Dynamic Model of Flexible Link Manipulators with Translational and Rotational Joints

Journal of Science & Technology 127 (2018) 022-028 22 Dynamic Model of Flexible Link Manipulators with Translational and Rotational Joints Bien Xuan Duong1,*, My Anh Chu1, Khoi Bui Phan2 1 Military Technical Academy, No. 236, Hoang Quoc Viet, Cau Giay, Hanoi, Viet Nam 2Hanoi University of Science and Technology, No. 1, Dai Co Viet, Hai Ba Trung, Hanoi, Viet Nam Received: August 10, 2017; Accepted: June 25, 2018 Abstract In this paper, two nonlinear dynamic models of flexible manipulator with translational and rotational joints are formulated based on Finite Element Method (FEM). Dynamic equations of manipulators are derived by using Lagrange equations. The new model TR (Translational-Rotational) is developed based on single flexible link manipulator with only rotational joint. The difference between flexible manipulators which have only rotational joints and others with translational joint is presented in RT model. Dynamic behaviors of flexible manipulators are analyzed through values of joints variable and elastic displacements at the end-effector point. The numerical simulation results are calculated by using MATLAB/SIMULINK toolboxes. The results of this study play an important role in modeling generalized planar flexible two-link robot, in designing the control system and basing on select the suitable structure robot with the same request. Keywords: Dynamic model, flexible link, translational joint, elastic displacements 1. Introduction1 Years ago, a number of researches focused on the flexible manipulator. Most of the investigations on the dynamic modeling of robot manipulators with elastic arms have been confined to manipulators with only revolute joints. Combining such systems with translational joints enables these robots to perform manipulation tasks in a much larger workspace, more flexibility and more applications. The flexible robot arm constructed with couples of rotational– translational joints (R-T/T-R joints) challenge the modeling and analysis for the robot. The two main methods for dynamic modeling of flexible link manipulators are the FEM and assumed mode method (AMM). Although the AMM has been used widely to model and analyze flexible links manipulators but most of studied have only considered the rotational joints. The FEM is a general method and it can be applied to the manipulators with complexly shaped links. This technique is used of this paper. Few authors have studied the manipulator with only translational joint. Wang and Duo Wei [1] presented a single flexible robot arm with translational joint. Dynamic model analysis is based on a Galerkin approximation with time dependent basis functions. They also proposed a feedback control law in [2]. Kwon and Book [3] present a single link robot which is described and modeled by * Corresponding author: Tel.: (+84) 166.7193.567 Email: xuanbien82@yahoo.com using assumed modes method (AMM). Other authors have focused on the flexible manipulator with a link slides through a translational joint with a simultaneous rotational motion (R-T robot). Pan et al [4] presented a model R-T with FEM method. The result is differential algebraic equations which are solved by using Newmark method. Yuh and Young [5] proposed the partial differential equations with R- T system by using AMM. Al-Bedoor and Khulief [6] presented a general dynamic model for R-T robot based on FEM and Lagrange approach. They defined a concept which is translational element. The stiffness of translational element is changed. The translational joint variable is distance from origin coordinate system to translational element. The number of element is small because it is hard challenge to build and solve differential equations. Khadem [7] studied a three-dimensional flexible n-degree of freedom manipulator having both revolute and translational joint. A novel approach is presented using the perturbation method. The dynamic equations are derived using the Jourdain’s principle and the Gibbs- Appell notation. Korayem [8] also presented a systematic algorithm capable of deriving equations of motion of N-flexible link manipulators with revolute- translational joints by using recursive Gibbs-Appell formulation and AMM. In this work, two nonlinear dynamic