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 pro
Trang 1MILITARY 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
Trang 2MILITARY 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
Trang 3orientation 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
Trang 4this 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
Trang 5 Angle between link i − and link i 1
, , ( )i i
n n t Number of links of robot, number of elements of link i
and joint variable of link i
x Arbitrary point on the element j of link i
Flexural displacement, slope displacement of node j
and node j +1 of element j of link i , respectively
r r Position vector of arbitrary point on the element j of
link i in the coordinate systems O XY and i i i O X Y 0 0 0
02r, 02f
r r Position vector of the end point of link 2 in cases of rigid
and flexible models in the coordinate system O X Y 0 0 0
Trang 6and elastic displacement vector of link i
( ), ( ), ( )
q q q Generalized elastic displacement vectors of the element
j, of the link i and of the system
m m m Mass per length unit of link i , mass of motor i , mass of
the tip load
, ,
ij i
T T T Kinetic energy of element j of link i , kinetic energy of
link i and kinetic energy of system
, ,
ie id p
T T T Elastic deforming kinetic energy of link i , kinetic
energy of motor driving link i and the tip load
e e Joint variable error vector, error vector in objective
function and Lyapunov function , ,
K K K Cross matrix of control parameters in PID controller
Trang 7CHAPTER 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
Trang 8Conclusion of chapter 4 115
CONCLUSION AND SUGGESION 116
LIST OF THE RESEARCH PAPERS OF THE AUTHOR 118
REFERENCES 121
APPENDICES 139
Trang 926
Table 3 1 The dynamic parameters of flexible robot type I (continuous) 65
Table 3 2 The mass ratios between the flexible links and tip load 67
Table 3 3 The maximum elastic displacements at the ending points of the links 67
Table 3 4 The parameters of the flexible robot type IV 70
Table 3 5 The length of the links in two cases 71
Table 3 6 The maximum values in two cases 74
Table 3 7 The parameters of flexible robot type III 75
Table 3 8 The parameters of the flexible robot type IV 92
Table 3 9 The parameters of the GA and the position PID controller 93
Table 3 10 The comparative results the control quality between two cases 94 Table 3 11 The parameters of the GA and the position PID controller 96 Table 3 12 The comparative results the control quality between two cases 97
Trang 10
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 P a 9
Figure 1 6 Translational joint type P b 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 P a [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
Trang 11Figure 2 12 Parts of matrix M 143 2j
Figure 3 1 The position of the element k and the robot type VII 60
Figure 3 2 The solving algorithm without the joint P b 63
Figure 3 3 The solving algorithm with the joint P b 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
Trang 12SIMULINK 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
Trang 13Figure 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
Trang 14P1 5 Flexural displacement (7 elements) 153
P1 6 Rotational joint variable (1, 3, 5, 7 elements) 154
P1 7 Flexural displacement (1, 3, 5, 7 elements) 154
P1 8 Rotational joint variable (1 element) 155
P1 9 Flexural displacement (1 element) 155
P1 10 Rotational joint variable (7 element) 155
P1 11 Flexural displacement (7 element) 155
P1 12 Rotational joint variable (1, 3, 5, 7 element) 155
P1 13 Flexural displacement (1, 3, 5, 7element) 155
P1 14 PID control law in SIMULINK 158
Trang 15PREFACE
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
Trang 16The 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
Trang 17are 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
Trang 18- 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
Trang 19Significant 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
Figure 0 1 The structure of the
dissertation
Figure 0 2 The order executing the
dissertation
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
Trang 20chapter 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
Trang 21CHAPTER 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)
Trang 22The 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
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 P (Fig 1.5) or translational joint type a P (Fig 1.6) Generally, the b
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 1
i −
Trang 23Figure 1 4 Rotational joint
Figure 1 5 Translational joint type P a
Figure 1 6 Translational joint type P b
Trang 24For the case in which the two links are connected by a rotational or a translational joint P (Fig 1.4 and Fig 1.6), the motion of link b i depends on the motion of the joint i and the elastic deformation at the distal end of the link 1
