MISS DISTANCE GENERALIZED VARIANCE NON-CENTRAL CHI DISTRIBUTION. Ken Chan ABSTRACT

Transcription

1 MISS DISTANCE GENERALIZED VARIANCE NON-CENTRAL CI DISTRIBUTION en Chan ABSTRACT In many current practical applications, the probability o collision is oten considered as the most meaningul criterion or determining the risk to a spacecrat. owever, when the miss distance is very small (say m or less), the collision probability may not comortably serve well as the only metric or measuring the risk. In these cases, it is also important to know the statistics o the miss distance distribution, particularly the conidence limits that bound the miss distance. One approach to determine the distribution o the miss distance is to perorm rather time-consuming Monte Carlo simulations on the two conjuncting objects by choosing random initial conditions at the epoch commensurate with the covariance. The orbit propagations must be o very high precision because accuracies down to the meter range are required. The simulated results are then itted with a eibull curve which, unortunately, does not bring out the salient eatures o the distribution. The analysis here ormulates the problem analytically. For this, we consider the miss distance as given by a non-central chi distribution with unequal variances. This method eliminates the need to perorm an inordinate amount o computation, thus reducing the desired results in a timely manner by several orders o magnitude. Moreover, this generalized variance non-central chi distribution possesses the requisite eatures o the miss distance statistics. INTRODUCTION In a short-term encounter between two orbiting objects, the nominal miss distance is determined by propagating their orbits rom initial conditions at some epoch. hen they are in the vicinity o the point o closest approach, the integration steps are smaller and the minimum separation is determined using interpolation and possibly some iteration. owever, this result provides only a nominal minimum separation at the nominal time o closest approach and not a distribution o what the miss distance would be. I we desire this inormation, then we would have to perorm Monte Carlo simulations by choosing random initial conditions at the epoch through the generation o Gaussian random numbers using the covariances associated with their orbit determinations. For each such choice o initial conditions, we would have to repeat the orbit propagations and subsequent minimum separation computations. This is obviously a very timeconsuming process because we need approximately one million runs to obtain a meaningul result. An alternative approach is to propagate the two nominal orbits to some instant o time when they are both in the encounter region. In this region, the rectilinear approximation is valid just as assumed in all the collision probability computations. Thus, with this knowledge o the ingress initial conditions, we may use the analytical methods o projective geometry to determine the minimum separation and also the nominal time when this occurs. ith the ingress covariances, we can then choose seeds to perorm the Monte Carlo simulations. Note that we have eliminated the orbit propagations which orm the bulk o the previous eort. For each choice o ingress initial conditions, we use the analytical ormulation to determine the minimum separation. Note

2 that, even though we do not have to compute the time o occurrence as a requirement, it is not the same or all the sample runs. In this way, we obtain a distribution o miss distances. Still another approach is to ormulate the entire problem analytically. Let us irst recall the simple case o the sum o the squares o independent random variables {X i ; i =,, n}, which are all Gaussian and have zero means and also equal variances. It is well-known [] that this sum, which is a random variable, does not possess a Gaussian probability density unction (pd), but rather has a chi squared distribution. The square root o this sum has a pd which is called a chi distribution. owever, what we need to consider or our application is the case o the square root o the sum o the squares o two Gaussian random variables each with a dierent mean and a dierent variance. This is an example o the non-central chi distribution with unequal variances. Another point to make is the ollowing: I we have two Gaussian random variables with dierent means and dierent variances, then their sum or dierence is also Gaussian. This result is proved in most books on probability and statistics such as Reerence []. owever, what is not readily proved is the case o vectors when their components are Gaussian random variables. (This generalization is obvious when the covariances have principal axes which are parallel; but it is not intuitively obvious when the principal axes are not parallel.) Reerence [] takes up the diicult and laborious task o proving this statement. owever, it can also be proved in a simpler way by using the concepts o aine transormations. Finally, it may be readily proved that the covariance o the sum (or dierence) o two independent Gaussian vectors is equal to the sum o the two individual covariances. For convenience, this sum is called the Combined Covariance. (owever, in contrast, the mean o the sum is the sum o the means and the mean o the dierence is the dierence o the means.) The two results (the dierence between two independent Gaussian vectors being Gaussian, and the covariance o the dierence o two independent Gaussian vectors being the sum o the two individual covariances) are used implicitly in the present day analyses o spacecrat collision probability. Next, let us consider the nominal encounter plane deined at the instant o nominal closest approach as discussed in detail in Reerence [3]. This plane is deined to contain the relative position vector and also be orthogonal to the relative velocity vector when the two orbiting objects are at their nominal minimum separation. e assume that the position uncertainty o each object is described by a three-dimensional Gaussian pd. ence, the uncertainty o their relative position vector also has a three-dimensional Gaussian pd involving the combined covariance, being the sum o the two individual covariances. Let and denote two orthogonal coordinates in the nominal encounter plane and let the origin be at the nominal position o one (Object ) o the two orbiting objects. Let the nominal relative position vector (o Object ) be denoted by (µ, µ ). In general, the pd o the relative position uncertainty in this plane is bivariate Gaussian with an o-diagonal component. owever, or simplicity, we choose the and axes to be along the principal directions. Then, the uncertainties o the two coordinates and have Gaussian pds with standard deviations which we denote as σ and σ. Thus, the pd o the relative position uncertainty is given by an unbiased Gaussian pd as shown in Figure.

