Computer Science and Engineering 2013, 3(2): 15-23
DOI: 10.5923/j.computer.20130302.01
Application of the Structural Analysis on a Three Axis
Manipulator Robot
Yasmine Derdour1,* , Hafid Haffaf1 , Kheireddine Bouazza2
1
Department of Computer, University of Oran, Faculty of Science, Oran, 31000, Algeria, Riir
Department of Computer, University of Oran, Faculty of Science, Oran, 31000, Algeria, Litio
2
Abstract The main objective is to show the efficiency of the structural analysis in the field of the mon itoring and the
fault detection by taking into account the constraints of the real world, and consequently, to know exact ly at wh ich level
and at which mo ment it will be necessary to intervene. The problem of fault detection (FD) is to check for the presence of
crash. When no fault is present in the system, it is in the safe mode, otherwise it is in the faulty mode. In the first case, the
analytical residual redundancy relations (ARRs) vanish, in the second case, they are different fro m zero if some fault arises,
and we have to decide among a nu mber of fau lty modes, wh ich mode is incriminated, this is the isolation process (FI) Th is
paper presents a three axis manipulator robot on which we apply the structural analysis for fault diagnosis purpose.
Structural analysis offers the opportunity to transform an analytical model into a structural model, so we can describe it
using an incidence matrix which allows representing components of the system. Fro m the structural graph obtained, we
apply the matching algorith m to extract analytical redundancy relations gathered in a signature failures table. Based on
“detectability” and “localizab ility” properties, the aim of the method is to identify the main variables which are influential
and dependent on the evolution of the system.
Keywords Structural Analysis, Monitoring, Fau lt Detection and Isolation, Matching, Bipart ite Graph, Incidence Matrix,
Table of Signature, Residue, Analytical Relations of Redundancies
1. Introduction
At present, the competition between manufacturers
became more and more rough to satisfy the requirements of
their customers in order to offer the best quality of products
and services which exists. This competition obliges
increasingly to integrate the concept of monitoring at every
level. In order to achieve this objective, manufacturers try
today more than ever to increase the productivity of their
companies; this by reducing the costs and production time
and by improving manufacturing quality. The ceaseless
modernizat ion of production tools makes industrial systems
increasingly co mplex and sophisticated. In parallel, a
greater request of reliability, availability, reconfigurability
and safety became real challenges in the third millenniu m
that industrial system ought to verify.
The development of co mputing, financial accessibility and
co mp u t in g p o wer o f co mp u t ers mad e p o s s ib le t h e
develop ment of diagnosis methods such as the structural
analysis. Ho wever, industrial systems are designed much
more co mp lex and the failure of one co mponent can lead to
dysfunction of the whole system. Th is can have major effects
* Corresponding author:
derdouryasmine@yahoo. fr (Yasmine Derdour)
Published online at http://journal.sapub.org/computer
Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved
on the availability and performance of the system so leading
industry efforts to failure. It can even cause damage
tomachines or hu man especially when these systems appear
in many processes with risk such as nuclear and chemical
ones. The improvement of their safety is based on the Fault
detection and isolation (FDI) procedures[1]. A fault in a
system may affect the actuators, the sensors or the
components. Different approaches for the design of FDI
procedures have been developed, on the basis of the kind of
knowledge used to describe the plant. The FDI methodology,
when applied to detect the failures by using the structural
analysis, is the use of the structural model. The later
represents only relationships between components
neglecting the values of variables and parameters[2][3]. The
structural model of a system is an abstraction of its
behavioural model in the sense that only the structure of the
system is important. Among those properties, we can cite:
controllability, observability, fault detectability or
monitorability, sensor placement objectives,[4]. The FDI
procedure is essentially based on generating Analytical
Redundancy Relations (ARRs) in order to compare the
actual behaviour with reference behaviour in monitoring
process[5]. In the other hand, control of robot is topical issue
in automat ic process, and the ability of monitoring sensitive
components is a main object ive in design process. For this
raison, the estimate of a process state quality and its
dependability are h ighly conditioned by the number and
16
Yasmine Derdour et al.: Application of the Structural Analysis on a Three Axis M anipulator Robot
distribution of measurements over it. The instru mentation
architecture design of a system represents a very important
step. The used analytical model can be given under structural
or state space equation form [6]. A bipart ite graph is then
generated from the mathematical model corresponding to the
three axis robot in order to perform structural analysis, which
in turn, boils down to matching characterizat ion.
The innovative interest of this paper is to apply this
structural method in robotics, a three axis manipulator robot
is considered, on which all the steps of structural analysis are
carried out: we apply the matching on a bipart ite graph, and
thus, generate ARRs fro m this matching which are gathered
on signature table. After giving the model of the robot
(section 5), the basic concepts of structural analysis are
recalled in section 6, and then in section 7 and 8, the
application of the analytical redundancy relations generation
steps.
These redundancy relations are then satisfied in the
normal mode, and not satisfied in the case of presence of
failure[6].
2. Monitoring Concepts
To detect locate errors, the anomalies and dysfunctions,
the surveillance includes all the tools allo wing to control the
evolution of the behavior of production system with regard
to its normal functioning. The surveillance is the base of an
excellent safety of functioning of technological processes, it
constitutes an interface between the operator and the
physical installat ion. Its role is to provide information on the
actual state (correct or incorrect) of mon itored devices, to
validate the in formation fro m the sensors and to locate the
failing co mponents. The FDI based methodology is divided
in two stages: design phase to obtain ARRs fro m the
structural (graphical) model, and the explo itation phase
where these residuals are on line analy zed.
Let us now introduce some basic concepts of
SURVEILLA NCE (SUPERVISION)[7]
Failure. (undesirable) : functional ano maly in a physical
system. A failure is a consequence of a risk that materializes.
Error or dysfunction : this is the part of the system that no
longer meets the need required, and can eventually cause
failure.
Breakdown :inability of a device to perform its function.
Safety of functioning : It is the technical and human
reliability of a process. The design of a system sure must
assess the risk of hardware failu re, software and human, and
environmental conditions.
