design and analysis of a 3-dof translational shock...
TRANSCRIPT
DESIGN AND ANALYSIS OF A 3-DOF TRANSLATIONAL SHOCK
ABSORBING PARALLEL MANIPULATOR
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY
BY
GENCAY YILDIZ
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR
THE DEGREE OF MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
FEBRUARY 2016
Approval of the thesis:
DESIGN AND ANALYSIS OF THE 3-DOF TRANSLATIONAL SHOCK
ABSORBING PARALLEL MANIPULATOR
Submitted by GENCAY YILDIZ in partial fulfillment of the requirements for the
degree of Master of Science in Mechanical Engineering Department, Middle
East Technical University by,
Prof. Dr. Gülbin Dural Ünver
Dean, Graduate School of Natural and Applied Sciences
Prof. Dr. R. Tuna Balkan
Head of Department, Mechanical Engineering
Prof. Dr. M. Kemal Özgören
Supervisor, Mechanical Engineering Dept., METU
Examining Committee Members:
Prof. Dr. Reşit Soylu
Mechanical Engineering Dept., METU
Prof. Dr. M. Kemal Özgören
Mechanical Engineering Dept., METU
Prof. Dr. Sıtkı Kemal İder
Mechanical Engineering Dept., CU
Assist. Prof. Dr. Erhan İlhan Konukseven
Mechanical Engineering Dept., METU
Assist. Prof. Dr. Yiğit Yazıcıoğlu
Mechanical Engineering Dept., METU
Date: 01.02.2016
iv
I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have full cited and referenced all
material and results that are not original to this work.
Name, Last name: Gencay YILDIZ
Signature :
v
ABSTRACT
DESIGN AND ANALYSIS OF THE 3-DOF TRANSLATIONAL SHOCK
ABSORBING PARALLEL MANIPULATOR
Yıldız, Gencay
M.S., Department of Mechanical Engineering
Supervisor: Prof. Dr. M. Kemal Özgören
February 2016, 89 pages
In defense industry, the environmental conditions play very critical and dangerous
roles for valuable and sensitive electronic equipments. One of the most dangerous
situations is high level shocks caused by explosions. In order to protect the valuable
assets from the dangerous effects of shock some special designed shock absorbers
are needed.
For some shock absorbing cases, standards products manufactured according to the
Military Standards are used and their implementation is considerable easy. Helical,
wire rope springs and dampers directly attached onto the platforms can be given as
an example of that systems. On the other hand, according to the device specifications
and its working principle, some special and custom designed shock absorbing
systems are required. Shock absorbing platform of naval radars mounted on the ship
structure is one example of this custom shock absorbers. At the same approach land
and air radars use special absorbing methods.
In this thesis, a 3-dof translational shock absorbing parallel manipulator for radar
systems is introduced and analyzed in terms of kinematics and dynamics by using
vi
rigid body assumption. For that purpose, a parallel manipulator which has only 3
translational degrees of freedom is selected among three candidates and shock
absorbing elements (spring, damper) are implemented on it. This special shock
absorbing platform is analyzed dynamically according to shock criteria given in
MIL-STD 810F [1].
Keywords: Parallel, Shock, Manipulator, Radar, Damper, Absorbing, Degree of
Freedom, Translation
vii
ÖZ
3 ÖTELEME SERBESTLİK DERECESİNE SAHİP BİR ŞOK
SÖNÜMLEYİCİ PLATFORM TASARIMI VE ANALİZİ
Yıldız, Gencay
Yüksek Lisans, Makina Mühendisliği Bölümü
Tez Yöneticisi: Prof. Dr. M. Kemal Özgören
Şubat 2016, 89 Sayfa
Savunma sanayiinde, çevresel koşullar hassas ve değerli elektronik teçhizatlar için
çok tehlikeli ve kritik rol oynamaktadır. Patlamalar sonucu ortaya çıkan şok benzeri
durumlar en tehlikeli çevresel koşullardan birini teşkil etmektedir. Şokun bu olası
zararlarını önlemek içi özel tasarlanmış şok sönümleyicilere ihtiyaç vardır.
Bazı şok sönümleyici uygulamalarında askeri standartlara göre üretilmiş olan
standart şok sönümleyiciler kullanılmaktadır ve bu ürünlerin montajı genellikle kolay
şekilde gerçekleştirilebilmektedir. Spiral ve kablo tipi yaylar ve amortisörler bu tarz
ekipmanlara örnek verilebilir. Diğer taraftan çalışma prensiplerine ve ürün
özelliklerine göre bağlı olarak bazı durumlar için özel tasarlanmış şok
sönümleyicilere ihtiyaç duyulmaktadır. Askeri gemilerde deniz radarları için
kullanılan şok sönümleyiciler özel tasarım uygulamalara örnek gösterilebilir. Bu
yaklaşımla bu tarz sönümleyicilerin kara ve hava radarlarında kullanıldığı durumlar
da mevcuttur.
Bu tez kapsamında, 3 öteleme serbestlik derecesine sahip bir şok sönümleyici paralel
platform modeli tanıtılmış ve rijit eleman varsayımı kullanılarak bu sönümleyicinin
kinematik ve dinamik analizleri parametrik olarak gerçekleştirilmiştir. Bu çalışmalar
sırasında kullanılan mekanizma 3 öteleme serbestlik derecesine sahip 3 aday
viii
arasından seçilmiştir. Bu mekanizmaya yay ve sönümleyici elemanlar entegre
edilmiştir. Daha sonra elde edilen şok sönümleyici platform MIL-STD 810F ‘de yer
alan şok profiline göre analiz edilmiştir.
Anahtar Kelimeler: Paralel, Şok, Yönlendirici, Amörtisör, Sönümleme, Serbestlik
Derecesi, Öteleme
ix
To Science and Fair Mankind
İlim ilim bilmektir
İlim kendin bilmektir
Sen kendini bilmezsin
Ya nice okumaktır
……….Yunus Emre
x
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my supervisor Prof. Dr. M.Kemal
ÖZGÖREN for his excellent supervision and leading guidance from beginning to end
of thesis work that made this study possible.
I am grateful to my family for their endless love and vulnerable support throughout
my life. I specially thank my aunt Firdes YILDIZ and my aunt’s husband Hayri
YILDIZ my brother Tolunay YILDIZ for their moral support.
And there are a lot of people that were with me in these three years. I would like to
thank Furkan LÜLECİ, my sister Özge ÇİMEN, Gökhan YAŞAR, my flate-mate
Hüseyin Gökcan SAYGILI, Evren KUTLU and my manager İsmail GÜLER for their
friendship and support.
Finally, I am also grateful to ASELSAN Inc. that has given lots of opportunities to
me to finish this study.
xi
TABLE OF CONTENTS
ABSTRACT ................................................................................................................. v
ÖZ .............................................................................................................................. vii
ACKNOWLEDGEMENTS ......................................................................................... x
TABLE OF CONTENTS ............................................................................................ xi
LIST OF TABLES .................................................................................................... xiv
LIST OF FIGURES ................................................................................................... xv
LIST OF SYMBOLS .............................................................................................. xviii
CHAPTERS
1 INTRODUCTION ................................................................................................ 1
1.1 Introduction to the Problem ........................................................................... 1
1.2 Literature Survey ........................................................................................... 4
1.3 Objective ....................................................................................................... 6
1.4 Scope of the Thesis ........................................................................................ 7
2 MECHANISM SELECTION ............................................................................... 9
2.1 Introduction ................................................................................................... 9
2.2 Mechanism Candidates .................................................................................. 9
2.3 Mechanism Candidates Evaluation ............................................................. 12
3 KINEMATIC ANALYSIS OF DELTA ROBOT .............................................. 15
3.1 Introduction ................................................................................................. 15
xii
3.2 Forward and Inverse Kinematics of Delta Parallel Robot ........................... 16
3.3 Conceptual Design for Shock Absorbing with Delta Robot Mechanism .... 21
4 DYNAMIC ANALYSIS OF THE SHOCK ABSORBER ................................. 25
4.1 Introduction ................................................................................................. 25
4.2 Mathematical Model of Shock Absorber ..................................................... 25
4.2.1 Method Selection for Equation of Motion Derivation ......................... 29
4.2.2 Equation of Motion of the Shock Absorber ......................................... 45
4.3 Dynamic Simulation of the Shock Absorber ............................................... 46
4.3.1 MATLAB®-Simulink ........................................................................... 47
4.3.2 Simulink Model of the Shock Absorber ............................................... 47
4.3.3 Dynamic Analysis Simulation .............................................................. 50
4.4 SIMULATION CASES ............................................................................... 53
4.4.1 CASE 1: Shock in Z direction .............................................................. 54
4.4.2 CASE 2: Shock in Y direction ............................................................. 61
4.4.3 CASE 3: Shock in X direction ............................................................. 68
4.5 ANALITIC MODEL COMPARISON WITH MSC ADAMS® ................. 75
4.5.1 MSC ADAMS® .................................................................................... 75
4.6 Simulation Comparison between Analytic Model and MSC ADAMS® ..... 76
4.6.1 Case 1 ................................................................................................... 79
4.6.2 Case 2 ................................................................................................... 80
4.6.3 Case 3 ................................................................................................... 80
xiii
CONCLUSION AND FUTURE WORK .................................................................. 83
REFERENCES ........................................................................................................... 87
APPENDIX ................................................................................................................ 89
xiv
LIST OF TABLES
TABLES
Table 1: Mechanism Candidates Evaluation Table .................................................... 13
Table 2: Parameters of Dynamic Simulation Scenarios ............................................. 50
Table 3: Dynamic Simulation Cases .......................................................................... 50
Table 4: Geometric and initial values ........................................................................ 51
Table 5: MSC ADAMS Simulation Cases ................................................................. 76
xv
LIST OF FIGURES
FIGURES
