DESIGN AND ANALYSIS OF A 3-DOF TRANSLATIONAL SHOCK...

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


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


Prof. Dr. Sıtkı Kemal İder

Mechanical Engineering Dept., CU

Assist. Prof. Dr. Erhan İlhan Konukseven


Assist. Prof. Dr. Yiğit Yazıcıoğlu


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


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 4.16 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





















xvii

Figure 4.40 Acceleration of moving platform in Z direction(C=600Ns/m) .............. 71







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



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

http://tureng.com/tr/turkce-ingilizce/equilateral%20triangle

http://tureng.com/tr/turkce-ingilizce/equilateral%20triangle

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.

https://en.wikipedia.org/wiki/Kinematic

https://en.wikipedia.org/wiki/Robot

https://en.wikipedia.org/wiki/Robot_end_effector

https://en.wikipedia.org/wiki/Kinematics

https://en.wikipedia.org/wiki/Robot_end_effector

https://en.wikipedia.org/wiki/Motion_planning

https://en.wikipedia.org/wiki/Actuator

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


[ 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


[ 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

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.

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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



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.

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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)


60



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.


62



63



64


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



66



67



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.


69



70



71


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



73



74



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

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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


in Y direction


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

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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.

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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


the same graph.


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


the same graph.


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.

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.

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.

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