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Because of the large applications, future potentials and challenges in modeling and controlling of the flexible robots, this dissertation has been trying to mention and solve some specif

TRAINING DEFENCE

MILITARY TECHNICAL ACADEMY

Duong Xuan Bien

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

ELEMENT METHOD

SUMMARY OF THE DOCTORAL DISSERTATION

Hanoi - 2019

MILITARY TECHNICAL ACADEMY – MINISTRY OF

NATIONAL DEFENCE

Science supervisors

Associate Prof Chu Anh My

Associate Prof Phan Bui Khoi

Reviewer 1: Prof Nguyen Dong Anh

Reviewer 2: Associate Prof Nguyen Phong Dien

Reviewer 3: Associate Prof Chu Duc Trinh

This doctoral dissertation will be defenced at the level Dissertation Assessment Council according to the Regular No 2, August-2019 of the Rector of Military Technical Academy (MTA) meets at the MTA at the time …h, August-2019

Academy-The dissertation can be found at

- Library of MTA

- National Library

PREFACE

In the past several years, lots of robots are designed and produced all over the world because of their important applications Using robots

is more and more popular in many different fields

The links of robots are mostly assumed rigid bodies in almost all previous studies to simplify calculation in system designing These systems with rigid links are called rigid robots In fact, the elastic deformation always exists on the links of robots in moving process 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 considering the effect of elastic deformation on links are called flexible robots

Researching on dynamic and control flexible robots have been mentioned for several recent decades 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 been trying 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

1 Motivation

Modern designing always aims at reducing mass, simplifying structure and reducing energy consumption of system especially robotic robots These targets could lead to lowering the material cost, manufacturing cost and increasing the operating capacity The best way

to optimize designing is optimal structures with longer length of the links, smaller and thinner links, more economical still warranting 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 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

In conclusion, analyzing above problems shows that 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

2 Scientific meaning

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

3 Practical meaning

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

4 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 have 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 Outline of the dissertation

Chapter 1 Literature review of flexible robot dynamics and control Chapter 2 Dynamic modeling of the planar flexible robots

Chapter 3 Dynamic analysis and position control of the planar flexible robots

Chapter 4 Experiment

CHAPTER 1 LITERATUTE 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 researching in our country and in the world is used to determine the problems which is focused and solved in this dissertation

Although 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 following problems as

- The general homogeneous transformation matrix is built to model the kinematic and dynamic of planar flexible robots which consist of different types of the joints and mention the order of these joints 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 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

CHAPTER 2 DYNAMIC MODELING OF THE PLANAR

FLEXIBLE ROBOTS

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

i i = n which can be the following three joint types: rotational joint

(R), translational joints P a and P b (figure 2.1)

Figure 2 1 Joint i

Define O XY i i i as the local coordinate system attached to the link i, where the origin O i is fixed to the proximal end of the link 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 Y0 0 0 is the referential coordinate system fixed to the base

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

w x t í the elastic displacement at the point x [12] The position vector

of arbitrary point on the element j of link i in coordinate system O X Y0 0 0

can be found as

VI (PaPb) (fig 2.3) have the first joint being the translational joint P a Last but not least, the structures VII (PbR), VIII (PbPa), IX (PbPb) (fig 2.3) have the first joint which is the translational joint P b

Figure 2 2 Structures with the first joint being the rotational joint

Figure 2 3 Structures with the first joint being the translational joint

a

P

Structure VII Structure VIII Structure IX

Figure 2 4 Structures with the first joint being the translational

joint P b

2.2 Dynamics of the planar flexible robots

2.2.1 Dynamic equations

1 The kinetic and potential energy of the element j of the link i

The kinetic energy of the element j is determined as [12]

2 0

2 The kinetic and potential energy of the link i

The kinetic energy of the link i is sum of kinetic energy of driving motor i and kinetic energy of all elements of link i The kinetic energy

of all elements of link i is given by

=

=  = 1

1 2

i n

T

ie ij i ie i j

The potential energy P i of link i is the sum of the elastic deforming potential energy and the gravitational potential energy of n i elements