i − Nevertheless, in the case of translational joint P (Fig 1.5), the motion a
of the link i does not depend on the elastic deformation at the distal end of the previous link i −1but depends on the elastic deformation of the sliding element
on the link i −1 This element varies along the length of the link i −1, with respect to time
For the cases of the rotational joint and the translational joint P (Fig 1.4 a
and Fig 1.5), it is usually assumed that the elastic displacements at the first node of the first element on the link i equal to zero However, for the case of
the translational joint Pb (Fig 1.6), the element of zero elastic deformation is the sliding element of the link I through the fixed translational joint Obviously, the elastic effects of links associated with the use of the three joint types should
be taken into account when working on the kinematic and dynamic modeling for a general flexible robot that consists of all three joint types There are some differences in solving the motion equations because of the differences between types of joints which are considered above
The single-link, two-link and multi-link flexible robots with only rotational joints are investigated in many studies for example [10], [12], [15], [24], [28], [34], [37], [66], [72], [73], [88], [100], [103], [136], … There are some studies mentioning single-link flexible robots with translational joint P or a P [13], b
[23], [29], [73], [116], [133] However, combining types of joints in flexible robots is not yet fully and clearly considered in modeling and controlling
Trang 251.3 Classifying flexible robots
The flexible robots are classified according to the number of joints and
links, types of joints and their structures
1.3.1 The flexible robots with regard to number of links and joints
1 The single-link flexible robots
The single-link flexible robots are clearly investigated [13], [14], [26], [36], [40], [48], [55], [64], [82], [101], [113], … The Fig 1.7 shows the single-link flexible robot with rotational joint and the Fig 1.8 describes the other with translational joint P b
Figure 1 7 The single-link flexible robot with rotational joint
Figure 1 8 The single-link flexible robot with translational joint
2 The two-link flexible robots
The two-link flexible robots are studied in [15], [18], [23], [25], [29], [48],
[65], [68], [75], [90], [95], [98], [116], 126], [135], [138], etc
Trang 26Figure 1 9 The two-link flexible
robots with only rotational joints
Figure 1 10 The two-link flexible
robots consist translational joints
The two-link flexible robots with only rotational joints are mainly studied (Fig 1.9) and a few others are mentioned consisting of the rotational and translational joints (Fig 1.10)
3 The planar serial multi-link flexible robots
The multi-link flexible robots (Fig 1.11) are studied in [10], [12], [15], [24], [28], [37], [66], [70], [73], [88], [100], [136], …
Figure 1 11 The planar serial multi-link flexible robots
1.3.2 Classifying the flexible robots according to structures
1 The series-link flexible robots
The flexible robots with series links are shown as Fig 1.11
Trang 272 The parallel-link flexible robots
The flexible robots with parallel links are described as Fig 1.12 [84]
Figure 1 12 The parallel-link flexible robots
The parallel robots are widely used in many applications such as entertainment, home services, flying machines, submarines, assembling robots, etc Compared with serial robots, parallel robots are provided with a series of advantages in terms of heavy payload, high positional accuracy and so on
3 The mobile flexible robots [85] (Fig 1.13)
Figure 1 13 The mobile flexible robots
Flexible robots with moving base such as macro-micro robots, space robots and underwater robotic vehicles can be used for extending the workspace in repair and maintenance, inspection, welding, cleaning, and machining operations
1.4 Modeling methods
In general, the flexible robots are the continuous systems characterized by unlimited degrees of freedoms It is difficult to accurately describe the system
Trang 28Therefore, these systems must be discretized into the finite elements to analyze the kinematic and dynamic
The AMM and FEM are mainly used in kinematic and dynamic modeling The single-link and two-link flexible models with only rotational joints are usually described by AMM because of its efficiency [10], [18], [26], [27], [28], [29], [30], [32], [33], [37], [38], [40], [47], [59], [66], [70], [73], [75], [88], [131], [136], [133], [135], [141], … There are many investigations using FEM
