Khảo sát động lực học và điểu khiển vị trí của tay máy hai khâu đàn hồi phẳng theo phương pháp phần tử hữu hạn tt

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


1

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.
1. 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


2

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.


3

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.
2. 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.
3. 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.
4. 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).


4

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.
5. Outline of the dissertation
Chapter 1. Literature review of flexible robot dynamics and control

Chapter 2. Dynamic modeling of the planar flexible robots
Chapter 3. Dynamic analysis and position control of the planar flexible
robots
Chapter 4. Experiment
CHAPTER 1. 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.


5

- 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 1
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 2. 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 − 1 is connected with a link i by a joint
i(i = 1  n) which can be the following three joint types: rotational joint
(R), translational joints Pa and Pb (figure 2.1).

Figure 2. 1. Joint i


6

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 − 1 .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
=
 0

 0

− sin
cos
0
0

0 i + ai cos 

0 u(i −1)f + ai sin 
, = i + u(i −1)s ,

1
0

0
1



(2.7)

where, the parameters i , u(i −1)s , u(i −1)f ,i ,ai are described in Tab. 2.1.
Table 2. 1. 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

ai

Note

u(i −1)(2n

0

(*): variable
joint

u(i −1)(2k +2)

u(i −1)(2k +1)

0

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 ) 0 1 ,



(2.11)

where x is the value of 0  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


7

r0ij = Hif0rij ,

(2.14)

f
Hif0 = H10f H21
...Hif(i −1)

(2.15)

with
2.1.3. Kinematic relationship of 32 = 9 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 = 9 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. 2. Structures with the first joint being the rotational joint

Structure IV
Structure V
Structure VI
Figure 2. 3. Structures with the first joint being the translational joint
Pa


8

Structure VII
Structure VIII
Structure IX
Figure 2. 4. Structures with the first joint being the translational
joint Pb
2.2. Dynamics of the planar flexible robots
2.2.1. Dynamic equations
1. 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

1
Tij = mi  r0Tij r0ijdx = qTij Mij qij ,
0
2
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
1
2
2

T
ijcv

q

T




(2.17)

The elastic deforming potential energy of element j is calculated as
[12]

2

 2w(x, t ) 
1 lie
1
Pije =  Eij I ij 
dx = qTij Kij qij ,

2
2 0
2
 x


(2.20)

The gravitational potential energy of the element j is described as
Pijg =



lie

0

0 −1 r dx ,
iAg
i 
 0ij

(2.21)

2. 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

=

1 T
q M q
2 i ie i

(2.22)


9

The potential energy Pi of link i is the sum of the elastic deforming
potential energy and the gravitational potential energy of

ni

elements.

It can be found as follows1
Pi =

ni

P
j =1

ij

=

ni

ni

P
j =1

ije

+  Pijg =
j =1

1 T
q K q + Pig ,
2 i i i

(2.26)

The gravitational potential energy of link i is shown as
Pig =

ni

P
j =1

(2.29)

ijg

3. The kinetic and potential energy of the flexible robots
The kinetic energy of the flexible robot is sum of kinetic energy of n
links Te and the tip load Tp and determined as
T = Te + TP =

1 T
q Mq ,
2

(2.30)

the generalized displacement vector of system q is constructed as
q = 1 qT1cv 2


qT2cv 3

qT3cv ... n

T

qTncv 


(2.31)

The potential energy of the flexible robot is given as
P=

1 T
q Kq + Pg ,
2

(2.35)

4. Dynamic equations
The Lagrange equations in matrix form is written as
T

T

d  L   L 
  −   = Qex ,
dt  q   q 

where,

Qex

(2.38)

is the generalized forces vector of system. The dynamic

equations of flexible robots can be obtained by deploying the Eq. (2.38)
M(q)q + C(q, q)q + Kq + G(q) = Qex ,
(2.40)
where, G(q) is the global gravitational potential energy and C(q, q) is the
Coriolis matrix and is determined from M(q) matrix following
Christoffel formula as [7]


