Dynamics and control analysis of a single flexible link robot with translational joints

Science & Technology Development Journal – Engineering and Technology, 3(4):588-595
Open Access Full Text Article Research Article
Le Quy Don Technical University, Ha
Noi, Vietnam
Correspondence
Bien Duong Xuan, Le Quy Don Technical
University, Ha Noi, Vietnam
Email: duongxuanbien@lqdtu.edu.vn
History
 Received: 14-4-2020
 Accepted: 22-12-2020
 Published: 31-12-2020
DOI : 10.32508/stdjet.v3i4.801
Copyright
© VNU-HCM Press. This is an open-
access article distributed under the
terms of the Creative Commons
Attribution 4.0 International license.
Dynamics and control analysis of a single flexible link robot with
translational joints
Bien Duong Xuan*
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ABSTRACT
Modern design always aims at reducing mass, simplifying the structure, and reducing the energy
consumption of the system especially in robotics. These targets could lead to lowing cost of the
material and increasing the operating capacity. The priority direction in robot design is optimal
structures with longer lengths of the links, smaller and thinner links, more economical still warrant-
ing 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 for kinematic, dynamic modeling, analyz-
ing, and controlling flexible robots. Because of the 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. 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 mod-
els consisting of rotational and translational joints will make the kinematic, dynamic modeling, and
control becomes more complex than models that have only rotational joints. This study focuses
on the dynamics model and optimal controller based on genetic algorithms (GA) for a single flexi-
ble link robot (FLR) with a rigid translational joint. The motion equations of the FLR are built based
on the Finite Element Method (FEM) and Lagrange Equations (LE). The difference between flexible
manipulators that have only rotational joints and others with the translational joint is presented
through boundary conditions. A PID controller is designed with parameters that are optimized
by the GA algorithm. The cost function is established based on errors signal of translational joint,
elastic displacements of the End-Point (EP) of the FLR. Simulation results show that the errors of
the joint variable, the elastic displacements (ED) are destructed in a short time when the system
is controlled following the reference point. The results of this study can be basic to research other
flexible robots with more joint or combine joint styles.
Keywords: flexible link robot, translational joint, elastic displacements, control, genetic algorithm
INTRODUCTION
Dynamics and control are fundamental problems in
the robotics field. The modeling and building of the
motion equations problems meet challenges with the
robot which has many degrees of freedom. Espe-
cially the robots take into account the ED factor of
links. There are many studies on the FLR. However,
the robot model mentioned in such works has only
rotational joints (R joint). The translational joint (T
joint) has almost not been considered yet. The inclu-
sion of the T joint into the FLR increases flexibility,
further enriching the application of this robot type.
Of course, the modeling complexity also increases.
Few authors have studied the FLR with only T joint.
A single FLR with T joint is presented in Wang and
Wei (1987)1. The Galerkin method is used to model
the robot. The authors also proposed a feedback con-
trol law in Wang and Wei (1987) 2. Kwon and Book3
present a single link robot which is described and
modeled by using assumed modes method (AMM).
Stable inversion method is studied for the same robot
configuration but the nonlinear effect is taken into ac-
count4. A new method to solve the inverse dynamics
of a FLR is described in5.
Some FLR with a T joint combining a R joint are con-
sidered in Pan et al. (1990)6, Yuh and Young (1994)7,
Bedoor and Khulief (1997)8. The main approach was
based on the Assumed mode method (AMM) and a
few works were based on FEM to model the system.
A FLR with R and T joints was presented in Pan et
al. (1990)6 based the FEM approach. The Newmark
method was used to solve the motion equations of the
robot. The Partial Differential Equations was estab-
lished in Yuh and Young (1994)7 for a FLR with R-T
joint by using AMM. A general dynamic model for R-
T robot was introduced in Bedoor and Khulief (1997)
[8] based on FEM and LE approach.
A Fuzzy Logic con ... of the form
M =266666666664
mle 0 0 0 0
0
13
35
mle
11
210
ml2e
9
70
mle  13420ml
2
e
0
11
210
ml2e
1
105
ml3e
13
420
ml2e 
1
140
ml3e
0
9
70
mle
13
420
ml2e
13
35
mle  11210ml
2
e
0  13
420
ml2e 
1
140
ml3e 
11
210
ml2e
1
105
ml3e
377777777775
(5)
The potential energy (PE) of element j includes two
components: elastic P j and gravity G j potential en-
ergy.
