INPE-8981-TDI/812/B
PARAMETERS IDENTIFICATION AND FAILURE DETECTION APPLIED TO SPACE ROBOTIC MANIPULATORS
Adenilson Roberto da Silva
Dissertation in Space Engineering and Technology in partial fulfillment of the requirements for Doctor degree, supervised by Dr. Luiz Carlos Gadelha de Souza and
Dr. Bernd Schäfer, approved in november the 11th , 2001.
INPE São José dos Campos
2002
629.7.062.2 SILVA, A. R. Parameters identification and failure detection applied to robotic maniputators / A. R. Silva. – São José dos Campos: INPE, 2001. 183p. – (INPE-8981-TDI/812/B). 1.Parameters identification. 2.System identification. 3.Robot dynamic. 4.Nonlinear systems. 5.Least square method. 6.Robots. 7.Fault detection. I.Título.
To everyone, which have shared with me the happiness of this victory,
I dedicate.
ACKNOWLEDGEMENTS
My sincere appreciation goes to my advisors Dr. Luiz C. G. de Souza and Dr. Bernd
Schäfer. Their challenging questions, insightful comments and stimulating discussions
always kept me thinking and learning.
I wish to express my gratitude to all of my doctorate professors, especially Drs. Marcelo
Lopes, Roberto Lopes for their incentive and fruitful comments during my research at
INPE. They have taught not technical lessons, but also how to be a critic and a minute
researcher.
I would like to thank all of examination group for their comments and suggestions that
have improved and enriched my work
My appreciation also goes to Prof. Luis S. Goes from ITA. His comments has always
improved my thought and helped me to solve problems during my research.
I thank DLR for propitiate an excellent environment and support during my stay in
Germany. My stay at DLR (and in Germany) was made much more enjoyable by the
company of various staff members. I wish also to express my gratitude to Rainer Krenn
for his help and friendship during all my stay. The pizza (after the work), the barbecue
and the mountain trekking will be always a fond memory. At this point, I wish to thank
Dr. Bernd Schäfer for his help and friendship. I am quite sure that may stay at Germany
would not be completely enjoyable without his friendship.
My appreciation goes to all of my colleagues for their friendship and support. My
special thanks to Evandro M. Rocco (who has shared a office for four years), Walkiria
Schulz, Ana P. Chiaridia, Cristina Tobler, Francisco Carvalho for their support and
friendship.
I express my gratitude also to Luiz R. R. Faria for his incentive in the beginning of my
education and also along my entire career.
I wish to thank my parents, Josué and Antonia, all my sisters, brothers, nephew and
nieces, at last all of my family (it will take to long to write down all names) for their
love and support throughout my life. I wish also to thank my girlfriend Keli for her love
and tolerance during my student lifestyle.
ABSTRACT
Physical parameters identification is useful in many applications, especially in aerospace and robotics fields. Aerospace and robotics system analysis normally requires accurate physical system models for control. On the other hand, the identification of physical parameters, besides the normal identification requirements (system excitation, for instance), involves several tasks: mathematical modeling and algorithm selection for instance. In this thesis, a detailed modeling of a robotic joint has been presented. The models are derived in an increasing degree of complexity (which means that, in theory, the mathematical representation is approaching to the real system), where the typical non-linear terms of a robotic joint have been taken into account. A new procedure to select suitable robotic trajectories based on the singular value decomposition (SVD) of measurement matrix is also presented. The identification task has been carried out by deriving (or improving) and implementing new algorithms. The strategies and algorithms have shown good performance in both: accuracy and also concerning computer load. In order to allow the inclusion of non-linear terms in the parameters vector, a new algorithm (TS -Two Step Algorithm) based on a modified version of Recursive Least Squares (mRLS) with a variable forgetting factor and MCS (Multi Level Coordinate Search) algorithms has been derived. The results have shown that the TS algorithm has excellent performance in identifying the unknown parameters vector by using both: real and simulated data. In addition, an integrated procedure for sensors failure detection and isolation (FDI) based on subspace theory is derived. The MOESP (MIMO Output Error State Space Model Identification) algorithm has been used to build a model, which serves as a reference for the FDI algorithm. The FDI algorithm has shown high reliability in detecting and isolating all the simulated failures in the sensors. Finally, the TS and the FDI algorithms have been integrated in a single environment to simulate an integrated situation where the system is time variant and the sensors also fail. The results have shown that reliable parameters are obtained even in case of multi failure. All derived models and algorithms have been tested by using data collected from IRJ (Intelligent Robotic Joint) experiment built at DLR (German Aerospace Centers) in Oberpfaffenhofen.
IDENTIFICAÇÃO DE PARÂMETROS E DETECÇÃO DE FALHAS APLICADA A MANIPULADORES ESPACIAIS
RESUMO
A identificação de parâmetros físicos é muito útil em muitas aplicações, especialmente na área aeroespacial e também na robótica. A análise de sistemas aeroespaciais e robôs normalmente requerem modelos matemáticos precisos os quais são utilizados pelo controle. Por outro lado, a identificação de parâmetros físicos, além dos requisitos normais de identificação (excitação do sistema, por exemplo), envolve tarefas adicionais, tais como modelagem matemática do sistema, seleção dos algoritmos de identificação, etc. Nesta tese, é mostrada uma detalhada modelagem matemática de uma junta robótica. Os modelos são mostrados numa ordem crescente de complexidade (o que significa, em teoria, que a representação matemática está mais próxima do sistema real), onde os típicos termos não-lineares da junta robótica foram considerados. Um novo procedimento para se selecionar trajetórias apropriadas (considerando o nível de excitação do sistema) baseada na decomposição em valores singulares da matriz de medidas é também apresentado. A tarefa de identificação foi realizada através da obtenção (ou melhora) e implementação de novos algoritmos. As estratégias e algoritmos mostraram bom desempenho em vários aspectos: precisão, confiabilidade e baixo esforço computacional. A fim de permitir a inclusão de termos não-lineares no vetor de parâmetros (na identificação recursiva), um novo algoritmo (TS – Algoritmo Duas Etapas) baseado numa versão modificada do algoritmo dos mínimos quadrados recursivos (mRLS) com um fator de esquecimento variável (variable forgetting factor) e no algoritmo Multi Level Coordinate Search (MCS) foi obtido. Os resultados mostraram que o algoritmo TS tem uma excelente performance na identificação dos parâmetros em ambos os casos: usando dados reais e simulados. Um procedimento integrado para detecção e isolamento de falhas (FDI) baseado na teoria de subespaço é também mostrado. O algoritmo MIMO Output Error State Space Model Identification (MOESP) foi usado para se obter um modelo matemático que serve como base para o algoritmo FDI. O algoritmo FDI mostrou elevada eficiência e confiabilidade na detecção e no isolamento das falhas em todos os casos simulados. Finalmente, os algoritmos TS e FDI foram integrados em um único ambiente a fim de simular uma situação onde o sistema a ser identificado é variante no tempo e vários sensores apresentam falhas. Os resultados indicam que parâmetros confiáveis podem ser obtidos mesmo no caso de múltiplas falhas. Todos os modelos e algoritmos obtidos foram testados utilizando-se dados coletados no experimento Intelligent Robotic Joint (IRJ) construído pelo Centro Espacial Alemão (DLR Oberpfaffenhofen).
SUMMARY
LIST OF FIGURES
LIST OF TABLES
LIST OF SYMBOLS
LIST OF ACRONYMS AND ABBREVIATIONS
CHAPTER 1.................................................................................................................. 25
INTRODUCTION ........................................................................................................ 25 1.1 - Motivation .............................................................................................................. 26 1.2 - Review of Literature............................................................................................... 27 1.2.1 - Identification Algorithms .................................................................................... 28 1.2.2 - Parameters Identification Applied in Robotics ................................................... 29 1.2.3 - Sensors Failure .................................................................................................... 32
CHAPTER 2.................................................................................................................. 35
MODELS AND MODELING ..................................................................................... 35 2.1 - Introduction ............................................................................................................ 35 2.2 - Black Box Models.................................................................................................. 36 2.2.1 - ARX Model ......................................................................................................... 36 2.2.2 - Armax Model ...................................................................................................... 37 2.2.3 - State Space Models ............................................................................................. 38 2.2.3.1 - Continuous Systems ......................................................................................... 38 2.2.3.2 - Discrete Systems .............................................................................................. 39 2.3 - Phenomenological Models ..................................................................................... 40 2.3.1 - Introduction ......................................................................................................... 40 2.4 - Modeling ................................................................................................................ 40 2.4.1 - Introduction ......................................................................................................... 40 2.4.2 - Harmonic Drive (HD) Gear Modeling................................................................ 41 2.4.2.1 - Work Principle of Harmonic Drive Gear ....................................................... 42 2.4.3 - Different Configuration in the HD Assembly ..................................................... 43 2.4.3.1 - Configuration 1 ................................................................................................ 43 2.4.3.2 - Configuration 2 ................................................................................................ 45 2.4.4 - Harmonic Drives Models .................................................................................... 46 2.4.4.1 - Ideal Transmission – Model 1.......................................................................... 46 2.4.4.2 - Transmission with Losses – Model 2............................................................... 47 2.4.4.3 - Transmission with Losses and Stiffness – Model 3 ......................................... 49 2.4.4.4 - Transmission with Friction Loss, Stiffness and Kinematic Error Model 4..... 50 2.5 - HD Configuration Used in the IRJ Experiment ..................................................... 52 2.5.1 - IRJ Experiment Description................................................................................ 52 2.5.2 - Joint Modeling – Kinematic and Dynamic Equations ........................................ 53 2.5.3 - Stiffness Torque .................................................................................................. 56
2.5.4 - Damping Torque ................................................................................................. 57
CHAPTER 3.................................................................................................................. 63
ALGORITHMS AND IDENTIFICATION STRATEGIES..................................... 63 3.1 - Introduction ............................................................................................................ 63 3.2 - Parameters Identification by Using RLS Algorithm.............................................. 64 3.3 - Non-linear Optimization - MCS Algorithm (Multilevel Coordinate Search) ....... 66 3.3.1 - MCS Algorithm................................................................................................... 67 3.3.1.1 - Initialization ..................................................................................................... 68 3.3.1.2 - Boxes Split ....................................................................................................... 69 3.3.1.3 - Local Search..................................................................................................... 72 3.3.1.4 - Coordinate Search ............................................................................................ 74 3.4 - Identification Models Based on the Subspace Theory (SMI) ............................... 77 3.4.1 - State Matrices Identification .............................................................................. 77 3.4.2 - Output Error State Space Model ......................................................................... 79 3.4.3 - Data Representation and Persistent Excitation Definition .................................. 80 3.4.4 - Persistent Excitation............................................................................................ 81 3.4.5 - MOESP Algorithm.............................................................................................. 82 3.5 - Proposed Strategy for On-line Identification ......................................................... 83 3.6 - Two Step identification Algorithm ......................................................................... 88 3.7 - Integrated Algorithm MRLS – MCS..................................................................... 89 3.8 - Trajectory Selection for Identification Purposes.................................................... 92 3.8.1 - Trajectory Description......................................................................................... 92 3.8.2 - Triangular Trajectory .......................................................................................... 93 3.8.3 - Sinusoidal Trajectory .......................................................................................... 94 3.8.4 - Trajectory Selection ............................................................................................ 96
CHAPTER 4................................................................................................................ 101
APPLICATIONS AND RESULTS ........................................................................... 101
4.1 - State Space Matrices Identification...................................................................... 101 4.1.1 - Obtaining the State Space Matrices .................................................................. 102 4.1.2 - MOESP Algorithm Accuracy Verification ....................................................... 103 4.1.3 - Model Order Selection ...................................................................................... 103 4.1.4 - Estimation Analysis - MOESP Algorithm ........................................................ 106 4.2 - Failure Detection in the Sensors .......................................................................... 109 4.2.1 - Introduction ....................................................................................................... 109 4.2.2 - Failure Detection Strategy................................................................................. 110 4.2.3 - Sensor Failure Simulation ................................................................................. 111 4.2.4 - Failure Detection – Comments.......................................................................... 117 4.3 - Parameters Identification...................................................................................... 117 4.3.1 - Off-line Parameter Identification...................................................................... 117 4.3.2 - Model 1: Without Stiffness ............................................................................... 120 4.3.2 - Model 2: Stiffness (2A) and Damping (2B)...................................................... 124 4.3.3 - On-line Parameter Identification....................................................................... 128 4.3.4 - Parameters Identification by Using Two –Step Algorithm............................... 130 4.3.4.1 - IRJ Experiment............................................................................................... 130 4.3.4.2 - Time Variant Systems .................................................................................... 136
4.4 - Identification under Failure Conditions ............................................................... 140 4.4.1 - Failure in Velocity Sensor................................................................................. 141 4.4.2 - Position Sensor Failure ..................................................................................... 142 4.4.2.1 - Alternative 1 – Reconfiguration of Identification Algorithm ........................ 143 4.4.2.2 - Alternative 2 – Reconfiguration of Algorithm and Freezing of Stiffness Coefficient .............................................................................................................. 145 4.4.3 - Simultaneous Failure in Position and Velocity Sensors. .................................. 149 4.5 - Summary .............................................................................................................. 152
CHAPTER 5................................................................................................................ 153
CONCLUSION AND RECOMMENDATIONS ..................................................... 153 5.1 - Conclusions .......................................................................................................... 153 5.2 - Recommendations for Future Works ................................................................... 155
BIBLIOGRAPHY....................................................................................................... 157 6.1 – Bibliography References ..................................................................................... 157 6.2 – Auxiliary Bibliography........................................................................................ 163
ANNEX A .................................................................................................................... 167 Accelerometer Calibration............................................................................................ 167 A.1 - Introduction ......................................................................................................... 167 A.2 - Static Calibration................................................................................................. 167
ANNEX B .................................................................................................................... 171 B.1 – Sensor Accuracy ................................................................................................. 171 B.2 – Additional Details of IRJ Experiment................................................................ 172
ANNEX C .................................................................................................................... 175 C.1 – Algorithms .......................................................................................................... 175
ANNEX D .................................................................................................................... 179 D.1 – Recursive Least Squares Algorithm – Derivation II.......................................... 179 D.2 – Recursive Least Squares Algorithm with Forgetting Factor .............................. 181
LIST OF FIGURES 2.1 - Internal view of Harmonic Drive .......................................................................... 42 2.2 - Work principle of Harmonic Drive ....................................................................... 43 2.3 - Configuration 1 ..................................................................................................... 44 2.4 - Configuration 2 ..................................................................................................... 46 2.5 - HD model 1– Ideal transmission........................................................................... 46 2.6 - HD model 2 - Incorporating friction losses........................................................... 47 2.7 - HD model 3 – Incorporating friction losses and stiffness..................................... 50 2.8 - HD model 4 – Transmission with friction stiffness, and kinematic error ............. 51 2.9 - Experimental configuration of IRJ for identification............................................ 54 2.10 - Dynamic representation of IRJ experiment......................................................... 55 2.11 - Harmonic Drive model used in IRJ experiment.................................................. 56 2.12 - Stiffness torque for HD type HFUC-25-160-2A-GR.......................................... 58 2.13 - Damping model type 3, Equation (2.77) ............................................................. 61 3.1 - Schematic representation of standard least squares .............................................. 66 3.2 - Schematic representation of MCS algorithm ........................................................ 68 3.3 - General view of the identification scheme............................................................ 80 3.4 - Typical behavior for gain in time invariant systems.( 95.)0(,99.00 == λλ ) ...... 86 3.5 - Behavior of variable gain – Polynomial adjust. .................................................... 87 3.6 - Behavior of variable gain – Exponential function. ............................................... 87 3.7 - Schematic representation of mRLS algorithm ...................................................... 88 3.8 - Schematic representation of the Integrated Algorithm ......................................... 91 3.9 - Triangular trajectory with constant velocity 45 deg/s........................................... 94 3.10 - Sinusoidal trajectory with T (period) = 0.9s, x (0) = 20 deg and v = 15 deg/s ... 95 3.11 - Sinusoidal trajectory with T (period) = 1.9s, x (0) = 20 deg and v = 15 deg/s ... 95 3.12 - Triangular trajectory – smallest singular value – full model. ............................. 96 3.13 - Triangular trajectory – smallest singular value – reduced model ....................... 98 3.14 - Sinusoidal trajectory – smallest singular value- full model. ............................... 98 3.15 - Sinusoidal trajectory – smallest singular value – reduced model. ...................... 99 3.16 - Sinusoidal trajectory – smallest singular value – full model φ normalized. ...... 99 4.1 - Singular values of measurement matrix .............................................................. 104 4.2 - Eigenvalue of matrix A (order 2) ........................................................................ 105 4.3 - Eigenvalue of matrix A (order 3) ....................................................................... 105 4.4 - Inputs for MOESP algorithm - Triangular trajectory ( motorθ& = 25 rad/s) ........... 107 4.5 - Output measurements and respective estimate.................................................... 108 4.6 - Relative position and velocity between input and output side ............................ 109 4.7 - Sensor position failure......................................................................................... 112 4.8 - Link position after measurement reconfiguration ............................................... 112 4.9 - Velocity sensor failure ........................................................................................ 113 4.10 - Link velocity after measurement reconfiguration ............................................. 113 4.11 - Position and velocity sensors failure at different instants ................................. 114 4.12 - Position and velocity after measurements reconfiguration ............................... 115 4.13 - Sensors failure simultaneously.......................................................................... 115 4.14 - Sensor temporary failure ................................................................................... 116 4.15 - Position and velocity after measurement reconfiguration................................. 116 4.16 - Model 1a decoupled – 1y according to Equation (4.15)..................................... 121
4.18 - Model 1a - Incorporating the Coulomb friction................................................ 123 4.19 - Model 2a - Motor side – Only stiffness term .................................................... 125 4.20 - Model 2a - Link side – Only stiffness term....................................................... 126 4.21 - Model 2b - Motor side – Including damping .................................................... 126 4.22 - Model 2b - Link side – Including damping....................................................... 127 4.23 - Stiffness and damping terms. ............................................................................ 129 4.24 - Damping coefficients ........................................................................................ 130 4.25 - Friction at low velocities ................................................................................... 132 4.26 - Non-linear parameters identification................................................................. 134 4.27 - Linear parameters.............................................................................................. 135 4.28 - Damping parameters ......................................................................................... 135 4.29 - Cyclic phase error ............................................................................................. 136 4.30 - Simulated damping torque ................................................................................ 138 4.31 - Parameters identified by RLS algorithm - Time variant case. .......................... 139 4.32 - Non-linear parameters – Time variant case....................................................... 140 4.33 - Viscous damping identification (link) by using MOESP estimation ................ 142 4.34 - Linear parameters – Using link velocity given by MOESP algorithm. ............ 143 4.35 - Parameters related to damping – Using MOESP estimation............................. 144 4.36 - Linear parameters – Using link position estimated by MOESP algorithm...... 146 4.37 - Damping coefficient– Using link position estimated by MOESP algorithm.... 146 4.38 - Link viscous damping – Using link position estimated by MOESP algorithm 147 4.39 - Linear parameters – Using link position estimated by MOESP algorithm ( 1k
and 2k frozen) .................................................................................................. 147 4.40 - Damping coefficient - Using link position estimated by MOESP algorithm
( 1k and 2k frozen)............................................................................................ 148 4.41 - Link viscous damping – Using link position estimated by MOESP algorithm
( 1k and 2k frozen)............................................................................................ 148 4.42 - Linear Parameters – Simultaneous failure in position and velocity sensor
(state estimated by MOESP) ............................................................................. 149 4.43 - Damping coefficient – Simultaneous failure in position and velocity sensor
(state estimated by MOESP) ............................................................................. 150 4.44 - Link viscous damping – Simultaneous failure in position and velocity sensor
(state estimated by MOESP) ............................................................................. 150 4.45 - Linear Parameters – Simultaneous failure in position and velocity sensor
( 1k and 2k frozen, state estimated by MOESP)............................................... 151 4.46 - Damping coefficient – Simultaneous failure in position and velocity sensor
( 1k and 2k frozen, state estimated by MOESP)............................................... 151 4.47 - Link viscous damping – Simultaneous failure in position and velocity sensor
( 1k and 2k frozen, state estimated by MOESP)............................................... 152 . A.1 - Static position for accelerometer calibration..................................................... 168 A.2 - Accelerometer 1 output – Positions: face up and face down ............................. 169 A.3 - Accelerometer 1 output – Positions: horizontal 1 and horizontal 2 ................... 169 A.4 - Accelerometers output – Positions: horizontal 1 for accelerometer 1 and
accelerometer 2 face up..................................................................................... 169 A.5 - Accelerometer 2 output – Positions: face down and horizontal 1...................... 170 B.1 Internal details of IRJ experiment........................................................................ 172 B.3 - Internal view of IRJ experiment ......................................................................... 173
B.4 - Seven joint robot................................................................................................. 174 C.1 - Simulink block diagram ..................................................................................... 175 C.2 - Failure detection interface . ................................................................................ 176 C.3 - Sub level 1 of failure detection and state estimation block. ............................... 177 C.4 - Sub level 2 of the failure detection and state estimation block .......................... 177
LIST OF TABLES
2.1 - Models comparison ............................................................................................... 35 2.2 - Conventional black box models ............................................................................ 38 2.3 - Data of HD type HFUC-20-160-2A-GR............................................................. 57 4.1 - Parameters Obtained from Catalog, Curve fitting or CAD design. .................... 120 4.2 - Parameters Identified for Model 1a (without damping) and 1b (with Coulomb
friction) Equation (4.15). ..................................................................................... 122 4.3 - Identified Parameters for Model 2B – Equation (4.17)....................................... 125 4.4 - Singular Value of Measurement Matrix φ ........................................................ 128 4.5 - Singular Value of Measurement Matrix φ Non-linear Parameters .................. 133 4.6 - Parameters Used in Equation (4.18).................................................................... 137 A2.1 - Sensors Accuracy and Resolution ................................................................... 171
LIST OF SYMBOLS GREEK SYMBOLS ℜ Real Numbers
π Degree of rank variability of MCS Algorithm
ρ Rank of matrix
Η Hankel matrix
Ξ Sub matrix
Σ Diagonal matrix – singular values
δ Precision factor
ω Angular velocity
ε Error
Θ Vector of parameters (to be identified)
φ Matrix of measurements
θ Angular Position
γ Phase of cyclic error
∏ Matrix formed by state space matrices
∆Θ Parameters variation
λ(t) Forgetting factor
η(t) Output of ideal sensors
(.)d Discrete state matrix
σi Singular values
LATIN SYMBOLS [u, v] Interval of search for non-linear optimization
|| . || Euclidian norm
A State matrix (dynamics)
A(q) Polynomial in q
Ai Amplitude of cyclic error (i = 1, 2, ...)
B State matrix (actuators location)
B(q) Polynomial in q
bi Damping coefficients
C State matrix (sensors location)
C(q) Polynomial in q
cs Circular Spline
D State matrix (direct feedback)
D(q) Polynomial in q
e(k) Residual error
f(x) Function to be minimized
fs Flexible Spline
G Hessian matrix
H(.,.) Transfer function
Ia Commanded current
J Index of performance
Jin Input moment of inertia (motor)
Jout Output moment of inertia (link)
k Discrete time index
k1 Linear stiffness coefficient
k2 Cubic stiffness coefficient
Km Motor constant
L(t) RLS gain
N Reduction ratio
R1 Model uncertainty
R2 Sensor uncertainty
s Number of levels in MCS algorithm
T Sampling time
Tg Gravity torque
Ti Torque (i= in, out,...)
u(k) Plant input in instant k
Uk Input vector (Batch estimation)
wg Wave generator (HD)
x(t) state
x* Optimal point
Xk State vector (Batch estimation)
y(k) Plant output at instant k
Z MCS split points
zk Plant output disturbed by noise
LIST OF ACRONYMS AND ABBREVIATIONS
AR Auto Regressive model
ARMAX Auto Regressive model input and Moving Average with exogenous
ARX Auto Regressive model with exogenous input
DLR German Aerospace Center.