Figure 1.1 Naval Radar example and its searching interface ....................................... 2
Figure 1.2 Example of sea mine explosion .................................................................. 2
Figure 1.3 Relation between rotational motion and target positioning error ............... 3
Figure 1.4 Example of special designed shock absorber for a radar ............................ 4
Figure 1.5 Simple Mass-Spring-Damper system ......................................................... 5
Figure 2.1 The Tripteron Mechanism and its kinematic view [2].............................. 10
Figure 2.2 3 dimensional view of the 3-PRS Parallel Manipulator [3] ...................... 11
Figure 2.3 Simple Schematic of Delta Robot [4] ....................................................... 11
Figure 2.4 Commercial Delta Robot (ABB® Company) ............................................ 12
Figure 2.5 Delta Robot pick and place application .................................................... 14
Figure 3.1 Illustration of forward and inverse kinematic [6] ..................................... 15
Figure 3.2 Delta Robot joints and link parameters [8] ............................................... 17
Figure 3.3 Intersection of three spheres at the point .................................................. 20
Figure 3.4 Conceptual 2D positioning of the springs-dampers ................................. 22
Figure 3.5 Spring-damper positioning in 3D on Delta Robot .................................... 23
Figure 3.6 Parallelogram chains in 3D on Delta Robot ............................................. 23
Figure 4.1 Input Joints of Delta Robot [9] ................................................................. 26
Figure 4.2 CGs and Reference Frames of Delta Robot [9] ........................................ 28
Figure 4.3 Reference Frames of Shock Absorber ...................................................... 29
Figure 4.4 Top view of base platform (B).................................................................. 33
Figure 4.5 Top view of moving platform (P) ............................................................. 34
Figure 4.6 40g-11ms Saw tooth Shock Profile (MIL-STD 810F) ............................. 48
Figure 4.7 Simulink Model of Equation of Motion of Shock Absorber .................... 49
Figure 4.8 Base excitation schematic of mass-spring-damper system ....................... 52
Figure 4.9 Vertical (in Z direction) shock applied to the base of absorber ................ 52
xvi
Figure 4.10 Lateral shock (in X-Y plane) in X or Y directions ................................. 53
Figure 4.11 Acceleration of moving platform in X direction(C=200Ns/m) .............. 54
Figure 4.12 Acceleration of moving platform in Y direction(C=200Ns/m) .............. 55
Figure 4.13 Acceleration of moving platform in Z direction (C=200Ns/m) .............. 55
Figure 4.14 Acceleration of moving platform in X direction(C=600Ns/m) .............. 56
Figure 4.15 Acceleration of moving platform in Y direction(C=600Ns/m) .............. 56
Figure 4.16 Acceleration of moving platform in Z direction(C=600Ns/m) ............... 57
Figure 4.17 Deflection of moving platform in X direction(C=200Ns/m) .................. 58
Figure 4.18 Deflection of moving platform in Y direction(C=200Ns/m) .................. 58
Figure 4.19 Deflection of moving platform in Z direction(C=200Ns/m) .................. 59
Figure 4.20 Deflection of moving platform in X direction(C=600Ns/m) .................. 59
Figure 4.21 Deflection of moving platform in Y direction(C=600Ns/m) .................. 60
Figure 4.22 Deflection of moving platform in Z direction(C=600Ns/m) .................. 60
Figure 4.23 Acceleration of moving platform in X direction(C=200Ns/m) .............. 61
Figure 4.24 Acceleration of moving platform in Y direction(C=200Ns/m) .............. 62
Figure 4.25 Acceleration of moving platform in Z direction(C=200Ns/m) ............... 62
Figure 4.26 Acceleration of moving platform in X direction(C=600Ns/m) .............. 63
Figure 4.27 Acceleration of moving platform in Y direction(C=600Ns/m) .............. 63
Figure 4.28 Acceleration of moving platform in Z direction(C=600Ns/m) ............... 64
Figure 4.29 Deflection of moving platform in X direction(C=200Ns/m) .................. 65
Figure 4.30 Deflection of moving platform in Y direction(C=200Ns/m) .................. 65
Figure 4.31 Deflection of moving platform in Z direction(C=600Ns/m) .................. 66
Figure 4.32 Deflection of moving platform in X direction(C=600Ns/m) .................. 66
Figure 4.33 Deflection of moving platform in Y direction(C=600Ns/m) .................. 67
Figure 4.34 Deflection of moving platform in Z direction(C=600Ns/m) .................. 67
Figure 4.35 Acceleration of moving platform in X direction(C=200Ns/m) .............. 68
Figure 4.36 Acceleration of moving platform in Y direction(C=200Ns/m) .............. 69
Figure 4.37 Acceleration of moving platform in Z direction(C=200Ns/m) ............... 69
Figure 4.38 Acceleration of moving platform in X direction(C=600Ns/m) .............. 70
Figure 4.39 Acceleration of moving platform in Y direction(C=600Ns/m) .............. 70
xvii
Figure 4.40 Acceleration of moving platform in Z direction(C=600Ns/m) .............. 71
Figure 4.41 Deflection of moving platform in X direction(C=200Ns/m) .................. 72
Figure 4.42 Deflection of moving platform in Y direction(C=200Ns/m) .................. 72
Figure 4.43 Deflection of moving platform in Z direction(C=200Ns/m) .................. 73
Figure 4.44 Deflection of moving platform in X direction(C=600Ns/m) .................. 73
Figure 4.45 Deflection of moving platform in Y direction(C=600Ns/m) .................. 74
Figure 4.46 Deflection of moving platform in Z direction(C=600Ns/m) .................. 74
Figure 4.47 3D Model of the Shock Absorber in ADAMS Environment ................. 78
Figure 4.48 Model verification toolbox of ADAMS ................................................. 78
Figure 4.49 Primitive joint toolbox of ADAMS ........................................................ 78
Figure 4.50 ADAMS and mathematical model result of Case 1................................ 79
Figure 4.51 ADAMS and mathematical model result of Case 2................................ 80
Figure 4.52 ADAMS and mathematical model result of Case 3................................ 81
xviii
LIST OF SYMBOLS
: Delta Robot input joint angle
l : Length of parallelogram link connected to moving platform
L : Length of parallelogram link connected to base platform
Ps : Edge length of equilateral triangle of moving platform
Bs : Edge length of equilateral triangle of base platform
BX : X component of base platform frame
BY : Y component of base platform frame
BZ : Z component of base platform frame
PX : X component of moving platform frame
PY : Y component of moving platform frame
PZ : Z component of moving platform frame
xV : Base platform input velocity in X direction w.r.t inertial frame
yV : Base platform input velocity in Y direction w.r.t inertial frame
zV : Base platform input velocity in Z direction w.r.t inertial frame
B : Origin point of the base frame
P : Origin point of the moving frame
O : Origin point of the inertial frame
/P BV : Velocity of moving platform w.r.t base platform frame
1m : Mass of parallelogram link connected to base platform
2m : Mass of parallelogram link connected to moving platform
M : Mass of the moving platform
C : Damping coefficient of the each damper
K : Stiffness coefficient of the each spring
KE : Kinetic energy of the whole system
xix
D : Dissipation function of the whole system
U : Potential energy of the whole system
p : Generalized momenta term of Lagrange equation
kQ : External force of Lagrange equation
sL : Length vector of each spring
0L : Initial length of each spring
CG : Center of gravity
DOF : Degrees of freedom
1
CHAPTER 1
1 INTRODUCTION
1.1 Introduction to the Problem
Parallel to advances in technology, military equipments are developing and getting
more sophisticated day by day. All these conditions make these equipments more
valuable and important assets for national defense purposes. Therefore, the
protection of these assets in the battle field is very important as much as taking
advantage of them. For that purpose, some protection tools are designed by engineers
according to the threat type and its destruction level.
Radar (RAdio Detecting and Ranging) system is one of the most advanced and
highly used devices for military purposes. They detect moving and stationary objects
around its location point in a predetermined margin. Figure 1.1 shows an example of
naval radar system with user interface.
2
Figure 1.1 Naval Radar example and its searching interface
While land radars are usually mounted on a fixed towers on the ground, naval radars
are mounted onto the superstructure of ships. Therefore, they are able to move
throughout the world. Because of this capability, they can be subject to some kind of
threats mainly mine explosion and torpedo attacks in the battle field.
Figure 1.2 Example of sea mine explosion
3
These kinds of threats are considerably dangerous for ship itself and its valuable
assets. As a very important device, radars on the ship should be protected against
these dangerous situations. For this problem, there two kinds of solution are
available. One is to make the foundation of the radar very massive in order to reduce
shock effects. Second one is to design a custom shock absorber which is quite lighter
than the foundation method but requires more effort to design. In this thesis, the
motivation is to come up with a solution like the second one. However, the main
challenge is about the shock absorber platform is that, it is supposed to have only
translational degrees of freedom, since rotational movements while ship is cruising
throughout seas can cause huge errors as the radar analyzing margin increases. The
error is calculated as follows:
Error distance (meter) = rotational degree (radian) x measuring distance (meter)
Figure 1.3 Relation between rotational motion and target positioning error
Therefore, a 3-DOF translational parallel mechanism criteria is essential for a radar
shock absorbing platform. For that criteria, some mechanism candidates found as a
result of literature survey are evaluated according to the some engineering design
parameters and the most suitable one is analyzed in terms of kinematic and dynamic
for a parametric shock absorber design.
4
1.2 Literature Survey
The literature survey is mainly focused on finding suitable mechanism candidates for
desired shock absorber design. The main keyword relating to survey is “parallel
manipulator architectures”, since it is an essential criteria for the shock absorber. For
that purpose, three mechanism candidates are found and analyzed. These candidates
are Tripteron mechanism [2], 3-PRS Parallel mechanism [3] and Delta robot [4],
respectively. They are evaluated according to some engineering design conditions.
