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Sighting-In on COLA Vaš Majer Integral Systems, Inc AIAA Space Operations Workshop 15-16 April 2008 5/8/2015 9:08 PM

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Introduction Hello

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Agenda Parametric Analysis of Two COLA (Collision Avoidance) Statistics Probability of Collision, pC Commonly Used COLA Statistic Risk of Collision, rC OASYS ™ COLA Statistic But First, a Discussion of GHRA...

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GHRA Ground-Hog Risk Assessment

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Background Nicole Keeps a Garden Ground-Hogs Like the Garden She Tried Sharing with Them Reasoning with Them Trapping Them

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It’s Come to This:.243 Varmint Rifle for Christmas Quarter-Size Pattern at 100 m After the Scope is Sighted-In Sighting-In Rifle Locked in Vise Target Set @ 100 m Scope Trained on Target at (0,0)

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Scope Trained on Origin (0,0) Target @ 100 m Cross-Hair @ (0,0) 2 cm Dia Ref Ring, S Rifle Locked in Vise

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Sighting-In Paradigm Truth, Z=(Z x,Z y ) Barrel Bore-Sight Location on Target @ 100 m Fixed but Unknown [Not a Random Variable] Observations/Measurements, z=(z x,z y ) Bullet Hole Coordinates on Target @ 100 m Subject to Dispersion Estimator/Predictor of Truth, U=(U x,U y ) Scope Cross-Hair Coordinates on Target @ 100 m Fixed and Known [Not a Random Variable] U = (0,0) in Sighting-In Set-Up Estimator Correction, u = (u x,u y ) Scope Cross-Hair Sighting-In Adjustment To Be Determined

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z,Z in Observation Space z = Z + w, w = Gauss(0,W) Z ~ barrel bore-sight coordinates z ~ bullet hole coordinates w ~ an instance of sample error W ~ bullet dispersion covariance Known or To Be Discovered 0 ~ mean bullet dispersion Z is Fixed, Unknown Bore-Sight Truth z is Random Variable on Bore-Sight Space

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u,U in Estimator Space u = Z-U U is Known [Cross-Hair] Z is Fixed But Unknown [Bore-Sight] u is Fixed And Also Unknown [Bias] Z,U,u are Deterministic Values [Truth] No Probabilities Are Involved Can’t Hit Anything Either

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The Connection z : u Sample z: {z k, k=1:n} Estimate u n Observation Model z k = U + u n + w k, k=1:n w k ~ Gauss(0,W) Making the Connection is a Model u,UDomain of Model z,ZRange of Model Objective: Align Scope ~ Bore-Sight

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The Confusion u n is Fixed, Deterministic, Computable For a Given Trial of n Samples, {z k, k=1:n} u n is a Random Variable On the Space of All Trials of n Samples, {z k,j, k=1:n, j=1:∞} u n Wears Two Hats

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3 Rounds Fired Rounds ~ Red Dots High and Right Need More Observations Cheap Rounds are $2 Each This Info Cost $6

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Triangulation “Two Rounds are Not Enough; Four are too Many” Barycenter of Triangle Scope Adjusters 1 click = 1 cm Could Sight-In Scope Now... But We Won’t

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100 Rounds Fired Calibrated Eyeball Suggests Bore Location ~ (1 cm, 3 cm) This Single Trial of 100 Rounds Cost $200 This Information Cost $200

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100 Sample Mean & Covariance u 100 ~ + W 100 ~ O R 100 ~ O

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Sample Mean & Covariance Mean of n Samples (n=100) + u n = (1/n) Σ k=1:n z k Covariance of n Samples (n=100) O W n = [1/(n-1)] Σ k=1:n (z k -u n )(z k -u n ) T W n = [ a n a n c n a n b n c n b n a n b n b n ] a n, b n > 0; -1 ≤ c n ≤ 1

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Sample Space Properties u n → u as n → ∞ u n ~ Finite Sample Mean u ~ Infinite Sample Mean aka Truth W n → W as n → ∞ W n ~ Observation Residual Covariance W ~ Dispersion of Rounds in Z-Space (u,W) Define a Metric on the Z-Space J(z; u,W) = (z-u) T W -1 (z-u) = ( - ) T ( - ) = W -½ z = W -½ u (Silly Putty Transform) Red Ellipse is Unit Sphere in (u n,W n ) Silly Putty Metric Scale in cm of 1 Unit of (u n,W n ) Metric Depends on Direction of (z-u n )

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Parameter Space Properties + u n = argmin v Σ k=1:n J(z k ; v,W) MinVariance/MaxLikelihood 0 R -1 n = Σ k=1:n W -1 Fisher Information R n ~ Dispersion of Estimators Over Many Trials of n Samples Covariance of u n when Wearing its Trial Hat R n = (1/n) W R n → 0 as n → ∞ (u,R) Define a Metric on the U-Space J(v; u,R) = (v-u) T R -1 (v-u) = ( - ) T ( - ) = R -½ z = R -½ u Green Ellipse is Unit Sphere in (u n,R n ) Silly Putty Metric Scale in cm of 1 Unit of (u n,R n ) Metric Depends on Direction of (v-u n ) And → 0 as n → ∞ ! (u n,R n ) Metric is Smaller Scale than (u n,W n ) Metric Reflects the Information Packed into R -1 n from n Samples