HD Harmonic Drive
IRJ Intelligent Robotic Joint experiment
MCS Multi Level Coordinate Search
MIMO Multiple Input and Multiple Output model
MOESP MIMO Output Error State Space Model Identification
RLS Recursive Least Squares
SISO Single Input and Single Output model
SVD Singular Values Decomposition
TS Two – Step Algorithm
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CHAPTER 1
INTRODUCTION The description of a process in terms of dynamic models is very useful in technological
as well as in scientific fields. Very often, the dynamic models are very important in the
analysis and management of such systems. Given a dynamic system, the goal of the
control system is to keep a pre-determined position even under disturbing effects or
track a reference. In order to design a control system with high performance, it is
essential that the mathematical model, which describes the systems, be accurate. The
proceeding of obtaining a mathematical model based on physical laws and relationships
that describe the behavior of the system is called modeling (Ljung and Söderström,
1983). However, in some special circumstances as in presence of non-linearity, the non-
complete knowledge of the system behavior or because the system has some
unpredictable characteristics, the direct modeling can be inconvenient or not possible. In
these situations, the system information obtained from sensors can be used to build a
mathematical model that describes the system under investigation. This process is called
identification (Ljung and Söderström, 1983). Thus, the main purpose of the
identification process is to obtain a mathematical model, which describes the static as
well as the dynamic characteristics of the system with fidelity. Using this model, tests
and simulations are carried out in order to design a control with high accuracy.
Space missions that use robots and automation have played an important role and
several projects have been developed or proposed in the last decades. Robots when
correctly designed are very effective and present high accuracy in performing their
tasks. Therefore, due to their special characteristics, the robots become very attractive
for applications in space projects. On the other hand, the parameters that describe the
robot dynamics are very sensitive and extremely dependent of environmental condition,
like gravity, temperature, etc. In space missions, the robot is exposed to micro gravity
conditions and also to very big temperature variations. Researches (Heimann, 1999)
have shown that the operating temperature has strong effects on the joint parameters,
damping for instance. These effects can and likely play an important role in the robot
dynamics. In this case, the important characteristics of the robot, which allow high
26
performance, that have been appropriately tested on ground may not fulfill all mission
requirements. The loss of performance occurs because, normally, the control strategy is
based on state feedback and the control parameters that have been optimized (on
ground) no longer will be an optimum. As a result, the whole performance of the robot
is affected. If the mission intends also to verify the behavior of the physical parameters
of a system, an additional task in the identification process shall be carefully selected:
the system structure (model). Thus, there exists a direct relationship between fidelity in
representing the system under investigation and the dynamic model (or model order, if
the state space approach is used). Therefore, when the goal is the identification of
physical parameters, the identification and modeling processes shall be harmonically
integrated.
1.1 - MOTIVATION
The identification of physical parameters is useful in many scientific applications,
especially in the aerospace field and robotics that always require accurate mathematical
models for control purposes; modeling and identification are very important for the
mission success. There exists big diversity of identification methods in time domain as
well as in frequency domain. Some methods use deterministic approaches (Algorithms
based on least squares, for instance), others stochastic approaches (Maximum
likelihood). Besides, these methods are divided into two groups: off-line and on-line.
Due to implementation and convergence problems, the number of on-line algorithms is
limited. In the robotic area, few works focusing on modeling and on-line identification
for physical parameters have been found. On the other hand, on-line algorithms are very
appropriate to be used in high performance control (essentially in adaptive control, for
instance) and also in the physical parameters behavior analysis. A big number of
aerospace systems present some kind of non-linearity, in the robotic systems; this non-
linearity appears due to damping and stiffness effects, which are typical in robots joints.
The problem of non-linearity brings big problem in the identification process, because it
excludes most of the existent identification algorithms. On the other hand, the non-
linear terms are very important when the goal is to study the physical parameters
behavior. The identification process may also not deliver accurate models (or
parameters) if the model used is not appropriate (wrong linearization for instance). This
27
illustrates the strong relationship between identification and modeling.
The robot dynamics is typically complex; this complexity is increased if the robot shall
operate in hostile environments, like in space applications. Under space operational
conditions, the robot dynamics will be affected mainly by big temperature variations,
micro-gravity, material degradation, etc. If the control laws use state information, the
robot performance will be also affected. This problem can be minimized by using an on-
line algorithm, which updates constantly the robot parameters, ensuring optimal
performance in all operational conditions.
Thus, this work represents a further step in the investigations to find new algorithms and
also in the improvement of the existent ones. The derived algorithms can identify both
linear and non-linear parameters. In addition, a detailed modeling of robot joint is
performed and applied to the IRJ experiment. During the research process, it turns out
that the failure situation shall be also considered. Therefore, a strategy and algorithms
that make this problem minimal have been exhaustively investigated and tested. As a
result, an algorithm has been derived that is able to detect and isolate failures in the
essential sensors. The main contributions of this thesis are the detailed modeling of the
robot joint, the investigation, development of news identification algorithms, detection
and failure isolation in sensors. Thus new findings have been observed in modeling,
identification and failure detection fields. All algorithms and strategies developed have
been tested by using real measurements from the IRJ experiment assembled in DLR
(Deutsches Zentrum für Luft - und Raumfahrt).
Finally, it is important to note that all algorithms and strategies derived in this research
for the specific system of a robot can be applied in the identification process of any
dynamic system.
1.2 - REVIEW OF LITERATURE
Many works using the off-line identification approach can be found in the literature.
The techniques presented can be used in the identification process of many dynamic
systems as well as in the robotic field. The majority of these methods are based on the
least squares (LS) approach. The procedures in the identification process are also
similar; the measurements are collected and after processed in order to identify the
28
unknown parameters. In the following, a summary of works related to this thesis is
presented.
1.2.1 - IDENTIFICATION ALGORITHMS
Using a least squares estimator, Fortescue et al (1981) designed an algorithm with a
variable forgetting factor. This algorithm has presented some improvements, avoiding
the explosion of covariance matrix and consequently the control instability. Canudas de
Wit and Carrillo (1990) have presented a modified version of EW-RLS (Exponentially
Weighted RLS) where the forgetting factor has been optimized by using the Lagrange
multipliers technique. The algorithm presented good performance in the applications
where the errors are known (bounded). On the other hand, the algorithm may stop and
do not work properly in some particular conditions. The parametric identification
problem of a system with uncertainty, priori information and bounded noise has been
studied by Tempo (1995) and a worst-case algorithm has been derived. The derived
algorithm belongs to the smoothing class and the main innovation is the computation of
the error by the SVD of the system model. Using the recursive incremental estimation,
the identification problem of a time variant system has been studied by Zhou and Cluet
(1996). In this approach, the system model is assumed to be not a constant but a time
variant one. This approach can be applied in the black box formalism, an ARX model,
for instance. In order to eliminate the bias in the LS estimates, Zhang and Feng (1997)
have proposed a procedure, which uses two filters that are used to filter the system input
and output signals respectively. After, an augmented system with known poles and
zeros is obtained. Using this procedure, the estimation process can be independent of
the noise model used. Based on set membership idea, Bai and Huang (2000) presented a
LMS (Least Mean Square Algorithm) and WRLS estimators focusing the problem of
tracking parameters variations and also decreasing the noise sensibility. In order to
improve track time variant systems, Lozano et al (2000), have introduced modification
in the standard LS algorithm. The modifications have been performed by introducing
additional conditions in the parameters update law as well as in the covariance matrix.
This algorithm gives a bounded estimation and maybe useful in the adaptive control
context, despite of its complexity.
29
1.2.2 - PARAMETERS IDENTIFICATION APPLIED IN ROBOTICS
The identification task applied to robotic systems has received expressive attention in
the last decades. The importance growth comes from the fact that the industries, in order
to fulfill the market requirements have used a lot of automation in the production
process. In the aerospace field, the use of robot becomes common and very attractive,
especially after the ISS (International Space Station) project, because the purposes of
ISS are almost impracticable without use of robots.
The use of identification techniques to improve the reliability in the modeling process
and also to increase the accuracy of the robot task has begun around two decades ago. In
the following, it is presented some works that directly focus the problem studied in this
thesis.
In order to identify the damping and the inertia of a robot equipped with rotary joints,
Olsen and Bekey (1985) have derived a formulation where the dynamic equations are
written in terms of linear combinations of measurements and unknown parameters.
Thus, the parameters can be identified by using a procedure based on a standard least
squares algorithm. Using the Newton-Euler equations to obtain a linear relationship
between measurements and inertial parameters, Atkeson et al (1986) have made some
comparisons between the real model (obtained by identification techniques) and models
obtained from CAD/CAM. The results have shown significant differences between the
models, the accuracy of identified model is expressive. The optimization process has
been carried out by using standard least squares. Using an industrial robot as test bed,
Spechet and Isermann (1988) applied the standard RLS in order to identify some
dynamic parameters like inertia, friction and gravitational force. The strategy used has
shown that the use of an integrated procedure control/identification results in a good
improvement in the robot performance and accuracy. The problem of identifying
parameters in a multi DOF (degree of freedom) robot has been studied by Canudas de
Wit and Aubin (1990). A sequential identification procedure has been proposed. In this
strategy, the identification process starts from the external joints (end effector) to the
internal ones. Therefore, in this process, the parameters that belong to the high level are
considered constant in the lower level. This idea can decrease the number of parameters
to be identified but the coupling effects are lost. The optimization processes can be done
30
by using an off-line (standard least square) or an on-line procedure (WRLS for
instance). In order to design a gain-scheduled control, Gomes and Chrétien (1992) have
written the dynamic equations in an appropriate way (linear combination of parameters
and measurements), making some linearization and using a friction model that is
linearly dependent in the measurement. The results have again shown that the
combination control/identification is very useful, bringing significant improvement in
the accuracy level. Using a simple model (decoupled) and its harmonic solution, Pfeiffer
and Hölzl (1995) have shown that it is possible to recover some dynamic parameters by
applying some static and dynamic torque in the system. This strategy allows that the
identification be simplified. Hanssen et al (2000) has developed a strategy that does not
make use of any force/torque sensor to identify big number manipulator wrist
parameters. In this modeling process, the system is considered as a rigid body and the
final model has been linearized.
In the situation where the goal is just to track the plant output, namely, the physical
meaning of the parameters are out of interest, there exist several ways to perform this
task. If the system has linear behavior or if the non-linearity is not too strong, the
methods ARX, ARMAX, BJ, etc. are able to give satisfactory results. If the system
under investigation has strong non-linearity, the NARMAX (Non-linear ARMAX)
approach should be used. An interesting solution is given by Blaszkowski et al (1998),
the discrete time deconvolution technique has been used to evaluate the system
parameters. Using this technique, the impulse response of a machine tool has been
estimated with good accuracy. By applying the non-linear filtering technique Elhami e
Brookfield (1997) have proposed a sequential identification process of Coulomb friction
and also viscous damping of a robot joint. The big complexity in the modeling process
of a robot joint has been also addressed and an asymmetric model for the Coulomb
friction has been also proposed. It is evidenced the difficulty in modeling (especially,
damping) the robot joint with high fidelity. One can find many works in tribology, the
study of surface contacts, which incorporates three main groups: friction, damage and
lubrication. Armstrong-Helouvry (1991, 1992) presented several empirical models that
try to describe the friction behavior with different levels of fidelity. These works can be
considered as a good starting point in the identification task of friction and damping.
Another relevant work has been performed by Olsson (1992), where several details of
31
damping and friction have been discussed. In addition, a new friction model has been
also proposed.
The effect of temperature variation in the friction behavior has been investigated by
Heimann (1999). It has been shown that the friction in the robot gears is extremely
dependent of operating temperature. The physical parameters have been identified by
using an off-line procedure (standard LS). When high accuracy is required, as in space
applications, it is important to take into account internal characteristics of robot gear.
Due to almost absence of backlash and also thanks to the high reduction ratio obtained
in a very compact mechanism, the Harmonic Drive (HD) has wide application in
robotics as well as in aerospace fields. The HD has a particular construction: it is very
simple from the mechanical point of view but complicated to model if the fidelity level
is high. The main problem in modeling the HD comes from the inherent non-linearity in
the friction part as well as in the stiffness behavior. Marilier and Richard (1989)
presented a simplified model to represent the dynamics of a robot joint. This model has
been used in the control of an industrial robot with success. An exhaustive HD
modeling work has been performed by Tuttle (1992); different HD models are proposed
and a comparative test is also shown. Seyfferth et al (1995) studying the HD dynamics
have proposed a model where the stiffness is represented by a quadratic function. The
hysteresis concept has been also introduced in order to improve the relationship
measurements/models.
The amount of work focusing the analysis and the dynamics of robot with space
purposes is not so big and few works directly related to this subject are found. In the
following, some works related with modeling, models and identification strategy are
presented. Gorter et al (1994) has presented a simplified model for HERA (robot)
reduction joint. The robot HERA originally was idealized to operate in the European
space shuttle HERMES. Several problems related to operations in space and the
inherent effect in the control has been addressed. Using the ETS – VI (Engineering Test
Satellite -VI) as test bed, Adachi et al (2000) have developed an experimental procedure
to check and validate the ETS – VII mathematical model that has been obtained on
ground. The experiment purpose is also to compare two different identification
techniques: a polynomial black box and the subspace approach. Using these two
32
approaches, the main goal is only to estimate the plant (satellite) output given a known
input. There is no interest in either modeling or behavior of physical parameters. In this
comparison, the subspace approach has presented better performance than the
polynomial black box models. The use of subspace methods in robot identification is
not so common due to the typical linear characteristic of these methods. Another
application of subspace identification methods in robotics is presented by Johansson et
al (2000). Several subspace-based methods have been tested and compared. In parallel,
a procedure to evaluate the friction force in the robot joint has been also proposed. Shi
et al (2000) developed an experiment to investigate the robot dynamics in operational
conditions similar to that ones encountered in space. The experiment consists in a two-
joint articulated manipulator horizontally assembled under effect of airflow. The friction
in the joint has been identified independently; only the HD has been used, in the test the
link has been removed. It has been shown that the identification is extremely useful in
the process of gathering precise mathematical models.
In the robot identification process, it is necessary to follow some standard steps; one of
them is the selection of the trajectory that will ensure that all parameters are properly
excited. There exist several procedures that allow one to obtain an optimal trajectory.
Swevers et al (1997) have proposed an interesting procedure that makes use of a series
of sine and cosine to design an optimal trajectory. This strategy is efficient if the gears
are considered as a rigid body, but can not be directly applied if a new degree of
freedom (stiffness) is introduced in the gear model. Another theory that is commonly
used to check the excitation levels is the minimization of the measurement matrix
condition number. This approach has been applied by Armstrong (1987) and Heimann
(1999).
1.2.3 - SENSORS FAILURE
The problem of detecting and isolating failures is extremely important in engineering
systems. In the literature, two approaches can be found: one that is based on hardware
redundancy and another one that uses filtering and analytical theory. Although this field
is very wide and offers different options to detect and isolate failures, this thesis
contributes with a simple and efficient algorithm based on state space identification. In
the following, some related works are focused.
33
The detection and failure isolation in linear systems has been studied by Caglayan
(1980). An algorithm based on LMS (Least mean squares) has been derived, where its
main characteristic is the reduced number of filters used in the detection and isolation
process. A good survey focusing different approaches and useful techniques in detection
and failure isolation is presented by Gertler (1988). Using the analytical redundancy
strategy, Yang et al (1988) have presented a procedure based on master and slave ideas
to detect failures in systems where the parameters are not completely known. The
implementation of this idea has been carried out by using a RLS estimator. In practice,
such procedure restarts the estimator when the errors are bigger than a specified value.
This strategy can be successfully applied in problems where the main goal is to estimate
the plant output, only. Using the state-space identification approach, Zimmerman and
Lyder (1993), presented a strategy to diagnose failures in sensors mounted in flexible
structures. The detection process is based on the analysis of the state-space matrices
rather than in the output of a bank of filters that are commonly used. This approach
associated to state-space identification becomes very interesting because it allows one to
detect and also to determine the possible failure characteristics. Applying the bilinear
theory, Yu and Shields (1996) proposed a bilinear filter that uses unknown inputs. Such
algorithm has been applied in a hydraulic machine and the results are satisfactory if
some restrictions in the error matrix are satisfied. Another survey focusing the problem
of failure detection and isolation has been presented by Leohardt and Ayoubi (1997). In
this survey, the use of Artificial Intelligence like neural network, fuzzy logic and
genetic algorithm in the failure detection problem has been also discussed. An extensive
study in the failure detection and isolation procedures is presented by Basseville (1997).
Several criteria to monitor the residues and to choose the decision rules are presented.
Using the minimax approach, Yin (1998) present a procedure to classify and detect
failures. The procedure is designed in such a way that a balance between robustness and
optimality is obtained. The minimax procedure is used to minimize the maximum
number of false failure that is previously declared. The problem of detecting and
isolating multiply failures has been studied by Keller (1999). A special form of Kalman
filter is used to detect and isolate failure that can occur sequentially or simultaneously.
When a failure is declared, the correspondent component is removed from the
estimation process, minimizing the effect in the states are to be estimated.
34
35
CHAPTER 2
MODELS AND MODELING 2.1 - INTRODUCTION
Normally, the parameters identification is based on models or structures that are
responsible for the representation of the physical system under investigation.
There exist several ways to represent a real system. This representation can be
performed by using differential equations that govern the system, recurrence
equations, state-space models, etc. The last two alternatives are commonly
known as black-box formalism. This idea comes from the fact that the main goal
is only to estimate the plant output given the inputs and the previous outputs. The
parameters meaning or the “dynamic” that govern the system is out of interest. In
such models, the estimation accuracy is obtained by adjusting the number of
previous past outputs to be used or by selecting the system order, in case of state-
space representation. The choice between models represented by differential
equations or by black box models depends exclusively on the requirements to be
fulfilled. The models where the parameters have physical meaning are called
phenomenological models, the models where the parameters do not have physical
meaning are called behavioral models. Table 2.1 shows the main characteristics
of these two classes of models.
Phenomenological models Behavioral models Parameters Have a concrete meaning Do not have concrete meaning
Simulation effort &
Processing time Hard and slow Easy and quick
Priori information Taken into account Not taken into account Validity Domain Wide (if the structure is
correct) Restricted
Although, the main purpose of this thesis is the identification of physical parameters,
namely phenomenological models, in the following some classical black box models are
TAB. 2.1 - MODELS COMPARISON
36
presented. This kind of identification approach (black box) is used in the failure
detection strategy to be presented later.
2.2 - BLACK BOX MODELS
2.2.1 - ARX MODEL
The ARX model is directly related with a transfer function of the system. This is one of
the simplest models of transfer function and this is the reason because it is widely used.
The ARX model is defined as a linear difference equation between inputs and outputs.
For a SISO system, the ARX model is written as
)()()()()()( 11 kenkubikubnkyaikyaky bbnaan +−++−=−++−+ LL (2.1) where k = time index
)( iky − = output in previous instant )( iku − = input in previous instant
)(ke = residual error an = number of a coefficients bn = number of b coefficients
Equation (2.1) can be written as
)()()()()( kekuqBkyqA += (2.2) where the polynomials )(qA and )(qB are defined in terms of the delay operator 1−q
anan qaqaqA −− +++= L1
11)(
bnbn qbqbqB −− ++= L1
1)(
The term )()( kyqA in Equation (2.2) corresponds to the AR parcel (Auto Regressive) of
ARX model and the term )()( kuqB corresponds to the external input. Equation (2.2) can
be re- written in transfer function form
37
)()(
1)()()()( ke
qAku
qAqBky += (2.3)
The term )()(
qAqB represents the plant discrete transfer function.
2.2.2 - ARMAX MODEL
The ARMAX model is similar to ARX model, which uses a sequence of past inputs that
are filtered by the model. However, the ARMAX model also filters the residual errors
aiming at a better disturbance characterization. In analogy with the ARX model, the
ARMAX model can be considered as transfer function of the system that is being
identified.
For SISO systems, the difference equations ARMAX is written as
)(
)()()()()()()( 111
ccn
bbnaan
nkec
ikeckenkubikubnkyaikyaky
−+
+−++−++−=−++−+ LLL
(2.4) where
)( ike − = is a white noise in the previous inputs
cn = is the number of c coefficients Similarly to ARX model, the ARMAX model is written as
)()()()()()( keqCkuqBkyqA += (2.5) where )(qA and )(qB are the same as already defined in ARX model and )(qC is
defined as
cncn qcqcqC −− ++= L1
1)(
The ARMAX model incorporates the additional term )()( keqC that is correspondent to
moving average part.
Writing Equation (2.5) in transfer function form one obtains
38
)()()()(
)()()( ke
qAqCku
qAqBky += (2.6)
Similarly to ARX model, the term )()(
qAqB is the discrete transfer function of the system
and )()(
qAqC works like a filter in the residual error.
In general the models can be written in a generalized form
)()()()(
)()()()( te
qDqCtu
qFqBtyqA += (2.7)
Depending of the polynomial used, different SISO black box models can be obtained.
Table 2.2 shows some common combinations
TAB. 2.2 - CONVENTIONAL BLACK BOX MODELS
2.2.3 - STATE SPACE MODELS
2.2.3.1 - CONTINUOUS SYSTEMS
In the state space formalism, the relationship between inputs, noise and output signals
are written as a set of first order differential equations or as difference equations using
an auxiliary vector )(tx . This formulation was widely used after the Kalman (1960)
work in the field of prediction and linear quadratic control.
The majority of physical parameters can be represented by continuous model rather than
discrete one because the most of the physical laws (Newton’s law, relationship in
electrical circuit, etc.) are expressed as continuous relationship. Thus the modeling task
POLYNOMIAL IN EQUATION (2.7) MODEL NAME B FIR (FINITE IMPULSE RESPONSE)
AB ARX ABC ARMAX AC ARMA
ABD ARARX ABCD ARARMAX
BF OE (OUTPUT ERROR) BFCD BJ (BOX-JENKINS)
39
normally leads to representation in the form
)()()()()( tuBtxAtx Θ+Θ=& (2.8) where the matrices BA and have appropriate dimensions ( nn × and mn × , respectively
for an n dimensional state and an m dimensional control vector). The differentiation
with respect to that time is represented by a dot over the variable, besides the vector Θ
typically corresponds to unknown physical parameters, material constants, etc.
Modeling, normally is performed in terms of state variables that have physical meaning
(position, velocity, etc.) and the output is normally a combination of the states. Defining
)(tη as the output of ideal sensors
)()( tCxt =η (2.9) defining p as differentiation operator, Equation (2.8) is written as
[ ] )()()()( tuBtxApI θθ =− (2.10) meaning that the transfer function between η and u in Equation (2.9) is
[ ] )()(),(
)(),()(1 θθθ
θη
BApICpH
tupHt−−=
= (2.11)
Thus, it is obtained a system transfer function model, which is parameterized in terms of
physical coefficients.
2.2.3.2 - DISCRETE SYSTEMS
Since most of the studies make use of computational process and also due to data
acquisition process, normally it is necessary to represent the systems in discrete form.
By assuming that the inputs are constant during one sampling time T
TktkTkTuutu k )1( ),()( +≤≤== (2.12) then Equation (2.8) is easily solved from kTt = to TkTt +=
)()()()()( kTuBkTxATkTx dd θθ +=+ (2.13)
40
where TAd eA )(θ=
τθτ
τθ dBeBT
Ad ∫
==
0
)( )(
Similarly to Equation (2.11), the system discrete transfer function model is represented
by
[ ] )()(),( 1 θθθ ddd BAqICqH −−= (2.14) 2.3 - PHENOMENOLOGICAL MODELS
2.3.1 - INTRODUCTION
The phenomenological models class is composed by models obtained by using physical
laws in the modeling process. Normally, such models are represented by a set of
differential equations. In this thesis, the main purpose is to study and characterize the
physical parameters of a robotic joint; the structure (model) to be adopted belongs to the
phenomenological models group. Namely, the models are obtained by applying physical
laws, which govern the dynamics of the system under investigation. Such models need
parameters that are partially or completely unknown; their behavior maybe also
unknown. An effective solution to determine these parameters is to use identification
methods. In this case, the parameters are obtained by using the information
(measurements) that comes from the sensors placed in the real system. In Chapter 3,
different identification methods are presented that can be used to solve this problem.
