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 hadhad 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
1
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
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
2.3
Dynamic Equation of Robot Manipulator
Mechanical Linkage
2.4
2.5.1
2.5.2
28
Dynamic Equation of 3 DOF Robot
Manipulator Mechanical Linkage
2.5.3
26
Dynamic Equation of the Electrohydraulic
Motors for the 3 DOF Robot Manipulator
31
Integrated Dynamic Model of the Three DOF
Electrohydraulic Robot Manipulator
3
23
Dynamic Model of 3 DOF Electrohydraulic
Robot Manipulators
2.6
21
Integrated Dynamic Model of the Electrohydraulic
Robot Manipulator
2.5
15
Summary
35
40
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
3.4
3.5
Large-Scale Uncertain System
45
Problem Formulation for Decentralized SMC
52
3.4.1
System Dynamics during Sliding Mode
54
3.4.2
Sliding Mode Tracking Controller Design
55
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
4.7
64
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
Summary
115
CONCLUSION
117
5.1
Conclusion
117
5.2
Suggestion For Future Work
118
4.6
5
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
1.1
TITLE
PAGE
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
2.2
16
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
4.5
Joint 2 tracking response (angle) using centralized SMC
tracking controller
4.6
68
Joint 3 tracking response (velocity) using centralized
SMC tracking controller 2
4.10
67
Joint 2 tracking response (velocity) using centralized
SMC tracking controller 2
4.9
67
Joint 1 tracking response (velocity) using centralized
SMC tracking controller 2
4.8
66
Joint 3 tracking response (angle) using centralized SMC
tracking controller
4.7
66
68
Joint 1 control input using centralized SMC
tracking controller
69
xiii
4.11
Joint 2 control input using centralized SMC
tracking controller
4.12
Joint 3 control input using centralized SMC
tracking controller
4.13
78
Joint 2 sliding surface using decentralized SMC
tracking controller
4.27
78
Joint 1 sliding surface using decentralized SMC
tracking controller
4.26
77
Joint 3 control input using decentralized SMC
tracking controller
4.25
77
Joint 2 control input using decentralized SMC
tracking controller
4.24
76
Joint 1 control input using decentralized SMC
tracking controller
4.23
76
Joint 3 tracking response (velocity) using decentralized
SMC tracking controller
4.22
75
Joint 2 tracking response (velocity) using decentralized
SMC tracking controller
4.21
75
Joint 1 tracking response (velocity) using decentralized
SMC tracking controller
4.20
74
Joint 3 tracking response (angle) using decentralized
SMC tracking controller
4.19
74
Joint 2 tracking response (angle) using decentralized
SMC tracking controller
4.18
71
Joint 1 tracking response (angle) using decentralized
SMC tracking controller
4.17
71
Joint 3 sliding surface using centralized SMC
tracking controller
4.16
70
Joint 2 sliding surface using centralized SMC
tracking controller
4.15
70
Joint 1 sliding surface using centralized SMC
tracking controller
4.14
69
79
Joint 3 sliding surface using decentralized SMC
tracking controller
79
xiv
4.28
Joint 1 tracking error (angle, 0kg) using Centralized
and Decentralized SMC
4.29
Joint 2 tracking error (angle, 0kg) using Centralized
and Decentralized SMC
4.30
85
Joint 2 tracking error (velocity, 10kg) using Centralized
and Decentralized SMC
4.39
85
Joint 1 tracking error (velocity, 10kg) using Centralized
and Decentralized SMC
4.38
84
Joint 3 tracking error (angle, 10kg) using Centralized
and Decentralized SMC
4.37
84
Joint 2 tracking error (angle, 10kg) using Centralized
and Decentralized SMC
4.36
83
Joint 1 tracking error (angle, 10kg) using Centralized
and Decentralized SMC
4.35
83
Joint 3 tracking error (velocity, 0kg) using Centralized
and Decentralized SMC
4.34
82
Joint 2 tracking error (velocity, 0kg) using Centralized
and Decentralized SMC
4.33
82
Joint 1 tracking error (velocity, 0kg) using Centralized
and Decentralized SMC
4.32
81
Joint 3 tracking error (angle, 0kg) using Centralized
and Decentralized SMC
4.31
81
86
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
4.47
91
Joint 2 tracking response (angle) with unsatisfied
controller parameter
92
xv
4.48
Joint 3 tracking response (angle) with unsatisfied
controller parameter
4.49
Joint 1 tracking error (velocity) with unsatisfied
controller parameter
4.50
101
Joint 3 tracking error (velocity) with different
sliding surfaces
4.64
100
Joint 2 tracking error (velocity) with different
sliding surfaces
4.63
100
Joint 1 tracking error (velocity) with different
sliding surfaces
4.62
99
Joint 3 tracking error (angle) with different
sliding surfaces
4.61
99
Joint 2 tracking error (angle) with different
sliding surfaces
4.60
97
Joint 1 tracking error (angle) with different
sliding surfaces
4.59
96
Joint 3 sliding surface with unsatisfied
controller parameter
4.58
96
Joint 2 sliding surface with unsatisfied
controller parameter
4.57
95
Joint 1 sliding surface with unsatisfied
controller parameter
4.56
95
Joint 3 control input with unsatisfied
controller parameter
4.55
94
Joint 2 control input with unsatisfied
controller parameter
4.54
94
Joint 1 control input with unsatisfied
controller parameter
4.53
93
Joint 3 tracking error (velocity) with unsatisfied
controller parameter
4.52
93
Joint 2 tracking error (velocity) with unsatisfied
controller parameter
4.51
92
101
Joint 1 tracking response (angle) with different
sampling time
103
xvi
4.65
Joint 2 tracking response (angle) with different
sampling time
4.66
Joint 3 tracking response (angle) with different
sampling time
4.67
108
Joint 2 tracking error (velocity) with different
closed-loop poles
4.75
108
Joint 1 tracking error (velocity) with different
closed-loop poles
4.74
107
Joint 3 tracking error (angle) with different
closed-loop poles
4.73
107
Joint 2 tracking error (angle) with different
closed-loop poles
4.72
105
Joint 1 tracking error (angle) with different
closed-loop poles
4.71
105
Joint 3 tracking response (velocity) with different
sampling time
4.70
104
Joint 2 tracking response (velocity) with different
sampling time
4.69
104
Joint 1 tracking response (velocity) with different
sampling time
4.68
103
109
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
4.80
Joint 2 tracking error (angle) using variation order of
Solver Ode14x
4.81
113
Joint 3 tracking error (angle) using variation order of
Solver Ode14x
4.82
113
113
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
4.84
114
Joint 3 tracking error (velocity) using variation order of
Solver Ode14x
115
xviii
LIST OF SYMBOLS
DESCRIPTION
SYMBOL
1.
UPPERCASE
A(*,*)
system matrix for the integrated direct drive robot arm
Δ A(*,*)
matrix representing the uncertainties in the system matrix
B(*,*)
input matrix for the intagrated direct drive robot arm
Δ B(*,*)
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
Sδ (t )
a continuous function used to eliminate chattering
TLi
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||
2.
Euclidean norm of (*)
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(*)
β ei
effective bulk modulus in ith actuator
θ&
joint displacement (rad)
θ&&
joint velocity (rad/s)
θ&&&
joint acceleration (rad/s2)
θ&d
desired joint angle (rad)
θ&&d
desired joint velocity (rad/s)
θ&&&d
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
DOF electrohydraulically driven revolute robot manipulator.
The robot control problem revolves around the computation of the required
actuator inputs in order to maintain the dynamic response of manipulator in
accordance with the desired response and performance. This control problem will
become complex due to the nonlinear 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.
In classical control theory, it is assumed that the control actions are
undertaken by a single controller that has all the available information about the
6
system and treat each joint of the manipulator as a simple linear servomechanism as
in most of industrial robots, whereby the simple controller like Independent Joint
Control (IJC), proportional plus derivative (PD), or proportional plus integral plus
derivative (PID) controllers are adopted. In these methods, the nonlinear, coupled
and time-varying dynamics of the mechanical linkage of the robot manipulator have
been excluded and completely ignored, or assumed as disturbances. Manipulators
that have been controlled by this method usually move at slow speeds with
unnecessary vibrations. In general, the method is only suitable for relatively slow
manipulation and limited precision tasks [11]. However, when the manipulator joints
are moving simultaneously and at high speed, the nonlinear coupling effects and the
interaction forces between the manipulator links may affect the performance of the
overall system and increase the tracking error. The disturbances and uncertainties
such as variable payload in a task cycle may also reduce the tracking quality of the
robot manipulator system [12].
Therefore, control strategies for high speed robotic system are of great
interest for both industrial and academic fields, whereby various advanced and
sophisticated control techniques have been proposed by numerous researchers in
providing the necessary tracking trajectory of robot manipulator and at the same
time guaranteeing the stability of the system. In general, these strategies can be
divided into two categories; the non-model based or usually known as Artifical
Intelligence approaches and model based approaches. While for the structures of
these controllers can be grouped into three categories; the centralized, decentralized,
and multilevel hierarchical. For the model based approaches, among the major
control approaches considered in the literature for the uncertainties nonlinear
systems are the Adaptive Control method [13], Lyapunov based control and
Variable Structure Control [14]. While for the non-model based approaches, where
the knowledge of the mathematical model is not needed, Fuzzy Logic system,
Genetic Algorithm and Neural Networks controls have become important research
topics [15].
In [8], PID and Computed Torque Control (CTC) strategies have been
implemented for the hydraulic robot and the performance of both methods did not
7
give satisfactory results. As mentioned earlier, the limitation with PID controllers is
that they are not meant for high speed robots and in situations that require precise
trajectory tracking. While for the CTC, it only can be applied to robots allowing
joint torque control. Moreover, the problem with CTC is essentially based on exact
robot arm dynamic model; the explicit use of an incorrect robot model will affect the
performance of control. He further synthesized the Kinematic Compensation Control
integrating a feedforward kinematic compensation and conventional regulatory
control techniques. Even though the system shows good experimental result, it still
lacks mathematical stability proof.
[16] has proposed a link-space pressure feedback controller for Stewart type
hydraulic manipulator. However, this approach lacks stability proofs that are
important from both theoretical and implementations points of view. [17] has
developed a generalized predictive control algorithm for hydraulic control system
but the limitation of this approach is that it relies heavily on online parameter
estimation and consequently, is computationally expensive. [18] on the other hand,
have adopted a Lyapunov based Model-based Adaptive Control for hydraulic
manipulator. Even though the technique possesses mathematical stability proof but it
needs persistent excitation and pressure feedback. A strategy based on Backstepping
approach has been developed by [19] to provide necessary position tracking for the
hydraulic robot. However, the proposed strategy also relies on online computation
for converting into task-space coordinates. Furthermore, it is said to be sensitive to
sensor noise, hence high-quality measurements are needed. To overcome this
drawback is by introducing heuristic limitation of time derivatives in the control law
so that the influence of sensor noise in the simulations reduces significantly, but may
deteriorate the control performance.
Non-adaptive control techniques such as computed torque methods and
optimal control have serious disadvantages, namely complex controller structure,
excessive on-line computation, and sensitivity to uncertainties and nonlinearities in
the system. These difficulties become more complicated by their being centralized
approaches. Adaptive control methods tackle the robot control problem fairly
effective, but the controller is greatly complicated. In much of the works on adaptive
8
control of manipulators, the justification for this added complexity is not addressed
[20]. Despite the fact that the centralized approaches have attained significant
achievements to improve the tracking performance of robots, but when the degree of
freedom (D.O.F) of robots is large, they stumble upon time consuming computations
and complexity of the controller structure. Furthermore, very few results could be
transferred into practice due to the high implementation cost. Besides, the
centralized approach treats the robot manipulator as a single entity plant, which is
unfavorable and impractical for implementation and maintenance of the controllers
[21]. Therefore control methods that reduce these problems by changing the problem
to control smaller scale subsystems by using the decentralized controllers have the
advantage of computation simplicity and low-cost hardware setup. For this reason,
improving the performance of the trajectory tracking problem of robot manipulator
through decentralized control is an interesting topic.
Variable Structure Control utilizes a high-speed switching control law to
drive the nonlinear plant’s state trajectory onto a specified and user-chosen surface
in the state space and to maintain the plant’s state trajectory on this surface for all
subsequent time [22]. This property of remaining on the switching surface once
intercepted is called a sliding mode. The plant dynamics will be restricted to this
surface to represent the controlled system’s behavior. Hence, it is suitable for
complex systems and is insensitive to parameters variations and uncertainties. SMC is
particularly well suited for the manipulator control problem for the following
reasons. First, the application of SMC does not require an exact knowledge of the
system dynamics, provided there is no un-modeled structural uncertainty. This
property is desirable since the complexity of the manipulator dynamics makes the
exact calculation of the dynamics infeasible if not impossible. Second, when the
SMC is applied, the performance of the system can be made insensitive to bounded
disturbances. This property is important in rejecting effects due to the Coulomb and
viscous frictions. It is also important when the manipulator is carrying payloads
because the payload exerts at the robot end-effector can be translated into forces or
disturbances at each of the joints. Thus, the application of the SMC technique results
in a performance that is robust with respect to disturbances and modeling errors,
while provides accurate tracking [21].
9
This project will extend the control strategies proposed in controlling two
coupled pendulums in [23] to provide trajectory tracking control of a three DOF
electrohydraulically driven revolute robot manipulator. The approach stabilizes a
system through a fast switching global control to render that the system to behave as
a second linear time invariant system which can be chosen to be stable. This method
incorporated the ideas of variable structure system with the technique of pole
placement in a decentralized nature. This is done by constructing a decentralized
sliding mode controller which will force the original nonlinear interconnected
uncertain subsystems to behave as a second linear interconnected uncertain
subsystem which has had its pole placed in the left half of the s-plane by pole
placement method.
1.3
Research Objective
The objectives of this project are:
i) To decompose the 3 degree of freedom (DOF) electrohydraulic robot
manipulator into interconnected uncertain subsystems;
ii) To design a decentralized robust controller based on Sliding Mode Control
(SMC) for the 3 degree of freedom (DOF) electrohydraulic robot
manipulator;
iii) To evaluate the performance of the system based on the proposed controller
and to compare the performance of the decentralized robust controller with a
centralized robust controller.
10
1.4
Scope of Research
The scope of work for this project includes:
i) To decompose the integrated nonlinear dynamic model of the 3 DOF
revolute electrohydraulic robot manipulator as described in [10] into a set of
interconnected subsystem by treating it as a large-scale system. The
interconnected model will be transformed into interconnected uncertain
subsystems models using the known parameters of the robot;
ii) To synthesize a decentralized tracking trajectory controller based on a
variant of the VSC approach, which is the SMC for large-scale uncertain
system. This proposed controller will be applied to the 3 DOF
electrohydraulically driven revolute robot manipulator models developed in
part (i);
iii) A centralized tracking trajectory controller will be designed based on the
deterministic approach for an uncertain system, where the transformationfree SMC will be utilized;
iv) Simulation studies will be done to investigate the performance of the
proposed controller and the effectiveness of the decentralized SMC as
compared to the centralized SMC in providing the necessary position
tracking control for the system. These will be done using SIMULINK and
MATLAB 7.0.
1.5
Research Methodology
The research work is undertaken in the following three developmental stages:
a) Development of an interconnected uncertain system models of a 3 DOF
revolute hydraulic robot manipulator. The steps taken in this stage are:
11
i)
Conduct literature review on the existing electrohydraulic robot
manipulator mathematical model in order to understand the
dynamics behavior of the system. Based on the information
obtained, the integrated model will be decomposed into three
interconnected subsystems in order to apply the decentralized
tracking controller;
ii)
Conduct literature review on the existing techniques for
decomposing
the
system
into
interconnected
uncertain
subsystems.
b) Application of a decentralized robust controller based on VSC technique to
the robotic system. The steps taken in this stage are:
i)
Conduct literature review on the existing control technique for
robotic systems;
ii)
Conduct literature review on the existing robust control technique
based on decentralized Sliding Mode Control algorithm;
iii)
Synthesize the decentralized SMC technique cited from [23] to
the robotic system. The procedures performed in this stage are:
•
Decomposition of the complete model into an large-scale
uncertain model
•
Definition of each local sliding surface
•
Determination of the system dynamics during Sliding Mode
•
Establishment of the final control law for each subsystem
c) Perform computer simulation of the proposed controller by using MATLAB
and SIMULINK to investigate the effectiveness of the decentralized SMC as
compared to a centralized SMC.
12
1.6
Structure and Layout of Thesis
This report is organized into five chapters. In Chapter 2, the formulation of
the mathematical modeling of the integrated electrohydraulic robot manipulator, in
which, first, a general dynamic model of a hydraulic actuator dynamics and a rigid
manipulator link are described separately. Then, based on these equations, an
integrated model in state space representation is presented. At the end of the chapter,
the complete integrated dynamic model of three DOF electrohydraulically driven
revolute robot manipulator is determined.
Chapter 3 deals with the controller design using decentralized sliding mode
control strategies with specific application to the plant described in Chapter 2.
Firstly, by treating the electrohydraulic robot manipulator as a large-scale system
and each joint as a subsystem, a linear interconnected uncertain system is developed
to represent the manipulator as interconnected subsystems. Secondly, based on the
known allowable range of operation of the manipulator and the maximum allowable
payloads, the nominal and bounded uncertainties of the system may be developed
and the manipulator can be represented as an interconnected uncertain subsystems.
