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MINISTRY OF EDUCATION
AND TRAINING

MINISTRY OF NATIONAL
DEFENCE

MILITARY TECHNICAL ACADEMY

DUONG XUAN BIEN

DYNAMIC MODELLING AND CONTROL OF TWO-LINK
FLEXIBLE ROBOTS BY USING FINITE ELEMENT METHOD

DOCTOR OF PHILOSOPHY

HA NOI, 2019


MINISTRY OF EDUCATION
AND TRAINING


MINISTRY OF NATIONAL
DEFENCE

MILITARY TECHNICAL ACADEMY

DUONG XUAN BIEN

DYNAMIC MODELLING AND CONTROL OF TWO-LINK
FLEXIBLE ROBOTS BY USING FINITE ELEMENT METHOD

Major: Technical mechanic
Code: 9.52.01.03

DOCTOR OF PHILOSOPHY

SCIENCE SUPERVISORS:
1.

Associate Prof, Dr Chu Anh My

2.

Associate Prof, Dr Phan Bui Khoi

HA NOI, 2019


ACKNOWLEDGMENTS
I would like to express my deepest gratitude to Professor Chu Anh My
and Professor Phan Bui Khoi for their support, dedicated guide and research
orientation on this work.
I wish to thank all my colleagues from Advanced Technology Center,
Faculty of Mechanical Engineering, Faculty of Aerospace in Military
Technical Academy and School of Mechanical Engineering in Hanoi
University of Science and Technology for the help they gave me in the many
different occasions.
The greatly appreciation is to my family for their love and support.
Last but not least, I would like to thank all the others that are not
mentioned and helped me on this thesis.


CONFIRMATION BY AUTHOR
I pled that this thesis is my own research work. The results presented in
this work are honest and has not been published by anyone in any other
works. The information cited in this thesis is clearly stated origins.

August, 2019

Duong Xuan Bien


LIST OF SYMBOLS AND ABBREVIATIONS

L
l

,

i
e

Length of link i , length of each element of link i

i

Angle between link i −1 and link i

i

Number of links of robot, number of elements of link i

)

and joint variable of link i

n, n ,

Arbitrary point on the element j of link i

(t
i

i

x
(x ), m
m

= 1
4

Shape functions of element j
Elastic displacement at arbitrary point on element j

of

link i

wij (t, x)
u

Flexural displacement, slope displacement of node j and

u

node j + 1 of element j of link i , respectively

u

Flexural and slope displacement at end point of link i
Flexural and slope displacement at node k

u

k + 1 of element k of link i −1

u
u(i −1) f ,u(i −1)s

H

r,r
ij

0ij

r ,r
02r

02f

and node


Flexural and slope
displacement at the
end point of link

i

−1

transforms from the coordinate system Oi XYii
coordinate system Oi −1X i −1Yi −1
Position vector of arbitrary point on the element

General

d

(t ), (t )
i i

q

Position vector of the end point of link 2 in cases of rigid
and flexible models in the coordinate system O0X 0Y0

C(q, q)
Q
F (t),
i
*

q

e

KP,KI ,KD
m,m

Tij ,Ti ,T
T,T,T

P
ije

M
M
K

j of

link i in the coordinate systems Oi XYii and O0X 0Y0

homogeneous
transformation
matrix which

ie

to the

id

p


Kinetic energy of element j of link i , kinetic energy of
Translational and
rotational joint
variable of link i
Elastic displacement
vector of the element
j
of link i
and elastic
displacement vector
of link i
Generalized elastic
displacement vectors
of the element

j , of the link i and
of the system
Mass per length unit
of link i , mass of
motor i , mass of
the tip load

link i and kinetic energy of system
Elastic deforming kinetic energy of link i , kinetic
energy of motor driving link i and the tip load
Elastic deforming potential and gravitational potential
energy of element j of link i , potential energy of link i
and the system
Mass matrix of element j , link i and system.
Mass matrices of the motor and the tip load
Stiffness matrix of element j , link i and system.
Coriolis matrix
Generalized force/torque vector of the
system Driving force, torque at the joint i
Joint variable error vector, error vector in objective
function and Lyapunov function
Cross matrix of control parameters in PID controller


