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.
Trang 14 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 Duong 1 , My Anh Chu 1 , Lac Van Duong 2 , Nghia Khanh Truong 1
1 Military Technical Academy; xuanbien82@yahoo.com
2 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
1 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],
Back-stepping [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 proportional-integrand-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
2 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 2 nodes For any nod assumes 2 variables, a flexural and slope displacement Concrete, element ( j) has
2 nodes The first nod ( j 1) has flexural displacement (u2j1) and slope displacement (u 2 j) The second nod ( j) has (u2j1) and (u2j2) which are elastic displacements, respectively The coordinate system XOY is the fixed frame
Figure 1 Schematic diagram of a single link flexible
manipulator
The coordinate system X O Y1 1 1 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
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sectional area (A) of link The symbol ( ) is the applied
torque at the joint and motor inertial moment is (I ) The h
vector from O to a point on ( j) element in the coordinate
system is (r ) Symbols ( 0 j m P) and (J P) are the mass and
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
elements are equal because the cross-sectional areas of links
are constant along length Position vector r on 1 j X O Y1 1 1
and vector r of 0 j j element on XOY are expressed as [5]
1
j
1 ,
j
j
j l x
x t
r
w (1)
1
0j 0 1j
r T r (2)
Where the total elastic displacement wj x t j, and
shape functions i x j ofj element with x j,y j
coordinate onX O Y1 1 1 and the transformation matrix T01
form X O Y1 1 1 to XOY and are given by [5]
j x t j, j x j j t
[ 1 2 3 4 ]
j x j x j x j x j x j
[ 2j 1 2j 2j 1 2j 2]T
1 0
Where shape function vector is Nj x j and Q j t is
the elastic displacement vector
Defining u1 and u2 are flexural and slope
displacement at the first point of link; u2n1 and u2n2
are flexural and slope displacement at the end point of
1 1 2 2n1 2n 2 T
the generalized coordinate overall system The kinetic T
and potential Penergy overall system are computed by
1 2
T
1
2
T
Where the kinetic energy of elastic Tdh is given
following from [5] with Mdh is the elastic mass matrix
1
1 2
n
T
j
2 01 0
1 2
j
t
Kinetic energy of rotor T and payload kinetic dc
energy T are determined as P
T
dc I q t h t dc t
2
2 0
2 2
( , )
1 2
j
T
m L t J q u t
t t
r T
(12)
Matrices Mdc and MP are determined from variables
Q vector, respectively The total inertial mass matrix is
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 Pj following from [5] with K is stiffness matrix of elastic on link
1
n j j
P P (13)
l j j
j
x t
x
2 2
j 2 0
, 1
2
w
2.2 Dynamic equations
The nonlinear dynamic equations of Lagrange for
t dt
L L
F Q
external generalized force F t while considering the coriolis forces and structural damping are summarized as follows
M Q Q + C Q,Q Q + DQ + KQ = F (15)
The coriolis force C Q,Q and structural damping ( )
D Q are calculated by using
,
2
T T
C Q Q Q M Q Q Q M Q Q
D M K (17) Where and are the damping ratios of system and are determined by experiences [5] The first node of
we have Q q u3 u4Tand F t t 0 0T vectors with u3and u4 are flexural and slope displacement of the end point of link Matrices Mand K
can be compacted by eliminating 2nd, 3rd rows and 2nd, 3rd
columns, respectively Matrices C and Dare determined
by using Eq (5) Elements of Mmatrix can be written as
Trang 36 Bien Xuan Duong, My Anh Chu, Lac Van Duong, Nghia Khanh Truong
2
2
Where SLA/ 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)
3 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 2
Figure 2 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 T d is the
control time andee1 e2 e3;uu pid 0 0, the
0
d
T
dt
J e e u u is used in PSO
The sequences of operation in PSO are described in Figure
3
Figure 3 Steps in PSO algorithm
Figure 4 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 x t v t i , i ,p t and i p g t , respectively The velocity v t serves as a memory of the previous i flight direction, can be seen as momentum At time step 1
t , the new position x t i 1 can be calculated based
on three components: momentum, cognitive and social component
Figure 4 The movement of a single particle
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:
1 2
1
v t kv t C rand P x t
C rand P x t
(18)
Where v t and i x t are the current velocity and i position of the i th particle in search space C1andC2are learning factors randis the random number between 0 and 1.kis 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
2
5
h
Parameters in PSO include 50 particles in a swarm, 50 searching steps for a particle and 3 optimization variables (K P,K I,K D) Cognitive and social acceleration are 2 The max and min inertia factor are 0.9 and 0.4, respectively Lower and upper bound of variables are 0 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
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Table 1 The reasonable parameters of PID controller
Cost K P K I K D
Pos AT X 0.366 0.046 0.53
PSO 3.45x10-9 5.379 6.0 1.79
Path 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 1 The
simulation results are shown in Figure 4, 5, 6 and Figure 7
for position control (q d 1rad) Figure 8, 9 and Figure
10 describe the simulation results for path control
sin
d
q t
Figure 5 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 5 Hub angles in AT and PSO for Position control
Figure 6 Errors of hub angle in AT and PSO
Figure 7 Flexural displacements in AT and PSO
Figure 8 Slope displacements in AT and PSO
Figure 6 describes position errors of AT and PSO As noted above, the error of hub angle in AT is higher than that in PSO Figure 7 and Figure 8 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 9 shows the simulation result for path control
in two modes which are AT and PSO
Figure 9 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 10 Errors of hub angle for path control
Trang 58 Bien Xuan Duong, My Anh Chu, Lac Van Duong, Nghia Khanh Truong
Figure 11 Flexural displacements for path control
Figure 12 Slope displacements for path control
4 Conclusion
This paper has presented the general nonlinear dynamic
model of single flexible link manipulators and controllers
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
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(The Board of Editors received the paper on 05/01/2017, its review was completed on 22/02/2017)