Finally, the decentralized SMC controller is designed for the system.
Chapter 4 outlines the performance of the proposed control system which is
evaluated by means of computer simulation study through MATLAB/SIMULINK.
The simulation begins with a pre-specified desired trajectory for each of the
manipulator joints in terms of joint displacement, velocities and accelerations. In
order to analyze the effectiveness of the decentralized SMC, centralized SMC
strategy is used as a mean of comparison. Conclusion on the robustness of the
approach in handling the nonlinearities, uncertainties and couplings effect present in
the system and the necessary trajectory tracking control of the plant is also made and
discussed based on the results obtained.
13
This thesis ends with Chapter 5, where the summary of the synthesized
approach while undertaking this project is described. Recommendations for future
work are also presented at the end of the chapter.
14
CHAPTER 2
MATHEMATICAL MODELING OF ELECTROHYDRAULIC ROBOT
MANIPULATOR
2.1
Introduction
The most important part in synthesizing a control technique for robot
manipulator system is to obtain a mathematical modeling of the manipulator as
complete and accurate as possible. The dynamic model, which describes the
relationship between the input and output, provides a better understanding of
behavior of the system [19]. The inevitability of such a model takes place due to the
proposed controller that must endure rigorous simulations before it is realized on a
real robot. The dynamic performance of a manipulator depends on the effectiveness
of the control algorithms and the dynamic model of the plant. Besides the
mechanical linkage dynamics, the dynamics of the actuator that drives each joint of
the robot also affect the dynamic model of the manipulator. Thus, the actuators
dynamics must be included into the robot manipulator dynamic equations. Based on
this model, suitable control laws are properly synthesized to drive the actuator to
achieve the desired position, velocity and acceleration in the robot’s joints and links.
Model-based robust control of a hydraulic arm has not been thoroughly
studied and only few results are available [18]. Most of the previous works in
literature concentrate only on the manipulator dynamics, whereas the actuator
dynamics are completely ignored [5].
This method works well in low speed
electrical actuated manipulator but when a hydraulic actuator is used, satisfactory
performance cannot be achieved.
There is limited number of research on
15
formulating the integrated mathematical model which incorporates the rigid body of
the plant and the actuators, as well as on the controller synthesis based on the
complete robot manipulator model. In general, the trajectory tracking performance
of a robotic manipulator can be enhanced by including a dynamic model of the robot
as well as the actuator dynamics in the controller, to compensate for dynamic forces
such as inertia, friction and gravity. This was successfully shown with many
electrically driven robots in the past [6].
This chapter deals mainly with the mathematical modeling of an integrated
dynamic model of an electrohydraulic robot manipulator outlined by [10]. Both the
manipulator dynamics and actuator dynamics are taken into account to represent a
more accurate dynamic behavior of the electrohydraulic robot manipulator, hence a
better model can be provided for the purpose of controller synthesis and analysis. In
the first part, general mathematical model of one DOF hydraulic actuator dynamics
and a mechanical linkage of the manipulator are described separately. In the second
part, based on these equations, a three DOF revolute robot manipulator and
hydraulic motor models are derived, and finally these dynamics equation are
integrated in state space representation.
2.2
Dynamic Equation and State Space Representation of Electrohydraulic
Actuator
Electrohydraulic actuators are widely used as the actuating mechanism for
robot manipulators and the physical model of this nonlinear electrohydraulic servo
control system is shown in Figure 2.1. The inertial-damping with a nonlinear torsion
spring system is driven by a hydraulic motor and the rotation of the motor is
controlled by a servo valve [24].
The equation is derived based on the assumptions that:
i)
the piston is centered such that the volume of the fluid trapped at the
sides of the actuator are equal
16
ii)
the valve is an ideal critical center valve with matched and symmetrical
orifices
iii)
return line pressure is zero
Figure 2.1: Physical Model of an Electrohydraulic Servo Control System [10]
The servo valve flow equation governed by the orifice law [25], [26] and
[24] can be described as:
Q L = K q X v − K c PL
(2.1)
where
QL
:
load flow
Xv
:
displacement of the spool in the servo valve
Kc
:
flow-pressure coefficient
PL
:
load pressure
Kq
:
flow gain
The flow gain Kq, which varies at different operating point is basically nonlinear and
can be expressed as:
K q = C d w ( P s − PL sgn( X v )) / ρ
where
Cd
:
discharge coefficient
w
:
area gradient
(2.2)
17
ρ
:
fluid mass density
Ps
:
supply pressure
The continuity equation to the cylinder is formulated as
V
Q L = Dmθ&m + C1 PL + t P&L
4β e
(2.3)
where
Dm
:
volumetric displacement
θ&m
:
angular velocity of the motor shaft
C1
:
total leakage coefficient of the motor
Vt
:
total compressed volume
βe
:
effective bulk modulus of the oil
Substituting equation (2.1) into equation (2.3) gives
V
K q X v = Dmθ&m + K t PL + t P&L
4β e
(2.4)
where K t = K c + C1 is total leakage coefficient of the hydraulic system.
The torque balance equation for the motor can be described as:
PL Dm = J mθ&&m + Bmθ&m + G (θ m + Gnθ m3 ) + TL
(2.5)
where
Jm
:
motor inertia
Bm
:
viscous damping coefficient
TL
:
load torque due to the joint of the manipulator on the motor
G
:
torsional spring constant
G n θ m3 :
nonlinear stiffness of spring
Combining equations (2.1)-(2.5) results in a single third-order differential equation
as follows:
18
⎛ 4β K
B ⎞
θ&&&m + ⎜⎜ e t + m ⎟⎟θ&&m
Jm ⎠
⎝ Vt
⎛ 4β K T
T&
+ ⎜⎜ e t L + L
Jm
⎝ Vt J m
⎛ 4β K B
4β K G
G 4β e Dm2 ⎞ &
⎟θ m + e t θ m
+ ⎜⎜ e t m +
+
⎟
Jm
Vt J m ⎠
Vt J m
⎝ Vt J m
⎞ 4 β e Dm
⎟⎟ =
(K q (t ) K vU (t )) + N ( X , t )
⎠ Vt J m
(2.6)
where
⎛ 4β K G G
⎞
3G G
N ( X , t ) = −⎜⎜ e t n x13 + n x12 x 2 ⎟⎟ ¸
Jm
⎝ Vt J m
⎠
T&L (t ) :
X v = K vU (t )
time derivative of the load torque due to the joint of the
manipulator on the motor
In most present robot manipulators, the motor shaft is mechanically coupled
to the manipulator link (load) through gears as shown in Figure 2.2 [12] in order to
reduce the nonlinearity effects. Assuming that the gear is ideal (frictionless), thereby
creating no power loss and inertialess; the relationship between the loading torque
T(t) and the load torque TL(t) on the motor shaft at the primary side can be written as
TL (t ) =
where n g
:
T (t )
,
ng
ng ≥ 1
(2.7)
inverse of the gear ratio.
ith Hydraulic Motor
Jm, Bm, G, Gn, Kt,,
Vt,, Dm, Kq, Kv , β e
ng
Figure 2.2: Diagram of the Gear Train Connecting the Motor and the Load [10]
19
In order to reflect the actuator dynamics to the manipulator side of the gearing
mechanism, the following identity is used:
θ m (t ) = n gθ (t )
(2.8)
From equations (2.6) and (2.7), after rearranging, the hydraulic actuator dynamics
then becomes;
⎛ 4 β e K t Bm
+
Jm
⎝ Vt
θ&&& + ⎜⎜
⎞ && ⎛ 4 β e K t Bm G 4 β e Dm2
⎟⎟θ + ⎜⎜
+
+
V
J
J
Vt J m
t
m
m
⎠
⎝
⎛ 4β K T
T&
+⎜ e t2 +
2
⎜V J n
⎝ t m g J m ng
⎞ & 4β e K t G
⎟θ +
θ
⎟
V
J
t
m
⎠
⎞ 4 β e Dm
⎟=
(K q (t ) K vU (t )) + N ( X , t )
⎟ VJ n
t m g
⎠
(2.9)
From equation (2.9), the state equation of the actuator dynamic model with ith
actuator angular position, velocity and acceleration, θ i (t ), θ&i (t ), θ&&i (t ) , as state
variable can be represented as:
X& i (t ) = AHi X i (t ) + BHiU i (t ) + FHi Ti (t ) + WHi T&i (t ) + N Hi
(2.10)
where
[
]
T
X i (t ) = θ i (t ) θ&i (t ) θ&&i (t )
(2.11)
From equation (2.9), AHi , B Hi , FHi , W Hi , N Hi can be written as
⎡
⎢
0
⎢
0
AHi = ⎢
⎢ 4β e K t G
⎢− V J
t m
⎣
⎤
⎥
.0
⎥
1
⎥
⎡ 4 β e K t Bm
⎡ 4 β e K t Bm ⎤ ⎥
4 β e Dm2 ⎤
G
−⎢
+
+
+
⎥ −⎢
⎥
Jm
Vt J m ⎦
J m ⎦ ⎥⎦
⎣ Vt
⎣ Vt J m
1
0
(2.12)
B Hi
⎡
⎢
0
⎢
=⎢
0
⎢ 4 β e Dm K q K v
⎢ VJ n
t m g
⎣
⎡
0
⎢
⎢
FHi = ⎢
0
4
β
e Kt
⎢−
⎢⎣ Vt J m n g2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(2.13)
⎤
⎥
⎥
⎥
⎥
⎥⎦
(2.14)
20
W Hi
⎡
⎤
⎢ 0 ⎥
= ⎢⎢ 0 ⎥⎥
⎢− 1 2 ⎥
⎢⎣ J m n g ⎥⎦
N Hi
⎡
⎤
⎢
⎥
0
⎢
⎥
=⎢
0
⎥
2
2
⎢ 4β e K t Gn Gn g 3 3Gn Gn g 2 ⎥
x1 +
x1 x 2 ⎥
⎢
Vt J m
Jm
⎣
⎦
(2.15)
(2.16)
where
X i (t ) :
3 x 1 state vector of the ith actuator
U i (t ) :
scalar input to the ith actuator = V i (t )
Ti (t ) :
the load torque acting on the ith actuator due to the
manipulator itself
T&i
:
the time derivative of the load torque Ti (t ) acting on the ith
actuator
AHi (t ) :
system matrix for the ith actuator
BHi (t ) :
input matrix for the ith actuator
FHi (t ) :
load distribution matrix for the ith actuator
W Hi (t ) :
rate of load distribution matrix for the ith actuator
N Hi (t ) :
nonlinear terms of the system
For N DOF robot manipulator (N actuators), the augmented dynamic equation of the
actuators can be written in the following compact form as follows [12]:
X& (t ) = AH X (t ) + BHU (t ) + FH T (t ) + WH T& (t ) + N H ( X , t )
where
[
]
X (t ) = X 1T (t ), X 2T (t ), K , X NT (t )
U (t ) = [U1 (t ) ,U 2 (t ) , K, U N (t )]
T
T (t ) = [T1 (t ) , T2 (t ) , K, TN (t )]
T
[
]
T
T& (t ) = T&1 (t ) , T&2 (t ) , K , T&N (t )
AH = diag [ AH 1 , AH 2 , K , AHN ]
B H = diag [B H 1 , B H 2 , K , B HN ]
T
(2.17)
21
FH = diag [FH 1 , FH 2 , K , FHN ]
W H = diag [W H 1 , W H 2 , K , W HN ]
and
X (t )
is a 3N × 1 vector, U (t ) is an
N × 1 input vector, T (t ) is the
N × 1 mechanical link torque, and T& (t ) is its time derivative.
From the structure of the input matrix BHi, the load distribution matrix FHi
and the derivative of the load distribution matrix WHi, it can be observed that these
matrices lie within the range space of the input matrix. Hence, the control input
which enters the system through the input matrix BHi or BH can affect and
compensate directly the nonlinear, coupled and time-varying effect due to the load
disturbances presented by the mechanical linkage through Ti(t) and T&i (t ) .
2.3
Dynamic Equation of Robot Manipulator Mechanical Linkage
By using Denavit-Hartenberg matrix representation and Lagrange-Euler
method, the dynamic equation describing the motion of an N DOF robot manipulator
with rigid mechanical links is governed by a second-order nonlinear differential
equation and can be written symbolically as:
M (θ (t ), ξ )θ&&(t ) + D (θ (t ), θ&(t ), ξ ) + G (θ (t ), ξ ) = T( t )
(2.18)
where
M (θ (t ), ξ )
: N × N inertia matrix of the manipulator
D (θ (t ), θ&(t ), ξ )
: N × 1 vector of coriolis and centrifugal forces
G (θ (t ), ξ )
: N × 1 vector of gravitational forces
T( t )
: N × 1 vector of generalized driving forces/ torques
applied by the actuators at the drive points on each
link of the manipulator
θ (t ), θ&(t ), θ&&(t )
: N × 1 f generalized joint displacements, velocities and
accelerations respectively
22
ξ
: a vector (with appropriate dimension) of parameters
of the mechanism such as payload
Since each element of D (θ (t ), θ&(t ), ξ )
is a complex function of both
θ (t ) and θ&(t ) , the dynamic equation of the mechanical links of the robot manipulator
can be rewritten as:
M (θ (t ), ξ )θ&&(t ) + D (θ (t ), ξ )Vˆ (θ&(t ))θ&(t ) + G (θ (t ), ξ ) = T(t )
(2.19)
where
θ (t ) = [ θ 1 (t ), θ 2 (t ) .... θ N (t )]T
(2.20)
T (t ) = [ T1 (t ), T2 (t )
(2.21)
D (θ (t ), ξ )
.... TN (t )]T
N
: N × ∑ i matrix related to D (θ (t ), θ&(t ), ξ ) vector
i =1
Vˆ (θ&(t ))
: 2N x 1 matrix related to coriolis and centrifugal velocity
Similarly, all the matrices M (θ (t ), ξ ) , D (θ (t ), ξ ) and G (θ (t ), ξ ) contain the
nonlinear elements. Therefore, the terms D (θ (t ), ξ )Vˆ (θ&(t ))θ&(t ) and G (θ (t ), ξ ) as
described in equation (2.19) can be written as
Dˆ (θ (t ), θ&(t ), ξ )θ&(t )
and
Gˆ (θ (t ), ξ )θ (t ) respectively. Hence, the manipulator dynamics can be presented as:
M (θ (t ), ξ )θ&&(t ) + Dˆ (θ (t ), θ&(t ), ξ )θ&(t ) + Gˆ (θ (t ), ξ )θ (t ) = T(t )
(2.22)
where
Dˆ (θ (t ), θ&(t ), ξ ) : N × N matrix related to D (θ (t ), θ&(t ), ξ ) vector
Gˆ (θ (t ), ξ )
: N x N matrix related to vector of gravitational forces
From equation (2.22), the derivative of the torque can be obtained by differentiating
it with respect to time, yields:
(
)
(
)
~
~
T& (t ) = M (θ (t ), ξ )θ&&&(t ) + C θ (t ), θ&(t ), ξ θ&&(t ) + D θ (t ), θ&(t ), ξ θ&(t )
(2.23)
where
(
)
~
C θ (t ), θ&(t ), ξ θ&&(t ) = M& (θ (t ), ξ )θ&&(t ) + D (θ (t ), ξ )V& (θ&(t ))
(2.24)
23
(
)
~
D θ (t ),θ&(t ),ξ θ&(t ) = D& (θ (t ),ξ )V (θ&(t )) + G& (θ (t ),ξ )
2.4
(2.25)
Integrated Dynamic Model of the Electrohydraulic Robot Manipulator
Even though the actuator dynamics represent an important part of the
complete robot manipulator system, the dynamics of the mechanical part plays a
significant role as well in determining the accuracy of the mathematical model of the
manipulator.