Reliability: ability to adequately perform the task in
normal conditions, during a given time interval.
Availability: (at any time) the availability is defined as
"the ability of an entity to be able to accomplish required
function in given conditions, at a given time or during a
given time interval, assuming the provision of external
resources necessary maintenance is assured.
Maintainability or possibility of restoring: according to
the AFNOR standard is "in the given conditions of use, the
ability of an entity to be maintained or restored, on a given
time interval, in a state in which it can perform a required
function, when maintenance is performed under given
conditions, with prescribed procedures and resources."
Security: the system's ability to avoid appear, in g iven
conditions, catastrophic events or the insurance system to
resist unauthorized or incorrect entrance and to be able to
report them.
3. Conception of a Monitoring System
The variables are of two kinds needed in the system:
a) The entrances
• Knowledge of the operation of the system to monitor
• Monitoring System Specifications
b) The outputs
• Detection, localization, diagnosis of failures
• Algorith ms variables that must be acquired to enable it
to fulfill the specifications
• Characterization of the system designed to assess its
performance[7].
In the monitored system, the decision-making between the
presence or absence of the errors is a problem of fau lt
detection (FD). When no error has been made in the system,
this last one is called " in normal mode ", otherwise it is
called " in degraded mode". The problem of fau lts isolation
is to determine the presence of degraded mode between
modes and possible states of the system.
4. Principle of Fault Detection and
Isolation (FDI)
For thirty years, the fault detection and isolation (FDI)
continues to grow in importance as a research topic,
especially within the co mmun ities of Automat ic Co mmand
and Artificial intelligence[8][9]. The FDI is based on the
following steps:
The diagnosis bases on an explicit model of the normal
behavior of the system.
The faults isolation is based on the analysis of sets
components (faults) involved in each detected incoherencies.
• A defect is detected fro m the incoherence between the
observed behavior and the behavior predicted by the model
• The faults isolation is based on the analysis of sets
components (faults) involved in each detected incoherence.
Nu merous methods are on the basis of the works of
detecting and locating defects, and there are constantly new
methods proposed in the literature[10]. Most of these
methods can be viewed as variations or imp rovements of
previous ones. This section is intended as a not exhaustive
review o f these methods, among which it is sometimes
difficult to determine which it is better necessary to use[7].
Nevertheless, we can retain three operations:
• Detection: decide if the system is faulty.
Computer Science and Engineering 2013, 3(2): 15-23
• Localization: to determine which part of the system is
affected.
• Identification: determine the extent of the defect.
The FDI methods differs not only in the way in wh ich
knowledge about the process being monitored is used but
also on[7] the nature the required knowledge. In general, the
methods are divided into two major families fo llo wing:
4.1. Approaches Based on Models
They are based on knowledge we get of the normal
behavior of the system represented by a specific model,
which can be quantitative or qualitative. The first model is
based on fundamental physical princip les expressed in the
form of mathemat ical equations, while the second is based
on the system structure and the links between co mponents,
expressed for examp le in the form of logical relat ions. The
principle of these approaches consists in comparing the
informat ion fro m the system in real time with those from the
model to detect differences.
4.2. Approaches wi th Historical Data (or wi thout
Models)
They are based on knowledge obrained from past
experiences (we shall also take into account superficial
knowledge based on the history story of the process). In
these approaches, an analysis is made on the records of the
past experiences in order to extract the characteristics of the
system to be monitored to ach ieve the observed symptoms
lin ked to the corresponding defects.
Techniques based model and techniques without model
make both use of redundancy information relative to the
system. This can be obtained fro m data of the system during
previous functioning in normal mode, either fro m the
knowledge wh ich we have on the system (analyt ical model).
The redundancy can result thus either fro m entrances. We are
interested in this article to structural analysis that is a
method-based model.
The structural analysis is a tool o f a collective reflect ion. It
offers the possibility of translating an analytical model into
structural model then it describes this model using an
incidence matrix linking all the co mponents of this system.
Fro m there, the structural graph can be obtained. We make
then the matching to be able to loosen analytical redundancy
relations which we shall classify in the signature failures
table. And at the end, we can apply the FDI. Leav ing of this
description, this method has for object to create the main
influential and dependent variables and thus the essential
variables in the evolution of the system.
We shall discuss the approaches which are interested in
the fault detection and isolation of the defects ( FDI) in a
system, especially the methods which use a behavioral
model. This model consists of a representation of system
behavior in the form of mathematical relations ( analytical
model).
17
5. Modelling of Dynamic Model of the
Robot[11]
The model is represented by the following equation:
𝛤𝛤 = 𝑀𝑀 (𝑞𝑞) 𝑞𝑞̈ + 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ )𝑞𝑞̇ + 𝐺𝐺 (𝑞𝑞) + 𝑓𝑓𝑣𝑣 𝑞𝑞̇ + 𝑓𝑓𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝑞𝑞̇ ) (1)
Where𝑞𝑞̈ is the n×1 accelerat ions vector, 𝑞𝑞̇ is the n×1
velocities vector, q is the n×1 coordinates vector and 𝛤𝛤 is
the n×1 external torques vector. 𝑀𝑀 (𝑞𝑞) represents the n×n
positive definite inertial matrix, 𝐶𝐶 (𝑞𝑞, 𝑞𝑞̇ ) is the n×n Coriolis
and centrifugal fo rces matrix and G(q) is the n×1
gravitational torques vector. fv is the n×n viscous frictions
diagonal matrix, and fs is the Coulo mb frictions diagonal
matrix, but these matrix will be ignored throughout our work.
