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MINISTRY OF EDUCATION
AND TRAINING
VIETNAM ACADEMY OF SCIENCE
AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
……..….***…………
NGUYEN DINH DUNG
INVERSE DYNAMICS AND MOTION CONTROL
OF DELTA PARALLEL ROBOT
Major: Engineering Mechanics
Code: 9 52 01 01
SUMMARY OF THE DOCTORAL THESIS
Hanoi – 2018
The thesis has been completed at Graduate University of Science and
Technology, Vietnam Academy of Science and Technology
Supervisor 1: Prof. Dr. Sc. Nguyen Van Khang
Supervisor 2: Assoc. Prof. Dr. Nguyen Quang Hoang
Reviewer 1: Prof. Dr. Dinh Van Phong
Reviewer 2: : Prof. Dr. Tran Van Tuan
Reviewer 3: : Assoc. Prof. Dr. Le Luong Tai
The thesis is defended to the thesis committee for the Doctoral Degree, at
Graduate University of Science and Technology - Vietnam Academy of
Science and Technology, on Date Month
Year 2018
Hardcopy of the thesis can be found at:
- Library of Graduate University of Science and Technology
- National Library of Vietnam
1
INTRODUCTION
The rationale for the thesis
Parallel robots are robots with closed kinematics structure in which the
links are connected by joints. Although the parallel robot has a complex
dynamic structure, and is difficult to design and control, but it has some
outstanding advantages over the serial robot: high load bearing capacity,
high rigidity due to configuration. They can perform complex operations and
operate with high accuracy. Therefore, study on the problem of dynamics
and control of the parallel robot in order to take advantage of it is a scientific
and practical matter.
2. The objective of the thesis
The objective of this thesis is to apply Lagrange equations with
multipliers to study dynamics and control of Delta parallel robots.
Particularly, mechanical model, mathematical model, and control algorithms
for Delta parallel robots are developed as a scientific basis for the research
and development of parallel Delta robots.
3. The object and the main content of the thesis
Research objects: Dynamics and control of two Delta parallel robots are
3RUS robots and 3PUS robots.
The main content of the thesis includes: Study of mathematical and
mechanical modeling problems, study of dynamics and control algorithms
for Delta parallel robot. The thesis does not study the problem of design and
manufacture of Delta parallel robots.
4. The outline of the thesis
The outline of the thesis contains Introduction, Four main chapters,
Conclusions and findings of the thesis.
Chapter 1: Overview of the study of dynamics and control of Delta
parallel robot in and outside the country is first presented. Since then, the
2
direction of the thesis has been selected to address scientific significance and
practical application.
Chapter 2: Presents the construction of mechanical models and
application of Lagrangian equations with multipliers to formulate
mathematical models for two Delta Parallel Robots. Each robot offers two
mechanical models for study and comparison.
Chapter 3: Presents some improvements in numerical methods to solve
the inverse kinematics and inverse dynamics of parallel robots. Inverse
kinematic problem is solved by applying the improved Newton-Raphson
method. Inverse dynamics problem is solved by reducing Lagrange
multipliers to calculate moments or driving forces in active joints.
Chapter 4: Presents tracking control of parallel robot manipulators
based on the mathematical model of parallel robots, which is a system of
differential – algebraic equations. The trajectory of serial robots described
by differential equations is often well studied. While the Delta parallel robot
trajectory is based on the mathematical model, the differential – algebraic
equations system is rarely studied. Control law such as PD control, PID
control, sliding mode control, sliding mode control using neural network are
studied in this chapter.
CHAPTER 1. OVERVIEW OF DYNAMICS AND CONTROL
PARALLEL ROBOT
1.1. Parallel robot
Parallel robots usually consist of a manipulator connected to a fixed
frame, driven in multiple parallel branches also called legs. The number of
legs is equal to the number of degrees of freedom. Each leg is controlled by
the actuator on a fixed frame or on the leg. Therefore, parallel robots are
sometimes referred to as platformed robots.