The complete robot manipulator model can be obtained by first defining the
following transformations [12]:
θ&&(t ) = Z B X (t )
θ&(t ) = Z B1 X (t )
θ (t ) = Z B 2 X (t )
(2.26)
where
ZB
Z B1
Z B2
⎡0
⎢
= ⎢⎢
⎢
⎢⎣
⎡0
⎢
= ⎢⎢
⎢
⎢⎣
⎡1
⎢
= ⎢⎢
⎢
⎢⎣
0
1
0
0
1
O
0
0
1
1
⎤
⎥
⎥
⎥
⎥
0 ⎥⎦
0
⎤ (2.27)
⎥
⎥
⎥
⎥
0 ⎥⎦
0
O
0
0
0
1
0
0
0
0
0
0
0
1
0
⎤
⎥
⎥
⎥
⎥
1 ⎥⎦
0
0
O
1
24
The integrated dynamic model of the electrohydraulically driven revolute
robot manipulator can be obtained by substituting equations (2.22) and (2.23) into
(2.10) and by utilizing equation (2.26):
X& (t ) = A( X , ξ , t ) X (t ) + B ( X , ξ , t )U (t ) + N ( X , ξ , t )
(2.28)
where
A( X , ξ , t ) = [I 3 N − WH M ( X , ξ , t ) Z B ]
−1
~
⎧ AH + [ FH M ( X , ξ , t ) + WH C ( X , ξ , t )]Z B ⎫
⎪⎪
⎪⎪
)
~
⎨+ [ FH D( X , ξ , t ) + WH D( X , ξ , t )]Z B1
⎬
)
⎪
⎪
⎪⎩+ FH G( X , ξ , t )Z B 2
⎪⎭
(2.29)
B ( X , ξ , t ) = [I 3 N − W H M ( X , ξ , t ) Z B ] B H
(2.30)
N ( X , ξ , t ) = [I 3N − WM( X , ξ , t )Z ] N H
(2.31)
−1
−1
with
I 3N
: 3N x 3N identity matrix
A( X , ξ , t )
: 3N x 3N system matrix for the integrated hydraulic robot
manipulator
B( X , ξ , t )
: 3N x N input matrix for the integrated hydraulic robot
manipulator
N ( X ,ξ , t)
: 3N x 1 nonlinear matrix for the integrated hydraulic robot
manipulator
However, it is difficult to synthesize a controller based on the above equation
due to the nonlinearity term. In order to make the development of controller easier,
the nonlinearity term in equation (2.16) can be factorized and included in the
hydraulic actuator system matrix AHi. The nonzero elements of the matrix NH in
equation (2.16) can be factorized as follows:
⎛ 4β ei K ti Gni Gi n gi2 3 3Gni Gi n gi2 2
⎞
xi xi +1 ⎟
xi +
−⎜
⎜
⎟
J ti
Vti J ti
⎝
⎠
⎞
⎛ 4β ei K ti Gni Gi n gi2 2 3Gni Gi n gi2
xi xi +1 ⎟ xi
xi +
= −⎜
⎟
⎜
J ti
Vti J ti
⎠
⎝
(2.32)
25
such that the matrix N Ni ( X , ξ , t ) is introduced where
⎡n Ni
N Ni ( X , ξ , t ) = ⎢⎢ 0
⎢⎣ 0
0 0⎤
0 0⎥⎥
0 0⎥⎦
(2.33)
And the nonzero elements of N Ni ( X , t ) is:
n Ni
⎛ 4β ei K ti Gni Gi n gi2 2 3Gni Gi n gi2
⎞
xi xi +1 ⎟ xi
xi +
= −⎜
⎜
⎟
J ti
Vti J ti
⎝
⎠
(2.34)
Therefore, the 3N x 1 nonlinear matrix N ( X , ξ , t ) for the augmented dynamic
equation of the hydraulic actuator in equation (2.17) is replaced by the modified 3N
x 3N nonlinear matrix N N ( X , ξ , t ) where
N N ( X , ξ , t ) = diag [ N N 1 ( X , ξ , t ) , N N 2 ( X , ξ , t ) , …. , N NN ( X , ξ , t ) ]
(2.35)
Summing both system matrix A as described in equation (2.17) and nonlinear matrix
N N ( X ,t)
as described in equation (2.35) yields a new system matrix
AN ( X , ξ , t ) where:
AN ( X , ξ , t ) = A( X , ξ , t ) + N N ( X , ξ , t )
(2.36)
Hence, the dynamics of the integrated electrohydraulic robot manipulator model can
be represented by
X& (t ) = AN ( X , ξ , t ) X (t ) + B( X , ξ , t )U (t )
A( X , ξ , t ) = [I 3 N − WH M ( X , ξ , t ) Z B ]
−1
(2.37)
~
⎧ AN + [ FH M ( X , ξ , t ) + WH C ( X , ξ , t )]Z B ⎫
)
~
⎪⎪
⎪⎪
⎨+ [ FH D( X , ξ , t ) + WH D( X , ξ , t )]Z B1
⎬
)
⎪
⎪
⎪⎩+ FH G ( X , ξ , t ) Z B 2
⎪⎭
B ( X , ξ , t ) = [I 3 N − W H M ( X , ξ , t ) Z B ] B H
−1
(2.38)
26
2.5
Dynamic Model of 3 DOF Electrohydraulic Robot Manipulators
A complete mathematical model of 3 DOF electrohydraulically driven
revolute robot manipulator as shown in Figure 2.3 is derived in this section based on
the approach presented in section 2.4. In this model, it is assumed that all the three
joints are actuated by an identical electrohydraulic motor and; the end-effector and
the variable load are lumped together as a single mass mL and is located at the end of
the third link. This load is assumed to vary between 0kg to 10kg. The range of
operations and parameters of the mechanical linkage as described in [12] are
tabulated in Table 2.1 and 2.2 and the parameters of the hydraulic actuators as
presented in [26] are tabulated in Table 2.3.
Figure 2.3: Three DOF Hydraulically Driven Robot Manipulator [10]
27
Table 2.1: Range of Operation of the Mechanical Linkage [10]
Physical Quantity
Link Number, i
1
Position, θ i (degree)
Velocity,
2
3
Min
Max
Min
Max
Min
Max
-160
160
-225
45
-46
225
0
120
0
110
0
110
θ&i (degree/s)
Table 2.2: Parameters of the Mechanical Linkage [10]
Parameter
Link Number, i
0
1
2
3
2.0
1.5
0.6
0.5
0.3
0.25
x-axis at center of gravity
0.06
0.03
y-axis at center of gravity
0.008
0.0025
0.06
0.03
Mass, mi (kg)
Length, li (m)
0.3
0.5
Position of the center of gravity
(m)
Moment of inertia Ii (kgm2)
with respect to:
z-axis at center of gravity
0.04
28
Table 2.3: Parameters of the Hydraulic Actuator [10]
Parameter
Actuator/
Value
Joint
Number, i
Flow gain, Kqi (m2 /s)
1, 2, 3
2.3 × 10 −7 Ps − sign( X v ) PL
Supply pressure, Psi (N/m2)
1, 2, 3
1.4 × 10 7
Effective bulk modulus, β ei ( N/m 2 )
1, 2, 3
3.5×107
Total compressed volume,
1, 2, 3
3.3 ×10−5
Total leakage coefficient, K ti ( m 2 /s )
1, 2, 3
2.3 ×10−11
Volumetric displacement,
1, 2, 3
1.6 × 10 −5
Motor inertia, J mi (kgm/s 2 )
1, 2, 3
5.8 ×10−3
Viscous damping coefficient,
1, 2, 3
0.864
1, 2, 3
0.5
Vti (m 3 /rad)
Dmi (m 5 /s/N)
B mi ( kgms/rad )
Servo valve gain, K vi (m/V)
2.5.1
Dynamic Equation of the Electrohydraulic Motors for the 3 DOF Robot
Manipulator
From equation (2.37), the augmented actuators dynamic equation for the 3
DOF robot manipulator can be rewritten as follows:
X& (t ) = AX (t ) + BU (t ) + FT (t ) + WT& (t ) + N ( X , t )
where
(2.39)
29
X (t ) =
[
x1
x2
x3
x4
x5
x6
x7
x8
x9
]
=
[
x11
x12
x13
x12
x 22
x 23
x31
x32
x33
]
=
[
θ1
θ&1
θ&&1
θ2
θ&2
θ&&2
θ3
θ&&3
]
⎡ 0
⎢ 0
⎢
⎢ a B 31
⎢
⎢
A =⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡0
⎢0
⎢
⎢bB1
⎢
⎢0
B =⎢ 0
⎢
⎢0
⎢0
⎢
⎢0
⎢0
⎣
⎡ 0
⎢ 0
⎢
⎢ wB1
⎢
⎢ 0
W =⎢ 0
⎢
⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 0
⎣
1
0
0
1
a B 32
a B 33
0
0
0
0
0
0
0
bB 2
0
0
0
0
0
0
0
0
wB 2
0
0
0
0
0
0
0
1
0
0
1
a B 64
a B 65
a B 66
0
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥,
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
bB 3 ⎥⎦
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥,
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
wB 3 ⎥⎦
θ&3
⎡ 0
⎢ 0
⎢
⎢ f B1
⎢
⎢ 0
F =⎢ 0
⎢
⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 0
⎣
0
1
0
0
a B 97
a B 98
0
0
0
0
0
f B2
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
1 ⎥
a B 99 ⎥⎦
T
T
T
(2.40)
(2.41)
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥,
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
f B 3 ⎥⎦
(2.42)
30
⎡ 0 ⎤
⎢ 0 ⎥
⎢ ⎥
⎢ n B1 ⎥
⎢ ⎥
⎢ 0 ⎥
N = ⎢ 0 ⎥,
⎢ ⎥
⎢n B 2 ⎥
⎢ 0 ⎥
⎢ ⎥
⎢ 0 ⎥
⎢n ⎥
⎣ B3 ⎦
⎡U 1 (t ) ⎤
U (t ) = ⎢U 2 (t )⎥,
⎢
⎥
⎢⎣U 3 (t ) ⎥⎦
⎡ T&1 (t ) ⎤
⎥
⎢
T& (t ) = ⎢T&2 (t )⎥
⎢T&3 (t ) ⎥
⎦
⎣
⎡ T1 (t ) ⎤
T (t ) = ⎢T2 (t )⎥,
⎢
⎥
⎢⎣T3 (t ) ⎥⎦
(2.43)
where
a B 31
⎡G
4β D 2
4β K B ⎤
a B 32 = − ⎢ 1 + e1 m1 + e1 t1 m1 ⎥
Vt1 J m1
Vt1 J m1 ⎦
⎣ J m1
− 4 β e1 K t1G1
=
Vt1 J t1
a B 64 =
− 4β e2 K t 2 G2
Vt 2 J t 2
a B 97 =
− 4 β e3 K t 3G3
Vt 3 J t 3
⎡G
4β D 2
4β K B ⎤
a B 65 = − ⎢ 2 + e 2 m 2 + e 2 t 2 m 2 ⎥
Vt 2 J m 2
Vt 2 J m 2 ⎦
⎣ J m2
⎡G
4β D 2
4β K B ⎤
a B 98 = − ⎢ 3 + e3 m 3 + e3 t 3 m 3 ⎥
Vt 3 J m3
Vt 3 J m3 ⎦
⎣ J m3
⎡B
4β e 2 K t 2 ⎤
a B 66 = − ⎢ m 2 +
⎥
Vt 2 ⎦
⎣ J m2
⎡B
4 β e1 K t1 ⎤
a B 33 = − ⎢ m1 +
⎥
Vt1 ⎦
⎣ J m1
⎡B
4β e3 K t 3 ⎤
a B 99 = − ⎢ m 3 +
⎥
Vt 3 ⎦
⎣ J m3
bB1 =
4β e1Dm1K q1 K v1
Vt1 J m1
bB 2 =
4β e 2 Dm 2 K q 2 K v 2
Vt 2 J m 2
bB 3 =
4β e 3 Dm3 K q 3 K v 3
Vt 3 J m3
f B1 = −
4β e1K v1
Vt1 J m1
f B2 = −
4β e 2 K v 2
Vt 2 J m 2
f B3 = −
4β e3 K v3
Vt 3 J m3
wB1 = −
1
J m1
wB 2 = −
1
J m2
wB 3 = −
1
J m3
31
⎛ 4β K G G
3G n1G1 2 ⎞
n B1 = −⎜⎜ e1 t1 n1 1 x13 +
x1 x 2 ⎟⎟
Vt1 J m1
J m1
⎝
⎠
⎛ 4β K G G
3G n 2 G 2 2 ⎞
n B 2 = −⎜⎜ e 2 t 2 n 2 2 x 43 +
x 4 x5 ⎟⎟
V
J
J
t 2 m2
m2
⎝
⎠
⎛ 4β K G G
3G n 3 G3 2 ⎞
n B 3 = −⎜⎜ e 3 t 3 n 3 3 x 73 +
x 7 x8 ⎟⎟
Vt 3 J m 3
J m3
⎝
⎠
(2.44)
2.5.2 Dynamic Equation of 3 DOF Robot Manipulator Mechanical Linkage
Using Lagrange-Euler method, the dynamic equations of a three DOF
revolute robot manipulator can be rewritten in the following form:
M (θ , ξ )θ&&(t ) + D (θ , ξ )Vˆ (θ&) + G (θ , ξ ) = T (t )
(2.45)
where
θ (t) = [θ1 (t) θ 2 (t) θ3 (t)]T
(2.46)
T (t ) = [T1 (t ) T2 (t) T3 (t)]
(2.47)
T
ξ = m L = varying payload mass carried by the manipulator
⎡ M 11
M (θ , ξ ) = ⎢ 0
⎢
⎢⎣ 0
⎡ 0
⎢
D (θ , ξ ) = ⎢ D12
⎢⎣ D 31
0
M 22
M 23
D12
0
0
0 ⎤
M 23 ⎥
⎥
M 33 ⎥⎦
D13
0
0
0
0
D 34
(2.48)
0
D 25
0
0 ⎤
⎥
D 26 ⎥
0 ⎥⎦
(2.49)
32
⎡ θ&1θ&1 ⎤
⎢& & ⎥
⎢θ 1θ 2 ⎥
⎢θ& θ& ⎥
Vˆ (θ&) = ⎢ 1 3 ⎥
& &
⎢θ 2θ 2 ⎥
⎢θ&2θ&3 ⎥
⎢& & ⎥
⎢⎣θ 3θ 3 ⎥⎦
(2.50)
⎡0⎤
G (θ , ξ ) = ⎢G 2 ⎥
⎢ ⎥
⎢⎣G 3 ⎥⎦
(2.51)
where the nonlinear elements of the matrices M (θ , ξ ), D (θ , ξ ) and G (θ , ξ ) are as
follows:
M 11 = A1 + A2 cos2 (θ 2 ) + A3 cos 2 (θ 2 + θ 3 ) + A4 cos(θ 2 ) cos(θ 2 + θ 3 )
M 22 = A5 + A4 cos(θ 3 )
M 23 = A6 + A7 cos(θ 3 )
M 33 = A8
D12 = − A2 sin(2θ 2 ) − A3 sin(2θ 2 + 2θ 3 ) − A4 sin(2θ 2 + θ 3 )
D13 = − A3 sin(2θ 2 + 2θ 3 ) − A4 cos(θ 2 ) sin(2θ 2 + θ 3 )
D25
D12
= 0.5 A2 sin(2θ 2 ) + 0.5 A3 sin(2θ 2 + 2θ 3 ) + 0.5 A4 sin(2θ 2 + θ 3 )
2
= − A4 sin(θ 3 )
D21 = −
D25
= −0.5 A4 sin(θ 3 )
2
D
D31 = − 13 = 0.5 A3 sin(2θ 2 + 2θ 3 ) + 0.5 A4 cos(θ 2 ) sin(2θ 2 + θ 3 )
2
D34 = − D26 = 0.5 A4 sin(θ 3 )
D26 =
G 2 = A9 cos( θ 2 ) + A10 cos( θ 2 + θ 3 )
G 3 = A10 cos( θ 2 + θ 3 )
where
(2.52)
33
1
3
+ I 222 + I 22
A1 = I 33
A2 = [0.25m2 + m3 + m L ]l 22 + I 112 − I 222
3
A3 = 0.25[m3 + 4m L ]l32 + I 113 − I 22
A4 = [m3 + 2m L ]l 2 l3
A5 = [0.25m2 + m3 + m L ]l 22 + 0.25[m3 + 4m L ]l32 + I 112 − I 113
A6 = 0.25[m3 + 4m L ]l32 + I 113
(2.53)
A7 = 0.5 A4 = 0.5[m3 + 2m L ]l 2 l3
A8 = 0.25[m3 + 4m L ]l32 + I 113
A9 = −[0.5m2 + m3 + m L ]gl 2
A10 = −[0.5m3 + m L ]gl3
g=9.8016m/s2
i
1
, and I 33
are the moment of inertia at the centre of gravity of the ith link
and, I 11i , I 22
with respect to the x-axis, y-axis and z-axis respectively.