3 Figure Miss Distance Approach On the other hand, by invoking the concept o the non-central chi distribution, we have a pd o a random variable chi χ ( + ) deined as the square root o the sum o the squares o two Gaussian variables and. These two random variables are biased rom the origin by the means µ and µ and have standard deviations denoted by σ and σ. This is illustrated in Figure. Figure Chi Distribution Approach 3

4 hile the ormulation to compute the chi distribution requires some eort, this approach circumvents the need to perorm the excessive amount o computation in Monte Carlo simulations and also oers insight into the problem rom an overall perspective. NOTATION AND PRELIMINARIES In conormance with current usage [], we shall denote the Cumulative Distribution Function (CDF), also called the Probability Distribution Function (PDF), by the upper case letter F(.). Moreover, we shall denote the random variable by the upper case Y and denote a value (also called the running variable ) by the lower case y. Furthermore, we shall denote the probability that Y is less than or equal to y by P{Y y}. Thus we have the deinition Y { } F(y) PY y. () In Equation (), we also use the subscript Y in the CDF because we shall be dealing with many random variables and do not want to introduce a dierent symbol such as G or the CDF o another random variable Z. Thus, the CDF or Z would be written as F Z (z) instead o G(z). Next, we shall denote the probability density unction (pd) o a random variable Y by the lower case (.). Thus, we have the relation (y) Y df (y) dy Y =. () Let Z be a random variable with pd Z (z). Let Y be another random variable deined in terms o Z by the relation Y = Z. (3) Then, we may show that, in general, the pd Y (y) or Y is given in terms o Z (z) by [] Y(y) = Z y + Z y y. (4) Equation (4) is valid or any pd. Next, suppose Z is a normal random variable N(,), i.e., a Gaussian random variable with zero mean and unit standard variation. Then, we may show that Equation (4) becomes [] (y) = Y e y/ π y. (5) I the Gaussian random variable Z has mean µ Z and standard variation σ Z, then we may painstakingly go through two changes o variables and show that the corresponding pd is y µ Z σz e Y(y) =. (6) πσ y Z 4

5 It must be noted that Equation (6) is not simply obtained rom (5) by merely substituting ( y µ Z ) or y and substituting σ Z or. e must cautiously go through the process o transorming pd using the standard equation involving the Jacobian or else we obtain the incorrect answer. Except or instances when it is imperative to distinguish between the random variable Y and the running variable y (e.g., as in Equation () or the CDF), we shall break away rom the conventional notation and simply use the upper case. Another compelling reason or this simpliication is that when we use the upper case as a subscript, it is almost indistinguishable rom the lower case in regular size or some letters. Examples are (w), X (x) and Z (z). This practice o not making a distinction between variables is common in calculus in which the integration variable also takes the same notation as the upper limit in indeinite integrals. opeully, this simpliication here will not introduce unnecessary conusion in an attempt to avoid undue complications, when the notation becomes very involved. ith this change, Equation (6) takes the simpliied orm Y µ Z σz e Y(Y) = πyσ Z. (7) TE CI SQUARED CUMULATIVE DISTRIBUTION FUNCTION Consider two orthogonal coordinates and in the encounter plane deined at the nominal instant o closest approach or two objects in a short-term encounter. Since one o the objects is at the origin, thereore the miss distance is given by x = +. (8) e Suppose is Gaussian with mean µ and standard variation σ, and is also Gaussian with mean µ and standard variation σ. Furthermore, assume that and are independent random variables. This means that there is no correlation term in the bivariate pd in the (, )-plane. That is, by this choice o orientation, the and axes are in the directions o the principal axes. Thereore, and are not deined as in Reerence [3] unless the combined covariance aligns with them as the principal axes. For convenience, let us use the notation x e X (9) Z. It ollows rom Equation (7) that we have e = = X (X) ( µ ) σ πσ () 5

