i
MODELLING OF SINGLE LINK FLEXIBLE MANIPULATOR
WITH FLEXIBLE JOINT
NOORFADZLI BIN ABDUL RAZAK
A project report submitted in partial fullfilment of the
requirements for the award of the degree of
Master of Engineering (Electrical - Mechatronic And Automatic Control)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
MAY 2007
iii
To my beloved family Razak , Habibah , Badrol , Haszni , Zila , Fizi , Fiqah , Haziq ,
Afifah and special person Ayu . Thank for all your support .
iv
ACKNOWLEDGEMENT
Alhamdullilah, thank to Allah, because of Him we are still here, breathing
His air, pleasuring His entire gift in this world. And most of all, for giving me
opportunities to learn His knowledge.
This work was supervised by Dr. Zaharuddin Mohamed at the Universiti
Teknologi Malaysia. I greatly appreciate all his help and guidance.
I am indebted to my parents, Abdul Razak Ahmad and Habibah Shaari, my
brother and sister, Noorazli Badrol, Noorazila and Noorhafizi, without whose help,
encouragement and patience I would never have gotten this thesis completed and
who made it all worthwhile.
I would also like to thank my fiancée, Noor Srirosayu, for her love, support
and encouragement and other help throughout. I am also grateful to my brilliant
friends, Takiyuddin , Buddiman , Prem , Rahimah and Norlela, who also gave me a
great deal of support and encouragement.
Finally, thank you to all the other people who have supported me during the
course of this work. Thank You! Thank You!
v
ABSTRACT
This paper presents a systematic approach to dynamic modelling of a single
link flexible manipulator with flexible joint for the case in which the link is oriented
vertically. Flexibility is attained by attaching the link to the motor shaft using a pair
of springs. The systems has two degrees of freedom, corresponding to the rotation of
the motor shaft with respect to a coordinate frame fixed to the base, and the rotation
of the flexible joint to which the link is attached with respect to the motor. The
output of the system is the tip angle, which is given the sum of the motor angle, θ
and the joint deflection α with respect to the motor shaft. A dynamic model of the
system is developed based on Lagrange’s equations of motion. The presence of the
gravity is accounted in this model which introduces a non-linearity into the system in
the form of a sinusoid, as a result of the potential energy due to gravity. The nonlinear model of the dynamics will be linearized by a reasonable assumption. The
resulting generalized model is validated through computer simulations and the
results will be validate with the existing results. The investigations on the dynamic
model in terms of time and frequency responses also are carried out. Besides, a
design technique for a vibration suppression of the system is also presented.
vi
ABSTRAK
Kertas ini mempersembahkan penyampaian sistematik untuk model dinamik
pengolah fleksibel satu lengan dengan sendi fleksibel di mana untuk kes ini lengan
berada di kedudukan menegak. Fleksibel diperoleh dengan menghubungkan lengan
kepada aci motor menggunakan sepasang spring. Sistem ini mempunyai dua darjah
kebebasan bersesuaian kepada putaran aci motor merujuk kepada rangka koordinat
bertempat kepada tapak dan putaran sendi fleksibel di mana lengan dihubungkan
merujuk kepada motor. Keluaran sistem adalah sudut hujung lengan di mana ianya
diberikan oleh jumlah sudut motor, θ dan penyimpangan sendi α merujuk kepada aci
motor. Model dinamik sistem ini dibangunkan berdasarkan persamaan pergerakan
Lagrange. Kewujudan graviti juga diambil kira di dalam model ini di mana ianya
menyebabkan ketidak linearan pada sistem dalam bentuk sinus akibat tenaga
keupayaan merujuk kepada graviti. Ketidak linearan dinamik model akan dilinearkan
dengan andaian yang munasabah. Keputusan menyeluruh model disahkan melalui
simulasi komputer dan keputusan tersebut akan dibandingkan dengan keputusan
sedia ada. Penyelidikan terhadap dinamik model dalam sambutan masa dan frekuensi
juga dilaksanakan . Selain itu , teknik rekaan untuk penyingkiran getaran untuk
sistem juga dipersembahkan.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xiv
LIST OF APPENDICES
xvi
INTRODUCTION
1.1
Introduction
1
1.2
Objective
4
1.3
Scope of Work
4
1.4
Outline of the Thesis
6
viii
2
LITERATURE REVIEW
2.1
Introduction
2.2
Overview of the Previous Researches About
Flexible Joint Manipulators
2.3
4
5
7
Overview on the Previous Researches About
Modelling Of Flexible Joint Manipulators
3
7
17
METHODOLOGY
3.1
Introduction
27
3.2
Development of the Dynamic Model
28
3.3
Dynamic Model
41
3.4
Conclusion
42
RESULTS AND DISCUSSIONS
4.1
Introduction
43
4.2
Time Responses Result
43
4.3
Simulation Result
48
4.4
Simulation Results with Compensator
52
4.5
Conclusion
61
CONCLUSION
5.1
Introduction
62
5.2
Conclusion
62
5.3
Future Development
64
ix
REFERENCES
65
APPENDICES
68
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
3.1
Plant parameters
32
4.1
Comparison of the system response with
and without compensator
60
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
Flexible joint
2
1.2
Flexible links
3
1.3
Flowchart represents the scope of work
5
2.1
Two revolute robot parameter
8
2.2
Motor and drive system model
8
2.3
Single link flexible manipulator with elastic joint
9
2.4
Single link flexible joint manipulator
13
2.5
Single link flexible manipulator with joint flexibility
14
2.6
Schematic diagram of two degrees of freedom
manipulator with flexible joints.
15
Experimental setup for two degrees of freedom
manipulator with flexible joints.
16
2.8
A spatial multi-link flexible manipulator system.
18
2.9
Schematic of flexible link and flexible joint assembly.
20
2.10
Joint drive model
21
2.11
Schematic diagram a rotating flexible arm driven by
a flexible joint
22
2.12
Deformed configuration of the arm-hub-joint system.
23
2.13
The flexible rotor beam element
24
2.7
xii
2.14
A planar three revolute manipulator
25
2.15
A flexible beam with a flexible joint
26
3.1
The schematic diagram of single link flexible
manipulator with flexible joint
28
3.2
Definition of generalized coordinates for the link
29
4.1
The root locus plot for G ( s ) =
4.2
The root locus plot for G ( s ) =
4.3
The root locus plot for G ( s ) =
4.4
Simulink model for single link flexible joint
manipulator
48
4.5
Input voltage
49
4.6
Tip angle (θ+α)
50
4.7
Angular position of motor (θ)
50
4.8
Joint deflection (α)
51
4.9
Angular velocity of motor ( θ& )
51
4.10
Joint deflection velocity ( α& )
52
4.11
Simulink model for system to obtain the
frequency responses
53
4.12
Joint deflection frequency spectrum
54
4.13
Joint deflection velocity frequency spectrum
55
4.14
Filter parameter for the system
56
4.15
Simulink model with and without compensator for the system
57
4.16
Tip angle (θ+α)
57
4.17
Angular position of motor (θ)
58
(θ + α )
Vin
θ
Vin
α
Vin
45
46
47
xiii
4.18
Joint deflection (α)
58
4.19
Angular velocity of motor ( θ& )
59
4.20
Joint deflection velocity ( α& )
59
xiv
LIST OF SYMBOLS
θ
-
angular position of the motor
α
-
angular displacement of the flexible joint
V
-
potential energy
K
-
kinetic energy
Vg
-
potential energy due to gravity
Vs
-
potential energy due to the springs
Kh
-
kinetic energy of the hub
Kl
-
kinetic energy of the load
m
-
mass of the shaft
g
-
gravity constant
h
-
height of the center of gravity of the link with
respect to the rest position
Jh
-
inertia at the motor output
Jl
-
inertia of the arm
Ks
-
spring stiffness
R
-
arm anchor point
d
-
body “y” anchor point
r
-
the body “x” anchor point
K
-
spring stiffness
Fr
-
spring restoring force
L
-
spring length at rest
Km
-
motor constant
xv
Kg
-
gear ratio
Rm
-
motor resistance
ω
-
angular velocity of the motor
i
-
armature current
ft
-
natural frequency
xvi
LIST OF APPENDICES
APPENDIX NO.
A.
TITLE
Matlab Programming (M-File)
PAGE
69
1
CHAPTER 1
INTRODUCTION
1.1
Introduction
Robotic manipulators are widely used to help in dangerous, monotonous, and
tedious jobs. Most of the existing robotic manipulators are designed and build in a
manner to maximize stiffness in an attempt to minimize the vibration of the end
effectors to achieve good position accuracy. This high stiffness is achieved by using
heavy material and a bulky design. Hence, the existing heavy rigid manipulators are
shown to be inefficient in terms of power consumption or speed with respect to the
operating payload.