10





C(q, q) = cij (q, q) , cij (q, q) =

1 n  mij mik mkj
+
−

2 k =1  qk
q j

qi


 qk



(2.41)

The dynamic equation Eq. (2.40) can be extended by mentioning the
Rayleigh damping factor [86] as
M(q)q + C(q, q)q + Dq + Kq + G(q) = Qex ,
(2.42)
where, D is the damping matrix and is not yet considered in this
dissertation.
Conclusion of chapter 2
The homogeneous transformation matrix is determined in this
chapter allowing the kinematics of planar flexible robot with links
connected rotationally and two types of translational joints are
analyzed. Following that, the dynamic equations of system are
constructed based on the kinetic analyzing and FEM combining the
extended assembly algorithms to build the global mass and stiffness
matrices. Besides, these results of this chapter can be used to take into
account the flexible robots which have many links or the variation of
length of links or varied boundary conditions with respect to time and
design control system.
CHAPTER 3. DYNAMIC ANALYSIS AND POSITION
CONTROL OF THE PLANAR TWO-LINK FLEXIBLE
ROBOTS
Two main problems are solved in this chapter. On the one hand, the

forward and inverse dynamic are considered to analyze the dynamic
behavior of flexible robots which are mentioned above under the
variation of payload, length of flexible links and boundary conditions.
On the other hand, the extended PID controller is designed to control
the position of planar flexible robots. The control law is determined and
stably proved based on Lyapunov’s theory. The parameters of


11

controller are found by using genetic algorithm. The flexible robots type
III and IV are used to illustrate as an example.
3.1. Boundary conditions
3.1.1. The flexible robots without the translational joint Pb
The flexible robots without the translational joint Pb are types I, II,
IV and V. The boundary conditions of these robots are

u11 = u12 = 0

and

u21 = u22 = 0 .

3.1.2. The flexible robots with translational joint Pb
The flexible robot types III, VI, VII, VIII, IX have the translational
joint Pb. The boundary conditions of these robots change with respect
to time because of the characteristics of the translational joint Pb . The
boundary conditions of these robots are u1(2k −1) = u1(2k ) = 0 . The value of
k is an integer.

3.2. Forward dynamic
3.2.1. The solving algorithm of the forward dynamic

Figure 3.2. The solving
algorithm without joint Pb

Figure 3.3. The solving algorithm
with joint Pb

3.2.2. The results of numerical calculations
1. The flexible robot type I
In this section, the dynamic behaviors of flexible robot type I (RR)
(fig 2.5) are analyzed under the variation of payload by solving the


12

forward dynamic problem. When the value of payload increases, the
values of the elastic displacements at the ending point of link 1
decreases and at the ending point of the link 2 increases. The
amplitude of vibration at the ending points is large. The time for the
elastic displacements values to reduce to zero is longer.

Figure 2. 1. Flexible robot type I

Figure 3. 1. Value of flexural
displacement at the end of link
2

Figure 3. 2. Value of slope

displacement at the end of link
2

Figure 3. 3. The position of the
end-effector in OX

Figure 3. 4. The position of the
end-effector in OY


13

The results of analyzing the effects of payload variation on the elastic
displacements at the end-effector point show that determining the load
capacity of robot in general and flexible robot in particular is very
important. The suitability of payload may be expressed through the
values of the elastic displacements and the time for these displacements
to drop to zero. On the other hand, the variation of driving
forces/torques and the length of flexible links is essentially studied
when solving the optimal structure problem.
2. The flexible robot type IV (PaR)
In this section, the influences of length of links ratio changing on the
value of elastic displacements at the end-effector are considered. The
results of this analysis can be used to select the suitable geometric
parameters of the links designing the flexible robots which combine the
different types of joints. The flexible robot type IV is shown as Fig.
3.15. The dynamic behaviors are analyzed varying the length of the
links in two cases which are described as tab 3.5.