TABLE OF CONTENTS
Pages
PREFACE...................................................................................................................................... 1
CHAPTER 1. LITERATURE REVIEW OF FLEXIBLE ROBOT DYNAMIC
AND CONTROL......................................................................................................................... 7
1.1. Applications of flexible robots....................................................................................... 7
1.2. Classifying joint types of flexible robots...................................................................... 8
1.3. Classifying flexible robots............................................................................................ 11
1.4. Modeling methods.......................................................................................................... 13
1.5. Differential motion equations...................................................................................... 14
1.6. Recent works on flexible robots.................................................................................. 15
1.7. Position accuracy of motion of flexible robots........................................................ 19
1.8. Comments and problems............................................................................................... 20
Conclusion of chapter 1....................................................................................................... 21
CHAPTER 2. DYNAMIC MODELING OF THE PLANAR FLEXIBLE
ROBOTS...................................................................................................................................... 22
2.1. Kinematic of the planar flexible robots..................................................................... 22
2.2. Dynamics of the planar flexible robots..................................................................... 38
Conclusion of chapter 2....................................................................................................... 58
CHAPTER 3. DYNAMIC ANALYSIS AND POSITION CONTROL OF THE
PLANAR TWO-LINK FLEXIBLE ROBOTS............................................................... 59
3.1. Boundary conditions...................................................................................................... 59
3.2. Forward dynamic........................................................................................................... 61
3.3. Inverse dynamic.............................................................................................................. 79
3.4. Position control system of the planar serial multi-link flexible robots..............86
Conclusion of chapter 3....................................................................................................... 99
CHAPTER 4. EXPERIMENT........................................................................................... 101
4.1. Objective and experimental model........................................................................... 101
4.2. Parameters, equipment and method of measuring............................................... 103
4.3. System connection diagram....................................................................................... 105


4.4. Experimental orders.................................................................................................... 107
4.5. Method of handling the measurement data............................................................ 108
4.6. Experimental results.................................................................................................... 110
Conclusion of chapter 4.................................................................................................... 115
CONCLUSION AND SUGGESION................................................................................ 116
LIST OF THE RESEARCH PAPERS OF THE AUTHOR..................................... 118
REFERENCES........................................................................................................................ 121
APPENDICES......................................................................................................................... 139


LIST OF TABLES

Table 2. 1. The parameters  i , u ( i  1)s , u (i 1)f , i ,ai depending on types of joints
.....................................................................................................................

Table 3.
Table 3.
Table 3. 3. The maximum elastic displacements at the ending points of the links

1. The dy
2. The m

.....................................................................................................................

Table 3. 4.
Table 3. 5.
Table 3. 6.
Table 3. 7.
Table 3. 8.
Table 3.
Table 3.
Table 3.
Table 3.

9.
10.
11.
12.

The pa
The len
The ma
The pa
The pa
The pa


LIST OF FIGURES
Figure 0. 1. The structure of the thesis............................................................................... 5
Figure 0. 2. The order executing the thesis....................................................................... 5
Figure 1. 1. Flexible robots.................................................................................................... 7
Figure 1. 2. The flexible robot in space............................................................................. 8
Figure 1. 3. Flexible robot in medicine.............................................................................. 8
Figure 1. 4. Rotational joint................................................................................................... 9
Figure 1. 5. Translational joint type Pa............................................................................. 9
Figure 1. 6. Translational joint type Pb............................................................................. 9
Figure 1. 7. The single-link flexible robot with rotational joint.............................11
Figure 1. 8. The single-link flexible robot with translational joint....................... 11
Figure 1. 9. The two-link flexible robots with only rotational joints....................12
Figure 1. 10. The two-link flexible robots consist translational joints................12
Figure 1. 11. The planar serial multi-link flexible robots......................................... 12
Figure 1. 12. The parallel-link flexible robots.............................................................. 13
Figure 1. 13. The mobile fiexlible robots........................................................................ 13
Figure 1. 14. Flexible planar closed mechanism [8]................................................. 15
Figure 1. 15. Spring-mass system [45]............................................................................ 16
Figure 1. 16. The single-link flexible robot with joint Pa [133]...........................17
Figure 1. 17. The two-link flexible robot Quanser...................................................... 17
Figure 1. 18. The two-link flexible robot with rotational joints.............................. 17
Figure 1. 19. The flexible robot with rotational and translational joints...........18
Figure 2. 1. A generalized schematic of an arbitrary pair of flexible links......23
Figure 2. 2. Structure I........................................................................................................... 29
Figure 2. 3. Structure II......................................................................................................... 30
Figure 2. 4. Structure III........................................................................................................ 31
Figure 2. 5. Structure IV........................................................................................................ 32
Figure 2. 6. Structure V.......................................................................................................... 33
Figure 2. 7. Structure VI........................................................................................................ 34
Figure 2. 8. Structure VII...................................................................................................... 35


