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Time Evolution of Risk COLA

This presentation discusses the critical assessment of collision risk in space operations using the OASYS COLA framework. It explores the time evolution of separation and risks associated with drifting satellites, focusing on estimating collision probabilities. The analysis encompasses uncertainty in estimators and highlights alarm levels for risk management in space environments. Emphasizing the importance of understanding how uncertainty grows over time, it presents methodologies for quantifying and evaluating collision risks and separation metrics crucial for safe space operations.

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Time Evolution of Risk COLA

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  1. Time Evolution of Risk COLA Vaš Majer Integral Systems, Inc AIAA Space Operations Workshop 15-16 April 2008 9/26/2014 12:01 PM

  2. Introduction Hello

  3. Agenda • OASYS COLA Risk Analysis • Drift-By Scenario • On-Station [Home] • Drifting [Visitor] • Time Evolution of • Separation • Risk

  4. COLA OASYS Collision Risk Assessment

  5. Given • t → u(t) = y(t) – x(t) • 3D Separation Vector Ephemeris • Vehicle Y with Respect to Vehicle X • u=0: Vehicle Y @ Vehicle X Center of Mass • t → R(t) • 3D Joint Uncertainty Covariance Ephemeris

  6. The Scenario u Separation Estimate R Covariance of Estimator d Radius of Hard Body Stay-Out Sphere, S z = (x,y) Any Trial Vector TRUTH, z=Z, is, As Always, Nowhere to be Seen, FixedBut Unknown

  7. The Definition • Collision • TRUTH, Z, is Inside Stay-Out Sphere S

  8. The Objective • Quantify Risk of Collision • For Estimators, t →u(t), t  T • In View of Uncertainty, R(t) • In View of Stay-Out Sphere, S • With a Scalar Function, t →r(t)

  9. Attributes of Risk Statistic, r • 0 < r ≤ 1 • r = 0 Lowest Possible Risk • r = 1 Highest Possible Risk • r is Conservative • r is Robust

  10. Conservative • Because Estimator, u... • Is Biased Relative to Truth, Z • Bias u-Z is Unknown • And Because Estimator Covariance, R... • Should be Centered on Truth, Z, which is Unknown • Is Notoriously Optimistic [Small] • Under-States Variance/Uncertainty • We Want Risk Statistic, r, Such That... • r is Upper Bound on Risk • r Threshold Levels Have MeaningIndependent of Scenario Geometry • r > 0; Risk Never Sleeps • r = 1 OK; Extreme Risk Deserves Notice

  11. Robust • r Conforms to Intuitive Notion of Risk • r increases as |u| decreases • r increases as |R| increases • r increases as d increases • r is Sensible for Limiting Scenarios • u in S implies r = 1 • u near S implies r ~ 1 • r makes sense even for d=0

  12. Risk of Collision, rC OASYS™ COLA Statistic

  13. Risk of Collision, rC • if (0 ≤ |u| ≤ d) rC = 1; • else v = d (u/|u|); V = {z | J(z; v,R) < J(u; v,R)} q = ∫V dp(z; v,R) rC = 1 – q;

  14. Risk of Collision Heuristic • Make the NULL Hypothesis: • u is a Trial Estimator of Truth Z=v, where • v = d (u/|u|); d = radius of S; and • Trial Estimators are z ~ Gauss(v,R) • v is the Point in S which is Closest to Estimator u • V is the (v,R) Metric Sphere of Radius |R-1/2(u-v)| Centered at v • Estimator u is on the Boundary of V • q is • the Probability Measure of the (v,R)-Sphere, V • the Probability that a Random Trial Estimator of Z=v Lies in V • rC = 1-q is • the Probability Measure of the Complement of V • an Upper Bound on the Probability that the NULL Hypothesis is TRUE

  15. Drift-By Scenario

  16. GEO Drift-By • SatX [HEX] on GEO Station • COV Epoch @ t=0 • SatY [WHY] in GEO Drift-By • COV Epoch @ t = 0 • Close Approach to HEX @ t ~ 10 hours

  17. COLA Analysis Controls • Hard-Body Sphere Around HEX • 100 m • Alarm Levels 

  18. Common Risks KSI: Killed or Seriously Injured

  19. Common GEO Separations : micros, 1e-6

  20. Time Evolution COLA 10 Day Span Centered on COV Epoch @ t=0

  21. Time Evolution of Separation

  22. Discussion of Separation • Near Linear Approach and Departure • Clear Point of CAP @ t ~ 10 hours • Alarm Level sepYEL=100 km Active • Alarm Level sepRED= 10 km InActive • sepMIN=10.023 km > sepRED = 10 km • Looks Safe Enough...

  23. Time Evolution of Risk

  24. Discussion of Risk • log10(1) = 0 • Periodic rskMAX ~= 1 • 12 hour Period • Risk Alarms Triggered • Well Before and Well After CAP • Risk Alarm Level Transitions Closely Spaced • High Risk Levels Despite Large Separations • COV Epochs @ t=0 • Uncertainty Grows Forward/Backward in Time • In Real Life... • COV Epochs are Many Revs Prior to CAP • Growth of Uncertainty is Significant

  25. Time Evolution of J,H Metrics

  26. Discussion of Metrics • The Whole Story Encapsulated • H ~ Squared Separation [cyan] • J ~ ChiSquared Separation [blue] • Minima of H ~ Minima of Separation • Apparently Benign Alarm Levels • Minima of J ~ Maxima of Risk • Alarm Levels Triggered Well in Advance • Risk Maxima Identified

  27. Summary • Useful • COL Risk Analysis • CAP Separation Analysis • Complementary Views of Close Encounter • Essential • Time Evolution Study of Risk/Separation • Acknowledge Growth of Uncertainty with Time • Myopic and Even Dangerous • Restrict COLA to Times of CAP • Restrict COLA to 2D  Relative Velocity

  28. The End

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