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Wait A Minute... Are not u,U and z,Z the Same Spaces? Yes, They Are All Fruit Vectors in the Target Plane No, They Are Not the Same Apples ~ Model Domain u,UCorrected/Prior Scope Cross-Hair Locations Oranges ~ Model Range z,ZObserved/True Bore-Sight Locations

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200 Sample Mean & Covariance u 200 ~ + W 200 ~ O R 200 ~ O This is a Virtual Trial only to Illustrate n=200 Another $400 for Ammo is Out of the Question

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More Information (n=200) u n,W n Adjust Slightly Centroid and Dispersion of Rounds are Revealed Slightly Better w/ 200 Rounds R n is Cut in Half (100/200) Reflects the Doubling of Information R ½ n Metric on u,U-Space is Cut By 1/√2 u 200 is NOT √2 Times More Accurate than u 100 Nicole...I Didn’t Fire 200 More Rounds

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Adjust Scope to 100 Sample Mean I Can Afford only 1 Trial of n=100 Samples u 100 is Best Estimator of Bore-Sight Location Given n=100 Samples Treat u 100 as if it WERE the True Bore-Sight Location Adjust Cross-Hair on u 100 Rifle Still Clamped in Vise

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But What if I... Had 99 More Boxes of 100 Rounds? Ran 99 More Trials of 100 Samples? u n,j, W n,j, R n,j n=100 for each Trial Trials j=1:m, m=100 10K Rounds

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Means for 100 Trials Each Trial having 100 Samples + Trial Means, u n,j * Mean over Trials, u n * O Sample Covariance over Trials, P O Estimator Covariance over Trials, R Scope is Trained on u 100 for Trial 1 Nicole, I Didn’t Spend $20K on Ammo

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Zoom-In On Mean of Means + u n,j for n=100 * u n * O Sample Covariance, P O Estimator Covariance, R Scope is Still Trained on u n,1

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What’s the Truth? Truth is Un-Obtanium Given... Finitely Many Samples (e.g., n=100) No u n,j is Truth It’s Simply A Different Trial of n=100 Samples Even u n * = Mean(u n,j ) = Mean of Means is Not Truth But u n * Converges to Truth as n → ∞ Just as u n,j Converges to Truth as n → ∞ for any fixed j In Real Life We Get Only 1 Trial Make n as Large as Affordable (Number of Rounds) Make W as Small as Affordable (Quality of Rounds) We Treat u n,1 As if It Were Truth ( Working Hypothesis) Fixed and Known, Deterministic not Probabilistic The Best Estimator/Predictor We Have or Can Afford The Practical Truth on Which We Base Operational Decisions

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For All Practical Purposes... Drop the n... u := u n Adjust the Scope... U := U+u = u Bore-Sight := Scope Cross-Hairs on... Z = U = u Fire 1 Round... z = u + w,w ~ (0,W) If z is not within Scope Reference Ring, S... Buy $3 Rounds with Smaller Dispersion, W

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Role of Estimator Covariance, R R is a Cruel Hoax Estimator Algorithm Yields (u,R) R is Centered on Z, Which is Unknown u-Z is Unknown Bias R is Notoriously Optimistic R → 0 ~ 1/n → 0, n = Sample Size No Comparable Convergence for Rate u → Z with n Having No Alternative... We Define Z := u(Working Hypothesis) Center R on u Consider v ~ Gauss(u,R) in U-Space v are Trial Estimators Every v in u,U-Space is a Possible Estimator u is MinVariance/Max Likelihood Estimator The Best Estimator [We Can Afford; Given the Observations]

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We Add to Confusion... By Suppressing/Discarding z,Z Once u,R Have Been Computed And then Switching Notation from v ~ Gauss(u,R), U = Truth v is Trial Variable in Estimator Space To z ~ Gauss(u,R), Z = Truth Now z is Trial Variable in Estimator Space

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Uses of Estimator Covariance, R R ~ Dispersion of Estimators, u With Respect to Truth, Z, over Many Trials R ~ Uncertainty Metric on u,U-Space R is NOT the Accuracy, |u-Z|, of u In Our Example: |u – Z| ~ 2 R-Units R Should be Constant across Trials Use R to QA Estimation Process Variation from Trial to Trial Signals that Trials are Not Comparable

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GHRA Summary v ~ Gauss(u,R)

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v ~ Gauss(u,R), v in U-Space u is Fixed U-Vector R is Fixed Covariance v is Trial or Dummy Variable J(v; u,R) is Squared Length of v-u w/r R-metric p = ∫ Q dp(v; u,R) is Probability-Weighed Measure of Set Q w/r R-metric dp(v; u,R) = [1/(2 π )|R| 1/2 ] exp[-J(v; u,R)/2] dv v is Variable of Integration over Q 0 ≤ p < 1 for Bounded Sets Q p = 1 for Q = Entire U-Space p is NOT Probability that Truth, Z, Lies in Q p is Probability that v Selected Randomly from Gauss(u,R) lies in Q A Measure of Set Q with Rapidly Decreasing Weight Centered @ u