And therefore we obtain the following model:[11]
𝛤𝛤 = 𝑀𝑀 (𝑞𝑞) 𝑞𝑞̈ + 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ )𝑞𝑞̇ + 𝐺𝐺 (𝑞𝑞)
(2)
Without loss of generality, the inert ial matrix of the robot
is given by:
0
𝑀𝑀11 𝑀𝑀12
0 �
𝑀𝑀 = �𝑀𝑀21 𝑀𝑀22
0
0
𝑀𝑀33
With:
▪ 𝑀𝑀11 = 𝑚𝑚1 𝑎𝑎21 + 𝑚𝑚2 (𝑙𝑙 21 + 𝑎𝑎22 + 2 𝑎𝑎2 𝑙𝑙 1 𝑐𝑐2) +
𝑚𝑚3 ( 𝑙𝑙 21 + 𝑙𝑙 22 + 2 𝑙𝑙 1 𝑙𝑙2 𝑐𝑐2 ) + 𝐼𝐼1𝑧𝑧𝑧𝑧 + 𝐼𝐼2𝑧𝑧𝑧𝑧 + 𝐼𝐼3𝑧𝑧𝑧𝑧
▪ 𝑀𝑀12 = +𝑀𝑀21 = 𝑚𝑚2 ( 𝑎𝑎22 + 𝑎𝑎2 𝑙𝑙1 𝑐𝑐2 ) + 𝑚𝑚3 (𝑙𝑙 22 +
𝑙𝑙1 𝑙𝑙2 𝑐𝑐 2+𝐼𝐼2𝑧𝑧𝑧𝑧+𝐼𝐼3 𝑧𝑧𝑧𝑧
▪ 𝑀𝑀22 = 𝑚𝑚2 𝑎𝑎22 + 𝑚𝑚3 𝑙𝑙 22 + 𝐼𝐼2𝑧𝑧𝑧𝑧 + 𝐼𝐼3𝑧𝑧𝑧𝑧
▪ 𝑀𝑀22 = 𝑚𝑚3
Where :
𝑚𝑚𝐼𝐼 : Mass of axis 𝑖𝑖
𝑠𝑠𝑖𝑖 : Sinus ( 𝑞𝑞𝑖𝑖 )
ci : Cosinus ( 𝑞𝑞𝑖𝑖 )
𝐼𝐼𝑖𝑖𝑖𝑖𝑖𝑖 : Inertia mo ment of joint 𝑖𝑖
The mat rix of Corio lis and Centrifugal forces is:
𝐶𝐶11 𝐶𝐶12 0
𝐶𝐶 = � 𝐶𝐶21 0 0 �
0
0 0
with
𝐶𝐶11 = −2 𝑙𝑙 1 𝑠𝑠2 ( 𝑚𝑚2 𝑎𝑎2 + 𝑚𝑚3 𝑙𝑙 2 )𝑞𝑞̇ 2
𝐶𝐶12 = −𝑙𝑙 1 𝑠𝑠2 (𝑚𝑚2 𝑎𝑎2 + 𝑚𝑚3 𝑙𝑙 2 )𝑞𝑞̇ 2
𝐶𝐶21 = 𝑙𝑙 1 𝑠𝑠2 (𝑚𝑚2 𝑎𝑎2 + 𝑚𝑚3 𝑙𝑙 2 )𝑞𝑞̇ 1
And the gravitational vector is: 𝑞𝑞̇
𝐺𝐺 = ( 0 0 − 𝑚𝑚3 𝑔𝑔) 𝑇𝑇
l1
a1
a2
l2
c2
c1
d2
Figure 1. The robot under study
18
Yasmine Derdour et al.: Application of the Structural Analysis on a Three Axis M anipulator Robot
In order to min imize, the nu mber of parameters (without
rank deficiency on observation matrix) and also to avoid the
probable cumulat ive errors in identification of each
parameter, we proceed to several grouping:[11]
𝑃𝑃1 = 𝑚𝑚1 𝑎𝑎21 + ( 𝑚𝑚2 +𝑚𝑚3 )𝑙𝑙 21 + 𝑚𝑚2 𝑎𝑎22 + 𝑚𝑚3 𝑙𝑙 22
+ 𝐼𝐼1𝑧𝑧𝑧𝑧 + 𝐼𝐼2𝑧𝑧𝑧𝑧 + 𝐼𝐼3𝑧𝑧𝑧𝑧
𝑃𝑃2 = 𝑚𝑚2 𝑎𝑎22 + 𝑚𝑚3 𝑙𝑙 22 + 𝐼𝐼2𝑧𝑧𝑧𝑧 + 𝐼𝐼3𝑧𝑧𝑧𝑧
𝑃𝑃3 = 𝑙𝑙 1 (𝑚𝑚2 𝑎𝑎2 + 𝑚𝑚3 𝑙𝑙 2 )
𝑃𝑃4 = 𝑚𝑚3
Then, we can write:
𝑝𝑝1 + 2𝑝𝑝3 𝑐𝑐2 𝑝𝑝2 + 𝑝𝑝3 𝑐𝑐2 0
𝑝𝑝2
0 � 𝑞𝑞̈ +
𝛤𝛤 = � 𝑝𝑝2 + 𝑝𝑝3 𝑐𝑐2
0
0
𝑝𝑝4
0
−2𝑝𝑝3 𝑠𝑠2 𝑞𝑞̇ 2 −𝑝𝑝3 𝑠𝑠2 𝑞𝑞̇ 2 0
� 𝑝𝑝3 𝑠𝑠2 𝑞𝑞̇ 1
(3)
0
0� 𝑞𝑞̇ + � 0 �
−𝑝𝑝4 𝑔𝑔
0
0
0
With the parameters presented in the table below:
Table 1. Robot parameters
p1
p2
p3
p4
19.1337
2.8522
4.1214
1.2568
• The Relati ons for Model of the Robot
The components of the robot are described by the
following relat ions:
−𝐶𝐶1 : 𝑞𝑞̈ 1 =
𝑎𝑎 𝛤𝛤1 + 𝑏𝑏 𝛤𝛤2 + 𝑐𝑐 𝛤𝛤3 + 𝑞𝑞̇ 1 (𝑎𝑎 8,2428 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥(3) 𝑥𝑥(4) −
𝑏𝑏 4,1214𝑠𝑠𝑖𝑖𝑛𝑛 𝑥𝑥3 𝑥𝑥 2+ 𝑐𝑐𝑚𝑚3𝑔𝑔 (4)
−𝐶𝐶2 : 𝑞𝑞̈ 2 = 𝑑𝑑 𝛤𝛤1 + 𝑒𝑒 𝛤𝛤2 + 𝑓𝑓 𝛤𝛤3 + 𝑞𝑞̇ 2
(𝑏𝑏 4,1214 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 (3) 𝑥𝑥 (4) + 𝑓𝑓𝑚𝑚3 𝑔𝑔)
(5)
(6)
−𝐶𝐶3 : 𝑞𝑞̈ 3 = 𝑔𝑔 𝛤𝛤1 + ℎ 𝛤𝛤2 + 𝑖𝑖( 𝛤𝛤3 +𝑚𝑚3 𝑔𝑔)
(7)
−𝑚𝑚1 : 𝑦𝑦1 (𝑡𝑡) = 𝑞𝑞1 (𝑡𝑡)
(8)
−𝑚𝑚2 : 𝑦𝑦2 (𝑡𝑡) = 𝑞𝑞2 (𝑡𝑡)
(9)
−𝑚𝑚3 : 𝑦𝑦3 (𝑡𝑡) = 𝑞𝑞3 (𝑡𝑡)
𝑑𝑑 𝑞𝑞1 (𝑡𝑡)
−𝑑𝑑 1 : 𝑞𝑞̇ 1 (𝑡𝑡) =
(10)
−𝑑𝑑 2 : 𝑞𝑞̈ 1 (𝑡𝑡) =
−𝑑𝑑 3 : 𝑞𝑞̇ 2 (𝑡𝑡) =
−𝑑𝑑 4 : 𝑞𝑞̈ 2 (𝑡𝑡) =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞̇ 1 (𝑡𝑡)
−𝑑𝑑 6 : 𝑞𝑞̈ 3 (𝑡𝑡) =
(12)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞3 𝑡𝑡)
(14)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞̇ 2 (𝑡𝑡 )
−𝑑𝑑 5 : 𝑞𝑞̇ 3 (𝑡𝑡) =
(11)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞2 (𝑡𝑡 )
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞̇ 3 (𝑡𝑡)
𝑑𝑑𝑑𝑑
(13)
(15)
The analytical model of the robot is a set of:
-Three constraints: 𝐶𝐶1 , 𝐶𝐶2 and 𝐶𝐶3
-Three measures: 𝑚𝑚1 , 𝑚𝑚2 and 𝑚𝑚3
-Six derivations : 𝑑𝑑 1 , 𝑑𝑑 2 , 𝑑𝑑 3 , 𝑑𝑑 4 , 𝑑𝑑 5 and 𝑑𝑑 6
This model connects nine unknown variables; it implies
that we can generate three relations of analytical redundancy.
It includes the follo wing co mponents:
-The inputs 𝛤𝛤1 , 𝛤𝛤2 and 𝛤𝛤3 of the three axis of robot.
-The position, the speed and acceleration ( 𝑞𝑞1 , 𝑞𝑞̇1 and 𝑞𝑞1̈ )
for axis1, and so on for axis 2 and axis 3.
- 𝑦𝑦: output,
- 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 , 𝑒𝑒, 𝑓𝑓, 𝑔𝑔, ℎ, 𝑖𝑖: coefficients,
- 𝑔𝑔 : gravity.
Industrial systems like those in Robotic much more
complex designed and the failure o f one component can lead
to the dysfunction of the whole system. This can have major
