MINITRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL 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 THE DOCTORAL DISSERTATION IS COMPLETED AT 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 Academylevel 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 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 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 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 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 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 Outline of the dissertation Chapter Literature review of flexible robot dynamics and control Chapter Dynamic modeling of the planar flexible robots Chapter Dynamic analysis and position control of the planar flexible robots Chapter Experiment CHAPTER 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 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 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 − 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 Pa and Pb (figure 2.1) Figure Joint i Define Oi XY as the local coordinate system attached to the link i , where i i the origin Oi is fixed to the proximal end of the link i and the axis Oi X i points in the direction of the link i Similarly, Oi −1Xi −1Yi −1 is defined for the link i − O0X 0Y0 is the referential coordinate system fixed to the base The general homogeneous transformation matrix Hif(i −1) which transforms from the coordinate system Oi XY to the coordinate system i i Oi −1Xi −1Yi −1 can be determined as Hf i ( i −1) cos  sin =    − sin cos 0 i + cos   u(i −1)f + sin  , = i + u(i −1)s ,    (2.7) where, the parameters i , u(i −1)s , u(i −1)f ,i ,ai are described in Tab 2.1 Table The parameters i , u(i −1)s , u(i −1)f ,i ,ai depending on types of joints Joints i i R Li −1 i* u(i −1)(2n Pa di* i Pb Li −1 i u(i −1)s u(i −1)f Note u(i −1)(2n (*): variable joint u(i −1)(2k +2) u(i −1)(2k +1) k  1,2, 3, , ni −1 u(i −1)(2n u(i −1)(2n + 2) i −1 i −1 + 2) +1) i −1 i −1   di* − Li +1) 2.1.2 Kinematics The position vector of arbitrary point on the element j of link i in the coordinate system Oi XY is determined as i i T rij = ( j − 1)lie + x wij (x , t ) 1 ,   (2.11) where x is the value of  x  lie and lie is the length of the element j wij (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 O0X 0Y0 can be found as r0ij = Hif0rij , (2.14) f Hif0 = H10f H21 Hif(i −1) (2.15) with 2.1.3 Kinematic relationship of 32 = structures of the arbitrary two flexible links The position vector of arbitrary point on all of links is determined based on the general homogeneous transformation matrix which is constructed above There are 32 = different structures of the arbitrary two flexible links when three types of joints (R, Pa, Pb) are combined The structures I (RR), II (RPa), III (RPb) (fig 2.2) have the first joint being the rotational joint Similarly, the structures IV (PaR), V (PaPa), VI (PaPb) (fig 2.3) have the first joint being the translational joint Pa Last but not least, the structures VII (PbR), VIII (PbPa), IX (PbPb) (fig 2.3) have the first joint which is the translational joint Pb Structure I Structure II Structure III Figure 2 Structures with the first joint being the rotational joint Structure IV Structure V Structure VI Figure Structures with the first joint being the translational joint Pa Structure VII Structure VIII Structure IX Figure Structures with the first joint being the translational joint Pb 2.2 Dynamics of the planar flexible robots 2.2.1 Dynamic equations The kinetic and potential energy of the element j of the link i The kinetic energy of the element j is determined as [12] lie 1 Tij = mi  r0Tij r0ijdx = qTij Mij qij , 2 (2.16) The generalized displacement vector of the element j is shown as [12] q ij = 1 u1(2n +1) u1(2n +2) 2 u2(2n +1) u2(2n + 2) i  1 2 T ijcv q T   (2.17) The elastic deforming potential energy of element j is calculated as [12]  2w(x, t )  lie Pije =  Eij I ij  dx = qTij Kij qij ,  2  x  (2.20) The gravitational potential energy of the element j is described as Pijg =  lie 0 −1 r dx , iAg i   0ij (2.21) 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 Tie = ni T j =1 ij = T q M q i ie i (2.22) 21 Figure 22 The control schematic PID with the GA 3.4.3 The position control of the flexible robots type III and IV In this section, the extended PID controller is used to control the position in joint space of the flexible robots type III and IV The optimal parameters of this controller are found by using the GA Two control cases are considered The first case is the flexible robot without considering the effects of the elastic displacements and the other case is the opposite The position control of the flexible robot type IV The values of translational joint variable in two cases are presented as Fig 3.59 The Fig 3.60 shows the values of the rotational joint variable in two cases Figure 23 The translational joint variable Figure 24 The rotational joint variable 22 The position accuracy in case is higher than in