In addition, the operation of high precision robots is severely limited by their
dynamic deflection, which persists for a period of time after a move is completed. The
settling time required for this residual vibration delays subsequent operations, thus
conflicting with the demand of increased productivity. These conflicting requirements
between high speed and high accuracy have rendered the robotic assembly task a
2
challenging research problem. Also, many industrial manipulators face the problem of
vibrations during high speed motion.
In order to improve industrial productivity, it is required to reduce the weight of
the links and/or to increase their speed of operation. Due to lightweight requirements
and high-speed operations, a comprehensive dynamic model that includes the link and/or
joint flexibilities is highly needed. Link flexibility is a consequence of the lightweight
constructional feature in manipulator arms that are designed to operate at high speed
with low inertia. Joint flexibility arises because of the elastic behaviour of the joint
transmission elements such as gears, chains and shafts. Compare to the conventional
heavy and bulky robots, by introducing link and/or joint flexibility in the mechanical
system of robots, its have the potential advantages of lower cost, larger work volume,
higher operational speed, greater payload-to-manipulator-weight ratio, smaller actuators,
lower energy consumption, better manoeuvrability, better transportability and safer
operation due to reduced inertia. Figure 1.1 and Figure 1.2 show some flexible joint and
flexible link devices which are used in the robotics application currently.
Figure 1.1: Flexible joint
3
Figure 1.2: Flexible link
However, the greatest disadvantage by introducing link and/or joint flexibilities
to the robotic mechanical system is the vibration problem due to low stiffness. If the
vibration problem cannot be solved, incorporate the flexible link and/or flexible joint in
the robotic mechanical system have not been favoured in production industries since this
problem will affect the accuracy of end point in response to input commands. In order to
overcome this problem, an accurate dynamic model that can characterize with link
and/or joint flexibility has to be developed. This is a first step towards designing an
efficient control strategy for these manipulators.
The present work is devoted towards establishing dynamic model for a single
link flexible manipulator with flexible joint for the case in which the link is oriented
vertically. The flexible joint is modelled as a pair of springs and used to attach the link
to the motor shaft. The output of the system is the tip angle, which is given the sum of
the motor angle, θ and the joint deflection α with respect to the motor shaft. The
dynamic model of the system is developed based on Lagrange’s equations of motion.
The presence of the gravity is accounted in this model, which introduces a nonlinearity into the system in the form of a sinusoid, as a result of the potential energy due
4
to gravity. The non-linear model of the dynamics will be linearized by assumption that
the gravity is equal to zero. The resulting generalized model is validated with the
existing results and simulate through computer simulations. The investigations on the
dynamic model in terms of time and frequency responses are carried out. Besides, a
design technique for a vibration suppression of the system is also presented.
1.2
Objective
The main objective of this project is to obtain a dynamic model of a single link
flexible manipulator with flexible joint. Besides, this project has a purpose to study the
dynamic characteristics of a single link flexible manipulator with flexible joint.
1.3
Scope of Work
The scope of work is clearly define the specific field of the research and ensure
that the entire content of this thesis is confined the scope. It is begun with the literature
review on flexible joint manipulator. The next step is to develop the dynamic equations
of motion for a single link flexible manipulator with flexible joint using Lagrange’s
equations of motion. At this stage, the mathematical model will be verified with the
existing results.
Then using Matlab and Simulink, the dynamic model will be simulated and
followed by the investigations of the dynamic model in terms of time and frequency
responses. Finally, a technique for a vibration suppression of the system is designed. The
scope of work can be described in terms of flowchart as shown in Figure 1.3.
5
Start
Literature Review On Flexible Joint Manipulator
Mathematical Model Of A Single Link Flexible Joint Manipulator Using
Lagrange’s Equations Of Motion
Model
Satisfied
No
Recheck Math
Model
Yes
Verification: Comparison Of Current Results With The Existing Result
Result
Satisfied
No
Recheck Model &
Programming
Yes
Simulation Analysis: The Dynamic Model Using Matlab / Simulink
Result
Obtain
No
Troubleshoot
Programming
Investigations: The Dynamic Model In Terms Of Time And Frequency
Responses
Design a technique for vibration suppression of the system.
Result
Satisfied
No
Recheck
compensator design
Yes
End
Figure 1.3: Flowchart represents the scope of work
6
1.4
Outline of the Thesis
The thesis presents the implementation or possibly implementation with the
improvement of the existing mathematical model of single link flexible manipulator with
flexible joint.
Chapter 2 focuses on the literature review, which introduces the overview of
flexible joint manipulators. The explanation begins with the previous researches on
flexible joint manipulators. This chapter is then described by related researches on
modelling of flexible joint manipulators, which is found to be related and facilitate to
this project.
Chapter 3 provides the methodology that is used through out the work of this
project. It covers the technical explanation of this project and derivation of the dynamic
model mathematical equations using related formula. The model also been verified with
the existing results.
Chapter 4 deals with the time response results and Matlab’s simulation results of
the dynamic model. The obtained results are being compared with the existing results
for model validation.
Chapter 5 presents the conclusions of the project as well as some constructive
suggestions for further development and the contribution of this project. The project
outcome is concluded in this chapter. As for future development, some suggestions are
highlighted with the basis of the limitation of the effectiveness mathematical equation
and simulation analysis executed in this project.
7
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
This chapter is arranged as the following sub-chapters and intentions:
i)
Overview of the previous researches on flexible joint manipulators
ii)
Overview of the previous researches on modelling of flexible joint
manipulators.
2.2
Overview of the previous researches on flexible joint manipulators
In this sub-chapter, a brief survey of previous researches about flexible joint
manipulators is highlighted.
8
Dado and Soni [1] carried out the investigation on dynamic response of a planar
two revolute robot with flexible joints. Servo stiffness and damping are modelled along
with the stiffness and material damping of the drive system. Figure 2.1 and Figure 2.2
show the two revolute robot parameters and the motor and drive system model.
Figure 2.1: Two revolute robot parameter
Figure 2.2: Motor and drive system model
9
This investigation reported that joint flexibility plays important role in the
dynamic behaviour of robots. The methodology they presented can predict the deviation
of the actual motion of the link from desired inputs to the motor. Besides the numerical
examples presented illustrate the effect of the damping ratio on the dynamic response
and show that the deviation at the base joint is greater than that at the second joint. In
this research, they assumed that the links are rigid and that the drive system is ideal for
which gear backlash and friction is ignored. They also suggested this studied should be
carried out to six axis industrial type robot for the purpose to arrive at optimum joint
properties that allow a robot to handle large payloads at high speeds.
Chen and Fu [2] presented a two stage control scheme for flexible joint
manipulator based on a simplified dynamic model. In this research, they assumed that
the mechanical stiffness of the elastic joint is not large for the purpose to formulate the
dynamic model of the manipulator system. Figure 2.3 shows the single link flexible
manipulator with elastic joint presented by Chen and Fu.
Figure 2.3: Single link flexible manipulator with elastic joint
10
They stated when the system parameters are exactly known, the first stage
controller is synthesized by a proportional derivative feedback mechanism whereas the
second stage controller applies the linearization scheme. However for the case where all
or partial system parameters are unknown prior to controller design, the second stage
control scheme was modified based on a Lyapunov method. As a result on their
research, the system is stable in the local sense and the output trajectory will follow the
desired trajectory asymptotically in spite of the existence of parameter uncertainties.
However they proposed an implication is the robustness of the control scheme to the
variation of payload or parameters so long as their rate of change is moderate. Moreover,
the convergence may be spent up by increasing the suitable controller gains.
Xi et al. [3] studied the coupling effects between the joint flexibility and the link
flexibility using the system’s natural frequencies. They offered two ratios which are the
inertia ratio and the stiffness ratio to quantify the coupling effect, wherein they have
studied the two limiting cases of having a rigid link with a flexible joint and a flexible
link with a rigid joint. However, due to the method followed in developing the equations
of motion and the need to develop a close form solution, some coupling terms between
the equations of motions have been dropped. Furthermore, the system reference
rotational equation of motion was not shown in the model with the associated effects
from the elastic deformations.
Smaili [4] analyzed a two revolute planar manipulator with rigid and compliant
joints using a three-node isoparametric finite beam element. In this formulation the joint
compliances, the shear deformation and rotary inertia and the coupling effects of
nonlinear gross motion of the manipulator links with their distributed flexibility and
mass properties are included. Instantaneous steady-state static response, modal analysis
and transient response are obtained.
11
Reshmin [5] applied a semi-analytical approach for a robot in case of the
dynamic control under the assumption of high stiffness of the joints and high gear ratios
of the actuators. The robots consists of n rigid links connected together and is driven by
n independent actuators. In this research, Reshmin obtained a constructive procedure for
dynamics simulation. As a result the integration of the nonlinear equations of motion can
be replaced by the integration of a simpler system of differential equations with large
step.
