DECENTRALIZED SLIDING MODE CONTROL FOR AN ELECTROHYDRAULIC ROBOT MANIPULATOR HASZURAIDAH ISHAK A project report submitted in partial fulfilment of the requirements for a award of the degree of Master of Engineering (Electrical-Mechatronics and Automatic Control) Faculty of Electrical Engineering Universiti Teknologi Malaysia MAY 2007
32
Embed
DECENTRALIZED SLIDING MODE CONTROL FOR AN …eprints.utm.my/id/eprint/5962/1/HaszuraidahIshakMFKE2007.pdfSliding Mode Control theory and the utilisation of Matlab / Simulink software.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
DECENTRALIZED SLIDING MODE CONTROL FOR AN
ELECTROHYDRAULIC ROBOT MANIPULATOR
HASZURAIDAH ISHAK
A project report submitted in partial fulfilment of the
requirements for a award of the degree of
Master of Engineering (Electrical-Mechatronics and Automatic Control)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
MAY 2007
iii
This thesis is especially dedicated to my dearest father, mother and family
for their love, blessing and encouragement ...
iv
ACKNOWLEDGEMENT
Alhamdulillah, I am greatly indebted to Allah SWT on His mercy and
blessing for making this project successful.
I would like to express my deepest gratitude and thanks to Professor Dr.
Johari Halim Shah Osman, my honourable supervisor, for his continuous guidance,
committed support and invaluable advice throughout my study.
I wish to thank Associate Professor Dr. Mohamad Noh Ahmad and
Associate Professor Dr. Yahaya Md Sam for their guidance and facilitation on the
Sliding Mode Control theory and the utilisation of Matlab / Simulink software.
I sincerely thank to all lecturers who have taught me, for the lesson that has
been delivered. Not to mention, to all my friends, thank you for sharing useful idea,
information and moral support during the course of study.
Last but not least, I would like to express my sincere appreciation and
gratitude to my parents, who are always there when it matters most.
v
PUBLICATION
The following paper, based on the work described in this thesis, has been submitted
to conference:
1. J.H.S Osman, H. Ishak. A Decentralized Sliding Mode Tracking Controller
for Hydraulic Robot Manipulators. The 1st International Conference on
Control, Instrumentation, and Mechatronics Engineering (CIM). 2007.
Johor, Malaysia. (Accepted for Oral Presentation and Publication).
2. S.Z Nordin, H. Ishak, J.H.S Osman. Sliding Mode Tracking Controller for
Hydraulic Manipulators with Numerical Analysis. 2ND International
Conference on Mathematical Sciences (ICoMS). 2007. Johor, Malaysia.
(Accepted for Oral Presentation and Publication).
vi
ABSTRACT
This thesis is concerned with the problems of modelling and controlling of a
3 DOF electrohydraulic robot manipulators. The control of electrohydraulic robot
manipulator is challenging due to the dependence of system parameters on variables
such as displacement and velocity, on the geometry and inertia of the links,
uncertainties associated with gravity, coriolis and centrifugal forces, variations in
payload handled by the manipulator, and environmental influences. To overcome
these problems, an integrated mathematical model of the 3 DOF electrohydraulic
robot manipulators is treated as a large-scale uncertain system models using the
known parameters of the robot. Decentralized control concept is used in this study
where the uncertain system is treated as large-scale system which composed of a set
of interconnected uncertain subsystems. A variable structure control (VSC) strategy
is utilized to overcome the inherent high nonlinearity in the system dynamics under
decentralized and centralized frameworks. In each of the approach, a variant of the
VSC known as the Sliding Mode Control (SMC) is adopted to ensure the stability of
the system dynamics during the sliding phase and to render that the system
insensitive to the parametric variations and disturbances. The performance and
robustness of the proposed controller is evaluated through computer simulation by
using Matlab and Simulink. The results proved that the controller has successfully
provided the necessary tracking control for the 3 DOF electrohydraulically driven
robot manipulator system.