case while considering the overshoot (14.6 %) The specific comparative results between two cases are presented in tab 3.10 Table The comparative results the control quality between two cases Link Comparative criteria Link Case Case Deviation Case 1.3 1.5 0.2 0.3 1.7 Settling time (s) 5.2 0.2 3.5 3.5 Overshoot (%) 25 22 14.6 14.6 State error 0 0 0 Rise time (s) Case Deviation The values of the elastic displacements at the end-effector point are described as Fig 3.61 and Fig 3.62 Figure 25 The flexural Figure 26 The slope displacement displacement The position control of the flexible robot type III The values of rotational and translational joint variables are given as Fig 3.67 and Fig 3.68 The Tab 3.12 shows the comparative results of the control quality between two cases in control Following that, the control results in two cases are almost similar Considering the overshoot criteria, this parameter in case reduces more than in case The value of the deviation is 26.3% 23 Figure 27 The rotational joint Figure 28 The translational joint variable variable The Fig 3.69 and Fig 3.70 describe the values of the flexural and slope displacements at the end-effector point Table 3 Comparative results the control quality between two cases Link Comparative criteria Link Case Case Deviation Case Case Deviation Rise time (s) 0.5 0.5 0.3 0.3 Settling time (s) 3.7 3.6 0.1 3.8 3.5 0.3 Overshoot (%) 28.3 29 0.7 43 16.7 26.3 0 0 0 State error Figure 30 The slope Figure 29 The flexural displacement displacement In summary, the position accuracy control of the flexible robot with translational joint Pb in case is better than in case (about 7%) 24 Conclusion of chapter The solving technique for dynamic equations of the flexible robots with the translational joint Pb has an important meaning in dynamic behaviors analyzing and expanding the applications of the flexible robots combining the different types of joints The values of the elastic displacements at the end-effector point of the system having joint Pb are smaller than in the system without joint Pb The results of dynamic analyzing of the flexible robot type I under the variation of the payload show the frequent existence of the range of suitable payload values which make the flexible robots having the minimum elastic displacements The different ratios of the length between links of the flexible robot type IV engender the different dynamic behaviors of this robot especially the length of the flexible link is longer than the length of rigid link The results of this analysis are very necessary in selecting the geometric dimensions of links in designing system process The control law of the extended PID controller warrants the position control quality of the flexible robots and is better than the control law without considering the effects of the elastic deformation of the links (about 7%) Besides, the control results show that using the popular PID controller combining the intelligent searching algorithm still well warrants the enhancement of the position accuracy of the flexible robots This is greatly advantageous in control designing because the PID controller is simple, cost saving and highly efficient 25 CHAPTER EXPERIMENT 4.1 Objective and experimental model 4.1.1 Objective of experiments This chapter focuses on executing the experiments to verify calculating results of the forward and inverse dynamic analyzing problems of the flexible robot type IV which are presented clearly in chapter and chapter The experimental results are obtained from real robot type IV Comparing the results between the simulation and real model is necessary to evaluate the rightness of the dynamic modeling, the solving motion equations method Determining the values of the joint variables and the flexural displacement at the end-effector point is the objective to achieve the purpose of the verification 4.1.2 Experimental model The flexible robot type IV is manufactured based on the parameters in tab 3.8 and is shown as Fig 4.1 Figure Experimental model DC motor; (2) Lead screw; (3) Encoder for step motor; (4) Step motor; (5) Flex sensor; (6) Encoder for DC motor; (7) Flexible link 26 The design drawing is in the appendix The translational joint Pa is driven by the DC motor (1) through the lead screw system (2) which are described as Fig 4.2 The encoder (6) measures directly the rotary angle of the DC motor The rotational joint is driven by the step motor (4) (Fig 4.3) The encoder (3) measures directly the rotary angle of the step motor The flex sensor (5) is fixed with flexible link (7) from the first element to next elements Figure Lead screw system Figure Step motor at the rotational joint The lead screw and motors are described as Fig 4.4, Fig 4.5 and Fig 4.6, respectively Figure 4 Lead screw Figure DC Figure Step motor GB37-3530 motor NEMA 17 4.2.3 Measurement method The motors, encoders and flex sensor are connected to the terminal board Arduino 2560 which is used to program the power supplies, 27 receive and handle the signals, show