2.4 - MODELING
2.4.1 - INTRODUCTION
The mathematical modeling of a system is an essential step in any scientific process,
since (normally) all operation and control planning are based on such models. Another
point that is related to the modeling task is the required fidelity degree in representing
the real system. There exists a direct relationship between fidelity, complexity and
accuracy. Therefore, it is necessary to find an “equilibrium” point between complexity
and accuracy. Specifically in this work, the derived models will be used later by the
identification algorithms. In this case, a complex model can introduce an excessive
number of parameters and also non-linear terms that make the identification task very
41
complicated and also may bring some inaccuracy on the results. Thus, in the following a
detailed modeling of a robotic joint is presented where the complexity level is gradually
increased aiming to get a reasonable degree of fidelity.
2.4.2 - HARMONIC DRIVE (HD) GEAR MODELING
The Harmonic Drive gear is a reduction system that due to its remarkable features such
as
• Compact size;
• Light weight;
• High reduction ratio;
• High performance;
• Good positioning and repeatability;
• High torque capacity;
• No backlash;
• High torsion stiffness;
• Reversible: it is possible to reduce or increase the velocity,
is widely used in the aerospace and robotics fields.
The Harmonic Drive gear is basically composed by three elements (Figure 2.1):
a) Wave Generator (wg)
The wave generator is thin bearing with elliptical shape working as an efficient torque
converter.
b) Flexspline(fs)
The flexspline is a flexible cylinder made of steel with outer teeth.
42
c) Circular Spline (cs)
The circular spline is a solid ring made of steel with internal teeth.
Fig. 2.1 - Internal view of Harmonic Drive
By combining these elements and the assembly configuration, several mathematical
models can be obtained, depending of details taken into account in the equations.
2.4.2.1 - WORK PRINCIPLE OF HARMONIC DRIVE GEAR
The fs has a diameter slightly smaller than the cs, this results in two teeth less in its
external circumference. It has elliptical shape due to the action of wg and its teeth its
teeth engage with the teeth on the cs across the major axis of the ellipse.
As soon as the ws starts to rotate clockwise, the zone of tooth engagement travels with
the major elliptical axis.
When the ws has turned through 180 degrees clockwise the fs has regressed by one
tooth relative to the cs.
Each turn of the ws moves the fs two teeth anti-clockwise on the cs. (Figure 2.2)
43
Fig. 2.2 - Work principle of Harmonic Drive In the following, four models are presented; starting from a very simple to more
complicated ones. It is also presented two different HD assembly configurations. These
configurations are different due to rotation direction: the first one, the input (motor side)
and output (link side) have the same rotation orientation. In the second, the input and
output rotate in opposite directions. Finally, the configuration used to represent the IRJ
(Intelligent Robotic Joint) to be described later is presented. The schematic
representation and model derivation follow similar direction presented by (Tuttle,1992).
The modeling procedure is based on the Newton laws, which are used to determine the
dynamic force interaction and to derive the equations of motion. The two HD different
configurations are presented respectively in Figure. 2.1 and Figure 2.2. In these figures,
the HD gear is considered as a black box (meaning that the gear is not modeled yet)
with three ports that transport the torque and rotation “felt” by the wave generator,
flexspline and circular spline. Later, this black box is replaced by one of the HD models
to be derived.
2.4.3 - DIFFERENT CONFIGURATION IN THE HD ASSEMBLY
2.4.3.1 - CONFIGURATION 1
In this configuration, the wave generator is attached to the motor shaft mounted in
circular spline and the flexspline is fixed. The output is delivered by circular spline.
Two different damping torques are considered, one in the input side and another one in
the output side, in accordance with Figure 2.3.
By applying the Newton’s law, the equation of motion of configuration 1 is written as
44
wginbminin TTTJ −−= _θ&& (2.15) and
inboutbcsoutout TTTJ __ +−=θ&& (2.16) Writing the torque expressions in terms of inθ , outθ and its corresponding velocities and
by assuming that the damping torque is a linear function of velocity, one obtains
)(_ outinininb NbT θθ && −= (2.17)
outoutoutb bT θ&=_ (2.18)
Fig. 2.3 - Configuration 1
The motor torque can be considered proportional to commanded current, thus
mtm IKT = (2.19) The torques wgT and csT are obtained by specifying the kinematic constraints on the
three HD model components to be presented later. Since each HD part is connected to
an inertia part or fixed, the kinematic constraints are represented as
45
inwg θθ = (2.20)
0 =fsθ (2.21)
outcs θθ = (2.22) 2.4.3.2 - CONFIGURATION 2
In this configuration the wave generator is attached to the motor shaft that is mounted in
the circular spline. The output is obtained through flexspline. Because the moving parts
are different from the configuration 1, a new model is derived. Similarly to
configuration 1, two damping torques has been also taken into account: one in the input
side, another on the output side.
According to Figure 2.4, the equations of motion for configuration 2 are
wginbminin TTTJ −−= _θ&& (2.23)
outbfsoutout TTJ _−−=θ&& (2.24) The damping torque on the output side is obtained in accordance with Equation (2.18)
and the input damping torque is given by
inininb bT θ&=_ (2.25) The kinematic constraints in the three HD ports, for this configuration, is determined by
inwg θθ = (2.26)
outfs θθ = (2.27) 0=csθ (2.28)
46
Fig. 2.4 - Configuration 2
2.4.4 - HARMONIC DRIVES MODELS
2.4.4.1 - IDEAL TRANSMISSION – MODEL 1
This is the simplest HD model; its configuration is shown in Figure 2.5. In this
configuration, the HD works exactly as a simple gear reduction box.
Fig. 2.5 - HD model 1– Ideal transmission
Based on manufacturer’s catalog values, using the reduction ratio N, the HD
transmission model is represented by
47
fscswg NN θθθ −+= )1( (2.29)
fscswg NN ωωω −+= )1( (2.30)
fscswg TN
TN
T 1)1(
1=
+= (2.31)
2.4.4.2 - TRANSMISSION WITH LOSSES – MODEL 2
In the HD gear, there exists considerable energy dissipation. This dissipation is
normally related to two different sources
• Loss in the wave generator bearing;
• Loss in the gear tooth meshing.
The main energy dissipation occurs due to friction in the flexspline and circular spline
teeth. Based on this observation, the following HD model is proposed: A dissipation
element is included between flexspline and the circular spline. This new configuration is
shown in Figure 2.6.
Fig. 2.6 - HD model 2 - Incorporating friction losses.
The addition of loss due to damping in the ideal transmission results in a new set
of equations
48
bfsnfs TTT −= _ (2.32)
bcsncs TTT −= _ (2.33) The torque fsnT _ is defined as positive in opposite direction of csnT _ . Using the
relationship for the torques in the ideal transmission model (Equation (2.31)), the torque
in the wave generator is related to csnT _ and fsnT _ by
fsncsnwg TN
TN
T __1
)1(1
=+
= (2.34)
Or by using the Equations (2.32) – (2.34),
NTT
NTT
T bfsbcswg
+=
++
=)1(
(2.35)
According to experimental results, (Tuttle, 1992), the friction in the HD gears have
typically three main components
• Velocity independent term;
• Non-linear term, velocity dependent;
• Cyclic friction torque depending of operational conditions.
Based on this assumption, a reasonable model for bT is
cyclicbdynamicbb TTTT __b_constant ++= (2.36) where
constantb_constant bT = (2.37) 3
__2__1_ )()( fsncsnfsncsndynamicb bbT ωωωω −+−= (2.38) ))(( ___ bfsncsnbcyclicb sinAT γθθ +−= (2.39)
Using the model kinematic constraints, one obtains the following relationship
; ; ; ____ fsfsncscsnfsfsncscsn θθθθωωωω ====
49
2.4.4.3 - TRANSMISSION WITH LOSSES AND STIFFNESS – MODEL 3
In a realistic representation, the HD gear can not be represented as a rigid body because
it presents a stiffness profile that is directly related with the applied torques. Thus, the
model derived in Chapter 2.4.4.2 brings some improvements if compared to ideal
transmission, but is unable to characterize a real system with an acceptable fidelity
degree. Therefore, in this section a new model is presented, which incorporates in the
model the stiffness component. The schematic representation of this model is shown in
Figure 2.7.
The majority of the equations derived for model 2 is still valid for model 3, thus, only
the new relationship related with the stiffness will be shown. By using the ideal
transmission relationship of Equation(2.31), kT can be represented as
fsncsnwgnwgk TN
TN
TTT ___1
)1(1
=+
=== (2.40)
In order to represent the HD stiffness, several functions can be used. Very often a cubic
model is used to incorporate the non-linear behavior,
3_2_1 )()( wgnwgwgnwgk kkT θθθθ −+−= (2.41)
where 21 and kk are the stiffness constants for the linear and cubic terms respectively.
To complete the model derivation, it is sufficient to note that the kinematic constraints
show that cscsnfsfsn θθθθ == __ and . By using this information, wgn _θ can be
obtained by using the ideal transmission (Equation(2.31))
fscswgn NN θθθ −+= )1(_ (2.42) Finally, using Equation (2.42) in Equation (2.41), the following relationship for the
stiffness torque is obtained
[ ]( ) [ ]( )321 )1()1( fscswgfscswgk NNkNNkT θθθθθθ −+−+−+−= (2.43)
50
Fig. 2.7 - HD model 3 – Incorporating friction losses and stiffness.
2.4.4.4 - TRANSMISSION WITH FRICTION LOSS, STIFFNESS AND KINEMATIC ERROR
MODEL 4
In applications that require high positioning accuracy, the kinematic errors that are
typical in the HD shall be taken into account. The kinematic errors originate from
different sources, but the majority of these errors appears due to manufacturing errors or
even in the assembly phase. In this section, a new HD model is given, which
incorporates the kinematic errors in the joint dynamic. The schematic representation for
this model is shown in Figure 2.8.
Using Equations (2.29), (2.30) and (2.31), the elements of ideal transmission can be
characterized by
fsncsnwgn NN ___ )1( θθθ −+= (2.44)
fsncsnwgn NN ___ )1( ωωω −+= (2.45)
fsncsnwgn TN
TN
T ___1
)1(1
=+
= (2.46)
The friction element in the model can be represented by the same relationship of model
2 (Equation (2.36)). According to Figure 2.6, the stiffness torque component is
described by
51
3_2_1 )()( outkwgoutkwgk kkT θθθθ −+−= (2.47)
And the kinematic error can be characterized by a harmonic series of three error
components
)4()2()( 332211 γθγθγθθ +++++= wgwgwgerfn sinAsinAsinA (2.48) Using Equation (2.48), the position constraint in this element can be specified as
erfnoutkwgn Nθθθ += __ (2.49) And the velocity constraint in this element is
dtd
N erfnoutkwgn
)(__
θωω += (2.50)
Using Equation (2.48),
)4cos(4)2cos(2)cos()(
332211 γθωγθωγθωθ
+++++= wgwgwgwgwgwgerfn AAA
dtd
(2.51)
erfnwgerfn
dtd
ωωθ
=)(
(2.52)
)4cos(4)2cos(2)cos( 332211 γθγθγθω +++++= wgwgwgerfn AAA (2.53)
Fig. 2.8 - HD model 4 – Transmission with friction stiffness, and kinematic error
52
Using Equation(2.52) in Equation(2.50), one obtains
erfnwgoutkwgn N ωωωω += __ (2.54) By applying the energy conservation law, the velocity constraint can be used to
determine the torque constraint in the position error element. Assuming
outin PP = (2.55)
wgnwgnoutkoutk TT ____ ωω = (2.56) And using Equation (2.54) in Equation (2.56)
)( ____ erfnwgoutkwgnoutkoutk NTT ωωωω += (2.57) The velocity variation along the spring is small if compared with the mean velocity,
outk _ω can be considered identical to wgω , then, Equation(2.56) is rewritten as
)1(__ erfnwgnoutk TT ω+= (2.58) The modeling task is completed by specifying the following relationship
bfsnfs TTT −= _ (2.59)
bcsncs TTT −= _ (2.60)
outkkwg TTT _== (2.61)
cscsn θθ =_ (2.62)
fsfsn θθ =_ (2.63) 2.5 - HD CONFIGURATION USED IN THE IRJ EXPERIMENT
2.5.1 - IRJ EXPERIMENT DESCRIPTION
The IRJ experiment has been designed aiming to increase the confidence in both
modeling and parameters identification of a robot gear for space application purposes.
Thus, there are two main goals: First to select dynamic models that offer a good
relationship between complexity and fidelity in representing the real system; second, to
develop identification algorithm and strategies suitable for identification in real time
(recursive algorithms).
The IRJ experiment consists in a single robotic joint (Figure 2.9) with multiple sensors
53
in both, input (motor side) and output (link side) of the joint. The robotic joint is based
on a special version of HD (lightweight unity), where the states are measured with very
high accuracy:
a) motor side
• position (analog hall sensor)
• velocity (analog differentiation)
• commanded current
b) link side
• position (opto-electronic sensor)
• torque (strain gauge system)
• accelerometers (placed on the top of the link)
These sensors give a high degree of intelligence to the joint. The experiment also uses a
new DC motor (brushless) that has been designed by DLR in order to provide hollow
axes where all cabling is fed through.
2.5.2 - JOINT MODELING – KINEMATIC AND DYNAMIC EQUATIONS
The main emphasis of identification experiments with space application purposes is to
gain confidence in the modeling process by accurate description of time variant
parameters that are very influent in the system dynamics The damping, for instance,
mainly the viscous one, friction and stiffness within the gears, all are expected to be of
strongly non-linear nature. Therefore, the following investigations have been restricted
to the modeling and understanding of the non-linear dynamics of one single intelligent
joint.
The model is based on Newton’s method that is used to determine the dynamic force
interactions and to derive the equations of motions of the joint. Since the primary goal is
to obtain a reliable joint dynamics model, much effort is given to detect the main
54
sources of modeling unknowns or even errors. These sources obviously are expected to
occur predominantly within the harmonic drive gear. Here, elastic effects, mainly
arising from the flexspline (fs), and non-linear types of viscous damping and dry friction
of Coulomb type have been identified as the critical ones.
The configuration used in the IRJ experiment is the following: the circular spline is
fixed, the wave generator is attached to the motor shaft and the flexspline is connected
to the link (output). Additionally, the torque sensor is connected to the flexspline.
Damping torques have been considered in both: input and output side. According to
Figures 2.10 and 2.11, and considering a rotary joint, the equations of motion for HD
are described by
wgindminin TTTJ −−= _θ&& (2.64)
loadfscsdoutdfsoutout TTTTJ +−−= __θ&& (2.65)
Fig. 2.9 - Experimental configuration of IRJ for identification
where amm IKT = is the applied torque with motor constant mK and commanded
current aI . inJ and outJ are the input (motor side) and output (link side) inertias, inθ
55
and outθ the respective angular position. The elasticity within the HD gear is given by
the stiffness torque stiffT , with wgdstiffwg TTT _+= on the input side of the gear
according to Figure 2.11, and wgcs TNT )1( += (Equation(2.68)). The various damping
torques are denoted by dT , attributed with appropriate indices. The applied load on the
link side is due to gravity and is given by outgload sinTT θ= (Figure 2.10) with the load
amplitude gT .
Several HD configurations can be obtained, with different kinematic constraints. In the
case of the IRJ experiment, one obtains
inwg θθ = , outfsnfsncsnwgn NNNN θθθθθ ⋅−=⋅−=⋅−⋅+= ____ )1( (2.66) and for the configuration used here:
0_ == cscsn θθ , outfsfsn θθθ ==_ (2.67)
Fig. 2.10 - Dynamic representation of IRJ experiment.
This means that the circular spline is fixed, the wave generator is attached to the motor
shaft and the flexible spline is the output, where the torque sensor is placed. In this
56
configuration, the rotation of input (motor) and the output (link) are opposite. Similarly,
the torque relationship is determined by
wgnwg TT _= , fsfsnwgn TN
TN
T ⋅=⋅=11
__ . (2.68)
Fig. 2.11 - Harmonic Drive model used in IRJ experiment.
2.5.3 - STIFFNESS TORQUE
Based on the information of HD manufacturer’s catalog, this type of gear typically
exhibits a non-linear stiffness profile. Normally, the dependency between the applied
torque and the relative angular position θ∆ is determined by continuous piecewise
functions, )(fTstiff θ∆= , depending of gear operation range. In order to use the
obtained models in the algorithms derived in this thesis, it is necessary to replace these
piecewise functions by a continuous function. Thus, as a first approximation, it is
sufficient to use a third order polynomial
321 )(kkTstiff θ∆θ∆ += (2.69)
with coefficients 1k and 2k to be identified. A typical example of HD gear stiffness
(type HFUC-20-160-2A-GR) it is shown in Figure 2.12. The data from manufacturer’s
catalog is shown in Table 2.3
57
K1 41.3 e Nm/rad K2 40.5 e Nm/rad K3 47.5 e Nm/rad T1 7 Nm T2 25 Nm
Torque range + 150 Nm Max Relative position + 0.15 0
Where Ti represent the torque knots according to Figure 2.10.
The third order polynomial coefficients are:
=)k,k( 21 ).,.( 3Nm/rad9101741Nm/rad4102784 ⋅⋅ .
In agreement with what is expected, the positive value of 2k represents a typical hard
spring in agreement with the increase of the relative position.
Besides, due to the mechanical nature of the system of torque measure with tension
measurement based on strain gauge, it can also be interesting to consider stiffness
related to such sensor’s systems. Thus, it is reasonable to model the torque sensor as an
additional spring connected in series with the flexspline. In total, this would result in a
combined softer spring, and it can be considered as a new stiffness constant that enter as
an unknown parameter in the identification algorithm.
2.5.4 - DAMPING TORQUE
To be more general, the prevailing damping torques dT on the input side, the output
side, and inside the HD gear are assumed to capture two facets of damping behavior, as
having been noticed in the experimental results of some authors (Tuttle,1992). These are
a viscous, highly non-linear part viscT , and a dry friction or Coulomb-type part fricT :
fricviscd TTT += (2.70) where, by experimental results, the viscous part is expected to be a strongly non-linear,
cubic relationship in the angular velocity,
TAB. 2.3 - DATA OF HD TYPE HFUC-20-160-2A-GR
58
321 θθ && bbTvisc += (2.71)
with the linearly depending coefficients 1b and 2b .
Fig. 2.12 - Stiffness torque for HD type HFUC-25-160-2A-GR
For the dry friction part a modified classical Coulomb friction model is taken, that
accounts for the well-known Stribeck effects. This observes the fact that for low
velocities the friction torque is normally decreasing continuously with increasing
velocity, not in a discontinuous manner(Armstrong, 1987 ). Here, three different
empirical mathematical descriptions will be followed within our identification process.
The most common is of the form
)()(1, ωδωω signeTTTT
SSCSCfric ⋅
⋅−+= − (2.72)
where Sω is called the Stribeck velocity and the exponential parameter Sδ is commonly
taken either as 0.5, 1 or even 2. Recalling the maximum Coulomb stiction torque by
NoS TT µ= and the classical Coulomb friction constant torque by NC TT µ= , with
the static friction coefficient oµ , the friction coefficient µ , and oµµ < , Equation
(2.72) gives
59
)()(1, ωµµµδωω signeTT
SSoNfric ⋅
⋅−+⋅= − (2.73)
NT is the normal torque that appears at the appropriate nodes as given in Figure 2.10.
For 0=Sδ Equation (2.73) goes to the classical standard form of Coulomb friction,
)(ωµ signTT Nfric ⋅= (2.74) where oµ has been replaced by µ . It may be noted, that according to several author's
reported results of experimental testing (Armstrong, 1992 and Tuttle, 1992), it may
prove necessary to also add some constant friction part constT , that does not depend upon
the normal torque,
)()( ωµ signTTT constNfric ⋅+= (2.75) The identification strategy to be proposed in Chapter 3 can deal with discontinuities at
zero velocity. However, for future use of more advanced schemes, in a first approach
we have almost circumvented stiction effects by applying a second type of modified
friction model that reduces the range of discontinuity at zero velocity
)()(tanh1
2, ωωω δωω signeTTTT
SSCSCfric ⋅⋅−+⋅= − (2.76)
⋅⋅−+⋅⋅= − )()(tanh
1ωµµ
ωωµ
δωω signeTSS
oN
or even avoids it totally as given by the third modified model by introducing
⋅+⋅⋅= − SS
Nfric eTTδωω
ωω
ωωµ
213, tanh (2.77)
with two more parameters 1ω and 2ω . Moreover, this functional relationship also
avoids the use of a sign-function. The first summation term accounts for the steady state
friction value for higher velocities, and the second term produces the desired 'overshoot'
of friction at very low velocities according to the Stribeck effect. The typical behavior
of the model given by Equation (2.77) is shown in Figure 2.13, where the parameters
60
have been manually adjusted (by selecting the desired friction coefficients, for instance,
130.=µ and 160.o =µ , the other parameters have been selected as 1=Sδ , 0101 .=ω
rad/s, 020.S =ω rad/s and 10802 .=ω rad/s using Equation (2.78). Since the stiction
coefficient oµ is no longer appearing in Equation (2.77), oµ can be recovered by
letting the maximum friction torque being as close as possible to the static friction
value. To avoid the solution of a transcendental algebraic equation for the non-linear
parameters, the first summation term in Equation (2.77) has been neglected for finding
the maximum value, giving the velocity at maximum friction torque approximately by
Smax ωω ≈ and finally the relationship
( ) oss
Nfric eTT µωω
ωω
µ =+⋅= −1
21max3, tanh (2.78)
where 1=Sδ has been set. Given the values for µ and oµ , this equation then can be
solved for one of the free parameters 1ω , 2ω or Sω . It may be noted here, that the
common identification schemes like RLS require linear parameter dependencies. The
friction models are strongly non-linear in the parameters, 1ω , Sω and Sδ , whereas
21 ω is considered a linearly depending parameter. In order to identify such parameters,
a new strategy based on two step algorithm is proposed on Chapter 3.
Special attraction is drawn to the modeling of the damping torque fscs_dT , that is acting
between the circular and the flexspline,
cyclicfricviscfscs_d TTTT ++= (2.79) in accordance with the literature (Tuttle,1992), these periodic have been introduced
variations in the frictional torque captured in a sinusoidal function,
( ) )sin())(sin __ cyclicoutcycliccyclicfsncsncycliccyclic AAT γθγθθ +=+−= (2.80)
As this relationship indicates, frictional torque fluctuations of amplitude cyclicA complete
one cycle every time the flexspline makes one complete rotation relative to the circular
61
spline. To match this model to experimental observations, a phase shift of cyclicγ is also
included. In order to obtain a linear dependency of the two parameters, Equation (2.80)
can be easily transformed to
outoutcyclic cosAsinAT θθ 21 += (2.81) with the new linearly depending parameters cycliccyclicAA γcos1 =
and cycliccyclicAA γsin2 = , from where cyclicA and cyclicλ can be recovered.
Fig. 2.13 - Damping model type 3, Equation (2.77)
Finally, by observing the kinematic constraints, the several torques of Equations (2.64)
and (2.65) can be written in terms of input and output positions, and its respectively
velocities
[ ] ),()()( __ ininindwgdstiffaminin TTTIKJ θθθθθ &&&& −∆+∆−= , (2.82)
[ ]outg
outoutfscsdoutoutoutdwgdstiffoutout
T
TTTTNJ
θ
θθθθθθθ
sinˆ),(),()()( ___ +−−∆+∆⋅= &&&&&
(2.83)
where
inoutwgwgn N θθθθθ −⋅−=−=∆ _ and inoutN θθθ &&& −⋅−=∆ (2.84) the relative angular position and measured speed between the input and the output.
