Voltar
Artigo Revisado por pares
BIBTEX RIS

Existence of Probability Density Function for Norm of Finite-Dimensional Random Vector

2022; American Institute of Aeronautics and Astronautics; Volume: 45; Issue: 6 Linguagem: Inglês

10.2514/1.g006551

ISSN

1533-3884

Autores

Ulises E. Núñez Garzón, E. Glenn Lightsey,

Tópico(s)

Particle Dynamics in Fluid Flows

Resumo

No AccessEngineering NotesExistence of Probability Density Function for Norm of Finite-Dimensional Random VectorUlises E. Núñez Garzón and E. Glenn LightseyUlises E. Núñez Garzón https://orcid.org/0000-0003-2179-0243Georgia Institute of Technology, Atlanta, Georgia 30332*Graduate Research Assistant, Guggenheim School of Aerospace Engineering; . Student Member AIAA (Corresponding Author).Search for more papers by this author and E. Glenn LightseyGeorgia Institute of Technology, Atlanta, Georgia 30332†Professor, Guggenheim School of Aerospace Engineering; . Fellow AIAA.Search for more papers by this authorPublished Online:21 Mar 2022https://doi.org/10.2514/1.G006551SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Klinkrad H., Space Debris: Models and Risk Analysis, Springer-Praxis Books in Astronautical Engineering, Springer–Verlag, Berlin, 2006, pp. 1–58, Chaps. 1, 2. https://doi.org/10.1007/3-540-37674-7 Google Scholar[2] Spencer D. B. and Madler R. A., “Orbital Debris—A Space Hazard,” Space Mission Engineering: The New SMAD, edited by Wertz J. R., Everett D. F. and Puschell J. J., Space Technology Library, Microcosm Press, Hawthorne, CA, 2011, pp. 139–147, Chap. 7.5. Google Scholar[3] Kessler D. J., “Collisional Cascading: The Limits of Population Growth in Low Earth Orbit,” Advances in Space Research, Vol. 11, No. 12, 1991, pp. 63–66. https://doi.org/10.1016/0273-1177(91)90543-S CrossrefGoogle Scholar[4] Pelton J. N., Space Debris and Other Threats from Outer Space, SpringerBriefs in Space Development, Springer–Verlag, New York, 2013, pp. 1–23, Chaps. 1, 2. https://doi.org/10.1007/978-1-4614-6714-4 Google Scholar[5] Chobotov V. A., Karrenberg H. K., Chao C., Miyamoto J. Y., Lan T. J., Kechichian J. A., Johnson C. G., Alekshun J. and Campbell E. T., “Space Debris,” Orbital Mechanics, edited by Chobotov V. A., AIAA Education Series, 3rd ed., AIAA, Reston, VA, 2011, pp. 301–334, Chap. 13. https://doi.org/10.2514/4.862250 Google Scholar[6] Letizia F., Colombo C. and Lewis H. G., “Collision Probability Due to Space Debris Clouds Through a Continuum Approach,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 10, 2016, pp. 2240–2249. https://doi.org/10.2514/1.G001382 LinkGoogle Scholar[7] Shan M., Guo J. and Gill E., “Review and Comparison of Active Space Debris Capturing and Removal Methods,” Progress in Aerospace Sciences, Vol. 80, Jan. 2016, pp. 18–32. https://doi.org/10.1016/j.paerosci.2015.11.001 CrossrefGoogle Scholar[8] Wander A., Konstantinidis K., Förstner R. and Voigt P., “Autonomy and Operational Concept for Self-Removal of Spacecraft: Status Detection, Removal Triggering and Passivation,” Acta Astronautica, Vol. 164, Nov. 2019, pp. 92–105. https://doi.org/10.1016/j.actaastro.2019.07.014 CrossrefGoogle Scholar[9] Scharf D. P., Hadaegh F. Y. and Ploen S. R., “A Survey of Spacecraft Formation Flying Guidance and Control (Part I): Guidance,” Proceedings of the 2003 American Control Conference, Vol. 2, Inst. of Electrical and Electronics Engineers, Denver, CO, June 2003, pp. 1733–1739. https://doi.org/10.1109/ACC.2003.1239845 Google Scholar[10] Alfriend K., Vadali S. R., Gurfil P., How J. and Breger L., Spacecraft Formation Flying: Dynamics, Control and Navigation, Elsevier Astrodynamics Series, Butterworth–Heinemann (Elsevier Science), Oxford, 2010, pp. 1–11, 329–330, Chaps. 