Pj =
1
2
Z l j
0
EI
"
¶ 2w j(x j; t)
¶x2j
#2
dx j
=
1
2
QTj (t)K jQ j(t)
(6)
Where, E and I are Young’s modulus and inertial mo-
ment of link. The stiffness matrix K j is shown as
K j =
EI
l3e
2666664
0 0 0 0 0
0 12 6le 12 6le
0 6le 4l2e 6le 2l2e
0 12 6le 12 6le
0 6le 2l2e 6le 4l2e
3777775 (7)
PE due to gravity can be given as G j =R l j
0 mg [0 1]x jdx j . The total elastic KE and the
PE of link are described as
T = ån1j=1Tj =
1
2
:
Q
T
(t)M
:
Q(t) (8)
P= ånj=1Pj+å
n
j=1G j =
1
2
QT (t)KQ(t)+G(t) (9)
The LEwith Lagrange function L= TP is shown as
d
dt
¶L
¶
:
Q(t)
!
 ¶L
¶Q(t)
= F(t) (10)
When KE and PE are known, the Eq. (10) can be
rewritten as
M(Q)
::
Q+D
:
Q+KQ+G = F(t) (11)
Vector Q(t) = [d(t) u1 u2 ::: u2n+1 u2n+2]T is gen-
eralized coordinate overall system. Vector F (t) =
Ft(t) 0 ::: 0 Fy 0
T is the external generalized force.
The matrices M and K are established from matrices
M j and K j . Vector G = ¶G¶Q(t) can be determined by
partial derivative G(t) = ånj=1G j .Structural damp-
ing matrix D= aM+bK is calculated as in Ge et al.
(1997)12. Symbols a and b are the damping ratios of
the system which are determined by experience. The
size ofM,K and Dmatrices is (2n3) (2n+3).
589
Science & Technology Development Journal – Engineering and Technology, 3(4):588-595
Figure 1: FLR with T joint
BOUNDARY CONDITIONS
The displacements of element k are zero because as-
sumed that the translational joint hub is treated as
rigid. The rows and columns (2k 1)th; (2k)th of
matricesM;K;D;G and F(t);Q(t) vectors are elim-
inated following FEM theory and values of these are
continually changed depending on time because of
changing of element k position. It is noteworthy to
mention that value of k depends on time. This bound-
ary condition is clearly different point between FLR
with only R joints and FLR with combine T joint
and R joint. So now, size of matrices M;K;D;G
is (2n+ 1) (2n+ 1) and size of F (t) and Q(t) is
(2n+ 1) 1. The k variable is continuously updated
for each time step in solving process. Vector gener-
alized coordinate Q(t) is rebuilt after each loop too
because of changing value k variable (Figure 2).
SYSTEM CONTROL
In this paper, a PID controller is constructed to con-
trol the position of an FLR. The position error of EP
of the elastic stitch is continuously reflected and min-
imized. GA is applied to find the optimal parameters
of the PID control system. The error values are used
to assess the fitness of each chromosome in the GA.
There are some main steps in this algorithm as selec-
tion, mutation, crossover, and reproduction. The re-
production step is stoppedwhen an optimumsolution
is found13. The sequences of operations involved in
GA are described in Figure 3.
Structural control of dynamic system is designed as
Figure 4.
From Figure 4, the objective is to tune the PID param-
eters with minimum consumable energy and mini-
mum errors which are translational joint variable er-
ror (e1 = d_re f d_real), flexural displacement u2n
as e2 and slope displacement u2n+1 as e3 of the EP of
the FLR. Symbol upid =Kpe1+KI
R Td
0 e1dt+KD
de1
dt is
the driving force and parameters KP;KI ;KD are pro-
portional gain, integral, derivative times, respectively.
Definition vectors e= [e1 e2 e3] and u=

upid 0 0

with time control and the objective function J used in
GA is defined as
J =
R Td
0

eT e+uT u


dt (12)
Cost function J is the linear quadratic regulator
(LQR)14. The optimum target is finding the mini-
mum cost of J function with values of respective pa-
rameters which are changed from lower bound to up-
per bound values. It means to reduce minimum driv-
ing energy of joint, error of T joint variable and elas-
tic displacements at the end-effector point. The op-
timum parameters KP;KI ;KD are given when mini-
mum value of J is found.