1.2. Comparison between Serial and Parallel Manipulators
3
Parallel robot has high rigidity and load bearing capacity due to load
sharing of each actuator operating in parallel. The accuracy of the position of
the parallel robot is high because there are no cumulative joint errors as the
serial robot. While kinematic chains create kinematic constraints and
workspace limitations, typical designs have low inertia characteristics. The
fields of parallel robot application include: CNC machine, high precision
machine, automation machine in semiconductor and high speed and high
acceleration electronics assembly industry. A comparison between parallel
and serial robots is given in the following table:
Table 1.1: Comparison between Serial and Parallel Manipulators
STT Features Serial robot Parallel robot
1 Accuracy Lower Higher
2 Workspace Large Small
3 Stiffness Low High
4 Payload Low High
5 Inertial Large Small
6 Speed Low High
7 Design/control
complexity
Simple complex
8 Singularity problem Some Abundant
1.3. Research on dynamics and control of parallel robots outside of
the country
1.3.1. Inverse dynamics of parallel robots
On the mechanical side, parallel robots are closed-loop multibody
system. Dynamic computation is essential to designing and improving the
control quality of parallel robots. The literature on the theory and calculation
method of robot dynamics is quite substantial [47, 73, 85-88, 96, 103]. The
methods of establishing the dynamic equations of closed-loop multibody
4
system are well documented in [88, 103]. The kinematics and dynamics
problems are then more specifically mentioned in the literature on parallel
robots [67, 96].
In the above studies on Delta parallel robots, the methods used to
establish equation of motion are Lagrange equations with multipliers, virtual
work principle, Newton-Euler equation, subsystem ... When establishing the
equation, the bar between the actuating link and the manipulator is modeled
with a uniform bar or with a zero-mass bar and two masses at the ends of the
bar. Up to now, there have been no comparative work on these two types of
models.
1.3.2. Tracking control of parallel robots
The documentation on robot control is very rich. There are various
approaches to controlling robots given by Spong and Vidyasagar [90],
Sciavicco and Siciliano [87]. However, these works are less focused on the
specific problems of parallel robots.
Recently, the works on improving the control quality of Delta robot was
also published quite a lot. These works develop control law based on the
equations of motion, which are obtained by simplifying the dynamics model
of each parallelogram by a zero-mass bar with two mass points at both ends.
Model linearization methods are used to establish simple control laws.
Hemici et al. [80-82] designed PID, H controllers based on linear models
to robustly control Delta robot. This model was also used by A. Mohsen [68]
to establish PD and PID control laws in combination with fuzzy supervision
to perform motion tracking control of manipulator.
These works use different controllers for the purpose of forcing the
movement of the manipulator to follow a desired trajectory. These
controllers partly meet the desired requirements. However, there is a lack of
5
comparative studies of controllers and recommendations on how to choose
an appropriate ones.
1.4. Studies in Vietnam
The research in Vietnam mainly focuses on solving the kinematics
problem, establishing the equation of motion and presenting the method of
solving the equations of motion. Control problems are little researched.
1.5 The research problem of the thesis
From the review and evaluation of the work that scientists have been
working on in Delta parallel robot, this thesis will investigate the following
issues:
Development of the solution for the inverse dynamics problem with the
aim of improving numerical accuracy.
Study and comparison of different dynamic models for a parallel robot,
the complexity of the models and their effect on the computational torque
moment. On that basis, it is advisable for the user to use a suitable model.
Design of direct control law based on differential – algebraic equations.
Research comparing the quality of the controllers using different
mechanical models.
Conclusions of chapter 1
Based on the results obtained from domestic and foreign researches, the
thesis has identified the need for in-depth research in order to improve the
quality of control for parallel robots, mechanical and mathematical modeling
and numerical algorithms for solving dynamic and control problems for two
parallel robots, 3RUS and 3PUS.
CHAPTER 2. BUILDING THE MECHANICAL MODEL AND
MATHEMATICAL MODEL FOR DELTA PARALLEL ROBOT
In this thesis, the new matrix form of the Lagrange equations with
multipliers [51] is used to establish the equation of motion of two parallel
6
robots, the 3RUS robot and the 3PUS robot. With the MAPLE or MATLAB
software, we obtain the analytic form of differential – algebraic equations
describing the movement of parallel robots.