The evaluation can be seen at Table 1. In addition to these candidates, commercial
market is also searched. The Thales ® Company which produces radar systems for
worldwide has a similar purpose shock absorbing platform for its radar products. It
can be seen in Figure 1.4. However, this design has a patent pending throughout
Europe. Hence, it is out of evaluation process because patent law regulations.
Figure 1.4 Example of special designed shock absorber for a radar
Thales-Smart-S® [5]
In this thesis, the other focusing point is the shock phenomenon that motivates
engineers to design shock absorbers to get rid of its dangerous effects. Alexander
(2009) defines this phenomenon as follows,
5
Shock: Sudden and huge change in the state of the motion of component parts or
particles of a body resulting from the sudden application of relatively large external
force such as explosion or impact [7].
In the dynamic analysis section of this thesis, 40g 11ms saw tooth profile shock
according to MIL-STD-810F is used for the mathematical model simulations.
In order to protect valuable assets from shock effects, some critical elements such as
spring and damper are needed. They store and dissipate the energy caused by shock
phenomena. For a basic shock absorbing solution, the system model that has only
one degree of freedom shown in Figure 1.5 is widely used for shock absorber design
and analysis in literature. The logic is such that spring (k) stores the shock energy
and the damper (c) dissipates it to ambient as heat.
Figure 1.5 Simple Mass-Spring-Damper system
Dynamic analysis of the mechanical systems is performed with two common
methods in general as Langrange equation and Newton-Euler method. While
Newton-Euler method deals with action- reaction force and moment equations,
Lagrange equation deals energy states of the system. Both approaches give the same
equation of motion for the same system. The one decides which methods is suitable
for his/her system. For this study, since the Delta Robot mechanism has
overconstraint characteristic according to Kutzbach equation [9], choosing Lagrange
equation method for the shock absorber dynamic analysis is more efficient.
6
Otherwise, it is going to need more effort to solve force-moment equations obtained
from Newton- Euler method. However, for the structural design of the shock
absorber, the forces and moments are needed to know. For this situation Newton-
Euler method is superior to Lagrange equation method. All in all, the method
selection for dynamic analysis depends on the user’s expectations and needs.
1.3 Objective
In the battle field, radars are critical and valuable assets that are used in order to
obtain information about the positions of the critical objects around. Because of that
reason, their protection is so essential. In the battle field, some shock cases like mine
explosion or torpedo attacks can cause some damages on the radar system as failure
or total destruction. In order to overcome the negative effects of the shock waves on
radar systems, some special shock absorbing platforms are needed. In addition to
helping to survival of radar system, the absorber should also maintain its parallelism
with respect to the platform on where it is mounted for high measuring accuracy of
the radar system while there is no threat. In other words, the platform should move
only in translational directions (X, Y and Z) not to reduce radar accuracy. Otherwise,
rotational motions even it is small, can cause huge errors in long distances.
Therefore, the main objective of the thesis is to design a proper shock absorbing
translational parallel platform for radar systems. Moreover, the thesis will also deal
with the kinematic and dynamic analysis of the proposed shock absorbing platform.
The dynamic analysis will focus on deriving parametric mathematical model to
analyze behavior of the platform according to shock specification mentioned in the
Military Standard 810F by using rigid body assumption. The mathematical model is
simulated with different spring constants and damping coefficients in order to
observe system behavior. Besides, some defined cases results are compared with
MSC ADAMS dynamic simulation software results in order to validate accuracy of
mathematical model.
7
1.4 Scope of the Thesis
The outline of the thesis is formed as follows:
In Chapter 2; firstly, mechanism candidates which have 3 translational DOF found as
result of literature survey are presented and they are shortly described. Lastly, they
are evaluated according to the some engineering design criteria in order to select best
candidate for shock absorbing platform. After selection, the detailed information is
given related to the selected mechanism.
In Chapter 3; the kinematic analysis of the mechanism is done by using symbolic
parameters. For kinematic analysis, there are two approaches. They are inverse and
forward kinematics, respectively. The both approaches are mathematically described
and introduced for the selected mechanism. Lastly, the conceptual design is made in
order to integrate shock absorbing elements such as springs and dampers to the
mechanism in order to give shock absorbing capability to selected parallel
manipulator.
In Chapter 4; the shock absorbing parallel mechanism is analyzed dynamically by
using rigid body assumption. For that dynamic analysis, 3-D model which planning
to manufacture according the selected mechanism architecture is used. The geometric
dimensions and values are obtained from the prototype. In this chapter, the equation
of motion of the shock absorbing platform is derived by using Lagrange equations of
motions. The derived equations are solved numerically with Simulink tool of
MATLAB software. The dynamic simulations are done with different spring
constants and damping coefficients. Transmitted shock to moving platform and the
deflection of it under this shock condition are observed with graphics in this chapter.
In addition to these observations, the derived mathematical model results are
compared with MSC ADAMS simulation software result for defined cases for
validation.
8
In Chapter 5, brief summary is given about work with recommendations and
discussions. The simulation results are commented with general interpretations. In
addition to the conclusion, some possible future works and design suggestions are
also mentioned in this chapter.
9
CHAPTER 2
2 MECHANISM SELECTION
2.1 Introduction
In this chapter, the mechanism candidates that have 3 translational DOFs found in
literature are proposed. In section 2.2, the candidates are presented by giving short
information about them. In section 2.3, the candidates are evaluated according to
some criteria. Mechanism that has the highest grade is selected as shock absorbing
mechanism. After selection, detailed information is given about it.
2.2 Mechanism Candidates
Tripteron Mechanism
Tripteron Mechanism [1] is simply a serial Cartesian robot mechanism. Each linear
actuator controls one translational motion independently. Since it has 3 independent
linear actuation joints in Cartesian coordinate, the mechanism has only 3
translational degrees of freedom. All joints used for this mechanism is revolute joint
type. The kinematic view of the mechanism is shown in Figure 2.1.
10
Figure 2.1 The Tripteron Mechanism and its kinematic view [2]
3-PRS Parallel Manipulator
The 3-PRS Parallel [2] mechanism consists of three prismatic, one revolute and one
spherical joint chain from base to end effector respectively. The base connection of
the chains is orientated in 120 degree with respect to each other around base platform
center axis and they are prismatic joint types. The connections joints on the end
effector are spherical type joints. This joint combination gives the end effector 3
degrees of freedom in translation directions.
11
Figure 2.2 3 dimensional view of the 3-PRS Parallel Manipulator [3]
Delta Parallel Robot
Delta robot [3] is a parallel robot which has 3 kinematic chains between the base and
end effector (moving platform). The chains consist of parallelograms which prevents
rotational movements of the end effector and maintain the parallelism of end effector
to base. Since the rotational motions are eliminated, the degrees of freedom are
reduced from 6 to 3. These degrees of freedom are all in translational axis.
Figure 2.3 Simple Schematic of Delta Robot [4]
12
Figure 2.4 Commercial Delta Robot (ABB® Company)
The main characteristic of mechanisms introduced above is that they permit the end
effector or in other words, moving platform to do only translational motions.
Therefore, they are counted to be all suitable candidates for desired shock absorbing
platform parallelism criteria.
2.3 Mechanism Candidates Evaluation
Found as a result of literature survey, these candidates are evaluated according to the
some criteria in Table 1 in order to determine the best one for the shock absorbing
platform. The evaluation is mainly based on general engineering design experiences.
The criteria are selected as follows:
Joint design simplicity
Easiness to integrate onto naval platforms
Spring-damper integration simplicity
Easiness to manufacture
All points are out of 10;
13
Table 1: Mechanism Candidates Evaluation Table
Tripteron
3-PRS
Parallel
Manipulator
Delta
Parallel
robot
Joint design simplicity 5 7 9
Easiness to integrate
onto naval platforms 2 6 8
Spring-damper
integration simplicity 2 5 8
Easiness to manufacture 6 6 8
RESULTS: 15 24 33
After evaluation of the mechanism candidates, the result reveals that Delta Parallel
Robot is the most suitable parallel manipulator among the candidates for shock
absorbing platform design for naval radars. Therefore, detailed study is done on delta
mechanism in this thesis.
Delta Parallel Robot
Delta robot was invented by Professor Reymond Clavel in the early 80’s. The basic
idea behind the mechanism is using parallelograms in order to eliminate rotational
motions. When it is compared to Steward Platform which has 6 degrees of freedom
in space, Delta Robot is capable of doing only translational motions. As an
architectural feature of the mechanism the links are very light compared to its
14
rigidity. Therefore, high accelerations for the end effector (moving platform) can be
obtained. That makes the mechanism a perfect candidate in robotic industry
especially for pick and place applications when mass production is concerned.
In this thesis, Delta Robot mechanism is used for designing a special shock absorber
architecture for military purposes as mentioned in introduction section in Chapter 1.
Figure 2.5 Delta Robot pick and place application
15
CHAPTER 3
3 KINEMATIC ANALYSIS OF DELTA ROBOT
3.1 Introduction
For the kinematic analysis of the mechanisms, in general, there are two approaches
as forward kinematics and inverse kinematics. In this chapter, Delta Robot
mechanism will be analyzed in terms of forward and inverse kinematic.
Figure 3.1 Illustration of forward and inverse kinematic [6]
16
Forward Kinematics
Forward kinematics refers to the use of the kinematic equations of a robot to
compute the position of the end-effector from specified values for the joint
parameters [6]. In other words, the position of the end-effector is calculated by using
joint parameters.
Inverse Kinematics
Inverse kinematics is defined as the use of the kinematics equations of a robot to
determine the joint parameters for obtaining a desired position of the end-effector
[6]. Specification of the movement of a robot so that its end-effector achieves a
desired task is known as motion planning. Inverse kinematics transforms the motion
plan into joint actuator trajectories for the robot.