Figure 2. 9. Structure VIII..................................................................................................... 36
Figure 2. 10. Structure IX..................................................................................................... 37
Figure 2. 11. The two-link flexible robot with rotational joints [64]....................56
Figure 2. 12. Parts of matrix M2j..................................................................................... 143
Figure 3. 1. The position of the element k and the robot type VII........................60
Figure 3. 2. The solving algorithm without the joint Pb............................................ 63
Figure 3. 3. The solving algorithm with the joint Pb.................................................. 63
Figure 3. 4. The schematic of the solving forward dynamic on SIMULINK .. 64
Figure 3. 5. The torque at joint 1....................................................................................... 65
Figure 3. 6. The torque at joint 2....................................................................................... 65
Figure 3. 7. The value of joint 1 variable........................................................................ 66
Figure 3. 8. The value of joint 2 variable........................................................................ 66
Figure 3. 9. The value of flexural displacement at the end of link 1.....................66
Figure 3. 10. The value of slope displacement at the end of link 1.......................66
Figure 3. 11. The value of flexural displacement at the end of link 2..................66
Figure 3. 12. The value of slope displacement at the end of link 2.......................66
Figure 3. 13. The position of the end-effector in OX.................................................. 67
Figure 3. 14. The position of the end-effector in OY.................................................. 67
Figure 3. 15. The flexible robot type IV........................................................................... 69
Figure 3. 16. Schematic of solving forward dynamic in SIMULINK....................69
Figure 3. 17. The driving force rule.................................................................................. 70
Figure 3. 18. The driving torque rule............................................................................... 70
Figure 3. 19. The value of translational joint................................................................ 71
Figure 3. 20. The value of rotational joint...................................................................... 71
Figure 3. 21. The value of flexural displacement......................................................... 71
Figure 3. 22. The value of slope displacement.............................................................. 71
Figure 3. 23. Position deviation in OX............................................................................ 72
Figure 3. 24. Position deviation in OY............................................................................. 72
Figure 3. 25. The value of translational joint................................................................ 73
Figure 3. 26. The value of rotational joint...................................................................... 73
Figure 3. 27. The value of flexural displacement......................................................... 73
Figure 3. 28. The value of slope displacement.............................................................. 73
Figure 3. 29. The position deviation in OX.................................................................... 73


Figure 3. 30. The position deviation in OY.................................................................... 73
Figure 3. 31. The flexible robot type III........................................................................... 75
Figure 3. 32. Schematic to solve the forward dynamic of the system in the
SIMULINK.................................................................................................................................. 76
Figure 3. 33. The rules of driving torque and force.................................................... 77
Figure 3. 34.The rotational joint variable displacement........................................... 77
Figure 3. 35. The translational joint variable displacement................................... 77
Figure 3. 36. The value of the flexural displacement................................................. 78
Figure 3. 37. The value of the slope displacement...................................................... 78
Figure 3. 38. The position of end-effector in OX......................................................... 78
Figure 3. 39. The position of end-effector in OY.......................................................... 78
Figure 3. 40. The solving inverse dynamic schematic in SIMULINK..................81
Figure 3. 41. The translational joint variable............................................................... 82
Figure 3. 42. The rotational joint variable..................................................................... 82
Figure 3. 43. The value of driving force.......................................................................... 83
Figure 3. 44. The value of driving torque....................................................................... 83
Figure 3. 45. The deviation of force between rigid and flexible models.............83
Figure 3. 46. The deviation of torque between rigid and flexible models..........83
Figure 3. 47. The flexural displacement value.............................................................. 83
Figure 3. 48. The slope displacement value................................................................... 83
Figure 3. 49. The rotational joint variable value......................................................... 84
Figure 3. 50. The translational joint variable value................................................... 84
Figure 3. 51. The driving torque value............................................................................ 84
Figure 3. 52. The driving force value............................................................................... 84
Figure 3. 53. The torque deviation value....................................................................... 85
Figure 3. 54. The force deviation value.......................................................................... 85
Figure 3. 55. The flexural displacement value.............................................................. 85
Figure 3. 56. The slope displacement value................................................................... 85
Figure 3. 57. Schematic of the GA..................................................................................... 88
Figure 3. 58. The control schematic PID with the GA............................................... 91
Figure 3. 59. The translational joint variable............................................................... 94
Figure 3. 60. The rotational joint variable..................................................................... 94
Figure 3. 61. The flexural displacement.......................................................................... 95
Figure 3. 62. The slope displacement............................................................................... 95
Figure 3. 63. The driving force........................................................................................... 95