The terms D (θ , ξ )Vˆ (θ&) and G (θ , ξ ) in equation (2.45) can be written
)
)
as D(θ , θ&, ξ )θ& and G (θ , ξ )θ respectively. Thus, the manipulator dynamics can be
rewritten as
)
)
M (θ , ξ )θ&&(t ) + D(θ , θ&, ξ )θ& + G (θ , ξ )θ = T (t )
(2.54)
where
⎡ 0
)
⎢
D (θ , θ&, ξ ) = ⎢ D 21θ&1
⎢ D31θ&1
⎣
⎡
⎢
⎢0 0
)
G2
G (θ , ξ ) = ⎢0
⎢
θ2
⎢
⎢0 0
⎣⎢
D12θ&1
0
D34θ&2
⎤
⎥
0 ⎥
0 ⎥
⎥
G3 ⎥
⎥
θ 3 ⎦⎥
⎤
D13θ&1
⎥
D25θ&2 + D 26θ&3 ⎥
⎥
0
⎦
(2.55)
(2.56)
From equation (2.54), the derivative of the torque can be obtained by
differentiate with respect to time, yields:
34
(
)
(
)
~
~
T& (t ) = M (θ (t ), ξ )θ&&&(t ) + C θ (t ), θ&(t ), ξ θ&&(t ) + D θ (t ), θ&(t ), ξ θ&(t )
(2.57)
where
(
)
(2.58)
(
)
(2.59)
~
C θ (t ), θ&(t ), ξ θ&&(t ) = M& (θ (t ), ξ )θ&&(t ) + D (θ (t ), ξ )V& (θ&(t ))
~
D θ (t ),θ&(t ),ξ θ&(t ) = D& (θ (t ),ξ )V (θ&(t )) + G& (θ (t ),ξ )
~
⎡C11
~
⎢~
C (θ , θ&, ξ ) = ⎢C 21
⎢C~31
⎣
~
C12
~
C 22
~
C 32
~
C13 ⎤
~ ⎥
C 23 ⎥
0 ⎥⎦
~
D12
~
D 22
~
D 32
⎡ 0
~
⎢~
D (θ , θ&, ξ ) = ⎢ D 21
~
⎢D
⎣ 31
~
D13 ⎤
~ ⎥
D 23 ⎥
~
D 33 ⎥⎦
(2.60)
(2.61)
~
~
and the elements of C (θ (t ), θ&(t ), ξ ) and D (θ (t ), θ&(t ), ξ ) matrices are as follows:
~
C11 = 2[D12θ&2 + D13θ&3 ]
~
C12 = D12θ&1
~
C13 = D13θ&1
~
C 21 = 2 D21θ&1
~
C22 = 2D25θ&3
~
C23 = D25 (θ&2 + θ&3 ) + D26θ&3
~
C31 = 2D31θ&1
~
C32 = 2D34θ&2 + D26θ&3
~
D12 = −2 A2θ&1θ&2 cos(2θ 2 ) − 2 A3θ&1 (θ&2 + θ&3 ) cos(2θ 2 + 2θ3 )
− A θ& (2θ& + θ& ) cos(2θ + θ )
4 1
2
3
2
3
~
D13 = −2 A3θ&1 (θ&2 + θ&3 ) cos(2θ 2 + 2θ 3 ) − A4θ&1θ&2 cos(2θ 2 + θ 3 )
− A θ& θ& cos(θ ) cos(θ + θ )
4 1 3
2
2
3
~
~
D 21 = −0.5 D12
~
D 22 = − A9 sin(θ 2 ) − A10 sin(θ 2 + θ 3 )
35
~
D23 = − A10 sin(θ 2 + θ3 ) − A4θ&3 (θ&2 + 0.5θ&3 ) cos(θ 3 )
~
~
D31 = −0.5D13
~
D32 = − A10 sin(θ 2 + θ 3 ) + 0.5A4θ&2θ&3 cos(θ 3 )
~
D33 = − A10 sin(θ 2 + θ 3 )
2.5.3
(2.62)
Integrated Dynamic Model of 3 DOF Electrohydraulic Robot
Manipulator
The complete dynamic model of three DOF electrohydraulic robot
manipulator model can be obtained by using the approach presented in Section 2.4.
First defining the following transformations [12]:
θ&&(t ) = Z B X (t )
θ&(t ) = Z B1 X (t )
θ (t ) = Z B 2 X (t )
(2.63)
where Z B , Z B1 , Z B2 are as follows:
ZB
Z B1
Z B2
⎡0
= ⎢⎢ 0
⎢⎣ 0
⎡0
= ⎢⎢ 0
⎢⎣ 0
⎡1
= ⎢⎢ 0
⎢⎣ 0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0⎤
0 ⎥⎥ ,
1 ⎥⎦
0⎤
0 ⎥⎥ ,
0 ⎥⎦
0⎤
0 ⎥⎥
0 ⎥⎦
(2.64)
36
Substituting equations (2.54) and (2.57) into (2.39) and utilizing (2.63), the
mathematical model of the integrated electrohydraulic robot manipulator has the
following form:
X& (t ) = A( X , ξ , t ) X (t ) + B ( X , ξ , t )U (t ) + N ( X , ξ , t )
(2.65)
where
A( X , ξ , t ) = [I 9 − WM ( X , ξ , t ) Z B ]
−1
~
⎧ A + [ FM ( X , ξ , t ) + WC ( X , ξ , t )]Z B ⎫
⎪⎪
⎪⎪
)
~
⎨+ [ FD( X , ξ , t ) + WD( X , ξ , t )]Z B1 ⎬ (2.66)
)
⎪
⎪
⎪⎩+ FG ( X , ξ , t ) Z B 2
⎪⎭
B ( X , ξ , t ) = [I 9 − WM ( X , ξ , t ) Z B ] B
(2.67)
N ( X , ξ , t ) = [I 9 − WM( X , ξ , t )Z ] N
(2.68)
−1
−1
and I9 is a 9 x 9 identity matrix.
The entries of the matrices are
⎡0
⎢0
⎢
⎢a31
⎢
⎢0
A( X , ξ , t ) = ⎢ 0
⎢
⎢0
⎢0
⎢
⎢0
⎢0
⎣
1
0
a32
0
0
a62
0
0
a92
0
1
a33
0
0
a63
0
0
a93
0
0
0
0
0
a 64
0
0
a94
0
0
a35
1
0
a 65
0
0
a95
0
0
a36
0
1
a66
0
0
a96
0
0
0
0
0
a67
0
0
a97
0
0
a38
0
0
a 68
1
0
a98
0 ⎤
0 ⎥⎥
a39 ⎥
⎥
0 ⎥
0 ⎥,
⎥
a 69 ⎥
0 ⎥
⎥
1 ⎥
a99 ⎥⎦
(2.69)
37
0
⎡0
⎢0
0
⎢
⎢b31 0
⎢
0
⎢0
B( X , ξ , t ) = ⎢ 0
0
⎢
⎢ 0 b62
⎢0
0
⎢
0
⎢0
⎢0 b
92
⎣
0⎤
0 ⎥⎥
0⎥
⎥
0⎥
0 ⎥,
⎥
b63 ⎥
0⎥
⎥
0⎥
b93 ⎥⎦
⎡ 0 ⎤
⎢ 0 ⎥
⎢ ⎥
⎢ n1 ⎥
⎢ ⎥
⎢ 0 ⎥
N ( X B ,ξ , t ) = ⎢ 0 ⎥
⎢ ⎥
⎢n 2 ⎥
⎢ 0 ⎥
⎢ ⎥
⎢ 0 ⎥
⎢n ⎥
⎣ 3⎦
(2.70)
(2.71)
However, in order to make the development and calculation of controller as
will be presented in the next chapter easier, the matrix N is factorized and included
in system matrix A. From equation (2.44) the nonzero elements of the nonlinear
terms of matrix N can be factorized as:
n1 = n B1
n2 = n B 2
n3 = n B 3
⎞
⎛ 4β e1 K t1Gn1G1 n g2 2 3Gn1G1 n g2
x1 x2 ⎟ x1
x1 +
= −⎜
⎟
⎜
J t1
Vt1 J t1
⎠
⎝
⎞
⎛ 4β e 2 K t 2 Gn 2 G2 n g2 2 3Gn 2 G2 n g2
x4 +
x 4 x5 ⎟ x 4
= −⎜
⎟
⎜
Vt 2 J t 2
Jt2
⎠
⎝
2
2
⎛ 4β e3 K t 3Gn3G3 n g 2 3Gn3G3 n g
⎞
x7 +
x7 x8 ⎟ x7
= −⎜
⎜
⎟
Vt 3 J t 3
J t3
⎝
⎠
such that the matrix N N = N ( X , t ) is introduced where
(2.72)
38
⎡ 0
⎢ 0
⎢
⎢ nN1
⎢
⎢ 0
N N= ⎢ 0
⎢
⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 0
⎣
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
nN 2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
nN3
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥ (2.73)
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
0 ⎥⎦
and the nonzero elements of N N are:
⎛ 4β K G G
⎞
3G G
n N 1 = −⎜⎜ e1 t1 n1 1 x12 + n1 1 x1 x 2 ⎟⎟
Vt 1 J t 1
J t1
⎝
⎠
⎛ 4β K G G
⎞
3G G
n N 2 = −⎜⎜ e 2 t 2 n 2 2 x 42 + n 2 2 x 4 x5 ⎟⎟
Vt 2 J t 2
Jt2
⎝
⎠
(2.74)
⎛ 4β K G G
⎞
3G G
n N 3 = −⎜⎜ e 3 t 3 n 3 3 x 72 + n 3 3 x 7 x8 ⎟⎟
Vt 3 J t 3
J t3
⎝
⎠
Summing both system matrix A as described in equation (2.41) - (2.43) and
nonlinear matrix N N ( X , t ) as described in equation (2.73) yields a new system
matrix AN ( X , ξ , t ) where:
AN ( X , ξ , t ) = A( X , ξ , t ) + N N ( X , ξ , t )
(2.75)
Then, the augmented actuators dynamic equation for a three DOF robot manipulator
as in equation (2.65) can be rewritten as:
X& (t ) = AN X (t ) + BU (t ) + FT (t ) + WT& (t )
(2.76)
where
AN = A + N N
(2.77)
Thus, the final form of the integrated dynamic of 3 DOF electrohydraulic driven
revolute robot manipulator model is represented by
X& (t ) = AN ( X , ξ , t ) X (t ) + B( X , ξ , t )U (t )
where
(2.78)
39
~
⎧ AN + [ FM ( X , ξ , t ) + WC ( X , ξ , t )]Z B ⎫
⎪⎪
⎪⎪
)
~
⎨+ [ FB D( X , ξ , t ) + WD( X , ξ , t )]Z B1 ⎬
)
⎪
⎪
⎪⎩+ FG ( X , ξ , t ) Z B 2
⎪⎭
AN ( X , ξ , t ) = [I 9 − WM ( X B , ξ , t ) Z B ]
−1
(2.79)
and B( X , ξ , t ) is similar as in equation (2.67).
The entries of system matrix of AN ( X B , ξ , t ) are equivalent to the entries of
A( X B , ξ , t ) as described in equation (2.69), with the following nonzero terms:
Let
ϕ1 = 1 − wB1 M 11
ϕ 2 = 1 − wB 2 M 22
ϕ 3 = 1 − wB 3 M 33
ϕ 4 = − wB 2 M 23
ϕ 5 = 1 − wB 3 M 23
ψ = ϕ 2ϕ 3 − ϕ 4ϕ 5
a 31 = (a B 31 + n N 1 ) / ϕ1
a 32 = a B 32 / ϕ1
(
(
(
)
(
(
)
{ (
{ (
)
)
~
a 33 = a B 33 + f B1 M 11 + wB1C11 / ϕ1
~
a 35 = f B1 D12 x 2 + wB1 D12 / ϕ1
~
a 36 = wB1C12 / ϕ1
~
a 38 = f B1 D13 x 2 + wB1 D13 / ϕ1
~
a 39 = wB1C13 / ϕ1
~
~
a 62 = ϕ 3 f B 2 D21 x 2 + wB 2 D21 − ϕ 4 f B 3 D31 x 2 + wB 3 D31 /ψ
~
~
a 63 = ϕ 3 wB 2 C 21 − ϕ 4 wB 3 C 31 /ψ
)
)
)
(
)}
(
)}
a 64 = {ϕ 3 (a B 64 + f B 2 G 2 / x 4 )}/ψ
~
~
a 65 = ϕ 3 a B 65 + wB 2 D22 − ϕ 4 f B 3 D34 x5 + wB 3 D32 / ψ
~
~
a 66 = ϕ 3 a B 66 + f B 2 M 22 + wB 2 C 22 − ϕ 4 f B 3 M 23 + wB 3 C 32 /ψ
{ (
{ (
)
(
)
)}
(
)}
a 67 = {− ϕ 4 (a B 97 + n N 3 + f B 3 G3 / x 7 )}/ψ
~
~
a 68 = ϕ 3 f B 2 (D25 x5 + D26 x8 ) + wB 2 D23 − ϕ 4 a B 98 + wB 3 D33 / ψ
~
a 69 = ϕ 3 f B 2 M 23 + wB 2 C 23 − ϕ 4 (a B 99 + f B 3 M 33 ) / ψ
{ (
{ (
)
)
(
}
)}
40
{ (
{ (
)
(
)}
~
~
a92 = ϕ 2 f B 3 D31 x 2 + wB 3 D31 − ϕ 5 f B 2 D21 x2 + wB 2 D21 /ψ
~
~
a93 = ϕ 2 wB 3C31 − ϕ 5 wB 2 C 21 /ψ
)
(
)}
a94 = {− ϕ 5 (a B 64 + n N 2 + f B 2 G2 / x4 )}/ψ
~
~
a95 = ϕ 2 f B 3 D34 x4 + wB 3 D32 − ϕ 5 a B 65 + wB 2 D22 /ψ
~
~
a96 = ϕ 2 f B 3 M 23 + wB 3C32 − ϕ 5 a B 66 + f B 2 M 22 + wB 2 C 22 /ψ
{ (
{ (
)
)
(
(
)}
)}
(2.80)
a97 = {ϕ 2 (a B 97 + n N 3 + f B 3G3 / x7 )}/ψ
~
~
a98 = ϕ 2 a B 98 + wB 3 D33 − ϕ 5 f B 2 (D25 x5 + D26 x8 ) + wB 2 D23 /ψ
~
a99 = ϕ 2 (a B 99 + f B 3 M 33 ) − ϕ 5 f B 2 M 23 + wB 2 C 23 /ψ
{ (
{
)
(
(
)}
)}
The nonzero elements of B( X , ξ , t ) are:
b31 = bB1 / ϕ1
b62 = (bB 2ϕ 3 ) / ψ
b63 = −(bB 3ϕ 4 ) / ψ
b92 = −(bB 2ϕ 5 ) / ψ
(2.81)
b93 = −(bB 3ϕ 2 ) / ψ
2.6
Summary
The derivation of the complete mathematical model for an integrated
dynamics of a three DOF electrohydraulically driven revolute robot manipulator has
been presented in this chapter. The integrated system consists of the manipulator
dynamics due to mechanical linkage as well as the hydraulic actuator dynamics that
drives each of the joints. However, it should be noted that this model are by no
means “complete and perfect” since there are other considerations that may
contribute to the robot dynamics such as un-modeled and unknown dynamics which
have been ignored. From equations (2.69) – (2.81) obtained, it can be observed that
the dynamics description of the robot manipulator is analytically complex. The
equations are highly nonlinear, time varying and coupled due to the nonlinear
mechanical linkage dynamics and dependence of the effective driving torque on
joint angle and nonlinearities that present in the electrohydraulic dynamics. These
equations also suffer from parameter uncertainties including the varying payload
41
mass. Each nonzero elements of the system and input matrix is a function of the
instantaneous position, velocity and payload mass of the manipulator. Due to these
properties of the system, it is a challenging task to design an appropriate controller.
Conventional controllers such as PID and Computed Torque Control cannot provide
good tracking control of the robot. Therefore, a more advanced and robust control
algorithm that can cater for all these problems is needed which will be described in
the next chapter.
42
CHAPTER 3
DECENTRALIZED SLIDING MODE CONTROLLER DESIGN
3.1
Introduction
Large-scale interconnected systems can be found in many real-life practical
applications such as electric power systems, nuclear reactors, aerospace systems,
economic systems, chemical process control systems, computer networks, and urban
traffic network, etc. Large-scale systems are often modeled as dynamic equation
composed of interconnection of lower dimensional subsystems. Robot may be
considered as a Large-Scale System (LSS) due to the complexity, nonlinearity,
couplings, multiple inputs and multiple outputs nature of its dynamics. The control
of a LSS is very complicated due to the high dimensionality and complexity of the
systems equations, interactions between the sub-systems, coupling of the inputs and
uncertainties resulted from the modeling errors and disturbances [21]. Therefore the
study of decentralized systems is critical when one attempts to design and apply
controller to large-scale system problem, especially in robotic system. Besides its
simplicity, the decentralized control is an effective way of controlling the large-scale
systems since each sub-system is independently controlled. Therefore, numerous
researchers have paid a great deal of attention on developing the decentralized
controller for the large-scale systems, such as [27], [28], [29], [30] and [31].
In this study, based on the argument that any system may be decomposed
into interconnected subsystems consisting smaller uncertain subsystems, a
decentralized controller is derived through Variable Structure Control (VSC) so that
certain outputs of the system asymptotically track a given reference inputs
43
independent of any external disturbances which may affect the system, and
independent of any variations in the plant parameters and gains of the system. The
VSC control is popularly adopted because it poses several attractive advantages in
the sliding mode such as good transient, fast response, easy realization, and
insensitivity to the variation of plant parameters and/or external disturbances [32].
3.2
Decentralized Sliding Mode Control Strategies
The problem of controlling uncertain nonlinear dynamical systems that are
subjected to external disturbances is a topic which is of great interest to control
researchers. One approach to solve this problem is by means of Variable Structure
Control (VSC) that is robust to parameter uncertainties and disturbances. The core
idea of designing VSC algorithms consists of enforcing sliding mode in some
manifold of system space. A sliding mode will exist for a system if in the vicinity of
the switching surface, the state velocity vector or the derivative of the state vector is
directed towards the surface. In other words, the property of remaining on the
switching or sliding surface once intercepted is called a sliding mode. Essentially,
VSC utilizes a high-speed switching control law to drive the nonlinear plant’s state
trajectory onto a specified and user-chosen surface in the state space and to maintain
the plant’s state trajectory on this surface for all subsequent time. This surface is
called the switching or sliding surface [22]. The major advantage of this controller is
its inherent insensitivity to uncertainties parameter and disturbances once a system is
in the sliding mode, thereby eliminating the necessity of exact modeling. Sliding
mode control enables separation of the overall system motion into independent
partial components of lower dimensions, and as a result reducing the complexity of
the control design [33].