vii
ABSTRAK
Tesis ini bertujuan untuk mengenengahkan model matematik dan teknik
kawalan bagi robot berkuasa hidro. Pengawalan robot berkuasa hidro adalah lebih
rumit disebabkan sifat sambungan mekanikal dan motornya yang tidak linear,
parameter yang berubah-ubah, ketidaktentuan beban dan kesan perangkai. Untuk
mengatasi masalah ini, model bersepadu untuk tiga darjah kebebasan robot berkuasa
hidro ditukar menjadi suatu sistem berskala besar tak pasti berdasarkan kepada had-
had parameter sistem. Reka bentuk kawalan berasaskan kepada pendekatan kawalan
ternyahpusat digunakan dalam simulasi ini dengan menganggap sistem robot sebagai
suatu sistem berskala besar, yang mempunyai model bersepadu yang boleh
dipecahkan menjadi beberapa sub-sistem yang saling terhubungkait. Strategi
kawalan struktur berubah (VSC) diadaptasi untuk mengatasi ciri-ciri rumit yang
terdapat dalam robot berkuasa hidro menggunakan konsep ternyahpusat dan sepusat.
Didalam setiap pendekatan ini, dua pengawal penjejakan menggunakan konsep
kawalan gelincir telah dicadangkan, di mana pengawal pertama menggunakan
konsep sepusat, manakala pengawal kedua menggunakan konsep ternyahpusat
Kawalan ragam gelincir (SMC) iaitu variasi daripada kawalan struktur berubah telah
dipilih untuk memastikan kestabilan sistem semasa fasa gelincir tanpa dipengaruhi
parameter yng berubah-ubah dan ketidaktentuan beban. Pencapaian kaedah ini
dinilai melalui simulasi komputer. Keputusan membuktikan bahawa, strategi
kawalan ini berjaya dalam membekalkan kawalan laluan/ trajektori yang diperlukan
untuk sistem lengan robot berkuasa hidro.
viii
CONTENTS
CHAPTER TITLE PAGE
TITLE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
PUBLICATION v
ABSTRACT vi
ABSTRAK vii
CONTENTS viii
LIST OF TABLES xi
LIST OF FIGURES xii
LIST OF SYMBOLS xviii
LIST OF ABBREVIATIONS xx
1 INTRODUCTION 1
1.1 Robot Manipulator System 1
1.2 Electrohydraulic Robot Manipulator 4
1.3 Research Objective 9
1.4 Scope of Research 10
1.5 Research Methodology 10
1.6 Structure and Layout of Thesis 12
ix
2 MATHEMATICAL MODELING OF ELECTROHYDRAULIC
ROBOT MANIPULATOR 14
2.1 Introduction 14
2.2 Dynamic Equation and State Space Representation
of Electrohydraulic Actuator 15
2.3 Dynamic Equation of Robot Manipulator
Mechanical Linkage 21
2.4 Integrated Dynamic Model of the Electrohydraulic
Robot Manipulator 23
2.5 Dynamic Model of 3 DOF Electrohydraulic
Robot Manipulators 26
2.5.1 Dynamic Equation of the Electrohydraulic
Motors for the 3 DOF Robot Manipulator 28
2.5.2 Dynamic Equation of 3 DOF Robot
Manipulator Mechanical Linkage 31
2.5.3 Integrated Dynamic Model of the Three DOF
Electrohydraulic Robot Manipulator 35
2.6 Summary 40
3 DECENTRALIZED SLIDING MODE
CONTROLLER DESIGN 42
3.1 Introduction 42
3.2 Decentralized Sliding Mode Control Strategies 43
3.3 Electrohydraulic Robot Manipulator as a
Large-Scale Uncertain System 45
3.4 Problem Formulation for Decentralized SMC 52
3.4.1 System Dynamics during Sliding Mode 54
3.4.2 Sliding Mode Tracking Controller Design 55
3.5 Summary 57
x
4 SIMULATION 58
4.1 Introduction 58
4.2 Trajectory Generation 58
4.3 Simulation Study 60
4.4 Controller Parameters Selection 61
4.5 Tuning Algorithm 62
4.6 Simulation using Centralized Sliding Mode
Controller 64
4.61 Numerical Computation and Selection
of Controller Parameters 64
4.7 Simulation Decentralized Sliding Mode
Controller 72
4.71 Numerical Computation and Selection
of Controller Parameters 72
4.7.2 Comparison of the Centralized SMC &
Decentralized SMC 80
4.7.3 Effect of Load Variation 87
4.7.4 Effect of the Controller Parameter 90
4.7.5 Effect of Sliding Surface Constant, C 97
4.7.6 Effect of Sampling Time 102
4.7.7 Effect of Closed-loop Poles 106
4.7.8 Control Input Chattering Suppression 110
4.6 Summary 115
5 CONCLUSION 117