SIMULATION RESULT AND
DISCUSSIONS
Parameters of the FLR are shown in Table 1.
It is noteworthy to mention that system has an elastic
displacement value at initial time as static state. This
initial displacement value is w(t=0)(0) =
Fy:d30
3:E:I .
Parameters are used in GA following Table 2.
The reference point and optimum parameters are
shown in Table 3.
The cost values of J function are described in Figure 5.
The minimum cost value is 0.0594. This value shows
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Science & Technology Development Journal – Engineering and Technology, 3(4):588-595
Figure 2: The dynamic model block diagram in SIMULINK
Figure 3: Steps in GAs
Figure 4: Structure of PID controller
that the control quality is effective. The generation can
be reduced.
The simulation results of T joint, error of joint is
shown in Figure 6 and Figure 7. The remaining fig-
ures are simulation results of their velocity. Consid-
ering T joint value, rise time is 0.4 (s). Settling time
is 1.1 (s), maximum overshoot is 1.5 (%), state error
is zero. The velocity of T joint variable is shown in
Figure 8. The maximum velocity of joint is 1.35 (m/s)
and fast reduces after that about 0.8 (s).
The ED values at the EP are fast reduced and show in
Figure 9 and Figure 10.
Maximum flexural displacement is 3.8 (mm). This
value reduced to displacement value at static state af-
ter 1.4 (s). The maximum slope displacement is 0.175
(rad). It reduced to displacement value at static state
after 1.4 (s) too. The velocities of ED are shown in
Figure 11 and Figure 12.
591
Science & Technology Development Journal – Engineering and Technology, 3(4):588-595
Table 1: FLR parameters
Property Symbol Value
Length of link (m) L 0.8
Width (m) b 0.005
Thickness (m) h 0.005
Number of elements n 20
Cross section area (m2) A=b.h 2.5.107
Mass density (kg/m3) r 7850
Mass per meter (kg/m) m=r .A 0.19625
Young’s modulus (N/m2) E 2.1010
Inertial moment of cross section (m4) I=b.h3/12 5.208x1011
Damping ratios a= b 0.005
External force (N) Fy 2
Time simulation (s) T 2
Table 3: Simulation data and results
Property Value
Reference position (m) 0.7
Initial position (m) 0.3
Simulation time (s) 2
Optimum values (Kp, Ki, Kd) 10.1; 0.2; 2.5
The minimum cost value of J function 0.0594
Table 2: GA parameters
Property Value
Maximum generation 50
Population size 50
Mutation rate 0.05
Fraction of population kept 0.5
Number of optimization variables 3
Lower bound of variables 0
Upper bound of variables 30
CONCLUSION
A dynamic model of a single FLR with T joint is con-
sidered based on FEM and LE approach. Considering
joint variable which is distance from element k at the
origin coordinate system to the EP of link is effective.
The difference between flexible manipulators which
have only R joint and others with T joint is presented
through boundary conditions. Control system is pro-
Figure 5: The cost value of J function
posed to reduce elastic displacement value at the EP
of link and joint error. Parameters of PID control are
optimized by using GA. The output search results are
successfully applied to control position. The approach
method and results of this study can be referenced to
research other flexible robot with more joint or other
592
Science & Technology Development Journal – Engineering and Technology, 3(4):588-595
Figure 6: Position control results of T joint variable
Figure 7: The error of T joint variable
Figure 8: Value of velocity of T joint
Figure 9: Value of flexural displacement at the EP
Figure 10: Value of slope displacement at the EP
Figure 11: Value of flexural displacement velocity
593
Science & Technology Development Journal – Engineering and Technology, 3(4):588-595
Figure 12: Value of slope displacement velocity
joint styles.
ACKNOWLEDGMENT
I am extremely grateful to anonymous reviewers for
valuable comments that helped to improve this article.
CONFLICT OF INTEREST
There are no potential conflicts of interest with respect
to the research, authorship, and publication of this ar-
ticle.
AUTHOR CONTRIBUTION
Theauthor is the only personwhomade the presented
study, conducted the numerical simulation, wrote the
manuscript, as well as formulated the statement to the
problem.