effects on the availab ility and performance o f the system
leading the industrialists’ efforts to failure. It can even cause
damage to machines or human.
The need for safety of functioning and monitoring became
crucial to meet the performance objectives and above all
safety of technological processes such as robots especially
for critical systems such as nuclear, aeronautical systems...
This is why there are several methods to overcome these
problems and ensure proper monitoring.
6. Structural Analysis
The structural analysis is a method of the pretreatment;
this pretreatment is effective even for the most co mplex
automated systems. This method is concerned with the
properties of the system structure model, and it’s considered
as a powerful tool which determinates these properties.
Thanks to graph theory, these properties are obtained
starting from the only knowledge of the existence of links
(constraints) between variables, without need the parameters
values[12].
The structural analysis, by the knowledge of the structural
conditions of observability or o f co mmandability, allo ws the
detection and localizat ion of the failures or the study of
reconfigurability.
The structural properties under study are[2]:
• The identification of the mon itorable part of the system
that is the subset of the system components of this system of
whose faults can be detected and isolated.
•The possibility to design the residuals which meet some
specific fau lt diagnosis requirements, namely which are
simp ly robust, (insensitive to disturbances and uncertainties),
and structured (sensitive to certain faults and insensitive to
others).
• The existence of reconfiguration possibilities in order to
estimate (respectively to control) some variable o f interest
in case of sensor, actuator or system component failures[2].
Answers to these questions are provided by the analysis of
the structural graph of the system.
The main purpose of representing the system as a
structural graph is to obtain the knowledge of the subsystem
with inherent redundant informat ion that exists within the
system. Th is part can be analy zed in detail, and redundant
informat ion can then be used for fault detection and isolation
(FDI) purpose.
6.1. Structural Model Representati on
The structural model of a system as a bi-partite graph
which represents the links between a set of variables and a
set of constraints . It is an abstraction of the behavior model,
because it merely describes which variab les are connected by
which constraints, but it doesn’t say how these constraints
look like. Hence, the structural model p resents the basic
Computer Science and Engineering 2013, 3(2): 15-23
features and properties of a system that are independent of its
parameters.
The system’s structural model is[13] represented by the
set of relations F= {f1 ,f2 ….fm } and the set of variables Z=
K∪X = {z1 ,z2 ….zn }.X is the set of unknown variables and
K=U ∪Y is the set of known variables where input/reference
signals (U),and measured signals(Y) [13].
The set of constraints F is separated into Fk, wh ich we only
apply to known variables, and Fx=F/Fk that is the set of those
constraints that include at least one unknown variable.
The types of variables in a diagnostic context are: the
known variab les corresponding to measurements and
controller input. The unknown variables, typically internal
states and unknown inputs should not influence the residual,
and the faults to be detected. Formally, the structural model
of the system is defined as follows:
R = {R1 , R2 ,..., Rm }a set of structural equations.