3.2 Forward and Inverse Kinematics of Delta Parallel Robot
Delta Robot consists of three kinematic chains that connect moving platform to base
platform. The main characteristic of the chains is that each of them includes
parallelogram which gives the Delta Robot parallelism feature. For a relatively
simpler kinematic analysis, all links in the Delta Robot are assumed to be rigid.
17
Figure 3.2 Delta Robot joints and link parameters [8]
Geometric parameters of Delta Robot kinematic [8]: 1 2, , , , ( 1,2,3)A B jL L r r j and
the joint angles 1 2 3, , ( 1,2,3)j j j j illustrated in Figure 3.2 .
The point P represents the moving platform centroid, and the body coordinate system
is attached to the base platform centroid.
The coordinate values with respect to body frame are as follows:
2 1 1 3 1 2 1 3cos ( cos cos cos( ) ) sin sinp j A j j j j B j jX r L L r L (3.1)
2 1 1 3 1 2 1 3sin ( cos cos cos( ) ) cos sinp j A j j j j B j jY r L L r L (3.2)
2 1 1 3 1 2sin cos sin( )p j j j jZ L L (3.3)
[ , , ]p p pX Y Z represents the coordinates of the point P with respect to body coordinate
system.
18
The equations (3.1), (3.2) and (3.3) are squared as follows,
2
2 1
2
1 3 1 2 1 3
[cos ( cos ) - ]
[cos ( cos cos( )) - cos sin )]
j j p
j j j j
r L X
L L
(3.4)
2
2 1
2
1 3 1 2 1 3
[sin ( cos ) ]
[sin ( cos cos( )) cos sin )]
j j p
j j j j
r L Y
L L
(3.5)
2 2
2 1 1 3 1 2[ sin ] [ cos sin( )]j p jL Z L (3.6)
Simply let’s say;
1 3 1 2cos cos( )jL A (3.7)
1 2 B (3.8)
Equations (3.4), (3.5) and (3.6) are summed together and shown in a simplified form
in Eqn.(3.10).
2 2
2 1 2 1
2
2 1
2 2 2 2 2
1 3 1 3
2 2 2 2 2
1 3
1 3
2 2
1 3
[cos ( cos ) ] [sin ( cos ) ]
[ sin ]
cos sin sin 2cos sin sin
sin cos sin
2sin cos sin
cos
j j p j j p
j p
j j j j j j
j j j
j j j
j
r L X r L Y
L Z
A L AL
A L
AL
L
(3.9)
2 2
2 1 2 1
2
2 1
2 2 2 2 2 2
1 3 1 3
[cos ( cos ) ] [sin ( cos ) ]
[ sin ]
sin cos sin
j j p j j p
j p
j j
r L X r L Y
L Z
A L L B
(3.10)
19
Where, 1 3 1 2cos cos( )jL A ;
Then the right hand side of the equation (3.10) become as follows,
2 2 2 2 2 2 2 2
1 3 1 3 1 3
2
1
cos cos sin cos sinj j jL B L L B
L
As a result:
2 2
2 1 2 1
2 2
2 1 1
[cos ( cos ) ] [sin ( cos ) ]
[ sin ]
j j p j j p
j p
r L X r L Y
L Z L
(3.11)
Where 1,2,3j and A Br r r ;
Forward Kinematic Model
In this model, the location of the point [ , , ]p p pP X Y Z is to be determined for given
joint angles 1 2 3, , ( 1,2,3)j j j j .
For known joint angles, equation (3.11) becomes,
2 2 2 2
1( ) ( ) ( )P j P j P jX X Y Y Z Z L (3.12)
Where,
2 1cosj jX r L
2 1cosj jY r L
2 1sinj jZ L
Eqn.(3.12) is a sphere equation centered in point [ , , ]j j j jS X Y Z with radius 1L .
The solution of the system of equations is shown by a point located at the
intersection of the three spheres as illustrated in Figure 3.3.
20
Figure 3.3 Intersection of three spheres at the point
Inverse Kinematics Model
In this model, [ , , ]p p pP X Y Z are known and 1 2 3, , ( 1,2,3)j j j j are to be
determined. Eqn.(3.13) shows the extended and simplified form of Eqn.(3.9)
2 2 2 1
2 1
2 2 2 2 2 2
2 1
(2 2 cos 2 sin )cos
2 cos 2 sin 2 sin
0
P j P J j
p j P j p J
P P P
rL L X L Y
rX L Z rY
X Z Y r L L
(3.13)
Eqn.(3.13) can be written in a form as below,
1 1cos sinj j j j jl m n (3.14)
Where,
21
2 2 2
2
2 2 2 2 2 2
2 1
2 2 cos 2 sin
2
2 cos 2 sin
j P j P J
j p
j p j p J
P P P
l rL L X L Y
m L Z
n rX rY
X Z Y r L L
The equation is valid if only if,
2 2 2
2 21 ( ) 0
j
j j j
j j
nn l m
l m
(3.15)
General rule:
sin cosa b c
Then,
2 2 2arctan 2( , ) arctan 2( , )a b a b c c
If the rule is applied to the Eqn.(3.13), the joint angles calculated as in Eqn.(3.16),
2 2 2
1 arctan 2( , ) arctan 2( , ), ( j 1,2,3)j j j j j j jl m l m n n where (3.16)
3.3 Conceptual Design for Shock Absorbing with Delta Robot Mechanism
After selecting appropriate 3-DOF translational parallel mechanism among
candidates found in literature, the next step is to provide it shock absorbing feature
by integrating spring and damper elements onto suitable places. Delta Robot
mechanism consists of three parallelogram chains oriented by 120 degree around the
base platform center axis. Therefore, for the sake this pattern, springs and dampers
are integrated onto the mechanism in the same orientation by using volume between
chains.
22
Positioning of the Spring-Damper System on Delta Robot Mechanism
On the Delta Robot Mechanism, it is considerably easy to allocate locations between
mechanism chains for spring-damper elements as it seen in Figure 3.4. These
locations are oriented by 120 degree around base platform center such that they are
placed between parallelogram chains. This helps the mechanism to sustain its
kinematic pattern as original Delta Robot architecture. Besides, taking advantage of
these location also help the one while assembling the absorber elements.
Figure 3.4 Conceptual 2D positioning of the springs-dampers
3D positioning of the springs and dampers
Three spring and damper elements are located in 120 degree angle pattern w.r.t base
center axis, connected to the base and moving platform as seen in Figure 3.5 and
Figure 3.6. The volume between parallelogram chains pattern is used for this
implementation. That situation gives easiness to whole system assembly process
since spring-dampers connection locations are independent from mechanical chains.
23
Figure 3.5 Spring-damper positioning in 3D on Delta Robot
Figure 3.6 Parallelogram chains in 3D on Delta Robot
24
25
CHAPTER 4
4 DYNAMIC ANALYSIS OF THE SHOCK ABSORBER
4.1 Introduction
In this chapter, the shock absorbing parallel platform will be analyzed dynamically
according to the naval shock profile stated in the MIL-STD-810F [1]. In that
dynamic analysis a parametric mathematical model of the shock absorber is derived
by using Lagrange equation of motion approach.
4.2 Mathematical Model of Shock Absorber
For a parametric shock absorber design and analysis, mathematical model of the
system is essential. In this thesis, for the mathematical model derivation, Lagrange
equation of motion approach is used. As a procedure of Lagrange equation, kinetic,
potential energy equations and dissipation function of the absorber are calculated. By
using partial differentiation method, equation of motion of the absorber is obtained.
Revolute Input Delta Robot
3-DOF Delta Robot, as shown in Figure 4.1, has three identical kinematic chains
located by 120° degree orientation. There are three input revolute joints at the base
26
platform. They are represented by ( 1, 2,3)i i . In robotic applications, these input
revolute joints are allocated for electrical motor connections. Since Delta Robot is
used as a shock absorber mechanism in this thesis, all these input joints are passive.
Figure 4.1 Input Joints of Delta Robot [9]
Delta Robot DOF Calculation
Delta Robot is well-known as a 3-DOF parallel robot. The DOF (degrees of freedom)
of the robot is calculated by using Kutzbach mobility equation. The general for of the
equation is shown in Eqn.(4.1),
1 2 36(N 1) 5 4 3M J J J (4.1)
M is number of degrees-of freedom
N is the total number of links, including ground
J1 is the number of one-dof joints
J2 is the number of two-dof joints
J3 is the number of three-dof joints
J1-one-dof joints: revolute and prismatic joints
27
J2-two-dof joints: universal joints
J3-three-dof joints: spherical joints
For the Delta Robot used as a shock absorber,
1
2
3
17
21 6(17 1) 5(21) 4(0) 3(0)
0 9
0
N
J M
J M dof
J
(4.2)
According to the Eqn.(4.2), Delta Robot is overconstrained, in other words, it has
statically indeterminate structure. However, in real it is not the situation, because the
geometry of Delta Robot has a very special. The robot would work kinematically
identical to the original one if one of the long parallel four-bar mechanism links is
removed with two revolute joints of it. Then the Kutzbach equation for new
configuration becomes as follows,
1
2
3
14
15 6(14 1) 5(15) 4(0) 3(0)
0 3
0
N
J M
J M dof
J
(4.3)
It is shown mathematically in Eqn.(4.3) that Delta Robot has 3-DOF.
Delta Robot (Reference Frames and CGs Indicated)
Delta Robot consists of two platforms. They are called base platform (B) fixed to the
ship structure and moving platform (P) which moves relative to base platform.
Therefore, there are two coordinate axes on the Delta Robot as shown in Figure 4.2.
28
Moreover, the center of gravity points of links in chains and moving platform are
indicated in same figure.