Readman and Belanger [6] studied the stabilization of the first mode of flexible
joint robots. In their research, the robot is assumed to be an open kinematic chain with
only revolute joints. They modelled each joint as a linear torsional spring. The model
equations consist of two coupled dynamic systems, one representing the usual rigid body
or slow dynamics and the other the fast dynamics introduced by the joint flexibility. The
model presented in this paper is in a form that brings out the influence on the fast
subsystem dynamics of the rigid body parameters and the robot geometry.
The model clearly shows the effect that link and drive parameters have on the
dynamics of the fast subsystem. Besides, it is shown that under certain assumptions there
exists a decentralized velocity control law that asymptotically stabilizes the fast
subsystem dynamics. For sufficiently small drive inertias, there always exists a fixed
decentralized control law that will asymptotically stabilize the fast dynamics. This is true
even for large drive ratios. For sufficiently large drive inertias, it may not be possible to
use a fixed decentralized control law.
.
12
Colbaugh and Glass [7] studied the motion control and compliance control
problems for uncertain rigid-link, flexible-joint manipulators, and presents new adaptive
task-space controllers as solutions to these problems. The motion control strategy is
simple and computationally efficient, requires little information concerning either the
manipulator or actuator/transmission models, and ensures uniform boundedness of all
signals and arbitrarily accurate task-space trajectory tracking.
In their research, they include an adaptive impedance control scheme in
compliant motion controllers, which is appropriate for tasks in which the dynamic
character of the end-effector/environment interaction must be controlled, and an
adaptive position/force controller, which is useful for those applications that require
independent control of end-effector position and contact force. The compliance control
strategies retain the simplicity and model independence of the trajectory tracking scheme
upon which they are based, and are shown to ensure uniform boundedness of all signals
and arbitrarily accurate realization of the given compliance control objectives. They
presented the capabilities of the proposed control strategies through computer
simulations with a robot manipulator possessing very flexible joints.
Tsaprounis and Aspragathos [8] studied an adaptive switching control scheme
for elastic joint robot manipulators. In their research, the characteristics of both flexible
and rigid subsystems are assumed unknown except the joint stiffness. An adaptive
estimator compensates the uncertainty due to the unknown robot characteristics. The
actuators and links position, velocity and an estimation of the link acceleration are used
as feedback to the control law. A filter is designed to estimate the links acceleration and
its accuracy is insured by the linear control theory. Lyapunov theory is used to verify the
asymptotic stability of the control law. The performance of the proposed control method
is tested using a simulated planar robot with two rotational degrees of freedom for high
and low joint stiffness.
13
Timcenko and Kircanski [9] described an advanced robot control system based
on the three parallel processor boards. They designed a controller as a general-purpose
robot control system dedicated to controlling industrial robots with up to six degrees of
freedom. The controller is based on a disk-oriented real-time operating system. The
entire set of robot parameters can be monitored and changed by the user on-line. The
main software layers include robot program interpreter, manipulator control facility,
robot kinematics, dynamic feed-forward compensation, and digital servos. Basis, hand
and joint coordinates are supported. Point-to-point and continuous path motions are
provided. Digital and analogue input/output modules are included for synchronization
with an environment.
Yim [10] designed a controller for flexible joint manipulator using an adaptive
output-feedback controller based on a back stepping design. Figure 2.4 shows the single
link robotic manipulator whose joint is modeled as linear torsional spring.
Figure 2.4: Single link flexible joint manipulator
14
He assumed that the parameters of system to be unknown and only motor
position and link position measurements are used for the synthesis of the controller. He
derived a canonical stare representation of the system and filters are designed to obtain
the estimates of the derivatives of motor angle and link angle. Then he derived adaptive
control law for the trajectory control of link displacement angle. Compared to the same
adaptive controller, which measures just a link displacement angle, the order of the
proposed controller becomes a half (order of 2) and complexity of control law can be
reduced dramatically. Besides, the proposed controller does not require the bounds on
uncertain parameters for the control law derivation compared other output adaptive
control law.
Upreti and Talole [11] presented a trajectory tracking controller formulation
based on the continuous time predictive control approach for a flexible joint robot
manipulator. They considered a single link manipulator with revolute joint actuated by
the DC motor and model the elasticity joint as a linear torsional spring with selected
stiffness as shown in Figure 2.5.
Figure 2.5: Single link flexible manipulator with joint flexibility
15
In their research, they able to establish closed loop stability of the controller and
its are shown that the well-known feedback linearizing controller represents a special
case of the predictive controller. They also reported, unlike many other controllers
including the linearizing one, this formulation explicitly accounts for penalty on control
input. Finally they presented the simulation results where it is observed that the
predictive controller offers a satisfactory tracking performance.
Kim and Oh [12] developed a framework under the design of robust control for
flexible joint manipulators that are uncertained and mismatched. In their research, they
used a two link flexible joint manipulator where the links are assumed rigid. All joints
are revolute or prismatic and are directly actuated by DC electric motors. Figure 2.6
shows the schematic diagram of two degrees of freedom manipulator with flexible joints
used in their research.
Figure 2.6: Schematic diagram of two degrees of freedom manipulator with
flexible joints.
16
They divided the total system into two subsystems. First, they implant virtual (or
implanted) control for the first subsystem. Then they introduce a state transformation via
virtual control. They discovered that based on this transformed system, a control scheme
is difficulty of control design in mismatched system. The control renders practical
stability for the transformed system and later the practical stability for the original
system is also investigated.
The control is applicable to both the constant uncertainty and time varying
uncertainty. Furthermore, the size of the uniform ultimate boundedness ball and uniform
stability ball can be made arbitrary small for the transformed system and for the original
system. The transformation is only based on the possible bound of uncertainty. This
scheme shows a major development in controlling flexible joint manipulators with
mismatched uncertainty and nonlinearity. They also carried out experiment for two- link
flexible joint manipulator as shown in Figure 2.7 to verify the control performance under
the proposed robust control. The robust control possesses most enhanced control
performance compared with the others.
Figure 2.7: Experimental setup for two degrees of freedom manipulator
with flexible joints.
17
2.2 Overview on the previous researches on modelling of flexible joint
manipulators
In this sub-chapter, the overview of the previous researches about modelling of
flexible joint manipulators is briefly discussed in order to clearly justify the necessitate
sources of previous researches to be referred for this project.
Spong [13] modelled the joint flexibility as a linear spring and proposed a globally
feedback linearizable rigid flexible joint robot model that reduces to the standard rigid
link robot model as the joint stiffness tends to infinity. This model has been widely used
by the robotic control research community for the design of robot controllers.
Bahrami and Rahi [14] presented an approach to study of the tip dynamic
response (displacement and velocity) of elastic joint manipulators subjected to a vertical
stochastic excitation of the base. In their research, the crossing analysis of maximum
displacement of the manipulator tip along a determined path is also investigated. The
dynamic equations of motion of an n-link articulated elastic joint manipulator subjected
to a vertical stochastic base excitation are derived by using the Euler-Lagrange equations
and then extended by Taylor series expansion.
The result dynamic equations are linearized with respect to links vibration. The
power spectral density representation is used to compute the second moment matrix of
angular displacement and velocity of the links and to compute the second moment
matrix of manipulator tip dynamic response. They also investigated the crossing analysis
and the probability that any maximum peak displacement values of manipulator tip
exceed from a determined level along a path. At the end of their journal and based on the
18
result, they found that the probability that the maximum peak displacement values of the
manipulator tip exceed from a determined level is changing along the path.
Ider and Ozgoren [15] studied spatial three revolute manipulators where the
flexible joints are modelled by torsional springs and dampers. It is shown that, in a
flexible-joint robot, the acceleration level inverse dynamic equations are singular
because the control torques does not have an instantaneous effect on the end-effector
accelerations due to the elastic media. Implicit numerical integration methods that
account for the higher order derivative information are utilized for solving the singular
set of differential equations. The trajectory tracking control law presented linearizes and
decouples the system and yields an asymptotically stable fourth order error dynamics for
each end-effector degree of freedom.
Farid and Lukaiewicz [16] developed an efficient finite element/Lagrangian
approach for dynamic modelling of lightweight multi-link spatial manipulators with
flexible links and joints. Figure 2.8 shows the manipulator system model used in this
paper where it consists of a chain of flexible links connected by revolute actuated joints.
Figure 2.8: A spatial multi-link flexible manipulator system.
19
They used Lagrange’s equations to derive the equations of motion of the system.