Figure 3. 64. The driving torque........................................................................................ 95
Figure 3. 65. The position of end-effector in OX......................................................... 95
Figure 3. 66. The position of end-effector in OY.......................................................... 95
Figure 3. 67. The rotational joint variable..................................................................... 97
Figure 3. 68. The translational joint variable............................................................... 97
Figure 3. 69. The flexural displacement.......................................................................... 98
Figure 3. 70. The slope displacement............................................................................... 98
Figure 3. 71. The position end-effector point in OX................................................... 98
Figure 3. 72. The position end-effector point in OY................................................... 98
Figure 3. 73. The driving torque........................................................................................ 99
Figure 3. 74. The driving force........................................................................................... 99
Figure 4. 1. Experimental model..................................................................................... 101
Figure 4. 2. Lead screw system........................................................................................ 102
Figure 4. 3. Step motor at the rotational joint............................................................ 102
Figure 4. 4. Lead screw....................................................................................................... 102
Figure 4. 5. DC motor GB37-3530................................................................................. 102
Figure 4. 6. Step motor NEMA 17................................................................................... 102
Figure 4. 7. Encoder LPD3806........................................................................................ 103
Figure 4. 8. Flex sensor...................................................................................................... 103
Figure 4. 9. Flex sensor FSL0095-103-ST................................................................... 105
Figure 4. 10. System connection diagram.................................................................... 105
Figure 4. 11. Principle diagram inside Arduino 2560............................................. 106
Figure 4. 12. LABVIEW diagram.................................................................................... 107
Figure 4. 13. Flex sensor circuit...................................................................................... 110
Figure 4. 14. Driving force................................................................................................ 111
Figure 4. 15. Driving torque............................................................................................. 111
Figure 4. 16. The value of translational joint variable........................................... 111
Figure 4. 17. The value of rotational joint variable................................................. 112
Figure 4. 18. The value of flexural displacement...................................................... 112
Figure 4. 19. The value of translational joint variable........................................... 113
Figure 4. 20. The value of rotational joint variable................................................. 114
Figure 4. 21. The value of flexural displacement...................................................... 114
P1. 1. Driving torque [64] .......................................................................... 149


P1.
P1.
P1.
P1.
P1.
P1.
P1.
P1.
P1.
P1.
P1.
P1.
P1.

2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.

Rotational joint variable (1 element) ............
Flexural displacement (1 element) ................
Rotational joint variable (7 elements) ..........
Flexural displacement (7 elements) ..............
Rotational joint variable (1, 3, 5, 7 elements)
Flexural displacement (1, 3, 5, 7 elements) ..
Rotational joint variable (1 element) ............
Flexural displacement (1 element) ................
Rotational joint variable (7 eleme
Flexural displacement (7 element)
Rotational joint variable (1, 3, 5, 7
Flexural displacement (1, 3, 5, 7el
PID control law in SIMULINK ....


1

PREFACE
In the past several years, lots of robots have been designed and produced
all over the world because of their important applications. Nowadays, using
robots is more and more popular in various fields.
In the literature, most of the designed robots are considered with an
assumption that all the links of the robots are rigid bodies. This is to simplify
the modelling, analysis and control for the robot systems. Such robotic
systems with rigid links are the so-called rigid robots.
In fact, the elastic deformation always exists on the links of robots during
the robot operation. This elastic factor has some certain effects on motion
accuracy of robots and these effects depend on the structure and characterized
motion of robots. The robots, of which the effect of elastic deformation on
links is taken into account, are called the flexible robots.
In recent decades, there have been several researches addressing the
dynamics and control of the flexible robots. The quality enhancement
modeling and controlling are mainly requested by researchers and designers.
Because of the large applications, future potentials and challenges in
modeling and controlling of the flexible robots, this dissertation has tried to
mention and solve some specific problems in kinematic, dynamic modeling
and position control of planar flexible robots based multi-bodies dynamic,
mechanically deformed body, finite element theory, control and numerical
computation method. The results of this research are referenced in designing
and producing the flexible robots used in some reality applications.
Motivation
Modern designing always aims at reducing mass, simplifying structure and
reducing energy consumption of system, especially in robotics. These targets
could lead to lowing cost of the material and increasing the operating capacity.