K = {k1 , k 2 ,..., k c } the set of the known variab les.
X = {x1 , x2 ,..., xn } the set of the unknown variables.
Z = X ∪ K is the set of all the variab les. Z = n + c A
constraint R impose a relation between variables and
parameters, belonging to
R j z1 , z 2 ,...z Z = 0
Z:
; j = 1, m.
These relations can represent dynamic, static, linear or
non-linear relat ions[13].
This model is then represented graphically by
bipartite graph or in an equivalent way, by an incidence
(
)
19
matrix.
6.2. Bi-Partite Graph
The structural model of the system (C, Z) is a bi-part ite
graph (C, Z, ℰ ) where ℰ ⊂ C × Z is the set of edges defined
by: �ci , zj � ∈ ℰ if the variable zj appers in the constraint ci .
Note that the bi-partite graph is an undirected graph,
which can be interpreted as follows: all variables and
parameters connected with a g iven constraint vertex has to
satisfy the equation or rule this vertex presents. This graph
allo ws representing the structure of the system rather general
models including both differential and algebraic constraints.
6.3. Inci dence Matri x
An incidence matrix is a matrix that shows the relat ionship
between two classes of objects. If the first class is C and the
second is 𝑍𝑍, the matrix has one row for each element of C
and one column for each element of 𝑍𝑍[14].The entry in row
c and column z is 1 if c and z are related (called incident in
this context) and 0 if they are not.
With C: Constraint, 𝑑𝑑 : Derivation and 𝑚𝑚: Measure
-The matching edges are indicated by gray boxes.
- 𝑥𝑥: is used to forbid the integral causality because the
initial values of variables are unknown.
-The arrows indicate the constraints not saturated by the
matching which are 𝐶𝐶1 , 𝐶𝐶2 and 𝐶𝐶3.
Table 2. Incidence matrix of the robot
𝐶𝐶1
𝐶𝐶2
𝐶𝐶3
𝑑𝑑1
Γ1
1
Γ2
1
Γ3
1
1
1
1
1
1
1
𝑞𝑞1
𝑞𝑞2
x
𝑞𝑞̈ 1
1
𝑞𝑞̈ 2
𝑞𝑞̈ 3
𝑦𝑦1
𝑦𝑦2
𝑦𝑦3
1
x
x
1
1
x
𝑑𝑑5
𝑚𝑚3
𝑞𝑞̇ 3
1
𝑑𝑑4
𝑚𝑚2
1
𝑞𝑞̇ 2
1
𝑑𝑑3
𝑚𝑚1
𝑞𝑞̇ 1
1
𝑑𝑑2
𝑑𝑑6
𝑞𝑞3
x
1
1
x
1
1
1
1
1
1
1
20
Yasmine Derdour et al.: Application of the Structural Analysis on a Three Axis M anipulator Robot
6.4. The Matching Concept[13]
The ultimate aim of representing the system in terms of
structured graph is to obtain knowledge about the
parts/subsystems with inherent redundant information that
exists within the system. These parts can be analyzed in
detail and the redundant informat ion can then be
man ipulated for FDI purpose[13]. A matching is a causal
assignment which associates unknown system variab les with
the system constraints fro m wh ich they can be calculated.
Unknown variables which cannot be matched cannot be
calculated. Variables which can be matched in several ways
can be determined in different (redundant) manners, which
provide a means for fault detection and possibility for
reconfiguration[2].
Definition[2]: Let (𝐶𝐶, 𝑍𝑍, ℰ ) be a bi-partite graph, 𝑒𝑒 ∈ ℰ ,
𝑒𝑒 = (𝛼𝛼, 𝛽𝛽) be an edge which lin ks the constraint 𝛼𝛼 and the
variable 𝛽𝛽, and 𝑝𝑝𝑐𝑐 and 𝑝𝑝𝑧𝑧 be two projections.
𝑝𝑝𝑐𝑐 : ℰ → 𝐶𝐶
𝑒𝑒 ⟼ 𝑝𝑝𝑐𝑐 (𝑒𝑒) = 𝛼𝛼
𝑝𝑝𝑧𝑧 : ℰ → 𝑍𝑍
𝑒𝑒 ⟼ 𝑝𝑝𝑍𝑍 (𝑒𝑒) = 𝛽𝛽
The projection of the edge on the constraint set is
𝑝𝑝𝑐𝑐 (𝑒𝑒 ) = 𝛼𝛼 (the constraint node of the edge 𝑒𝑒 ) and the
projection of the edge on the variable set is
𝑝𝑝𝑍𝑍 (𝑒𝑒) = 𝛽𝛽 (the variable node of the edge 𝑒𝑒);
A matching 𝑀𝑀 is a subset of such that the restriction of
𝑝𝑝𝑐𝑐 and 𝑝𝑝𝑧𝑧 to 𝑀𝑀 are inject ive,
∀ 𝑒𝑒1 , 𝑒𝑒2 ∈ 𝑀𝑀: 𝑒𝑒1 ≠ 𝑒𝑒2 ⇒ 𝑝𝑝𝑐𝑐 ( 𝑒𝑒1 ) ≠ 𝑝𝑝𝑐𝑐 ( 𝑒𝑒2 ) ⋀𝑝𝑝𝑍𝑍 ( 𝑒𝑒1 )
≠ 𝑝𝑝𝑧𝑧 ( 𝑒𝑒2 )
This means that a matching is a subset of edges such that
any two edges have no common node (neither in 𝐶𝐶nor in 𝑍𝑍).
There are two types of matching maximal and co mplete.
is only one in the co lu mn). It is obviously possible to define
matchings, maximal matchings, and complete matchings by
considering either the whole structure of the system or only
subgraphs of its structural graph, i.e. subsets of the
constraints and variables instead of the whole set. As know
variables need not to be determined by some constraint, the
matching is accomp lished in the following for the subgraph
containing all unknown variab les rather than the whole
structure graph.
𝛤𝛤1
𝛤𝛤2
𝛤𝛤3
𝑦𝑦1
6.4.1. Maximal Matching
A maximal matching is a matching 𝑀𝑀 such that ∀𝑁𝑁 ∈
2𝜀𝜀 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑀𝑀 ⊂ 𝑁𝑁, 𝑁𝑁 is not a matching.