in modeling the flexible robots [12], [15], [23], [24], [48], [64], [77], [103], [130], …The authors in [39], [80] show that FEM is more suitable than AMM for modeling the flexible robots which are combined rotational and translational joints or have many links with varied cross section area In FEM, each flexible link is divided into finite elements The kinetic and potential of elements are determined The mass and stiffness matrices are also calculated based on variables of the joints and the elastic displacements, respectively The number of elements on each link are different leading to the size of these matrices being different Therefore, the global mass and stiffness matrices are constructed with largely calculated volumes, complex transformations and assembly especially difficult for the flexible robots which are combined rotational and translational joints or have many links with varied cross section area These global matrices are used to determine the dynamic equations of system
1.5 Differential motion equations
The differential motion equations can be described as
- The Newton-Euler equations [13], [15], [29], [103], [115], [135],
- The Lagrange-Euler equations [10], [12], [14], [18], [23], [24], [25],
[28], [33], [37], [48], [64], [66], [68], [72], [88], [90], [98], [136], …
- The Gibbs-Appel equations [73], [100], [105], [116], …
Trang 29- The Kane equations [68]
The AMM can be combined with Lagrange-Euler equations [10], [28], [37], [66], [88], [98], [101], [136], with Newton-Euler equations [13], [29], [40], [47], [135] and with Gibbs-Appel equations [73], [100] However, these combinations are executed for single-link or two-link flexible robots with only
rotational joints
1.6 Recent works on flexible robots
In our country, flexible structures were studied a few decades ago [1], [2], [3], [4], [8] Khang and Khiem [2], [3], [4] had numerically evaluated the vibration of elastic connecting rods in a six-link mechanism The conditions of dynamic stability were checked by using numerical method Khang and Nam [8] studied on the vibration of planar mechanism with an elastic link (Fig 1.14) based on multi-body dynamics theory The AMM and FEM are used to build the differential motion equations of four-link and six-link planar closed mechanisms The linearized method is proposed to analyze these mechanisms The PD control is designed to reduce the vibration of the flexible link Hoang [9] presented the inverse dynamic of a two-link flexible robot consisting the translational and rotational joints by using the FEM and Lagrange equations Each flexible link has only an element The effects of the variety of laws of variable joints on driving laws are solved
Figure 1 14 Flexible planar closed mechanism [8]
Trang 30The dynamic modeling and controlling of the flexible robots attract many researchers over the world The background of flexible robots is described in
some particular studies such as in [39], [76], [80], [113], [127], [128], [130]
1 The single-link flexible robots
The single-link flexible robots are clearly presented in [13], [14], [26], [36], [40], [48], [55], [64], [82], [101], [113], …Kalker [14] investigated building the nonlinear dynamic equations and designing the control system for a single-link robot Kwon and Book [36] addressed the inverse dynamic of system in the time domain The dynamic behaviors of a single-link robot with tip load are analyzed by using FEM and Lagrange equations in [44], [48], [64] The dynamic modeling is studied by using AMM in [40], [55], [101] Trautt [56] also developed the inverse dynamic of flexible single-link robot considering the Coulomb friction force and backflash factor The loop algorithm Newmark
is applied to solve inverse dynamic in the frequency domain Gee [42] used the genetic algorithm to optimize the parameters of the position control system for the single-link robot based on Lyapunov’s stable theory Zhu [45] used the backstepping method to control the position of end-effector point with tip load The authors applied previous work to design the feedback nonlinear control presented in [50] The flexible link is lumped to a spring-mass system (Fig 1.15)
Figure 1 15 Spring-mass system [45]
Trang 31Kuo and Lin [69] focused on designing the control system by using fuzzy controller The Neural network algorithm is applied to control the system in [79] The elastic displacements in three dimensions are mentioned in [129] Ju and Li [133] studied the single-link flexible robot with translational joint P a
(Fig 1.16) and base m The system is driven by force ( ) b F t T
In summary, the single-link flexible robots with rotational joint are mainly mentioned using AMM and FEM The linearized methods are applied in almost
studies to reduce the complexity of models
Figure 1 16 The single-link flexible robot with joint P [133] a
2 The two-link and multi-link flexible robots