2.1. Dynamic model of Delta parallel robot
2.1.1. Dynamic model of Delta parallel robot 3RUS
From realistic models of robots from Figure 2.1, it can be seen that the
parallelogram will make the kinematic and dynamic computation on the
robot quite complex. For simplicity we build two models of robot dynamics
based on real model as follows:
Figure 2.1: Delta parallel robot 3RUS
Model 1: The parallelogram mechanisms is modeled by a bar with a
uniformly distributed mass over the length of the bar. The mass and length
of the bars correspond to the mass and length of the parallelogram.
Model 2: The parallelogram mechanisms is modeled by a weightless
bar with a concentric mass at both ends, the mass of each bar end equals half
the mass of the parallelogram.
R1
m1,L1, I1
mp
m2, L2, I2
r
7
2.1.2. Dynamic model of 3PUS Delta parallel robot
Spatial 3PUS Delta robot is a variant of the 3RUS robot when
replacing rotary actuation joints linear actuation joints as shown in Figure
2.4. The 3PUS robot is also equipped with two dynamic models similar to
the 3RUS.
Figure 2.4: Delta parallel robot 3PUS
2.2. Establish equations of motion of the Delta parallel robot
Applying the new matrix form of the Lagrange equation with
multipliers [4, 51], the equation of motion of two 3RUS and 3PUS robots is
the differential - algebraic equations of the following general form:
, T
s M s s C s s s g s Φ s λ τ (2.20)
f s 0 (2.58)
2.3 Compare the equations of motion of robot models
From the equation of motion of model 1 and model 2 of each robot we
have the comparison table as follows:
8
Table 2.1. Compare the equations of motion of Models 1 and 2
Number of.. Model 1. Model 2.
Degree of freedom 3 3
generalized
coordinates 3x3 + 3 = 12 3 + 3 = 6
Constrained equations 9 3
Lagrangian multipliers 9 3
equations 21 9
Matrices M and C ( ), ( , )M M s C s s 0 ( ) , ( , ) 0constM s C s s
From Table 2.1 we find that the equation of motion of model 2 is
simpler and easier to establish than model 1, but the inertia effect is not
clear.
2.3. Conclusions of chapter 2
The establishment of analytical equation of the equation of motion of Delta
parallel robot is a very complex problem. Using the symbolic programming
technique, this thesis has achieved some new results as follows:
Using the new matrix form of equations Lagrange with multipliers [51],
the differential - algebraic equations describing the motion of the two kinds
Delta parallel robot (robot 3RUS and robot 3PUS) has established
analytically.
In addition to establishing equations of motion in view of rigid body,
the thesis also provides a simple equation for motion equation by replacing
the parallelogram mechanisms by two mass points. These mechanical
models are the basis for computational dynamics and control of parallel
robots 3RUS and 3PUS.
9
CHAPTER 3. NUMERICAL SIMULATION OF INVERSE
KINEMATICS AND INVERSE DYNAMICS FOR DELTA
PARALLEL ROBOT
Based on the explicit analytical form of the differential - algebraic equations
description of the motion of the Delta parallel robot set up in Chapter 2, this
chapter applies and develops numerical algorithms to solve the inverse
kinematic and inverse dynamic problem for parallel robots 3RUS and 3PUS.
3.1 Calculation of inverse kinematic parallel robot by improved
Newton-Raphson method
The constrained equations of robot are rewritten in vector form as follows:
( ) ( , )f s f q x 0 (3.1)
where: , , r n mf q x
Contents of the inverse dynamics problem: Given the motion law of the
manipulator, it is necessary to find the law of motion of the driving joints.
Here, we will present an improved Newton-Raphson method [4] to solve the
inverse kinematic problem:
Step 1: Correct the increment of the vector of generalized coordinates at
time t0 = 0. First, we can determine the approximate vector 0q by drawing
method (or experiment). Then apply Newton - Raphson methods to find a
better solution of 0q from nonlinear equations (3.1).