models of flexible manipulator with translational and rotational joints are presented based on using FEM and Lagrange equations. The flexible link is assumed Euler-Bernoulli beam. The first model is shown in Journal of Science & Technology 127 (2018) 022-028 23 fig. 1 (called T-R model). The second model is presented in fig. 2 (called R-T model). Flexible link slides through the base of translational joint which rotates at the fixed point. Both motion on horizontal plane and ignore the effect of gravity and structure damping. The boundary conditions of both are differently. The differences between flexible manipulators which have only rotational joints and others with translational joint are mentioned. Furthermore, T-R model has not been mentioned yet. 2. Dynamic modeling 2.1. General equations of motion In both model, the coordinate system XOY is the fixed frame. The translational joint variable ( )d t is driven by force ( )TF t . The rotational joint variable ( )q t is driven by torque ( )t . Both joints are assumed rigid. Flexible link is divided n elements. Elements are assumed interconnected at certain points which are known as nodes. Each element has two nodes. Each node of element j has 2 elastic displacement variables which are flexural ( 2 1ju − ) and slope displacements ( 2 ju ). Fundamentally, the dynamic equations motion relies on the Lagrange equations with Lagrange function L T P= − are given by [9] ( ) ( ) ( ) T T d L L t dt tt      − =          F QQ (1) Where, T and P are kinetic and potential energy of system. External generalized forces vector is defined as ( ) ( ) ( )1 2 0 . . 0 0 T t F t F t=   F (2) Specific generalized coordinate vector is given by ( ) ( ) ( )1 2 1 2 1 2 2. . T n nt q t q t u u u+ +=   Q (3) Elastic displacements vector of element j is ( ) 2 1 2 2 1 2 2 T j j j jj t u u u u− + + =  Q (4) In T-R case, driving forces and joints variables are ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 1 2; ; ;TF t F t F t t q t d t q t q t= = = = Kinetic and potential are T RT T −= and T RP P −= . In R-T case, respectively, 1 2( ) ( ); ( ) ( )TF t t F t F t= = , 1 2( ) ( ); ( ) ( )q t q t q t d t= = and , R T R TT T P P− −= = . When kinetic and potential energy are known, it is possible to express Lagrange equations as shown ( ) ( ) ( )tM Q Q+C Q,Q Q+KQ = F (5) Coriolis force are calculated as ( ) ( ) ( ) 1 , . . . 2 T = −     C Q Q Q M Q Q Q MQ Q (6) Structural damping is ignored in this paper. Mass matrix M and stiffness matrix K are calculated by using FEM theory in each detail case, respectively. Size of matrices ,M K and C is (2 4) (2 4)n n+  + and size of ( )tF and ( )tQ is (2 4) 1n+  . 2.2. The T-R configurations The T-R model is shown as fig. 1. The translational joint moves along its axis. The translational joint variable ( )d t is distance from origin point of fixed coordinate to position of translational joint. Coordinate system 1 1 1X OY is attached to end point of link 1. Coordinate system 2 2 2X O Y is attached to first point of link 2. Link 1 with length 1L is assumed rigid and link 2 with length 2L is assumed flexibility. Symbol tm is mass of payload on the end-effector point. Coordinate 01r of end point of link 1 on XOY is computed as ( )01 1 T L d t=   r . Coordinate 2 jr of element j on 2 2 2X O Y can be found as ( ) ( )2 j1 , ;0 T j e j j j ej l x w x t x l = − +   r . Where, length of each element is 2 e L l n = and ( ) ( ) ( ),j j j j jw x t x t=N Q is total elastic displacement of element. The vector of shape function ( )j jxN is presented in [9]. Coordinate of element j on 1 1 1X OY is 1 21 2 2.j j=r T r . Where, ( ) ( ) ( ) ( ) 1 2 cos sin sin cos q t q t q t q t  −  =     T is transformation matrix from 2 2 2X O Y to 1 1 1X OY . Fig. 1. T-R model Journal of Science & Technology 127 (2018) 022-028 24 Coordinate 02 jr of element j on XOY is 02 01 21j j= +r r r . Coordinate 0Er of end point of flexible link 2 on XOY can be computed as ( ) ( ) ( ) ( ) ( ) 1 2 2 1 0 2 2 1 cos sin sin cos nT R E n L L q t u q t d t L q t u q t +− + + −  =   + +   r (7) If assumed that robot with all of links are rigid, coordinate 0 _ T R E rigid − r on XOY is given as ( ) ( ) ( ) 0 _ 1 2 2 cos sin E rigid TT R L L q t d t L q t− = + +  r (8) The kinematic energy of rigid link 1 can be computed as 1 1 01 01 1 . . 