5.1 Conclusion 117
5.2 Suggestion For Future Work 118
REFERENCES 120
xi
LIST OF TABLES
TABLE NUMBER TITLE PAGE
2.1 Range of Operation of the Mechanical Linkage 27
2.2 Parameters of the Mechanical Linkage 27
2.3 Parameters of the Hydraulic Actuator 28
3.1 Value of Elements in Matrices 50
4.1 Order of Extrapolation for Chattering Suppression 110
xii
LIST OF FIGURES
FIGURE NUMBER TITLE PAGE
1.1 Schematic diagram of a hydraulic system and its
components 2
1.2 Schematic diagram of a spool valve in a neutral position 3
2.1 Physical Model of an Electrohydraulic Servo Control
System 16
2.2 Schematic diagram of the Gear Train Connecting the
Motor and the Load 18
2.3 Three DOF Hydraulically Driven Robot Manipulator 26
4.1 Desired Joint Position Profile 59
4.2 Desired Joint Velocity Profile 60
4.3 Simulation Flow Chart 61
4.4 Joint 1 tracking response (angle) using centralized SMC
tracking controller 66
4.5 Joint 2 tracking response (angle) using centralized SMC
tracking controller 66
4.6 Joint 3 tracking response (angle) using centralized SMC
tracking controller 67
4.7 Joint 1 tracking response (velocity) using centralized
SMC tracking controller 2 67
4.8 Joint 2 tracking response (velocity) using centralized
SMC tracking controller 2 68
4.9 Joint 3 tracking response (velocity) using centralized
SMC tracking controller 2 68
4.10 Joint 1 control input using centralized SMC
tracking controller 69
xiii
4.11 Joint 2 control input using centralized SMC
tracking controller 69
4.12 Joint 3 control input using centralized SMC
tracking controller 70
4.13 Joint 1 sliding surface using centralized SMC
tracking controller 70
4.14 Joint 2 sliding surface using centralized SMC
tracking controller 71
4.15 Joint 3 sliding surface using centralized SMC
tracking controller 71
4.16 Joint 1 tracking response (angle) using decentralized
SMC tracking controller 74
4.17 Joint 2 tracking response (angle) using decentralized
SMC tracking controller 74
4.18 Joint 3 tracking response (angle) using decentralized
SMC tracking controller 75
4.19 Joint 1 tracking response (velocity) using decentralized
SMC tracking controller 75
4.20 Joint 2 tracking response (velocity) using decentralized
SMC tracking controller 76
4.21 Joint 3 tracking response (velocity) using decentralized
SMC tracking controller 76
4.22 Joint 1 control input using decentralized SMC
tracking controller 77
4.23 Joint 2 control input using decentralized SMC
tracking controller 77
4.24 Joint 3 control input using decentralized SMC
tracking controller 78
4.25 Joint 1 sliding surface using decentralized SMC
tracking controller 78
4.26 Joint 2 sliding surface using decentralized SMC
tracking controller 79
4.27 Joint 3 sliding surface using decentralized SMC
tracking controller 79
xiv
4.28 Joint 1 tracking error (angle, 0kg) using Centralized
and Decentralized SMC 81
4.29 Joint 2 tracking error (angle, 0kg) using Centralized
and Decentralized SMC 81
4.30 Joint 3 tracking error (angle, 0kg) using Centralized
and Decentralized SMC 82
4.31 Joint 1 tracking error (velocity, 0kg) using Centralized
and Decentralized SMC 82
4.32 Joint 2 tracking error (velocity, 0kg) using Centralized
and Decentralized SMC 83
4.33 Joint 3 tracking error (velocity, 0kg) using Centralized
and Decentralized SMC 83
4.34 Joint 1 tracking error (angle, 10kg) using Centralized
and Decentralized SMC 84
4.35 Joint 2 tracking error (angle, 10kg) using Centralized
and Decentralized SMC 84
4.36 Joint 3 tracking error (angle, 10kg) using Centralized
and Decentralized SMC 85
4.37 Joint 1 tracking error (velocity, 10kg) using Centralized
and Decentralized SMC 85
4.38 Joint 2 tracking error (velocity, 10kg) using Centralized
and Decentralized SMC 86
4.39 Joint 3 tracking error (velocity, 10kg) using Centralized