Step 2: Correct the increment of of the vector of generalized coordinates
at time tk+1. The approximate initial value of qk+1 is approximated by the
formula:
2
1
1( )
2k k k kt tq q q q
In the robot kinematics computation [87], the infinitesimals of order
n≥2 are often neglected in the initial approximation of Newton-Raphson. In
this thesis, we take into account the second order infinitesimals, neglecting
10
the infinitesimals of order 3 and taking the formula (3.14) as the
approximation of the original Newton-Raphson loop.
After each step of calculating the coordinates of the joints using the
improved Newton-Raphson method, the generalized velocity and
acceleration of the joints are calculated by the following formulas:
1
q xq J J x (3.4)
1q q x xq J J q J x J x (3.6)
3.2 Numerical method for solving the inverse dynamics problem of
parallel robots
3.2.1 Inverse dynamics problem
The general equations of motion of the robot is as follows:
TsM(s)s +C(s,s)s +g(s)+Φ (s)
(3.20)
( )f s 0 (3.21)
Let f
a q be the vector of coordinates of active joints, rz is the
vector of redundant coordinates (including passive coordinates and end-
effector coordinates). Symbol:
, , , , ,
TT T n f r
a a n f r s q z s q z
The inverse dynamics problem of the parallel robot is expressed as: The
equation of motion of the robot is known as in (3.20), (3.21), given the
motion law of the operation , mt x x x . Determine f
a τ the
driving momen / force required to produce the desired motion.
3.2.2 Solving the inverse dynamics problem by eliminating the Lagrange
multipliers [4]
Through the inverse kinematic with the given trajectory of the mobile
platform center, we have found the vector , , t t ts s s . From this, mass
matrices, centrifugal inertia and Coriolis matrices, matrices sΦ , as well as
the vector g(s) have been completely determined. Thus, Equation (3.20) is a
linear algebraic equation with unknown driving torque vector aτ and
11
Lagrange multipliers λ with equal numbers of equations and numbers.
Thus, we can directly solve this system of equations and then separate the
resulting momen.
In this thesis, we will not directly solve equation (3.20) but try to
eliminate Lagrange multipliers λ , transforming the system of differential -
algebraic equations (3.20), (3.21) into the system of equations of only
unknowns of only joint moments aτ as follows:
We put the symbol [4, 47]:
1( , )az q
sE
R R q zΦ Φ
(3.24)
where E is the unit matrix size f f and , a
z af f
Φ Φz q
Left multiplying both sides of (3.20) by matrix T sR and simplying it
yields
,T T Tas s sR M s s R C s s s R g s τ (3.29)
The expression in the left-hand side of equation (3.29) are known from the
results of the inverse kinematics problem. Thus, active joint moments are
calculated according to this equation.
3.3. Numerical simulation of inverse kinematics and inverse dynamics of
Delta parllel robot
3.3.1. Numerical simulation of 3RUS inverse kinematics of robot
To evaluate the correctness of algorithms and calculations of the thesis,
we computed the inverse dynamics problem of 3RUS robot with the
DELTA-IMECH program developed based on MATLAB software. For
comparison, the robot parameter data and manipulation motion are given in
[61] of Y. Li and Q. Xu.
Using the DELTA-IMECH program we obtain the results of the
numerical simulation of inverse kinematics and have the following
comparison table:
12
The results of the thesis The results of work [61]
Figure 3.11: Comparison of the results of the inverse kinematic problems against
the literature [61]
0 0.5 1 1.5 220
40
60
80
100
t[s]
[deg
ree]
Joint1
Joint2
Joint3
0 0.5 1 1.5 2-2
-1
0
1
2
t[s]
[ra
d/s
]
0 0.5 1 1.5 2-4
-2
0
2
4
6
8
t[s]
[rad/s
2]
13
Comment: Figure 3.11 shows that the results of the inverse kinematics of
the thesis are consistent with the results of the paper [61].
