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Biokinematic Analysis of Human Body

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Biokinematic Analysis of Human Body BIOKINEMATIC ANALYSIS OF HUMAN BODY A Thesis Submitted to the Graduate School of Engineering and Sciences of İzmir Institute of Technology in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in Mechanical Engineering by Erkin GEZGİN March 2011 İZMİR We approve the thesis of Erkin GEZGİN ___________________________ Prof. Dr.Sc. Rasim ALİZADE Supervisor ___________________________ Prof. Dr.Sc. Rafail ALİZADE Committee Member ___________________________ Assist. Prof. Dr. H. Seçil ARTEM Committee Member ___________________________ Assist. Prof. Dr. M. İ. Can DEDE Committee Member ___________________________ Assist. Prof. Dr. Fatih Cemal CAN Committee Member 25 March 2011 ___________________________ ___________________________ Prof. Dr. Metin TANOĞLU Prof. Dr. Durmuş Ali DEMİR Head of the Department of Dean of the Graduate School of Mechanical Engineering Engineering and Sciences ii Anyone who has never made a mistake has never tried anything new… Albert Einstein iii ACKNOWLEDMENTS I would like to express my deepest gratitude to my supervisor Prof. Dr. Sc. Rasim ALIZADE, my family and my dearest friends for their instructive comments, valued support throughout the all steps of this study and patience to my questions. This thesis would not have been possible without them. Also, it is an honor for me to offer my regards to all of those, who supported me in any respect during the completion of the thesis. iv ABSTRACT BIOKINEMATIC ANALYSIS OF HUMAN BODY This thesis concentrates on the development of rigid body geometries by using method of intersections, where simple geometric shapes representing revolute (R) and prismatic (P) joint motions are intersected by means of desired space or subspace requirements to create specific rigid body geometries in predefined octahedral fixed frame. Using the methodical approach, space and subspace motions are clearly visualized by the help of resulting geometrical entities that have physical constraints with respect to the fixed working volume. Also, this work focuses on one of the main areas of the fundamental mechanism and machine science, which is the structural synthesis of robot manipulators by inserting recurrent screws into the theory. After the transformation unit screw equations are presented, physical representations and kinematic representations of kinematic pairs with recurrent screws are given and the new universal mobility formulations for mechanisms and manipulators are introduced. Moreover the study deals with the synthesis of mechanisms by using quaternion and dual quaternion algebra to derive the objective function. Three different methods as interpolation approximation, least squares approximation and Chebyshev approximation is introduced in the function generation synthesis procedures of spherical four bar mechanism in six precision points. Separate examples are given for each section and the results are tabulated. Comparisons between the methods are also given. As an application part of the thesis, the most important elements of the human body and skeletal system is investigated by means of their kinematic structures and degrees of freedom. At the end of each section, an example is given as a mechanism or manipulator that can represent the behavior of the related element in the human body. v ÖZET İNSAN VÜCUDUNUN BİYOKİNEMATİK ANALİZİ Tezin ilk aşamasında, döner (R) ve doğrusal (P) mafsal hareketlerini temsil eden basit geometrik şekillerin, önceden belirlenmiş sabit hacim içerisinde özel katı cisim geometrilerinin oluşturulması için, istenen uzay ve altuzay gereksinimleri göz önüne alınarak kesiştirildiği kesişimler metoduna yoğunlaşılmıştır. Sistematik yaklaşım kullanılarak, oluşturulan sabit çalışma hacmine göre fiziksel sınırlamalara sahip geometrik cisimlerin yardımıyla uzay ve altuzay hareketleri açık olarak canlandırılmıştır. Ayrıca, bu çalışma, makinalar ve mekanizmalar bilimdalının en önemli dallarından biri olan robot manipülatörlerin yapısal sentezi konusunda çalışmalar içermektedir. Çalışma içerisinde kinematik mafsallar ve vida teorisi üzerine odaklanılmıştır. Her bir kinematik mafsalın ardışık vidalar ile hem fiziksel hem de kinematik gösterimleri verildikten sonra robot manipülatörler için yeni genel serbestlik derecesi formülü belirtilmiştir. Tezin ileriki aşamalarında “quaternion” ve “dual quaternion” cebiri kullanılarak mekanizmaların sentezlenmesi üzerine çalışılmıştır. Interpolasyon yaklaşımı, en küçük kareler yaklaşımı ve Chebyshev yaklaşımı olmak üzere üç farklı teknik kullanılarak küresel dört çubuk mekanizmasının altı dizayn noktasında fonksyon üretme sentezi yapılmış, her bölüm için farklı örnekler verilmiş ve sonuçlar tablolarda gösterilmiştir. Çalışma içerisinde kullanılan metodların karşılaştırılmaları ayrıca yapılmıştır. Tezin son bölümlerinde insan vücudu ve iskelet sisteminde bulunan önemli elemanlar kinematik yapıları ve serbestlik derecelerine göre incelenmiş, her bölümün sonunda incelenen elemanın davranışını gösterebilecek mekanizma veya manipulatör örnekleri verilmiştir. vi TABLE OF CONTENTS LIST OF FIGURES .......................................................................................................... x LIST OF TABLES.........................................................................................................xiii CHAPTER 1. INTRODUCTION..................................................................................... 