the results The program codes are presented clearly in the appendix 4.3 System connection diagram The Fig 4.10 describes the diagram of the system which connects the equipment executing the experimental requests Figure System connection diagram Figure Principle diagram inside Arduino 2560 4.4 Experimental orders The experiments are executed following steps below 28 Step Test and evaluate the stability of the system Step Execute the experiments Step Handle the signals to display on LABVIEW software 4.5 Method of handling the measurement data 4.5.1 Calculating the input signals The input signals include the voltage into DC motor, the pulse signals into step motor Calculating the input signal into DC motor GB37 3530 Calculating the input signals into step motor NEMA-17 4.5.2 Handling the measurement signals Handling the signals of encoder of DC motor Handling the signals of encoder of step motor Handling the measurement of the flex sensor 4.6 Experimental results 4.6.1 Forward dynamic experiment The values of joint variables between the calculations and experiments are given in Fig 4.16 and Fig 4.17 The maximum deviation of the translational joint variables is 4.35% and the rotational joint variables is 7.5% in two cases These differences can be explained following the approximation in calculation and in actuality such as the friction factor, the small deformation, the homogenous material, etc The Fig 4.18 describes the value of flexural displacement at the end-effector point between two cases The deviation value of these displacements is about 7.5% in maximum These results show that the dynamic equations and the solving derivative equations method are correct and suitable in reality 29 Figure The value of translational joint variable Figure 10 The value of rotational joint variable Figure 11 The value of flexural displacement 4.6.2 Inverse dynamic experiment The laws of joint variables are given as    d1 = 0.012 cos(t − )(m);q2 = cos(t − )(rad);  = 2 (rad ) s The values of joints variables are described as Fig 4.19 and 4.20 between the calculation and experiment The deviation value of the translational joint variable is about 8.3% and the rotational joint variable is about 1.15% Noted that the deviation value in responding time is sizable about 17.8% This difference can be explained from the time delay of transmitting and receiving the pulses 30 Figure 12 The value of translational joint variable Figure 13 The value of rotational joint variable Figure 14 The value of flexural displacement The values of flexural displacement at the end-effector point between the calculation and the experiment are shown as Fig 4.21 These displacements are quite similar with the deviation value being about 11.5% Conclusion of chapter Although there are certain deviations about the joint variables and the flexural displacements, the experimental results of the flexible robot type IV also reflect the suitability between calculation and experiment Following that, the extended assembly algorithm, the building and solving dynamic equations method can warrant the reliability These are the basis of applying to continue researching other flexible robots 31 CONCLUSIONS AND RECOMMENTDATIONS FOR FUTUREWORK Conclusion On the one hand, flexible robots have many advantages over heavy and rigid robots such as lower energy consumption, small actuators to move robot arm, higher payload to robot weight ratio On the other hand, flexible robots also satisfy the requirement of structure optimization However, the rigidity of flexible robots is reduced because of their characteristic structure Considering the elastic displacements faces many challenges in kinematic and dynamic modeling of the flexible robots, especially combining the different types of joints Furthermore, the designing control system for flexible robots is more difficult than rigid robots because of the complexity of modeling This dissertation has solved some of the issues mentioned above Specifically results can be shown as below The homogeneous transformation matrix is established to study on kinematics and dynamics of the planar serial multi-link flexible robots which have different types of joints (rotational joint, translational joint Pa and translational joint Pb) This is the reliable basis to build the kinematic equations of these flexible robots or other flexible robots Generalization using the FEM and proposed extended assembly algorithm which has high reliability made a great contribution to building the dynamic equations of the planar serial multi-link flexible 32 robots which have many links, a lot of elements of each link even varied cross-section area The results of forward and inverse dynamic analyzing of the flexible robot under the variation of payload or length of links, especially the variation of boundary conditions demonstrates not only efficiency of modeling and solving methods but also plays an important role in designing and selecting the optimal structures of flexible robots The extended