3.3.1. Numerical simulation of inverse dynamics of robot 3PUS
The robot parameter data and movement of the manipulator as follows:
1 20.242, 0.16, 0.029( ), 0.12, 2 0.15, 0.2(kg)PL R r m m m m
Numerical simulation results were computed based on models 1 and 2 of the
3PUS robot using the DELTA-IMECH program.
Model 1 Model 1
T = 1 (s), (Fast motion manipulator)
T = 10 (s), (Slow motion manipulator)
Figure 3.22: Results of numerical simulation inverse dynamics robot 3PUS
2 20.05cos ; 0.05sin ; 0.5 ( )P P Px t y t z m
T T
0 0.2 0.4 0.6 0.8 1-6
-5.5
-5
-4.5
-4
t[s]
[N]
0 0.2 0.4 0.6 0.8 1-5.2
-5
-4.8
-4.6
-4.4
t[s]
[N]
0 2 4 6 8 10-7
-6
-5
-4
-3
t[s]
[N]
0 2 4 6 8 10-7
-6
-5
-4
-3
t[s]
[N]
14
Comment: When motion of the manipulator is fast, the results of the two
models are different. When motion of the manipulator is slow, the results of
the two models are the same.
Conclusions of chapter 3
The contribution of the thesis in this chapter is:
1. Develop a program, called DELTA-IMECH program, to calculate
numerically inverse kinematics and inverse dynamics problems of 3RUS and
3PUS. The results computed by this program are consistent with the
literature [61], [92]. This proves that the equations of motion of robots that
have been established and the algorithms and programs in DELTA-IMECH
are correct.
2. The numerical simulation results show that when the movement of the
manipulator is not fast, a simple robot model can be used to compute the
dynamics of two types of research robots. However, when using simple
models the inertial effects of spatial rigid bodies are not reflected in the
equation. That is the limitation that should be considered.
CHAPTER 4. TRAJECTORY TRACKING CONTROL OF THE
DELTA PARALLEL ROBOT BASED ON MECHANICAL MODELS
The use of inverse dynamic methods to control position of serial robot has
been discussed extensively in engineering [1, 87]. In this chapter, based on
the differential - algebraic equations written explicitly in Chapter 2 and the
numerical method for solving the inverse dynamics problem in Chapter 3,
the PD, PID, Sliding mode control, Sliding mode control using neural
network controller is built for 3RUS and 3PUS Delta parallel robots.
4.1. Overview of the tracking control of the manipulator
The task of the trajectory tracking control problem of the manipulator:
To guarantee that the end-effector moves along the desired trajectory in the
15
work space. Given the desired trajectory d(t)x , it is required to control the
actual trajectory x to satisfy the following condition:
d|| ||x x (4.1)
4.2. Trajectory tracking control of the parallel robots in joint space
based on Lagrange equations with multipliers
4.2.1. Background of dynamics of closed-loop multibody systems
Using the Lagrange equations with multipliers, equation of motion of
parallel robots in the form of (3.20) and (3.21) and equation (3.20) is
transformed into (3.29), we proceed to modify this equation in the following
form:
( , ),a a aq q d s sM s C s s g s τ (4.12)
Where :
Equation (4.12) is the basis for establishing the control law for parallel
robots.
4.2.2. Development of control algorithms
In this thesis the trajectory tracking control algorithms are built based
on the differential - algebraic equations describing the motion of parallel
robots. The stability and tracking properties of the control algorithms are
well proven.