62
63
CHAPTER 3
ALGORITHMS AND IDENTIFICATION STRATEGIES 3.1 - INTRODUCTION
In this Chapter, the theoretical base related with the identification algorithms is
presented. It is composed by several methods and identification algorithms, algorithms
for failure detection and techniques for selection of trajectories. The algorithms used for
identification of parameters with linear dependence in the measurement are based on
Least Squares approach in both: on-line and off-line versions. In order to adapt the
techniques to the needs of the project (on-line identification, robustness against failure,
etc.), modifications are introduced in the algorithms as well as in the strategies. In the
identification process of non-linear parameters, several algorithms have been tested and
finally due to the good robustness and convergence feature of the algorithm MCS (Multi
Level Coordinate Search), which will be presented further, has been selected. In order to
guarantee that the algorithms would continue working even in adverse situations like
sensor failure, a strategy for failure detection and estimation based on state-space
identification approach has been implemented. Thus, the algorithm MOESP (Verhaegen
and Dewilde, 1992) will be also described. On the other hand, any identification process
is extremely dependent not only of the quality (use of accurate sensor) of measurement
but also of the system excitation level. Thus, in this work, it is also presented a new
technique that can not be considered optimal (since no optimization method has been
used), but offers a fast and easy alternative in the choice of the trajectories that best
excite the system to be investigated.
The appropriate identification method for the experiment purposes should guarantee
good performance and to adapt to several requirements. The main points are:
• capability of on-line estimation during data collection phase;
• performing a data pre-processing task to eliminate measurement of bad quality
by using filtering techniques and detrending approaches;
• to eliminate high and low disturbances;
64
• to use phemenological models instead of behavioral models in order to attribute
physical meanings to the parameters
• work not only with linear parameters but also non-linear ones
• to provide efficient algorithms with low computational load (on-line version)
• be robust against sensor failures.
In order to obtain a mathematical model based on the information (measurement) of
input/output of a plant, it is necessary to perform the identification of the system. The
system can be fully identified (system realization – Identifying all the state space
matrices)) by using subspace approach or only identifying some unknown parameters
(parameter identification). In the parameter identification, most of methods are based on
Least Squares approach. In this thesis, both approaches are focused: the LS based
methods are used to identify the parameters and the subspace theory is used in the
failure detection procedure. In the following, a modified version of RLS is presented,
where a variable forgetting factor is introduced.
3.2 - PARAMETERS IDENTIFICATION BY USING RLS ALGORITHM
By assuming a time discrete model
)()()()()()( )1( 1111 tttubtubtxatxaty Tnnnn Θ=+++++=+ φLL (3.1)
where
][ 11 nnT bbaa LL=Θ (3.2)
is the unknown parameters vector, and
][ 11 nnT uuxx LL=φ (3.3)
is the measurement vector. An adjustable predictor model can be described by
)(ˆ)()1(ˆ ttty T Θ=+ φ (3.4)
65
The RLS goal is to minimize the quadratic error
∑=
ΘΘ=t
i
TiiN
tJ1
),(),(211)( εε
(3.5) where Θ−=Θ )()(),( iiyi tφε . The optimal value of )(tΘ based on LS approach using all data available until instant t
is
−
−−=Θ ∑∑
=
−
=
t
i
t
i
T iyiiit1
1
1)()1()1()1()(ˆ φφφ (3.6)
Defining a )(tR matrix and a vector )(tf
∑=
−−=t
i
TiitR1
)1()1()( φφ , ∑=
−=t
iiyitf
1)()1()( φ
Then Equation (3.6) can be written as
)()()(ˆ 1 tftRt −=Θ (3.7) in the recursive form
TtttRttR )()()()()1( φφλ +=+ (3.8) )1()()()()1( ++=+ tyttfttf φλ (3.9)
)1()1()1(ˆ 1 ++=+Θ − tftRt (3.10) where )(tλ is the forgetting factor that will weight the identification process based on
the previous estimation performance.
Applying Equations (3.8) and (3.9) in Equation (3.10) and using Equation (3.6), one
obtains
),ˆ()()1()(ˆ)1(ˆ 1 tttRtt Θ++Θ=+Θ − εφ (3.11) By applying the matrix inversion lemma in Equation (3.8), Equation (3.11) can be written as
66
),ˆ()1()(ˆ)1(ˆ ttLtt Θ++Θ=+Θ ε (3.12) where
)()()()()()()1(
ttPttttPtL T φφλ
φ+
=+ , 1)()( −= tRtP
[ ])()()1()()(
1)1( tPttLtPt
tP Tφλ
+−=+ .
Assuming that )(ˆ tΘ is time variant and that the sensors accuracy are known, the parameters update is given by
)()()()()()()()1(
2 ttPttRtttPtL T φφλ
φ+
=+ (3.13)
[ ])()()()1()()(
1)1( 1 tRtPttLtPt
tP T ++−=+ φλ
(3.14)
where the covariance matrices )(1 tR and )(2 tR represent the model and sensors
uncertainty respectively. A schematic representation of the standard least squares
algorithm is shown in Figure 3.1
Fig. 3.1 - Schematic representation of standard least squares
3.3 - NON-LINEAR OPTIMIZATION - MCS ALGORITHM (MULTILEVEL COORDINATE SEARCH)
In this section, the MCS algorithm (Huyer and Neumaier, 1998) is presented. This
algorithm has been used in the integrated identification process where the linear and
non-linear parameters are identified by different algorithms. In the following, the main
structure of MCS algorithm is presented (Figure 3.2).
67
The algorithm MCS is an intermediary algorithm between the methods that are purely
heuristic and methods that allow an evaluation of the quality of the obtained minimum.
The method guarantees convergence if the objective function is continuous in the
neighborhoods of the global minimum. No other continuity property is necessary, in
contrast with many stochastic methods that only operate at global level, being therefore
very slow. The method MCS possesses local searches which results in a fast
convergence, once the global part of the method has already determined a base point for
the global minimum.
Considering an unconstrained minimization problem
[ ]vuxxf
,)(min
∈ (3.15) with infinite or finite bounds and by using the rectangular box notation, one obtains
[ ] { }nivxuxvu iii ,,1,/:, L=≤≤ℜ∈= Ν
With u and v n-dimensional vectors belong to { }∞∞−ℜ=ℜ ,: and ii vu ≤ for ni L,1= ,
meaning that only points with finite components are considered as elements of the box
[ ],, vu although the bounds can be infinite. If the bounds are infinite, the optimization
process is an unbounded one.
3.3.1 - MCS ALGORITHM
The algorithm MCS finds the minimum (or maximum) through the division of the space
of search in small boxes. Each box contains a distinct point, denominated base point,
where the value of the function is known. The partition process is not uniform, but the
parts where small values of the function are expected, have a larger preference. The
algorithm combines the global search (dividing the boxes in great areas not explored)
and local search (dividing the boxes with the best values of the function). The multilevel
search theory is based on the balance between the global search and the local search.
Denoting the amount of times that the box is processed by { },,1,0 maxss L∈ the boxes
with level maxs are considered small for posterior divisions and the level 0=s indicates
that the box has been already divided and it can be ignored. When a level box
68
( )max0 sss << is divided, its level is defined as zero and its descendant receives the
level 1+s . In each step, the division is made just in the direction of one coordinate.
The algorithm MCS without local search places the base points and the value of the
function of the level maxs in a variable called " shopping basket " (that contains the
useful points). The algorithm with local search accelerates the convergence process by
starting a local search, having as starting point those points before placing them in the
shopping basket.
The algorithm with local search essentially consists in building a local quadratic model
by using triple searches; defining a search direction by means of the minimization of
this quadratic model. The schematic representation of MCS algorithm is shown in
Figure 3.2.
Fig. 3.2 - Schematic representation of MCS algorithm 3.3.1.1 - INITIALIZATION
The algorithm initialization process occurs through the definition of an initial group of
boxes. The divisions of the boxes happen in 3 or more parts that can be defined by the
user as lix (where the value of the function is calculated) and in some intermediary
points, where at least 4 sub boxes are obtained. Namely, in the points
niLvxxu iiLii
lii LL ,1,3 =≥<<<≤
to be defined.
69
First the function is evaluated in the initial point 0x and also defining 0xx =∗ . Then for
ni ,,1L= the function f is evaluated in the 1−L of [ ]vu, . Thus, one obtains
iL couples ( ) ( )il
il Llfx ,,1, L= with
( )( )ll
i
klk
xff
ikxx
=
≠= ∗
(3.16) and lixx =∗ for any il . The point with smaller function value is renamed as ∗x before
to repeat the process for the other coordinate. The numbers ili Lx ,,1L= and the indexes
il are stored in the initialization list. By using the initialization list and the function
values an initial set of boxes is obtained in the following way: the main box is formed
by [ ] [ ]vuyxB ,, = , with 0xx = as a base point and y as corner of [ ]vu, far from x . It is
important to note that x does not necessarily represent the border and that also some or
all y can be infinite. Thus, the actual box is divided along of the n coordinate in iL2
sub intervals, having one of lix as an end point.
The additional divisions are made by using the relationship
( ) i
lli
kli
li LlxxqxZ L,111 =−+= −−
(3.17)
where ( )1521
−=kq is known as golden ratio and 1 equalset is k or 2 . The k values
are selected such that the part of with the lower function values gets the big part of the
interval.
3.3.1.2 - BOXES SPLIT
In the following, the splitting rules are described. First the splitting by rank and after the
splitting rule by gain. The split rule by rank is used to check if all coordinates have been
dived in proper manner.
70
• Splitting by rank
If
( ),1min2 +> njns (3.18) where s is the spilt level, nj is the minimum of times that the coordinate j has been
divided. This means that although the box has already reached a high level, there exists
at least one coordinate along the box that was not divided very often. Then, the index i
among the indexes i with smaller in is chosen as being the smallest iπ (π measures the
variability of the rank). The name “Division by rank” refers to the ranking of the
coordinates in and iπ .
If 0=in , the division is made in agreement with the initialization list in the points
nli Llx ,,1L= and in the points defined by the golden search. The boxes with the
smallest fraction of the golden search (corresponding to the biggest function value)
receive the level ( )max,2min ss + and the other ones the level 1+s .
If ,0>in the n-th component varies between ix and iy and the values of the division
are selected as
( )iii xxZ −+= ''32 ξ (3.19)
where
( )( )
><
><
==
otherwiseyxyxifxysign
yxifysign
yx ii
1000,11000)(10
1000,11000
,subint''ξ
The box is then divided into the points iZ and into the points defined by the golden
search. A third box is obtained with an additional calculation of the function in the point
71
'x obtained by changing the n coordinate of x by iZ . The smallest fraction of golden
search receives the level ( )max,2min ss + and the other two ones, the level .1+s The
division by rank is made in order to reduce the size of the search interval. Thus, the
value of iZ is pre-determined by the box borders. The factor 32 appears due to the fact
that the box is divided in 3 parts, where the second division is made between ix and iZ .
• Splitting by gain
If, ( )1min2 +≤ njns , is the biggest index of split and the split position is determined
from a local quadratic model, which actually is a simple local approximation of .f
Assuming that
( ) ( ) ( )∑=
+=n
iiiexfe
1ξξ
(3.20)
the separable model of ( )ξf is obtained by interpolation in x and in the n2 additional
points (the first two points of each coordinate). For each coordinate i it is defined an
expected gain ie in the function value calculated in the points obtained by changing the
coordinate of the base points. Two cases are possible:
Case 1:
If ,0=in the division is made in agreement with the initialization list in the points
already known and the gain is computed as
{ } ilii
lii fLlfe −== ,,1/minˆ L (3.21)
Case 2:
If ,0>in the n-th component varies between ix and ,iy and by using the quadratic
function of partial correction
72
( ) ( ) ( )2iiiiiiii xxe −+−= ξβξαξ (3.22) an approximation for the maximum expected gain can be calculated just by changing
the value of ix . The division happens in the interval { }''' ,ξξ with ( ) .10/'''ii xx −+= ξξ
Then, the expected gain is calculated as
( )ni
fexff besti
≤≤
<+=
1
ˆminexp
(3.24) with the minimum found in bestf . If the best expected value of the function satisfies
where bestf is the best value of the function at the moment (including the values found
by the local search), it is expected that the box contains a better point and a new division
is made by using the indexes of the components with ie nominal. The condition given
by Equation (3.24) avoids the calculation of functions through the division of boxes
with base points where function values are bad.
3.3.1.3 - LOCAL SEARCH
The local search is performed by using a triple search procedure. The triple search
procedure is described in the following
Considering that the quadratic model
( ) ( ) ( ) ( )bestTbestbestT xxGxxxxgfxQ −−+−+=21 (3.25)
is a good local approximation of .f The triple search is used not only to build the
quadratic model, but also to reduce the function value.
Assuming that 3 vectors rml xxx << are given, the function values is calculated in the
points x with { }ri
mi
lii xxxx ,,∈ (This is the reason for the name triple search). If bestx
represents the current best point in the triple search, denote ( )1,ix and ( )2,ix the points
obtained from bestx by changing its nth coordinate for the other two values in
73
{ }ri
mi
li xxx ,, and by ,, ikxik < the points obtained by the nth and kth coordinates by those
with smallest ( )( ).ixQ . This strategy drives to the following algorithm
( )bestxff =
for i=1 to n
( )( ) ( )( ) iiiii Ggxfxf e Compute ; e Compute 2,1,
updatenot dobut ,Store bestbestnew xx
for k = 1 to i - 1
( ) ( )( )( )( )
ik
bestbestnew
ik
kk
Gxx
xf
xqexQ
Compute updatenot do , update
Compute
Compute 2,1,
end forend if.gfxxxif
end for
ibestbestbestnew , e, update ,
:1≠
When for l,1,k1,-i,1,l, and for ionapproximat the,1, LL ==≤≤ lkl Ggnii are
already computed by the interpolation in the points, which differs only in the first 1−i
components, and bestx is the best point of the triple search in the moment, the
approximation for iii Gg and are obtained by means of quadratic polynomial.
( ) ( ) ( ) ( )221 best
iiibestii
best xGxgxfp −+−+= ξξξ (3.26)
If ( )( ) ( )( ) ( ){ } bestbestii xxfxfxf ,,min 2,1, < is not updated, but this new point is stored
as .newbestx
Assuming that the approximations for 11,, −≤≤ klG li using a 11, −≤≤ ikk has been
74
already calculated, it is possible to compute
( )( ) ( ) ( )( ) ( )( ).2,1
21 2,,,
=
−+−+=
j
xxGxxgxfxQ bestk
jkkkk
bestk
jkkk
bestjk (3.27)
By using the quadratic model Q one obtains
( )( ) ( )( ) .2,1,,, == jxfxQ jiji
Selecting kiik GG = such that
( ) ( ) ( ) ( )( ) ( )( )
( )22
21
21
besti
ikiii
besti
iki
bestk
ikkik
bestk
ikikk
besti
ikii
bestk
ikkk
bestik
xxG
xxxxGxxG
xxgxxgxfxf
−
+−−+−
+−+−+=
(3.28) is satisfied
Again bestx is not updated, but iknewbest xx if is, represents an improvement in the
function value.
By the end of the loop ,k the model is re-expanded around the new optimal point by
( )∑=
=−+=i
l
bestl
newbestlklkk ikxxGgg
1,,1, L
(3.29) ( )., newbestnewbestbest xffxx ==
In case of a quadratic function, the exact value of the index of performance is
represented by ( ).xq
3.3.1.4 - COORDINATE SEARCH
In order to find the rml xxx and , used in the triple search, sometimes a coordinate
search is used based on a line search routine. The first search is initialized with the
candidates of the “shopping basket”. After the first search, the best point and its two
75
close neighbors are stored as{ }.,, 111rml xxx . The next searches are made by using the best
point found in the previous searches. At the end of the search of coordinate i , there are
stored the best point, the initial point and if it is possible the closest neighbor of the best
point as { }.,, ri
mi
li xxx .
• Local Search
The local search can be divided into 6 different steps
Step 1: By using the shopping basket candidates, it is performed a full triple search,
where rml xxx ,, are determined by coordinate search. This procedure gives a new
point x , the function value ,f an approximation for the gradient g and an
approximation for the Hessian .G
Step 2: By assuming that ( )( ) 00 where125.0,,min: xxxuxxvd −+−−= is the lowest
absolute value in the interval [ ]vu, and minimizing the quadratic function
( )[ ].d-d,
hGhhgfhq TT
interval thein : ++= (3.30)
Then, a line search is performed along of pα+x , where p is the solution for the
minimization problem. α can assume values 1 or 0. The new point found by the line
search is denoted by x and the function value by .f Making
( ) ( )redepoldold ffffr −−= /: where oldf is the function value of actual point at the
beginning of step 2 and predf is the function value predicted by the quadratic model.
Since that , if 1 predffr == the “distance” of 1 from r measures the estimation
quality of the quadratic model.
Step 3: If the number of iteration of step 3 or number of function calls is exceeded, the
process is stopped. The process is also stopped if the actual x lies on the border or if the
criterion defined in step 4 is satisfied.
Step 4: If some component of x lies on the bounds and the stopping criteria is satisfied,
76
a line search along this coordinate is performed. If the function value does not improve,
the process is stopped.
Step 5: If 25,01 >−r or if the stopping criteria has been fulfilled in step 3, a full triple
search is performed. Otherwise, a diagonal search is performed. This search is
performed only along the coordinates that do not lie in the bounds.
Step 6: If 25,0<r , the quadratic model has been not adequate, therefore, the
minimization boxes are decreased where .2/dd = On the other hand, if 75,0>r , the
direction of search by using the quadratic model was adequate, therefore, make the
adjustment .2dd =
The quadratic model Q is minimized in the interval ( ) ( )[ ]xvdmixud −−− ,,,max and
a line search along of pα+x is performed. A new r is evaluated where oldf is the
function value at the beginning of step 6 and goes back to step 3.
• Shopping basket
The local search is performed only by the end of each sweeping. Thus, all candidates of
shopping basket (all base points of the box with level maxs ) that have been stored in the
sweeping are sorted in crescent order (related to the function value). After, the
properties any wxf point any and between is verified. This is performed in four
steps.
Step 1: It is verified if a local search from x has been already performed (very often
one point belongs to two boxes). If the checks are true, takes the next x .
Step 2: The function value in the points ( ) ( ) ( ) xxfxfxwxx , If .31 '' >−+= does not
lies in attraction domain of .w Therefore, a new w is used and go back to step 2.
Step 3: Evaluate the function value at the point
77
( ) .32'' xwxx −+=
( ) ( ) ( )( ) '''' denote ,max If xxwfxfxf => ,
( ) ( )xfxf <' If use the next w and go back to step 2.
If ( ) ( )( ) ( ),,min ''' wfxfxf < the four points seem to lie in the same valley. However, x
is not discarded for a local search, denote ''' or xxxx == (choose the point with the
lowest function value), use a new w and go back to step 1.
Step 4: If one point x “survives” to the loops of step 2 and 3, then a local search is
performed. After, a similar procedure to steps 2 and 3 is applied to the x that has been
obtained by the local search in order to verify if a new point has been found. Only in
this situation, the point is placed in the shopping basket. Finally, the point of the
shopping basket that corresponds to the lowest function value represents the global
minimum in the interval ],[ vu .
The algorithm convergence is guaranteed when ∞→maxs , according the theorems
presented in (Huyer and Neumaier, 1998).
3.4 - IDENTIFICATION MODELS BASED ON THE SUBSPACE THEORY (SMI)
The identification problem of time invariant MIMO (Multiple-Input Multiple-Output)
systems from measurements of inputs and output has relevant importance in the system
analysis, design and control. Basically, the identification based on the subspace theory
can be defined as the mapping between the available data of the input and output and
the unknown parameters in a defined class of models, the state space model, for
instance.
3.4.1 - STATE MATRICES IDENTIFICATION
Considering a time invariant system
78
kkk
kkkDuCxyBuAxx
+=+=+
1
(3.31) The identification problem consists in to determine the state matrices A, B, C, and D. If
it is assumed that kkk xuy and , are available, the state space matrices are easily found
by using a least squares approach.
Rewriting the set of equations given by Equation (3.31) one obtains
=
+
+
k
k
k
kUX
DCBA
YX
1
1
UY XX Π= (3.32)
where
=
+
+
1
1
k
kY Y
XX ,
=Π
DCBA
and
=
k
kU U
XX
Thus, Equation (3.32), having as unknown parameter Π is solved by LS approach as
( ) YUUU XXXX Τ−Τ=Π1
(3.33) The solution given by Equation (3.33) is unique. The main problem comes from the fact
that the states are not known; therefore, the problem is not easily solved. It is necessary
to apply the subspace theory, but in this case, the solution is no longer unique.
According to the similarity transformations for linear systems (Kailath, 1980), the A
(unknown) matrix and the idA (obtained by subspace approach) have the same
eigenvalues (by discarding the numerical error), being one of the connections between
the real system and the identified one.
Because of this characteristic, such identification methods are also classified as black
box type: given an input u , the system matrices give an output iy (that should be very
close to the measured information y ), the states of the system are not visible to the user.
Several state space identification methods can be found in the literature: ERA
(Eigensystem Realization Algorithm) (Juang, 1988), ERA/DC (ERA with Data
Correlation), SRIM (System Realization Based on Information Matrix) (Juang,1997),
MOESP (MIMO Output Error State Space Model Identification) (Verhaegen, Dewilde,
79
1992), etc.
In this thesis, in the failure detection procedure, the MOESP algorithm has been used to
recover the states and also the outputs. The choice is based on many reasons: accuracy,
easy use/implementation and also by the good results of the previous applications
(Verhaegen et al, 1994).
Thus, in the following, the main steps of MOESP algorithm is presented.
3.4.2 - OUTPUT ERROR STATE SPACE MODEL
Considering a deterministic linear time invariant system P (Figure 3.3) being
represented by the set of Equations. (3.31), where mku ℜ∈ (is the input vector),
lky ℜ∈ (is the output vector – measurements) and n
kx ℜ∈ (is the state vector). The set
of unknown matrices DCBA and,, has appropriate dimensions. It is also assumed that
the system is minimal (Kailath, 1980), namely, completely controllable and observable.
In accordance with the dashed box of Figure 3.3, the deterministic system output is
disturbed by the noise kυ . Thus, the output is written as
kkk yz υ+= (3.34) When the stochastic process kυ is assumed to be an arbitrary stochastic process with
zero mean, the model given by Equation (3.31) incorporates a wide variety of
parametric or not parametric models, such as ARMAX, ARX and OE (Ljung, 1987) or
stochastic state space models (Kailath, 1980).
80
Fig. 3.3 - General view of the identification scheme
3.4.3 - DATA REPRESENTATION AND PERSISTENT EXCITATION DEFINITION
It is assumed that the signals used in the identification process are a finite segment of
ergodic stochastic realization process. Namely, for ∞→N there exists a ergodic
stochastic process lk
mj ℜ∈ℜ∈ vu and such that
[ ] [ ]1111 and −++−++ jnjjjnjj vvvuuu LL (3.35) are realization of kj vu and , where the expression
[ ]Tkj
N
i
Tikij
Nvu
NvuΕ=∑
=−+−+
∞→ 111
1lim (3.36)
and [ ]T
kj vuΕ represents the expected mathematical operator .
Considering a batch of measurements, Equation (3.31) can be written as
81
[ ] 11
121
21
11
+
=
−++
−−+++−+
+++
−++
Njjj
iiNjijij
Njjj
Njjj
xxx
CA
CAC
yyy
yyyyyy
LMMOM
L
L
−+++−+
+++
−++
− 21
21
11
2
00000
iNjijij
Njjj
Njjj
i uuu
uuuuuu
DBCA
DCBCABDCB
D
L
MOM
L
L
L
MOM
L
L
L
(3.37)
Defining ii H and Γ as
1
−iCA
CAC
Me 0
0000
2
− DBCA
DCBCABDCB
D
i L
MOM
L
L
L
,
respectively, Equation (3.37) can be written in the compact form as
NijiNjiNij U ,,,,, Η+ΧΓ=Υ (3.38) In similar way, the output kz is expressed as
NijNijiNjiNij VU ,,,,,,, +Η+ΧΓ=Ζ (3.39) where NijNij V ,,,, and Ζ are Hankel matrices build by using kz and kν respectively .