It can be found as follows1

1 2

ig ijg j

3 The kinetic and potential energy of the flexible robots

The kinetic energy of the flexible robot is sum of kinetic energy of n

links T e and the tip load T p and determined as

1 2

where, G q( ) is the global gravitational potential energy and C q q( , ) is the Coriolis matrix and is determined from M q( ) matrix following Christoffel formula as [7]

 

1

1 ( , ) ( , ) , ( , )

CHAPTER 3 DYNAMIC ANALYSIS AND POSITION

CONTROL OF THE PLANAR TWO-LINK 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 The flexible robots type III and IV are used to illustrate as an example

3.1 Boundary conditions

3.1.1 The flexible robots without the translational joint P b

The flexible robots without the translational joint P b are types I, II,

IV and V The boundary conditions of these robots are u11 =u12 = 0 and

21 22 0

3.1.2 The flexible robots with translational joint P b

The flexible robot types III, VI, VII, VIII, IX have the translational

to time because of the characteristics of the translational joint P b The boundary conditions of these robots are u1 2( k−1) =u1 2( )k = 0 The value of

k is an integer

3.2 Forward dynamic

3.2.1 The solving algorithm of the forward dynamic

Figure 3.2 The solving

algorithm without joint P b

Figure 3.3 The solving algorithm

with joint P b

3.2.2 The results of numerical calculations

1 The flexible robot type I

In this section, the dynamic behaviors of flexible robot type I (RR) (fig 2.5) are analyzed under the variation of payload by solving the

forward dynamic problem When the value of payload increases, the values of the elastic displacements at the ending point of link 1 decreases and at the ending point of the link 2 increases The amplitude of vibration at the ending points is large The time for the elastic displacements values to reduce to zero is longer

Figure 2 1 Flexible robot type I

Figure 3 1 Value of flexural

displacement at the end of link

The results of analyzing the effects of payload variation on the elastic displacements at the end-effector point show that determining the load capacity of robot in general and flexible robot in particular is very important The suitability of payload may be expressed through the values of the elastic displacements and the time for these displacements

to drop to zero On the other hand, the variation of driving forces/torques and the length of flexible links is essentially studied when solving the optimal structure problem

2 The flexible robot type IV (P a R)

In this section, the influences of length of links ratio changing on the value of elastic displacements at the end-effector are considered The results of this analysis can be used to select the suitable geometric parameters of the links designing the flexible robots which combine the different types of joints The flexible robot type IV is shown as Fig 3.15 The dynamic behaviors are analyzed varying the length of the links in two cases which are described as tab 3.5

Figure 3 5 The flexible robot type IV

Table 3 1 The length of the links in two cases

a Simulation results of case 1

The Fig 3.21 and Fig 3.22 describe the values of the elastic displacements at the end-effector point The variations of the length

of the rigid link 1 do not have much effect on the values of the effector point

end-Figure 3 6 The value of

flexural displacement

Figure 3 7 The value of slope

displacement

b Simulation results of case 2

In this case, the mass of the flexible link 2 increases gradually as the length of this link increases The values of the elastic displacements at the end-effector point are described in Fig 3.27 and Fig 3.28 and have changed dramatically These values increase as the length of flexible link 2 increases

Figure 3 8 The value of

3 The flexible robot type III

The flexible robot type III is shown as Fig 3.31 In this section, the dynamic behaviors of robot type III are considered under changing of the boundary conditions The Fig 3.32 shows the schematic to solve the forward dynamic of the system with the translational joint P b in the SIMULINK toolbox

Figure 3 10 The flexible robot type III

Figure 3 11 Schematic to solve the forward dynamic of the system in

the SIMULINK

The values of the elastic displacements at the end-effector point are displayed as Fig 3.36 and Fig 3.37 These displacements are very

slope displacement is 6.10-3(rad) then quickly reduces

Figure 3 12 The value of the

flexural displacement

Figure 3 13 The value of the

slope displacement

3.3 Inverse dynamic

3.3.1 The solving inverse dynamic algorithm

In this section, the inverse dynamics problem of the flexible robot types III and IV can be solved based on model with rigid links [25] and direct from dynamic equations [35], [36] in the time domain The desired position and path of joints of rigid models are used as inputs data for solving inverse dynamics problem of flexible models The forces and torques of joints can be found directly from the dynamic