Figure 4.2 CGs and Reference Frames of Delta Robot [9]
Figure 4.3 shows the 3D view of the shock absorber with the reference frames (base
frame, moving frame and inertial frame) that is analyzed dynamically. Moreover, the
base and moving platform are clearly indicated on this figure.
29
Figure 4.3 Reference Frames of Shock Absorber
4.2.1 Method Selection for Equation of Motion Derivation
While deriving the equation of the motion of the mechanical systems there two
common methods in general. These are Newton-Euler method and Lagrange
equations method, respectively. Newton-Euler method deals with joint reaction
forces and moments in order to model the system dynamic. When joints forces or
joint torques are needed, this method is very useful. Since all joints reactions are
supposed to be calculated in this method, the number of equations is directly
dependent on the joint numbers and types. Newton-Euler method is very helpful and
widely used in the case of making structural link and joint desing for the systems to
achieve desired task.
On the other hand, Lagrange equations use energy equations to provide mathematical
model of the mechanical system. It doesn’t focus on joint reactions and directly gives
overall system characteristic with the help of energy states. However, since the
30
Lagrange equations include partial differentiation of energy equations, it is
mathematically inefficient method for the system that has lots of degrees of freedom.
According to the Kutzbach equation, Delta Robot mechanism has overconstraint
architecture, in other words, it is statically indeterminate system. For those kinds of
systems, joint reaction calculations cannot be found since the number of unknowns
and the number of equations is not equal each other. For this situation, some other
solution techniques such as elasticity theorem are needed. Besides, for the
mathematical model of the shock absorber, there is no need for joint forces and
moments. Because, all joints on the absorber are passive type. Therefore, in order to
overcome overconstraint situation of Delta Robot, Lagrange equations of motion
method is decided to be used in this thesis to derive mathematical model of the shock
absorber system.
While formulating Lagrange equation of motion of the shock absorber, firstly, kinetic
energy, potential energy and dissipation function of the system are obtained. Then
these equations are partially differentiated in terms of generalized coordinates. The
number of the generalized coordinates are equal to the number of degrees of
freedom. Since Delta Robot has 3 degrees of freedom, Lagrange equation should
have 3 generalized coordinates. In this case, the generalized coordinates of our
system are:{x, y,z} .
4.2.1.1 Dynamic Model Assumptions
In engineering calculations, some assumptions are made in order to simplify the
problem. This helps the one to handle mathematical calculations easily. For this
study, assumptions below are made for mathematical model formation;
31
All links are rigid.
All joints are frictionless.
All springs and dampers are linear.
Gravity is not considered in dynamic model.
Rotational motions of the shock absorber with respect to inertial frame are
not considered.
The mathematical model of the shock absorber in this thesis is derived based on
these assumptions.
4.2.1.2 Kinetic energy of the system
As a first step of the Lagrange equation, kinetic energy of the whole system is
needed to be calculated. For kinetic energy calculation, velocity of each element
must be determined with respect to inertial frame (O).
The velocity of the base platform with respect to inertial frame,O
B
Vx
V Vy
Vz
The velocity of the moving (P) platform with respect to base frame (B),B
P
x
V y
z
The absolute velocity of the moving platform (P), /p P B B
x Vx
V V V y Vy
z Vz
By using velocity relations above, the total kinetic energy calculation formula
becomes as in Eqn.(4.4).
32
3
1 2
1
2 2 2
1 1 1 ,COG 1 ,COG
2 2 2 ,COG 2 ,COG
( )
,
1[(x ) (y Vy) (z Vz) ]
2
1K [( V V ) .( V V )]
2
1K [( V V ) .( V V )]
2
p i i
i
p
B O T B O
i i i B i B
B O T B O
i i i B i B
KE K K K
where
K M Vx
m
m
(4.4)
11,COG 11,COG 1 1
12,COG 12,COG 2 2
13,COG 13,COG 3 3
V V ( , )
V V ( , )
V V ( , )
B B
B B
B B
(4.5)
Where,
3
1 2
1
( )i i
i
K K
is the kinetic energy of the Delta Robot links [9],
pK is the kinetic energy of the moving platform(P) with mass M,
In order to find the velocity of the each links, time derivative of position equations of
Delta Robot is needed. The position equations are obtained from loop closure
equations [8]. Eqn.(4.6) shows Delta Robot loop closure equation.
33
B B B B B
i i i P iB L l P P (4.6)
B B B B B
i i P i i il l P P B L (4.7)
22 2 2 2
,
B
i i ix iy izl l l l l
where
(4.8)
1,2,3i
In order to specify the connection points of each chain and spring on both base and
moving platform easily with respect to body frames, the shock absorber is analyzed
with top views of each platform as in Figure 4.4 and Figure 4.5.
Figure 4.4 Top view of base platform (B)
34
Figure 4.5 Top view of moving platform (P)
The vector B
iL is dependent on base joint (input) angles, 1 2 3, ,T
2 3
1 1 2 2 3 3
12 3
3 3cos cos
2 201 1
cos cos cos2 2
sinsin sin
B B B
L L
L L L L L L
LL L
2 3
1 1 2 2 3 3
12 3
3 3cos cos
2 2
1 1cos cos cos
2 2sin
sin sin
B B B
x L b x L b
x
l y L l y L c l y L c
z Lz L z L
35
Where,
3
2 2
1
2
B p
p
B
P B
a w u
sb w
c w w
All there equations above yield,
2 2 2 2 2 2
1 1
2 2 2 2 2 2 2
2 2
2 2 2 2 2 2 2
3 3
2 ( ) cos 2 sin 2
0
( 3(x b) y c) cos 2 sin 2 2
0
( 3(x b) y c) cos 2 sin 2 2
0
L y a zL x y z a L ya l
L zL x y z b c L xb yc l
L zL x y z b c L xb yc l
(4.9)
Analytically, the position equations are treated as following form in Eqn.(4.10).
cos sin 0 1,2,3i i i iE F G i (4.10)
36
1
1
2 2 2 2 2 2
1
2
2
2 2 2 2 2 2 2
2
2
2
2 2 2 2 2 2 2
2
,
2 (y a)
F 2
2
( 3(x b) y c)
F 2
2 2
( 3(x b) y c)
F 2
2 2
where
E L
zL
G x y z a L ya l
E L
zL
G x y z b c L xb yc l
E L
zL
G x y z b c L xb yc l
(4.11)
Tangent Half-Angle Method for the solution,
2
2 2
1 2tan cos sin
2 1 1
i i ii i i
i i
t tt
t t
(4.12)
2
2 2
2 2
1 20 1 2 1 0
1 1
i ii i i i i i i i i
i i
t tE F G E t F t G t
t t
(4.13)
1,2
2 2 2
2(G E ) t (2F ) t (G E ) 0 ti i i i
i i i i i i i i
i i
F E F G
G E
(4.14)
12tan (t )i i (4.15)
37
It is seen in Eqn.(4.15) that all ( 1,2,3)i i can be calculated in terms of , ,x y z
which are generalized coordinates, by using kinematic equation above.
Since ( 1,2,3)i i and ( 1,2,3)i i can be expressed in terms of , ,x y z and , ,x y z ,
the velocity equations of link CGs in Eqn.(4.16) can be also expressed in terms of
, ,x y z and , ,x y z .
11,COG 11,COG
12,COG 12,COG
13,COG 13,COG
V V ( , , , , , )
V V ( , , , , , )
V V ( , , , , , )
B B
B B
B B
x y z x y z
x y z x y z
x y z x y z
(4.16)
All links located on the delta robot chains have very small mass compared to main
mass which is located on the moving platform ( 1 2,i im m M ). Hence, their masses
are assumed to be zero.
1
2
0
0
i
i
m
m
(4.17)
In this situation, the kinetic energy of the each link except the moving platform (P)
becomes zero. Hence, detailed velocity analysis related to the links is unnecessary
effort. As a result of this condition, kinetic energy terms of chain links becomes zero
as stated in Eqn.(4.18).
1 1 1 ,COG 1 ,COG
2 2 2 ,COG 2 ,COG
1K [( V V ) .( V V )] 0
2
1K [( V V ) .( V V )] 0
2
B O T B O
i i i B i B
B O T B O
i i i B i B
m
m
(4.18)
38
Then total kinetic energy of the system forms as in Eqn.(4.19),
2 2 2
,
1[(x ) (y Vy) (z Vz) ]
2
p
p
KE K
where
K M Vx
(4.19)
4.2.1.3 Potential energy of the system
As a second step of Lagrange equation for the shock absorber, potential energy of the
whole system must be calculated. Based on the assumptions, gravity acceleration in
the analysis is neglected since it has very small effect on the system compared to 40g
11ms sawtooth shock profile. Therefore, whole potential energy is only stored on 3
springs on the shock absorber. The potential energy formulation of the shock
absorber is stated in Eqn.(4.20).
2 2 2
1 0 2 0 3 0
2 2 2
1
2 2 2
2
2 2 2
3
1[(L L ) (L L ) (L L ) ]
2
,
3 3L (x wb) (y sp) (z)
2 2 2 6
3L (x) (y sp wb) (z)
3
3 3L (x wb) (y sp) (z)
2 2 2 6
s s s
s
s
s
U K
where
sp wb
sp wb
(4.20)
39
0L Initial length of the springs (They are all identical springs)
4.2.1.4 Dissipation function of the system
The third step of Lagrange equation related to the shock absorber is to formulate the
general dissipation function of the system. The formula is stated in Eqn.(4.21).