The constraint equations representing kinematical relations among different coordinates
due to the connectivity of the links are added to the equations of motion of the system by
using Lagrange multipliers. This leads to a mixed set of ordinary differential equations
and nonlinear algebraic equations with coordinates and Lagrange multipliers as
unknown variables. The resulting system of differential algebraic equations is converted
to a set of differential equations by substituting the constraints with their double time
derivatives. The system is solved numerically to predict the dynamic behaviour of the
manipulator.
The dynamic model they proposed is free from assumption of a nominal motion
and takes into account the coupling effects among the rigid body motion of the system,
the bending and torsional deflections of the links, and the flexibility of the joints. Due to
these couplings as well as the time variation in the effective inertia of the system, the
model is highly nonlinear and coupled. They shown the validity of the model and
illustrated the effect of link and joint flexibilities by some case examples. They found
out that the torsional deflections have more significant effect than the bending
deflections and joint deformations. In addition, it is shown that the effect of joint
flexibility is significant when the links are flexible too. Finally, they concluded that the
interaction among various flexibilities plays an important role in the dynamic behaviour
of the system.
Subudhi and Morris [17] presented a dynamic modelling technique for a
manipulator with multiple flexible links and flexible joints, based on a combined Euler–
Lagrange formulation and assumed modes method. The model presented in this paper
for a multi-link manipulator with both link and joint flexibility is a generalized one. This
is very useful for the study of manipulators with multiple links and joints that are all
flexible. Figure 2.9 shows the schematic of flexible link and flexible joint presented by
them.
20
Figure 2.9: Schematic of flexible link and flexible joint assembly.
Subudhi and Morris affirmed that the proposed model they developed is
complete in the sense that it considers the effects of payload and structural damping of
the links. The general model formulation can be exploited to obtain the closed-form
dynamic models for practical flexible manipulators with any number of links. They
verified the model equations using bang–bang torque inputs in a two-link manipulator,
and the model responses have been discussed.
However , they found out controlling such a manipulator is more complex than
controlling one with rigid joints because only a single actuation signal can be applied at
each joint and this has to control the flexure of both the joint itself and the link attached
to it. In order to resolve the control complexities associated with such an under-actuated
flexible link/flexible joint manipulator, they formulated a singularly perturbed model
and used to design a reduced-order controller. This is shown to stabilize the link and
joint vibrations effectively while maintaining good tracking performance.
21
Dado and Eljabali [18] developed a dynamic simulation model for mixed-loop
planar robots with flexible joint drive and servo-motor control. Figure 2.10 shows the
joint model presented by Dado and Eljabali.
Figure 2.10: Joint drive model
The motion of the links is coupled with the deflection of the drive shaft at the
joints. They used virtual work method for the derivation of the mathematical model. In
their research, the drive signal at the motor is based on the error between the desired and
actual motions using proper position and velocity gains. Different motion programs are
considered in the simulation for the time histories of the angular displacements and
velocities of the links and motors. They also computed the driving torques and the total
error produced at the end-effector. From the simulation results for a five-link, three
degrees of freedom manipulator, it shows that the model presented is capable of
simulating the coupling effect of joint flexibility and rigid body motion for planar
robots.
22
Bedoor and Almusallam [18] presented a dynamic model of a flexible-arm and
flexible-joint manipulator carrying a payload with rotary inertia .A schematic diagram of
a rotating flexible arm driven by a flexible joint presented by Bedoor and Almisallam is
shown in Figure 2.11.
Figure 2.11: Schematic diagram a rotating flexible arm driven by a flexible joint
The joint is taken to have small torsional deformations. The hub is assumed rigid
and the flexible link is attached radially to the hub. They assumed that the link to be
inextensible and they adopted the Euler-Bernoulli beam theory. Figure 2.12 shows the
coordinate systems used by them to develop the dynamic model.
23
Figure 2.12: Deformed configuration of the arm-hub-joint system.
They employed the Lagrangian approach in conjunction with the finite element
method in deriving the equations of motion. All the dynamic coupling terms between the
system reference rotational motion, joint torsional deformations and arm bending
deformations are accounted for. The dynamics of a payload with rotary inertia are
incorporated in the model in a consistent manner. Furthermore, they included the effects
of axial shortening due to beam and bending deformations and motions induced inertial
forces as well as the effects of gravity in the model.
Based on the resulting model and simulation results they obtained, the found out
that the joint flexibility has a pronounced effect on the dynamic behavior of rotating
flexible arms that should not be simply neglected. The effect is shown to be due to the
nonlinear dynamic interaction between the joint torsional deformations, the arm bending
deformations and the system reference rotational motion. They compared simulation
results of the nonlinear and the linearized models and the results are discussed. Besides,
24
they found out that the effects of the payload are shown to be increasing the elastic
deformations amplitudes and reducing the frequency of oscillations.
They concluded that due to the model nonlinearity, different combinations of
system parameters are expected to develop different effects where this makes the
proposed model valuable in the design process as well as in the performance evaluations
of such systems. They proposed that more studies on the effects of coupling on the
system's natural frequencies and mode shapes as well as on the stability of such a system
are needed.
Shigang and Yueqing [19] presented a flexible rotor beam element to study the
dynamic behaviour of planar manipulators with multiple flexible links and joints. They
incorporated the link and joint flexibility together by using the element. Figure 2.13
shows the flexible rotor beam element presented by Shigang and Yueqing.
Figure 2.13: The flexible rotor beam element
25
In their research, the effects of longitudinal loads on lateral vibration,
gravitational body force, internal damping, actuator and payload masses are all
considered at the same time. The flexible joint is studied by introducing an additional
degree of freedom into the element. The system of differential equations of a flexible
manipulator is the combination of a finite element dynamic model for links and a
torsional spring model for joints, and all the coupling terms due to link and joint
flexibility are included in the equations. The effect of each flexible joint on link
deformations is investigated through the solution of these combined equations. They
analyzed a plane three revolute manipulator with flexible links and joints as an example.
Figure 2.14 shows the planar three revolute manipulator. Based on the results of
numerical simulation, they found out that the significant role of joint, as well as link,
flexibility in the dynamic characteristics of flexible manipulator.
Figure 2.14: A planar three revolute manipulator
26
Li and Zu [20] presented a systematic approach to dynamic modelling and mode
analysis of a single-link flexible robot, which has a flexible joint and a hub at the base
end and a payload at the free end. Figure 2.15 shows the flexible beam with a flexible
joint presented by Li and Zu.
Figure 2.15: A flexible beam with a flexible joint
In this research, they able to derived the exact mode shapes of a constrained
flexible beam supported by a flexible joint where payload and hub are taken into
account. These modes are useful in the modelling of multiple link robots with both link
flexibility and joint flexibility. Besides, they adopted analytical approach to establish the
equations of motion of the system. As results of their research, they discovered that even
a small joint flexibility could significantly affect the system frequencies and the
fundamental frequency is not sensitive to the hub inertia or payload inertia. They also
found out that for a given flexible system, the fundamental frequency is mainly affected
by the payload mass, while the second frequency is mainly affected by the payload
inertia.
27
CHAPTER 3
METHODOLOGY
3.1
Introduction
This section is focused on the development of the dynamic model and the
mathematical model of the single link flexible joint manipulator with the flexible joint.
Firstly, the derivation of the dynamic model using the formulation of kinetic energy,
potential energy and the Lagrange’s equation of motion is presented. At this stage, the
model will be verified with the existing published papers. The next part is to obtain the
mathematical model where the plant parameter for the system is obtained from existing
papers. The model is expressed in state – space form in order to get the system responses
by using computer simulation.
28
3.2
Development of the Dynamic Model
The schematic diagram single link flexible joint manipulator is shown in Figure
3.1. The single link flexible manipulator with flexible joint is considered operating on a
vertical plane. The mathematical model of the dynamics of the link with flexible joint in
vertical position is easily obtained from Lagrange's equations of motion. It is clear that
the systems has two degrees of freedom, corresponding to the rotation of the motor shaft
with respect to a coordinate frame fixed to the base, and the rotation of the flexible joint
to which the link is attached with respect to the motor.
Side View
Spring
Front View
Figure 3.1: The schematic diagram of single link flexible manipulator with flexible joint
The generalized coordinates are therefore the angular position of the motor θ and
the angular displacement of the flexible joint α as shown in Figure 3.2.