1 1.1. Research Statement................................................................................. 1 1.2. Literature Survey .................................................................................... 3 CHAPTER 2. QUATERNION AND DUAL QUATERNION ALGEBRA .................. 14 2.1. Quaternion Preliminaries ...................................................................... 14 2.2. Quaternion Addition and Equality........................................................ 15 2.3. Quaternion Multiplication..................................................................... 16 2.4. Conjugate of the Quaternion................................................................. 19 2.5. Norm of the Quaternion........................................................................ 19 2.6. Inverse of the Quaternion...................................................................... 20 2.7. Dual Quaternion Preliminaries ............................................................. 21 2.8. Dual Quaternion Multiplication............................................................ 22 2.9. Conjugate of the Dual Quaternion ........................................................ 22 2.10. Norm of the Dual Quaternion ............................................................. 23 2.11. Inverse of the Dual Quaternion........................................................... 23 CHAPTER 3. SCREW THEORY .................................................................................. 24 3.1. Introduction to Screws .......................................................................... 24 3.2. Motor Screw.......................................................................................... 25 3.3. Particular Cases..................................................................................... 26 3.4. Intersections of Two Screws................................................................. 27 3.5. Transformation Unit Screw Equations.................................................. 29 3.6. Transformation Unit Screw Equations by Using Denavit Hartenberg Notations............................................................................................... 32 CHAPTER 4. RIGID BODY MOTIONS IN SPACE AND SUBSPACES BY USING METHOD OF INTERSECTIONS .............................................................. 33 4.1. Rigid Body Motions.............................................................................. 33 vii 4.2. Method of Intersections ........................................................................ 35 CHAPTER 5. STRUCTURAL SYNTHESIS OF ROBOT MANIPULATORS BY USING SCREW THEORY ......................................................................... 38 5.1. Kinematic Pairs and Mathematical Models.......................................... 38 5.2. Mobility Equations ............................................................................... 45 CHAPTER 6. KINEMATIC SYNTHESIS OF MECHANISMS BY USING QUATERNION AND DUAL QUATERNION ALGEBRA ...................... 49 6.1. Objective Function (Spherical Four Bar).............................................. 49 6.2. Interpolation Approximation ................................................................ 52 6.2.1. Numerical Example ........................................................................ 54 6.3. Least Square Approximation ................................................................ 55 6.3.1. Numerical Example ........................................................................ 57 6.4. Chebyshev Approximation ................................................................... 58 6.4.1. Numerical Example ........................................................................ 60 6.5. Discussion (Spherical Four Bar)........................................................... 62 6.6. Dual Quaternions in Rigid Body Rotations and Translations............... 64 6.7. Objective Functions .............................................................................. 66 6.7.1. Numerical Example ........................................................................ 72 CHAPTER 7. BIOKINEMATIC
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