Insuring the existence of a sliding mode on the switching surface is a key
necessity in VSC design while designing the proper surface is the complementary
key problem. Therefore, there are two major phase involved in VSC design, which
are the sliding phase and the reaching phase. The first phase or sliding phase entails
44
the construction of the switching surface so that the original system will be restricted
to the surface responds in a desired manner. While the second phase or reaching
phase design is the development of discontinuous control law with varying control
structures, and then force the trajectory of the system state to a certain predefined
surface, known as sliding or switching surface. This control law ensures
convergence to the sliding surface in the phase space. Once the plant reaches the
switching plane, its response depends thereafter on the gradient of the switching
plane and remains insensitive to variations of system parameters and external
disturbances such that the system dynamics are only determined by the dynamics of
the sliding surface. The system state is adaptively altered to slide along the
switching plane [33] provided certain matching conditions are satisfied. This
condition states that the disturbances and uncertainties present in the system must be
in the range space of the input matrix B [32].
Sliding mode control facilitates separation of the overall system motion into
independent partial components of lower dimensions, and as a result reducing the
complexity in the control design. The potential of control resources may be used to
the fullest extent within the construction of nonlinear control methods since the
actuator limitations and other performance specifications may be included in the
design procedure. Many efforts have been made by numerous researchers around the
world to take advantage on the robustness of the decentralized variable structure
control in controlling the robotic systems [31], [12] and [20]. Generally, there are
two classes of decentralized control, namely, the local approach and the global
approach [34]. The local approach is based on the state information of each subsystem only. On the other hand, the global approach utilizes the extra feedback
information from the states of neighboring sub-systems beside the information from
the local sub-system. The study work presented in this chapter falls into the local
control category due to the argument that the local approach imitates a truly total
decentralized control as compared to the global one.
This chapter presents a systematic approach to the design of a robust tracking
controller based on decentralized sliding mode control techniques with the
application to the three DOF revolute electrohydraulic robot manipulator plant
45
detailed in Chapter 2. In this study, the typical structure of a sliding-mode controller
is composed of a nominal part and additional terms to deal with model uncertainty
and disturbances. For applying the decentralized control, the system may be
decomposed into interconnected uncertain subsystems. The way decentralized SMC
deals with interconnected uncertain subsystem is to drive the plant’s state trajectory
onto a sliding surface and maintain the error trajectory on this surface for all
subsequent times. Standard formulations and some basic assumptions are described
in the following sections.
3.3
Electrohydraulic Robot Manipulator as a Large-Scale Uncertain System
The first step in applying the tracking controller based on the decentralized
sliding mode control to the robot manipulator is to decompose the complete
integrated dynamics of the 3 DOF electrohydaulic robot manipulators given by
equation (2.15) into a set of interconnected subsystems.
Based on the structure of the system matrix AN ( X , ξ , t ) and input matrix
B ( X , ξ , t ) defined by the equation (2.79) and (2.67) respectively, they can be
partitioned into the following form:
⎡ A11 ( X , ξ , t )
AN ( X , ξ , t ) = ⎢⎢ A21 ( X , ξ , t )
⎢⎣ A31 ( X , ξ , t )
A12 ( X , ξ , t )
A22 ( X , ξ , t )
A13 ( X , ξ , t ) ⎤
A23 ( X , ξ , t )⎥⎥
A33 ( X , ξ , t ) ⎥⎦
(3.1)
⎡ B11 ( X , ξ , t ) B12 ( X , ξ , t ) B13 ( X , ξ , t ) ⎤
B N ( X , ξ , t ) = ⎢⎢ B21 ( X , ξ , t ) B22 ( X , ξ , t ) B23 ( X , ξ , t )⎥⎥
⎢⎣ B31 ( X , ξ , t ) B32 ( X , ξ , t ) B33 ( X , ξ , t ) ⎥⎦
(3.2)
A32 ( X , ξ , t )
From equations (3.1) and (3.2), then the complete integrated electrohydraulic
robot manipulator as described in equation (2.1) can be decomposed into a set of
interconnected subsystems as below:
46
X& (t ) = Ai ( X , ξ , t ) X i (t ) + Bi ( X , ξ , t )U i (t )
+
3
∑A
j = 1, j ≠ i
ij
( X , ξ , t ) X j (t ) +
3
∑B
j = 1, j ≠ i
ij
(X , ξ , t )U j (t )
(3.3)
with i, j = 1, 2, 3, where
[
]
T
X i (t ) = θ i (t ) θ&i (t ) θ&&i (t )
Ui (t) = [U1 (t) U2 (t) U3 ]T
X i (t )
: ith component of the X (t ) state vector
X j (t )
: jth component of the X (t ) state vector
Ai ( X , ξ , t )
: 3 x 3 ith diagonal sub-matrix of AN ( X , ξ , t )
Aij ( X , ξ , t )
: 3 x 3 ijth off-diagonal sub-matrix of AN ( X , ξ , t )
Bi ( X , ξ , t )
: 3 x 1 ith diagonal sub-matrix of B N ( X , ξ , t )
Bij ( X , ξ , t )
: 3 x 1 ijth off-diagonal sub-matrix of B N ( X , ξ , t )
Ui (t)
: ith row of input vector U (t )
U j (t)
: jth row of input vector U (t )
For non-direct drive robot manipulators, the magnitudes of the non-zero
elements in the off-diagonal sub-matrices in Bij ( X , ξ , t )
are often very small
compared to the elements of the diagonal sub-matrices Bi ( X , ξ , t ) . Therefore, the
off-block diagonal sub-matrices can often be assumed to be negligible and can be
ignored and thus in these situations, the integrated input matrix B ( X , ξ , t ) may be
treated as in block diagonal form, which is
B ( X , ξ , t ) = diag [B1 ( X , ξ , t ), B2 ( X , ξ , t ), B3 ( X , ξ , t )]
(3.4)
Then, in this case, the interconnected electrohydraulic robot manipulator
model can be rewritten as follow without the last term on the right hand side of
equation (3.3):
X& (t ) = Ai ( X , ξ , t ) X i (t ) + Bi ( X , ξ , t )U i (t ) +
3
∑
Aij ( X , ξ , t ) X j (t )
j = 1, j ≠ i
(3.5)
47
Basically, equation (3.5) describes the interconnection between the ith
subsystem with the other subsystems through Aij ( X , ξ , t ) X j (t ) . The structure of the
matrices Ai ( X , ξ , t ) , Aij ( X , ξ , t ) and Bi ( X , ξ , t ) are as shown below:
⎡0 1 0 ⎤
Ai ( X , ξ , t ) = ⎢⎢0 0 1 ⎥⎥ ’
⎢⎣# # #⎥⎦
⎡0 ⎤
Bi ( X , ξ , t ) = ⎢⎢0⎥⎥
⎢⎣#⎥⎦
⎡0 0 0 ⎤
Aij ( X , ξ , t ) = ⎢⎢0 0 0⎥⎥
⎢⎣# # #⎥⎦
(3.6)
(3.7)
where the term # the indicates nonlinear, coupled element i.e. the term with the
function of state variable X, varying payload carried by the manipulator, ξ and time,
t.
In order to apply the decentralized sliding mode control to the
electrohydraulic robot manipulator, it is necessary to present the interconnected subsystems dynamics of 3 DOF robot manipulators given by equation (3.5) into an
interconnected uncertain dynamical system as follows:
X& i (t ) = [ Ai + ΔAi ( X , ξ , t )] X i (t ) + [ Bi + ΔBi ( X , ξ , t )]U i (t )
+
3
∑ [A
j = 1, j ≠ i
ij
(3.8)
+ ΔAij ( X , ξ , t )] X j (t )
where Ai , Aij , B i and Bij which are time-invariant component representing the
nominal value of the matrices Ai ( X , ξ , t ) ,
Aij ( X , ξ , t )
and
Bi ( X , ξ , t )
respectively, with
Ai =
Aij =
Bi =
AiMAX + AiMIN
2
(3.9)
AijMAX + AijMIN
2
BiMAX + BiMIN
2
(3.10)
48
On the other hand, the element of the matrices ΔAi ( X , ξ , t ) , ΔAij ( X , ξ , t ) and
ΔBi ( X , ξ , t ) are uncertainty value of Ai ( X , ξ , t ) , Aij ( X , ξ , t ) and Bi ( X , ξ , t )
respectively, with
ΔAi ( X , ξ , t ) = Ai − AiMIN
(3.11)
ΔAij ( X , ξ , t ) = Aij − AijMIN
ΔBi ( X , ξ , t ) = Bi − BiMIN
(3.12)
In view of the physical parameters of the manipulators and actuators as
tabulated in Table 2.1 and Table 2.2, as well as the range of operation and the
maximum load, the dynamics of electrohydraulic robot manipulator as described in
equation (2.67)-(2.69) can be decomposed into a linear interconnected uncertain
system of equation (3.8). The nominal matrices Ai , Aij and B i as well as the bounds
on the nonzero elements of the matrices ΔAi ( X , ξ , t ) , ΔAij ( X , ξ , t ) and
ΔBi ( X , ξ , t ) may be computed off-line only once. Some of the results are shown
below and the rest are summarized in Table 3.1:
0
1
0
⎡
⎤
⎢
⎥’
0
0
1
A1 = ⎢
⎥
⎢⎣− 110.03e6 − 108.44e3 − 176.43⎥⎦
0
⎡
⎤
⎢
⎥
B1 = ⎢
0
⎥
⎢⎣236.87e6⎥⎦
0
1
0 ⎤
⎡
⎢
ΔA1 ( X , ξ , t ) = ⎢
0
0
1 ⎥⎥
⎢⎣110.02e6 87.96e3 66.05⎥⎦
0
⎡
⎤
⎢
⎥
ΔB1 ( X , ξ , t ) = ⎢
0
⎥
⎢⎣192.14e3⎥⎦
49
0
1
0
⎡
⎤
⎢
⎥
0
0
1
A2 = ⎢
⎥
⎢⎣− 231.07e6 − 107.99e3 − 175.14⎥⎦
0
⎡
⎤
⎢
⎥
B2 = ⎢
0
⎥
⎢⎣209.75e6⎥⎦
0
1
0 ⎤
⎡
ΔA2 ( X , ξ , t ) = ⎢⎢
0
0
1 ⎥⎥
⎢⎣230.99e6 60.66e3 45.73⎥⎦
0
⎡
⎤
⎢
⎥
ΔB 2 ( X , ξ , t ) = ⎢
0
⎥
⎢⎣117.72e3⎥⎦
0
1
0 ⎤
⎡
⎢
0
0
1 ⎥⎥
A3 = ⎢
⎢⎣− 240.66e6 − 139.97e3 − 199.9⎥⎦
0
⎡
⎤
⎢
⎥
B3 = ⎢
0
⎥
⎢⎣271.77e6⎥⎦
0
1
0 ⎤
⎡
⎢
ΔA3 ( X , ξ , t ) = ⎢
0
0
1 ⎥⎥
⎢⎣240.37e6 54.92e3 40.15⎥⎦
0
⎡
⎤
⎢
⎥
ΔB3 ( X , ξ , t ) = ⎢
0
⎥
⎢⎣106.63e3⎥⎦
(3.13)
50
Table 3.1: Value of Elements in Matrices
Variable
Maximum
Minimum
Nominal
Uncertainty
a31
-19480.0000
-220050000.0000
-110034740.0000
110015260.0000
a32
-20475.0000
-196400.0000
-108437.5000
87962.5000
a33
-110.3800
-242.4800
-176.4300
66.0500
a35
585.2200
-578.1300
3.5450
581.6750
a36
5.8602
-5.6811
0.0896
5.7707
a38
800.8800
-815.6900
-7.4050
808.2850
a39
7.8799
-8.1227
-0.1214
8.0013
a62
214.8600
-219.3200
-2.2300
217.0900
a63
4.3334
-4.4740
-0.0703
4.4037
a64
-84720.0000
-462060000.0000
-231072360.0000
230987640.0000
a65
-47337.0000
-168660.0000
-107998.5000
60661.5000
a66
-129.4100
-220.8700
-175.1400
45.7300
a67
14453000.0000
-35481000.0000
-10514000.0000
24967000.0000
a68
64513.0000
-21301.0000
21606.0000
42907.0000
a69
47.1600
-15.5530
15.8035
31.3565
a92
343.6600
-333.4900
5.0850
338.5750
a93
6.9526
-6.6770
0.1378
6.8148
a94
176730000.0000
-58286000.0000
59222000.0000
117508000.0000
a95
64539.0000
-21300.0000
21619.5000
42919.5000
a96
47.8150
-15.8230
15.9960
31.8190
a97
-299100.0000
-481030000.0000
-240664550.0000
240365450.0000
a98
-85043.0000
-194890.0000
-139966.5000
54923.5000
a99
-159.7500
-240.0500
-199.9000
40.1500
b31
429010.0000
44723.0000
236866.5000
192143.5000
b62
327470.0000
92037.0000
209753.5000
117716.5000
b63
41308.0000
-125260.0000
-41976.0000
83284.0000
b92
41308.0000
-125260.0000
-41976.0000
83284.0000
b93
378400.0000
165140.0000
271770.0000
106630.0000
51
Substituting the nominal values into the whole system and input nominal matrices,
yields
0
1
⎡
⎢
0
0
⎢
⎢ − 1.100e9 − 1.084e5
⎢
0
0
⎢
⎢
A=
0
0
⎢
− 2.230
0
⎢
⎢
0
0
⎢
0
0
⎢
⎢
0
5.085
⎣
0
0
0
1
0
0
− 176.4
0
3.545
0
0
1
0
0
0
− .07030 − 2.310e8 − 1.080e5
0
0
0
0
0
0
.1378
5.922e7
2.162e4
0
0
0
0
1
− 175.1
0
0
16.00
0
0
0
0
− 7.405
0.08955
0
0
0
0
− 1.051e7 2.161e4
0
1
0
0
− 2.407e9 − 1.400e5
0
0
− .1214
0
0
15.80
0
1
− 199.9
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(3.14)
0
⎡
⎢
0
⎢
⎢ 2.369e5
⎢
0
⎢
B = ⎢
0
⎢
0
⎢
⎢
0
⎢
0
⎢
⎢
0
⎣
⎤
⎥
0
0
⎥
⎥
0
0
⎥
0
0
⎥
⎥
0
0
⎥
2.098e5 − 4.198e4⎥
⎥
0
0
⎥
0
0
⎥
− 4.198e4 2.718e5 ⎥⎦
0
0
(3.15)
The uncertainties for the system and input matrices can be obtained by
substituting the uncertainty value into equations (3.11) and (3.12), gives
1
⎡ 0
⎢ 0
0
⎢
⎢1.100e9 8.796e5
⎢
0
⎢ 0
⎢
ΔA =
0
0
⎢
217.1
⎢ 0
⎢ 0
0
⎢
0
⎢ 0
⎢
338.6
⎣ 0
0
0
0
0
0
0
1
0
0
0
0
0
66.06
0
0
0
581.7
1
0
0
5.771
0
808.3
0
0
0
0
1
0
0
4.404
0
0
6.815
2.310e8 6.066e5
0
0
0
0
1.175e8 4.292e4
45.73
0
0
31.82
2.497e7 4.291e4
0
0
1
0
2.404e9 5.492e4
⎤
0 ⎥⎥
8.001 ⎥
⎥
0 ⎥
0 ⎥
⎥
31.36 ⎥
0 ⎥
⎥
1 ⎥
⎥
40.15 ⎦
0
(3.16)
52
0
0 ⎤
⎡ 0
⎢ 0
0
0 ⎥⎥
⎢
⎢1.921e5
0
0 ⎥
⎢
⎥
0
0 ⎥
⎢ 0
ΔB = ⎢ 0
0
0 ⎥
⎢
⎥
1.177e5 8.328e4⎥
⎢ 0
⎢ 0
0
0 ⎥
⎢
⎥
0
0 ⎥
⎢ 0
⎢ 0
8.328e4 1.066e5 ⎥⎦
⎣
(3.17)
Equation (3.8) can also be decomposed into [23];
X i = [ Ai + ΔAi ( X , ξ , t )] X i (t ) − [ Bi + ΔBi ( X , ξ , t )] f i + [ Bi + ΔBi ( X , ξ , t )]U i (t )
(3.18)
where
3
∑L
fi =
j =1
(
T
Lij = Bi Bi
3.4
(3.19)
Xj
ij
)
−1
(3.20)
T
Bi Aij
Problem Formulation for Decentralized SMC
Consider the dynamics of the 3 DOF electrohydraulic robot manipulator as
an uncertain system as described by equation (3.18)
Define the state vector of the system as
[
T
T
T
X (t ) = X 1 (t ), X 2 (t ), X 3 (t )
]
T
(3.21)
Let a continuous function X d (t ) ∈ R 3 N be the desired state trajectory, where Xd(t) is
defined as:
[
T
T
T
X d (t ) = X d 1 (t ), X d 2 (t ), X d 3 (t )
]
T
Define the tracking error, Zi(t) as
Z i (t ) = X i (t ) − X di (t )
(3.22)
53
In this study, the following assumptions are made:
i)
The state vector Xi(t) can be fully observed;
ii)
There exist continuous functions Hi(t) and Ei(t) such that for all
X i (t ) ∈ R N and all t:
iii)
ΔAi (t ) = Bi H i ( X , ξ , t ) ;
H i ( X , ξ , t) ≤ αi
(3.23)
ΔAij (t ) = Bi H ij ( X , ξ , t ) ;
H i ( X , ξ , t ) ≤ α ii
(3.24)
ΔBi (t ) = Bi Ei ( X , ξ , t ) ;
Ei ( X , ξ , t ) ≤ β i
(3.25)
There exist a Lebesgue function Ω i (t ) ∈ R , which is integrals on
bounded interval such that
•
X di (t ) = Ai X di (t ) + Bi Ω i (t )
(3.26)
where Ai and Bi are the i-th subsystem nominal system and input
matrices, respectively;
iv)
The pair ( Ai , Bi ) is controllable.
