Journées des Thèses 2011 Branche TIS (DCSD/DCPS/DTIM) 26 Janvier à Palaiseau et 7-8 Février à Toulouse Estimating satellite versus debris collision probabilities via the adaptive splitting technique Rudy PASTEL ([email protected]) ONERA–DPRS, Palaiseau Rennes I, Rennes Thèse encadrée par : Jérôme MORIO (ONERA–DPRS, Palaiseau) et François LEGLAND (INRIA-Rennes) Résumé On February 10th 2009, satellites Iridium and Cosmos collided though their confi- guration had been reported safe. This triggered the quest for a better assessment methodology and therefore a better collision probability estimation. As a matter of facts, rare event probability and extreme quantile estimations arise more and more frequently in the industrial and engineering worlds, be it for safety requirement or performance objectives. As usual Monte Carlo technique can not cope, a new dedicated tool is needed. Using the spacecraft collision real context as a sup- port, the Adaptive Splitting Technique is introduced and compared with Crude Monte Carlo it is shown to outperform. th A. INTRODUCTION these two satellites collided on February 10 2009, tough their configuration did not appear trouble- The constantly growing population of active spa- some [Kel09]. We then try to estimate the collision cecrafts now has to face the threat of colliding probability through Crude Monte Carlo (CMC) in with the debris bequeathed by their predecessors sectionC., in vain. Next, in sectionD. we present [NAS10, Gla09] : due to their high speeds, even small the Adaptive Splitting Technique (AST), the spe- left behind bolts can seriously damage a satellite. cial rare event probability estimation technique we Collision avoidance procedures were hence desi- advocate to our purpose, and show it can provide gned for improved satellite safety such as described valuable accurate information in this framework. in [Smi02]. Active spacecraft managing teams, when a potential collision occurs, now have to decide whe- B. SPACECRAFT ENCOUNTER AND ther to start a collision avoidance maneuver. This NOISED STATE MEASUREMENT extra move sure clears away the danger but also costs extra energy, thereby shortening the satellite’s On February 10, 2009, a commercial Iridium activity timespan. A trade-off between safety and communications satellite and a defunct Russian sa- lifetime expectation therefore has to be found. tellite Cosmos collided, though their configuration To support the maneuver decision, a dedicated was not reported dangerous [Kel09]. We will esti- tool is the probability of collision between the de- mate in this article what was the probability it hap- bris and the satellite. A great deal of effort has been pened. spent on how to estimate its very low value. In his In order to understand the predicament an ac- work at the NASA [Cha08], Chan concludes through tive spacecraft managing team can find itself in, we a numerical integration after a sequence of approxi- will describe the geometrical issue at hand and then mations and hypothesis. To avoid those and improve the main source of randomness. the estimation reliability, we choose the Monte Carlo method. As Crude Monte Carlo (CMC) fails to de- 1. A basic geometry problem liver when faced with a rare event, we aim at es- Consider two satellites orbiting around the Earth timating this very low collision probability via a in a Galilean frame of reference with our planet as rare event probability estimation tool : the Adap- origin and equipped with the Euclidean distance. tive Splitting Technique (AST) proposed by Cerou This three-body problem will be considered a double and Guyader et alii in [CDFG09]. two-body problem : each satellite interacts through In this paper, we first introduce the space- gravity only with the Earth and not with the other craft collision issue via the Iridium-Cosmos case : satellite. Besides, the Earth and the satellites are assumed to be homogeneous spheres with radii dE, Things would be all that easy and deterministic, d1 and d2. The collision distance is therefore dc = had randomness not barged in. d1 + d2. We wonder about the relative position of the two satellites : might they collide during a given 3. Random measurements lead to time span I = [ts, te] ? uncertainty In real life, the states are not monitored around- 2. A convenient model of dynamics the-clock : they are merely measured from times to To keep things simple, orbital mechanics being times, by a radar. The Two Line Elements (TLE) not our topic, we will use Kepler mechanics. One can provided by NORAD sum up