2 2 2
1 2 3
0
1
1
2
2
3
3
1[(L ) (L ) (L ) ]
2
, 0
,
3 3 1(x wb) x (y wb)
2 2 6 2
3x (y wb)
3
3 3 1(x wb) x (y wb)
2 2 6 2
s s s
s
s
s
s
s
s
D C
Since L
where
spsp y zz
LL
x sp y zz
LL
spsp y zz
LL
(4.21)
40
4.2.1.5 General Equation of Motion (Lagrange Equation)
After kinetic, potential energy and dissipation function formulas of the shock
absorber is derived, they are partially differentiated according to the general form of
Lagrange equation shown in Eqn.(4.22)
Since there is no external force on the shock absorber, 0kQ
k k
k k k
K U Dp Q
q q q
(4.22)
, x, y,z , x, y,z (1,2,3) 0k k k k
k
Kp q q where k and Q
q
4.2.1.6 Generalized momenta
The generalized momenta terms of Lagrange equation are derived as in Eqn.(4.23).
(x Vx)
(y Vy)
( Vy)
x
y
z
KEp M
x
KEp M
y
KEp M z
z
(4.23)
Time derivative of generalized momenta terms in Eqn. (4.23) is stated in Eqn.(4.24).
41
(x ax)
(y ay)
(z az)
x
y
z
p M
p M
p M
(4.24)
4.2.1.7 Partial Differentiation of Potential Energy
As a second procedure of Lagrange equation stated in Eqn.(4.22), potential energy
formula is partially differentiated with respect generalized coordinates. The
differentiation steps of potential energy formula are shown in Eqn.(4.25).
1 2 31 0 2 0 3 0
1 2 31 0 2 0 3 0
1 2 31 0 2 0 3 0
L L L[(L L ) (L L ) (L L ) ]
L L L[(L L ) (L L ) (L L ) ]
L L L[(L L ) (L L ) (L L ) ]
s s ss s s
s s ss s s
s s ss s s
UK
x x x x
UK
y y y y
UK
z z z z
(4.25)
42
11
1
22
2
33
3
11
1
22
2
33
3
11
1
22
2
33
,
3(x wb)
L 2 2
L
3(x wb)
L 2 2
3 1(y wb)
L 2 2
3(y wb)
L 3
3 1(y wb)
L 2 2
L
L
L
ss x
s
ss x
s
ss x
s
ss y
s
ss y
s
ss y
s
ss z
s
ss z
s
ss z
where
sp
Lx L
xL
x L
sp
Lx L
sp
Ly L
sp
Ly L
sp
Ly L
zL
z L
zL
z L
L
3s
z
z L
43
4.2.1.8 Partial Derivative of Dissipation Function
The third procedure of Lagrange equation is to differentiate dissipation function of
the shock absorber partially with respect to time derivative of generalized
coordinates. The differentiation steps are stated in Eqn.(4.26).
1 2 31 2 3
1 2 31 2 3
1 2 31 2 3
L L L[(L ) (L ) (L )]
L L L[(L ) (L ) (L )]
L L L[(L ) (L ) (L )]
s s ss s s
s s ss s s
s s ss s s
DC
x x x x
DC
y y y y
DC
z z z z
(4.26)
1
1
2
2
3
3
,
3 3 1(x wb) x (y wb)
2 2 6 2
3x (y wb)
3
3 3 1(x wb) x (y wb)
2 2 6 2
s
s
s
s
s
s
where
spsp y zz
LL
x sp y zz
LL
spsp y zz
LL
44
11
1
22
1
33
3
11
1
22
2
33
3
11
1
22
2
3
3(x wb)
2 2
3(x wb)
2 2
3 1(y wb)
6 2
3(y wb)
3
3 1(y wb)
6 2
sx
s
sx
s
sx
s
sy
s
sy
s
sy
s
sz
s
sz
s
sz
spL
Dx L
L xD
x L
spL
Dx L
spL
Dy L
spL
Dy L
spL
Dy L
L zD
z L
L zD
z L
LD
3
3s
z
z L
45
4.2.1.9 Partial Derivative of Kinetic Energy
Since, there is no dependency of the whole kinetic energy formula of the shock
absorber on , ,x y z terms, partial differentiation of kinetic energy with respected
these generalized coordinates are zero. The mathematical representation of this
condition is shown in Eqn.(4.27).
0
0
0
KE
x
KE
y
KE
z
(4.27)
4.2.2 Equation of Motion of the Shock Absorber
All differentiated equations (4.24),(4.25),(4.26) and (4.27) are substituted into the
general Lagrange equation (4.28). The obtained equations in Eqn.(4.28) represent the
equation of the motion of the shock absorber.
1
2
3
x
y
z
U DQ p
x x
U DQ p
y y
U DQ p
z z
(4.28)
46
1 0 1 2 0 2 3 0 3
1 0 1 2 0 2 3 0 3
1 0 1 2 0 2 3 0 3
1 1 2 2 3 3
1 1 2 2
,
[(L L )L (L L )L (L L )L ]
[(L L )L (L L )L (L L )L ]
[(L L )L (L L )L (L L )L ]
[L D L D L D ]
[L D L D
s s x s s x s s x
s s y s s y s s y
s s z s s z s s z
s x s x s x
s y s y
where
UK
x
UK
y
UK
z
DC
x
DC
y
3 3
1 1 2 2 3 3
1 2 3
L D ]
[L D L D L D ]
, , 0
s y
s z s z s z
DC
z
Q Q Q
They are 3 set 3 unknown homogenous nonlinear differential equations which
represent the general mathematical model of the shock absorber.
4.3 Dynamic Simulation of the Shock Absorber
Motion equation of the shock absorber is obtained analytically by using Lagrange
equation of motion approach based on the assumptions. After conducting necessary
mathematical steps, 3 set of differential equations are needed to be solved
47
simultaneously with respect to 3 unknowns , ,x y z in order to analyze the motional
characteristic of the shock absorber. For the differential equations solution in this
thesis, Simulink [11] tool of the MATLAB® is used.
4.3.1 MATLAB®-Simulink
MATLAB® is a commercial and widely used software developed for mathematical
calculation on computer environment. Simulink [11] is a sub calculation tool of the
MATLAB. While MATLAB mainly uses scripts for model construction, Simulink
uses block diagrams in order to construct and solve the mathematical models. The
solution method for MATLAB and Simulink is same but the difference is only the
user interface.
4.3.2 Simulink Model of the Shock Absorber
With the help of Simulink, motion equations of the shock absorber obtained
analytically are implemented into computer environment in order to conduct
numerical calculations. Numerical differential solver of MATLAB is used for
simultaneous solution of motion equations. All these calculation is conducted by
constructing a compact Simulink model that composed block diagrams. The
overview of the model can be seen in Figure 4.7. For dynamic analysis, the input is,
as it predetermined, saw tooth 40g-11ms acceleration shock as in Figure 4.6 that is
applied to the base of shock absorber.
48
Figure 4.6 40g-11ms Saw tooth Shock Profile (MIL-STD 810F)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
5
10
15
20
25
30
35
40
Time [s]
Accele
ration [
g]
Mechanical Shock Type
40g-11ms sawtooth
49
Figure 4.7 Simulink Model of Equation of Motion of Shock Absorber
50
4.3.3 Dynamic Analysis Simulation
The scenario for dynamic analysis simulation is to apply the predetermined
acceleration shock profile onto the base platform of the absorber in X, Y and Z
directions independently by using base frame axis. Since the mathematical model is
parametric in terms of mass and spring-damper coefficients, the number of
simulation scenarios can be easily increased. In this thesis study, 3 simulations are
conducted according to the values define in the Table 2.
Table 2: Parameters of Dynamic Simulation Scenarios
M=40 kg
Scenario 1 Scenario 2
C=200 Ns/m K=20000 N/m C=600 Ns/m K=20000 N/m
C=200 Ns/m K=30000 N/m C=600 Ns/m K=30000 N/m
C=200 Ns/m K=40000 N/m C=600 Ns/m K=40000 N/m
Table 3: Dynamic Simulation Cases
Simulation Cases
Case 1 Shock in purely Z direction
Case 2 Shock in purely Y direction
Case 3 Shock in purely X direction
51
The geometric and initial values of the mathematical model simulation are defined at
Table 4.
Table 4: Geometric and initial values
Dimensional Parameters
bs =0.848m
ps =0.258m
0L =0.188m
Bw =3
6bs
0Z =162mm (Initial height of moving
platform w.r.t base platform)
0
0
, , 0,
, , 0,
, 0,
x x x
y y y
z z
The simulation is mainly based on general base excitation of the shock absorber
platform in X, Y and Z directions as Figure 4.8. All mathematical model simulations
are done according to purely base excitation case.
52
Figure 4.8 Base excitation schematic of mass-spring-damper system
Base excitation of the shock absorber can be illustrated as a 2D form in Figure 4.9
and Figure 4.10. While vertical shock represents shock in Z direction, lateral shock
represents shock in X and Y directions, independently. 40 kg mass (useful load) is
located on the moving platform.
Figure 4.9 Vertical (in Z direction) shock applied to the base of absorber
53
Figure 4.10 Lateral shock (in X-Y plane) in X or Y directions
4.4 SIMULATION CASES
The simulations of the mathematical model are done according to the some
predetermined scenarios. The number of simulation with different parameters can be
easily increased since the mathematical model is parametric. In this thesis, specified
values for the simulation are specified at Table 2. The shock absorber platform is
observed in terms of transmitted shock acceleration and deflection of the moving
platform in X, Y and Z direction when the base of the shock absorber platform is
subject to 40g-11ms saw tooth acceleration in X, Y and Z directions independently.
Two different damping coefficients and three different spring coefficients are
selected for simulations.
54
4.4.1 CASE 1: Shock in Z direction
At this case, the acceleration shock is applied purely in Z direction of the base
reference frame. The acceleration and deflection of the moving platform with respect
to base reference frame is observed.