Figure 3. 5. The flexible robot type IV

14

Table 3. 1. The length of the links in two cases
The length of the links
Case 1
Case 2
L1 (m)

0.2 0.4 0.6 0.8 0.6 0.6 0.6

L2 (m)

0.6 0.6 0.6 0.6 0.6 0.8 1.0

Ratio L1/L2

1:3 2:3 1:1 4:3 1:1 3:4 3:5

a. Simulation results of case 1
The Fig. 3.21 and Fig. 3.22 describe the values of the elastic
displacements at the end-effector point. The variations of the length
of the rigid link 1 do not have much effect on the values of the endeffector point.

Figure 3. 6. The value of
flexural displacement
b. Simulation results of case 2

Figure 3. 7. The value of slope

displacement

In this case, the mass of the flexible link 2 increases gradually as
the length of this link increases. The values of the elastic
displacements at the end-effector point are described in Fig. 3.27 and
Fig. 3.28 and have changed dramatically. These values increase as
the length of flexible link 2 increases.


15

Figure 3. 8. The value of
flexural displacement
c. Summary

Figure 3. 9. The value of slope
displacement

The results of two case are summarized in tab 3.6 which shows the
maximum values of the joint variables and the elastic displacements at
the end-effector point.
3. The flexible robot type III
The flexible robot type III is shown as Fig. 3.31. In this section, the
dynamic behaviors of robot type III are considered under changing of
the boundary conditions. The Fig. 3.32 shows the schematic to solve
the forward dynamic of the system with the translational joint Pb in the
SIMULINK toolbox.

Figure 3. 10. The flexible robot type III

16

Figure 3. 11. Schematic to solve the forward dynamic of the system in
the SIMULINK
The values of the elastic displacements at the end-effector point are
displayed as Fig. 3.36 and Fig. 3.37. These displacements are very
small. The maximum value of flexural displacement is 8.10-4 (m) and
slope displacement is 6.10-3(rad) then quickly reduces.

Figure 3. 12. The value of the
flexural displacement
3.3. Inverse dynamic

Figure 3. 13. The value of the
slope displacement

3.3.1. The solving inverse dynamic algorithm
In this section, the inverse dynamics problem of the flexible robot
types III and IV can be solved based on model with rigid links [25] and
direct from dynamic equations [35], [36] in the time domain. The
desired position and path of joints of rigid models are used as inputs
data for solving inverse dynamics problem of flexible models. The
forces and torques of joints can be found directly from the dynamic


17

equations even though link 2 is deformed. There are some different
points between this dissertation and studies before. Firstly, the flexible

link is divided into more than an element. Secondly, the matrices
M,C, K are obtained by using the extended assembly algorithm. Lastly,
the inverse dynamic of the flexible robot with translational joint Pb is
solved based on handling the boundary conditions regarding time.
3.3.2. The numerical calculations
1. The flexible robot type IV
The input paths of joint variables are described as






d1 = 0.05cos(t − )(m);q2 = cos(t − )(rad);  =  (rad / s )
2
4
2

(3.17)

Figure 3. 14. The deviation of
force between rigid and flexible
models

Figure 3. 15. The deviation of
torque between rigid and
flexible models

Figure 3. 16. The flexural
displacement value

Figure 3. 17. The slope
displacement value


18

The deviations of force and torque between rigid and flexible robots are
shown as Fig. 3.45 and Fig. 3.46. The values of the elastic
displacements at the end-effector point are shown as Fig. 3.47 and Fig.
3.48.
2. The flexible robot type III
The desired paths of joint variables are given as


q1 = 1.5 sin(t )(rad );d2 = 0.4 + 0.3 sin(t − )(m);  =  (rad/ s)
2

Figure 3. 18. The torque
deviation value

(3.18)

Figure 3. 19. The force
deviation value

Figure 3. 20. The flexural
Figure 3. 21. The slope
displacement value
displacement value

The deviations values of the torque and force between two models are
displayed as Fig. 3.53 and Fig. 3.54. The elastic displacements values
at the end-effector point are presented as Fig. 3.55 ad Fig. 3.56. There
are differences of the driving torque after each change cycle of the joint