and Decentralized SMC 86
4.40 Joint 1 tracking error (angle) with mass variation 87
4.41 Joint 2 tracking error (angle) with mass variation 88
4.42 Joint 3 tracking error (angle) with mass variation 88
4.43 Joint 1 tracking error (velocity) with mass variation 89
4.44 Joint 2 tracking error (velocity) with mass variation 89
4.45 Joint 3 tracking error (velocity) with mass variation 90
4.46 Joint 1 tracking response (angle) with unsatisfied
controller parameter 91
4.47 Joint 2 tracking response (angle) with unsatisfied
controller parameter 92
xv
4.48 Joint 3 tracking response (angle) with unsatisfied
controller parameter 92
4.49 Joint 1 tracking error (velocity) with unsatisfied
controller parameter 93
4.50 Joint 2 tracking error (velocity) with unsatisfied
controller parameter 93
4.51 Joint 3 tracking error (velocity) with unsatisfied
controller parameter 94
4.52 Joint 1 control input with unsatisfied
controller parameter 94
4.53 Joint 2 control input with unsatisfied
controller parameter 95
4.54 Joint 3 control input with unsatisfied
controller parameter 95
4.55 Joint 1 sliding surface with unsatisfied
controller parameter 96
4.56 Joint 2 sliding surface with unsatisfied
controller parameter 96
4.57 Joint 3 sliding surface with unsatisfied
controller parameter 97
4.58 Joint 1 tracking error (angle) with different
sliding surfaces 99
4.59 Joint 2 tracking error (angle) with different
sliding surfaces 99
4.60 Joint 3 tracking error (angle) with different
sliding surfaces 100
4.61 Joint 1 tracking error (velocity) with different
sliding surfaces 100
4.62 Joint 2 tracking error (velocity) with different
sliding surfaces 101
4.63 Joint 3 tracking error (velocity) with different
sliding surfaces 101
4.64 Joint 1 tracking response (angle) with different
sampling time 103
xvi
4.65 Joint 2 tracking response (angle) with different
sampling time 103
4.66 Joint 3 tracking response (angle) with different
sampling time 104
4.67 Joint 1 tracking response (velocity) with different
sampling time 104
4.68 Joint 2 tracking response (velocity) with different
sampling time 105
4.69 Joint 3 tracking response (velocity) with different
sampling time 105
4.70 Joint 1 tracking error (angle) with different
closed-loop poles 107
4.71 Joint 2 tracking error (angle) with different
closed-loop poles 107
4.72 Joint 3 tracking error (angle) with different
closed-loop poles 108
4.73 Joint 1 tracking error (velocity) with different
closed-loop poles 108
4.74 Joint 2 tracking error (velocity) with different
closed-loop poles 109
4.75 Joint 3 tracking error (velocity) with different
closed-loop poles 109
4.76 Control input for each joint using Solver Ode14x Set 1 111
4.77 Control input for each joint using Solver Ode14x Set 2 111
4.78 Control input for each joint using Solver Ode14x Set 3 112
4.79 Joint 1 tracking error (angle) using variation order of
Solver Ode14x 113
4.80 Joint 2 tracking error (angle) using variation order of
Solver Ode14x 113
4.81 Joint 3 tracking error (angle) using variation order of
Solver Ode14x 113
4.82 Joint 1 tracking error (velocity) using variation order of
Solver Ode14x 114
xvii
4.83 Joint 2 tracking error (velocity) using variation order of
Solver Ode14x 114
4.84 Joint 3 tracking error (velocity) using variation order of
Solver Ode14x 115
xviii
LIST OF SYMBOLS
SYMBOL DESCRIPTION
1. UPPERCASE
A(*,*) system matrix for the integrated direct drive robot arm
ΔA(*,*) matrix representing the uncertainties in the system matrix
B(*,*) input matrix for the intagrated direct drive robot arm
ΔB(*,*) matrix representing the uncertainties in the input matrix
Bmi viscous damping coefficient of ith actuator
C constant matrix of the PI sliding surface
Dmi volumetric displacement of ith actuator
Jmi ith motor inertia
Ii moment of inertia of ith link
