/ / Uncertainty Optimization Applied to the Monte Carlo Analysis of Planetary Entry Trajectories Final Report NGT-1-52163 NASA Langley Research Center For a NASA Graduate Student Fellowship Program (GSRP) Training Grant originally entitled A Design-Oriented Model for Conceptual Rocket Engine Simulation Submitted by: Dr. John R. Olds (PI) Dr. David Way (thesis author) Georgia Institute of Technology Atlanta, GA 30332-0150 July 31, 2001 TABLE OF CONTENTS Table of Contents ....................................................... vi List of Tables •eoo• .......................eo•eeoe• e• •e••••••••J•.J•e•• •••••el .•e••e•e•e ee•••e•e• oee•*•o• •oo*•e•• e••*••o•ee•e*e•oe• ix List of Figures ................................................................................................................... xi ,,. List of Symbols ................................................................................................ xm Nomenclature .................................................................................................................. xv Summary ........................................................................................................................ xvii CHAPTER 1 1 Introduction ....................................................................................................................... 1 1.1 History and Exploration of Mars ........................................................................ I 1.2 The 4 t_ Rock from the Sun ................................................................................... 5 1.3 Future Exploration of Mars ................................................................................ 7 CHAPTER 2 12 Research Summary ......................................................................................................... 12 2.1 Motivation ......................................................................................................... 12 2.2 Goals ................................................................................................................. 16 2.3 Objectives .......................................................................................................... 18 2.4 Approach ........................................................................................................... 19 CHAPTER 3 23 Monte Carlo in Trajectory Analysis .............................................................................. 23 3.1 METEOR (1995) ............................................................................................... 24 3.2 Pathfinder (1997) .............................................................................................. 25 3.3 Stardust (1999) .................................................................................................. 26 3.4 Mars Surveyor (2001) ....................................................................................... 27 3.5 Genesis (2001) .................................................................................................. 30 3.6 MUSES-C (2002) .............................................................................................. 31 3.7 Mars Ascent Vehicle (2005) .............................................................................. 31 CHAPTER 4 33 Optimization in Design ................................................................................................... 33 4.1 Multi-objective vs. Single-objective .................................................................. 33 4.2 Analysis vs. Metamodel ..................................................................................... 34 4.3 Many Variables vs. Screening ........................................................................... 36 4.4 Constrained vs. Unconstrained Optimization ................................................... 37 4.5 Probabilistic vs. Deterministic .......................................................................... 38 vi CHAPTER 5 41 Random Number Generators ......................................................................................... 41 5.1 Random Numbers .............................................................................................. 41 5.2 Random Number Generators ............................................................................ 42 5.3 Linear, Congruential Generators ...................................................................... 43 5.4 The "Ran1" Random Number Generator ......................................................... 45 5.5 Tests for Random Number Generators .............................................................. 46 CHAPTER 6 60 Landing Footprints ......................................................................................................... 60 6.1 3-Sigma Radius ................................................................................................. 61 6.2 Rayleigh ............................................................................................................. 62 6.3 Weibull .............................................................................................................. 64 6.4 3-Sigma Down-range and Cross-range ............................................................ 67 6.5 Bivariate Normal ............................................................................................... 68 6.6 Comparisons ...................................................................................................... 80 CHAPTER 7 87 Trajectory Simulation ..................................................................................................... 87 7.1 POST ................................................................................................................. 87 7.2 Monte Carlo POST ............................................................................................ 88 7.3 Automation ........................................................................................................ 95 CHAPTER 8 97 Mathematical Problem Formulation ............................................................................. 97 8.1 ConstrahTts ........................................................................................................ 99 8.2 Objective .......................................................................................................... 105 8.3 Optimization Problem ..................................................................................... 112 8.4 Method of Lagrange Multipliers ..................................................................... 112 8.5 Newton-Raphson Iteration .............................................................................. 114 CHAPTER 9 116 Proof-of-Concept (2D Problem) ................................................................................... 116 9.1 Problem Statement .......................................................................................... 116 9.2 Results ............................................................................................................. 119 CHAPTER 10 132 "Real World" Example ................................................................................................ 132 10.1 MSP 2001 Lander. ........................................................................................... 132 10.2 Monte Carlo Simulation .................................................................................. 134 10.3 Baseline Monte Carlo Results ......................................................................... 139 10.4 Optimized Solution .......................................................................................... 143 CHAPTER 11 159 Comments and Conclusions ......................................................................................... 159 11.1 Methodology Summary .................................................................................... 159 11.2 Objectives ........................................................................................................ 163 vii 11.3 Comments ........................................................................................................ 164 11.4 Research Accomplishments ............................................................................. 169 APPENDIX A 172 Mathematics Review ..................................................................................................... 172 A. 1. Method of Lagrange Multipliers ..................................................................... 172 A.2. Newton-Raphson Method ................................................................................ 175 A.3. PLU Matrix Decomposition ............................................................................ 178 A.4. Cholesky Matrix Decomposition ..................................................................... 180 APPENDIX B 181 Source Code ................................................................................................................... 181 B. 1. "mcp " Perl Module ......................................................................................... 181 B.2. "mcp " Executive ............................................................................................. 186 B.3. "RandDist" Perl Module ................................................................................ 197 References ...................................................................................................................... 201 Vita .................................................................................................................................