3.4.4 - PERSISTENT EXCITATION
Definition : (Ljung, 1987)
An ergodic sequence mk ℜ∈u gives a persistent excitation of order i from an instant j ,
if and only if
82
miUUN
TNijNij
N=
∞→,,,,
1limρ (3.39)
Equation (3.39) can be written as
miUUN NN
TNijNij =
Ε+ 11
,,,,1 ερ (3.40)
Thus, when N is large enough, the lowest singular value ( )TNijNij UU
N ,,,,1 is bigger
than 1Nε , the definition of persistent excitation of ku sequence can be interpreted for
the case of finite data as the rank condition
( ) miU Nij =,,ρ (3.41) 3.4.5 - MOESP ALGORITHM
The MOESP algorithm can be represented by the following steps
Given
1- An estimation of the system order n
2- A parameter i , satisfying ni >
3- The data sequence: input and output
[ ] [ ]miN
yyyuuu iNiN>>
−+−+ with
,,, and ,,, 121121 LL
Perform the following steps
Step1: Build the Hankel matrices NiNiU ,,1,,1 and Υ , defined in Equation (3.37).
Step 2: Compress the data via RQ factorization
83
−−
−−−−−−=
−−−
2
1
2221
11
,,1
,,1
|
0|
Q
Q
RR
R
Y
U
Ni
Ni
(3.42)
Step 3: Compute the singular value decomposition of 22R matrix (Theorem 4,
Verhaegen and Dewilde, 1992)
[ ]
−−
−−−−−−=
⊥
⊥
Tn
Tnn
nnV
V
S
SUUR
)(
|0
0| |
2
22 (3.43)
Step 4: Compute idididid DCBA and ,, matrices from the set of equations
( ) ( )21nidn UAU = (3.44)
( ):,:1 lUC nid = (3.45)
( )
( )
( )( )
( ) ( )( )( )
( )( )
+−
++−
=
+−Ξ
+ΞΞ
⊥
⊥
⊥⊥
00:,:11000:,2:1
:,:11:,:1
:11:,
2:1:,:1:,
L
MM
L
L
MT
in
Tn
Tin
Tn
lilU
llUlilUlU
miim
mmm
( )
id
id
n
lBD
UI
100
(3.46)
where
( )1nU is the sub matrix composed by the first ( )li 1− rows of matrix nU
( ) 11121: −⊥=Ξ RRU
Tn
When the set of Equations. (3.44) - (3.46) has a non-compatible solution, it is possible
to obtain a solution by applying the LS method.
3.5 - PROPOSED STRATEGY FOR ON-LINE IDENTIFICATION
Since a model that represents the robotic joint dynamic with desired accuracy is
available, an identification scheme can be designed. In the adopted one, the
84
measurements are stored by a short period (7 s). The measurement matrix φ is
continuously monitored in order to ensure that all initial conditions can be obtained with
a desired reliability level. The adopted procedure is also very efficient to detect a linear
combination in the information matrix. There exist several ways to verify the quality of
the signal used in the identification process or to detect a linear combination. A good
alternative is to use the singular value decomposition technique (SVD). Thus, the
measurement matrix is decomposed as
TVUΣ=φ (3.47) where U and V are isometric matrices and ∑ is a diagonal matrix
),,,( 21 mdiag σσσ L=Σ containing the singular values (positive square root of eigenvalues) of φ .
If some states are not adequately excited or if there exists some linear combination in
the φ , the correspondent singular value iσ will have very low magnitude, close to
machine precision.
After the check of matrix φ , the initial conditions for the recursive algorithm are
obtained by applying the LS method
( ) ktt
initial yφφφ1ˆ −
=Θ (3.48) Using Equation (3.47) in (3.48), one obtains
kTT
initial yUVˆ −Σ=Θ (3.49) The standard procedure (Ljung, 1983) to identify time variant systems is based on a
forgetting factor ( 1<λ ) that marginally weights the previous estimation. For time
invariant systems, normally, a gain sequence is used that exponentially grows to 1:
)1()1()( 00 λλλλ −+−= tt (3.50) where 0λ is the growing rate. Usually, the value of )0(λ is chosen smaller than 1 in
85
order to have strong corrections (small time constant) in the parameters at the beginning
of estimation. After some time (depending of 0λ value), the gain grows to 1 (which
represents an infinite memory). The behavior of Equation (3.50) is shown on Figure 3.4.
The corrections given by Equation (3.50) are satisfactory for time invariant systems or
for short process intervals. After some iterations, the gain grows to 1 and the parameters
tracking is no longer effective. Since we are searching for an algorithm to be used in the
IRJ experiment (that may operate in space environment), it is likely that extremely
environmental changes can occur, therefore some parameters can present time variant
behavior. If the system is considered inherently time variant (namely, using λ < 1,
constant), and some parameters are not time variant, the algorithm likely will present
instability and does not converge. On the other hand, if it is assumed that the system is
time invariant, (namely, using a gain sequence represented by Equation (3.50)) and
some parameters exhibit variations, the algorithm will not be able to follow the
parameters variations, it will take only the mean of the variations. Face to this problem,
a strategy based on the relative error between estimation and measurement is used to
adjust the gain value. Thus, one obtains an adaptive gain sequence: If the system is time
invariant, λ is close to 1. If the system presents time variant characteristics, the gain is
automatically adjusted in order to minimize the identification error.
Based on this idea, it is necessary to find a function, which relates the relative error with
the gain value. A function that presents the inverse behavior of Equation (3.50) has been
searched. This function shall present, at the beginning, an exponential decay and at the
end, a constant a value.
The function represented by Figure 3.5 is a 17th polynomial function in ε (relative
error), where the coefficients have been obtained by a LS fit. The polynomial order has
been selected in order to have a good balance between complexity and accuracy.
Other functions have been also tested, the bell (Gauss) function for instance,
2 )1()( εβααελ e+−= (3.51)
that also offer a good approximation for the desired function shape (Figure 3.6).
86
Fig. 3.4 - Typical behavior for gain in time invariant systems.( 95.)0(,99.00 == λλ )
Observing the Figures 3.4 and 3.5, one can note that when the estimation presents small
errors, the gain ( )(ελ ) gets value close to 1, meaning that the value of )(tP decreases
(see Equation (3.14)) and consequently the correction rate in the parameters. On the
other hand, if the relative error increases, the gain value decreases by following one of
the functions used. Decreasing the λ values means an increment on the value of )(tP ,
which makes the corrections in the parameters strong. It is important to note that in the
two different functions tested, the smallest value of )(ελ is set to 0.95. This is a safe
action to avoid an explosion on )(tP .
87
Fig. 3.5 - Behavior of variable gain – Polynomial adjust.
Fig. 3.6 - Behavior of variable gain – Exponential function.
In the strategy selected for the recursive algorithm, it is also possible to reset the
covariance matrix )(tP in case of λ gets low values for a long period. This situation
can occur if the algorithm is in the steady state operation for a long period (The )(tP
88
has already converged to the final value) and suddenly the system presents some
variations. In this case, the single gain adjustment can not be able to relieve the
algorithm. An algorithm reset is than necessary. A schematic representation for the
strategy described above is shown in Figure 3.7
Fig. 3.7 - Schematic representation of mRLS algorithm
3.6 - TWO STEP IDENTIFICATION ALGORITHM
In this section, the idea and conception of two step identification algorithm is presented.
It is evident that not all parameters required by the friction model described by Equation
(2.77) (without linearization) can be identified by any algorithm based on LS approach.
It is necessary to apply a non-linear optimization methods to find those parameters.
Usually, the non-linear algorithm is used to identify all parameters, inclusive those ones
that have linear dependency on the measurements. As a result, it is necessary to find
satisfactory initial conditions for all parameters and in addition, the optimization process
becomes time consuming. Besides, a loss in accuracy (usually) occurs when a new
parameter is introduced in the parameters vector. The proposed alternative is to separate
the parameters into two different groups: one group with linear dependency in the
measurements (they will be called linear thereafter) and another one with non-linear
dependency in the measurements (called non-linear). Once this separation is made, one
can use two different algorithms (that work sequentially) to identify the parameters.
89
There exist several advantages in doing this procedure: it is not necessary to find
feasible initial condition for all parameters, the non-linear algorithm has a fast
convergence (tests have shown that such a strategy can be 20 times faster than the a
procedure where all the parameters are identified by the non-linear algorithm only).
In the non-linear optimization algorithm selection, several methods available in the
MATLAB software package have been tested. The main problem in using such methods
comes from the fact that such algorithms are local minimizer, which are extremely
dependent of the initial conditions. Sometimes such algorithms do not converge or
deliver only mathematical solution (parameters negatives, for instance) for the problem,
rather solutions with physical meaning. Thus the algorithm MCS has been selected to
identify the non-linear parameters that appear in Equation (2.77). The MCS algorithm
has a good feature because it does not require an initial condition, but only a range
where the parameters shall lie. In case of no information about the parameters is
available, the bounds are set to ∞± .
As is usual, the MCS algorithm requires that a function (IP – Index of Performance) to
be minimized should be given. There exist several ways to define a criteria for the IP. In
this thesis, two criteria have been tested: a quadratic function of the error
[ ][ ]TyyyyIP ˆ ˆ21
−−= (3.52)
and the function absolute value of the error
yyIP ˆ−= (3.53) where y is the measured information, and y is the estimation of y regarding the linear
parameters optimized by the linear algorithm and ⋅ represents the Euclidean norm.
3.7 - INTEGRATED ALGORITHM MRLS – MCS
In order to solve the identification problem characterized by Equation (2.77), an
integrated algorithm using RLS and MCS has been derived. The adopted strategy can be
divided into two different operational phases: an initial procedure and a normal
operation mode. In the initial procedure, the measurements are collected and stored in a
90
batch with pre selected length. The measurements are continuously updated, working
like a time variant moving window. Given an initial value for the non-linear parameters,
the linear parameters are estimated by part of the algorithm based on LS. After, the
linear parameters are passed to MCS algorithm, which is responsible to identify the non-
linear parameters. This process is repeated until the convergence criteria be satisfied,
namely, when the norm of the error is smaller than a determined threshold
) and ( minΘδδ . When the convergence criteria is fulfilled, the recursive algorithm is
started and the non-linear parameters are kept frozen while the norm of the error does
not reach the specified threshold. When this situation occurs, the MCS algorithm is
activated in order to make a correction in the non-linear parameters. In case, a very
strict δ is set, the integrated algorithm will perform a pre-defined number of iterations.
If the requirement is not fulfilled, a full optimization process using a local minimizer
(simplex algorithm) is performed.
The integrated algorithm can be summarized in the following steps:
Initial procedure
1- Input: u, v (boundaries for the non-linear parameters ) and an initial value for
NLΘ ;
While yyNerr ˆ−= > δ and min
)( 1 Θ− >Θ−Θ=∆Θ δkk
2- Store the measurements;
3- Evaluate φ ;
4- Verify the rank of φ (SVD);
5- Estimate LΘ (LS based );
6- Estimate NLΘ(
( NLΘ(
means global minimum in the interval [ ]vu , ) by using the
MCS algorithm;
7- Evaluate errN and ∆Θ ;
91
8- If errN < δ and minΘ<∆Θ δ , stop and keep constant NLΘ .
Normal mode
9- Using Θ (estimated in the initial procedure), initialize the recursive algorithm;
10- Check errN
11- If errN > δ , activate the MCS algorithm and using the last measurement window
available update NLΘ(
;
else NLΘ(
can be considered a minimum (variations in the non-linear parameters
did not occur);
12- Take the next measurement .
Here Θ is the parameters vector to be identified and NLΘ is a subset of Θ that
represents the non-linear parameters.
Thus, the proposed algorithm is able to continuously update the linear parameters and
make periodic corrections in the non-linear parameters (only when the norm of error
increases). The schematic representation of the integrated algorithm is presented in
Figure 3.8.
Fig. 3.8 - Schematic representation of the Integrated Algorithm
92
3.8 - TRAJECTORY SELECTION FOR IDENTIFICATION PURPOSES
Experimental identification of robots deals with the problem in estimating the model
parameters from measurements collected during the realization of a determined
experiment. It is known that an accurate, reliable and efficient identification process
requires some additional procedures, the trajectory selection, for instance. During the
identification experiment design of a manipulator it is necessary to check if the
excitation level is sufficient to ensure a fast and efficient identification even under some
disturbing effects (noise). An alternative is to use the condition number of the
measurement matrix to select the best trajectory for the identification process. Thus, it is
necessary to give priority to the trajectories that maximizes the smallest singular value,
since the biggest singular value is not sensible to the disturbances.
3.8.1 - TRAJECTORY DESCRIPTION
In the experimental part, two families of trajectories are available: a triangular trajectory
with constant velocity and infinite acceleration (in theory) at the bounds of the
workspace. In this kind of trajectory, there is just one single parameter to be adjusted:
the magnitude of angular velocity. The second type of trajectory is obtained by using a
combination of periods of constant velocity and constant acceleration. Thus, two
parameters (magnitude and period of constant velocity) can be adjusted to get the
desired the excitation level.
In the IRJ experiment, the task of finding appropriate trajectories to identify the desired
parameters can not be performed by applying standard procedures (Swevers et al, 1997
), since the joint stiffness shall be also be dynamically identified. This requirement
increases the degree of freedom of the system. Thus, the term outin θθ − and its
derivative appear in the measurement matrix φ .
By considering a system with linear and cubic stiffness, damping in the input and output
side, Equations (2.82) and (2.83) can be written in matrix form
Θ= φY (3.54)
93
where
−−−−∆∆∆
−−∆−∆−∆−=
outoutoutout
inininout
NN
signsignN
T
θθθθθθθ
θθθθθθφ
cossin000
0000
33
3
&&&
&&&&
and Θ the parameters vector to be identified.
Since the measurement matrix also requires the information of output side, the complete
system dynamics shall be simulated. The strategy proposed here can be summarized in
four steps
1. Generate the trajectory;
2. Integrate the system;
3. Build the φ matrix;
4. Evaluate the condition number of φ matrix (Equation (3.54)).
This procedure is repeated until the complete operation range is covered. In the case of
triangular trajectories, the angular velocity ranges from 15 to 45 deg/s. In the case of
sinusoidal trajectories, the angular velocity range is the same and the periods range from
0.5 to 6 s. The simulation time used is 10 s for the triangular trajectory and 15 s for the
sinusoidal one. Since the condition number is evaluated, a plot where the behavior of
the singular values is obtained.
3.8.2 - TRIANGULAR TRAJECTORY
This trajectory is obtained by only changing the velocity signal at the position bounds,
which means that in this point an impulsive acceleration is required. By using this
trajectory type, the only variation possible is the velocity magnitude. Figure 3.9 shows
the position and velocity profile. In this trajectory, the initial point shall be around zero
in order to ensure that all workspace ( ± 80 0) can be used.
94
3.8.3 - SINUSOIDAL TRAJECTORY
In this trajectory, the maxim amplitude occurs at the point of zero velocity, therefore,
being dependent of the velocity magnitude and period. In order to use all workspace, the
motion shall not start around zero position (like in the triangular trajectory) but the
periodic motion shall start at the point of zero velocity, namely, in the point of
maximum amplitude. By using these requirements, an initial procedure has been
developed in order to ensure that given the period and the desired velocity, the link is
driven to the point of zero velocity from any starting point (within the workspace). In
this phase the link experiences an exponentially decreasing velocity until reaching the
target (point of zero velocity). This phase is called “rendezvous”. By using this
procedure, it is ensured that all the workspace can be used. Figures 3.10 and 3.11
present some examples of this procedure.
Fig. 3.9 - Triangular trajectory with constant velocity 45 deg/s
95
0 5 10 15 20 25 30 35 40-20
-15
-10
-5
0
5
10
15
20
Time [s]
Pos
[deg
] and
Vel
[deg
/s]
Period T = 0.9
Fig. 3.10 - Sinusoidal trajectory with T (period) = 0.9s, x (0) = 20 deg and v = 15 deg/s
0 5 10 15 20 25 30 35 40-30
-20
-10
0
10
20
30
Time [s]
Pos
[deg
] and
Vel
[deg
/s]
Period T = 1.9
Fig. 3.11 - Sinusoidal trajectory with T (period) = 1.9s, x (0) = 20 deg and v = 15 deg/s
96
3.8.4 - TRAJECTORY SELECTION
In order to verify the excitation level of the measurement matrix, the smallest and the
greatest singular value of φ have been evaluated. First, the trajectories are generated.
After, using these trajectories as a reference, the system is simulated (in the tests, the
stiffness value is obtained from catalog, and the damping torques are estimated from an
off-line identification) and the states are used to build the measurements. Finally a
singular value decomposition (SVD) is performed.
It has been noted that some parameters have strong impact on the matrix conditioning.
In the IRJ experiment, it is evident that the term related to the non-linear stiffness is
dominant (in terms of sensitivity). Thus, two different tests have been performed: one
considering all terms that appears in Equation (3.54), called full model and another
(called reduced model) where the second column of φ matrix has been eliminated. In
order to illustrate this fact, a plot of velocity versus singular values is shown for
triangular trajectories and a contour plot for sinusoidal trajectory is presented next.
Fig. 3.12 - Triangular trajectory – smallest singular value – full model.
97
By observing Figures 3.12 and 3.13, it is evident the impact of non-linear stiffness in
the condition number of φ matrix. The smallest singular value jumps from 10-7 to 10-1.
On the other hand, the greatest singular value almost does not change when the reduced
model is used.
For sinusoidal trajectories, a contour plot of velocity and period can be obtained. By
inspecting this plot, it is possible to select the period and constant velocity that will
maximize the smallest singular value of measurement matrix. Two different cases can
be obtained: the full model and the reduced one.
The straight line in the contour plot means that the periodic motion could not be
complete a cycle in the simulation period. Therefore, these values of period and velocity
are not adequate for a simulation of 10s. The measurements shall be collected in longer
interval. The effect of the term related with cubic stiffness is expressive, a big jump of
the smallest singular value has been again verified, when this term has been removed
from the measurement matrix. Comparing Figures 3.14 and 3.15, it is evident that the
measurement matrix of the reduced model is better conditioned then the matrix of the
full model. An alternative to minimize this problem, in case an off-line identification is
appropriate, is to normalize (to divide each column by its respective norm) the
measurement matrix. This technique will improve the conditioning number of φ . After
the parameters have been identified, the true values can be retrieved by using the
inverse operation, namely, by multiplying each parameter by its respective norm. The
effect of φ normalization is shown in Figure 3.16.
98
Fig. 3.13 - Triangular trajectory – smallest singular value – reduced model
Fig. 3.14 - Sinusoidal trajectory – smallest singular value- full model.
99
Fig. 3.15 - Sinusoidal trajectory – smallest singular value – reduced model.
Fig. 3.16 - Sinusoidal trajectory – smallest singular value – full model φ normalized.
The procedure presented in this section can not be considered optimal (due to the
100
requirement of identifying the stiffness and relative damping, no optimization procedure
has been used). But it gives good indicators to select from a set of trajectories the best
one, which is adequate for the identification process. These indicators are easily
extracted from the obtained plots. In addition, the improvements in normalizing the
measurement matrix has been also focused.
101
CHAPTER 4
APPLICATIONS AND RESULTS
In this Chapter, the models derived in Chapter 2 as well as the methods and algorithm
derived in Chapter 3 are used and tested by using measurements collected from the IRJ
experiment. It will be shown that the developed theory is extremely useful and can be
used to solve general identification problems. They are used to identify complete
models (state space matrices), physical parameters of robotic joints and also to detect
sensor failures.
The Chapter is organized in the following way
• First, the state space matrices identification process is described (by using the
MOESP algorithm);
• The sensor failure detection process is presented;
• The integrated algorithm is tested in both simulated and real data;
• Finally, the effects of sensors failure on the identification algorithm and the
algorithm reconfiguration is presented.
It is important to note that during the research development (in Brazil as well as in
Germany), several dynamic models, algorithm and strategies have been developed
aiming not only to solve the identification general problem, but also fulfill specific
needs of the IRJ experiment. In this Chapter, not all cases will be presented, but only the
interesting and expressive cases from practical point of view.
4.1 - STATE SPACE MATRICES IDENTIFICATION
The parameters identification is based on system information, namely, in the collected
measurements. If some measurement is not available or if some sensor presents failure,
the related parameters no longer can be identified. The idea to be presented further, aims
to minimize this problem by estimating the failed measurement. Several theories
(Kalman filters, black box models, etc.) can be used to estimate the states
102
(measurements) under the suspect of failure.
A procedure for state estimation based on subspace theory is proposed in the following.
There are several reasons for this choice, the main ones are: easy model selection, good
performance (accuracy) and also the possibility to use the state space matrices to
develop a failure detection procedure.
In order to estimate the state space matrices, the MOESP algorithm has been selected.
This algorithm has been selected due to its versatility and reliability. The MOESP
algorithm has already been successfully tested in the identification process of a
helicopter by using real data (Verhaegen et al, 1994).
4.1.1 - OBTAINING THE STATE SPACE MATRICES
In the performed tests, the input measurements (motor side) have been used to estimate
the output measurement (link side). It is assumed that the relationship between the
measurements of input side and output side is linear or has low non-linearities. Thus, the
algorithm MOESP can be used to determine the mathematical model
)(+)(= )()(+)(=)1+(
kDukCxkykBukAxkx
(4.1)
where )(kx are the states
)(ku are the inputs, namely, motormotor θθ &, and motorθ&&
)(ky are the output : outθ and outθ&
CBA ,, and D are the unknown state space matrices to be determined.
The commanded current could also be used as an input for the algorithm, but due to big
magnitude difference (the current has been measured in increments) in the signals, most
of times an unstable model has been obtained. Thus, the identification process based on
motormotor θθ &, and motorθ&& presented excellent results.
103
4.1.2 - MOESP ALGORITHM ACCURACY VERIFICATION
The state space matrices have been obtained by using all set of measurements available,
namely, a set of 12 trajectories have been used. This procedure tries to ensure that the
obtained set of matrices are able to deliver a satisfactory estimation even in the case
when a different trajectory is used.
The estimation accuracy has been verified in a statistical way by using as an indicator
the vaf (variance accounted for) , which is represented by
%100*)var(
)ˆvar(1
−−=
yyyvaf
(4.2) where
})]({[)var( 2xExEx −= is the x variance.
It is noted that when value of vaf is high (close to 100), the better is the estimation
quality. In case of zero error, vaf = 100 %.
4.1.3 - MODEL ORDER SELECTION
The identification process based on subspace theory does not require special attention
with the type of models (physical) that best describe the system, since it works like a
black box estimator. On the other hand, it is necessary to select the model order,
namely, the size of matrix A. Normally, the system order is chosen by inspection of the
singular values of the measurement matrix (Hankel form), evaluating vaf of several
models and also verifying the stability of the model obtained. The order that best fulfills
all the requirements is then selected as the “true” order of the system. Sometimes, the
model is unstable, but offers a good (high value of vaf) approximation of the measured
data. In this case, it is possible to improve the model stability by implementing a
predictor, which makes use of a Kalman gain. The Kalman correction is based on the
past measurements. This feature becomes a problem for the application in the failure
detection algorithm. In this situation, the measurements are not available or their values
104
are not reliable. Thus, in the system order selection, the procedure of improving the
model stability by using a Kalman gain has been not used. It is shown in Figure 4.2 –
4.3 just as an illustration.
Figure 4.1 shows the singular values magnitude of the measurement matrix in
logarithmic scale. The magnitude of the singular values gives an indicator of the order
of the system. The values smaller than a specified threshold can be considered as noise,
and hence, being, discarded.
Fig. 4.1 - Singular values of measurement matrix
Inspecting Figure 4.2, one observes that the maximum system order, at which the
system still remains stable without compensation (Kalman gain), is two. The third order
system (Figure4.3) is apparently unstable (it has an eigenvalue outside of unity circle).
This fact has happened for all other superior orders. Thus, due to the proposed goals
(state estimation and failure detection), the system order has been selected as 2.
105
Fig. 4.2 - Eigenvalue of matrix A (order 2)
Fig. 4.3 - Eigenvalue of matrix A (order 3)
106
4.1.4 - ESTIMATION ANALYSIS - MOESP ALGORITHM
The state space matrices have been estimated by using a set of 12 different trajectories.
The data incorporate triangular trajectories as well as sinusoidal ones. The ideal
situation is when the obtained matrices are able to estimate any kind of trajectory with
good accuracy. In order to fulfill this requirement, in case where the data is more
diversified, a more general model is the identified model. Thus the ideal case is when an
arbitrary input (amplified noise, for instance) is used. But in case of a robotic joint, this
idea may not be applicable, because the experiment can be damaged.