2 T R TT m− = r r . Where, symbol 1m is the mass of first link. Kinetic energy of element j is determined as ( ) ( ) ( ) 02 02 2 2 0 2 1 2 1 2 e T l j jT R j j T T R T R T R T R j jg j jg T m dx t t T t t − − − − −      =          =  r r Q M Q (9) Where, 2m is mass per meter of link 2. Generalized elastic displacement vector of element j is given as ( ) ( ) ( ) 2 1 2 2 1 2 2 T T R jg j j j jt d t q t u u u u − − + +  =  Q (10) Each element of inertial mass matrix T R j − M can be computed as ( ) 02 022 0 , ; , 1,2,..,6 e T l j jT R j j js je s e m dx Q s e Q −      =             =  r r M (11) Where, jsQ and jeQ are the , th ths e element of ( )T Rjg t − Q vector. It can be shown that T R j − M is of the form rr rfT R j fr base −   =     M M M M M (12) Where, matrix baseM is the base mass matrix which is given as 2 2 2 2 2 2 2 3 2 3 2 2 2 2 as 2 2 2 2 2 2 2 3 2 3 2 2 2 2 13 11 9 13 35 210 70 420 11 1 13 1 210 105 420 140 9 13 13 11 70 420 35 210 13 1 11 1 420 140 210 105 e e e e e e e e b e e e e e e e e e m l m l m l m l m l m l m l m l m l m l m l m l m l m l m l m l   −     −   =    −      − − −   M ( 3) Matrix 11 12 21 22 rr m m m m   =     M has elements which are computed as ( ) ( ) ( ) ( ) ( ) ( ) 11 2 12 21 2 2 2 12 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 2 2 22 2 2 2 1 2 . ; ; sin sin 1 12cos 6sin 12 6sin 6cos 210 ( 1) (2 3 2 ) 22 ( )1 210 13 ( e e j e j e e j j e e e j j j j e j j j j e e j j j m m l m m q l u q l u m m l q jl q u q u q l l j j l u u u u l u u u u m m l l u u u + − + + + − + + + = =  −   = − − +    + +  − + − + + − = + − 21 2 2 2 2 2 1 2 1 2 1 2 1 ) 70 78( ) 54 j e j j j j u l u u u u − + − + − +         +    + + +  (14) Matrix 13 14 15 16 23 24 25 26 T rf fr m m m m m m m m   = =     M M has elements which are computed as 2 13 15 2 14 2 2 16 14 23 2 3 2 24 2 25 2 3 26 2 1 1 cos ; cos ; 2 12 1 ; (10 7); 20 1 1 (5 3); (10 3); 60 20 1 (5 2); 60 e e e e e e m m m l q m m l q m m m m l j m m l j m m l j m m l j = = = = − = − = − = − = − − (15) Total elastic kinetic energy of link 2 is yielded as ( ) ( ) ( )2 1 1 2 n T T R T R T R T R T R e j e j T T t t− − − − − = = = Q M Q (16) The mass matrix T R e − M is constituted from matrices of elements follow FEM theory, respectively. The generalized coordinate vector of system is defined as ( ) ( ) ( ) 1 2 1 2 2. . TT R n nt d t q t u u u − + +=   Q (17) Kinetic energy of payload is ( )0 0 1 . . 2 T T R T R T R P t E ET m − − −= r r and kinetic energy of system is determined as ( ) ( ) ( ) 1 1 2 T R T R T R T R e P T T R T R T R T R T T T T T t t − − − − − − − − = + + = Q M Q (18) Matrix T R− M is total mass matrix. The gravity effects can be ignored as the robot movement is confined to the horizontal plane. Defining E and I are Young’s modulus and area moment of inertia link 2. Elastic Journal of Science & Technology 127 (2018) 022-028 25 potential energy of element j with stiffness matrix baseK which is presented as [9] is given as ( ) 2 2 20 ,1 2 1 ( ) ( ) 2 el j jT R j j j T R T T R j j j j w x t P EI dx x P t t − − −    =    =  Q K Q (19) Stiffness matrix of element j is defined as 0 0 0 0 0 0 0 0 T R j base −    =      K K (20) Where, 3 2 3 2 2 2 as 3 2 3 2 2 2 12 6 12 6 6 4 6 2 12 6 12 6 6 2 6 4 e e e e e ee e b e e e e e e ee e EI EI EI EI l l l l EI EI EI EI l ll l EI EI EI EI l l l l EI EI EI EI l ll l   −      −   =   − − −      −     K (21) Total elastic potential energy of system is yielded as ( ) ( ) ( ) 1 1 2 n T T R T R T R T R j j P P t t− − − − = = = Q K Q (22) Total stiffness matrix T R− K is constituted from matrices of elements follow FEM theory similar T R− M matrix, respectively. 2.3. The R-T configurations The R-T model is shown as fig. 2. Flexible link 2 is divided n elements. The position of elements k continuously changes through translational joints. The angle 0 is the angle between 1 1O X and 2 2O X . The translational joint variable ( )d t is the distance from element k at the origin coordinate system to the end point of link 2. The value of k depends on time. The position of other elements is changed, too. Total length of flexible link 2 is 2L . Length of each element is 2 e L l n = . Element k is inside the hub of the translational joint and we have contact formulas ( )2ek.l = L - d t . Effect of the length of hub is ignored. Fig. 2. R-T model It is noteworthy