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Tạp chí Phát triển Khoa học và Công nghệ – Kĩ thuật và Công nghệ, 3(4):588-595
Open Access Full Text Article Bài Nghiên cứu
Trường Đại học Kỹ thuật Lê Quý Đôn,
Hà Nội, Việt Nam
Liên hệ
Dương Xuân Biên, Trường Đại học Kỹ thuật
Lê Quý Đôn, Hà Nội, Việt Nam
Email: duongxuanbien@lqdtu.edu.vn
Lịch sử
 Ngày nhận: 14-4-2020
 Ngày chấp nhận: 22-12-2020 
 Ngày đăng: 31-12-2020
DOI : 10.32508/stdjet.v3i4.801 
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International license.
Phân tích động lực học và điều khiển rô bốt cómột khâu đàn hồi
với khớp tịnh tiến
Dương Xuân Biên*
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TÓM TẮT
Thiết kế hiện đại luôn hướng tới mục tiêu giảm khối lượng, đơn giản hóa cấu trúc và giảm tiêu thụ
năng lượng của hệ thống, đặc biệt trong lĩnh vực robot. Các mục tiêu này giúp giảm chi phí vật
liệu và nâng cao năng suất hoạt động. Hướng ưu tiên trong thiết kế rô bốt là các cấu trúc tối ưu
với chiều dài các khâu dài, nhỏ và mỏng hơn, tiết kiệm vật liệu, giảm khối lượng mà vẫn đảm bảo
khả năng làm việc. Tuy nhiên, các thành phần trong kết cấu rô bốt loại này bị giảm độ cứng và
độ chính xác chuyển động do ảnh hưởng của yếu tố biến dạng đàn hồi. Do đó, việc xem xét ảnh
hưởng của yếu tố đàn hồi là rất cần thiết khi phân tích động học, động lực học và điều khiển. Do
tính chất phức tạp của việc mô hình hóa và điều khiển, rô bốt đàn hồi có 1 khâu và 2 khâu với chỉ
một loại khớp quay được xem xét và nghiên cứu là chủ yếu trong hầu hết các công trình đã công
bố. Dễ nhận ra rằng, việc kết hợp các loại khớp khác nhau cho rô bốt đàn hồi có thể mở rộng các
ứng dụng, tính linh hoạt và tính đa dạng về kết cấu của chúng. Tuy nhiên, các rô bốt bao gồm
khớp quay và khớp tịnh tiến sẽ làm cho việc phân tích động học, động lực học và điều khiển trở
nên phức tạp hơn so với các rô bốt đàn hồi chỉ có khớp quay. Bài báo này trình bày kết quả phân
tích động lực học và điều khiển tối ưu vị trí dựa trên thuật toán di truyền (GA) cho rô bốt có một
khâu đàn hồi (FLR) với khớp tịnh tiến được giả thiết cứng tuyệt đối. Mô hình động lực học được xây
dựng và phân tích dựa trên phương pháp phần tử hữu hạn. Hệ phương trình mô tả chuyển động
được xây dựng từ hệ Lagrange. Điểm khác biệt giữa mô hình FLR chỉ với khớp quay và mô hình
FLR với khớp tịnh tiến cũng được xem xét dựa trên các điều kiện biên. Hệ điều khiển PID với các
thông số được tìm tối ưu bằng thuật toán GA được thiết kế nhằm giảm tối đa sai lệch biến khớp
và sai lệch do chuyển vị đàn hồi (ED) tại điểm thao tác cuối (EP). Kết quả mô phỏng cho thấy, sai
lệch biến khớp, giá trị ED đã được triệt tiêu trong thời gian ngắn ứng với yêu cầu ban đầu. Kết quả
của bài báo có thể làm nền tảng để tiếp tục nghiên cứu các rô bốt có khâu đàn hồi khác với các
loại khớp khác nhau hoặc có sự kết hợp giữa chúng.
Từ khoá: Khâu đàn hồi, khớp tịnh tiến, chuyển vị đàn hồi, điều khiển, thuật toán di truyền
Trích dẫn bài báo này: Biên D X. Phân tích động lực học và điều khiển rô bốt có một khâu đàn hồi 
với khớp tịnh tiến. Sci. Tech. Dev. J. - Eng. Tech.; 3(4):588-595.
595