Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2012 Spacecraft Collision Probability Estimation for Rendezvous and Proximity Operations Michael R. Phillips Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Aerospace Engineering Commons Recommended Citation Phillips, Michael R., "Spacecraft Collision Probability Estimation for Rendezvous and Proximity Operations" (2012). All Graduate Theses and Dissertations. 1398. https://digitalcommons.usu.edu/etd/1398 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. SPACECRAFT COLLISION PROBABILITY ESTIMATION FOR RENDEZVOUS AND PROXIMITY OPERATIONS by Michael R. Phillips A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Aerospace Engineering Approved: Dr. David K. Geller Dr. R. Rees Fullmer Major Professor Committee Member Dr. Stephen A. Whitmore Dr. Mark R. McLellan Committee Member Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2012 [This page intentionally left blank] iii Copyright c Michael R. Phillips 2012 All Rights Reserved [This page intentionally left blank] v Abstract Spacecraft Collision Probability Estimation for Rendezvous and Proximity Operations by Michael R. Phillips, Master of Science Utah State University, 2012 Major Professor: Dr. David K. Geller Department: Mechanical and Aerospace Engineering The topic of this thesis is on-board estimation of spacecraft collision probability for orbital rendezvous and proximity operations. All of the examples shown in this work assume that the satellite dynamics are described by the Clohessy-Wiltshire equations, and that the spacecraft are spherical. Several collision probability metrics are discussed and compared. Each metric can be placed into one of three categories. The first category provides an estimate of the instantaneous probability of collision, and places an upper bound on the total probability of collision. The second category provides an estimate of total collision probability directly. The last category uses Monte Carlo analysis andanovel Pseudo Monte Carlo analysis algorithm to determine total collision probability. The metrics are compared and their accuracy is determined for a variety of on-orbit conditions. Lastly, a method is proposed in which the metrics are arranged inahierarchy such that those metrics that can be computed quickest are calculated first. As the proposed algorithm progresses the metrics become more costly to compute, but yield more accurate estimates of collision probability. Each metric is compared to a threshold value. If it exceeds the limits determined by mission constraints, the algorithm computesamore accurate estimate by calculating the next metric in the series. If the threshold is not reached, it is assumed there is a tolerable collision risk and the algorithm is terminated. In this way the algorithm is capable of adapting to the level of collision probability, and can be sufficiently accurate without needless calculations being performed. This work shows that collision probability can be systematically estimated. (111 pages) [This page intentionally left blank] vii Public Abstract Spacecraft Collision Probability Estimation for Rendezvous and Proximity Operations by Michael R. Phillips An increasing number of missions require spacecraft to fly in close proximity. Despite the technological advances that have made these types of missions increasingly possible there remains a limit to how accurately the trajectory of a spacecraft can be predicted. As such, one of the main risks to spacecraft in close proximity is the threat of a collision. Many methods exist for computing the probability of collision between spacecraft. These methods incorporate the uncertainty in predicting the spacecrafts motion when computing an estimate. The accuracy and required computation time of these methods very greatly. A poor estimate of collision risk will not only increase the likelihood of a collision occurring, it can also carry other negative consequences. One possible side effect is additional fuel being spent to reduce the threat of a collision which is estimated to be high but in reality is low. This thesis explores several methods for estimating spacecraft collision probability and addresses wether each can be used effectively in real time while the spacecraft are on-orbit. Each of the methods can be placed into one of three categories. The first category of methods do not directly estimate collision probability. Instead they place an upper limit on what the actual collision probability can be. The second category are methods which are capable of directly estimating collision probability over a time interval. The last category uses a common statistical analysis tool known as Monte Carlo to determine the probability of collision over an interval of time. This category also includes a novel method called Pseudo Monte Carlo. Each of the methods are compared and their accuracy are evaluated for a variety of orbit conditions. Finally, an algorithm is proposed in which the methods are arranged in a hierarchy so that those methods which can be computed quickest are calculated first. As the proposed algorithm progresses the methods become more costly to compute, but yield more accurate estimates of collision probability. The result from each method is compared to a threshold value. If it exceeds the limits determined by mission constraints, the algorithm computes a more accurate estimate by utilizing the next method in the series. If the threshold is not reached, it is assumed there is a tolerable collision risk and the algorithm is terminated. In this way the algorithm is capable of adapting to the level of collision probability, and can be sufficiently accurate without needless calculations being performed. This work shows that collision probability can be systematically estimated. [This page intentionally left blank] ix Acknowledgments I would like to acknowledge the many people who have made this thesis possible. The Air Force Research Lab has provided key support to this research, and I would like to thank Frank Chavez and Nathan Stastny. There are many people in the Guidance, Navigation, and Control Research Group at USU who deserve thanks for the support they have given me. Shane Robinson in particular has been a great friend and help throughout the course of my graduate studies. I would like to give special thanks to my major professor, Dr. David Geller, who has provided me with countless hours of instruction and support. Without his help and input I would not have been able to complete this thesis. I would also like to thank the other members of my committee, Rees Fullmer and Tony Whitmore, who have provided me with guidance throughout my graduate studies. Bonnie Ogden and Christine Spall, my departmental advisors, have also played key roles in bringing me to this point. I give special thanks to my family, for their encouragement, support, and patience as I worked my way through school. Last, I wholeheartedly thank my dear wife. Her support and kind- ness have been so appreciated. She has been with me through the whole process of completing this thesis, and without her I could not have come this far. Michael R. Phillips [This page intentionally left blank] xi Contents Page Abstract ............................................. v Public Abstract ........................................ vii Acknowledgments ....................................... ix List of Tables .......................................... xiii List of Figures ......................................... xv Acronyms ............................................ xxi 1Introduction........................................ 1 2ProblemStatement.................................... 5 3LiteratureSurvey..................................... 7 3.1 InstantaneousMetrics .................................. 7 3.2 TotalProbabilityofCollision . 7 4SimulationModels..................................... 13 4.1 CoordinateFrames .................................... 14 4.1.1 Earth Centered Inertial . 14 4.1.2 Local Vertical Local Horizontal . 14 4.1.3 Quasi Inertial Local Vertical Local Horizontal . 15 4.1.4 EncounterFrame................................. 15 4.2 RelativeStateandCovariancePropagation . 16 4.2.1 RelativeOrbitElements . 18 4.3 InertialStateandCovariancePropagation . 19 4.4 MonteCarloAnalysis .................................. 20 4.4.1 StateandStateCovarianceInitialization. 21 4.4.2 CollisionDetection ................................ 21 4.4.3 ConfidenceInterval................................ 23 5Metrics............................................ 25 5.1 InstantaneousMetrics .................................. 25 5.1.1 Vertical Projection Instantaneous Metric, dx .................. 28 5.1.2 Horizontal Projection Instantaneous Metric, dy ................ 30 5.1.3 Line-of-Sight Projection Instantaneous Metric . 32 5.1.4 MahalanobisDistance .............................. 33 5.1.5 MaximumInstantaneousProbability . 34 5.2 TotalProbabilityMetric-Patera’sAlgorithm . 35 5.3 TotalProbabilityMetric-PseudoMonteCarlo . 39 5.3.1 AltitudeCheck .................................. 40 5.3.2 Cross-TrackCheck ...............................