Figure 4.11 Acceleration of moving platform in X direction(C=200Ns/m)
55
Figure 4.12 Acceleration of moving platform in Y direction(C=200Ns/m)
Figure 4.13 Acceleration of moving platform in Z direction (C=200Ns/m)
56
Figure 4.14 Acceleration of moving platform in X direction(C=600Ns/m)
Figure 4.15 Acceleration of moving platform in Y direction(C=600Ns/m)
57
Figure 4.16 Acceleration of moving platform in Z direction(C=600Ns/m)
The acceleration of the moving platform in the case of purely Z direction shock, can
be interpreted by analyzing the graphs above. Figure 4.11, Figure 4.12, Figure 4.14
and Figure 4.15 indicates that there is no acceleration in X or Y direction while the
shock absorber is subject to only shock in Z direction of base frame as expected
(Small errors caused by numerical calculations are counted to be zero). The
acceleration of moving platform in Z direction in this case on the other hand is
dependent on the spring constant and the damping coefficient of the system. The
stiffer spring and the more damping, the more acceleration is transmitted to moving
platform. These deduction can be observed by looking at Figure 4.13 and Figure
4.16.
58
Figure 4.17 Deflection of moving platform in X direction(C=200Ns/m)
Figure 4.18 Deflection of moving platform in Y direction(C=200Ns/m)
59
Figure 4.19 Deflection of moving platform in Z direction(C=200Ns/m)
Figure 4.20 Deflection of moving platform in X direction(C=600Ns/m)
60
Figure 4.21 Deflection of moving platform in Y direction(C=600Ns/m)
Figure 4.22 Deflection of moving platform in Z direction(C=600Ns/m)
61
Since there is no acceleration or initial velocity of moving platform in X and Y
direction in case 1, the deflection of it in these axis is zero as it seen in Figure 4.17,
Figure 4.18, Figure 4.20 and Figure 4.21 . However, moving platform has deflection
in Z direction since it has acceleration component in this direction. It is observed that
the value of deflection has a contradiction with transmitted acceleration. Figure 4.19
and Figure 4.22 show that as spring and damping coefficients gets higher, the
deflection of the moving platform gets smaller.
4.4.2 CASE 2: Shock in Y direction
At this case, the acceleration shock is applied purely in Y direction of the base
reference frame.
Figure 4.23 Acceleration of moving platform in X direction(C=200Ns/m)
62
Figure 4.24 Acceleration of moving platform in Y direction(C=200Ns/m)
Figure 4.25 Acceleration of moving platform in Z direction(C=200Ns/m)
63
Figure 4.26 Acceleration of moving platform in X direction(C=600Ns/m)
Figure 4.27 Acceleration of moving platform in Y direction(C=600Ns/m)
64
Figure 4.28 Acceleration of moving platform in Z direction(C=600Ns/m)
The acceleration of the moving platform in the case of purely Y direction shock on
the base platform, can be interpreted by analyzing the graphs above. Figure 4.23 and
Figure 4.26 indicates that there is no acceleration in X direction while the shock
absorber is subject to only shock in Y direction of the base frame (Small errors
caused by numerical calculations are counted to be zero). Since the positions of
springs on the base and moving platform w.r.t base frame, this result is expected. The
acceleration of moving platform in Y and Z direction in this case on the other hand is
dependent on the spring constant and the damping coefficient of the system. The
stiffer springs and the more damping, the more acceleration is transmitted to moving
platform. These deduction can be observed by looking at Figure 4.24, Figure 4.25,
Figure 4.27 and Figure 4.28. Moreover, because the special geometry of the shock
absorber, a coupled motion is observed in Y and Z direction in case 2. In other
words, while base shock input is in pure Y direction, the moving platform has
acceleration component in Y and Z directions.
65
Figure 4.29 Deflection of moving platform in X direction(C=200Ns/m)
Figure 4.30 Deflection of moving platform in Y direction(C=200Ns/m)
66
Figure 4.31 Deflection of moving platform in Z direction(C=600Ns/m)
Figure 4.32 Deflection of moving platform in X direction(C=600Ns/m)
67
Figure 4.33 Deflection of moving platform in Y direction(C=600Ns/m)
Figure 4.34 Deflection of moving platform in Z direction(C=600Ns/m)
68
Since there is no acceleration or initial velocity of moving platform in X direction in
case 2, the deflection of it in this axis is zero as it seen in Figure 4.29 and Figure
4.32. However, moving platform has deflection in Y and Z direction. It is observed
that the value of the deflection has a contradiction with transmitted acceleration.
Figure 4.30, Figure 4.31, Figure 4.33 and Figure 4.34 show that as spring and
damping coefficients gets higher, the deflection of the moving platform gets smaller
like case 1. As indicated acceleration section of case 2, special geometry of the shock
absorber cause coupled deflection in Y and Z direction in shock case 2.
4.4.3 CASE 3: Shock in X direction
At this case, the acceleration shock is applied purely in X direction of the base
reference frame.
Figure 4.35 Acceleration of moving platform in X direction(C=200Ns/m)
69
Figure 4.36 Acceleration of moving platform in Y direction(C=200Ns/m)
Figure 4.37 Acceleration of moving platform in Z direction(C=200Ns/m)
70
Figure 4.38 Acceleration of moving platform in X direction(C=600Ns/m)
Figure 4.39 Acceleration of moving platform in Y direction(C=600Ns/m)
71
Figure 4.40 Acceleration of moving platform in Z direction(C=600Ns/m)
The acceleration of the moving platform in the case of purely X direction shock, can
be interpreted by analyzing the graphs above. It is obvious that the moving platform
has acceleration component in every direction in case 3. This situation occurs
because of the special geometry of the system and shock direction w.r.t base frame.
The acceleration of moving platform in every direction show same characteristic as
in other cases. The stiffer springs and the more damping, the more acceleration is
transmitted to moving platform. These deduction can be observed by looking at
Figure 4.35, Figure 4.36, Figure 4.37, Figure 4.38, Figure 4.39, and Figure 4.40.
72
Figure 4.41 Deflection of moving platform in X direction(C=200Ns/m)
Figure 4.42 Deflection of moving platform in Y direction(C=200Ns/m)
73
Figure 4.43 Deflection of moving platform in Z direction(C=200Ns/m)
Figure 4.44 Deflection of moving platform in X direction(C=600Ns/m)
74
Figure 4.45 Deflection of moving platform in Y direction(C=600Ns/m)
Figure 4.46 Deflection of moving platform in Z direction(C=600Ns/m)
75
Since moving platform has acceleration component in all direction of inertial frame,
it has a deflection value in these directions as well. The deflection values in case 3
can be analyzed by using Figure 4.41, Figure 4.42, Figure 4.43, Figure 4.44, Figure
4.45 and Figure 4.46. It is also seen that like other cases, although stiffer springs and
more damping coefficients reduces the deflection of moving platform, this situation
increases acceleration transmitted moving platform with respect to base platform. In
the design phase of the shock absorber, this situation should be considered carefully
in order to get desired shock absorber.
4.5 ANALITIC MODEL COMPARISON WITH MSC ADAMS®
In engineering analysis, for the systems, some mathematical models are generated or
proposed in order to simulate desired conditions. After deriving mathematical model
of the system, this model should be justified by some independent methods. Making
experiments or comparison with some other studies are examples of this justification
processes. In this thesis, the mathematical model derived for the shock absorber is
compared with the result of MSC ADAMS software for the same simulation cases in
order to check result consistency. Since MSC ADAMS is widely used in commercial
product design and dynamic analysis phases worldwide, the comparison can be
counted as an acceptable consistency test method for the analytical model of the
shock absorber.
4.5.1 MSC ADAMS®
MSC ADAMS (Advanced Dynamic Analysis of Mechanical Systems) is a numerical
solution based computer software that is capable of doing dynamic analysis of
multibody mechanical systems in 3 dimensions by using rigid and flexible body
assumption [10]. The software is widely used in aircraft and automotive industry.
ADAMS software uses Newton-Euler methods for mathematical model derivation to
analyze dynamic systems. During simulation preparation, projects are imported into
76
preprocessor interface of the software to define input parameters like joints, mass,
motion, force etc. After simulation is done, user obtain and observe desired outputs
like acceleration, required torque etc. in the postprocessor interface. All generated
data can be exported from this interface to outside for other purposes.
4.6 Simulation Comparison between Analytic Model and MSC ADAMS®
In this thesis, justification of the mathematical model related to the shock absorber
platform which is formulated by using Lagrange Equations is conducted with
ADAMS software results for similar simulation scenarios. For that study, the
simulation cases are defined in Table 5.
Table 5: MSC ADAMS Simulation Cases
Case 1 Case 2 Case 3
K=30000 N/m K=30000 N/m K=30000 N/m
C=600 Ns/m C=600 Ns/m C=600 Ns/m
Input 40g 11ms shock
in Z direction
Input 40g 11ms shock
in Y direction
Input 40g 11ms shock
in X direction
Observed
OutputDeflection of
moving platform in Z
direction
Observed
OutputDeflection of
moving platform in Y
direction
Observed
OutputDeflection of
moving platform in X
direction
Simulation model preparation of the shock absorber is done in preprocessor
environment of ADAMS, by using 3D parasolid model prepared with CAD software.
After importing 3D model, the necessary input parameters are defined such as joint
definitions, springs, dampers and masses, shock profile and its action location etc. by
using necessary tools of ADAMS. Picture of 3D model is shown in Figure 4.47.