2

The priority direction in robots design is optimal structures with longer length
of the links, smaller and thinner links, more economical still ensuring ability
to work. However, all of these structures such as flexible robots are reducing
rigidity and motion accuracy because of the effect of elastic deformations.
Therefore, taking the effects of elastic factor into consideration is absolutely
necessary in kinematic, dynamic modeling, analyzing and controlling flexible
robots.
Because of complexity of modeling and controlling flexible robots, the
single-link and two-link flexible robots with only rotational joints are mainly
mentioned and studied by most researchers. A few others considered the
single-link flexible robot with translational joint. It is easy to realize that
combining the different types of joints of flexible robots can extend their
applications, flexibility and types of structure. However, the models
consisting of rotational and translational joints will make the kinematic,
dynamic modeling and controlling become more complex than models which
have only rotational joints.
There are two main modeling flexible robot methods which are assumed
modes method (AMM) and finite element method (FEM). Most studies used
AMM in modeling the single-link and two-link flexible robots with only
rotational joints because of simplicity and high accuracy. The FEM is recently
mentioned because of the strong development of computer science. This
method has shown the high efficiency and generality in modeling flexible
robots which have more than two links, varying cross section of links, varying
boundary conditions and controlling in real time especially combining
different types of joints.
The control of flexible robots is the most important problem in warranting
the robots moves following position or trajectory requests. The errors of motion


3

are appeared by errors of joints and elastic deformations of the flexible links.
Therefore, developing the control system for flexible robots is necessary,
especially for models with combining different types of joints.
The above raised critical issues and problems lead to the motivations of
developing a new kinematic and dynamic formulation for the multi-link
flexible robots. It is necessary to establish generalized kinematic modeling
method for planar flexible robots which have links connected in series and
consist rotational and translational joints by using FEM. The dynamic
equations can be built on that basis. Dynamic behaviors of these robots are
considered based on dynamic analyzing under varying payload, length of
flexible link and boundary conditions. Furthermore, position control system is
designed warranting requirement.
Objective of the dissertation
The first objective is to formulate the kinematic and dynamic model for a
planar flexible robot arm which consists of the rotational and translational
joints, by using the FEM/Langrangian approach.
The second objective is to investigate the position control for the flexible
robot arm with respect to the deformation of the robot links.
Main contents of the dissertation
The main contents of the dissertation are the followings.
-

The general homogeneous transformation matrix is built to model the

kinematic and dynamic of planar flexible robots. FEM and Lagrange’s
equations are used to build the dynamic equations. Extended assembly
algorithm is proposed to create the global mass matrix and global stiffness
matrix.


4

-

The forward and inverse dynamic will serve to analyze the dynamic

behavior of flexible robots which are mentioned above under varying payload,
length of flexible links and boundary conditions.
-

The extended PID controller is designed to control the position of planar

flexible robots. The control law is determined and stably proved based on
Lyapunov’s theory. The parameters of controller are found by using genetic
algorithm.
-

A flexible robot is designed and produced. The results of forward and

inverse dynamic experiments are used to evaluate results of calculations.
The contents can be shown as Fig. 0.1.
Methodology
The researching theory, numerical calculation and experimental method
are used to execute the contents of dissertation. The order of executing the
dissertation is shown as Fig. 0.2.
Contributions of the dissertation
Fistly, this dissertation presents the generalized kinematic, dynamic
modeling and building the motion equations of planar flexible robots with
combining rotational and translational joints.
Secondly, forward and inverse dynamic analyzing for these flexible robots
under varying payload, length of flexible links and boundary conditions.
Building the position control PID system which has parameters found by
using optimal algorithm (Genetic algorithm - GA).
Thirdly, designing and producing a planar flexible robot with the first joint
is traslational joint and the other is rotational joint. The results of experiments
are used to evaluate results of calculations.