1, 14. Google Scholar[11] Slater G. L., Byram S. M. and Williams T. W., “Collision Avoidance for Satellites in Formation Flight,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, 2006, pp. 1140–1146. https://doi.org/10.2514/1.16812 LinkGoogle Scholar[12] Bevilacqua R., Lehmann T. and Romano M., “Development and Experimentation of LQR/APF Guidance and Control for Autonomous Proximity Maneuvers of Multiple Spacecraft,” Acta Astronautica, Vol. 68, No. 7, 2011, pp. 1260–1275. https://doi.org/10.1016/j.actaastro.2010.08.012 CrossrefGoogle Scholar[13] Newman L. K., Frigm R. C., Duncan M. G. and Hejduk M. D., “Evolution and Implementation of the NASA Robotic Conjunction Assessment Risk Analysis Concept of Operations,” 2014 Advanced Maui Optical and Space Surveillance Technologies Conference Proceedings, GSFC-E-DAA-TN17386, Curran Assoc., Red Hook, NY, Sept. 2014, pp. 1–14. Google Scholar[14] Núñez Garzón U. E. and Lightsey E. G., “Relating Collision Probability and Miss Distance Indicators in Spacecraft Formation Collision Risk Analysis,” 2020 AAS/AIAA Astrodynamics Specialist Conference, Univelt, Inc., San Diego, CA, Aug. 2020, pp. 3853–3872. Google Scholar[15] Núñez Garzón U. E. and Lightsey E. G., “Relating Collision Probability and Separation Indicators in Spacecraft Formation Collision Risk Analysis,” Journal of Guidance, Control, and Dynamics, Vol. 45, No. 3, 2022, pp. 517–532. https://doi.org/10.2514/1.G005744 LinkGoogle Scholar[16] DeMars K. J., Cheng Y. and Jah M. K., “Collision Probability with Gaussian Mixture Orbit Uncertainty,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 3, 2014, pp. 979–985. https://doi.org/10.2514/1.62308 LinkGoogle Scholar[17] Chan F. K., Spacecraft Collision Probability, Aerospace Press and AIAA, El Segundo, CA, and Reston, VA, 2008, pp. 1–97, 237–271, Chaps. 1–5, 13, 14. https://doi.org/10.2514/4.989186 LinkGoogle Scholar[18] Alfano S., “Satellite Conjunction Monte Carlo Analysis,” 2009 AAS/AIAA Space Flight Mechanics Meeting, AAS/AIAA, Savannah, GA, 2009, pp. 2007–2024. Google Scholar[19] Coppola V. T., “Including Velocity Uncertainty in the Probability of Collision Between Space Objects,” 2012 AAS/AIAA Space Flight Mechanics Meeting, AAS/AIAA, Charleston, SC, Jan. 2012, pp. 2159–2178. Google Scholar[20] Patera R. P., “Satellite Collision Probability for Nonlinear Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 5, 2003, pp. 728–733. https://doi.org/10.2514/2.5127 LinkGoogle Scholar[21] Alfano S., “A Numerical Implementation of Spherical Object Collision Probability,” Journal of the Astronautical Sciences, Vol. 53, No. 1, 2005, pp. 103–109. https://doi.org/10.1007/BF03546397 CrossrefGoogle Scholar[22] Zhang S., Fu T., Chen D. and Cao H., “Satellite Instantaneous Collision Probability Computation Using Equivalent Volume Cuboids,” Journal of Guidance, Control, and Dynamics, Vol. 43, No. 9, 2020, pp. 1757–1763. https://doi.org/10.2514/1.G004711 LinkGoogle Scholar[23] Alfano S. and Oltrogge D., “Probability of Collision: Valuation, Variability, Visualization, and Validity,” Acta Astronautica, Vol. 148, July 2018, pp. 301–316. https://doi.org/10.1016/j.actaastro.2018.04.023 CrossrefGoogle Scholar[24] Núñez Garzón U. E. and Lightsey E. G., “Sensitivity of Separation Indicators in Spacecraft Formation Collision Risk Analysis,” 2021 AAS/AIAA Astrodynamics Specialist Conference, Univelt, Inc., San Diego, CA, Aug. 2021, pp. 1–20. Google Scholar[25] Heil C., Introduction to Real Analysis, Graduate Texts in Mathematics, Springer International Publishing, Cham, Switzerland, 2019, pp. 15–288, Chaps. 1–7. https://doi.org/10.1007/978-3-030-26903-6 Google Scholar[26] Dovgoshey O., Martio O., Ryazanov V. and Vuorinen M., “The Cantor Function,” Expositiones Mathematicae, Vol. 24, No. 1, 2006, pp. 1–37. https://doi.org/10.1016/j.exmath.2005.05.002 Google Scholar[27] Billingsley P., Probability and Measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1995, pp. 1–326, 400–481, Chaps. 