29
θ
α
Figure 3.2: Definition of generalized coordinates for the link
The potential and kinetic energies of the system are given respectively by:
a)
Potential Energy, V
V = V g + Vs
(3.1)
where
Vg
=
mgh cos(θ + α )
(3.2)
Vs
=
1
K sα 2
2
(3.3)
30
By substituting equation (3.3) into equation (3.1), yields
mgh cos(θ + α ) +
1
K sα 2
2
V
=
b)
Kinetic Energy, K
K
=
K h + Kl
(3.5)
Kh
=
1 &2
J hθ
2
(3.6)
Kl
=
1
J l ( θ& + α& ) 2
2
(3.7)
(3.4)
where
Then, substituting equations (3.6) and (3.7) into (3.5), yields
K
=
1 &2 1
J hθ + J l ( θ& + α& ) 2
2
2
=
2
1 &2 1
J hθ + J l θ& 2 + 2θ&α& +α&
2
2
(
)
(3.8)
31
where
Vg = the potential energy due to gravity
Vs = the potential energy due to the springs
K h = the kinetic energy of the hub
K l = the kinetic energy of the load
m = the mass of the shaft
g = the gravity constant
h = the height of the center of gravity of the link with respect to the rest position
J h = the inertia at the motor output
J l = the inertia of the arm
K s = the spring stiffness
The value of the constant K s depends on the way the springs are attached to the
link. Following the Quanser manual, the expression for K s is
Ks
=
⎛ 32
⎞ ⎞
2 R ⎛⎜
2
⎜ D d − DLd + Rr 2 L ⎟ K ⎟
Dd
−
Rr
F
+
r
3 ⎜
⎜
⎟ ⎟
⎝
⎠ ⎠
D2 ⎝
D
=
r 2 + (R − d )
(
)
2
where
R = the arm anchor point
d = the body “y” anchor point
r = the body “x” anchor point
K = spring stiffness
Fr = spring restoring force
L = spring length at rest
(3.9)
(3.10)
32
The values of the constants can be found in Table 3.1. The plant parameters of
the system are obtained from published paper written by Kevin Groves and Andrea
Serrani [21].
Table 3.1: Plant parameters
Parameter
Symbol
Value
Spring length at rest
L
0.0318 [m]
“Y” anchor point
d
0.0318 [m]
“X” anchor point
r
0.0318 [m]
Arm anchor point
R
0.0762 [m]
Spring restoring force
Fr
1.33 [N]
Spring stiffness
K
368 [N/m]
Inertia of hub
Jh
0.0021
[Kgm2]
Link mass
m
0.403
[Kg]
Gravity constant
g
-9.81
[N/m]
Height of C.M.
h
0.06
[m]
Motor constant
Km
0.00767 [N/rad/s]
Gear ratio
Kg
70
Load inertia
Jl
0.0059
[Kgm2]
Motor resistance
Rm
2.6
[Ω]
33
The Lagrarian is given by
L
=
K −V
(3.11)
By substituting equations (3.4) and (3.8) into equation (3.11), yields
L
=
(
)
2
1 &2 1
1
J hθ + J l θ& 2 + 2θ&α& +α& - mgh cos(θ + α ) - K sα 2
2
2
2
(3.12)
The Lagrange equations of motion is defined as equations (3.13) and (3.14)
d dL
dL
dt dα& dα
=
0
(3.13)
d dL dL
dt dθ& dθ
=
τ
(3.14)
Then, differentiating equation (3.12), yields
dL
dα
=
mgh sin(θ + α ) - K sα
(3.15)
dL
dα&
=
J lθ& + J lα&
(3.16)
d dL
dt dα&
=
J lθ&& + J lα&&
(3.17)
dL
dθ
=
mgh sin(θ + α )
(3.18)
dL
dθ&
=
J hθ& + J lθ& + J l α&
(3.19)
d dL
dt dθ&
=
(J h + J l )θ&& + J l α&&
(3.20)
34
After that, by substituting equations (3.15) and (3.17) into equation (3.13), yields
J lθ&& + J l α&& - mgh sin(θ + α ) + K sα
=
0
(3.21)
Lastly, by substituting equations (3.18) and (3.20) into equation (3.14), yields
(J h + J l )θ&& + J l α&& -
mgh sin(θ + α )
=
τ
(3.22)
In equation (3.22), τ is the torque produced by the motor. The torque is
commanded by the voltage applied to the armature, which represent the input to the
system. The relationship between torque and the applied voltage is given in Quanser lab
manual as:
v
=
iRm + K m K g ω
(3.23)
v KmK g
−
ω
Rm
Rm
(3.24)
This gave
i
=
where
ω = the angular velocity of the motor
i = the armature current
Rm = the motor resistance
K m = constant parameter
K g = constant parameter
35
Since
i
=
θ&
=
τ
(3.25)
KmKg
ω
(3.26)
And by substituting equations (3.25) and (3.23) into equation (3.24), yields
τ
KmKg
=
KmKg
v
−
θ&
Rm
Rm
(3.26)
Consequently, the desired relationship is obtained as
τ
=
KmK g
Rm
2
2
Km K g &
θ
v−
Rm
(3.27)
By defining the state variables as
x1 = θ
(3.28)
x2 = α
(3.29)
x3 = θ&
(3.30)
x 4 = α&
(3.31)
36
Differentiating the state variables, yields
x&1 = θ&
(3.32)
x& 2 = α&
(3.33)
x& 3 = θ&&
(3.34)
x& 4 = α&&
(3.35)
Equation (3.32) can be written as x&1 = x3 while equation (3.33) can be written as
x& 2 = x 4 . The following are the steps to obtain the x& 3 and x& 4 in term of state x .
a)
x& 3 = θ&&
Substituting equation (3.21) into equation (3.22), yields
(J h + J l )θ&& + J lα&& − mgh sin (θ + α )
=
τ
J hθ&& + J lθ&& + J l α&& - J lθ&& - J l α&& - K sα
=
τ
J hθ&& - K sα
=
τ
(3.36)
Then by substituting equations (3.27), (3.29) and (3.34) into equation (3.36), yields
J h x& 3 - K s x 2
=
KmKg
Rm
2
v−
Km K g
Rm
2
x3
2
x& 3
=
2
Km Kg
K s x2 K m K g
+
v−
x3
Rm J h
Rm J h
Jh
(3.37)
37
b)
x& 4 = α&&
Substituting equations (3.29), (3.30) and (3.31) into equation (3.21), yields
J lθ&& − J l α&& + K sα - mgh sin(θ + α )
=0
J l x& 3 + J l x& 4 + K s x 2 - mgh sin( x1 + x 2 ) = 0
x& 4
= - x& 3 -
K s x 2 mgh
+
sin( x1 + x 2 )
Jl
Jl
2
2
Km Kg
K x KmKg
K x
mgh
=- s 2 v+
x3 - s 2 +
sin( x1 + x 2 )
Rm J h
Rm J h
Jh
Jl
Jl
(3.38)
The system can be written as
x&1
=
x3
x& 2
=
x4
=
Km K g
K s x2 K m K g
+
v−
x3
Rm J h
Rm J h
Jh
=
Km Kg
K x
K s x2 K m K g
mgh
v+
x3 - s 2 +
sin( x1 + x 2 )
Rm J h
Rm J h
Jl
Jh
Jl
x& 3
x& 4
2
2
2
2
Letting u = v and choosing the tip position y = [θ + α , θ , α] = (x1 + x2 , x1 , x2)
as the output, the system can now be represented the usual state space form as
x&
=
Ax + Bu
(3.39)
y
=
Cx
(3.40)
38
where
Ax
B
Cx
=
x3
⎛
⎞
⎜
⎟
x4
⎜
⎟
⎜
⎟
2
2
K s x2 K m K g
⎜
⎟
−
x3
⎜
⎟
Jh
Rm J h
⎜
⎟
2
2
⎜
⎟
⎛ 1 1 ⎞ Km K g
mgh
x3 +
sin ( x1 + x2 )⎟⎟
⎜⎜ − K s x2 ⎜⎜ + ⎟⎟ +
Rm J h
Jl
⎝ J h Jl ⎠
⎝
⎠
(3.41)
=
0
⎛
⎜
0
⎜
⎜ K K
⎜ m g
⎜ Rm J h
⎜
⎜ − KmK g
⎜ R J
m h
⎝
(3.42)
=
⎛ x1 + x2 ⎞
⎜
⎟
⎜ x1 ⎟
⎜ x ⎟
⎝ 2 ⎠
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.43)
However, the state space form above is nonlinear. In order to make it become
linear, it is reasonable to assume that the gravity constant, g is equal to zero. Under this
assumption, the approximation
mgh
sin( x1 + x 2 ) = 0 can be used.