In accordance with Equation (4.1), the system to be identified is a MIMO one with three
( motormotor θθ &, and motorθ&& ) inputs and two outputs ( outθ and outθ& ):
)()( )()()()1(
kuDkxCkykuBkxAkx
nn
nn
+=+=+
(4.3) The numerical values of the identified matrices are
=
1.0041 0.0756 0.3232- 0.9550
nA ,
=
0.4048 1.8935- 0.0023-0.1894- 0.9562- 0.0277-
nB
=
0.4464- 0.3263-0.0088- 0.0058
nC ,
=
0.9836- 5.3887 0.0068 0.0323 0.7962 0.0003-
nD
where the index n means that the matrices have been obtained by using all type of
trajectory available, thus corresponding to the nominal model.
In order to illustrate the identification process, the inputs and outputs in the MOESP
algorithm are shown in Figures 4.4 and 4.5.
107
Fig. 4.4 - Inputs for MOESP algorithm - Triangular trajectory ( motorθ& = 25 rad/s)
Figure 4.5 shows the outθ , outθ& measured and its respective estimate outθ and outθ& . In
this figure, the two lines are almost indistinguishable, which shows the excellent result
of the estimation process. This is confirmed by the vaf indicator
% 100=posvaf , % 99.9898=velvaf (4.4) where vafpos is related to the position and vafvel is related to the angular velocity.
A critical point in the parameters identification process (to be described further) of IRJ
experiment is the stiffness coefficient. The problem arises from the small magnitude of
the relative position ( θ∆ ) between motor side and link side, which is used to identify
the stiffness coefficient. Thus, it is reasonable to verify the accuracy precision not only
in terms of states, but also considering θ∆ . Figure 4.6 shows the relative position and
velocity between input and output. The vaf indicator stays also within the expected
range
% 95.2262_ =posrelvaf
108
This result fully satisfies almost all control applications, but may not be sufficiently
accurate for the parameters identification requirements. This problem is focused in
details in the parameters identification section.
Fig. 4.5 - Output measurements and respective estimate
In this section, it has been shown that the model obtained by using the subspace
identification theory (MOESP algorithm) presents excellent (concerning accuracy and
reliability) results being a good alternative for state estimation in case of sensor failure.
Concerning the accuracy, it has been observed that the vaf indicator gets satisfactory
values, 100 % for the position estimation and almost 100 % for velocity. Another good
feature in using the SMI approach to obtain the nominal model is due to the fact that the
mathematical model is automatically obtained by only using the system measurements.
No other knowledge is required. It is also important to note that in the results to be
shown further, the measurements of the input side have been used to estimate the
measurements of the output side. The inverse procedure, namely, to use the
measurement of the output side to estimate the measurement of the input side is also
109
technically possible, but this may not make sense because the input measurements are
fed into the control laws. If some input sensor fails, the information for the control shall
be also reconfigured or the experiment may not succeed.
Fig. 4.6 - Relative position and velocity between input and output side
4.2 - FAILURE DETECTION IN THE SENSORS
4.2.1 - INTRODUCTION
The failure detection (and isolation) in sensors and actuators is extremely relevant in
systems engineering. Much effort in developing algorithms and strategies that
efficiently detect and isolate failure in sensors has already been done (Basseville, 1997).
Normally, two theories are applied: one using equipment redundancy and another using
analytical redundancy. The equipment redundancy theory uses the information from two
or more sensors to declare a failure of another one. The main characteristics of this
theory are low computer effort and simple algorithms. These features result in heavy
systems and are also more expensive (it is necessary to use redundant equipment). The
two characteristics sometimes make the failure detection process based on redundancy
110
almost unfeasible. On the other hand, the failure detection based on analytical
redundancy is attractive due to low costs (financial) and also from the design point of
view.
Normally, these techniques are very complex and demands high computational efforts.
Different methods have been applied to analytically detect and isolate sensor failure.
The majority of these methods employ filtering techniques, Kalman filter for instance.
The main problem to use these methods is the complexity and consequently, the high
computational load. The heavy computational load appears because, normally, it is
necessary to compute a bank of filters, whose increases proportionally to the number of
sensor to be monitored. In this thesis, it is not intended to develop complex algorithms
or sophisticated techniques, since the failure detection mechanism shall work in parallel
with the parameters identification algorithm. Thus, the goal is to elaborate an algorithm
that monitors the sensors output and also gives an alternative solution in case of failure.
The technique to be presented further uses the subspace identification approach to
derive the state space matrices, which are used in the failure detection algorithm.
4.2.2 - FAILURE DETECTION STRATEGY
Using the SMI, it is possible to obtain a set of matrices that will estimate the states with
a known degree of accuracy given the output of some sensors. For the case studied, the
output of link position and velocity sensors are estimated given the information of input
side (motor side).
It is important to note that the matrices given by Equation (4.3) are obtained before any
failure occurs, thus they are able to estimate all states with known statistical property.
The measurements to be monitored, outθ and outθ& , are written in vector form
[ ]outoutty θθ &=)( (4.5) The relative percental error between measurement and estimation is given by
% 100)(ˆ
)()(ˆ)( ∗
−=
tytytyteε (4.6)
111
for min)(ˆ δty ≥ . Where minδ is the lower bound in the plant output.
When )(teε is bigger than a specified value given by
)(100max vafδε ee −+= (4.7) where vaf is given by Equation (4.2) and eδ is an estimation tolerance factor, and when
a significant variation ( )(tεio ) between motor sensor and link sensor occurs
−=
)()()()(
tytytytε
in
outinio , min)( δtyin ≥ (4.8)
a failure in the respective sensor will be declared. After this instant, the sensor output is
replaced by the estimation obtained from MOESP algorithm. If )(teε has increased, but
)(tεio lies in the acceptable range, a sensor failure cannot be declared. This means that
the state space matrices are no longer valid. This situation can occur in cases where the
system is experiencing large environmental changes, big temperature variations for
instance. In this case, the nominal set of matrices shall be recalculated. This task is
easily done by using the MOESP algorithm.
4.2.3 - SENSOR FAILURE SIMULATION
The failure detection mechanism described above has been tested by simulating
different cases and situations. The cases differ due to the trajectories used and the
situations due to the instant of failure: simultaneously or not.
Figure 4.7 shows the simulation of position sensor failure by using a triangular
trajectory. The sensor has presented a failure at instant t = 5s, but the velocity sensor did
not present a failure. The failure is simulated by replacing the real data by a random
noise. It can be noted that the algorithm instantaneously detects the bad measurements
delivered by the position sensor and replaces them by the corresponding estimation
(Figure 4.8). Because of the good accuracy, it is almost impossible to note the transition
from real data to estimated one only by inspecting the plots.
The failure in the velocity sensor is shown in Figure 4.9. At time t = 6s, the measured
112
data are replaced by random noise, the algorithm detects a variation in )(tεe , then a
failure is declared. The measurement is replaced by the estimation and a estimated
signal (Figure 4.10) is available for the parameters identification algorithm
Fig. 4.7 - Sensor position failure
Fig. 4.8 - Link position after measurement reconfiguration
113
Fig. 4.9 - Velocity sensor failure
Fig. 4.10 - Link velocity after measurement reconfiguration
Carefully observing Figure 4.10, it can noted that there is a subtle difference between
the estimated velocity and the real one: the estimated velocity has small oscillation in
period of constant velocity.
114
The situation where multiple failures occur has been also tested. In the following two
different cases are presented: In the first one, the failures occur in different instants. In
the second one, the failures occur simultaneously. In these tests, a sine trajectory has
been used. Figure 4.11 shows the situation where the position measurement is replaced
by the random noise and 2 seconds later the velocity information is also replaced by a
random noise. Figure 4.12 shows the outputs of failure detection and isolation
algorithm. The efficiency is clear: Instantaneously, the noise signal is replaced by the
MOESP algorithm estimation.
The case where the simultaneous failures occur is show in Figure 4.13. Similarly to
cases already presented, the failures have been efficiently detected and isolated. Finally,
the case where a temporary failure occurs has been also tested. In this situation the
sensor for some reasons outputs bad measurements and some time later, the sensor
delivers a good measurement (Figure 4.14). The strategy here proposed is also able to
detect that the sensor returned to operate in normal condition. Then, the estimated data
are replaced by the real ones (Figure 4.15). This is possible because the measurement
check is performed continuously.
Fig. 4.11 - Position and velocity sensors failure at different instants
115
Fig. 4.12 - Position and velocity after measurements reconfiguration
Fig. 4.13 - Sensors failure simultaneously
116
Fig. 4.14 - Sensor temporary failure
Fig. 4.15 - Position and velocity after measurement reconfiguration
117
4.2.4 - FAILURE DETECTION – COMMENTS
The proposed failure and detection algorithm presents very low computer effort since
that it requires to run only a linear state space integrator. The states obtained are used in
the failure detection as well as an estimation of the measurement if the failure has
occurred declared. The method efficiency and reliability have been observed for all
tests. It is important to note that the same state space matrices have been used for
different trajectories. There is no necessity to recalculate (run the MOESP algorithm)
the state space matrices.
The measurements are treated in such way that it is possible to detect sensor failures or
variations due to environmental changes. If the set of state space matrices does not
deliver a compatible estimation with the sensor information, a failure cannot be
declared, instead the state space matrices shall be recalculated by using a new batch of
measurement. Thus, the algorithm is able to incorporate the new environmental
condition and an acceptable estimation is expected.
Another feature in using the state space approach is the possibility to detect a failure in
sensors with non-predictable outputs (the output of an accelerometer mounted in a
flexible structure for instance) (Zimmerman,1993). The state space matrices can be
directly exploited not only to detect but also to classify the failure. In this case, the
failure cannot be continuously detected (off-line method).
4.3 - PARAMETERS IDENTIFICATION
In this section, are presented the results that are obtained in the identification process for
the parameter required to characterize the IRJ dynamic. As already mentioned, a large
number of different models has been derived. Therefore, there are several parameters
associated with these models. Thus, the results are presented by using the models in
increasing order of complexity. First are presented the parameters related to the simplest
model, after the parameter related to the models using non-linear parameters and finally,
the results in situations where a sensor failure is simulated.
4.3.1 - OFF-LINE PARAMETER IDENTIFICATION
In the parameter identification process of a phenomenological model a mathematical
118
model is required (normally described in terms of differential equations), which
describes the system characteristics. The models to be used have been already derived in
Chapter 2, therefore, here, only the final equations needed for the identification task are
given.
The IRJ dynamic is described (Equation (2.82) – (2.84)) by
[ ] ),()()( __ ininindwgdstiffaminin TTTIKJ θθθθθ &&&& −∆+∆−= , (4.9)
[ ]outg
outoutfscsdoutoutoutdwgdstiffoutout
T
TTTTNJ
θ
θθθθθθθ
sinˆ),(),()()( ___ +−−∆+∆⋅= &&&&&
(4.10)
where
inoutwgwgn N θθθθθ −⋅−=−=∆ _ and inoutN θθθ &&& −⋅−=∆ (4.11) is the relative angular position and velocity between input and output.
A big effort in the identification process has been dedicated in determining the
appropriate model that best describes the HD gear dynamics. This specifically means to
find an appropriately friction and stiffness model for the robotic joint. The dynamic
identification of 1k and 2k required by Equation (2.69) is a complicated task, since it is
required that both input (motor side) and output (link side) position be measured. Even
though these measurements are available, the difference between them (measurements)
is about 3 order of magnitude smaller than a single measurement (considering the gear
ratio). Due to this problem, the identification task can be complicate and the results
might be not reliable if the system is not well excited. Therefore, in a first
approximation, aiming to minimize the influence of the stiffness coefficients in the
identification process, the equations of motion Equation (4.9) and (4.10) are rearranged
in such way that the stiffness torque and also wgdT _ are eliminated; these terms couple
the two differential equations. Multiplying Equation (4.9) by N and adding to Equation
(4.10), one obtains
119
[ ]),(),(),(
sinˆ
___ outoutfscsdoutoutoutdininind
outgamoutoutinin
TTTN
TIKNJJN
θθθθθθ
θθθ&&&
&&&&
++⋅
−+⋅=+⋅ (4.12)
In Equation (4.12) and in the following ones, the transmission ratio has been set to N=1,
meaning that inθ and its derivatives are already translated to the link side by N. The
same hypothesis is valid for the parameters shown in Table (4.1). Similarly to Equation
(3.4)
Θ= )()( tty Tφ (4.13)
where the plant output y in the link side is composed by the measured torque
measoutT _ and the torque due to the load )sin(ˆ outgload TT θ= in case of gT and outθ
are known. In the motor side, inθ and the commanded torque amIK are measured. In
Table (4.1), the following parameters are considered known
• inJ (from manufacture catalog);
• outJ (from CAD design);
• mgsTg =ˆ (s the center of gravity for the load, m is the mass of the load and g is
the gravity acceleration).
The velocity in input side is obtained from analog differentiation and the acceleration
from numeric differentiation. Both signals are pre processed by using appropriate filters
(8th order Butterworth filter with cutoff frequency of 15 Hz). Thus, one component of
the plant output is given by
outgoutoutmeasout TJT θθ sinˆ, −= && (4.14)
120
Parameter N mKN ⋅ (Nm/A)
inJN ⋅2 (kg m2)
outJ (kg m2)
gT (Nm)
1k (Nm/rad)
2k ( Nm/rad3)
Value 160 0.00746 4.447 10.277 105.18 4.278•104 1.174•109
In the following, the results obtained by using several models that differs by
incorporating or not the stiffness terms and also by including different damping terms
are presented.
4.3.2 - MODEL 1: WITHOUT STIFFNESS
This set of models is characterized by disregarding the stiffness terms in the parameters
vector. This model is used initially to test the algorithm performance and reliability
where 1y is obtained from Equation (4.12) and 2y from Equation (4.14)
[ ]
−++⋅−+−
=
−⋅=
=
outgoutout
outoutfscsdoutoutoutdininindoutgoutout
measout
aminin
TJTTTNTJ
TIKJN
yy
y
θθθθθθθθθθ
θ
sinˆ),(),(),(sinˆ
)(
___
,2
1
&&
&&&&&
&&
(4.15)
During the tests of different models, a good fit between measured data and estimated
one was not satisfactory. By combining different measured signals in forming the
vectors 1y and 2y , it was observed that the catalog motor constant value mK shall be
corrected. Thus, the term am IK has been transported to the right side of Equation
(4.15) giving a new parameter ( mK ) to be identified. A new value
Nm/A00884.0=mK bigger (18 %) the previous one has been used in all the tests.
The models denominated 1, are decoupled (concerning the identification task), namely,
the identification process is considered independent for 1y and 2y . Besides, damping
has not been taken into account.
In the first tests (model 1a), the damping torque, which appears in the equation for 1y ,
TAB. 4.1 - PARAMETERS OBTAINED FROM CATALOG, CURVE FITTING OR CAD DESIGN.
121
has been neglected, resulting, therefore, in 12 yy −= . The equations are still coupled by
the terms related to the inertia outJ and gT . However, aiming an analysis more
simplified, the two equations have been considered decoupled, meaning that there exist
four parameters to be identified. Thus, in this situation, there exist more freedom in the
identification process and consequently the better performance in fitting the measured
data. The obtained results in this test are shown in Table (4.2). Figures 4.16 – 4.18 show
that the result is very good in fitting 2y and not so good for 1y due to the lack of
damping in the system. When the parameters are considered coupled in the
identification task, the coupling parameters outJ and gT assuming values, which are
almost the mean of the values obtained in the decoupled case.
Fig. 4.16 - Model 1a decoupled – 1y according to Equation (4.15)
122
outJ (kg m2) gT (Nm) inconstT _ ( Nm) Model n 0
1y 2y 1y 2y ( Coulomb Friction)
decoupled 8.7143 9.9550 108.570 112.672 1a
coupled 9.3346 110.621
___
decoupled 8.4032 9.9550 111.277 112.672 28.1358 1b coupled 9.1791 111.975 28.1442
Figure 4.17 Model 1a decoupled – 2y according to Equation (4.15)
TAB. 4.2 - PARAMETERS IDENTIFIED FOR MODEL 1A (WITHOUT DAMPING) AND 1B (WITH COULOMB FRICTION) EQUATION (4.15).
123
Fig. 4.18 - Model 1a - Incorporating the Coulomb friction In model 1b (Table (4.2)), several empirical damping models (Schäfer and Silva,2000)
have been considered. A good improvement has been achieved in adjusting 1y by only
incorporating the term related to Coulomb friction. Due to the fact that there is no
damping in Equation (4.15) for 2y , it is expected that the parameters identified from
2y in the decoupled model be very close to those from CAD design (Table (4.1)). In
fact, the parameter related to the inertia outJ present an error smaller than 3 % of the
CAD values, compared to 18 % for 1y considering the Coulomb friction. However, in
the estimation for gT the results for the decoupled case are very similar (7 % for 2y
and 5.8 % for 1y considering Coulomb friction). The errors between CAD data and
estimation show the potential difference between the design phase and the real
equipment. It can be concluded that in the tests, an excellent agreement between the
expected and estimated values has been obtained, showing the efficiency and the
reliability of the algorithms used. Based on this, the next step is to consider a more
complicated model given by Equation (4.9) and (4.10).
124
4.3.2 - MODEL 2: STIFFNESS (2A) AND DAMPING (2B)
This model considers the coupling of the differential equations by both, stiffness torque
and damping. To the author knowledge, this is the first approach to identify the stiffness
of a harmonic drive gear within a robotic joint by just using input and output position
signals. According to Equations (4.9) and (4.10) the output plant equation y reads now
[ ][ ]
−∆+∆⋅−∆+∆−
=
−=
=
),()()(),()()(
__
,,__
,2
1
outoutoutdwgdstiff
ininindwgdstiff
measout
amininTTTNTTT
TIKJ
yy
yθθθθ
θθθθθ&&
&&&&(4.16)
The damping torque fscsdT _ has been neglected since it did not show any remarkable
improvement on the output plant estimation. In the first test, all damping torques that
appears in Equation (4.16) have been neglected, only the stiffness torque has been taken
into account. In this situation, the estimated stiffness coefficient is
Nm/rad102815.5 41 ⋅=k , which represents a factor of 1.23 bigger than the catalog
value. The apparent good agreement between measured data and estimated one is shown
in Figure 4.19. The estimation does not improve when the non-linear stiffness
coefficient ( 2k ) is incorporated in the parameters vector. This fact shows that the
damping torques effect can not be neglected.
By using different damping laws (model 2b), the estimation has presented some
improvement, but not enough to fulfill all the accuracy requirements. The results get a
remarkable improvement when an additional stiffness term has been included in the
parameters vector. This means that besides the typical internal stiffness of HD, there
exist an additional stiffness in the system, acting between input (motor side) and output
(link side). Thus, Equation (4.16) is written as
[ ][ ]
−+−∆+∆⋅−−−∆+∆−
=
−=
=
outOutconstoutoutinastiffstiff
inInconstinininastiffstiff
measout
aminin
signTbkTTNsignTbkTT
TIKJ
yy
y
θθθθθθθθθθ
θ
&&
&&
&&
_3
_3
,2
1
)()()()(
(4.17)
Equation (4.17) also takes into account the Coulomb friction in both sides(input and
125
output). In the following, two different cases are presented: first the damping losses are
not considered, second, a more sophisticated model is used, considering an additional
spring. The results are shown in Figures 4.20 – 4.22.
Tab (4.3) shows the parameters values identified according to Equation (4.17). The
estimation accuracy has been checked by using the Vaf indicator as well as the error’s
norm: 1y = 99.5065%, 2y = 99.4152% (Vaf) and 8 % respectively
1k (Nm/rad) 2k (Nm/rad3)
ak (Nm/rad)
inconstT _ (Nm)
inb (Nm s/rad)
OutconstT _ (Nm)
outb (Nm s/rad)
2.1513e4 3.3543e8 86.2313 21.5057 23.6303 0.4124 9.7285
Fig. 4.19 - Model 2a - Motor side – Only stiffness term
TAB. 4.3 - IDENTIFIED PARAMETERS FOR MODEL 2B – EQUATION (4.17)
126
Fig. 4.20 - Model 2a - Link side – Only stiffness term
Fig. 4.21 - Model 2b - Motor side – Including damping
127
Fig. 4.22 - Model 2b - Link side – Including damping
By observing Table (4.3), one can note that the Coulomb friction has significant
importance on the motor side, but it is almost negligible in the link side. On the other
hand, the viscous damping is relevant in both sides. The estimation indicators show that
the plant output ( 1y and 2y ) are very close to the measured data, which represents the
good estimation accuracy.
It is important to note that the parameters have been obtained by using a batch of
measurements (off-line procedure) and Equation (4.17) as a model. This procedure is
used to determine the initial conditions for the on-line algorithm to be presented later.
Considering the obtained results, one can conclude that the derived mathematical model
represents the system with very good fidelity, and reliable initial conditions for the
recursive algorithm are obtained. This feature helps the on-line algorithm to have a fast
convergence to the expected values. Since a basic model has been validated by using
batch estimation, the next step is to consider a more complex model (including non-
linear parameters) and also to applying an on-line identification procedure.
128
4.3.3 - ON-LINE PARAMETER IDENTIFICATION
First, in order to verify the consistency and convergence of an recursive algorithm, the
basic model described by Equation (4.17), whose off-line results are shown in Table
(4.3) is used. It is evident that the Coulomb term does not have big influence on link
side, but in order to have a coherent comparison, this parameter has been included in the
on-line identification.
Figure 4.23 shows the stiffness coefficient ( 1k , 2k and ak ) and also Coulomb term
(link side) behavior. The measurements are collected by 7 s, and then the off-line
procedure is used to evaluate the initial conditions for the recursive algorithm. It can be
noted that the algorithm has a fast convergence to the values obtained by the off-line
procedure. The parameter related to the stiffness term has a slow convergence. This fact
is predicted in the preliminary analysis (during the initial condition evaluation process)
of the measurement matrix φ . The singular values of matrix φ are shown in Table (4.4)
Parameter 1k 2k ak inconstT _ inb outb OutconstT _
Sing. Val. 0.0338 1.33e-7 18.0883 36.0278 15.6012 15.6400 36.0278
By observing the singular values shown in Table (4.4), one can note (as already
mentioned) that the terms related to stiffness are the critical ones in the identification
sense. The parameter 2k is extremely sensitive to both: model and measurement quality.
Thus, during the initial conditions evaluation process, a procedure that monitors the
singular values magnitude has been implemented. If some of them (singular values)
have magnitude smaller than a minimum (1e-9), an alarm is indicated and the
correspondent parameter is eliminated from the unknown variable vector. This action is
taken in order to avoid degradation in the identification process.
Figure 4.24 shows the values related with damping in both, input and output side. It can
TAB. 4.4 - SINGULAR VALUE OF MEASUREMENT MATRIX φ
129
be noted that all parameters have fast convergence to the expected values. The Coulomb
term on the output side is negligible, thus, in the next tests it will be removed from
parameters vector. The oscillations that appears in the plots are related to the trajectory
used. The triangular trajectory has high accelerations at the end, this high accelerations
influences the identification process. But after some iterations, the algorithm is adjusted
and the oscillations almost disappear.
Fig. 4.23 - Stiffness and damping terms.
By observing the obtained results, one can note that both algorithm and strategy have
presented excellent results in accuracy and also in convergence to the expected values.
The accuracy indicator vaf is 99.5026 % and 99.3924 % for the estimation of 1y and
2y respectively.
130
Fig. 4.24 - Damping coefficients
4.3.4 - PARAMETERS IDENTIFICATION BY USING TWO –STEP ALGORITHM
4.3.4.1 - IRJ EXPERIMENT
Since the results obtained are very satisfactory, the next step is to use more
sophisticated models to represent the robotic joint. Thus, a model that incorporates not
only a non-linear term in the stiffness, but also a non-linear damping model to better
describe the friction at low velocities has been used. The effect of friction at low
velocities presents serious inconvenience (Olsson, 1995 and Armstrong,1992). These
inconvenience becomes relevant if an accurate control is desired. In the IRJ experiment,
the effect of friction at low velocities is evident when trajectories with periods of
constant velocity and acceleration have been used. Figure4.25 presents the time
behavior of the angular velocity. In the designed trajectory, the velocity shall just cross
the zero point. But, this does not occur, for some seconds (detail in Figure 4.25), the
angular velocity has experienced small oscillations around zero and increases later. It is
evident that any linear friction model cannot describe this behavior. Thus, the
131
application of models like Equation (2.77) becomes interesting.