In view of equations (3.23)-(3.26), equation (3.18) can be written as an error
dynamic system:
Z& i (t ) = [ Ai + Bi H i ( X , ξ , t )]Z i (t ) + Bi H i X di − Bi Ω i − [ Bi + Bi E i ( X , ξ , t )] f i
+ [ Bi + Bi E i ( X , ξ , t )]U i (t )
(3.27)
Define the local sliding surface as described by [231;
σ i (t ) =
t
3
∑c
j =1
ij
z ij −
∫
f i ( q ) dq
(3.28)
−∞
where cij is a constant value, which is the element of matrix Ci. The structure of the
matrix Ci is as follows:
⎡ 0
C i = ⎢⎢ 0
⎢⎣− ci1
1
0
− ci 2
0 ⎤
1 ⎥⎥
− ci 3 ⎥⎦
(3.29)
54
The matrix Ci is chosen such that C i Bi ∈ R 3×1 is nonsingular. The elements of
matrix Ci can be determined by pole placement technique with prespecified poles
locations Pi:
Pi = [ p1i
p 2i
p 3i ]
(3.30)
pji is the jth desired pole of the ith subsystem. For this class of system, the sliding
manifold can be described as
σ (t ) = [σ 1 σ 2 σ 3 ]T
(3.31)
The control problem is then to design a controller using a decentralized
control law Ui(t) for each subsystem Si(t) using the local sliding surface given by
equation (3.28) such that the system state trajectory Xi(t) tracks the desired state
trajectory Xdi(t) as closely as possible for all t in spite of the uncertainties and nonlinearities present in the system. In view of the error dynamic, the tracking problem
has become the problem of stabilizing the error system of equation (3.27).
3.4.1
System Dynamics during Sliding Mode
Differentiating equation (3.28) gives:
σ& i ( t ) = C i Z& i ( t ) − f i ( t )
(3.32)
Substituting equation (3.27) into equation (3.32) gives:
σ& i (t ) = Ci [ Ai + Bi H i ( X , ξ , t )]Z i (t ) + Ci Bi H i X di − Ci Bi Ωi
− Ci [Bi + Bi Ei ( X , ξ , t )] f i (t ) + Ci [Bi + Bi Ei ( X , ξ , t )]U i (t ) − f i (t )
(3.33)
Equating equation (3.33) to zero gives the equivalent control, Ueq(t):
U eq (t ) = −(C i Bi + C i Bi E i ( X , ξ , t )) −1 {[C i Ai + C i Bi H i ( X , ξ , t )]Z i (t )
+ C i Bi H i ( X , ξ , t ) X di − C i Bi Ω i − [C i Bi + C i Bi E i ( X , ξ , t )] f i(t ) − f i (t )}
(3.34)
55
Noting that
[Ci Bi + Ci Bi Ei ( X , ζ , t )]−1 = [(Ci Bi )(I ni + Ei ( X , ζ , t ))]−1
= [I ni + Ei ( X , ζ , t )]−1 (Ci Bi ) −1
(3.35)
The system dynamics during sliding mode can be found by substituting the
equivalent control (3.34) into the system error dynamics (3.27).
Z& ( t ) = {[ I ni − B i ( C i B i ) −1 C i ] Ai } Z i ( t ) + B i ( C i B i ) −1 f i ( t )
(3.36)
Define
P si Δ ( I ni − Bi (C i Bi ) −1 C i )
≡
(3.37)
where Psi is a projection operator and satisfies the following two equations [10]:
C i P si = 0
and
P si Bi = 0
(3.38)
with the matching condition is satisfied (equations (3.24) and (3.25) hold),by
assuming that fi (t) as disturbance signal, equation (3.36) can be reduced as
Z& i ( t ) = {[ I ni − B i ( C i B i ) −1 ] Ai } Z i ( t )
(3.39)
Hence if the matching condition is satisfied, the system’s error dynamics
during sliding mode as described by equation (3.39) is independent of the system
uncertainties and couplings between the inputs, and, insensitive to the parameter
variations, and can be determined through a proper adjustment of the value of C,
which can be obtained through a proper selection of the desired closed loop poles
locations.
3.4.2 Sliding Mode Tracking Controller Design
In this section, a set of local sliding mode will be constructed to drive the
nonlinear plant’s state trajectory of each subsystem to the corresponding
56
surfaces σ i (t ) , so that once the state trajectory hits the surface, it remains there for
all subsequent time.
The control for each component is chosen to be
3
U i = ∑ψ ij z ij
(3.40)
j =3
Based on rules
⎧γ ij
ψ ij = ⎨
⎩β ij
if
if
σ i (t ) zij > 0
σ i (t ) zij < 0
(3.41)
To determine the value of control Ui(t) that will drive a solution trajectory, first
differentiates the local sliding mode given in equation (3.28) with respect to t. This
produces
σ& i (t ) = C i Z& i (t ) − f i (t )
(3.42)
Using the fact that
z& ij = z ij + 1
for j = 1,2
z i3 = f i + U i +
3
∑a
j =1
ij
z ij
(3.43)
By defining ci 0 = 0 produces,
3
σ& i (t ) = ∑ (cij −1 + a ij + ψ ij ) z ij
(3.44)
1
By multiplying equation (3.44) with σ i (t ) , and noting that
0.5 σ& i (t ) = σ (t )σ& (t )
2
(3.45)
we obtain
3
σ& i 2 (t ) = 2∑ (c ij − 1 + a ij + ψ ij ) z ij σ i (t )
(3.46)
j =1
In the above equation, if we choose
cij −1 + aij + γ ij < 0
or
γ ij < min (−cij −1 − aij )
(3.47)
57
cij −1 + aij + β ij > 0
or
β ij > max (−cij −1 − aij )
(3.48)
Then each term in equation (3.47) and (3.48) will be negative forcing and this
implies the motion of the state trajectories is always toward the surface.
3.6
Summary
A robust controller, based on decentralized SMC approach to provide the
tracking trajectory control of the electrohydraulic robot manipulator described
previously has been proposed in this chapter. The design task is divided into two
phase. The first phase entails the development of a corresponding controller to drive
the system dynamics to the predefined sliding surface. The second phase aims at
deriving a control signal to maintain the system dynamics on this particular surface
for all subsequent time. However, only the controller designed in the second phase is
viable in the real controller. The controller is designed based on the assumption that
the system uncertainties, the nonlinearities and the interactions between the subsystems are bounded and its values are known. It is shown mathematically that the
proposed controller does not only make the system insensitive to parameter
variations, uncertainties and couplings; but also guarantees stability in large.
58
CHAPTER 4
SIMULATION
4.1
Introduction
This chapter deals with the steps taken to carry out the simulation work in
SIMULINK, an extension to MATLAB programming tool. The software is chosen
since it offers a rich set of capabilities and resources for engineering problems [21].
The preliminary simulation results are also discussed and presented.
4.2
Trajectory Generation
The robust controller is required to make the robot track a pre-specified
reference trajectory as close as possible. A smooth continuous trajectory is normally
used in simulation works as well as in real robot applications. Therefore, the desired
joint angle is generated by using the cycloidal function, as follows [12]:
Δ 2πt
2πt
⎧
⎪θ i (0) + i [
)],
− sin(
θ di (t ) = ⎨
2π τ
τ
⎪⎩ θ i (τ ),
0 ≤ t ≤τ
τ ≤t
(4.1)
where,
Δ i = θ i ( τ) − θ i (0), i = 1, 2,3
τ = 2s
.
(4.2)
59
The joint trajectories for the three joints are set to start at the initial position
of [θ1 (0)
of [θ1 (τ )
θ2 (0) θ3 (0)]T = [−0.8
− 1.5 − 0.5]T radians, to a desired final position
θ 2 (τ ) θ3 (τ )]T = [1
0.2 1.2]T radians in time τ = 2 seconds [28].
Figure 4.1 below shows the desired joint angle trajectories for the three joints.
DESIRED JOINT ANGLE TRAJECTORY
2
1.5
Joint Angle (rad)
1
0.5
0
-0.5
-1
Joint 1
Joint 2
Joint 3
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.1: Desired Joint Position Profile
Differentiating equation (4.1) with respect to time, t gives the smooth bell
shaped desired velocity profile as shown in the Figure 4.2.
2πt
⎧Δi
⎪ [1 − cos( )],
&
θ di (t ) = ⎨ τ
τ
⎪⎩ 0
0 ≤ t ≤τ
τ ≤t
(4.3)
60
DESIRED JOINT VELOCITY TRAJECTORY
2
1.8
Joint 1
Joint 2
Joint 3
1.6
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.2: Desired Joint Velocity Profile
4.3
Simulation Study
The simulation is performed in the routine as described in the flow chart in
Figure 4.4. The control signal generated from each controller is implemented on the
robot model. Fourth order Runge Kutta method is used to solve the differential
equation representing the model formulation of the robotic system due to its
effectiveness in terms of accuracy and minimum computing time [21]. In the cases
where the output trajectory does not exhibit satisfactory results, controller
parameters can be tuned based on the tuning algorithm as described in the following
sections for each controller.
61
Figure 4.3: Simulation Flow Chart
4.4
Controller Parameters Selection
The following three values will determine the shape of the plant output in
response to the desired input trajectory.
a. The Desired Poles Location
62
The desired poles location can be placed anywhere on the left half plane
(LHP) of the s-plane to guarantee stability during the sliding phase.
However, if the locations of the desired closed-loop poles are placed too far
on the LHP of the s-plane, high gain will be produced and will somehow
affect the shape of the sliding surface of equation (3.27).
b. Sliding Surface Constant, Ci
The constant c n , in equation (3.27) will determine the magnitude of the ith
input, U i (t ) , while the constants c1 , c 2 ,L c ni −1 will determine the shape of
the trajectories during the reaching phase.
c. Sliding Mode Controller Parameters, γ ij and
β ij
The values of the controller parameters γ ij and β ij must be large enough to
accommodate for the constraint as stated in equations (3.41) and (3.42) but
not too large to avoid excessive magnitude of the control input U i (t ) .
These three parameters need to be tuned correctly since it not only affects the
magnitude of the input and shape of the trajectories during reaching phase but also
influences the shape of decentralized sliding surface.
4.5
Tuning Algorithm
Although the conditions to achieve pre-specified desired output response are
completely stated as in section 4.4.1, a specific tuning rule for the controller
parameters is required to compensate any constraints that may occur due to the
physical limitations of the elements of the system. Besides, the task of choosing the
right controller parameters to get the satisfactory tracking response can be time
consuming and in some cases very exhaustive. The conditions presented above can
be used to develop a tuning algorithm, which, for each setting of the controller
parameters, determines in a systematic way whether the output tracking performance
63
is satisfactory and at the same time assurance the control input U i (t ) stays within the
stipulated limit. The algorithm can be stated as follows:
Step 1. Input data: Numerical values for each ith subsystem Ci =[c1, c2 ...
cni ],
and
cij − 1 + a ij + γ ij < 0
or
cij − 1 + a ij + β ij > 0
Step 2. Check if the sliding mode exists and whether the output tracking
response is satisfactory. If the condition does not hold then try other
combinations. If the condition holds, proceed to Step 3.
Step 3. Check if all of the control inputs U i (t ) are within the admissible
range. If the condition does not hold then increase the value of c n ,
and place the desired poles closer to the origin until sliding mode
exist and the control input Ui(t) is within the admissible limit. If the
condition holds, then proceed to Step 4.
Step 4. Check if the output trajectories are satisfactory during the reaching
phase. If the conditions do not hold then adjust the values of c1, c2
... cni until satisfactory shape of the output trajectories are
achieved. If the conditions hold, then proceed to Step 5.
Step 5. Check if the tracking errors of the output trajectories are satisfactory.
If the conditions do not hold then increase the values of γ ij or β ij
until satisfactory tracking errors are achieved. These values should
not be too large to guarantee that the control input Ui(t) is within the
admissible limit. If the conditions hold, then go to Step 6.
Step 6. Finish.
The tuning algorithm described above not only guarantees that the desired
tracking response is achieved, but it also assures that the system control input Ui(t) is
within the permissible range of operation.
64
4.6
Simulation using Centralized Sliding Mode Controller
In this section, the proposed control law described by equation (3.40) will be
applied to the electrohydraulic robot manipulator in centralized approach. This
simulation study is meant as a comparison purpose to the proposed Decentralized
SMC controller.
4.6.1
Numerical Computation and Selection of Controller Parameters
In this section, the controller parameters C are selected to obtain the
controller parameters γ o r β . These numerical values are then substituted in
equation (3.35) and used in developing the simulation block diagram in SIMULINK.
The parameters are calculated and selected as follows:
1) Desired poles location
The desired poles location can be placed anywhere on the left half plane
(LHP) of the s-plane to guarantee stability during the sliding phase. However, in this
simulation, the poles are set as described in (4.4).
Joint 1: λ1 = {−0.5, − 0.51, − 0.52)
Joint 2: λ 2 = {−0.6, − 0.61, − 0.62)
Joint 3: λ3 = {−0.7, − 0.71, − 0.72)
(4.4)
2) C
Matrix C can be obtained from the closed-loop poles equations:
0
0
0
0
0
0 ⎤
⎡78.02 153 100
⎢
C=⎢ 0
0
0 111.62 183 100
0
0
0 ⎥⎥
⎢⎣ 0
0
0
0
0
0 151.22 213 100 ⎥⎦
(4.5)
65
3) Controller Parameters
The controller parameters for each of the centralized controller as described
by equations (3.41) and (3.42) can now be computed and the results are tabulated
below:
⎡ − 110.04e6 − 108.37e3 − 23.5⎤
γ ≥ ⎢⎢− 231.073e6 − 107.89e3 − 7.9 ⎥⎥
⎢⎣ − 240.67e6 − 139.82e3 − 27.1⎥⎦
(4.6)
⎡ 110.04e6 108.37e3 23.5⎤
β ≥ ⎢⎢231.073e6 107.89e3 7.9 ⎥⎥
⎣⎢ 240.67e6 139.82e3 27.1⎥⎦
(4.7)
For the simulation, the following controller parameters have been used
⎡ - 110.04e6 - 108.37e3 - 23.5⎤
γ = ⎢⎢- 231.073e6 - 107.89e3 - 7.9 ⎥⎥
⎢⎣ - 240.67e6 - 139.82e3 - 27.1⎥⎦
⎡ 110.04e6 108.37e3 23.5⎤
β = ⎢⎢231.073e6 107.89e3 7.9 ⎥⎥
⎢⎣ 240.67e6 139.82e3 27.1⎥⎦
(4.8)
(4.9)
The system was simulated to study the performance of the controller. The
manipulator is desired to follow the trajectory as outlined in equations (4.1) and
(4.2). The study was conducted while the robot in no load, half load and full load
condition with the pole locations as in equation (4.4), sliding surface constant as in
equation (4.5) and controller parameters as in equation (4.8)-(4.9).
The simulation results of the three joints by using the Centralized SMC are
shown in Figures 4.4 - 4.9 when the robot is operated with no load, half load and full
load. From these results, it can be observed that the joints track the desired trajectory
at all times under the control of Centralized SMC in spite of parameter variations
present in the system. In fact, the control input generated for all joints switch fast
enough (Figures 4.10 - 4.12) to ensure that all the states are directed toward the
switching surfaces (Figures 4.13 – 4.15). Hence, the system is robust and
insensitivity to uncertainties and couplings.