Jl
39
The state space form now can be defined as follows:
x&
=
Ax + Bu
(3.39)
y
=
Cx
(3.40)
=
x3
⎛
⎞
⎜
⎟
x4
⎜
⎟
⎜
⎟
2
2
K s x2 K m K g
⎜
⎟
−
x3
⎜
⎟
Jh
Rm J h
⎜
⎟
2
2
⎜
⎟
⎛ 1 1 ⎞ Km K g
x3 ⎟⎟
⎜⎜ − K s x2 ⎜⎜ + ⎟⎟ +
Rm J h
⎝ Jh Jl ⎠
⎝
⎠
(3.44)
=
0
⎛
⎜
0
⎜
⎜ K K
⎜ m g
⎜ Rm J h
⎜
⎜ − KmK g
⎜ R J
m h
⎝
(3.45)
=
⎛ x1 + x2 ⎞
⎜
⎟
⎜ x1 ⎟
⎜ x ⎟
⎝ 2 ⎠
where
Ax
B
Cx
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.46)
40
The state space in matrix form is written as follows:
⎛
⎜
⎜
⎜
⎜
⎜
⎝
x&1 ⎞
⎟
x&2 ⎟
x&3 ⎟
⎟
x&4 ⎟⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
0
0
0
1
0
0
Ks
Jh
0
⎛ 1 1⎞
− Ks ⎜⎜ + ⎟⎟
⎝ Jh Jl ⎠
0
0
0
0
1
0
0
Ks
Jh
Km Kg
0
⎛ 1 1⎞
− Ks ⎜⎜ + ⎟⎟
⎝ Jh Jl ⎠
2
−
Km Kg
2
RmJh
2
Km Kg
2
RmJh
0⎞
⎟
1 ⎟ ⎛⎜
⎟⎜
0⎟ ⎜
⎟
⎟ ⎜⎜
⎟⎝
0⎟
⎠
⎛
0
⎞
⎜
⎟
x1 ⎞
⎜ 0 ⎟
⎟
x2 ⎟
⎜ KK ⎟
⎜ m g⎟
x3 ⎟ + ⎜ RmJh ⎟ u
⎟
⎜
⎟
x4 ⎟⎠
⎜− KmKg ⎟
(3.47)
⎜ RJ ⎟
⎝ m h⎠
where
A
B
C
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
2
−
2
RmJh
2
Km Kg
RmJh
2
0⎞
⎟
1⎟
⎟
0⎟
⎟
⎟
⎟
0⎟
⎠
(3.48)
=
⎛ 0 ⎞
⎜
⎟
⎜ 0 ⎟
⎜ KK ⎟
⎜ m g⎟
⎜ RmJh ⎟
⎜
⎟
⎜− KmKg ⎟
⎜ RJ ⎟
⎝ m h⎠
(3.49)
=
⎛ 1 1 0 0⎞
⎜
⎟
⎜ 1 0 0 0⎟
⎜ 0 1 0 0⎟
⎝
⎠
(3.50)
The dynamic model had been compared with the dynamic model from the
existing result [21] where similar results have been obtained.
41
3.3
Dynamic Model
The model of the single link flexible manipulator with flexible joint has been
obtained in previous section as given in equations (3.47) to (3.50). In order to obtain the
mathematical model, primarily the value for spring stiffness K s must be calculated. By
substituting the plant parameter value as tabulated in Table 1 into equations (3.9) and
(3.10), the value of K s is given as 1.61. Thus, by repeating using the spring stiffness
value, K s and plant parameter values into equations (3.49) to (3.50), the system for the
single link flexible manipulator with flexible joint may be obtained as
⎛
⎜
⎜
⎜
⎜
⎜
⎝
x&1 ⎞
⎟
x&2 ⎟
x&3 ⎟
⎟
x&4 ⎟⎠
y
=
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
0
0
0
0
0
767.05
−1040.1
⎛ 1 1 0 0⎞
⎟
⎜
⎜ 1 0 0 0⎟
⎜ 0 1 0 0⎟
⎠
⎝
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
0
− 52. 795
52.795
0⎞⎛
⎟⎜
1⎟ ⎜
0⎟ ⎜
⎟⎜
0⎟⎠ ⎜⎝
x1 ⎞
⎟
x2 ⎟
x3 ⎟
⎟
x4 ⎟⎠
x1 ⎞
⎛ 0
⎞
⎜
⎟
⎟
x2 ⎟
⎜ 0
⎟
+ ⎜
u
⎟
98.333⎟
x3
⎟
⎜
⎟
⎜ −98.333⎟
x 4 ⎟⎠
⎝
⎠
(3.51)
(3.52)
where
A
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
0
0
0
0
0
767.05
−1040.1
1
0
− 52. 795
52.795
0⎞
⎟
1⎟
0⎟
⎟
0⎟⎠
(3.53)
42
=
⎛ 0
⎞
⎜
⎟
⎜ 0
⎟
⎜ 98.333⎟
⎜
⎟
⎜ −98.333⎟
⎝
⎠
(3.54)
C
=
⎛ 1 1 0 0⎞
⎟
⎜
⎜ 1 0 0 0⎟
⎜ 0 1 0 0⎟
⎠
⎝
(3.55)
3.4
Conclusion
B
The dynamic model of the single link flexible manipulator with flexible joint had
been obtained using the formulation of kinetic energy, potential energy and the
Lagrange’s equation of motion. For the model validation, the dynamic model of the
system is compared with the existing result [21]. As a result, the dynamic model is
approved. However, there is a problem where the model obtain is a non – linear model.
The non-linear model is linearized by assuming that the gravity constant is equal to zero.
The linear model is presented in state space form for the use in computer simulation to
observe the responses. The plant parameters from the system are obtained from the
published paper [21]. Substituting the plant parameters into the dynamic model yield the
system mathematical model. Therefore, the model is ready to be simulated.
43
CHAPTER 4
RESULTS AND DISCUSSIONS
4. 1
Introduction
This chapter is divided into 3 sub chapters; - i) time response result ii) simulation
result iii) simulation result with compensator. In subchapter (i) the transfer function and
root locus plot is presented where these results are used for model validation. Subchapter
(ii) show the Matlab’s simulation result of the model. For subchapter (iii), the
compensator design is presented where the compensator is used to improve the system
responses. The Matlab’s simulation result for the model with compensator also
presented in this part.
4.2
Time Responses Result
The transfer function can be computed using the transfer function matrix
equation as stated in equation (4.1).
44
G(s)
C (sI – A)-1 B + D
=
(4.1)
Therefore, the value of matrix A, B, C and D are substituted into equation (4.1) and the
command ss2tf is used in Matlab, the transfer function may be obtained as stated below.
The root locus plot for the system is shown in Figure 4.1, Figure 4.2 and Figure 4.3
respectively.
a)
G(s) =
Input voltage (Vin ) with respect to tip angle (θ+α)
7.105 x 10 -15 s 3 - 1.819 x 10 -12 s 2 - 2.547 x 10 -11 s + 2.685 x 10 4
s 4 + 52.8 s 3 + 1040 s 2 + 1.441x 10 4 s
From the transfer function, the poles and zeros are:
Poles
Zeros
0
7.788 x 105 + 13.489 x 105 i
-34.8126
7.788 x 105 - 13.489 x 105 i
-8.9913 +18.2538i
-8.9913 -18.2538i
-15.574 x 105
45
Imaginary Axis
The root locus for the system is shown in Figure 4.1
Real Axis
Figure 4.1: The root locus plot for G ( s ) =
b)
(θ + α )
Vin
Input voltage (Vin) with respect to angular position of motor (θ)
G(s) =
- 1.421 x 10 - 14 s 3 + 98.33 s 2 - 3.092 x 10 - 11 s + 2.685 x 10 4
s 4 + 52.8 s 3 + 1040 s 2 + 1.441 x 10 4 s
46
From the transfer function, the poles and zeros are:
Poles
Zeros
0
0
-34.8126
0
-8.9913 +18.2538i
6.92 x 1015
-8.9913 -18.2538i
Imaginary Axis
The root locus for the system is shown in Figure 4.2
Real Axis
Figure 4.2: The root locus plot for G ( s ) =
θ
Vin
47
c)
Input voltage (Vin ) with respect to joint deflection (α)
G(s)
- 4.263 x 10 -14 s 3 - 98.33 s 2 - 5.639 x10 -11 s
s 4 + 52.8 s 3 + 1040 s 2 + 1.441x 10 4 s
=
From the transfer function, the poles and zeros are:
Poles
Zeros
0
0
-34.8126
0
-8.9913 +18.2538i
-2.3065 x 1015
-8.9913 -18.2538i
Imaginary Axis
The root locus for the system is shown in Figure 4.3
Real Axis
Figure 4.3: The root locus plot for G ( s ) =
α
Vin
48
Those three transfer functions are also similar as the published results [21].
Referring to the three root locus plots, the poles locations for the three transfers are on
the left half s plane. Thus, the system is stable.
4.3
Simulation Results
In order to obtain the responses of the system, the dynamic model of single link
flexible manipulator with flexible joint is simulated by using Simulink. Figure 4.4 shows
the simulink model for the system.
Figure 4.4: Simulink model for single link flexible joint manipulator
In this work, a pulse signal as shown in Figure 4.5 is used as input voltage to
rotate the link an angle θ equal to 450 in 1.2 seconds. The pulse signal has a positive
(acceleration) for 1 second and the negative (deceleration) for 0.2 seconds allowing the
manipulator to, initially, accelerated and then decelerated and eventually stopped at the
target location.
