Time Evolution of Risk COLA

Vaš MajerIntegral Systems, Inc

AIAA Space Operations Workshop15-16 April 2008

04/21/23 03:45

Introduction

Hello

Agenda OASYS COLA Risk Analysis

Drift-By Scenario On-Station [Home] Drifting [Visitor]

Time Evolution of Separation Risk

COLA

OASYS Collision Risk Assessment

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

The Scenario

0

0

Collision Avoidance Scenario

x

y

separation,u

sphere,S

covariance,R a

b

d

u Separation EstimateR 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

The Definition

Collision TRUTH, Z, is Inside Stay-Out Sphere S

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)

Attributes of Risk Statistic, r

0 < r ≤ 1 r = 0 Lowest Possible Risk r = 1 Highest Possible Risk r is Conservative r is Robust

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 Meaning

Independent of Scenario Geometry r > 0; Risk Never Sleeps r = 1 OK; Extreme Risk Deserves Notice

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

Risk of Collision, rC

OASYS™ COLA Statistic

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;

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

Drift-By Scenario

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

COLA Analysis Controls Hard-Body Sphere Around HEX

100 m

Alarm Levels 

Alarm Risk Separation

RED 1x10-3 1 km

YELLOW 1x10-6 10 km

GREEN 1x10-9 100 km

Common Risks

Event Risk

Winning Lottery 1/100M= 1e-8

Car Crash KSI 6/1000 = 6e-3

Ace of Hearts 2/104 = 2e-2KSI: Killed or Seriously Injured

Common GEO Separations

Description Separation

1 deg/day Relocation 89 km

50 mdeg Slot Half-Width 37 km

100 Eccentricity 08 km

: micros, 1e-6

Time Evolution COLA

10 Day Span Centered on COV Epoch @

t=0

Time Evolution of Separation

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...

Time Evolution of Risk

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

Time Evolution of J,H Metrics

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

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

The End