66
JOINT 1 TRACKING RESPONSE (Angle)
1
0.8
Joint Angle (rad)
0.6
0.4
0.2
0
-0.2
-0.4
Desired
Actual (0 kg)
Actual (5 kg)
Actual (10 kg)
-0.6
-0.8
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.4: Joint 1 tracking response using centralized SMC tracking controller
JOINT 2 TRACKING RESPONSE (Angle)
0.4
0.2
0
Joint Angle (rad)
-0.2
-0.4
-0.6
-0.8
-1
Desired
Actual (0 kg)
Actual (5 kg)
Actual (10 kg)
-1.2
-1.4
-1.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.5: Joint 2 tracking response using centralized SMC tracking controller
67
JOINT 3 TRACKING RESPONSE (Angle)
1.2
1
Joint Angle (rad)
0.8
0.6
0.4
0.2
0
Desired
Actual (0 kg)
Actual (5 kg)
Actual (10 kg)
-0.2
-0.4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.6: Joint 3 tracking response using centralized SMC tracking controller
JOINT 1 TRACKING RESPONSE (Velocity)
2
Desired
Actual (0 kg)
Actual (5 kg)
Actual (10 kg)
1.8
1.6
Joint Angle (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.7: Joint 1 tracking response using centralized SMC tracking controller
68
JOINT 2 TRACKING RESPONSE (Velocity)
1.8
Desired
Actual (0 kg)
Actual (5 kg)
Acual (10 kg)
1.6
Joint Angle (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.8: Joint 2 tracking response using centralized SMC tracking controller
JOINT 3 TRACKING RESPONSE (Velocity)
1.8
Desired
Actual (0 kg)
Actual (5 kg)
Actual (10 kg)
1.6
Joint Angle (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.9: Joint 3 tracking response using centralized SMC tracking controller
69
JOINT 1 CONTROL INPUT
100
50
0
Control Input (V)
-50
-100
-150
-200
-250
-300
-350
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.10: Joint 1 control input using centralized SMC tracking controller
JOINT 2 CONTROL INPUT
900
800
700
Control Input (V)
600
500
400
300
200
100
0
-100
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.11: Joint 2 control input using centralized SMC tracking controller
70
JOINT 3 CONTROL INPUT
400
200
Control Input (V)
0
-200
-400
-600
-800
-1000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.12: Joint 3 control input using centralized SMC tracking controller
JOINT 1 SLIDING SURFACE
1
0.8
0.6
Sliding Surface, S1
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.13: Joint 1 sliding surface using centralized SMC tracking controller
71
JOINT 2 SLIDING SURFACE
1
0.8
0.6
Sliding Surface, S2
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.14: Joint 2 control input using centralized SMC tracking controller
JOINT 3 SLIDING SURFACE
2
1.5
Sliding Surface, S3
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.15: Joint 3 control input using centralized SMC tracking controller
72
4.7
Simulation using Decentralized Sliding Mode Controller
In this section the simulation is carried out using the proposed Decentralized
SMC controller as described by equation (3.40).
4.7.1
Numerical Computation and Selection of Controller Parameters
In this section, the controller parameters Ci are selected to obtain the
controller parameters γ ij or β ij . These numerical values are then substituted in
equation (3.40) and used in developing the simulation block diagram in SIMULINK.
The parameters are calculated and selected as follows:
1) Desired poles location
The desired poles location can be placed anywhere on the left half plane
(LHP) of the s-plane to guarantee stability during the sliding phase. However, in this
simulation, the poles are set as described in (4.10).
Joint 1: λ1 = {−0.5, − 0.51, − 0.52)
Joint 2: λ 2 = {−0.6, − 0.61, − 0.62)
Joint 3: λ3 = {−0.7, − 0.71, − 0.72)
(4.10)
2) Ci
Matrix Ci can be obtained from the closed-loop poles equations:
Subsystem 1: C1 = [78.02 153 100]
Subsystem 2: C 2 = [111.62 185 100]
Subsystem 3: C 3 = [151 .22 213 100 ]
(4.11)
73
3) Controller Parameters
The controller parameters for each of the decentralized controller as
described by equations (3.41) and (3.42) can now be computed and the results are
tabulated below:
Controller 1:
γ 11 ≥ −110.04e6, γ 12 ≥ −108.37e3, γ 13 ≥ −23.5
β11 ≥ 110.04e6, β12 ≥ 108.37e3, β13 ≥ 23.5
Controller 2:
γ 21 ≥ −231.073e6, γ 22 ≥ −107.89e3, γ 23 ≥ −7.9
β 21 ≥ 231.073e6, β 22 ≥ 107.89e3, β 23 ≥ 7.9
Controller 3:
γ 31 ≥ −240.67e6, γ 32 ≥ −139.82e3, γ 33 ≥ −27.1
β 31 ≥ 240.67e6, β 32 ≥ 139.82e3, β 33 ≥ 27.1
(4.12)
The system was simulated to study the performance of the controller. The
manipulator is desired to follow the trajectory as outlined in equations (4.1) and
(4.2). The study was conducted while the robot in no load, half load and full load
condition with the pole locations as in equation (4.10), sliding surface constant as in
equation (4.11) and controller parameters as in equation (4.12).
The simulation results of the three joints using Decentralized SMC are
shown in Figures 4.16 - 4.21 when the robot is operated with no load. From these
results, it can be observed that the joints track the desired trajectory at all times
under the control of Decentralized SMC. The results prove that each decentralized
controller is capable to render that the system is robust against parameter variations
present in the system although only the local states of the subsystem are used. The
control input generated for all subsystems switch fast enough (Figures 4.22 - 4.24)
due to the fact that all the states are directed toward the linear sliding surfaces
(Figures 4.25 – 4.27). Hence, the system is robust and insensitivity to uncertainties
and couplings.
74
JOINT 1 TACKING RESPONSE (Angle, 0 kg)
1
0.8
Joint Angle (rad)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Desired
Actual
-0.8
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.16: Joint 1 tracking response using decentralized SMC tracking controller
JOINT 2 TACKING RESPONSE (Angle, 0 kg)
0.4
0.2
0
Joint Angle (rad)
-0.2
-0.4
-0.6
-0.8
-1
-1.2
Desired
Actual
-1.4
-1.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.17: Joint 1 tracking response using decentralized SMC tracking controller
75
JOINT 3 TACKING RESPONSE (Angle, 0 kg)
1.2
1
Joint Angle (rad)
0.8
0.6
0.4
0.2
0
-0.2
Desired
Actual
-0.4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.18: Joint 3 tracking response using decentralized SMC tracking controller
JOINT 1 TRACKING RESPONSE (Velocity, 0 kg)
2
Desired
Actual
1.8
1.6
Joint Angle (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.19: Joint 1 tracking response using decentralized SMC tracking controller
76
JOINT 2 TRACKING RESPONSE (Velocity, 0 kg)
1.8
Desired
Actual
1.6
Joint Angle (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.20: Joint 2 tracking response using decentralized SMC tracking controller
JOINT 3 TRACKING RESPONSE (Velocity, 0 kg)
1.8
Desired
Actual
1.6
Joint Angle (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.21: Joint 3 tracking response using decentralized SMC tracking controller
77
JOINT 1 CONTROL INPUT
100
50
0
Control Input (V)
-50
-100
-150
-200
-250
-300
-350
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.22: Joint 1 control input using decentralized SMC tracking controller
JOINT 2 CONTROL INPUT
900
800
700
Control Input (V)
600
500
400
300
200
100
0
-100
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.23: Joint 2 control input using decentralized SMC tracking controller
78
JOINT 3 CONTROL INPUT
400
200
Control Input (V)
0
-200
-400
-600
-800
-1000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.24: Joint 3 control input using decentralized SMC tracking controller
JOINT 1 SLIDING SURFACE
1
0.8
0.6
Sliding Surface, s1
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.25: Joint 1 sliding surface using decentralized SMC tracking controller
79
JOINT 2 SLIDING SURFACE
1
0.8
0.6
Sliding Surface, s2
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.26: Joint 2 sliding surface using decentralized SMC tracking controller
JOINT 3 SLIDING SURFACE
2
1.5
Sliding Surface, s3
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.27: Joint 3 sliding surface using decentralized SMC tracking controller
80
4.7.2
Comparison of the Centralized SMC & Decentralized SMC
In this section, the Centralized SMC and the Decentralized SMC tracking
controllers are applied to the 3 DOF electrohydraulic robot manipulator for
comparison purposes.
Figures 4.28 – 4.39 show the tracking errors of each joint under the two
above-mentioned controllers for no load and full load condition. Among the two
control methods, the simulation results show that the performance of the Centralized
SMC tracking controller is slightly better than the Decentralized SMC method.
This is due to the fact that the Centralized SMC tracking controller was developed
based on the complete robot information and all states at all joints are directly used
to compute the controls.
On the other hand, the Decentralized SMC tracking
controller exploits only the local states for each subsystem and disregarded the
information from other subsystems for the control law. Hence the Decentralized
controller does not compensate for the interconnection between the subsystems.
However, the difference error between Centralized SMC and Decentralized SMC is
very small and may be negligible. Thus, the decentralized SMC is capable to cater
the manipulator’s nonlinearities, parameter variations and coupled effects
comparable to the Centralized SMC.
81
-4
2
JOINT 1 TRACKING ERROR (Angle, 0 kg)
x 10
Centralized
Decentralized
Joint Angle (rad)
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.28: Joint 1 tracking error using Centralized and Decentralized SMC
-4
2
JOINT 2 TRACKING ERROR (Angle, 0 kg)
x 10
Centralized
Decentralized
Joint Angle (rad)
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.29: Joint 2 tracking error using Centralized and Decentralized SMC
82
-5
5
JOINT 3 TRACKING ERROR (Angle, 0 kg)
x 10
Centralized
Decentralized
4
3
Joint Angle (rad)
2
1
0
-1
-2
-3
-4
-5
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.30: Joint 3 tracking error using Centralized and Decentralized SMC
-3
2
JOINT 1 TRACKING ERROR (Velocity, 0 kg)
x 10
Centralized
Decentralized
1.5
Joint Velocity (rad/s)
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.31: Joint 1 tracking error using Centralized and Decentralized SMC
83
JOINT 2 TRACKING ERROR (Velocity, 0 kg)
0.2
Centralized
Decentralized
0.15
Joint Velocity (rad/s)
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.32: Joint 2 tracking error using Centralized and Decentralized SMC
JOINT 3 TRACKING ERROR (Velocity, 0 kg)
0.05
Centralized
Decentralized
0.04
0.03
Joint Velocity (rad/s)
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.33: Joint 3 tracking error using Centralized and Decentralized SMC
84
-4
2
JOINT 1 TRACKING ERROR (Angle, 10 kg)
x 10
Centralized
Decentralized
Joint Angle (rad/s)
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.34: Joint 1 tracking error using Centralized and Decentralized SMC
-4
4
JOINT 2 TRACKING ERROR (Angle, 0 kg)
x 10
Centralized
Decentralized
3
Joint Angle (rad/s)
2
1
0
-1
-2
-3
-4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.35: Joint 2 tracking error using Centralized and Decentralized SMC
85
-4
4
JOINT 3 TRACKING ERROR (Angle, 10 kg)
x 10
3
Joint Angle (rad/s)
2
1
0
-1
-2
-3
-4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.36: Joint 3 tracking error using Centralized and Decentralized SMC
-3
4
JOINT 1 TRACKING ERROR (Velocity, 10 kg)
x 10
Centralized
Decentralized
3
Joint Angle (rad/s)
2
1
0
-1
-2
-3
-4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.37: Joint 1 tracking error using Centralized and Decentralized SMC
86
JOINT 2 TRACKING ERROR (Velocity, 10 kg)
0.25
Centralized
Decentralized
0.2
0.15
Joint Angle (rad/s)
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.38: Joint 2 tracking error using Centralized and Decentralized SMC
JOINT 3 TRACKING ERROR (Velocity, 10 kg)
0.04
Centralized
Decentralized
0.03
Joint Angle (rad/s)
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.39: Joint 3 tracking error using Centralized and Decentralized SMC
87
4.7.3 Effect of Load Variation
In the following simulations, the effect of payload mass variation on the
system using Decentralized tracking controller is studied. The simulation is repeated
with the same controller parameters and simulation settings, but this time, the mass
of the payload is varied between no load, half load and full load.
The tracking errors shown in Figures 4.40-4.45 reveal that the system
performance maintains as during the robot operating with no load, half load and full
load condition. This proves that Decentralized SMC has successfully rendered the
system to track the specified trajectory even though the mass of the load is varied.
Thus, it can be deduced that the Decentralized SMC does not only efficiently
compensates the hydraulic robot’s parameter variations, nonlinearities, coupled
effects, but also is robust against its uncertainty (payload mass).
-4
2
x 10
JOINT 1 TRACKING ERROR WITH PAYLOAD MASS VARIATION (Angle)
0 kg
5 kg
10 kg
Joint Angle (rad)
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
Figure 4.40: Joint 1 tracking error with mass variation
2
88
4
-4
x 10 JOINT 2 TRACKING ERROR WITH PAYLOAD MASS VARIATION (Angle)
0 kg
5 kg
10 kg
3
Joint Angle (rad)
2
1
0
-1
-2
-3
-4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.41: Joint 2 tracking error with mass variation
-4
2
x 10
JOINT 3 TRACKING ERROR WITH PAYLOAD MASS VARIATION (Angle)
0 kg
5 kg
10 kg
1.5
1
Joint Angle (rad)
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
Figure 4.42: Joint 3 tracking error with mass variation
2
89
-3
1
x 10
JOINT 1 TRACKING ERROR WITH PAYLOAD MASS VARIATION (Velocity)
0 kg
5 kg
10 kg
0.8
0.6
Joint Velocity (rad/s)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.43: Joint 1 tracking error with mass variation
JOINT 2 TRACKING ERROR WITH PAYLOAD MASS VARIATION (Velocity)
0.25
0 kg
5 kg
10 kg
0.2
0.15
Joint Velocity (rad/s)
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
Figure 4.44: Joint 2 tracking error with mass variation
2
90
JOINT 3 TRACKING ERROR WITH PAYLOAD MASS VARIATION (Velocity)
0.04
0 kg
5 kg
10 kg
0.03
Joint Velocity (rad/s)
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.45: Joint 3 tracking error with mass variation
4.7.4 Effect of Controller Parameter
This section studies the effect of varying the decentralized controller
parameters. The simulation is conducted with the robot in no load condition, using
the same simulation settings, except that the controller parameters are modified to
dissatisfy the requirement of equations (3.47) – (3.48) or equation (4.12). The values
of the controller parameter chosen are:
γ 1 = [- 110.04e3 - 108.37 23.5]
γ 2 = [- 231.073e3 - 107.89 22.9]
γ 3 = [- 240.67e3 - 139.82 77.1]
β 1 = [110.04e3 108.37 - 23.5]
β 2 = [231.073e3 107.89 - 22.9]
β 3 = [240.67e3 139.82 - 27.1]
(4.13)
(4.14)
91
The tracking response for each joint is demonstrated in Figures 4.46 – 4.51
respectively, when the robot handles no load with the unsatisfied controller
parameters. The results show that the controller fails to track the desired positions in
the case where the controller parameters do not fulfill the requirement of equations
(3.47) – (3.48). Figures 4.52 – 4.57 illustrate the joints control input and sliding
surfaces respectively.
The simulations confirms the theory that the will not be negative forcing and
this implies the motion of the state trajectories is not toward the surface if the
controller requirements (equations (3.47) – (3.48)) are not fulfilled. Thus sliding
mode does not occur in this case causing the control inputs unable to switch fast
enough to provide the necessary tracking trajectory control.
JOINT 1 TRACKING RESPONSE WITH UNSATISFIED CONTROLLER PARAMETER (Angle)
1
0.8
Joint Angle (rad)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Desired
Actual
-0.8
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.46: Joint 1 tracking response with unsatisfied controller parameter
92
JOINT 2 TRACKING RESPONSE WITH UNSATISFIED CONTROLLER PARAMETER (Angle)
0.4
0.2
0
Joint Angle (rad)
-0.2
-0.4
-0.6
-0.8
-1
-1.2
Desired
Actual
-1.4
-1.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.47: Joint 2 tracking response with unsatisfied controller parameter
JOINT 3 TRACKING RESPONSE WITH UNSATISFIED CONTROLLER PARAMETER (Angle)
1.4
1.2
1
Joint Angle (rad)
0.8
0.6
0.4
0.2
0
-0.2
Desired
Actual
-0.4
-0.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.48: Joint 3 tracking response with unsatisfied controller parameter
93
JOINT 1 TRACKING RESPONSE WITH UNSATISFIED CONTROLLER PARAMETER (Velocity)
2.5
Desired
Actual
Joint Velocity (rad/s)
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Time (s)
1.2
1.4
1.6
1.8
2
Figure 4.49: Joint 1 tracking response with unsatisfied controller parameter
JOINT 2 TRACKING RESPONSE WITH UNSATISFIED CONTROLLER PARAMETER (Velocity)
2
Desired
1.8
Actual
1.6
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Time (s)
1.2
1.4
1.6
1.8
2
Figure 4.50: Joint 2 tracking response with unsatisfied controller parameter
94
JOINT 3 TRACKING RESPONSE WITH UNSATISFIED CONTROLLER PARAMETER (Velocity)
2
Desired
1.8
Actual
1.6
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Time (s)
1.2
1.4
1.6
1.8
2
Figure 4.51: Joint 3 tracking response with unsatisfied controller parameter
JOINT 1 CONTROL INPUT WITH UNSATISFIED CONTROLLER PARAMETER
100
50
0
Control Input (V)
-50
-100
-150
-200
-250
-300
-350
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.52: Joint 1 control input with unsatisfied controller parameter
95
JOINT 2 CONTROL INPUT WITH UNSATISFIED CONTROLLER PARAMETER
900
800
700
Control Input (V)
600
500
400
300
200
100
0
-100
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.53: Joint 2 control input with unsatisfied controller parameter
JOINT 3 CONTROL INPUT WITH UNSATISFIED CONTROLLER PARAMETER
400
200
Control Input (V)
0
-200
-400
-600
-800
-1000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.54: Joint 3 control input with unsatisfied controller parameter
96
JOINT 1 SLIDING SURFACE WITH UNSATISFIED CONTROLLER PARAMETER
1
0.8
0.6
Sliding Surface, s1
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.55: Joint 1 sliding surface with unsatisfied controller parameter
JOINT 2 SLIDING SURFACE WITH UNSATISFIED CONTROLLER PARAMETER
1
0.8
0.6
Sliding Surface, s1
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.56: Joint 2 sliding surface with unsatisfied controller parameter
97
JOINT 3 SLIDING SURFACE WITH UNSATISFIED CONTROLLER PARAMETER
2
1.5
Sliding Surface, s3
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.57: Joint 3 sliding surface with unsatisfied controller parameter
4.7.5
Effect of Sliding Surface Constant, C
In this set of simulations, the effect of different sliding surfaces for the
decentralized controller performance is studied. The simulation is conducted with
the robot in full load (10 kg) condition with satisfied controller parameters values.