Representing the damping on motor side by
in_dT
S
Sin
inN
inNinin eTTb
δ
ωθ
ωθ
ωθµθ
&
&&&
−⋅⋅+
⋅⋅+=
21tanh (4.18)
and the damping on the output of HD gear by
fscs_dT outoutoutfscsoutout AAsignbb θθθθ cossin)( 2111 +++= && (4.19)
Similarly to Equation (4.17), one obtains
TY Θ⋅= φ (4.20)
where
−=
−−
=
=
out
aminin
outgoutout
amininT
IKJTJ
IKJyy
Yθ
θθθ &&
&&
&&
sinˆ2
1
−−−−−∆⋅∆⋅
−⋅−
−∆−∆−=
−
)cos()sin()(000/
0000tanh
31
3
outoutoutoutin
inSin
inin
in
signNNN
e
θθθθθθθ
θθωθ
θθθ ωθ
φ&&
&&&
&
The parameters vector Θ is represented by
[ ] ToutCoutina AAbbbCCkkk 212121=Θ (4.21)
where
µTC N ⋅=1 and 2
2 ωT
C N= . The exponential coefficient Sδ is assumed to be 1.
132
Fig. 4.25 - Friction at low velocities
It is important to note that the parameters vector represented by Equation (4.21) also
incorporates the cyclic term that normally is related to assembly imperfections.
Comparing Equation (4.17) and (4.21), it is evident that their parameter can not be
identified by the same algorithm (due to non-linear terms). In order to identify the
parameter required by Equation (4.21), the two-step algorithm that is described in
Section 3.5 has been used. In the following, the results obtained with the application of
TS algorithm to IRJ experiment is presented.
The analysis of the measurement matrix φ during the initialization process, has
presented similar results as those obtained with Equation (4.17): The stiffness
coefficients are sensitive to the measurement quality. Besides, the parameter related to
the Stribeck effect also presented high sensitivity. This problem occurs because the
Stribeck term is multiplied by the angular velocity, which is also used to identify the
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viscous damping. The matrix rank is preserved, but the sensitivity is affected. The
singular value of measurement matrix is shown in Table (4.5).
Parameter 1k 2k ak 1C 2C inb outb
Sing. Val. 0.0338 1.33e-7 18.0883 35.8555 0.0272 15.5951 15.6339
Parameter 1A 2A
Sing. Val. 12.3118 33.8440
Figure 4.26 shows the optimization process by using the MCS algorithm. It can be
noted that after a few iterations, the MCS algorithm fulfills all the convergence
requirements. After the identification of non-linear parameters, the on-line identification
of the linear parameters is initialized. In this period, the non-linear parameters kept
constant and the MCS algorithm is waiting for operating if the norm error increases.
Using this strategy, one can obtain an on-line update in the linear parameters and a
random correction (only when the error norm increases) in the non-linear parameters.
This procedure offers a remarkable gain in the computer effort, allowing the inclusion
of non-linear terms in the parameters vector for on-line identification.
In the figures to be presented, the constant dashed line represents the parameters value
obtained by applying off-line methods using all measurements available. Figure4.27
shows the identification results related to the stiffness terms and also related to 1C . It
can be noted that the linear stiffness term has fast convergence to the off-line value
despite of its high sensibility. The cubic stiffness term is more sensitive and presented
more oscillations. However, the algorithm presented enough robustness to overcome
these critical points and converges to the expected value. The parameter correspondent
to the additional spring and to 1C , in agreement with the singular values analysis, are
very stable and presented fast convergence.
TAB. 4.5 - SINGULAR VALUE OF MEASUREMENT MATRIX φ NON-LINEAR PARAMETERS
134
Fig. 4.26 - Non-linear parameters identification Figure 4.28 shows the behavior of 2C , viscous damping (in both sides) and also the
amplitude of cyclic error (normally related to assembly imperfections). In agreement
with the preliminary analysis, the 2C behavior has presented sensitivity bigger than
other parameters. However, the recursive identification after some seconds converges to
the value obtained in the off-line procedure. The viscous damping in the motor side also
presented oscillations and after a short transient converges to the expected values. The
parameter related to the cyclic error presented also stable behavior and its magnitude is
around 3.7 Nm. The phase γ can be considered zero (0.050). The phase behavior is
shown in Figure 4.29. It is important to note that the identification of cyclic phase error
may not be an easy task. This parameter is related to values obtained from trigonometric
functions, which may present small variations depending of robot workspace. However,
It can be noted that the recursive identification always asymptotically converges to the
reference values.
135
Fig. 4.27 - Linear parameters
Fig. 4.28 - Damping parameters
136
Fig. 4.29 - Cyclic phase error
The results have shown the robustness and efficiency of the TS algorithm in
identifying the stiffness and damping parameters of IRJ the experiment. It can be
noted that even in the case where not all parameters have good sensitivity, the
algorithm always deliver coherent results that agree with the values obtained
from other methods. It is also important to stress that in the presented results all
measurements have been obtained from sensors placed in the IRJ experiment. No
simulated data have been used.
4.3.4.2 - TIME VARIANT SYSTEMS
Many times, the system under investigation presents variations, mainly due to
environmental changes. Thus, it is interesting to test the ability of the TS algorithm in
tracking the non-linear parameters, since the linear parameters are continuously updated
by the RLS algorithm. In order to perform these tests, hybrid (measured and simulated)
data is used. The simulated data is used to represent variations in the non-linear
137
parameters. The angular velocity has been taken from the experiment and the stiffness
torque has been calculated by defining values for the parameters that appear in Equation
(4.18). The parameter Sδ is set to 1 and the other ones are given in Table (4.6)
Parameter Value
inb 9 Nm.s.rad-1
µTN ⋅|| 29 Nm 1
2|| −⋅ωTN 480Nm
1ω 0.0595 rad.s-1
Sω 0.0251 rad.s-1
By using the values from Table (4.6) and the measured velocity, the friction
torque( indT _ ) has been simulated. The torque behavior is shown in Figure 4.30.
Using indT _ as a reference for the TS algorithm, a system that experiences variations in
the non-linear parameters has been simulated. At time t = 16 s, the reference torque has
been recalculated and the linear parameters have been increased by 30 % of their
nominal values (Table4.6). The non-linear parameters have been modified at time t = 27
s, when their values has been increased by 20 %. The changes in the nominal values try
to simulate system operational changes.
TAB. 4.6 - PARAMETERS USED IN EQUATION (4.18)
138
Fig. 4.30 - Simulated damping torque
Figure 4.31 shows the linear parameters behavior during identification process. It can be
noted that during the starting procedure when both (linear and non-linear) parameters
are being optimized, the linear parameters presented a typical transient oscillation,
converging after to the nominal values. When the linear parameters are changed (t = 16
s), the RLS algorithm efficiently updates the parameters and converges to the new
nominal values. This is represented by the small jumps in Figure 4.31. The changes in
the linear parameters do not affect the non-linear ones (Figure 4.32). This is possible
because the RLS has fast response, keeping the errors below the value required to
activate the MCS algorithm. On the other hand, the change in the non-linear parameters
affects the linear ones. This occurs because the errors are bigger than the required value
to activate the MCS. The MCS algorithm does not have fast response like RLS. In fact,
in this period, the algorithm behaves in similar manner than in the starting procedure;
the only difference is because the linear parameters are now almost optimized.
139
Fig. 4.31 - Parameters identified by RLS algorithm - Time variant case.
It can be noted that the TS algorithm offers the possibility to identify non-linear
parameters (recursively) at extremely low computational cost. The non-linear algorithm
is activated only in the starting procedure or in case of an expressive change in the
operational conditions. This allows one to obtain a real-time procedure even in cases
where non-linear terms are included in the identification process. The recursive
identification is not possible only in short time interval, when the non-linear
optimization algorithm is activated. Another point to be stressed is related to the initial
conditions. Non-linear optimization algorithms are extremely depending on the initial
conditions, the number of variables to be optimized and even on the profile of the
objective function. As a result, the local optimizer many times found false solutions and
global methods (like MCS) may become excessively slow and sometimes, solution
degradation occurs. Thus, one can conclude that the TS algorithm fulfills most of the
inherent problems of non-linear systems identification, such as
• Computer load;
• Possibility of recursive identification;
140
• Reduction of the parameters to be identified by the non-linear algorithms;
Fig. 4.32 - Non-linear parameters – Time variant case
4.4 - IDENTIFICATION UNDER FAILURE CONDITIONS
The identification techniques require not only a minimal number of measurements but
also that these measurements are of good quality. In specific cases the IRJ experiment,
may operate also in space environment. This fact shows that a special attention to the
sensors and actuators shall be dispensed. Thus, during this research strategies and
alternatives have been developed in order to investigate the special requirement for this
task. An important detail to be considered is related to failure in sensors. Thus, aiming
to maximize the available experiment information in case of failure occurrence, two
solutions are presented:
• Algorithm reconfiguration;
• Stiffness isolation.
141
It is important to note that the solutions proposed always try to maintain the low
computer effort and also to keep the algorithm reconfiguration ability. The first
requirement is needed because there exists the possibility to have on-board processing ,
normally the on-board computer has limited characteristics if compared to computers
used on ground. The second requirement is needed because there is a high possibility of
the system present time variant behavior. Therefore, the main requirements for the
algorithm and strategy to be developed are: adaptation capability, reconfiguration and
low computer effort.
4.4.1 - FAILURE IN VELOCITY SENSOR
In the following, the parameters obtained by using TS algorithm under link velocity
sensor failure (simulated) are presented. In the detection, isolation and estimation
process, the procedure described in Section 4.2 has been used. The model used is
described by Equation (4.21), namely, the model incorporates the non-linear terms and
cyclic errors.
The failure in the link velocity sensor is programmed to occur at t = 20s, and the
corresponding measurement signal has been replaced by random noise (0,0.1). From
this moment, the information given by this sensor is replaced by the MOESP estimation.
The behavior of the identification by using MOESP estimation is shown in Figure 4.33.
It is observed that the accuracy given by the vaf indicator is confirmed. No oscillations
have been observed after replacing the measured data by the estimated ones. The dashed
line in Figure 4.33 represents the values obtained by he off-line identification using only
measured data. It can be noted that the TS algorithm converges to the expected value
using estimated state. This ensures that mission success is guaranteed even in the case
of link velocity sensor failure.
142
Fig. 4.33 - Viscous damping identification (link) by using MOESP estimation
In Figure 4.34 the stiffness and damping parameters are also shown. It can be noted that
all parameters have similar behavior as those presented in Figure 4.27 and 4.28. This
means that the estimation given by MOESP algorithm has no big effect in the global
identification process.
4.4.2 - POSITION SENSOR FAILURE
The coefficient related to the system stiffness is extremely sensitive (this is already
shown by the correspondent singular value) to the measurement quality. This fact is
directly related to the magnitude of measurements to be used. The linear stiffness
coefficient for instance, is related to measurements with magnitude 10-3; the cubic
coefficient is related to measurement of magnitude of 10-9. Although, the MOESP
algorithm gives an excellent state estimation, it is almost impossible to maintain the
estimated data equal or better than the measured one. In the face to this problem, two
solutions have been found that are aiming to minimize the state estimation effect in the
stiffness and 2C . These alternatives are presented in the following..
143
Fig. 4.34 - Linear parameters – Using link velocity given by MOESP algorithm.
4.4.2.1 - ALTERNATIVE 1 – RECONFIGURATION OF IDENTIFICATION ALGORITHM
The first alternative is just to keep the algorithm that identifies the parameters using the
MOESP estimation. The only change is in the term 2R (confidence in the sensors
output) of the RLS algorithm. In this case, as expected, after failure detection and
isolation process, the parameters slide from the nominal values and converge to a new
value, which is assumed as “nominal”.
144
Fig. 4.35 - Parameters related to damping – Using MOESP estimation The link position sensor failure has been scheduled to occur at t = 20s. After the alarm,
immediately all identification process is reconfigured where the following actions have
been made:
• Replace the output of sensor under sensor failure by the correspondent MOESP
estimation;
• Automatic change in the value of 2R of RLS algorithm.
In Figure 4.36 – 4.38 the identification results are shown in the case where the MOESP
estimation has been used. Considering the severe requirements (measurements of
magnitude 10-9) for stiffness parameters, one can conclude that the results are excellent.
The parameters present small changes in comparison with the values obtained by off-
line estimation using only measured data. The variations in the parameters range from
0.5 % (additional spring) to 10.5 % (term related to the Stribeck effect). The variation in
2C is justified by the high sensitivity of this term indicated in the singular value
145
analysis. Once, the identification process is reconfigured, and considering that the
estimation is very good, there exist still a difference between measured and estimated
ones. This difference is naturally reflected more sensitively in the parameters.
4.4.2.2 - ALTERNATIVE 2 – RECONFIGURATION OF ALGORITHM AND FREEZING OF STIFFNESS COEFFICIENT
Besides the actions taken in Section 4.4.2.1, in this situation, the parameters related to
the stiffness term have been frozen in their last value acquired before the failure alarm
happens. In this case, the reconfiguration action also eliminates the parameters 1k and
2k from the vector Θ .
The results obtained by using this strategy are presented in Figures 4.39 – 4.41. A
similar behavior of the strategy presented in Section 4.4.2.1 can be noted: all parameters
present variations from 0.5 % to 10.5%. The difference is because here, the parameters
1k and 2k are kept constant. The impact on the identification process can be described
in the following manner: By one side, when the two elements are removed from the
parameters vector, the system degree of freedom has decreased. This decreases the
model ability in tracking the reference value. On the other hand, when the number of
parameters to be identified is reduced, the algorithm performance increases. Therefore,
in the case studied, these two situations, , almost annihilate the effect one of the other.
This justifies the similarity with the strategy presented in Section 4.4.2.1.
146
Fig. 4.36 - Linear parameters – Using link position estimated by MOESP algorithm
Fig. 4.37 - Damping coefficient– Using link position estimated by MOESP algorithm
147
Fig. 4.38 - Link viscous damping – Using link position estimated by MOESP algorithm
Fig. 4.39 - Linear parameters – Using link position estimated by MOESP algorithm ( 1k
and 2k frozen)
148
Fig. 4.40 - Damping coefficient - Using link position estimated by MOESP algorithm
( 1k and 2k frozen)
Fig. 4.41 - Link viscous damping – Using link position estimated by MOESP algorithm
( 1k and 2k frozen)
149
4.4.3 - SIMULTANEOUS FAILURE IN POSITION AND VELOCITY SENSORS.
The case where both sensors, position and angular velocity, present failure have been
also studied. In accordance with the analysis already presented for isolated sensor
failure, the results are very similar to the case where the position sensor is failed. This is
expected, since the failure in velocity sensor has no big impact on the identification
process. Using the same strategy shown in the case of isolated failure, two alternatives
have been tested: one where the stiffness coefficients are identified after algorithm
reconfiguration and another where the stiffness coefficients are frozen in their last value
acquired before the failure alarm happens. The failure have been adjusted to occur at
time t = 20 s. After this instant, the measurements of link position and velocity are
replaced by random noise (0,0.1), all identification process is reconfigured and the
position and velocity information is given by the MOESP algorithm. Figures 4.42 –
4.44 show the results obtained after the algorithm reconfiguration. The stiffness terms
are also identified. The case where the stiffness parameters are frozen is shown in
Figure 4.45 – 4.47.
Fig. 4.42 - Linear Parameters – Simultaneous failure in position and velocity sensor
(state estimated by MOESP)
150
Fig. 4.43 - Damping coefficient – Simultaneous failure in position and velocity sensor
(state estimated by MOESP)
Fig. 4.44 - Link viscous damping – Simultaneous failure in position and velocity sensor
(state estimated by MOESP)
151
Fig. 4.45 - Linear Parameters – Simultaneous failure in position and velocity sensor ( 1k
and 2k frozen, state estimated by MOESP)
Fig. 4.46 - Damping coefficient – Simultaneous failure in position and velocity sensor
( 1k and 2k frozen, state estimated by MOESP)
152
Fig. 4.47 - Link viscous damping – Simultaneous failure in position and velocity sensor
( 1k and 2k frozen, state estimated by MOESP) 4.5 - SUMMARY
In this Chapter, the applications of the strategies and algorithms derived in the previous
Chapters are presented. A big number of dynamic models has been used and tested. The
complexity of such models has been continuously increased and the effect of each term
has been analyzed. A failure detection and isolation procedure, which requires low
computer effort has been also presented.
All proposed algorithms and strategies developed presented excellent results concerning
accuracy as well as computer effort. The TS algorithm shows efficiency in the
identification process using measured data (IRJ experiment) as well as in case where
time variant systems have been simulated.
The integrated process (identification and failure isolation) show efficiency in the
failure detection and isolation of the data under failure suspect. Due to critical
conditions of the identification of stiffness coefficients, one can conclude that the results
obtained in identification process by using MOESP estimates are remarkable. Based on
these results, applications for those strategies can be immediately suggested, in the
adaptive control are design for instance.
153
CHAPTER 5
CONCLUSION AND RECOMMENDATIONS
5.1 - CONCLUSIONS
The modeling process is extremely important in the system analysis as well as in the
control. Normally, the complexity of the model increases proportionally to the level of
the desired fidelity in system modeling. The more complex the model is, in theory, the
closer we are to the real system. On the other hand, complex models normally present
non-linear terms, which add severe restrictions to the identification process.
In this thesis, a detailed mathematical modeling of a robot joint is presented, where not
only non-linear terms related to the stiffness have been incorporated but also related to
different damping models. The modeling process is started by assuming very simple
models. The model complexity level has been analyzed aiming at a good balance
between accuracy and the impact that the addition of the new term brings to the
identification process. All derived models have been used in analyzing and studying of
the IRJ experiment, where excellent results have been obtained.
When the mathematical model has been assumed satisfactory, the next step was to
choose the trajectory type that improves the excitation of the physical parameters to be
identified. If the robot joint is considered a rigid one, the task to obtain an optimal
trajectory is not complicated, since there are procedures that analytically deliver the
optimal trajectory. On the other hand, when the system is considered flexible (each joint
has two degree of freedom) such procedures are not readily applicable. In face of the
this problem, a strategy based on the singular values analysis has been derived. This
strategy enable ones to select from a determined set of trajectories, the best one which
excites the system.
The identification task has been divided in different steps that aim to give support (and
basis) to the next ones. This methodological strategy has been adopted to ensure that the
next step is only started when the actual one is completely studied and analyzed.
154
Thus, the identification process has been started by using simplified models and also
off-line methods. The off-line methods have been used in the beginning due to its
inherent characteristics (accuracy, stability, etc), which are more stable than the on-line
version. In this phase (off-line identification) excellent results have already been
obtained, where the values present very good agreement with the available reference
data (CAD, catalog, etc.). Since the off-line identification has presented excellent
results (showing that adopted mathematical model represents the real system with good
fidelity), the on-line identification is started; this is the main goal of this thesis. The
recursive identification is very important when the goal is to monitor the system
dynamic behavior and also to control it with high accuracy. Aiming to fulfill these
requirements, two mechanisms have been developed that improve and give an excellent
versatility to RLS algorithm: automatic evaluation of initial condition and adjustment of
forgetting factor. These improvements allow the algorithm to exhibit high performance
at low computer cost.
The next step is to use more complex models, which necessarily incorporate non-linear
terms in the parameters vector. At this point, the TS algorithm, which is based on RLS
(modified) and MCS has been derived. It is important to note that the non-linear
identification process can be performed by any non-linear optimization algorithm. The
MCS algorithm has been selected because it does not requires any initial condition and
belongs to the global optimizer class. The TS algorithm has presented excellent results
not only in identifying the IRJ experiment parameters, but also in the case when a time
variant system has been simulated. The efficiency and accuracy are verified in all
performed tests, correctly identifying the linear and non-linear parameters.
Finally, a failure detection technique based on SMI approach has been developed.
Excellent results have been obtained in both detection (and isolation) and state
estimation given by MOESP algorithm. It has been shown that even under adverse
conditions (two sensors failed) the identification process was kept, giving results that
can be considered very satisfactory. Inclusive, this allows the use of the identified
parameters in an adaptive control, since the maximum error (in the very sensitive
parameter) is in the maximum around 10 %. The very good features of the presented
technique are low computer effort, high accuracy and integrated failure detection and
155
state estimation action.
Finally, it can be concluded that this thesis fulfills all goals proposed, giving innovator
solutions to several typical problems of modeling and identification of robotic systems.
These contributions are cited below:
• Extensive robotic joint modeling;
• Dynamic identification of robotic joint stiffness coefficient (linear and non-
linear);
• Automatic forgetting factor evaluation;
• Development of technique to verify the parameters excitation level;
• Implementation of TS algorithm;
• Development of failure detection and isolation technique;
• Development of techniques for parameters identification under failure situations.
5.2 - RECOMMENDATIONS FOR FUTURE WORKS
The main recommendations for future works are:
• Extend the modeling process to systems with more degrees of freedom, if
possible to a complete robot (DLR light weight robot, for instance);
• Test the identification algorithm in systems with more degree of freedom;
• Verify the TS algorithm performance in case the non-linear part is optimized by
algorithms originating from evolutionary computation (genetic algorithm, neural
network, Fuzzy logic, etc.);
• Develop an adaptive control strategy that makes use of the identification
procedures for friction torque (and force) compensation.
156
• Use the identification algorithm to identify other multibody space system. A
satellite solar panel deployment for instance.
157
BIBLIOGRAPHY
6.1 – BIBLIOGRAPHY REFERENCES
Adachi, S.; Yamaguchi, I.; Kida, I.; Sekiguchi, I.; Yamada, K.; Chida, Y. On-orbit
system identification experiments on engineering Test Satellite – VI. Control
Engineering Practice v.7,n. 7, p. 831 – 841, 1999.
Armstrong, B. On finding exciting trajectories for identification experiments involving
systems with non linear dynamics. IEEE – Transactions on Robotics and
Automation. v. 8, n. 6, p. 1131 – 1139, 1987.
Armstrong, H., B. Friction lag and stick-slip. In: IEEE – International Conference on
Robotics and Automation. Proceedings 1992, v. 2, p. 1448 – 1453.
Atkeson, C. G.; NA, C. H.; Hollerback, J. M. Estimation of inertial parameters of
manipulator loads and links. International Journal of Robotics Research. v. 5, n. 3, p.
101 – 119, 1986.
Bai, E. W.; Huang, Y. Variable gain parameters estimation algorithm for fast tracking
and smooth steady state. Automatica. v. 36, n. 7, p. 1001 – 1008, 2000.
Basseville, M. Information criteria for residual generation and fault detection and
isolation. Automatica. v. 33, n. 5, p. 783 – 803, 1997.
Blaskowiski, M.; Szafarczyk, M.; AL – Taleb, M. A New technique for identification
of servo drivers of machine tools and robots. Journal of Materials Processing
Technology. v. 76, n. 1, p. 36 – 41, 1998.
Caglayan, A. K. Simultaneous failure detection and estimation in linear systems. IEEE
– Transactions on Robotics and Automation. n. 2, p. 1038 – 1041, 1980.
Canudas de Wit, C.; Aubin , A. robot parameter identification via sequential hybrid
estimation algorithm. In: IEEE International Conference on Robotic and Automation.
Sacramento,1991 Proceedings IEEE Service Center, Piscataway, 1991, v. 2, p. 952 –
158
957.
Canudas de Wit, C.; Carrillo, J. A modified EW-RLS algorithm for systems with
bounded disturbances. Automatica. v. 26 n. 3 p. 559 –609, 1990.
Elhami, M. R.; Brookfield, D. J. Sequential identification of coulomb and viscous
friction in robot drivers. Automatica. v. 33 n. 3, p. 393 – 401, 1997.
Fortescue, T. R.; Kershenbam, L. S.; Ydstie, B. E. Implementation of self tuning
regulators with variable forgetting factor. Automatica. v. 17 – n. 6 p. 831 – 835, 1987.