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COLA Collision Risk Assessment

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Given t → u(t) 3D Separation Vector Ephemeris Vehicle Y with Respect to Vehicle X u=0 @ Vehicle X Center of Mass t → R(t) 3D Joint Uncertainty Covariance Ephemeris

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Common Practices Reduce Dynamic to Static Restrict Attention to Time[s] of Closest Approach Reduce 3D to 2D Remove Dimension Along Velocity Vector

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The Scenario Looks Familiar 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, Fixed But Unknown

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

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The Objective Quantify Risk of Collision For Given Estimator, u In View of Uncertainty, R In View of Stay-Out Sphere, S With a Single Number, r

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

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Conservative Because Estimator, u... Is Biased 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

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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

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COLA Statistic Definitions Two Statistics

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Probability of Collision, pC Commonly Used COLA Statistic

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Probability of Collision, pC pC = ∫ S dp(z; u,R) dp(z) = [1/(2 π ) n/2 |R| 1/2 ] exp[-J(z; u,R)/2] dz pC = Integral over S of Gauss(u,R) Density dp(z) = Gauss(u,R) Density S = Sphere of Radius d Centered @ Origin Here n = 3, Dimension of 3-Space

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Probability of Collision Heuristic * dp(z) = Probability Density of True Separation, Z, w/r to u pC = Probability that True Separation, Z, is Inside Sphere S * R.P. Patera, General Method for Calculating Satellite Collision Probability, AIAA J Guidance, Control and Dynamics, Vol 24, No 4, July-August 2001, pp 716-722.

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Risk of Collision, rC OASYS ™ COLA Statistic

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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;

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Risk of Collision Heuristic Make the NULL Hypothesis: u is a Trial Estimator of 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

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Parametric Comparison pC and rC Over a Family of Scenarios

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The Scenarios R = [a 2 0 0 b 2 ] Diagonal u = [0 u]Vertical u = [-4 : 0.1 : 4] a = [1.5, 1, 1/1.5] b = [1.5, 1, 1/1.5] d = [1, ½, ¼]

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Nominal Reference Case u = [-4 :.1 : 4] a = 1 b = 1 d = 1

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Scenario Set A pC, rC for Fixed a=1, b=1 and Decreasing Sphere Radius, d

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Fixed a,b=1; Decreasing d

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Scenario Set A Discussion d is Radius of Stay-Out Sphere pC(u,d) < 1 for Every (u,d) pC(u,d) → 0 as d → 0 for Fixed u pC(u,d) = 0 for EVERY u when d = 0 !? rC(u,d) Contracts Congruently as d → 0 rC(u,d) = 1 for |u|

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Scenario Set B pC, rC for Fixed a=1, d=1 and Increasing Vertical Uncertainty, b

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Fixed a,d=1; Increasing b

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Scenario Set B Discussion b is Vertical Uncertainty (|| to u) pC(u,b) < 1 for Every (u,b) pC(u,b) → 0 as b → ∞ for |u| 0 pC(u,b) → 1 as b → ∞ for d+δ<|u| !? rC(u,b) = 1 for |u|

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Scenario Set C pC, rC for Fixed b=1, d=1 and Increasing Horizontal Uncertainty, a

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Fixed b,d=1; Increasing a

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Scenario Set C a is Horizontal Uncertainty ( to u) pC(u,a) < 1 for Every (u,a) pC(u,a) → 0 as a → ∞ for Fixed u !? rC(u,a) = 1 for |u|<1 and Every a rC(u,a) is Independent of a

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Conclusions

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pC pC is Not Conservative Understates Risk pC << 1 When u=0 pC << 1 When u in S pC is Not Robust Variation of Risk Level with Uncertainty Defies Common Sense Variation of Risk Level with Hard-Body Size Defies Common Sense

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rC rC is Conservative rC is Robust rC is a Practical Indicator of Risk This Surprised Me as Much as it Surprises You

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The End Take It Easy

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COLA @ Close Approach Bonus Hidden Tracks

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COLA @ Close Approach In Reality u = u(t), R=R(t) Evolve with Time, t Common Practice Restrict COLA pC/rC Analysis to Times of Close Approach Compress 3D to 2D By Removing Direction || to du(t)/dt

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Unfortunately Risk @ Time of Close Approach Not Necessarily Maximum Risk Can Be Lower than Alarm Level Risk Before or After Close Approach Can Be Higher than Alarm Level Will Be Missed by @ Close Approach COLA

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Time Evolution of Risk Applies to Both pC and rC Next Example Uses rC

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Corner in Winslow, Arizona

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Time Evolution of Risk

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Risk Assessment She Didn’t Slow Down

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The Real End Really

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