to mention that value of k is the number of position element k . So, it must be take raw value of k in computing process. Coordinate jr on 2 2 2X O Y of element j is ( ) ( ) ( )2 j1 , T j e j jj l L d t x w x t = − − + + r and on XOY is 0 2 0 01 1 1. .j j= +r r T T r . Transformation matrix 1 0T between XOY and 1 1 1X OY is ( ) ( ) ( ) ( ) 0 1 cos sin sin cos q t q t q t q t  −  =     T . Transformation matrix 2 1T between 1 1 1X OY and 2 2 2X O Y is 0 02 1 0 0 cos sin sin cos     −  =     T . Coordinate of end point of link 1 on XOY is  01 1 1.cos ( ) .sin( ) T L q t L t=r . Coordinate of end-effector is given as ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 0 1 2 1 .cos .sin .sin .cos nR T E n d t L q t u q t d t L q t u q t +− +  + −    =  + +    r (23) If assumed that link 2 is rigid, coordinate 0 _ R T E rigid − r on XOY is given as ( ) ( ) ( ) ( ) 1 0 _ 1 .cos .sin R T E rigid d t L q t d t L q t −  +   =   +     r (24) Kinetic and potential energy of R-T system are calculated by using the same method in T-R case ( ) ( ) ( ) ( )0 0 1 1 . . 2 2 R T R T R T e P T T R T R T R T R T R T R T e t E E T T T T t t m − − − − − − − − − = + = +Q M Q r r (25) Journal of Science & Technology 127 (2018) 022-028 26 ( ) ( ) ( ) 1 1 2 n T R T R T R T R T R T j j P P t t− − − − − = = = Q K Q (26) Where, R T eT − is the elastic kinetic and R T PT − is kinetic energy of payload. Mass matrix R Te − M and stiffness matrix R T− K can be calculated by using FEM theory following mass and stiffness matrices of all elements. Mass matrix of element j with mass per meter 2m is given as T R j − M at recipe (12). It noted that baseM is the same but others elements are difference. They are computed as ( ) ( )( ) ( )( ) ( )( ) ( ) 11 12 2 2 2 2 2 1 2 1 13 2 1 2 2 14 2 1 2 15 2 1 2 2 16 2 1 2 ( , , , , ( ), ( )); 1 6 6 12 1 10 10 10 10 7 20 1 5 5 5 5 3 60 1 10 10 10 10 3 20 1 5 5 5 5 2 60 j e e j e j j j e e e e e e e e e e e m f m le j L d t Q t m m l l u l u u u m m l jl d t L L l m m l jl d t L L l m m l jl d t L L l m m l jl d t L L l + − + = = − − + + = + + − − = + + − − = + + − − = − + + − −( ) 22 23 24 25 26. ; 0; e em m l m m m m= = = = = (27) Matrices R T base − M and R T j − K have format as T R base − M , T R j − K . 3. Boundary conditions In T-R case, rotational joint of link 2 is constrained so that elastic displacements of first node of element 1 on link 2 can be zero. Thus variables 1 2,u u are zero. The rows and columns 3 th, 4th of matrices , ,M K C and ( ) ( ),t tF Q vectors are eliminated as presented in FEM theory. In R-T case, the displacements of element k are zero because assumed that the translational joint hub is treated as rigid. The rows and columns ( ) ( )2 1 ; 2 th th k k− of matrices , ,M K C and ( ) ( ),t tF Q vectors are eliminated and values of these are changed, too. It is noteworthy to mention that value of k depends on time. This boundary condition is clearly different point between flexible link manipulator with only rotational joints and flexible link manipulator with combine translational joint and rotational joint. So now, size of matrices ,M K,C is ( ) ( )2 2 2 2n n+  + and ( )2 2 1n+  is size of ( )tF and ( )tQ vector. The k variable is continuously updated for each time step in solving process. Vector generalized coordinate ( )tQ is rebuilt after each loop too because of changing value k variable. 4. Simulation results Simulation specifications of two flexible models are given by Table. 1. Table 1. Parameters of dynamic models Property T-R model R-T model Length of link 1 (m) L1=0.1 L1=0.2 Mass of link 1 (kg) m1=1.4 m1=0.9 Flexible link parameters Length of flexible link (m) L2=0.3 L2=0.8 Width (m) b=0.03 b=0.01 Thickness (m) h=0.003 h=0.002 Number of element n=5 Cross section area (m2) (A=b.h) A=9.10-5 A=2.10-5 Mass density (kg/m3) =7850 Mass per meter (kg/m) (m2=.A) 0.7 0.157 Young’s modulus (N/m2) E=2.1010 Inertial moment of cross section (m4) I=b.h3/12=1.67.10-12 Mass of payload (g) mt=100 mt=20 Initial value at t=0 (m) 0 d0=0.4 β0 0 Simulation time (s) T=10 Dynamic equations (5) for both models are solved by using MATLAB/SIMULINK toolboxes with boundary conditions. The simulation results of T-R model are shown in fig. 3, fig. 4 and fig. 5. Driving force and torque of joints are built using Bang-Bang rule which are shown fig. 