Kqi flow gain in ith actuator
Kti total leakage coefficient in ith actuator
Kvi servo valve gain
N number of joints
Psi supply pressure in ith actuator )(tSδ a continuous function used to eliminate chattering
iLT load torque for ith motor
U(*) N x 1 control input vector for a N DOF robot arm
Vti
total compressed volume in ith actuator
X(*) state vector for the integrated electrohydraulic manipulator
Z(*) error state vector between the actual and the desired states of
the overall system
(*)T transpose of (*)
xix
||(*)T|| Euclidean norm of (*)
2. LOWERCASE
aij ijth element of the integrated system matrix A(*,*)
bij ijth element of the integrated input matrix B(*,*)
g gravity acceleration (m.s2)
li length of the ith manipulator link (m)
mi mass of the ith manipulator link (kg)
t time (s)
3. GREEK SYMBOLS
α norm bound of continuous function H(*)
β norm bound of continuous function E(*)
eiβ
effective bulk modulus in ith actuator
θ& joint displacement (rad)
θ&& joint velocity (rad/s)
θ&&& joint acceleration (rad/s2)
dθ& desired joint angle (rad)
dθ&& desired joint velocity (rad/s)
dθ&&& desired joint acceleration (rad/s2)
σ Integral sliding manifold
τ time interval for arm to travel from a given initial position to a final
desired position (seconds)
xx
LIST OF ABBREVIATIONS
CTC Computed Torque Control
DC Direct Current
DOF Degree of Freedom
LHP Left Half Plane
PID Proportional Integral Derivative
SMC Sliding Mode Control
1
CHAPTER 1
INTRODUCTION
1.1 Robot Manipulator System
Industrial robots have gained a wide popularity as essential components for
the realization of automated manufacturing systems. Reduction of manufacturing
costs, increase of productivity, improvement of product quality standards and last
but not least, the possibility of eliminating harmful or alienating tasks for the human
operator in manufacturing system, represent the main factors that have spearhead the
spreading of robotics technology in a wide range of applications in manufacturing
industry [1].
An industrial robot is constituted by a mechanical structure or manipulator
that consists of sequence of rigid bodies (links) connected by means of articulation
whether revolute or prismatic joints and this manipulator is characterized by an arm
that ensures mobility, a wrist that confers dexterity and an end effector that performs
the task required of the robot, actuators that set the manipulator in motion through
actuation of the joints while the motors employed are typically electric and
hydraulic, and occasionally pneumatic, sensors that measure the status of the
manipulator and if necessary the status of the environment and a control system
(computer) that enables control and supervision of manipulator motion [1].
Although electrically driven robots are widely used in an increasing number
of applications, there are many industrial tasks where hydraulic actuators can be
used advantageously. For special applications such as for very large robots and civil
2
service robots, hydraulic actuator may be an appropriate choice [2]. Machinery in
construction, forming, mining, forestry industry, heavy load motion control and
mobile equipment applications as well as large flight simulators take the advantage
of the high power to weight ratio, possible speed reversals and continuous operation,
the stiffness and short response time of hydraulic drives.
Electrohydraulic system uses low power electrical signals for precisely
controlling the movements of large power pistons and motors. The interface between
the electrical equipment and the hydraulic (power) equipment is called ‘hydraulic
servo valve’. These valves used in the system must respond quickly and accurately.
A schematic diagram of a typical hydraulic system is shown in Figure 1.1.