Four sets of constant matrix C of the PI sliding Surfaces as described below are
considered in this study;
Set 1
C1 = [0.7802 1.53 1], C 2 = [1.1162 1.85 1], C 3 = [1.5122
2.13 1]
(4.15)
C1 = [7.802 15.3 10 ], C 2 = [11.162 18.5 10 ], C 3 = [15.122 21.3 1 0]
(4.16)
Set 2
98
Set 3
C1 = [78.02 153 100 ], C 2 = [111 .62 185 100 ], C 3 = [151 .22 213 1 00 ]
(4.17)
Set 4
C1 = [780 .2 1530 1000 ], C 2 = [1116 .2 1850 1000 ], C 3 = [1512 .2 2130 1000 ]
(4.18)
The third element of the sliding surface constant matrix Ci, is set to unity
since it will only affect the magnitude of the respective joint control input while the
first two elements Ci are set arbitrarily.
Figures 4.58 – 4.63 illustrate the tracking error for each joint, respectively,
for all of the above cases, indicated that the shape of the joints trajectory are greatly
influenced by the sliding surface constant, matrix Ci. Therefore, besides adjusting
the satisfied value of the controller parameter, proper selection of the elements Ci are
also important in forcing the manipulator to track the pre-specified trajectory.
99
-4
2
x 10 JOINT 1 TRACKING ERROR WITH DIFFERENT SLIDING SURFACES (Angle)
C1
C2
C3
C4
Joint Angle (rad)
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.58: Joint 1 tracking error with different sliding surfaces
JOINT 2 TRACKING ERROR WITH DIFFERENT SLIDING SURFACES (Angle)
0.01
0
-0.01
Joint Angle (rad)
-0.02
-0.03
-0.04
-0.05
-0.06
C1
C2
C3
C4
-0.07
-0.08
-0.09
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.59: Joint 2 tracking error with different sliding surfaces
100
JOINT 3 TRACKING ERROR WITH DIFFERENT SLIDING SURFACES (Angle)
0.01
0
-0.01
Joint Angle (rad)
-0.02
-0.03
-0.04
-0.05
C1
C2
C3
C4
-0.06
-0.07
-0.08
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.60: Joint 3 tracking error with different sliding surfaces
-3
2
x 10 JOINT 1 TRACKING ERROR WITH DIFFERENT SLIDING SURFACES (Velocity)
0
Joint Angle (rad/s)
-2
-4
-6
-8
-10
C1
C2
C3
C4
-12
-14
0
0.2
0.4
0.6
0.8
1
Time (s)
1.2
1.4
1.6
1.8
Figure 4.61: Joint 1 tracking error with different sliding surfaces
2
101
Joint Angle (rad/s)
JOINT 2 TRACKING ERROR WITH DIFFERENT SLIDING SURFACES (Velocity)
1
C1
C2
C3
0.5
C4
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.62: Joint 2 tracking error with different sliding surfaces
Joint Angle (rad/s)
JOINT 3 TRACKING ERROR WITH DIFFERENT SLIDING SURFACES (Velocity)
0.6
C1
0.4
C2
C3
C4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.63: Joint 3 tracking error with different sliding surfaces
102
4.7.6
Effect of Sampling Time
This section studies the effect of different sampling intervals on the tracking
performance of the decentralized control system. The simulation is done with the
robot in full load (10 kg) condition with satisfied controller parameters values.
Three set of sampling times are considered in this case, that is 0.01s, 0.001s and
0.0001s.
The tracking response for each joint is shown in Figures 4.64 - 4.69
respectively. From the simulation results, it can be observed that the system fails to
track the desired trajectory when the sampling interval 0.01s and 0.001s is used.
The relatively large step size or sampling interval causes inadequate data being
captured since the dynamics of the robot changes rapidly. Many data are loss in
between the sampling times, as a result leading to imprecise calculation of the
Runge Kutta numerical algorithm in simulating the system.
The results also reveal that by using 0.0001s sampling interval, the tracking
responses overlap with the desired trajectory for all the joints. In this case, more
data are being captured rapidly since the step size is smaller, thus providing fast and
enough information for the Runge Kutta numerical algorithm to compute the robot
dynamics precisely. In general, the smaller time interval will result in a more
accurate simulation. On the other hand, if the time interval is taken to be too small,
a longer computation time is required.
103
JOINT 1 TRACKING RESPONSE WITH DIFFERENT SAMPLING TIME (Angle)
1
0.8
Joint Angle (rad)
0.6
0.4
0.2
0
-0.2
-0.4
Desired
0.01s
0.001s
0.0001s
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.64: Joint 1 tracking response with different sampling time
JOINT 2 TRACKING RESPONSE WITH DIFFERENT SAMPLING TIME (Angle)
0.4
0.2
0
Joint Angle (rad)
-0.2
-0.4
-0.6
-0.8
-1
Desired
0.01s
0.001s
0.0001s
-1.2
-1.4
-1.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.65: Joint 2 tracking response with different sampling time
104
JOINT 3 TRACKING RESPONSE WITH DIFFERENT SAMPLING TIME (Angle)
1.4
1.2
1
Joint Angle (rad)
0.8
0.6
0.4
0.2
0
Desired
0.01s
0.001s
0.0001s
-0.2
-0.4
-0.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.66: Joint 3 tracking response with different sampling time
Joint Velocity (rad/s)
JOINT 1 TRACKING RESPONSE WITH DIFFERENT SAMPLING TIME (Velocity)
2.5
Desired
0.01s
0.001s
2
0.0001s
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.67: Joint 1 tracking response with different sampling time
105
JOINT 2 TRACKING RESPONSE WITH DIFFERENT SAMPLING TIME (Velocity)
2
Desired
1.8
0.01s
0.001s
1.6
0.0001s
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.68: Joint 2 tracking response with different sampling time
JOINT 3 TRACKING RESPONSE WITH DIFFERENT SAMPLING TIME (Velocity)
2
Desired
1.8
0.01s
0.001s
1.6
0.0001s
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.69: Joint 3 tracking response with different sampling time
106
4.7.7 Effect of Closed-loop Poles
In this section, the effect of the variation in the desired closed-loop poles on
the tracking performance of the decentralized control system is studied.
The
simulation is done with the robot in full load (10 kg) condition. Three set of desired
closed-loop poles are considered in this case, that is;
Set 1
λ1 = {−0.5, − 0.51, − 0.52}, λ 2 = {− 0.6,−0.61,−0.62}, λ3 = {− 0.7,−0.71,−0.72}
(4.19)
Set 2
λ1 = {−0.3 + j 0.6, − 0.3 − j 0.6, − 0.4}
λ2 = {−0.25 + j 0.5, − 0.25 − j 0.5, − 0.35}
λ3 = {−0.35 + j 0.7, − 0.35 − j 0.7, − 0.45}
(4.20)
Set 3
λ1 = {−1, − 1.2, − 1.4}, λ 2 = {−2, − 2.2, − 2.4}, λ3 = {−3, − 3.2, − 3.4}
(4.21)
Figures 4.70 - 4.75 demonstrate the tracking response for each joint
respectively for all of the above cases, indicated that the shapes of the joints
trajectory are greatly influenced by the desired closed-loop poles. From the results
obtained, it can be observed that the system fails to track the desired trajectory at the
beginning for Set 2 and Set 3. The desired closed-loop poles location can be placed
anywhere on the left half plane of the s-plane to guarantee stability during the
sliding phase. However, the locations situated too far on the left half plane of the splane will somehow affect the time taken for the system to tracks the pre-specified
trajectory. In addition, if the complex closed-loop poles were chosen, it will
somehow affect the system to be oscillating at the beginning before it tracks the
desired trajectory.
107
JOINT 1 TRACKING RESPONSE WITH DIFFERENT CLOSED-LOOP POLES (Angle)
1
0.8
Joint Angle (rad)
0.6
0.4
0.2
0
-0.2
-0.4
Desired
Set 1
Set 2
Set 3
-0.6
-0.8
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.70: Joint 1 tracking response with different closed-loop poles
JOINT 2 TRACKING RESPONSE WITH DIFFERENT CLOSED-LOOP POLES (Angle)
0.4
0.2
0
Joint Angle (rad)
-0.2
-0.4
-0.6
-0.8
-1
Desired
Set 1
Set 2
Set 3
-1.2
-1.4
-1.6
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.71: Joint 2 tracking response with different closed-loop poles
108
JOINT 3 TRACKING RESPONSE WITH DIFFERENT CLOSED-LOOP POLES (Angle)
1.2
1
Joint Angle (rad)
0.8
0.6
0.4
0.2
0
Desired
Set 1
Set 2
Set 3
-0.2
-0.4
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.72: Joint 3 tracking response with different closed-loop poles
JOINT 1 TRACKING RESPONSE WITH DIFFERENT CLOSED-LOOP POLES (Velocity)
2
Desired
1.8
Set 1
Set 2
1.6
Set 3
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.73: Joint 1 tracking response with different closed-loop poles
109
JOINT 2 TRACKING RESPONSE WITH DIFFERENT CLOSED-LOOP POLES (Velocity)
2
Desired
1.8
Set 1
Set 2
1.6
Set 3
Joint Velocity (rad/s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.74: Joint 2 tracking response with different closed-loop poles
Joint Velocity (rad/s)
JOINT 3 TRACKING RESPONSE WITH DIFFERENT CLOSED-LOOP POLES (Velocity)
1.8
Desired
1.6
Set 1
Set 2
Set 3
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.75: Joint 3 tracking response with different closed-loop poles
110
4.7.8
Control Input Chattering Suppression
In a practical SMC system, there is always the presence of time delays for
control computation, added with the switching frequency of physical actuators. In
other word, the switching operation as described in equation (3.41) can not be
infinite. Therefore, the trajectory of the system never lies on the switching surface
but they lie within the neighboring region of the surface. This non ideal
characteristic amplifies the emergence of chattering.
In this section, a control input chattering suppression simulation performed
on electohydraulic robot manipulator is presented. The simulation procedure is
similar to those described in Section 4.7.1, with the solver type is Ode14x
(extrapolation) as follows;
Table 4.1: Order of Extrapolation for Chattering Suppression
Extrapolation Order
Set 1
Order 2
Set 2
Order 3
Set 3
Order 4
The simulation results by using solver Ode14x (extrapolation) type are
shown in figures 4.76 – 4.84 when the robot is operated in full load condition.
Figure 4.76 – 4.78 shows the control input of each joint, is employed in the control
respectively, as for Set 1, Set 2 and Set 3. It can be observed that the control input
for Set 3 has completely eliminated the chattering effect. This shows that the
chattering effect can be successfully suppressed by using this solver type, with
proper selection order of integration. The tracking errors as shown in figures 4.79 –
4.84 reveal that the continuous control inputs maintain an accurate tracking result for
all the joints.
111
CONTROL INPUT USING SOLVER SET 1
700
600
Control Input (V)
500
400
300
200
100
Joint 1
Joint 2
Joint 3
0
-100
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.76: Control input for each joint using Solver Ode14x Set 1
CONTROL INPUT USING SOLVER SET 2
700
600
Control Input (V)
500
400
300
200
100
Joint 1
Joint 2
Joint 3
0
-100
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.77: Control input for each joint using Solver Ode14x Set 2
112
CONTROL INPUT USING SOLVER SET 3
700
600
Control Input (V)
500
400
300
200
100
Joint 1
Joint 2
Joint 3
0
-100
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.78: Control input for each joint using Solver Ode14x Set 3
1
-5
x 10 JOINT 1 TRACKING ERROR USING SOLVER ODE14x (Angle)
Set 1
Set 2
Set 3
0.8
0.6
Joint Angle (rad)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.79: Joint 1 tracking error using variation order of Solver Ode14x
113
1
-5
x 10 JOINT 2 TRACKING ERROR USING SOLVER ODE14x (Angle)
Set 1
Set 2
Set 3
0.8
0.6
Joint Angle (rad)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.80: Joint 2 tracking error using variation order of Solver Ode14x
1
-5
x 10 JOINT 3 TRACKING ERROR USING SOLVER ODE14x (Angle)
Set 1
Set 2
Set 3
0.8
0.6
Joint Angle (rad)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.81: Joint 3 tracking error using variation order of Solver Ode14x
114
-3
2
x 10
JOINT 1 TRACKING ERROR USING SOLVER ODE14x (Velocity)
Set 1
Set 2
Set 3
1.5
Joint Angle (rad/s)
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.82: Joint 1 tracking error using variation order of Solver Ode14x
JOINT 2 TRACKING ERROR USING SOLVER ODE14x (Velocity)
0.02
Set 1
Set 2
Set 3
0.015
Joint Angle (rad/s)
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.83: Joint 2 tracking error using variation order of Solver Ode14x
115
-3
5
x 10
JOINT 3 TRACKING ERROR USING SOLVER ODE14x (Velocity)
4
3
Joint Angle (rad/s)
2
1
0
-1
-2
-3
Set 1
Set 2
Set 3
-4
-5
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 4.84: Joint 3 tracking error using variation order of Solver Ode14x
4.8
Summary
Decentralized SMC tracking controller has been proposed for the tracking
trajectory control of electrohydraulically driven revolute robot manipulators. The
controller is designed based on the assumption that the system uncertainties, the
nonlinearities and the interactions between the sub-systems are bounded and its
values are known. Simulation studies on the performance of the proposed
decentralized controller constructed in Chapter 3 is presented in this chapter. The
simulation begins with a pre-specified desired trajectory for each of the manipulator
joints. From the simulation results, it can be concluded that the proposed
decentralized SMC controller with suitable controller parameters, will successfully
overcome the nonlinearities, uncertainties and couplings present in the plant and
thus provide the necessary tracking control of the system. In order to investigate or
116
verify the robustness of the controllers in future, the payload mass is varied between
0kg to 10kg. The developed decentralized SMC controller is compared to
centralized SMC technique in terms of effectiveness and robustness of each
controller. In term of design effort, the decentralized SMC controller is simpler than
the centralized SMC controller and requires less on-line computation time due to
smaller matrices dimension involved, at the expense of less tracking accuracy as
compared to the centralized SMC approach.
117
CHAPTER 5
CONCLUSION
5.1
Conclusion
In this thesis, the decentralized sliding mode control strategy has been
implemented to the electrohydraulically driven revolute robot manipulator system. A
3 DOF electrohydraulic robot manipulator model is used to investigate the tracking
trajectory performance under the proposed controller. A mathematical modeling of a
3 DOF electrohyraulic robot manipulator has been presented in this study.
The system model developed represents a more realistic representation of the
electrohydraulic robot manipulator system due to both of the manipulator link
dynamics and the hydraulic actuator dynamics taken into consideration.
Furthermore, the dynamic equations of the hydraulic actuator in this study comprise
most of the parameters present in the hydraulic actuator. From the model, it clearly
can be observed that the plant is highly nonlinear, contains uncertainty parameters
and suffers from coupling effects. Hence, an appropriate control technique is needed
to overcome this problem.
Based on variable structure control theory, a decentralized sliding mode
control (SMC) has been developed as described in [23]. This controller is utilized
for the tracking control of the robot arm. The aim of this controller is to provide a
tracking control for the arm so that it follows a pre-specified trajectory as closely as
118
possible in spite of the nonlinearity, uncertainties and coupling effects present in the
system. Unlike centralized Sliding Mode Control (SMC), the proposed controller
avoids the complexity of the controller structure and reduces computational cost. As
a result, the proposed controller is more effective in terms of computational or
economic point of view.
The performance of the proposed controller mathematical formulation is
verified by computer simulations. In this analysis, centralized Sliding Mode Control
(SMC) is used as a comparison. The simulation results demonstrated that the
decentralized Sliding Mode Control is successful, effective and capable in
compensating the manipulator’s inertia, gravitational, coriolis and centrifugal forces
and hydraulic actuator dynamics with zero steady state error, and robust against
varying payload comparable to the centralized Sliding Mode Control.
5.2
Suggestions for Future Work
In this project, the performance of the proposed controller is verified by
computer simulations alone. Although it performs very satisfactorily during the
simulation studies on digital computer, implementation of the control algorithms on
a real hardware industrial robot is the way where the research may proceed. The
performance of the proposed robot controller under real practical environment can
be investigated by experimental studies.
Furthermore, new insights into the
problems may also be improved beside the verification of the theoretical results. It
should be noted that there are issues that can only be studied through experimental
studies.
Although the integrated dynamic model of the robot manipulator comprise
most of the parameters exist in the linkage and hydraulic actuator, it is actually not
representing a complete model of the robotic system. This is so since the drive
system nonlinearities such as Coulomb friction and various sources of flexibility
such as defections of the link under load and vibrations are not incorporated in the
119
formulation of the integrated dynamic model. Thus, further study on the
performance and practicality of the proposed controller to deal with these
nonlinearities and uncertainties is required.
120
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