Thus,a maximal matching is a matching such that no edge
can be added without violating the no co mmon node
property. Since the set of matchings 𝑀𝑀 is only part ially
ordered, it fo llo ws that there is in general more than one
maximal matching[2].
6.4.2. Co mp lete Matching[2]
𝑦𝑦2
𝑦𝑦3
C1
d1
C2
d2
C3
d3
m1
d4
m2
d5
m3
𝑞𝑞 1
𝑞𝑞̇ 1
𝑞𝑞̈ 1
𝑞𝑞 2
𝑞𝑞̇ 2
𝑞𝑞 2̈
𝑞𝑞 3
𝑞𝑞̇ 3
A matching is called co mplete with respect to 𝐶𝐶 if
|𝑀𝑀| = |𝐶𝐶 | holds. A matching is called complete with respect
𝑞𝑞 3̈
d6
to 𝑍𝑍 if |𝑀𝑀| = |𝑍𝑍| holds.
For a co mplete matching 𝑀𝑀 on 𝐶𝐶(respectively on 𝑍𝑍),each
constraint (respectively each variab le) belong to exactly one
edge of the matching:
∀ 𝑐𝑐 ∈ 𝐶𝐶: ∃𝑧𝑧 ∈ 𝑍𝑍 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 (𝑐𝑐, 𝑧𝑧) ∈ 𝑀𝑀
Figure 2. The graph of the structural model of the robot
∀ 𝑧𝑧 ∈ 𝑍𝑍: ∃𝑐𝑐 ∈ 𝐶𝐶 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 (𝑐𝑐, 𝑧𝑧) ∈ 𝑀𝑀
However, the incidence matrices and the graphical
A matching can be represented by selecting at most one”1”
in each row and each column in the incidence matrix of the representations are usually given for the co mplete structure
bi-partite graph. Each selected “1” represents an edge of the graph.
-The structural graph corresponding to our matrix of
matching. No other edge should contain the same variab le
(thus the only one in the row) or the same constraint (thus it incidence is shown in the Fig 2. Every colu mn of the matrix
Computer Science and Engineering 2013, 3(2): 15-23
corresponds to a circle-vertex and every row to a bar-vertex.
This matching is maximal and co mplete with respect to the
variables. For FDI considerations i.e to obtain a signature
table that could give all the colu mns disjoint two by two, we
tried to obtain the "best" matching. If we cannot obtain this
property, the system is not structurally monitorable and we
have to place some additional sensors [15].
The chosen matching is drawn by dark lines.
6.5. Relations after Matching
𝐶𝐶1 : 0 = 𝑎𝑎 𝛤𝛤1 + 𝑏𝑏 𝛤𝛤2 + 𝑐𝑐 𝛤𝛤3 + 𝑞𝑞̇ 1 (𝑎𝑎 8,2428 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 (3) 𝑥𝑥 (4) −
𝑏𝑏 4,1214𝑠𝑠𝑖𝑖𝑛𝑛 𝑥𝑥3 𝑥𝑥 2+ 𝑐𝑐𝑚𝑚3𝑔𝑔− 𝑞𝑞1
(16)
𝐶𝐶1 (𝑞𝑞̇ 1 , 𝑞𝑞̈ 1 ) → 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 = 𝑑𝑑 1 � 𝑚𝑚1 (𝑦𝑦1 ) , 𝐶𝐶1 �𝑑𝑑 1 �𝑚𝑚1 (𝑦𝑦1 )��� (17)
𝐶𝐶2 : 0 = 𝑑𝑑 𝛤𝛤1 + 𝑒𝑒 𝛤𝛤2 + 𝑓𝑓 𝛤𝛤3 + 𝑞𝑞̇ 2
(4,1214 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥(3) 𝑥𝑥(4) + 𝑓𝑓𝑚𝑚3 𝑔𝑔) − 𝑞𝑞̈ 2
(18)
𝐶𝐶2 (𝑞𝑞̇ 2 , 𝑞𝑞̈ 2 ) → 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ⟹ 𝑑𝑑 3 � 𝑚𝑚2 ( 𝑦𝑦2 ), 𝐶𝐶2 � 𝑑𝑑 3 �𝑚𝑚2 (𝑦𝑦2 ) ��� (19)
𝐶𝐶3 : 0 = 𝑔𝑔 𝛤𝛤1 + ℎ 𝛤𝛤2 + 𝑖𝑖 ( 𝛤𝛤3 +𝑚𝑚3 𝑔𝑔) − 𝑞𝑞̈ 3
𝐶𝐶3 (𝑞𝑞̈ 3 ) → 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ⟹ 𝐶𝐶3 �𝑑𝑑 6 (𝑑𝑑 5 �𝑚𝑚3 (𝑦𝑦3 ) ��
𝑚𝑚1 : 𝑞𝑞1 (𝑡𝑡) = 𝑦𝑦1 (𝑡𝑡) ⟶ 𝑚𝑚1 ( 𝑦𝑦1 ) ⟶ 𝑞𝑞1
𝑚𝑚2 : 𝑞𝑞2 (𝑡𝑡) = 𝑦𝑦2 (𝑡𝑡) ⟶ 𝑚𝑚2 (𝑦𝑦2 ) ⟶ 𝑞𝑞2
𝑚𝑚3 : 𝑞𝑞3 (𝑡𝑡) = 𝑦𝑦3 (𝑡𝑡 ) ⟶ 𝑚𝑚3 (𝑦𝑦3 ) ⟶ 𝑞𝑞3
𝑑𝑑 𝑞𝑞 (𝑡𝑡 )
𝑑𝑑 1 : 𝑞𝑞̇ 1 (𝑡𝑡) = 1
⟶ 𝑑𝑑 1 (𝑞𝑞1 ) ⟶ 𝑞𝑞̇ 1
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞̇ 1 (𝑡𝑡)
𝑑𝑑 2 : 𝑞𝑞̈ 1 (𝑡𝑡) =
𝑑𝑑 3 : 𝑞𝑞̇ 2 (𝑡𝑡) =
𝑑𝑑 4 : 𝑞𝑞̈ 2 (𝑡𝑡) =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞2 (𝑡𝑡)
𝑑𝑑 5 : 𝑞𝑞̇ 3 (𝑡𝑡) =
𝑑𝑑 6 : 𝑞𝑞̈ 3 (𝑡𝑡) =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞̇ 2 (𝑡𝑡)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞3 𝑡𝑡)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑞𝑞̇ 3 (𝑡𝑡 )
𝑑𝑑𝑑𝑑
⟶ 𝑑𝑑 2 (𝑞𝑞̇ 1 ) ⟶ 𝑞𝑞̈ 1
(20)
(21)
(22)
(23)
(24)
(25)
(26)
⟶ 𝑑𝑑 3 (𝑞𝑞2 ) ⟶ 𝑞𝑞̇ 2
(27)
⟶ 𝑑𝑑 5 (𝑞𝑞3 ) ⟶ 𝑞𝑞̇ 3
(29)
⟶ 𝑑𝑑 4 (𝑞𝑞̇ 2 ) ⟶ 𝑞𝑞̈ 2
⟶ 𝑑𝑑 6 ( 𝑞𝑞̇ 3 ) ⟶ 𝑞𝑞̈ 3
21
values.