The two-link planar flexible robots are investigated in [15], [18], [23], [25], [29], [48], [65], [68], [75], [90], [95], [98], [116], 126], [135], [138], etc The kinematic and dynamic of the two-link and multi-link flexible robots are analyzed based on the methods which are also used for the single-link models The two-link flexible robots with only rotational joints are the major objectives (Fig 1.17 and Fig 1.18) The number of researches such as [65], [98], [119], [121], [141] are presented using AMM, the others [12], [43], [93], [97], [104] are studied by FEM Usoro [12] presented the FEM and Lagrangian approach for the mathematical modeling of the two-link flexible robot with rotational
Trang 32joints This paper is one of the first studies regarding flexible robots The comparison between the rigid and flexible models is discussed in [58]
Figure 1 17 The two-link flexible
Figure 1 19 The flexible robot with rotational and translational joints
Al-Bedoor and Khulief [48] proposed the transition element to model these structures This element includes two parts The first part is rigid and inside the translational joint, the other is flexible and outside the joint The length of this element is considerable to warrant clear difference between two parts (the length of flexible link is 3.6m, the length of each element is 0.9m) The recursive equations Gibbs-Appel and AMM are applied constructing the motion equations of the flexible robots in [116]
Trang 33There are many investigations on the controlling problems for two-link flexible robots such as [43], [50], [58], [69], [79], [121], [126], [129], [141], etc Gee and Lee [43] showed a class of robust stable controllers to control the tip position of a multi-link flexible robots The controllers are derived by using
a basic relationship of system energy and are independent of the system dynamics The PID and PD controllers are developed in [58] The optimal mixed sensitivity algorithm H2/H∞ is considered and executed in MATLAB [121] Kherraz [126] designed the control system combining the sliding mode and fuzzy logic methods Lochan [141] used the AMM and neural network controller to model and control the flexible robots
The multi-link planar flexible robots are developed in [10], [12], [15], [24], [28], [37], [66], [70], [73], [88], [100], [136], [143], [144], [145], [146], [147], [148], etc However, all of studies focused on the flexible robots with only rotational joints The simulated and calculated results of these researches are presented for the two-link flexible robots Some of the studies mentioned the planar flexible parallel robots [84], mobile robots [85], flexible joints [31], [46], [61], [81], [102] and the effects of stiffness of flexible links [71], [102], [140]
A few researches provided the results of experiments [11], [16], [53], [67]
In summary, the two-link and multi-link flexible robots with only rotational joints are mainly focused with respect to modeling and controlling A few studies consider the flexible robots consisting of the translational joint P but b
not yet mention the order of joints
1.7 The position accuracy of motion of flexible robots
Finding solutions to warrant the accuracy of motion of robots in general and flexible robots in particular is extremely important because the errors always exist in technique These solutions are usually proposed based on the dynamics modeling and control designing The motion errors appear because
Trang 34of many reasons such as the payload, the inertia, the manufacturing errors, the friction joints and the elastic deformations, etc The effects of elastic deformations on motion accuracy of flexible robots are sizable The accuracy level of the movement equations and the correction of control law decide the efficiency of ensuring position accuracy of flexible robots Therefore, the dynamic modeling based on FEM or AMM and using intelligent control system
are popular in almost studies as their targets
1.8 The comments and the problems
A few main comments can be discussed below after considering a number
of studies on flexible robots in the past
Firstly, the number of motion equations and the variables are increased when mentioning the effects of elastic deformations The derivative equations solving method and control designing are more complex than rigid robots Secondly, the single-link, two-link and multi-link flexible robots with only rotational joints are studied a lot on modeling and controlling The link flexible robots with translational joint are also considered in dynamic modeling There
is not any research investigating generally and clearly the flexible robots combining the types of joints (rotational joint, translational joint P and a
translational joint P ) and their order b