Gertler, J. J. Survey of model-based failure detection and isolation in complex plants.
IEEE – Control System Magazine. v. 8, n. 6, p. 3 – 11. Dec, 1988.
Goes, L. C. S.; Costa, A., Grandineti, F. J. and Soares, A. M. S. Nonlinear dynamic
modeling and control of an underactuated flexible structure mounted micromanipulator
System. In: International Symposium on Dynamics Problems of Mechanics – Diname
Florianópolis, 2001, Proceedings Rio de Janeiro 2001, Sociedade Brasileira de Ciências
Mecânicas (SBCM). p. 225 –230.
Gomes, S. C. P.; Chrétien, J. P. Dynamic modeling and friction compensated control of
a robot manipulator joint. In: IEEE – International Conference on Robotics and
Automation .Nice, 1992. Proceedings IEEE Service Center, Piscataway, 1992, p. 1429
– 1435.
Gorter, R. J. A.; Modeling and Control of HERA Joint. Delft 593 p. Master Thesis of
Delft University of Technology, 1992.
Heimann, B.; Friction and rigid body identification of robot dynamics. In: Pan
American Congress of Applied Mechanics PACAM, Rio de Janeiro 1999 Proceedings
American Academy of Mechanics, Philadelphia 1999, v. 8 p.1557-1560.
Hanssen, S.; Houland, G. E.; Brogardh. Verification of physical parameters in a rigid
manipulator – Wrist Model. In: IMACS Symposium on Mathematical Modeling .Viena
2000 Proceedings IMACS, New Brunswick 2000, p. 849 – 855.
159
Huyer, W. and Neumeier, A. global optimization by multilevel coordinate search.
Journal of Global Optimization. v.14, n. 4, p.331 – 355, 1999.
Johansson, R.; Robutsson, A.; Nilsson, K. and Verhaegen, M. State-space identification
of robot manipulator dynamics. Mechatronics. v. 10, n. 3 p. 403 – 418, 2000
Juang, J. N., System realization using information matrix. Journal of Guidance,
Control, and Dynamics. v. 20, n. 3, p. 492 – 500, 1997.
Juang, J. –N. System realization using information matrix. Journal of Guidance,
Control and Dynamics. v. 20, n. 3, May-June 1997.
Keller, J. Y. Fault isolation filter design for linear stochastic systems. Automatica. v.
35, n. 10, p. 1701 – 1706, 1999.
Leohardt, S.; Ayoulvi, M. Methods of fault diagnosis. Control Engineering Practice.
v. 5 n. 5, p. 683 – 692, 1997.
Ljung, L. System identification theory for the user, New Jersey: Englewood
Prentice-Hall Cliffs, 1987. 519 p..
Ljung, L.; Soderstron, T. Theory and practice of recursive identification.
Massachusetts: MIT Press, 1983. 530 p.
Lozano, R. Dimigianopoulos, D.; Mahory, R. Identification of linear time-varying
systems using a modified least squares algorithm. Automatica. v. 36, n. 7, p. 1009 –
1015, 2000.
Marilier, T.; Richard, J. A. Non linear mechanic and electric behavior of a robot axis
with a Harmonic Drive gear. Robotics and computer Integrated Manufacturing. v. 5,
n 2/3, p. 129 – 136, 1989.
Olsen, H. B.; Bekey, G. A. Identification of parameters in models of robots with rotary
joints. In: IEEE – International Conference on Robotics and Automation, St Louis 1985
Proceedings IEEE Service Center, Piscataway, 1985 p. 1045 – 1049.
160
Olsson, H. Control systems with friction. Lund. 172 p. PhD Thesis of Lund Institute
of technology, 1996,.
Pfeiffer, F. and Hoelzl, J. Parameter identification for industrial robots. IEEE –
Transactions on Robotics and Automation.. v.2, p. 1468 – 1476, 1995.
Schaefer, B.; Silva, A. R. Joint modeling and parameters identification for elastic space
manipulators in micro-gravity long term missions. In: International Conference on
Nonlinear Dynamics, Chaos Control and their Applications in Engineering Sciences,
Campos do Jordão, 2000 Proceedings ABCM Rio de Janeiro 2000,p. 16.
Schaefer, B.; Silva, A. R. Space robotics experiments for increasing dynamics
modeling fidelity. In: European Congress on Computational Methods in Applied
Sciences and Engineering , Barcelona , 2000 Proceedings Eccomas Barcelona 2000 ,p.
20.
Seyfferth, W.; Magzal, A. J.; Angeles, J. Nonlinear modeling and parametric
identification of harmonic drive robotic transmissions. IEEE – Transactions on
Robotics and Automation., v. 3, p. 3027 – 3032,1995.
Shi, P.; Mcphee, J.; Heppler, G. Design and control of an experimental facility for
emulating space-based robotic manipulators In: 31st International Symposium on
Robotics. Montreal, 2000. Proceedings IFR Montreal 2000, p. 481 – 486..
Silva, A. R.; Schaefer, B.;. Harmonic drive modeling and physical parameters
identification for robotic joints . In: International Symposium on Dynamics Problems of
Mechanics – Diname Florianópolis, 2001, Proceedings Rio de Janeiro 2001, Sociedade
Brasileira de Ciências Mecânicas (SBCM). p. 213 – 218.
Silva, A. R.; Schaefer, B.; Souza, L. C. G.; Fonseca, R. A. Space robotics joints non
linear modeling and on-line parameters identification. In: 31st International Symposium
on Robotics. Montreal, 2000. Proceedings IFR Montreal 2000, p. 461 – 467.
Silva, A. R.; Schaefer, B.; Souza, L. C. G.; Fonseca, R. A. Integrated linear and
nonlinear identification for space robots dynamics. IAF – 51st International
161
Astronautical Congress. Rio de Janeiro ,Oct. 2000.
Spechet, R.; Isermann. On-line identification of inertia friction and gravitational Forces
applied to an industrial robot. IFAC Proceedings series, v. 10 p. 219 – 224. 1988.
Swevers, J.; Ganseman C.; Tukel, D. B.; Schutter, J. Optimal robot excitation and
identification. IEEE – Transactions on Robotics and Automation. v. 13 n. 5 p. 730 –
740, 1997
Tempo, R.; Worst Case. Optimality of smoothing algorithm for parametric system
identification. Automatica, v. 31, n. 5, p. 759 – 763,1995.
Tuttle, T. D. Understanding and modeling the behavior of harmonic drive gear
transmission. , Cambridge: 314 p Master Dissertation. MIT – Massachusetts Institute
of Technology. 1992.
Verhaegen, M., Varga, A.; Grubel, G. Some experience with MOESP class of
subspace model identification methods in identifying the B 105 helicopter-
Oberpfaffenhofen, DLR, 1994, 26 p.
Verhaegen, M.; Dewilde E, P. Subspace model identification Part 2: Analysis of
elementary output-error state-space model identification algorithm. International
Journal of Control, 1992 – v. 36, n. 5, p 1211-1241.
Verhaegen, M.; Dewilde E, P. Subspace model identification Part 3: Analysis of
ordinary output-error state-space model identification algorithm. International Journal
of Control, 1993 – v. 58, n. 5, p 555-586.
Yin, K. Minimax Methods for fault isolation in the directional residual approach.
Chemical Engineering Science. v. 53, n. 16, p. 2921 – 2931, 1998.
Yu, D. and Shields, D. N. A bilinear fault detection observer. Automatica. v.. 32, n.
11, p. 1597 – 1602, 1996.
Zhou, Q. G.; Clevet, W. Recursive identification of time-varying systems via
incremental estimation. Automatica. v. 32 n. 10, p. 1427 – 1431, 1996.
162
Zimmerman, D. C.; Lyde, T. L. Sensor failure detection and isolation in flexible
structures using system realizations redundancy. Journal of Guidance, Control and
Dynamics. v. 16, n. 3, p. 490 – 497, 1993.
Zang, Y.; Feng, C. Unbiased parameter estimation of linear system with colored noise.
Automatica. v. 33 n. 5, p. 969 – 973, 1997.
163
6.2 – AUXILIARY BIBLIOGRAPHY
Astrom, K. J.; Soderstrom, T.; Trends in system identification, Automatica, v.31, n. 12,
Dec., p. 1689-1900, 1995.
Caccavale, F.; Chiacchio Identification of dynamic parameters and feedforward control
for a conventional industrial manipulator – Control Engineering Practice, v.2, n.6, p.
1039-1050, 1994.
Chen, C. W.; Huang, J. K.; Phan, M.; Juang, J. N. Integrated system identification and
state estimation for control of flexible space structures. – Journal of Guidance,
Control, and Dynamics. v.15, n.1, Jan-Feb, p. 88-95, 1992.
Cooper, J. E., On-line version of the eigensystem realization algorithm using data
correlation – Journal Of Guidance, Control, and Dynamics. v. 20, n.1, Jan-Feb, p.
137-142, 1997.
Cooper, J. E.; Wright, J. R. Spacecraft in-orbit identification using eigensystem
realizations methods. Journal of Guidance, Control, and Dynamics. v.15, n.2, Mar-
Apr, p. 352-359, 1992.
Dépincé, Ph. Parameters identification of flexible robots In: IEEE – International
Conference on Robotics & Automation. Leuven 1998. Proceedings IEEE Service
Center, Piscataway, 1998, p. 1116-1121.
Ho, B. L.; Kalman, R. E., Effective construction of linear state-variable models from
input/output, Proceedings of the 3rd Annual Allerton Conference on Circuit and
System Theory, 1965, p. 445-459.
Huang, J. K.; Hsiao, M. H.; Cox, D. E. Indirect identification of linear stochastic
systems with known feedback dynamics. – Journal of Guidance, Control, and
Dynamics. v.19, n.4, Jul-Aug, p. 836- 841, 1996.
Inman, D. J. Vibration with control, measurement, and stability, Englewood Cliffs:
Prentice Hall, 1989, 350 p.
164
Johansson, R. System modeling identification, Englewood Cliffs: Prentice Hall, 1993,
512 p.
Juang, J. N. System realization using information matrix. Journal of Guidance,
Control, and Dynamics. v.20, n.3, May-Jun, p. 492-500, 1997.
Juang, J. N. Unification of several system realization algorithms. – Journal of
Guidance, Control, and Dynamics. v. 20, n.1, Jan-Feb, p. 67-73, 1997.
Juang, J. N.; Pappa R. S. An eigensystem realization algorithm for modal parameter
identification and model reduction. – Journal of Guidance, Control, and Dynamics.
v. 8, n.5, Set-Oct, p. 620-627, 1985.
Juang, J. N.; Pappa R. S. Effects of noise on modal parameters identified by the
eigensystem realization algorithm – Journal of Guidance, Control, and Dynamics. v.
.9, n.3, p. 294-303, 1986.
Landau, I. D. System identification and control design. Englewood Cliffs: Prentice-
Hall, 1990, 253 p.
Ljung, L.; Guo, L. Role of model validation for assessing the size of the unmodeled
dynamics IEEE Transactions on Automatic Control, v.42, n. 9, Sept., p. 1230-1239,
1997.
Maciejowski, J. M. Multivariable feedback design, Wokingham: Addison-Wesley,
1989, 424 p.
Medina, E. A.; Irwin, R. D.; Mitchell, J. R. ; Bukley, A. P. Mimo identification using
frequency response data. Proceedings of 15th Annual Rocky Mountain Guidance and
Control Conference. p. 457-473, 1992.
Mook, D. J.; Lew, J. S. A robust algorithm for system realization/identification – The
Journal of the Astronautical Sciences. v. 38, n.2, Apr-Jun., p. 229-243, 1990.
Norris, M. A. Constrained eigensystem realization algorithm for lightly damped
distributed structures. Journal of Guidance, Control, and Dynamics. v. 14, n.5, Sept-
165
Oct, p. 1014-1020, 1991.
Oshman, Y and Mendlboim, T. Maximum likelihood identification and realization of
stochastic systems. Journal of Guidance, Control, and Dynamics. v.17, n.4, Jun-Aug,
p. 692-700, 1994.
Peterson, L. D. Efficient computation of the eigensystem realization algorithm. –
Journal of Guidance, Control, and Dynamics. v. 18, n.3, May-Jun., p. 395-403, 1995.
Ramirez, W. F. Process control and identification. San Diego: Academic Press, 1994,
424 p.
Readlea, J. C. , Henryb, L. M. Intelligent on-line system identification using a
blackbord system. Engineering Applications of Artificial Intelligence, v. 11, n.6,
Dec., p. 759-770, 1998.
Reyes, F., Kelly, R. On parameter identification of robot manipulator. In: IEEE –
International Conference on Robotics and Automation. Albuquerque, 1997
Proceedings, IEEE Service Center, Piscataway 1997, v. 03, p. 1910-1915.
Söderström, T. System identification. Englewood Cliffs: Prentice-Hall, 1989, 612p.
Unbehaunen, H.; Rao, G. P. Identification of continuous systems. Amsterdam: North-
Holland, 1987, 378 p.
Wang, Shih-Ho; Lee, T.F.; Zachery, R. System identification via singular value
decomposition. Electronics Letters, v. 32, n. 1, Jan., p. 76-78, 1996.
166
167
ANNEX A
ACCELEROMETER CALIBRATION
A.1 - INTRODUCTION
In the accelerometer calibration task, several procedures can be used to eliminate the
static influence (offset) on the measurement:
1. static procedures;
2. mean removal (detrending);
3. differentiation;
4. filtering.
The static procedure consists in using specific position to determine the zero point and
offset of the sensor. The mean removal procedure (detrending) consists in subtract the
mean value of the measurement. This can be interpreted in the following way: If a
bounded process oscillates around the mean value, this is the equilibrium point. When
the mean value is subtracted, the equilibrium point is translated (in theory) to zero. The
differentiation process can be viewed as kind of filtering. After the differentiation
process, the data is integrated using zero initial condition. Finally, a high pass filter that
has gain close to zero can be also used to remove static effects on the measurements. To
remove the offset of the measurement from accelerometers a static procedure has been
used. Thus, in the following, the static calibration procedure is presented.
A.2 - STATIC CALIBRATION
In this procedure, the measurements have been collected in five different static positions
(Figure A.1), which have been used to eliminated the offset of the measurements.
168
Fig.. A.1 - Static position for accelerometer calibration.
The accelerometers output in the positions of Figure A.1 are shown in Figures A.2 –
A.4. The static effect presented in the accelerometers measurement can be determined
by using measurement shown in A.2 – A.4 and the relationship
downmeandownmeanupmean
offsetc _2
__−
+= (A.1)
where offsetc represents the calculated offset.
By using A.1, the offset of accelerometers 1 and 2 has been determined respectively by
V 372.0 1 =coffset
V 396.02 −=coffset
Another way to evaluate the offset is to use measurements obtained in horizontal
position:
V 390.0 1 =moffset
V 406.02 −=moffset
By using these values, the calibration factor for the two accelerometers has been
determined as
V 2.401 1 1 =g
V 537.2 1 2 =g
where g is the gravity constant.
169
Fig. A.2 - Accelerometer 1 output – Positions: face up and face down
Fig. A.3 - Accelerometer 1 output – Positions: horizontal 1 and horizontal 2
Fig. A.4 - Accelerometers output – Positions: horizontal 1 for accelerometer 1 and
accelerometer 2 face up
170
Fig. A.5 - Accelerometer 2 output – Positions: face down and horizontal 1
171
ANNEX B
B.1 – SENSOR ACCURACY
The accuracy of the sensors used in the IRJ experiment are shown in Table B.1
Sensors Accuracy/Resolution Accelerometers 0.00098 g (9.6e-3 m/s^2) Torque Sensor 50 mNm Encoder 0.0036 0
Hall Sensor 0.0010
Current 15 bits
TAB. B.1 - SENSORS ACCURACY AND RESOLUTION
172
B.2 – ADDITIONAL DETAILS OF IRJ EXPERIMENT
Fig. B.1 Internal details of IRJ experiment
173
Fig. B.2 - Internal view of IRJ experiment
174
Fig. B.3 - Seven joint robot
175
ANNEX C C.1 – ALGORITHMS
The algorithms derived in this thesis have implemented by using the MATLAB© and
SIMULINK© package. In the following, to illustrate the software flow, some blocks and
schemes are shown.
Ndata
T o Workspace1
fdata
T o Workspace
Selector
Selector Scope
SelectS
S-Function
Inputs Meas
RLS Identi fi ca tion
Double cl ickhere before
each new run
r_m easT .m at f rom f ile good m eas
Fai lure S im ula tion
Fig. C.1 - Simulink block diagram
The software has the following structure:
a) First, the data are read from file, after the states are estimated and the failure
detection is also implemented (the interface for this part is shown in Figure
C.2).
b) After, the measurements are sorted in a proper way to start the identification
process;
c) Since the data are sorted, all process explained in Chapter 3 is initialized. It
is important to note that the blocks shown in Figure C.1 are parent blocks,
namely, all of them have sub level (children block). In addition, the block
type MATLAB function or S function are used in order have more
flexibility in changing the number of parameter to be identified.
176
Fig. C.2 - Failure detection interface .
Figure C.2 shows the interface estimation block interface. It can be noted that all
process is easy configurable for different tests. It is possible to change the gear ratio,
freeze the stiffness coefficient, etc.
Just as illustration, Figures C.3 – C.5 show some sub levels of the state estimation and
failure detection block..
177
Fig. C.3 - Sub level 1 of failure detection and state estimation block.
Fig. C.4 - Sub level 2 of the failure detection and state estimation block
178
179
ANNEX D
D.1 – RECURSIVE LEAST SQUARES ALGORITHM – DERIVATION II
Having as a starting point the model described by
)(ˆ)1()( kkky T ξφ +Θ−= (D.1)
and a data sequence, represented by )(ky and L,1 , )1( =− kkφ , for which, there is no
length limit. It is desired to estimate Θ in recursive way. For this, the recursive least
squares estimator is derived in the following way
−
−−=Θ ∑∑
=
−
=
k
i
k
i
T iyiii1
1
1)()1()1()1(ˆ φφφ (D.2)
By using the notation
1
1)1()1(
−
=
−−= ∑
k
i
Tk iiP φφ
)1()1()1()1(1
1 −−+
−−= ∑
=
− kkiiP Tk
i
Tk φφφφ
)1()1( 11 −−+= −
− kkP Tk φφ (D.3)
The basic idea of the equations above is to express the quantities at instant k as a
function of previous instants. This is the principle of the recursive algorithms. Equation
(D.2) can be rewritten as
−+−=Θ ∑
−
=
1
1)()1()()1(ˆ
k
ikk kykiyiP φφ (D.4)
180
Writing Equation (D.2) for instant k-1, one obtains
−=Θ
−− ∑∑
−
=−
−
=
1
11
1
1)()1()1()1(
k
ik
k
i
T iyiii φφφ (D.5)
by observing that the left side of Equation (D.5) can be represented in a compact form
as 111
ˆ−
−− ΘkkP . Replacing this result in Equation (D.4),
[ ]( )[ ]
[ ])(ˆ
ˆ)1()()1(ˆ
)()1(ˆ)1()1(
)()1(ˆˆ
1
11
11
111
keL
kkykP
kykkkPP
kykPP
kk
kT
kk
kT
kk
kkkk
+Θ=
Θ−−−−Θ=
−+Θ−−−=
−+Θ=Θ
−
−−
−−
−−−
φφ
φφφ
φ
(D.6)
where )1( −= kPL kk φ is a gain matrix and 1ˆ)1()()( −Θ−−= kT kkyke φ is called
innovation at instant k.
Equation (D.6) shows that to use the recursive algorithm, it is necessary to evaluate Pk..
The direct use of Equation (D.3) is inadequate since this equation requires a matrix
inversion in each algorithm iteration. By applying the matrix inversion lemma
( ) 11
111 )1(1)1()1()1( −−
−−− −+−−−−= kT
kT
kkk PkkPkkPPP φφφφ (D.7)
and using Equation (D.7), one can note that the term to be inverted is only a scalar (for
models with only one output). Finally, using Equation (D.7) , the gain matrix Lk is
defined as
1)1()1(
)1(
1)1()1(
)1()1()1()1(
1
1
1
111
+−−
−=
+−−
−−−−−=
−
−
−
−−−
kPk
kPkPk
kPkkPkPL
kT
k
kT
kT
kkk
φφ
φφφ
φφφφ
(D.8)
181
Ordering the Equations D.6 – D.8 in a proper way and setting )1( −= kk φφ , one
obtains
[ ]11
11
1
1
ˆ)(ˆˆ
1
−−
−−
−
−
−=
Θ−+Θ=Θ
+=
kTkkkk
kTkkkk
kkTk
kkk
PLPP
kyL
P
PL
φ
φ
φφ
φ
(D.9)
that is the recursive least squares estimator.
D.2 – RECURSIVE LEAST SQUARES ALGORITHM WITH FORGETTING FACTOR
In the following, the derivation of a recursive algorithm with a forgetting factor is
presented. The staring point is
∑∑=
−
=−
−−=Θ
k
ii
k
i
Tik iyikwiikw
1
1
1)()1()()1()1()(ˆ φφφ (D.10)
The difference related to the algorithm derive in the previous section is the gain
sequence w. Normally, the gain sequence must follow the following constraints
)1( )(1)(
−==
kwkwkw
ii
kλ
(D.11)
namely, the biggest weigh corresponds to the last value and is equal to 1. When a new
data comes, all weighs are multiplied by a factor υ , which normally receives values in
the range 99.095.0 ≤≤ λ . It is important to note that λ can vary in each iteration, in
this case, it is indicated as kλ . Another way to understand the factor λ is letting it be a
ratio between two consecutive weigh for the same data, namely, )(/)(1 kwkw ii−=λ .
Because this, λ is also known as a forgetting factor.
From the first parcel of Equation (D.10),
182
)1()1(
)1()1()1()1()1(
)1()1()()1()1()(
11
1
1
1
1
1
−−+=
−−+−−−=
−−+−−=
−−
−
=
−
=
−
∑
∑
kkP
kkiikw
kkkwiikwP
Tk
k
i
TTi
k
i
Tk
Tik
φφλ
φφφφλ
φφφφ
(D.12)
where the properties of (D.11) has been used in the derivation. The second parcel of
Equation (D.10) results in
)()1(
)()1()()()1()(
1
1
1
kykF
kykkwiyikwF
k
k
ikik
−+=
−+−=
−
−
=∑
φλ
φφ (D.13)
Rewriting Equation(D.11) and replacing Equation(D.12) and Equation (D.13), one
obtains
[ ][ ]
[ ]11
1
1
111
1
ˆ)1()()1(ˆ
)()1(ˆ)1()1(
)()1(ˆ
)()1(
ˆ
−−
−
−
−−−
−
Θ−−−+Θ=
−+Θ
−−−=
−+Θ=
−+==Θ
kT
kk
k
Tk
k
kkk
kk
kkk
kkykP
kykkkP
P
kykPP
kykFPFP
φφ
φλ
φφλ
φλ
φλ
(D.14)
In Equation (D.14) the adaptive gain of the estimated vector (usually known as a
Kalman gain) is )1( −= kPL kk φ . By using again the matrix inversion gain in Equation
(D.12),
[ ]
+−−
−−−=
−−−−−=
−
−−−
−−−
−−
λφφφφ
λ
λλφ
φφφλλ
)1()1()1()1(1
)1()1()1()1(
1
111
111
11
kPkPkkP
P
PkkPkk
PPP
kT
kT
kk
kT
kTkk
k
(D.15)
Finally,
183
λφφφ
λφφφφφ
φλ
φ
+−−
−=
+−−
−−−−−=
−=
−
−
−
−−−
)1()1()1(
)1()1()1()1()1(
)1(1
)1(
1
1
1
111
kPkkP
kPkkPkkP
kP
kPL
kT
k
kT
kT
kk
kk
(D.16)
Thus, the recursive least squares estimator with forgetting factor λ is
[ ]
+−=
Θ−+Θ=Θ
+=
−
−−−
−−
−
−
λφφφφ
λ
φφ
λφφφ
kkTk
kTkkk
kk
kTkkkkk
kkTk
kkk
PPP
PP
kyP
PP
L
1
111
11
1
1
1
ˆ)(ˆˆ
(D.17)
where )1( −= kk φφ