3. Driving time is 0.5(s). Dynamic behaviors of system are described through values of joint variables and elastic displacements in fig. 4 and fig. 5. Maximum values of translational and rotational joint are 0.075(m) and 0.24(rad). These values are shown in fig. 4. Elastic displacements at the end-effector point are presented in fig. 5. Maximum values of these displacements are 0.14(m) and 0.25(rad) at 0.6(s). These displacements are fast reduced after that with 0.02(m) and 0.035 (rad). The simulation results of R-T model are shown from fig. 6 to fig. 8. The Bang-Bang rule in fig. 6 is used for driving torque and force of joints. Fig. 7 shows values of joint variables. It note that value of rotational joint continuously increases while maximum value of translational joint increases from initial value 0.4(m) to 0.51(m) at 0.5(s). Values different of rotational joint between both models can Journal of Science & Technology 127 (2018) 022-028 27 be explained by effect of higher nonlinearity in R-T model, difference of boundary condition and direction of motion of flexible link when value of translational joint increase over time. Elastic displacements at the end-effector point are shown in fig. 8. Values of these displacements are smaller than T-R case. Fig. 3. Driving force and torque at joints of T-R model Fig. 4. Values of joint variables in T-R model Fig. 5. Value of elastic displacements at the end- effector point in T-R model Fig. 6. Driving torque and force at joints in R-T model Fig. 7. Values of joint variables in R-T model Fig. 8. Value of elastic displacement at the end- effector point in R-T model 5. Conclusion Mathematical model of two flexible manipulators with translational and rotational joints has been presented based upon finite element method and Lagrange equations. T-R model is the new model which is developed from single flexible link manipulator with only rotational joint. R-T model shows difference points between translational joint and rotational joint. These difference points are Journal of Science & Technology 127 (2018) 022-028 28 boundary conditions. The final models derived for the flexible robot are nonlinear and complex. Simulation results show that the response of flexible robots is unstable especial without effect of structure damping and needs to control. Moreover, the effects of structure damping and varying payload on dynamic characteristic of models have been studied and discussed. The results in this paper are helpful and important in development of modeling generalized planar flexible two-link robot which combines translational and rotational joints and designing control system. References [1] P. K. C Wang and Jin Duo Wei, Vibration in a moving flexible robot arm, Journal of Sound and vibration. 116 (1987) 149-160. [2] P. K. C Wang and Jin Duo Wei, Feedback control of vibrations in a moving flexible robot arm with rotary and prismatic joints, Proceedings of the IEEE International Conference on Robotics and Automation, Raleigh, North Carolina, March (1987) 1683-1689. [3] D. S. Kwon and W. J. Book, A time-domain inverse dynamic tracking control of a single link flexible manipulator, Journal of Dynamic Systems, Measurement and Control. 116 (1994) 193–200. [4] Pan. Y. C, Scott and R. A. Ulsoy, Dynamic modeling and simulation of flexible robots with translational joints, J. Mech. Design. 112 (1990) 307–314. [5] Yuh. J and Young. T, Dynamic modeling of an axially moving beam in rotation: simulation and experiment, Trans. ASME J. Dyn. Syst. Meas. Control. 113 (1991) 34–40. [6] Al-Bedoor. B. O and Khulief. Y. A, General planar dynamics of a sliding flexible link, Sound and Vibration. 206 (1997) 641–661. [7] S. E. Khadem and A. A. Pirmohammadi, Analytical development of dynamic equations of motion for a three-dimensional flexible manipulator with revolute and prismatic joints, IEEE Trans. Syst. Man Cybern. B Cybern. 33 (2003) 237–249. [8] M. H. Korayem, A. M. Shafei and S. F. Dehkordi, Systematic modeling of a chain of N-flexible link manipulators connected by revolute–prismatic joints using recursive Gibbs-Appell formulation, Springer- Verlag, Berlin Heidelberg, (2014). [9] M. O. Tokhi and A. K. M. Azad. Flexible robot manipulators-Modeling, simulation and control, The Institution of Engineering and Technology. London, UK (2008).