4.2.2.1. PD control
, ,at s s su M ν C s sq dsg
(4.18)
with: da D a P aν q K e K e
(4.19)
: ( ) ( )
: ( ) ( , ) ( )
: ( )
( , ) : ( ) ( , )
, ,
T
T
T
T
s s
s s s s
s
d s s s d s s
M s R M s R
C s s R M s R C s s R
g s R g s
R
16
1 2 1 2, , , , , , , , 0 , 0P P P Pna D D D Dna Pi Didiag k k k diag k k k k kK K
4.2.2.2. PID control
, ,at s s su M ν C s sq dsg
(4.26)
With: 0
( )t
da a a I aD P deν q K e K e K
(4.27)
1 2 1 2 ,, , , , , , ,P P P Pna D D D Dnadiag k k k diag k k kK K
1 2, , ,I II Inadiag k k kK
where: 0, 0; 1,2,..., 0, ,0
aDi Pi Ii Di Pi Ii i nk k k k k k
4.2.2.3. Sliding Mode control
( ) ( , ) ( )d d
a a at s q s su M C q g d s ΛM e (4.45)
( , ) D Sa P signC e Ks s Λ Kν ν
where: 1 2, , ,T
nasign sign v sign v sign vν (4.46)
1 2, , , , 0PD PD PPD Dna PDidiag k k k kK (4.47)
1 2, , , , 0S S SnS a Sidiag k k k kK
4.2.2.4. Sliding mode control using neural network
( ) ( , ) ( (
)
)
,
1
d da a a as q s s su M C q g d M Λe C es s Λ ν
ν
ν
K
Wσ
(4.62)
iiw ν (4.63)
in which K is positive definite symmetric matrix of size ,an and 0 ,
0 .
4.3. The numerical simulation of the control law of the Delta parallel
robot based on the mechanical models
4.3. 2. The numerical simulation of the control laws of the robots 3RUS
and 3PUS Delta parallel
17
Table 4.1: Comparison of robot trajectory errors
Using model 1 in controller Using model 1 in controller
PD control, the robot errors and disturbance
PID control, the robot errors and disturbance
Sliding mode control, the robot errors and disturbance
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8
-505
x 10-4
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8
-5
0
5x 10
-4
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8-4-202
x 10-3
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8-2
0
2x 10
-3
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8-2
0
2x 10
-3
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8-4
-2
0
2x 10
-3
18
Sliding mode control using neural network, the robot errors and disturbance
4.3.3. Comments on numerical simulation results
It is easier to design the control law based on the model 1 than on the
model 2 because the model 1 has larger number of equations and is more
complicated than model 2 (see Table 2.1)
When using the PD, PID to control the real robot with exact knowledge
of the dynamic parameters and no disturbance, the use of model to design
the control law produces less accurate results compared to when using model
1.
When using PD and PID control laws to control real robots without
knowing exactly the dynamic and disturbance parameters, both models give
inaccurate results (~10-3 m see Figures 4.11, 4.12). , 4.19 and 4.20).
When using the sliding mode controller, controller based on the
principle of sliding using neural network for the robot, with exact knowledge
of the dynamic parameter and without disturbances, and without exact
knowledge of the dynamic parameter and with disturbances, same good
results are obtained (accuracy of ~10-4 m is shown in Figure 4.27 to Figure
4.36).
So when using the sliding mode control law and sliding mode control
using neural network, we only need to use simple model to design the
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8-2
0
2x 10
-4
0 0.2 0.4 0.6 0.8 1-0.01
-0.005
0
0.005
0.01
0.015
t[s]
[m]
ex
ey
ez
0.6 0.7 0.8-4
-2
0x 10
-4
19
control rules. The control law will be very simple but still give good results
like when we use complex models.
The comments for control law design for the 3PUS robot are similar to
those of the 3RUS robot when simulating the tracking trajectory.
Conclusions of chapter 4
The contribution of the thesis in this chapter is:
1. To prove theoretically the stability of PD, PID, sliding mode control
and sliding mode control using neural networks laws of the parallel robot
based on the differential - algebraic equations describing the motion of the
robot.
2. When the mechanical model of a robot is correctly constructed and
there is no force disturbance during operation, the PD and PID control laws
can be used but must be set up from a complex mechanical model (model 1),
so that it still ensures the desired trajectory of operation.
3. When the mechanical model of the robot is not properly constructed
and in the presence of disturbance during the working process, modern
control rules such as sliding mode control sliding mode control using neural
network are used. It is required to design the controller only from a simple
mechanical model (model 2) but the controller still ensures the desired
traction of the operation.