this information and m m use more advanced models such as SGP4 [Miu09] if feed the models with the (~si , ti ) pairs. However, wanted. The discussed probability estimation me- TLEs are inaccurate and their inaccuracy is unk- thodologies are independent of the method. nown. This uncertainty is our very issue. To cope At time t, the satellites will be represented by with this and to better reflect the reality, we added their states ~s1(t) and ~s2(t) i.e. their positions ~r1(t) independent, identically distributed (iid) noises ~1 m and ~r2(t) and their speeds ~v1(t) and ~v2(t) such that and ~2, with density f~, to the models’ inputs si . ~s = (~r ,~v ). i i i ~ In our setting, the speeds evolve according to E~ = 1 ~ (5) the same well-known Ordinary Differential Equa- 2 f (~e) = f (~ε ) × f (~ε ) tions defining the two body problem E~ ~ 1 ~ 2 The collision issue can not be answered in a cut- d~vi ~ri and-dried way anymore as it has to be rephrased in ∀t, = −a 3 (1) dt ri a probabilistic fashion itself : what is the probability of collision between the two satellites ? where a is a positive constant given by physics. Via the random counterpart of our deterministic This ordinary differential equation (ode) is ana- geometrical problem lytically solved in many textbooks and its solution ~ m m ~ ~ depends continuously and in a bijective fashion on i ∈ {1, 2}, ∀t ∈ I, Si = Φ(~si + ~i, ti , t) = (Ri(t), Vi(t)) m ~ ~ the given so called initial conditions : its value ~si ∆ = mint∈I {kR2 − R1k(t)} m m m m m at ti , the measurement time, through Φ the ode’s Ξ = ξ(~s1 + ~1, t1 , ~s2 + ~2, t2 ,I) resolvant i.e. its solution map : (6) this question is equivalently stated as m m i ∈ {1, 2}, ∀t ∈ I, ~si = Φ(~si , ti , t) (2) P[{ The satellites collide during I}] = E[Ξ] (7) At this point, there is a natural way to clear out We now face a plain expectation estimation pro- the collision issue using blem. δ = min{k~r2 − ~r1k(t)} (3) t∈I C. THE CRUDE MONTE CARLO t ∈ I 7→ k~r − ~r k(t) experimental convexity, figure 2 1 As an explicit analytical way to calculate [Ξ] 1, makes δ available through numerical optimisation E is unlikely to be found, one will most likely make and its associated test : do with an estimation. Crude Monte Carlo (CMC) 1 if δ ≤ d is a very convenient and reliable way to reach it, if ξ(~sm, tm, ~sm, tm,I) = c (4) 1 1 2 2 0 otherwise the sought probability is not too low, indeed. CMC can not handle rare event probability estimation ef- eventually closes the deal. ficiently. 1. A basic set of tools and notations Fig. 1 – Iridium-Comos distance on collision day accor- ding to TLE. CMC’s estimators for Ξ’s expectation E[Ξ] and Iridium−Kosmos Distance on February 10th 2009 variance V[Ξ] are defined respectively as µ, the em- 6000 pirical mean and σ2, the empirical variance of n iid i 5000 tests, Ξ , i ∈ {1, ··· , n} : n 4000 1 X µ(Ξ, n) ≡ Ξi (8) n 3000 i=1 n Distance (m) 2000 1 X σ2(Ξ, n) ≡ (Ξi − µ(Ξ, n))2 (9) n 1000 i=1 0 16.5 16.55 16.6 16.65 16.7 16.75 As the empirical mean of iid random variables, Hour (h) µ(Ξ, n) is a random variable as well and we hope 2 its variance is as small as possible with respect to the work is divided in estimating a sequence of easier its estimated mean. To measure this, m iid µ(Ξ, n) probabilities and eventually calculating the sought throws are made in order to calculate µ(Ξ, n)’s em- value as a plain product. This is the very purpose of pirical relative deviation (Erd) estimator ρ this Bayesian formulation of our problem. σ(µ(Ξ, n), m) ρ(µ, Ξ, n, m) ≡ (10) K 1 Pm µ(Ξ, n)i Y m i=1 P[∆ ≤ dc] = P[∆ ≤ li|∆ ≤ li−1] (13) as a way to measure its accuracy as the ratio of its i=1 standard deviation over its empirical mean. where l0 = ∞ ≥ l1 ≥ · · · ≥ lK = dc form a de- creasing sequence of thresholds to be defined later. 2. Why CMC can not cope with Hopefully, we have just reformulated our hard to rare events estimate expectation as the product of easy to esti- Using the independence of the Ξi and the fact mate conditional expectations ! 2 that Ξ = Ξ for it is a binary 1 or 0 mapping, one The Adaptive Splitting Technique (AST) is a can write the following easily. way of making this wish come true in an iterative three step way. Start with a sample of iid throws of E[µ(Ξ, n)] = E[Ξ] (11) ∆ known to be under threshold l . With i = 1, this [Ξ](1 − [Ξ]) i [µ(Ξ, n)] = E E (12) is only performing a plain CMC simulation. Then, V n and until the threshold is less than dc, do as follows : If the sought probability E[Ξ] is 10−4, to be ac- curate up to a tenth of the real value i.e.