77
During simulation model preparation, the critical section is model verification before
simulation run. On simulation run window seen in Figure 4.48 , ADAMS analyze the
system according to the Grübler’s Equation in order to determine redundant
constraints. The redundant constraints can cause some miscalculations especially
when joints reaction forces or moments are important. To avoid these redundant
constraints, “primitive joints” toolbox shown in Figure 4.49 can be used. These joints
mathematically help ADAMS solver to obtain accurate results without changing
system kinematics. ADAMS users should aim to get zero redundant constraint for
their systems while doing model verification based on Grübler’s Equation for a more
accurate simulation. In this thesis, in ADAMS simulation phase, nonzero redundant
constraint situation is faced after model verification is performed. ADAMS model
verification tool states that there are 3 redundant constraints caused by revolute joints
on 3 parallelograms in Delta Robot chains. In order to eliminate these redundant
constraints, primitive joint toolbox is used. By defining “inline primitive joint”
instead of one revolute joint on each parallelogram, the redundant constraint number
is reduced to zero. “Inline primitive joint” has 2 dof and revolute joint has 1 dof. In
this condition, 3 “inline primitive joints” add 3 extra dof to system without changing
the kinematic of the shock absorber. These 3 extra dof reduce the redundant
constrain number to zero. For detailed information about primitive joints and
redundant constraints subject, ADAMS Help [12] document can be used. After
model preparation is done, simulation tool is used and the desired outputs are
analyzed.
78
Figure 4.47 3D Model of the Shock Absorber in ADAMS Environment
Figure 4.48 Model verification toolbox of ADAMS
Figure 4.49 Primitive joint toolbox of ADAMS
The first procedure of comparison study is to export output results of ADAMS as
numerical data into MATLAB environment in order to manipulate them with
mathematical model outputs for the same simulations cases. The second step is to
79
plot the ADAMS and mathematical result on the same graph to analyze result for
better comparison.
4.6.1 Case 1
In this case, the main shock profile is applied in the Z direction of the base reference
frame. The deflection of the moving platform in Z direction is obtained from
mathematical model and ADAMS simulations separately and they are compared on
the same graph.
Figure 4.50 ADAMS and mathematical model result of Case 1
While the red color represents ADAMS result, the black color represents
mathematical model result. As it seen in Figure 4.50 , the results are consistent with
each other. This situation justifies the accuracy of the mathematical model for case 1.
80
4.6.2 Case 2
In this case, the main shock profile is applied in the Y direction of the base reference
frame. The deflection of the moving platform in Y direction is obtained from
mathematical model and ADAMS simulations separately and they are compared on
the same graph.
Figure 4.51 ADAMS and mathematical model result of Case 2
While the red color represents ADAMS result, the black color represents
mathematical model result. As it seen in Figure 4.51, the results are consistent with
each other. This situation justifies the accuracy of the mathematical model for case 2.
4.6.3 Case 3
In this case, the main shock profile is applied in the X direction of the base reference
frame. The deflection of the moving platform in X direction is obtained from
81
mathematical model and ADAMS simulations separately and they are compared on
the same graph.
Figure 4.52 ADAMS and mathematical model result of Case 3
All cases above are studied for validation of the mathematical model by checking the
accuracy of the results with ADAMS simulation results. The number of cases can be
easily increased with different parameter values. The comparison results are quite
consistent with each other and that situation validates the mathematical model of the
shock absorber platform.
82
83
CHAPTER 5
CONCLUSION AND FUTURE WORK
In this thesis, a 3-DOF translational shock absorbing parallel manipulator for military
purposes is designed and proposed. The shock absorber architecture is constructed
according to the most suitable mechanism among three candidates found in literature.
After evaluation of three parallel manipulator candidates, Delta Robot mechanism is
counted to be most suitable architecture for the shock absorber.
Further step, after mechanism selection, is to analyze the shock absorber in terms of
kinematic and dynamic. Chapter 3 deals with the inverse and forward kinematic of
Delta Robot mechanism, Chapter 4 deals with dynamic analysis the shock absorber.
All dynamic analysis is done by using equations of motion of the shock absorber
derived from Lagrange equation of motion approach. For this derivation, kinetic and
potential energy and dissipation function of the shock absorber are calculated
symbolically. While determining the velocities as a requirement of kinetic energy
calculation, time derivate of the position equations are used. The relation between
kinematic equations and velocity of each link for the shock absorber mechanism is
basically shown in this chapter. However, since mass of each link is very small
compared to useful load (M) on moving platform, their kinetic energy is assumed to
be zero. For that reason, the velocity calculation with kinematic equations is shown
only conceptually without doing deeper derivations. For a more detailed dynamic
analysis, these kinetic energy terms shouldn’t be ignored. After doing necessary
mathematical calculations according to Lagrange equation procedures, equations of
84
motion of the shock absorber are obtained. They are 3 unknowns, 3 set of nonlinear
homogenous differential equations. These equations are solved numerically and
simultaneously with the help of Simulink tool of MATLAB® software. In order to
observe motional characteristic of the shock absorber, 3 simulation cases are
specified. In fact, the number of simulation cases can be easily increased by
manipulating parameter values. Acceleration of moving platform with respect to
base platform and the deflection change between base and moving platform are
observed for each simulation case as an observable output.
It is interpreted that, for selecting shock absorber parameters, making tradeoff
between transmitting acceleration to the moving platform and the deflection of it is
essential. It is deduced that the higher spring coefficient, the more acceleration is
transmitted to moving platform. On the other hand, it is observed that the deflection
value gets smaller with stiffer springs or vice versa. Moreover, selecting proper
damping coefficient is also very important for settling time of the shock absorber. It
is seen that the system that has higher damping coefficient settles faster compared to
which has lower. All these comparisons can be interpreted by analyzing graphics in
Chapter 4.
At the final section of the Chapter 4, the mathematical model results for specified 3
case are compared with MSC ADAMS dynamic simulation software in order to
check the accuracy of the mathematical model. For this study, a computer model in
ADAMS environment is prepared according to assumptions made in the thesis. The
results are plotted on the same graph by using MATLAB® for an easy comparison.
These graphs show that the mathematical and ADAMS results are very consistent
with each other.
While doing kinematic and dynamic analysis, all links are assumed to be rigid for
relatively simpler calculations. However, in real world, there is no rigid material.
Therefore, for a realistic shock absorber design, elasticity of materials should be
85
considered. Some design solution can be also offered to realistic design phases. For
example, the links can be designed very thick to reduce elastic deformations if the
weight is not critical. The distance between parallelogram links can be increased in
order to reduce moment stresses while environmental conditions are forcing the
shock absorber mechanism to do rotational motions. The manufacturing tolerances
on revolute joints should be very tight for a very small parallelism errors of moving
platform according to base platform.
As a future work, the dynamic mathematical model of the shock absorber platform
can be treated as an optimization problem for the determining of spring and damper
coefficients according to the desired outputs.
In this study, a special shock absorber for naval radar systems is introduced and the
mathematical model of it is derived by using Lagrange equation of motion to perform
a dynamic analysis. The mathematical model can be parametrically used in order to
design a desired 3-dof translational shock absorber by observing output characteristic
of it in terms of transmitting acceleration and deflection values based on the shock
input.
86
87
REFERENCES
[1] Department of Defense, USA. (2008). DEPARTMENT OF DEFENSE TEST
METHOD STANDARD. MIL-STD-810F.
[2] Gosselin, C. M., & Masouleh, M. T. (2007). Parallel Mechanisms of the
Multipteron Family: Kinematic Architectures and Benchmarking.
doi:10.1109/ROBOT.2007.363045.
[3] Xu, Y. C., Li, B., & Zhao, X. H. (2013). Influence upon Kinematics
Performance of a Family of 3-PRS Parallel Mechanisms Affected by Kinematic
Chain Layout. AMM, 321-324, 37-41. doi:10.4028/www.scientific.net/amm.321-
324.37
[4] R. Clavel, Une nouvelle structure de manipulation paralle’le pour la robotique
le’ge’re, R.A.I.R.O.APII 23(6) (1986).
[5] Indonesia Military News & Discussion Thread | Page 54. Retrieved from
http://defence.pk/threads/indonesia-military-news-discussion-
thread.229571/page-54. (Accessed Octeber 2015).
[6] OpenStax CNX.Retrieved from http://cnx.org/contents/BDDH_rPS@12/Protein-
Inverse-Kinematics.(Accessed December 2015).
[7] J.E.Alexander (2009),’’Shock Response Spectrum-A Primer’’,Sound and
Vibration.
88
[8] Laribi, M., Romdhane, L., & Zeghloul, S. (2006). Mechanism and Machine
Theory. Analysis and dimensional synthesis of the DELTA robot for a prescribed
workspace.
[9] Williams, R. L. (2015). The Delta Parallel Robot: Kinematics Solutions.
[10] Brinker, J. (2015). The 14th IFToMM World Congress, Taipei, Taiwan, October
25-30, 2015. A Comparative Study of Inverse Dynamics based on Clavel’s Delta
robot.
[11] SIMULINK. Retrieved from http://www.mathworks.com/products/simulink/
.(Accessed September 2015)
[12] ADAMS View Help. Retrieved from
http://www.mscsoftware.com/product/ADAMS .(Accessed December 2015)
89
APPENDIX
Useful Sources
Ivan J.Baiges-Valentin (1996). Dynamic Modeling of Parallel Manipulators.
Garcia, J.M. (2010). Inverse-Forward Kinematics of a Delta Robot.
Dragos A., Marius P.,Lucian M.(2012). Determining the Workspace Shape of
A Robot with Delta 3D of Parallel Structure.
Khalil W. (2010). Dynamic Modeling of Robots Using Recursive Newton-
Euler Techniques.
Ocak O.,Oysu C.,Bingül Z.(2010).Otomatik Kontrol Ulusal Toplantısı. Delta
Robot Tasarımı ve Simülasyonu.
Kunt E.,Khalil I.,Naskali A.,Fidan K.,Sabanovic A.(2010). Otomatik Kontrol
Ulusal Toplantısı.Yüksek Hassasiyetli Montaj İşlemleri İçin Minyatür Delta
Robot Tasarımı, En İyilemesi ve Denetimi
Poppeova V.,Rejda R.,Uricek J.,Bulej V.(2012). Journal of Trends in the
Development Machinery and Associated Technology. Vol. 16, p.p195-198.
The Design and Simulation of Training Delta Robot.