3.2 the CORDIC Algorithm

3.2 the CORDIC Algorithm

UC San Diego UC San Diego Electronic Theses and Dissertations Title Improved VLSI architecture for attitude determination computations Permalink https://escholarship.org/uc/item/5jf926fv Author Arrigo, Jeanette Fay Freauf Publication Date 2006 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California 1 UNIVERSITY OF CALIFORNIA, SAN DIEGO Improved VLSI Architecture for Attitude Determination Computations A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical and Computer Engineering (Electronic Circuits and Systems) by Jeanette Fay Freauf Arrigo Committee in charge: Professor Paul M. Chau, Chair Professor C.K. Cheng Professor Sujit Dey Professor Lawrence Larson Professor Alan Schneider 2006 2 Copyright Jeanette Fay Freauf Arrigo, 2006 All rights reserved. iv DEDICATION This thesis is dedicated to my husband Dale Arrigo for his encouragement, support and model of perseverance, and to my father Eugene Freauf for his patience during my pursuit. In memory of my mother Fay Freauf and grandmother Fay Linton Thoreson, incredible mentors and great advocates of the quest for knowledge. iv v TABLE OF CONTENTS Signature Page...............................................................................................................iii Dedication … ................................................................................................................iv Table of Contents ...........................................................................................................v List of Figures.................................................................................................................x List of Tables...............................................................................................................xiv Acknowledgements ......................................................................................................xv Vita and Publications..................................................................................................xvii Abstract ……............................................................................................................xix 1 INTRODUCTION.......................................................................................1 1.1 Motivation ..........................................................................................................1 1.2 Overview of Chapter 2 .......................................................................................2 1.3 Overview of Chapter 3 .......................................................................................3 1.4 Overview of Chapter 4 .......................................................................................4 1.5 Overview of Chapter 5 .......................................................................................5 1.6 Overview of Chapter 6 .......................................................................................6 1.7 Overview of Chapter 7 .......................................................................................7 2 BACKGROUND..........................................................................................8 2.1 Introduction ........................................................................................................8 2.2 Attitude Parameterization methods ....................................................................9 2.2.1 Direction Cosine Representation......................................................................10 2.2.2 Euler Angle Representation............................................................................122 2.2.3 Euler Axis/Angle Notation...............................................................................13 2.2.4 Quaternion Mathematics ..................................................................................14 2.3 The Kinematic Equation for Rotation ..............................................................18 2.3.1 Methods of Integration .....................................................................................19 2.3.1.1 Direct Integration..............................................................................................19 2.3.1.2 Local Linearization Method .............................................................................20 v vi 2.3.1.3 Exponential Matrix Function............................................................................21 2.3.1.4 Closed Form Approximation (Constant Angular Rate)....................................22 2.3.2. Closed Form Approximation Errors.................................................................25 2.3.2.1 Coning Error....................................................................................................25 2.3.2.2 Orthogonality Error .........................................................................................27 2.4 Conclusion........................................................................................................28 References ........................................................................................................29 3 IMPLEMENTATION ..............................................................................30 3.1 Introduction ......................................................................................................30 3.2 The CORDIC Algorithm ..................................................................................32 3.3 Parallel Design of the Attitude Update System................................................37 3.3.1 Limitations on the Attitude Update Computation ............................................44 3.3.2 ASIC Synthesis.................................................................................................44 3.3.3 Accuracy Analysis............................................................................................45 3.4 Serial Design of the Attitude Update System...................................................47 3.4.1 CORDIC Operations........................................................................................47 3.4.2 Computation of the Magnitude of the Angular Displacement .........................48 3.4.3 Calculation of the Cross Product......................................................................49 3.4.3.1 Scaling ..............................................................................................................50 3.4.4 System Operation and Scheduling....................................................................51 3.4.5 ASIC Synthesis.................................................................................................54 3.4.6 Attitude Update Computation Performance Analysis ......................................55 3.4.6.1 Boundary Analysis ...........................................................................................55 3.4.6.2 Accuracy Analysis...........................................................................................56 3.5 Serial System Design with Parallel Scale Factor Compensation .....................57 3.5.1 Scale Factor Compensation ..............................................................................58 3.5.2 Vectoring Mode Normalization........................................................................59 3.5.3 System Architecture .........................................................................................60 3.5.4 System Operation and Scheduling....................................................................64 3.6 Conclusion........................................................................................................67 vi vii References ........................................................................................................68 4 PARAMETRIC ANALYSES FOR CORDIC BASED ALGORITHM AND COMPARISON TO OTHER APPROXIMATION METHODS ....................................................................70 4.1 Introduction ......................................................................................................70 4.2 Parametric Analysis of the CORDIC Based Algorithm...................................72 4.2.1 Accuracy as a Function of CORDIC Processor Bit Width and Iterations........73 4.2.2 Orthogonality as a Function of Bit Width and the Number of Iterations.........78 4.2.3 Area, Update Rate and Power Consumption as a Function of CORDIC Processor Bit Width and Iterations...............................................................................81 4.2.4 Sensor Errors and Output Accuracy.................................................................85 4.3 Taylor Series Approximation ...........................................................................87 4.3.1 Comparisons of Accuracy ................................................................................89 4.3.2 Comparisons of Orthogonality .........................................................................92 4.4 Wie-Barba Method ...........................................................................................93 4.4.1 Comparisons of Accuracy ................................................................................94 4.4.2 Comparisons of Orthogonality .........................................................................94 4.5 Other Integration Methods ...............................................................................95 4.5.1 Obrechkoff Integration.....................................................................................95

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