Dynamic modeling and control in joint space of a single flexible link manipulator using particle swarm optimization algorithm

In this article, the nonlinear dynamic modeling and tip control methodology for a single flexible link manipulator are presented. In Lagrange approach, the nonlinear modeling is built based on finite element method (FEM) so that the elastic displacements effects of elements of the whole dynamic system can be included.

4 Bien Xuan Duong, My Anh Chu, Lac Van Duong, Nghia Khanh Truong DYNAMIC MODELING AND CONTROL IN JOINT SPACE OF A SINGLE FLEXIBLE LINK MANIPULATOR USING PARTICLE SWARM OPTIMIZATION ALGORITHM Bien Xuan Duong1, My Anh Chu1, Lac Van Duong2, Nghia Khanh Truong1 Military Technical Academy; xuanbien82@yahoo.com Hanoi University of Science and Technology Abstract - In this article, the nonlinear dynamic modeling and tip control methodology for a single flexible link manipulator are presented In Lagrange approach, the nonlinear modeling is built based on finite element method (FEM) so that the elastic displacements effects of elements of the whole dynamic system can be included The PID controller is designed in joint space with parameters which are optimized by Particle Swarm Optimization (PSO) algorithm The research results play an essential role in modeling and analysis for the design and control of real industrial flexible manipulators The control quality in PSO is better than in auto tuning mode for single flexible link manipulators The results can be a foundation for selection of reasonable controllers and optimization algorithm while control designing for manipulators with serial flexible links Key words - Dynamic modeling; manipulator; flexible link; control; particle swarm optimization Introduction Recently, manipulators with flexible links are used frequently in space technology, nuclear reactors, medical and many other applications Flexibility, small mass, high speed, and low power consumption are advantages over the conventional rigid manipulator The considering elastic displacements effects on robot motion make dynamic modeling and control become complicated by highly nonlinear characteristics On the one hand, a number of researchers tried to reduce complexity of system by using linearization methods [1-5] which are assumed as small deflections, small hub angle while building dynamic models Besides, the structure damping, coriolis and centrifugal forces are neglected These made models are not general and practical On the other hand, there are many researchers who focused on intelligent control system development to end-effectors control as Fuzzy Logic [1], Neural Network [2], PSO [3], Backstepping [4] and Genetic Algorithm [5] As noted above, linearization methods are used in most of these studies PSO algorithm is optimization technique by social behavior of bird flocking [3] Optimum solution is found by sharing information in the search space The main strength of PSO is that it is easy to implement and fast convergent.PSO has become robust and widely applied in continuous and discrete optimization for engineering applications This is a population based search algorithm which is initialized with the population of random solutions, called particles and the population is known as swarm [6] This paper presents a general dynamic model of a single flexible link manipulator based on finite element method in Lagrange approach Significant dynamics associated with the system such as hub inertia, payload, structural damping, coriolis and centrifugal forces are incorporated to obtain the accurate dynamic model The coulomb friction and gravity effects are ignored as the manipulator movement is confined to the horizontal plane Controller are built by using PSO algorithms to optimize the parameters of the proportionalintegrand-derivative (PID) with input signal is reference hub angle Fitness function is built based on signals of hub angle, flexural and slope displacement of the end point of flexible link manipulators Dynamic modeling 2.1 Finite element method In Finite Element Method (FEM) approach, the flexible link is considered as an assemblage of a finite number of small elements The elements are assumed interconnected at certain points, known as nodes In this work, a single link flexible manipulator which motions on horizontal plane as depicted in Figure1 is concerned In Figure 1, the symbol ( q ) is the angle of rotation at the hub The link is assumed as Euler-Bernoulli’s beam It can be divided into such elements along the length of link and any element has nodes For any nod assumes variables, a flexural and slope displacement Concrete, element ( j ) has nodes The first nod ( j  ) has flexural displacement ( u2 j 1 ) and slope displacement ( u j ) The second nod ( j ) has ( u2 j 1 ) and ( u2 j  ) which are elastic displacements, respectively The coordinate system XOY is the fixed frame Figure Schematic diagram of a single link flexible manipulator The coordinate system X 1O1Y1 is attached to link Symbols ( E, I ) and (  ) are mass density, Young’s modulus, include inertial moment of area of link and total length of link ( L ), thickness ( h ), width ( b ) and cross ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(115).2017 sectional area ( A ) of link The symbol ( ) is the applied torque at the joint and motor inertial moment is ( I h ) The vector from O to a point on ( j ) element in the coordinate system