19

variables because of the effect of the variation of the boundary
conditions with respect to time.
3.4. The position control system of the planar serial multi-link flexible
robots
This section focuses on designing the extended PID controller to
control the position of the planar serial multi-link flexible robots in the
joint space. The optimal parameters of this controller are developed by
using the genetic algorithm. The control law is proved following the
Lyapunov’s theory. The flexible robots type III and IV are used to
illustrate the efficiency of the extended controller.
3.4.1. The PID controller and genetic algorithm
The PID controller is a popular control system, which is used a lot in
industry because of its simplicity, stability and ability to be controlled
in various working conditions. The GA is an optimization and search
technique based on the principles of genetics and natural selection.
3.4.2. The position control law and fitness function
1. The position control law
Minimizing the errors of the joint variables and the elastic
displacements at the end-effector point is the goal of designing the
position control system in joint space. The extended control PID law is
proposed as
Qrr = KP e*(t ) + KD

t
d *
e (t ) + KIen (2n +1)(t ) e*( )en (2n +1)( )d ,
0
dt

(3.21)

The conventional PID control law is described as
(Qrr )* = KP e*(t ) + KD

t
d *
e (t ) + KI  e*( )d
0
dt

the position errors of joint variables which are described as

(3.35)


20

T

e* = e1 e2 . . . en  ,



(3.22)

The stability of the position control law can be proved by using the
Lyapunov’s function which is presented as
1
1
V (t ) = T + P + (e* )T KP (e* ) + AT KI A
2
2

(3.23)

where, T , P are the kinetic and potential energy of system and denoted
A=



t

0

e*( )en (2n +1)( )d

.

2. The objective function
Define the vector e(t ) in the objective function. It is described as
T

e(t ) = e1(t ) e2 (t )....en (t ) en (2n +1)(t ) en (2n +2)(t )



,

(3.32)

This vector includes the errors of the position joints ei (t ), i = 1 ÷ n and
the elastic displacements en(2n +1)(t ),en(2n +2)(t ) at the end-effector.
Defining the driving force/torque vector is shown as
T

uF (t ) = 1(t ) 2 (t ) ... n (t ) 0 0 



(3.33)

Noted that i (t ) is the driving force (in case of translational joint) or the
driving torque (in case of rotational joint) of the joint i . The objective
function is given as
J =



Tf

0

(eTQ1e + uTFQ2u F )dt ,

(3.34)

The control schematic in SIMULINK is described as Fig. 3.58.


21

Figure 3. 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.
1. 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 3. 23. The translational joint
variable

Figure 3. 24. The rotational joint
variable


22

The position accuracy in case 2 is higher than in case 1 while
considering the overshoot (14.6 %). The specific comparative results
between two cases are presented in tab 3.10.
Table 3. 2. The comparative results the control quality between two
cases
Link 1

Comparative
criteria

Link 2

Case 1

Case 2

Deviation

Case 1

1.3

1.5

0.2

0.3

2

1.7

Settling time (s)

5

5.2

0.2

3.5

3.5

0

Overshoot (%)

25

22

3

14.6

0

14.6

State error

0

0

0

0

0

0

Rise time (s)

Case 2 Deviation

The values of the elastic displacements at the end-effector point are
described as Fig. 3.61 and Fig. 3.62.

Figure 3. 25. The flexural
Figure 3. 26. The slope
displacement
displacement
2. 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 2 reduces more than in
case 1. The value of the deviation is 26.3%.


23

Figure 3. 27. The rotational joint Figure 3. 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. 3. Comparative results the control quality between two cases
Link 1

Comparative
criteria

Link 2

Case 1

Case 2

Deviation

Case 1

Case 2

Deviation

Rise time (s)

0.5

0.5

0

0.3

0.3

0

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

0

0

0

State error

Figure 3. 30. The slope
Figure 3. 29. The flexural

displacement
displacement
In summary, the position accuracy control of the flexible robot with
translational joint Pb in case 2 is better than in case 1 (about 7%).