Thirdly, the flexible links in almost all studies are usually assumed only an element in FEM to reduce the number of variables and the complexity of assembly the global matrices and solving the dynamic equations However, the generalization and the advantages of FEM are not yet expressed especially considering the flexible robots with many links, varying cross section area or determining the values of elastic displacements at the arbitrary point on any flexible link
Trang 35Although there are many problems which must be studied on modeling and controlling for the flexible robots in general and these robots combining the types of joints in particular, this dissertation only focuses on some problems as follow
- The kinematic and dynamic modeling of multi-link planar flexible robots which consist of different types of the joints and mention the order of these joints The dynamic behaviors of these robots are analyzed under the varying
of payload, the ratios of the length of links and boundary conditions
- The extended position control system is designed based on classic PID controller with its parameters optimized by using the genetic algorithm
- A specific flexible robot is designed and manufactured to execute some experiments The results of these experiments are used to evaluate results of calculations
Conclusion of chapter 1
This chapter determined the objectives and contents of the dissertation based on reviewing modeling and controlling of the flexible robots in our country and over the world
Trang 36CHAPTER 2 DYNAMIC MODELING OF PLANAR FLEXIBLE ROBOTS
In this chapter, kinematic and dynamic modeling of the planar flexible robots are presented The links are connected in series via a rotational joint or
a translational joint which has two types (Pa and Pb) Note that, the order of joints is considered FEM and Lagrange’s equations are used to build the mechanical model and dynamic equations Extended assembly algorithm is proposed to create the global mass matrix and global stiffness matrix This algorithm is proved accurately by calculation following previous research Kinematic and dynamic modeling are analyzed with some assumptions such as
- The links are elastic beam, homogenous material and constant cross section area
- Elastic deformation of links is small The shear deformation of the beam is neglected
- Each node of element only has the flexural and slope displacement
- The tip load is concentrated mass
- The effect of length of translational joint is not considered yet
- The joints are rigid
2.1 Kinematic of the planar flexible robots
2.1.1 The general homogeneous transformation matrix
Let us consider the flexible planar robot consisting of n n( Z) links and
n joints The arbitrary link i −1 is connected with a link i by a joint
( 1 )
i i = n which can be the following three joint types: rotational joint (R),
translational joints P and a P (Figure 2.1) The link b i with a length L is i
divided into n elements of the equal length i l Each element ie j of the link i
has two nodes which are j and j + 1 Node j has a flexural displacement
Trang 37(2 1)
i j
u − and a slope displacement u i(2 )j Similarly, node j + 1 has a flexural displacement u i(2j+1) and slope displacement u i j(2 +2)
Figure 2 1 A generalized schematic of an arbitrary pair of flexible links
Let us define O XY as the local coordinate system attached to the link i i i i, where the origin O is fixed to the proximal end of the link i i and the axis O X i i
points in the direction of the link i Similarly, O X Y i−1 i−1 i−1 is defined for the link i −1.O X Y is the referential coordinate system fixed to the base Define 0 0 0
matrix is determined by executing in order of the below steps
Step 1 Translate the coordinate system O X Y i−1 i−1 i−1 along i in the direction
Trang 39It is note that from step 1 to step 4 as
If the joint i is the rotational joint, the parameters i,u( 1)i− f,u( 1)i− s,i are in turn the length L i−1, the flexural displacement
− + at the end of link i −1 and the joint variable i
If the joint i is the translational joint P , the parameters b i,u( 1)i− f,u( 1)i− s,iare in turn the length L i−1, the flexural displacement
Step 5 Translate O X Y i−1 i−1 i−1, at the previous location, a long a in the i
direction O X If the joint i i i is the rotation joint or translational joint P , the b
value of a is equal to zero If the joint i i is the translational joint P , the value a
of a is i d i − The general transformation matrix L i T( )a i of this step is
established as
1 0 0
0 1 0 0 ( )
Trang 40The general homogeneous transformation matrix which transforms from the coordinate system O X Y i−1 i−1 i−1 to the coordinate system O XY can be i i i
where, the parameters i,u( 1)i− s,u( 1)i− f, ,i a i are described in Tab 2.1
Table 2 1 The parameters i,u( 1)i− s,u( 1)i− f, ,i a i depending on types of joints