is ( r0 j ) Symbols ( m P ) and ( J P ) are the mass and Kinetic energy of rotor Tdc and payload kinetic energy TP are determined as Tdc  inertial moment of payload on the end point of link The material of link is assumed homogeneity Link is divided ( n ) elements, the length of any element ( l j ) The lengths of (1) r0 j  T01r j (2)    Where the total elastic displacement w j x j , t shape functions i  x j  of j Matrices M dc and M P are determined from variables element with and M  M dh  M dc  M P The total potential energy P of system due to the elastic displacement of link by neglecting the effects of the gravity can be computed by summing over all the potential energy of each element P j following from [5] with K is stiffness matrix of elastic on link xj, yj  P  coordinate on X 1O1Y1 and the transformation matrix T01 form X 1O1Y1 to XOY and are given by [5]     N j  x j   [ 1  x j  2  x j  3  x j  w j x j , t  N j x j Q j t  Q j t   [u2 j 1 cos q1 T01    sin q1 4  x j  ] (5)  sin q1   cos q1     Where shape function vector is N j x j and Q (4) j j  j 1 Tj   T Q  t  M dh Q  t  lj (9)  r01 j   dx j  t   A   (14) j 2.2 Dynamic equations The nonlinear dynamic equations of Lagrange for d  L  L  F  t  with L  T  P and the systems   dt  Q  Q  t  is the generalized coordinate overall system The kinetic T and potential P energy overall system are computed by (7) T  Tdh  Tdc  TP  QT  t  MQ  t  P  QT  t  KQ  t  Where the kinetic energy of elastic Tdh is given following from [5] with M dh is the elastic mass matrix T coriolis forces and structural damping are summarized as follows u2n1 u2n2  which represents n    dx external generalized force F  t  while considering the T Tdh  (13) j j 1 (6) the elastic displacement vector Defining u1 and u2 are flexural and slope displacement at the first point of link; u n 1 and u2 n  are flexural and slope displacement at the end point of link we have vector Q  q1 u1 u2 n  P   2 w x , t lj j j Pj   EI   x j  (3) u2 j 1 u2 j 2 ]T u2 j (11) Q vector, respectively The total inertial mass matrix is  j  1 l j  x j   r1 j    wj xj ,t     1 I h q  t   QT  t  Mdc Q  t  2  r0 j  1 TP  m p  ( L, t )   J p  q  u n   2  t  (12) T TP  Q  t  M P Q  t  elements are equal because the cross-sectional areas of links are constant along length Position vector r1 j on X 1O1Y1 and vector r0 j of j element on XOY are expressed as [5] (10)   M  Q  Q + C Q,Q Q + DQ + KQ = F  t  (15)  The coriolis force C Q,Q  and structural damping D(Q) are calculated by using T 1  T  C Q, Q Q  M  Q  Q   Q M Q  Q   Q    D  M   K (16) (17) Where  and  are the damping ratios of system and are determined by experiences [5] The first node of link has elastic displacements which are zero link modeled by one element, u1  t   0; u2  t   As the(8) we have Q  q u3 u4  and F  t     t  0  vectors with u3 and u4 are flexural and slope T T displacement of the end point of link Matrices M and K can be compacted by eliminating 2nd, 3rd rows and 2nd, 3rd columns, respectively Matrices C and D are determined by using Eq (5) Elements of M matrix can be written as Bien Xuan Duong, My Anh Chu, Lac Van Duong, Nghia Khanh Truong  m  J P  Ih Figure shows the movement of a single particle i at the time step t in space search At time step t , the position, velocity, personal best and global best are indicated as xi  t  , vi  t  , pi  t  and pg  t  , respectively Where S   LA / 420 and L is the total length of link The dynamic nonlinear equations of one-link flexible manipulator can be derived as follows Eq (15) The velocity vi  t  serves as a memory of the previous PID controller and PSO algorithm The PID controller has been widely used in the industry but it is hard to determine the optimal or near optimal PID parameters using classical tuning methods as Ziegler Nichols This paper presents the PSO algorithm to find the suitable parameters of the PID controller Structural control of dynamic system is designed as in Figure on three components: momentum, cognitive and social component m11  4SL2u42  156Su32  44Su3u4  140SL  L  u32 P m12  m21  147 S  mP  L; m13  m31  21SL2  J P m22  156S  mP ; m33  4SL2  J P ; m45  m54  22SL flight direction, can be seen as momentum At time step t  , the new position xi  t  1 can be calculated based Figure The movement of a single particle Figure Structural control in MATLAB/SIMULINK From Figure 2, the objective is to tune the PID parameters with minimum consumable energy and minimum errors which are hub angle error ( e1 ), flexural ( e2 ) and slope ( e3 ) displacement of the end point of flexible link manipulator Symbol u pid is the PID control law and parameters K P , K I , K D are proportional gain, integral, derivative times, respectively With Td is the control time and e  e1 e2 e3 ; u  u pid 0 , the Td objective function J   e e  u u  dt T T is used in PSO The sequences of operation in PSO are described in Figure After finding the personal best and global best, particle is then accelerated toward those two best values by updating the particle position and velocity for the next iteration using the following set of equations: vi  t   kvi  t  1  C1.rand  Pi  xi  t  1   C2 rand Pg  xi  t  1  xi  t   xi  t  1  vi  t  (18) Where vi  t  and xi  t  are the current velocity and position of the ith particle in search space C1 and C2 are learning factors rand is the random number between and k is the inertia serves as memory of the previous direction, preventing