5

Significant impacts of the dissertation
Kinematic, dynamic and control problems of planar flexible robots with
combining different types of joints and varying joints order are solved based
on multi-bodies dynamic, mechanically deformed body, finite element theory,
control and numerical computation method.
The results of this research allow determining the values of elastic
displacements at the arbitrary point on flexible links and evaluating the effect
of these values on position accuracy of flexible robots. Furthermore, this
dissertation can be referenced in designing and producing the flexible robots
which can be used in some practical applications.

Outline of the dissertation
The dissertation organization includes abstract, four chapters, conclusions,
recommendations, references and appendices.
Chapter 1. Literature review of flexible robot dynamics and control
The background information of flexible robots such as their applications,
characteristics, classification, and modeling methods are presented in this


6

chapter. The status of research in our country and in the world is taken into
account to determine the problems focused and solved in this dissertation.
Chapter 2. Dynamic modeling of the planar flexible robots
This chapter focuses on kinematic, dynamic modeling of planar flexible
robot with combining different types of joints. The general homegeneous
transformation matrix is established. FEM and Lagrange’s equations are used
to build the dynamic equations. Extended assembly algorithm is proposed to
create the global mass matrix and global stiffness matrix. This algorithm is
proved accurately by comparing with previous research.
Chapter 3. Dynamic analysis and position control of the planar flexible
robots
Two main problems are solved in this chapter. On the one hand, the
forward and inverse dynamic are considered to analyze the dynamic behavior
of flexible robots which are mentioned above under the variation of payload,
length of flexible links and boundary conditions. On the other hand, the
extended PID controller is designed to control the position of planar flexible
robots. The control law is determined and stably proved based on Lyapunov’s
theory. The parameters of controller are found by using genetic algorithm.
Chapter 4. Experiments
This last chapter presents designing and producing a planar two-link
flexible robot in which the first joint is translational joint and the second
joint is rotational joint. The results of forward and inverse dynamic
experiments are used to evaluate results of calculations.


7

CHAPTER 1. LITERATURE REVIEW OF FLEXIBLE ROBOT
DYNAMICS AND CONTROL
The background information of flexible robots such as their applications,
characteristics and classifying, modeling methods is presented in this chapter.
The background of research in our country and all over the world is used to
determine the problems which are focused and solved in this dissertation.
1.1. Applications of flexible robots
Researching on flexible robots (Fig. 1.1) has been started since 1980 [76],
[80], [113], [127], [128], [130], [131]. Applications of flexible robots can be
seen in [34], [86], [91], [137], [138]. The major applications of these robots
are in space, medicine and nuclear technology.

Figure 1. 1. Flexible robots
The Figure 1.2 describes a flexible robot used in space technology. Energy
consumption is decreased radically when flexible robots are catapulted into
the space because of a small number of these robots. The workspace of
flexible robots is extended based on increasing the length of flexible links.
The control system is less complex because there are only a few links. For
example, the Remote Robot System (RMS) [34] is used to serve many
important tasks in space by NASA agency. This flexible robot is executed in
space with low frequency about 0.04 (Hz) to 0.35 (Hz), the angle velocity is
about 0.5 (degree/second). The mass of RMS is 450 (kg). The mass of tip load
is 27200 (kg).


8

The flexible robots are also used in microsurgery in medicine dealing with
small and narrow position of human body. These surgeries are extremely hard
difficult for doctors in a long time such as neurosurgery, neck and heart
surgery (Fig. 1.3).

Figure 1. 2. The flexible robot in Figure 1. 3. Flexible robot in space
medicine
The flexible robots are suitable for some important and dangerous tasks in
nuclear field [137]. These robots are used to bring and assemble radiative
rods, reduce radiative with small driving energy, small interactional force with
surroundings, flexibility and high accuracy. The other applications of flexible
robots are in army, machining and construction, etc.
1.2. Classifying joint types of flexible robots
The classification of flexible robots becomes easier based on determining
the main types of joints used to design the robots.
Considering the robot with n flexible links. The arbitrary link i − 1 and link

i

are connected by joint i which is rotational joint (Fig. 1.4) or
translational
joint type Pa (Fig. 1.5) or translational joint type Pb (Fig. 1.6). Generally, the
kinematics of a flexible link i depend on the motion of joint which connects
the link i with the previous link i − 1 and the elastic deformation on the link


i − 1.


9

Figure 1. 4. Rotational joint

Figure 1. 5. Translational joint type Pa

Figure 1. 6. Translational joint type Pb


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