• Structured: this insures that in the presence of a given
fault, only a subset of the ARRs is not satisfied, th is allows to
recognize (satisfied and not satisfied ARRs subset) the faulty
mode.
Applying the previous method in our case by
characterizing the ARRS in the bipartite graph, three
Analytical redundancy relations are obtained for this model.
• The first ARR model results following constraints:
C1 , d1 , d2 and m1.
𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 : 0 =
d q (t )
𝑎𝑎𝛤𝛤 1+𝑏𝑏𝛤𝛤 2 +𝑐𝑐𝛤𝛤 3+ 1 (𝑎𝑎8.2428 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 (3) 𝑥𝑥 (4) −
dt
𝑏𝑏4.1214 𝑠𝑠𝑖𝑖𝑛𝑛𝑥𝑥3 𝑥𝑥 2+𝑐𝑐𝑚𝑚3 𝑔𝑔− dq1(t)dt
(31)
− d1 (m1 ( y 1 ), C1 (d1 (m1 (y 1 )))) ⟶ Zéro. (32)
• The second ARR model results following
constraints: C2 , d3 , d4 and m2 .
ARR2 : 0 =
𝑑𝑑 𝑞𝑞2 (𝑡𝑡)
(4.1214𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥(3) 𝑥𝑥(4) + 𝑓𝑓𝑚𝑚3 𝑔𝑔) −
𝑑𝑑𝛤𝛤1+𝑒𝑒𝛤𝛤 2 +𝑓𝑓𝛤𝛤 3+
𝑑𝑑𝑑𝑑
d q̇ 2 (t )
(33)
(34)
− d3 (m2 (y 2 ) , C2 (d3 (m2 ( y 2 ))))⟶ Zéro.
• The third ARR model results following constraints:
C3 , d5 , d6 and m3
𝑑𝑑𝑞𝑞̇ (𝑡𝑡)
ARR3 : 0 = 𝑔𝑔 𝛤𝛤1 + ℎ 𝛤𝛤2 + 𝑖𝑖( 𝛤𝛤3 +𝑚𝑚3 𝑔𝑔) − 3 (35)
𝑑𝑑𝑑𝑑
(36)
− C3 (d6 (d5 (m3 (y 3 ) ))) ⟶ Zéro.
dt
7.1. Defini tion of Residue
(28)
(30)
The form, under which the equations are presented above,
translate exactly the matching selected in the incidence
matrix.
7. Analytical Redundancy Relations of
Robot (ARRs)[2]
Analytical redundancy relations are static or dynamic
constraints which link the time evolution of the known
variables when the system operates according to its normal
operation model.
An analytical relation of redundancy (ARR) is a relat ion
resulting fro m the equations of the nominal model (without
disturbances, failures), building the input output variables
and the successive derivative of these variables until a given
order.
Moreover, in order to the fault diagnosis procedure works
properly, ARR should have the follo wing properties:[2]
• Robust, i.e. insensitive to unknown input and unknown
parameters. Th is insures that they are satisfied when no fault
is present.
• Sensitive to faults: this insures that they are not satisfied
when faults are present, so that there is no missed detection.
At least in this case, one of the residual differs fro m vanishes
A residue is a signal conceived as indicator of functional
or behavioral anomalies. The princip le of a residue fo r the
supervision is that it connects only what is known[2].
With regard to a given matching, Analytical Redundancy
Relation is a relat ion not saturated by the matching and
which is either a relation wh ich is connected only to known
variables (sensors, actuators) or a relation which is
connected to variables unknown but already saturated by the
matching (Or both at the same time).
When the occurrence of a defect is detected, the procedure
of localization or insulation is used to determine the origin of
this defect (the defective component).
In order to establish the insulation, it is necessary to use a
set (or a vector) of residues, which must react differently and
predefined manner to various defects.
For this, two methods are possible[16]:
• The direct ional residues.
• The structured residues.
a) Directional residues
The idea of this method[16] is to generate a residue as a
vector of standard ideally zero in the case of the proper
functioning of the system. In case of defect, this vector is
directed in a direct ion which depends on the defect in the
system. The faults insulation step is then to determine which
of the different predefined directions which is the closest to
the residue observed. So the objective of design of the
generator of residue, in this approach, is to predefine the
directions of the most distinct defects possible to obtain good
22
Yasmine Derdour et al.: Application of the Structural Analysis on a Three Axis M anipulator Robot
insulation performance thereafter.
b) Structured residues
This approach is widely used in the field of the defects
insulation because the principle o f the method[16] is very
simp le and applies to a wide variety of systems (linear, not
linear ..). The principle of this method consists in having a
group of residues in which each residue is sensitive to a
subset of defects monitored.
