basis
.In particular, the elements
of the basis
(originally, a formal notation) can be
understood directly as the velocity vectors of the coordinate
lines
(all coordinates but
are fixed).
Since we now know how to handle velocities in arbitrary
coor
dinates, the best way to treat the differential of a map
is by its action on the velocity vectors. By
definition, we set
Now
is a linear map that takes vectors attached to a
point
to vectors attached to the point
In particular, for the differential of a function we always have
Where
are arbitrary coordinates. The form of the
differential does not change when we perform a change of
coordinates.
Example 1
.3
Consider a 1

form in
given in the
standard coordinates:
In the polar coordinates we will have
, hence
Substituting into
, we get
Hence
is the formula for
in the polar
coordinates. In particular, we see that this is again a 1

form, a
linear combination of the differentials of coordinates with
functions as coefficients. Secondly, in a more conceptual way,
we can define a 1

form in a domain
as a linear function on
vectors at every point of
:
If
, where
. Recall that the
differentials of functions were defined as linear funct
ions on
vectors (at every point), and
at every point
.
Theorem 1.9
.
For arbit rary 1

form
and path
, the
integral
does not change if we change parametrizat ion of
provide the orientation remains the same.
Proof:
Consider
and
As
=
Let
be a rational prime and let
We write
for
or this section. Recall that
has degree
over
We wish to show that
Note that
is a root of
and thus is an algebraic
integer; since
is a r
ing we have that
We
give a proof without assuming unique factorizat ion of ideals.
We begin with some norm and trace computations. Let
be
an integer. If
is not divisible by
then
is a primit ive
root of unity, and thus its conjugates are
Therefore
If
does divide
then
so it has only the one
conjugate 1, and
By linearity of the
trace, we find that
We also need to compute the norm of
. For this, we use
the factorization
Plugging in
shows that
Since the
are the conjugates of
this shows
that
The key result for determining the
ring of integers
is the following.
LEMMA 1.
9
Proof.
We saw above that
is a mult iple of
in
so the inclusion
is immediate.
Suppose now that the inclusion is strict. Since
is an ideal of
containing
and
is
a maximal ideal of
, we must have
Thus we can write
For some
That is,
is a unit in
COROLLARY 1.1
For any
PROOF. We have
Where the
are the complex embeddings of
(which we
are really viewing as automorphisms of
) with the usual
ordering. Furthermore,
is a multiple of
in
for every
T
hus
Since the trace is also a
rational integer.
PROPOSITION 1.4
Let
be a prime number and let
be the
cyclotomic field. Then
Thus
is an
integral basis for
.
PROOF. Let
and write
With
Then
By the linearity of the trace and our above calculations we find
that
We also have
so
Next consider the
algebraic integer
This is an
algebraic in
teger since
is. The same argument as
above shows that
and continuing in this way we find
that all of the
are in
. This completes the proof.
Example
1.
4
Let
, then the local ring
is simply
the subring of
of rational numbers with denominator
relatively prime to
. Note that this ring
is not the
ring
of

adic integers; to get
one must complete
. The usefulness of
comes from the fact that it h
as
a particularly simple ideal structure. Let
be any proper ideal
of
and consider the ideal
of
We claim
that
That is, that
is generated by the
elements of
in
It is clear from the definition of an
ideal that
To prove the other inclusion,
let
be any element of
. Then we can write
where
and
In particular,
(since
and
is an ideal), so
and
so
Since
this implies that
as claimed.We can use this
fact to determine all of the ideals of
Let
be any ideal
of
and consider the ideal factorization of
in
write it as
For some
and some ideal
relat ively prime to
we claim first that
We now find that
Since
Thus every ideal of
has the form
for some
it follows immediately that
is noetherian. It i
s also now
clear that
is the unique non

zero prime ideal in
. Furthermore, the inclusion
Since
this map is also surjection, since the
residue class of
(with
and
) is
the image of
in
which makes sense since
is
invertible in
Thus the map is an isomorphis m. In
particular, it is now abundantly clear that every non

