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Sighting-In on COLA Vaš Majer Integral Systems, Inc AIAA Space Operations Workshop April /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 100 m Scope Trained on Target at (0,0)

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Scope Trained on Origin (0,0) 100 m (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 100 m Fixed but Unknown [Not a Random Variable] Observations/Measurements, z=(z x,z y ) Bullet Hole Coordinates on 100 m Subject to Dispersion Estimator/Predictor of Truth, U=(U x,U y ) Scope Cross-Hair Coordinates on 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 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 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 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

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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 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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Close Approach Bonus Hidden Tracks

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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 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 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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