CONCLUSIONS AND FINDINGS OF THE THESIS
1. The findings of the thesis
1) Applying the new matrix form of the Lagrangian Equation with
multipliers establishes differential - algebraic equations describing the
movement of 3RUS and 3PUS Delta parallel robots. Equations are explicit
analytical equations but are quite complex.
20
2) Transformation the differential - algebraic equations is transformed
into a system of ordinary differential equations based on the idea of W.
Schiehlen and colleagues [28]. Then the driving moment/ force is calculated.
This method is used to solve the inverse dynamic problem of parallel robot
3RUS and 3PUS. The simulation results obtained according to the proposed
algorithm are consistent with known results.
3) Transformation the differential- algebraic equations of motion of
parallel robots into the ordinary differential equation with redundant
coordinates the new mass matrices )M(s , the inertial force matrix and the
new coriolis and centrifugal matrix ( , )C s s are obtained. The PD, PID,
sliding mode control, and sliding mode control using neural network
algorithms developed for serial robots then can be applied to control parallel
robots. Stability of PD, PID, sliding mode control, sliding mode control
using neural network is proven based on mathematical model of robot which
is the system of differential- algebraic equations.
4) Writing a program named DELTA-IMECH for calculating inverse
kinematics, inverse dynamics, and motion control of two parallel robots
3RUS and 3PUS. Some examples calculated by the DELTA-IMECH
program are consistent with known results. The algorithm and program of
DELTA-IMECH are correct and reliable.
5) In engineering, the rigid body moving in space is sometimes replaced
by a model of mass points. In this thesis, the dynamic and control
calculations with this model are presented for the Delta parallel robot 3RUS
and 3PUS. Caution should be taken, because in the simplest model the
inertial effect of the system is not reflected accurately.
2. Suggestions
Passive-based control of parallel robots.
Dynamics and control of parallel robots taking into account the elasticity of
the links.
LIST OF PUBLISHED WORKS
1. Nguyen Van Khang, Nguyen Quang Hoang, Nguyen Duc Sang, Nguyen
Dinh Dung (2015), “A comparison study of some control methods for
Delta spatial parallel robot”. Journal of computer science and
cybernetics, VAST, Vol. 31, pp 71-81.
2. Nguyen Van Khang, Nguyen Quang Hoang, Nguyen Dinh Dung,
Nguyen Van Quyen (2016), “Model-based Control of a 3-PRS Spatial
Parallel Robot in The Space of Redundant Coordinates”. Journal of
Science and Technology Technical Universities, Vol. 112, pp. 49-53.
3. Nguyen Quang Hoang, Nguyen Van Khang, Nguyen Dinh Dung (2015),
Influence of models on computed torque of delta spatial parallel robot.
Proceedings of the 16th Asia Pacific Vibration Conference, Hanoi, pp.
791-798
4. Nguyen Dinh Dung, Nguyen Van Khang, Nguyen Quang Hoang (2016),
Modelling and sliding mode control based models of a 3RUS spatial
parallel. Proceedings of International Conference on Engineering
Mechanics and Automation (ICEMA4), Ha Noi, pp. 198-205.
5. Nguyen Van Khang and Nguyen Dinh Dung (2013), Về một dạng thức
mới phương trình chuyển động của robot song song, The 2nd Vietnam
Conference on Control and Automation VCCA-2013, Da Nang, pp. 457-
466.
6. Nguyen Van Khang, Nguyen Quang Hoang, Nguyen Dinh Dung, Mai
Trong Dung (2015), Xây dựng mô hình cơ học cho robot song song
Delta không gian 3PUS. National Conference on Engineering
Mechanics, Da Nang, pp. 398-406.
7. Nguyen Van Khang, Nguyen Dinh Dung, Nguyen Van Quyen (2016),
Điều khiển bám quỹ đạo robot song song Delta không gian 3-PRS dựa
trên mô hình hệ các phương trình vi phân-đại số. The Vietnam
Conference on Mechatronics 2016 (VCM 2016), Can Tho, pp. 830 –
840.