49
The system responses that have to be observed are tip angle, angular position of
the motor, angular displacement of the flexible joint, angular velocity of motor and
angular displacement velocity of the flexible joint. All the responses are monitored for
duration of 5 seconds.
Amplitude (V)
The simulated responses of the system are presented in Figure 4.6 to Figure 4.10.
Time (second)
Figure 4.5: Input voltage
Radian
50
Time (second)
Radian
Figure 4.6: Tip angle (θ+α)
Time (second)
Figure 4.7: Angular position of motor (θ)
Radian
51
Time (second)
Rad/s
Figure 4.8: Joint deflection (α)
Time (second)
Figure 4.9: Angular velocity of motor ( θ& )
Radian
52
Time (second)
Figure 4.10: Joint deflection velocity ( α& )
Based on the system responses, the tip angle experienced about 23.5 % overshoot
and some oscillatory behaviour. The oscillation only settled at ts equal to 1.37 seconds.
For the motor angular position, it also experienced 16.7 % overshoot and oscillation.
Besides during the motor rotated to the desired angle, there is a vibration occurred. The
joint deflection for the system is around ± 0.07 radian or ± 40.
4.4
Simulation Results with Compensator
In order to reduce overshoot and eliminate oscillation, a compensator must be
introduced to the system. However, the major problem is to determine what type of
compensation would be effective to the system to achieve this goal.
53
The goal to introduce compensator in the system is to improve the transient
response system where to reduce overshoot and eliminate oscillation. Therefore a low
pass filter could be used as the compensator. The low pass filter will be used to filter the
undesired frequency from the input voltage in order to get better system responses.
The first step to design the filter is to find the undesired frequency or natural
frequency that exists in the input voltage. In order to find out the natural frequency, the
frequency responses of the system are carried out. The frequency responses of the
system are obtained by transforming the time domain representation of the system into
the frequency domain using Fast Fourier Transform analysis. Figure 4.11 shows the
simulink model for the system in order to obtain the system frequency responses.
Figure 4.11: Simulink model for system to obtain the frequency responses
The frequency responses of the system are shown in Figure 4.12 and Figure 4.13
respectively.
Rad/s
54
Frequency (Hz)
Rad/s
(a) Zoom factor = 1
ft
Frequency (Hz)
(b) Zoom factor = 10
Figure 4.12: Joint deflection frequency spectrum
Rad/s
55
Frequency (Hz)
Rad/s
(a) Zoom factor = 1
ft
Frequency (Hz)
(b) Zoom factor = 10
Figure 4.13: Joint deflection velocity frequency spectrum
56
The natural frequency of the system is a place where the frequency has the
highest amplitude. Based on the frequency responses of the system, the natural
frequency ft is 2.4 Hz. The low pass filter is used to remove out the input voltage with
frequency less than 2.4 Hz. The parameter for the filter is shown in Figure 4.14.
Figure 4.14: Filter parameter for the system
The pass band frequency can be calculated by using equation as stated in equation (4.2)
and for this system the pass band frequency is 15 rad/s.
ω = 2πf
(4.2)
Figure 4.15 shows the simulink model with and without compensator for the
system and the responses of the system are shown in Figure 4.16 to Figure 4.20
57
Radian
Figure 4.15: Simulink model with and without compensator for the system.
Legend
without filter
with filter
Time (second)
Figure 4.16: Tip angle (θ+α)
Radian
58
Legend
without filter
with filter
Time (second)
Radian
Figure 4.17: Angular position of motor (θ)
Legend
without filter
with filter
Time (second)
Figure 4.18: Joint deflection (α)
Radian
59
Legend
without filter
with filter
Time (second)
Radian
Figure 4.19: Angular velocity of motor ( θ& )
Legend
without filter
with filter
Time (second)
Figure 4.20: Joint deflection velocity ( α& )
60
Based on the system responses with compensator, the tip angle still experienced
overshoot, however the overshoot has reduced to 16.7 % and the oscillatory behaviour is
eliminated. The settling time for the tip angle is 1.44 seconds. The motor angular
position still experienced overshoot, however it has reduced to 12.3%. There is no
oscillation and vibration exists when motor rotated to the target angle. The joint
deflection for the system also reduced to ± 0.03 radian or ± 1.70. Table 4.1 shows the
comparison of system responses with and without compensator.
Table 4.1: Comparison of the system responses with and without compensator
Responses
Without filter
Overshoot
Tip angle
Motor angular
position
Joint deflection
: 23.5 %
With filter
Overshoot
: 17.4 %
Settling time : 1.37 s
Settling time : 1.54 s
Oscillation occurred
Oscillation removed
Overshoot
Overshoot
: 16.7 %
: 12.3 %
Settling Time : 1.44 s
Settling Time : 1.54 s
Oscillation occurred
Oscillation removed
Vibration existed
Vibration eliminated
± 0.07 radian or ± 4.00.
± 0.03 radian or ± 1.70
However there is a disadvantage of introducing a compensator to the system
where the response of the system becomes slower. This can be seen from the settling
time of the system with and without compensator as shown in Table 4.1.
61
4.5
Conclusion
Based on the transfer function of the system, the model is approved since the
transfer function of the system is equal with the transfer function presented in the
published papers. From the root locus plotted, the poles and zeros locations of the
system can be determined. Since all the poles are in the left half s-plane, thus the system
is stable.
The model is simulated by using Matlab and Simulink in order to get the system
responses. Based on the responses, the tip able to arrive at the desired location,
conversely there are overshoot and oscillation occurred. The motor also experienced
overshoot and oscillation when it rotated to the desired angle. Additionally there is a
vibration when the motor rotated. The joint also experienced deflection.
A compensator is required to improve the transient responses of the system. By
introducing a low pass filter to the system, the response of the system become better
where for the tip angle, the overshoot reduced and oscillation eliminated. For motor
angle, the overshoot is reduced while oscillation and vibration are eliminated. The
deflection for the joint also reduced. However the system response become slower
compared to system responses without compensator.
62
CHAPTER 5
CONCLUSION
5.1
Introduction
In this chapter, the conclusion of the project as well as some constructive
suggestions for further development and the contribution of this project will discussed.
The project outcomes are concluded in this chapter. As for future development, some
suggestions are highlighted with the basis of the limitation of the effectiveness
mathematical equation and simulation analysis executed in this project. The aim of the
suggestions is no other than to improve the study.
5.2
Conclusion
The dynamic model of single link flexible joint manipulator with flexible joint is
recognized. The dynamic model is validated with the dynamic model from the published
papers based on the dynamic model itself and the time responses result. In order to attain
63
the responses of the model, the dynamic model is simulated by using Matlab and
Simulink. From the responses of the system, it seems that the tip angle and angular
position of motor experienced overshoot, oscillation and vibration when it moved to the
target location. For the joint, there is a huge deflection occurred.
In order to improve the system responses, a compensator is needed. For this
system, low pass filter is chosen as the compensator. This low pass filter is used to filter
out the undesired frequencies that exist in the input system. The model with
compensator is simulated again in order to get the responses. From the responses, the
oscillation and vibration that exist in tip angle and motor angular position earlier is
removed. Besides, the overshoot also reduced but somehow it still exists. For the joint,
the deflection decrease to satisfied value.
Based on the simulation result for model with compensator, it can be concluded
that a filter only able to eliminate the vibration and oscillation problem that existed in
the system responses. For the overshoot and joint deflection, it only able to minimize the
value, not eliminate its. Therefore, the system required another compensator to be
introduced with it in order to make the system responses better.
64
5.3
Future Development
The future development of this dynamic model is suggested to be
chronologically executed as per the following recommendation:
i) Design another compensator using other methods.
ii) Investigate the system responses when the configuration of the springs and the
spring types change.
The objective of these recommendations is to improve the system responses when it
moves to the exact desired destination.
65
References
Journals
1.
M.H.F. Dado and A.H. Soni, Dynamic response analysis of 2-R robot with
flexible joints, Proceedings of the IEEE International Conference on Robotics
and Automation 4 (1987), pp. 479–483.
2.
K.P. Chen and L.C. Fu, Nonlinear adaptive control for a manipulator with
flexible joints, Proceedings of the IEEE International Conference on Robotics
and Automation (1989), pp. 1201–1206.
3.
F. Xi, R.G. Fenton, B. Tabbarok, Coupling effects in a manipulator with both
flexible link and joint, Journal f Dynamic Systems, Measurements and Control
116 (1994) 826-831.
4.
A.A. Smaili, A three node finite beam element for dynamic analysis of planar
manipulators with flexible joints, Mechanism and Machine Theory 28 (1993) (2),
pp. 193–206.