zero
prime ideal of
is maximal.
To show that
is a
Ded
ekind domain, it remains to show that it is integrally
closed in
. So let
be a root of a polynomial with
coefficients in
write this polynomial as
With
and
Set
Multiplying by
we find that
is the root of a monic polynomial with coefficients in
Thus
since
we have
. Thus
is integrally close in
COROLLARY
1.
2. Let
be a number field of
degree
and let
be in
then
PROOF. We assume a bit more Galois theory than usual for
this proof. Assume first that
is Galois. Let
be an
element of
It is clear that
since
this shows
that
. Taking the product
over all
we have
Since
is
a rational integer and
is a free

module of rank
W
ill have order
therefore
This completes the proof. In the general case, let
be the
Galois closure of
and set
F.
Control
for Climbing Motion
The robot can move in two different modes. The first
mode is
a stepping mode. In the stepping mode, the order
of the
grippers is constant. For example, the upper gripper
can be
moved but it will stay above the other gripper. The
second
mode is a flipping mode. For example, the robot is
flipping
and the upper gripper is moved downwards
passing the other
gripper.
The control software is programmed as a state
machine.
Every stepping mode is implemented as state. As
described before, the cal
culation of the kinematics is
different
due to the gripper, which is fixed on the
substrate. When both
grippers are locked on the substrate,
another state is needed.
There are state for a flipping
motion, for a stepping motion, a
state for the two

gripperl
ocking,
and an initializing state.
G.
Functional Tests
First qualitative test have been done. The climbing
motion is
very stable. The robot climbs ascending and
descending in a
flipping mode and a stepping mode.
Those modes could be
interpreted as gaits. One
gait is a
stepwise climbing as
describes before. The other gait is a
flipping motion pattern
.
In
the flipping gait the power consumption is about 14
Watts at a
climbing speed of 0.7m/s. The maximum step
length is about
510 mm. The robot is able to overste
p
obstacles with a height
of 100 mm.
V.
CONCLUSION
The modular robot system consists of different
heterogenic
types of modules, passive connector elements,
control
hardware, power supply. Due to the modular
concept it is
possible to configure different set
ups of the
robot.
In this way,
the mechanics of the robot can be adapted
to the requirements
of the climbing task. In comparison to
other climbing robots, a
generalist system is realized. The
system is remote

controlled
by the user via game pad. Its
mass d
epends on the
configuration and is in the range of
m = 1...2 kg. A sensory
system is capable of being
integrated for detecting the contact
between robot and
substrate. Safety and robustness of the
locking on the
substrate can be controlled.
A reference sys
tem
is built, capable to climb pipe

like
substrates. Servo drives are
suitable for this design of
robots. In future the system will be
enhanced, to be able to
climb not only on pipe

like structures,
but also on flat
surfaces. Different modules could be com
bined
to
climbing robots, optimized to different applications. In
this
way, robot configuration could be tested for service
robots, for
industrial robots or for scientific robots.
A.
Authors and Affiliations
Dr Akash Singh is working with IBM Corporation a
s an IT
Architect and has been designing Mission Critical System and
Service Solutions; He has published papers in IEEE and other
International Conferences and Journals.
He joined IBM in Jul 2003 as a IT Architect which
conducts research and design of High
Performance Smart Grid
Services and Systems and design mission critical architecture
for High Performance Computing
Platform
and Computational
Intelligence and High Speed Communication systems. He is a
member of IEEE (Institute for Electrical and Electron
ics
Engineers), the AAAI (Association for the Advancement of
Artificial Intelligence) and the AACR (American Association
for Cancer Research). He is the recipient of numerous awards
from World Congress in Computer Science, Computer
Engineering and Applied
Computing 2010, 2011, and IP
Multimedia System 2008 and Billing and Roaming 2008. He is
active research in the field of Artificial Intelligence and
advancement in Medical Systems. He is in Industry for 18
Years where he performed various
roles
to provide t
he
Leadership in Information Technology and Cutting edge
Technology.
VI.
REFERENCES
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Yamato,
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microactuator powered by
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T. Kitamori, “An actuated pump on

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