Figure 1.1: Schematic diagram of a hydraulic system and its components
A servo valve is created when a servomotor is attached to the spool valve
and the servo valve. The servo valve together with the hydraulic actuator will form
the hydraulic servomotor. A simple movement of the spool valve controls the
motion of the actuator. As the spool moves up and down, it opens the supply and
returns to the port through which the fluid travels to the cylinder or return to the
reservoir as shown in Figure 1.2. The amount of the supply fluid and the
displacement of the cylinder can be controlled by adjusting the size of these ports
3
opening. Similarly, the flow rate of the supply fluid and the velocity of the cylinder
can be controlled by adjusting the rate of the ports opening [2].
Figure 1.2: Schematic diagram of a spool valve in a neutral position
In electrohydraulic manipulator system, each joint is driven by a hydraulic
servomotor. A higher control input voltage will produce larger valve flow from the
servo valve into the hydraulic motor. This will eventually results in a faster
rotational motion of the motor and thus the path (position and the orientation) of the
manipulator can be done in a specified time. The timed path is called trajectory of
the manipulator’s end effector [3]. However, it is difficult to precisely control the
position of electrohydraulic robot both theoretically and practically due to the
nonlinearities and coupling effects present in the system.
The control variable for the DC motor is either motor voltage or current that
is proportional to the actuation torque. The hydraulic actuator on the other hand, the
control voltage or current signal to the valve of hydraulic actuator controls the speed
of the actuator rather than its force or torque. The system also has large extent of
parametric uncertainties due to the large variations of inertial load and the change of
bulk modulus caused by the entrapped air or change of temperature. Besides, the
system may also have large extent of lumped uncertain nonlinearities including
external disturbances and unmodeled friction forces. Therefore, the
4
electrohydraulically driven revolute robot manipulator dynamics are more complex
than electrically driven manipulator dynamics. All of these factors make the
modeling and control of such system a challenging task [4].
In all industrial robot applications, completion of a generic task requires the
execution of a specific motion that prescribed by the desired path trajectory and
performance. However, when faster trajectories are demanded, the performance of
the manipulator will be worst. Therefore, the correct execution is entrusted to the
control system which shall provide the joint actuators of the manipulator with the
command consistent with the desired motion trajectory. Thus, an accurate analysis
of the characteristics of the manipulator dynamics is therefore a necessary premise
to finding motion control strategies. In general, motion control problem consists of
obtaining the dynamics model of the electrohydraulic robot manipulator in which
these models will be used to determine the control laws or strategies to achieve the
desired system response and performance.
1.2 Electrohydraulic Robot Manipulator
Nowadays, hydraulic robots are widely used in the construction and mining
industries. However, majority of earlier work in the design of the control laws for
manipulators deal with electrically actuated manipulators. In terms of hydraulic
actuators, comparatively less work has been done [5]. However, by taking the
dynamics of the actuator alone is not sufficient to represent the dynamic of hydraulic
manipulator, since it does not take into account the arm dynamic forces such as the
inertia forces, the coriolis and centrifugal effects and also the gravity effects that will
affect the performance of the controller. Tracking performance of the system can be
improved by implementing mechanical linkage dynamic model in the controller
design since it is part of the overall control system. This approach has been
successfully shown in many electrical robots in the past [6].
5
[7] has used 6 DOF hydraulic robot and the models incorporate manipulator
dynamics. The limitation with this project is that it did not include the hydraulic
motor nonlinear spring stiffness, viscous damping and inertia. Same goes with [8]
who has incorporated manipulator dynamics in developing 2 DOF hydraulic robot
arm models, but the hydraulic motor nonlinear spring stiffness, viscous damping and
inertia are not taken into account in the design. [9] on the other hand, have used
hydraulic cylinders with the application to robot manipulators. The problem with the
study is that it did not incorporate the mechanical linkage dynamics in the design
model.
[5] has incorporated both rigid body dynamics and hydraulic actuator
dynamics into the design. However, the mathematical model presented was not in
state-space form. For some recent development on mathematical model derivation of
this type of manipulator, [10] has developed the mathematical model in state-space
form for the electrohydraulic robot manipulator that integrated both the manipulator
dynamics and hydraulic actuator dynamics including the hydraulic motor nonlinear
spring stiffness, viscous damping and inertia.
The mathematical model derived in [10] will be used in this project to
synthesize different control laws in providing the trajectory tracking control of the 3