the particle from drastically changing direction The information details of PSO can be considered as [3, 6] Simulation specifications of model are given as L  1(m), b  0.02(m), h  0.003(m), A  b.h(m )   2710(kg / m ), E  7.11 1010 ( N / m ) I  4.5  1011 (m ), I h  5.86  105 (kg m ) mP  0.1(kg ),   0.76,   5.6  105 , qd  1( rad ) qd  sin(t ), Td  2 ( s) Figure Steps in PSO algorithm Parameters in PSO include 50 particles in a swarm, 50 searching steps for a particle and optimization variables ( K P , K I , K D ) Cognitive and social acceleration are The max and inertia factor are 0.9 and 0.4, respectively Lower and upper bound of variables are and 6, respectively The values of bound are considered from auto tuning mode in MATLAB/SIMULINK The control qualities are compared between Auto tuning mode (AT) in MATLAB software and Particle Swarm Optimization for parameters of PID controller ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(115).2017 Table The reasonable parameters of PID controller AT Pos Path Cost KP KI KD X 0.366 0.046 0.53 PSO 3.45x10-9 5.379 6.0 1.79 AT X 1.254 0.332 0.92 PSO 5.65x10-7 3.496 2.69 3.06 The reasonable parameters in AT and PSO for position and path control are described in Table The simulation results are shown in Figure 4, 5, and Figure for position control ( qd  1 rad  ) Figure 8, and Figure 10 describe the simulation results for path control qd  sin  t  Figure shows the hub angles in AT and PSO The value of undershoot is zero and the overshoot value of AT (14.5%) is higher PSO (13.5%) State error in PSO is zero different in AT (0.02rad) The rise time is the same but the settling time in PSO is 2.8(s) and in AT is 4(s) Figure Slope displacements in AT and PSO Figure describes position errors of AT and PSO As noted above, the error of hub angle in AT is higher than that in PSO Figure and Figure show the elastic displacement at the end point of link The maximum flexural displacement in AT (0.13m) is smaller than that in PSO (0.18m) but the slope displacement is the same (0.3rad) Figure shows the simulation result for path control in two modes which are AT and PSO Figure Hub angles in AT and PSO for Position control Figure Hub angles in AT and PSO for Path control The path errors are described in Figure 10 The maximum error value is 0.095rad in PSO Elastic displacement is shown in Figure 11 and Figure 12 The maximum and minimum values of flexural displacement are0.093m and 0.025m The value of slope displacement in AT is bigger than slope displacement in PSO The control quality in PSO is better than AT for single flexible link manipulators in position and path control with parameters of algorithm which are used Figure Errors of hub angle in AT and PSO Figure Flexural displacements in AT and PSO Figure 10 Errors of hub angle for path control Bien Xuan Duong, My Anh Chu, Lac Van Duong, Nghia Khanh Truong with parameters which are optimized by using PSO algorithm The research results show that the state error in position control is zero in short time and error of hub angle in path control is small The elastic displacements in two control cases are fast reduced The control quality in PSO is better than in auto tuning mode for single flexible link manipulator control The results can be a foundation for selection of reasonable controllers and optimization algorithm while control designing for manipulators with serial flexible links REFERENCES Figure 11 Flexural displacements for path control Figure 12 Slope displacements for path control Conclusion This paper has presented the general nonlinear dynamic model of single flexible link manipulators and controllers [1] Kuo Y K and J Lin, Fuzzy logic control for flexible link robot arm by singular perturbation approach Applied Soft Computing 2, pp 24–38 (2002) [2] Tang Yuan-Gang, Fu-Chun Sun and Ting-Liang Hu, Tip Position Control of a Flexible-Link Manipulator with Neural networks, International Journal of control automation and systems (2006), vol.4, No.3, Pg 308-317 (2006) [3] Yatim H M and I Z Mat Darus, Swarm Optimization of an Active Vibration Controller for Flexible, Control and Signal Processing, ISBN: 978-1-61804-173-9 (2010) [4] Huang J W, Jung-Shan Lin, Back-stepping Control Design of a Single-Link Flexible Robotic Manipulator, Proceedings of the 17th World Congress The International Federation of Automatic Control, Seoul, Korea (2008) [5] Tokhi M O, A K M Azad, Flexible robot manipulators (modeling, simulation and control), The Institution of Engineering and Technology, London, United Kingdom, ISBN 978-0-86341-448-0 (2008) [6] Kennedy J and R Eberhart, Particle Swarm Optimization, Proceedings of IEEE International Conference on Neural Networks, Perth, 27 November-1 December 1995, pp 1942-1948 (The Board of Editors received the paper on 05/01/2017, its review was completed on 22/02/2017) ... Steps in PSO algorithm Parameters in PSO include 50 particles in a swarm, 50 searching steps for a particle and optimization variables ( K P , K I , K D ) Cognitive and social acceleration are... control quality in PSO is better than AT for single flexible link manipulators in position and path control with parameters of algorithm which are used Figure Errors of hub angle in AT and PSO... paper has presented the general nonlinear dynamic model of single flexible link manipulators and controllers [1] Kuo Y K and J Lin, Fuzzy logic control for flexible link robot arm by singular