PID controllers are designed to control the position of planar serial multi-link flexible robots in general and two specific flexible robots in particular The simulation results show that these controllers warrant the control requests with the control quality being 10% higher than the conventional PID controller Furthermore, these results show the ability to apply the extended PID controllers in reality in such an economical way The dynamic experiments executing on specific flexible robot show the experimental results which are suitable for the calculating results This is the basis for applying to continue researching other flexible robots Future works Researching on the space or parallel flexible robots applying the modeling and building methods in this dissertation Studying on different and intelligent control methods to find the optimal control solution 33 LIST OF THE RESEARCH PAPERS OF THE AUTHOR Chu Anh My, Duong Xuan Bien, Le Chi Hieu, Michael Packianather, (2018), “An efficient finite element formulation of dynamics for a flexible robot with different type of joints”, Mechanism and Machine Theory (SCI - Q1), Elsevier, 134, pp 267-288 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2018), “Controller design for enhancement position accuracy of a rigidflexible links robot by using Particle Warm Optimization algorithm”, The fifth national conference on mechanical science and technology -VCME2018, pp 1289-1298 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2018), “Extended assembly algorithm in finite element method in building dynamic equations process of flexible robot”, The fifth national conference on mechanical science and technologyVCME2018, pp 1299-1308 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2018), “Dynamic modeling and control of a flexible links robots with translational and rotational joints”, VNU Journal of Science: Mathematics-physics, Vietnam National University, Hanoi, 34(1), pp 52-66 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2017), “Dynamic model of flexible link robots with translational and rotational joints”, Journal of Science and Technology Technical universities, (127), pp 022-028 34 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2017), “Modeling and control of general planar two links flexible robot”, The tenth national conference on mechanics 2017 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2017), “Analysis of dynamic of flexible robot arm with translational and rotational joints under varying length of links”, The tenth national conference on mechanics 2017 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2017), “Dynamic analysis of two-link flexible robot considering the link length ratio and the payload”, Vietnam Journal of Mechanics, VAST, 39(4), pp 315-325 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2017), “Inverse dynamic analyzing of flexible link robots with translational and rotational joints”, Science and Technology Development Journal, Vietnam National University-Ho Chi Minh City, 20(2), pp 42-50 10 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2018), “Dynamic modeling and control of a single flexible link robots with translational joint”, Science and Technology Development Journal, Vietnam National University-Ho Chi Minh City, ISSN 1859-0128 (Accepted) 11 Duong Xuan Bien, Chu Anh My, Phan Bui Khoi, (2017), “Dynamic behaviors of a single flexible link robot under different driving rules”, Journal of Science and Technology, the University of DaNang, 12, pp 06-10 35 12 Duong Xuan Bien, Chu Anh My, Truong Khanh Nghia, (2017), “Dynamic modeling and control in joint space of a single flexible link robot using particle swarm optimization algorithm”, Journal of Scien ce and Technology, the University of DaNang, 6(115), pp 04-09 13 Dương Xn Biên, (2016), “Mơ hình động lực học hệ tay máy có khâu đàn hồi ứng xử hệ tải thay đổi”, Tạp chí Khoa học Giao thơng Vận tải, 55, tr 20-24 14 Duong Xuan Bien, Chu Anh My, (2019), “Inverse dynamic analysis and experiment study for flexible robot consisting rotational and sliding translational joints”, 9th IFAC conference MIM 2019 on Manufacturing modeling, management, and control, Berlin, Germany (Accepted) 15 Chu Anh My, Duong Xuan Bien, (2018), “New development of the dynamic modeling and inverse dynamic analysis for flexible robot”, International Journal of Advanced Robotic Systems (Waiting review) 16 Chu Anh My, Duong Xuan Bien, (2018), “A new mathematical model to analyze the dynamics of flexible robots”, Applied Mathematical Modelling (Waiting review) ... 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... joints order are solved based on multi-bodies dynamic, mechanically deformed body, finite element theory, control and numerical computation method Practical meaning The results of this research... translational joints Pa and Pb (figure 2.1) Figure Joint i Define Oi XY as the local coordinate system attached to the link i , where i i the origin Oi is fixed to the proximal end of the link i and the