1–4, 6. Google Scholar[28] Mathai A. M. and Provost S. B., Quadratic Forms in Random Variables: Theory and Applications, Statistics: Textbooks and Monographs, Marcel Dekker, New York, 1992, pp. 25–242, Chaps. 3–5. Google Scholar[29] DeMars K. J., Bishop R. H. and Jah M. K., “Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 4, 2013, pp. 1047–1057. https://doi.org/10.2514/1.58987 LinkGoogle Scholar[30] Valli M., Armellin R., Di Lizia P. and Lavagna M. R., “Nonlinear Mapping of Uncertainties in Celestial Mechanics,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 1, 2013, pp. 48–63. https://doi.org/10.2514/1.58068 LinkGoogle Scholar[31] Vishwajeet K., Singla P. and Jah M., “Nonlinear Uncertainty Propagation for Perturbed Two-Body Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 5, 2014, pp. 1415–1425. https://doi.org/10.2514/1.G000472 LinkGoogle Scholar[32] Horwood J. T., Aragon N. D. and Poore A. B., “Gaussian Sum Filters for Space Surveillance: Theory and Simulations,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 6, 2011, pp. 1839–1851. https://doi.org/10.2514/1.53793 LinkGoogle Scholar[33] DeMars K. J. and Jah M. K., “Probabilistic Initial Orbit Determination Using Gaussian Mixture Models,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 5, 2013, pp. 1324–1335. https://doi.org/10.2514/1.59844 LinkGoogle Scholar[34] Vittaldev V. and Russell R. P., “Space Object Collision Probability Using Multidirectional Gaussian Mixture Models,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 9, 2016, pp. 2163–2169. https://doi.org/10.2514/1.G001610 LinkGoogle Scholar[35] Jones B. A. and Doostan A., “Satellite Collision Probability Estimation Using Polynomial Chaos Expansions,” Advances in Space Research, Vol. 52, No. 11, 2013, pp. 1860–1875. https://doi.org/10.1016/j.asr.2013.08.027 CrossrefGoogle Scholar[36] Nguyen T., “N-Dimensional Quasipolar Coordinates - Theory and Application,” Master’s Thesis, Univ. of Nevada, Las Vegas, NV, May 2014. Chap. 2, pp. 11–22. Google Scholar[37] Hubbard R. G. and Hubbard B. B., “Solving Equations,” Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 5th ed., Matrix Editions, Ithaca, NY, 2015, pp. 159–282, Chap. 2. Google Scholar[38] Núñez Garzón U. E. and Lightsey E. G., “Mahalanobis Shell Sampling (MSS) Method for Collision Probability Computation,” AIAA SciTech 2021 Forum, AIAA, Nashville, TN, Jan. 2021, pp. 1–25. https://doi.org/10.2514/6.2021-1855 Google Scholar Previous article Next article FiguresReferencesRelatedDetailsCited byStochastic Convergence of Sobol-Based Mahalanobis Shell Sampling Collision Probability ComputationUlises E. Núñez Garzón and E. Glenn Lightsey9 March 2023 | Journal of Guidance, Control, and Dynamics, Vol. 0, No. 0 What's Popular Volume 45, Number 6June 2022 CrossmarkInformationCopyright © 2022 by Ulises Eduardo Núñez Garzón. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsApplied MathematicsGeneral PhysicsMathematical AnalysisPlanetary Science and ExplorationPlanetsSpace MissionsSpace Science and TechnologyStatistical Analysis KeywordsCumulative Distribution FunctionSpacecraft Formation FlyingSpacecraft MissionsComputingSpace Flight MechanicsSensitivity AnalysisOrbit DeterminationEarthAcknowledgmentsThe authors would like to thank Konstantin Tikhomirov and Christopher Heil at the School of Mathematics at Georgia Tech, specifically, for their valuable insights on the existence of pdfs of transformed random variables, and on the connections between absolute continuity of functions and absolute continuity of measures.PDF Received2 November 2021Accepted18 February 2022Published online21 March 2022

Referência(s)