5.
S.A. Reshmin, Control of robots with flexible joints, Proceedings of 2nd
International Conference on Control of Oscillations and Chaos 1 (2000), pp.
177–178.
6.
M.C. Readman and P.R. Belanger, Stabilization of the first modes of a flexible
joint robot, The International Journal of Robotics Research 11 (1992) (2), pp.
123–134.
7.
R. Colbaugh and K. Glass, Adaptive task-space control of flexible-joint
manipulators, Journal of Intelligent and Robotic Systems 20 (1997) (2–4), pp.
225–249.
66
8.
J. Tsaprounis and N.A. Aspragathos, Adaptive tracking controller for rigid-link
elastic-joint robots with link acceleration estimation, Journal of Intelligent and
Robotic Systems 27 (2000) (1–2), pp. 67–83.
9.
Timcenko, A. and Kircanski, N., Multiprocessor control system for industrial
robots , in Proc. IEEE Int. Conf. Robotics and Automation, Vol. 722, 1992.
10.
Woosoon Yim, Adaptive Control of a Flexible Joint Manipulator, Proceedings of
the 2001 IEEE International Conference on Robotics & Automation Seoul,
Korea. May 21-26, 2001
11.
Vinay Upreti and S.E.Talole, Predictive Control of Flexible Joint Manipulator,
Journal of Guidance , Control and Dynamics, Vol. 17, No.3 , 1994.
12.
Dong Hwan Kim and Wan Ho Oh , Robust Control Design for Flexible Joint
Manipulators, International Journal of Control, Automation, and Systems, vol. 4,
no. 4, pp. 495-505, August 2006
13.
M.W. Spong, Modelling and control of elastic joints robots, ASME Journal of
Dynamic Systems, Measurement, and Control 109 (1987) (6), pp. 310–319.
14.
M. Bahrami and A. Rahi, Tip dynamic response of elastic joint manipulators
subjected to a stochastic base excitation, JSME, Series C 46 (2003) (4), pp.
1502–1508.
15.
S.K. Ider and M.K. Ozgoren, Trajectory tracking control of flexible-joint robots,
Computers and Structures 76 (2000), pp. 757–763.
16.
M. Farid and S.A. Lukaiewicz, Dynamic modelling of spatial manipulators with
flexible links and joints, Computers and Structures 75 (2000), pp. 419–437.
67
17.
B. Subudhi and A.S. Morris, Dynamic modelling, simulation and control of a
manipulator with flexible links and joints, Robotics and Autonomous Systems 41
(2002) 257–270.
18.
Mohammad H. F. Dado, and A. Karim Eljabali, Dynamic simulation model for
mixed-loop planar robots with flexible joint drives , Mechanism and Machine
Theory Volume 36, Issue 4, 1 April 2001, Pages 547-559.
19.
B.O. Al-Bedoor and A.A. Almusallam, Dynamics of flexible-link and flexiblejoint manipulator carrying a payload with rotary inertia, Mechanism and
Machine Theory 35 (2000) 785-820.
20.
Degao Li and Jean W. Zu, Dynamic modelling and mode analysis of flexible
link, flexible joints robots, Mech. Mach. Theory Vol. 33, No. 7, pp. 1031-1044,
1998
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K.Groves and A.Serrani, Modelling and non-linear control of a single-link
flexible joint manipulator.
68
APPENDICES
69
APPENDIX A1
(MATLAB PROGRAMMING: M - FILE)
70
% State Space Model Of A Single Link Flexible Manipulator With Flexible Joint
% Parameter value
L
= 0.0318
d
= 0.0318
r
= 0.0318
J_hub = 0.0021
J_load = 0.0059
R
= 0.0762
Km
= 0.00767
Kgi
= 14
Kge
=5
Fr
= 0.83
K
= 220
Rm
= 2.6
% Spring Stiffness formula
D = r^2 + (R-d)^2
Ks = (2*R/D^(3/2))*((D*d-R*r^2)*Fr + (D^(3/2)*d - D*L*d + R*r^2*L)*K)
Kg = Kgi*Kge
% Matrix Conversion
a = Ks/J_hub
b = - (Km^2*Kg^2) / (Rm*J_hub)
c = - Ks*(J_hub + J_load)/(J_hub*J_load)
d = (Km^2*Kg^2) / (Rm*J_hub)
e = (Km*Kg) / (Rm*J_hub)
f = -(Km*Kg) / (Rm*J_hub)
71
% Steady - State Representation
A = [ 0 0 1 0;0 0 0 1; 0 a b 0;0 c d 0 ]
B = [ 0;0;e;f ]
C =[1100]
D =0
% A program to generate a bang-bang input
sampling_time=0.002;
t=sampling_time;
for i = 1 : (0.2/sampling_time)+1
u(i,1)=0;
end
for i = (0/sampling_time)+1 : (0.2/sampling_time)+1
u(i,1)=0.5238;
end
for i = (0.2/sampling_time)+1: (0.24/sampling_time)+1
u(i,1)=-0.5238;
end
for i = (0.24/sampling_time)+1: (1/sampling_time)+1
u(i,1)=0;
end
figure(1);set(gca,'fontsize',13);
T = 0:0.01:5;
plot(u)
xlabel('Time (second)','fontsize',14);
ylabel('Amplitude(Nm)','fontsize',14);
title('Figure 1 : Torque')
grid on;
72
% Linear Simulation Result
% (Tip Angle)
figure(2);set(gca,'fontsize',13);
T = 0:0.01:5;
% simulation time = 5 seconds
sys = ss(A,B,C,D);
% construct a system model
[Y2, Tsim, X] = lsim(sys,u,T);
plot(Tsim,Y2)
% simulate
% plot the output vs. time
xlabel('Time (second)','fontsize',14);
ylabel('Radian','fontsize',14);
title('Figure 2 : Tip Angle')
grid on;
% (Angular Position Of Motor)
C =[1000]
figure(3);set(gca,'fontsize',13);
T = 0:0.01:5;
% simulation time = 5 seconds
sys = ss(A,B,C,D);
% construct a system model
[Y3, Tsim, X] = lsim(sys,u,T);
plot(Tsim,Y3)
% simulate
% plot the output vs. time
xlabel('Time (second)','fontsize',14);
ylabel('Radian','fontsize',14);
title('Figure 3 : Angular Position Of Motor')
grid on;
73
% (Angular Displacement Of Flexible Joint)
C =[0100]
figure(4);set(gca,'fontsize',13);
T = 0:0.01:5;
% simulation time = 5 seconds
sys = ss(A,B,C,D);
% construct a system model
[Y4, Tsim, X] = lsim(sys,u,T);
plot(Tsim,Y4)
% simulate
% plot the output vs. time
xlabel('Time (second)','fontsize',14);
ylabel('Radian','fontsize',14);
title('Figure 4 : Joint Deflection')
grid on;
% (Angular Velocity Of Motor)
C =[0010]
figure(5);set(gca,'fontsize',13);
T = 0:0.01:5;
% simulation time = 5 seconds
sys = ss(A,B,C,D);
% construct a system model
[Y5, Tsim, X] = lsim(sys,u,T);
plot(Tsim,Y5)
% simulate
% plot the output vs. time
xlabel('Time (second)','fontsize',14);
ylabel('Rad/s','fontsize',14);
title('Figure 5 : Angular Velocity Of Motor')
grid on;
74
% (Angular displacement velocity of flexible joint)
C =[0001]
figure(6);set(gca,'fontsize',13);
T = 0:0.01:5;
% simulation time = 5 seconds
sys = ss(A,B,C,D);
% construct a system model
[Y6, Tsim, X] = lsim(sys,u,T);
plot(Tsim,Y6)
% simulate
% plot the output vs. time
xlabel('Time (second)','fontsize',14);
ylabel('Rad/s','fontsize',14);
title('Figure 6 : Joint Deflection Velocity ')
grid on;
% (Angular Displacement Of Flexible Joint) - Frequency Response
figure(7);
Y1 = fft(Y4,500);
Pyy = Y1.* conj(Y1) / 500;
f = 100*(0:256)/500;
plot(f,Pyy(1:257)),grid
title('Figure 7 :Frequency Content Of Joint Deflection')
xlabel('Frequency (Hz)','fontsize',14)
ylabel('Radian','fontsize',14)
75
% (Angular displacement velocity of flexible joint) - Frequency Response
figure(8);
Y1 = fft(Y6,500);
Pyy = Y1.* conj(Y1) / 500;
f = 100*(0:256)/500;
plot(f,Pyy(1:257)),grid
title('Figure 8 : Frequency Content Of Joint Deflection Velocity')
xlabel('Frequency (Hz)','fontsize',14)
ylabel('Rad/s','fontsize',14)