8. The Signature Failures Table
In the signature failure table, every line corresponds to a
residue and every column to a failure. The analysis of the
signature table of the residuals enables us to check the
structural detectability and localizability property[17],[12].
A failure is detectable if its signature comprises at least
one ‘’1’’, two failures are localizable, or identifiab le if their
signatures are different.
The structural analysis gives only the structural
proprieties.
Fro m the signature failures table we can deduct the
structural properties.
The signatures surrounded by a circle are different fro m
the others.
Table 3. The signature failures
- Concerning the isolability, only the violation of variables
associated in (𝑞𝑞̇ 3 , 𝑞𝑞̈ 3 ) are structurally isolable.
- All other defects possess the same signature, a fact that
makes them not structurally localizab le.
- Concerning the Γ1 , Γ2 , Γ3 commands possess the same
signature thus they are neither mon itorable nor detectable.
- As for the measures y 1 , y 2 , y 3 they are impossible to
monitor (not detectable and not isolable).
- Since the positions q1 , q2 , q3 have the same
structuralization then they are not monitorable and not
isolable. We can’t detect failu re if it affect one of these
variables; so some additional sensors have to be placed in the
system to make it monitorab le.
rules, look-up tables)·
This methodology helps to highlight the monitorable parts
of system by using informat ion included in the model itself,
without extra or physical redundancy requirement and also
allo ws building the strong structured residues for the FDI.
When applied to the model robot, we have been able to see
which of the state variables are mon itorable and which are
not. The approach presented, when applied to the robot
model, it provides a powerful tool for analy zing systems at
any stage of the design process, providing the designer to
correct in early stage, the system (robot) by adding
sensors[18].
Analyzing disturbances and errors of estimates will be the
subject of future work. It will be interesting to compare these
results with the diagnosis approach on the basis of observer
theory.
In future work, the co mbinatorial problem related to
which is the best matching for monitoring, will be
investigated as well as how to make the reconfigurat ion by
structural analysis.
REFERENCES
[1]
H.Haffaf, B. Ould Bouamama, “Cycle Algorithm in Tripartite
Graph for FDI.”.Journal of the Franklin Institute, Vol 349
(Isuue 1), pages:112-125.M arch 2008.
[2]
M .Blanke, M .Kinnaert, J.Lunze et M . Staroswiecki
“Diagnosis and Fault-Tolerant Control”,2nd edition with
contributions by Jochen Schroder,Berlin Heidelberg,may
2006.
[3]
D. Dustgor . “Aspects Algorithmics of the Structural Analysis
for the M onitoring”, PhD of the University of the Sciences
and the Technologies of Lille,2005.
[4]
M . Krysander,E.Frisk, E. Sensor placement for fault
diagnosis, IEEE systems and humans, p 1398-1410,
1083-4427 vol.38 NO.6,2008.
[5]
V.de Flaugergues, V.Cocquempot, M . Bayart,M.Pengov.
“Structural Analysis for FDI, a M odified, Invertibility,based
Canonical Decomposition”. Proceedings of the 20th
International Workshop on principles of Diagnosis, p59—66,
2009.
[6]
D. Dustegor and al. Automated graph-based methodology for
fault detection and location in power systems, IEEE
Transactions on power delivery, pp-638-646, VOL.25,
NO.2,2010.
[7]
M . Belaguid . Optimization Algorithms Generating RRAS in
Structural Analysis for M onitoring, M agister Thesis in
Computer Science University of es Senia, October 2004.
[8]
J.Ph. Cassar and M . Staroswiecki . A Structural Approach for
the Design of Failure Detection and Identification Systems.
Proceedings IFAC-IFIP-IM ACS Conference on Control of
Industrial Processes (Berfort, France), Année 1997
[9]
R. Carpentier, Litwak, and J.Ph. Cassar .Criteria Evaluation
of FDI Systems Application to Sensors Location. Proceedings
9. Conclusions
The results show the efficiency and simp licity of structural
analysis for fau lts detection and isolation for either a linear or
nonlinear system because it bases on the graph theory. The
structural analysis displays the relations between constraints
and variables, and identifies the observable and monitorab le
parts in terms of decomposition. Th is method applies to
complex systems with hundreds of variables and equations.
Thanks to the structural analysis we can represent different
types of modelling (qualitat ive, quantitative, static, dynamic,
Computer Science and Engineering 2013, 3(2): 15-23
of IFAC/Safeprocess, Année 1997.
[10] M . KRYSANDER.,M .NYBERG, Structural Analysis for
Fault Diagnosis of DAE Systems Using M SS Sets, IFAC
World Congress, 2002
[11] K.E
Bouazza,
M .Ouali,
M .Boutayeb.”M odelling,
Identification and Control of Three A xis M anipulator robot”.
Journal of Automation and System Engineering, Volume3,
Issue4,2009.
[12] H.Haffaf, M .Belaguid.”Algorithm of the Cycles in Structural
Analysis for the M onitoring”. M aghrebian conference ,
Agadir, M orocco ,2007.
[13] R.l.Zamanabadi.“Structural Analysis Approach to Fault
Diagnosis with Application to fixed-wing Aircraft M otion”.
Department of control engineering ,Intelligent Autonomous
Systems(IAS).Aalborg University, Denmark,2002.
23
[14] D.Reinhard,
Graph Theory, Graduate Texts
in
M athematics,173(3rd ed),Springer-Verlag,ISBN 3-540-2618
3-4,2005.
[15] M . Khemlich , B.Ould Bouamama and H. Haffaf Sensors and
actuators A,«Sensor placement for Component diagnosability
using Bond-graph». Elsevier pp 547-556. January 2006
[16] J. Gertler. Analytical Redundancy M ethods in Fault Detection
and Isolation, Survey and Synthesis. IFAC Fault Detection,
Supervision end Safety for Technical Processes, Germany,
pp.43-55. 26, Année 1991.
[17] M .Gondran , M.M inoux. “Graphs and Algorithms“ . Edition
EROYLLES ,1995.
[18] D. Dustegor .Automated graph-based methodology for fault
detection and location in power systems, IEEE Transactions
on power delivery, pp-638-646, Vol.25, No.2,april 2010.