6 e = = Z(Z) ( µ ) σ πσ. () ence, Equation (8) takes the orm = X+ Z. () Because X and Z are non-negative and their sum must be equal to, it ollows that the integration with respect to (wrt) one o them, say X, must go rom zero to the value on the line X = Z. The second integration wrt Z must go rom zero to. Thus, the integration wrt to X and Z is perormed over the isosceles triangle bounded by the lines X =, Z = and X + Z =. Thereore, the CDF o is given by X+ Z w ww z X X X+ Z w w z = X Z Z Z { } F (w) = F (x+ z) = P X+ Z w = = (x) (z)dxdz (x) (z)dxdz (x)dx (z)dz. (3) The pd o is given by dierentiation o Equation (3) w w z df (w) d (w) = = X(x)dx Z(z)dz dw dw. (4) To perorm this dierentiation, we need to invoke Leibnitz s Rule (given, or example, in Reerence [4]) or dierentiation o integrals whose integrand and limits o integration involve a parameter: I(u) b(u) a(u) b(u) a(u) g(y, u)dy di(u) db(u) da(u) = g(y, u)dy g( b(u), u) g( a(u), u ). du + u du du (5) It is noted that F (w) in Equation (3) contains a double integral with two levels o upper integration limits involving the parameter w. Thus, i we apply Equation (4) to obtain the derivative o F (w), we would have to perorm two consecutive operations with Leibnitz s Rule. Even though the last term on the RS o Equation (5) vanishes because a(u) =, still there is much work involved. e shall next cast Equation (3) or computing the CDF in a slightly dierent orm. Instead o perorming the integration in terms o the variables X and Z over the isosceles triangle, we shall obtain the same result by integrating in terms o the variables and over a circle o radius centered at the origin. In this case, we use the Gaussian pds or and given by 6

7 ( µ ) σ = e πσ (6) ( µ ) σ = e. πσ (7) Thereore, we have F () = e e d d πσ σ e recall that the error unction is deined by [3] ( µ ) σ ( µ ) σ. (8) x t er (x) = e dt π. (9) Then, Equation (8) may be written as where ( µ ) σ F () = e er Z + er Z d () πσ Z µ σ Z +µ σ. () In turn, Equation () may be written as ( µ ) σ πσ F () = e er Z + er Z d ( µ ) σ + e er ( Z) + er ( Z) d πσ ( µ ) σ ( +µ ) σ = e + e er Z + er Z d. πσ () in which, or convenience, we have used the random variable instead o the running variable w. Unlike Equation (3), it is noted that Equation () expresses the CDF F () as a single integral. e note that, by deinition, we have { } F w P w { } F w P w. (3) 7

8 It ollows that F w F w. (4) A word o caution is that the unctional orm on both sides o this equation are dierent because the subscripts are dierent. That is, F (.) F (.). e may prove this result more laboriously by perorming the actual substitutions into the equation or F (w) as ollows w F w w dw. (5) I we change the variable rom w to w, then the new pd w ( w) is given by dw = = = = w w w w. (6) w w w w d w ence, we obtain the desired result: w F w w dw w w = = w w d w w w = w d w F w (7) TE CI PROBABILITY DENSITY FUNCTION e shall next obtain the Chi distribution or the generalized variance non-central Chi distribution. Let us re-write the CDF in Equation () compactly as where F () = D α(, ) β(, )d. (8) D πσ α() e + e ( µ ) σ ( +µ ) σ β(,) er Z + er Z Z µ σ Z +µ σ. (9) Thereore, rom Equation (4), the pd o the Chi squared distribution is given by 8

9 () df () d = (3) in which, or convenience, we have used the random variable instead o the running variable w. By applying Leibnitz s Rule given by Equation (5), we obtain e note that (,) β d () = D α d+ D α β(,) +. (3) d = ( µ ) σ ( +µ ) σ µ µ α β(,) = e e er er = + + =. (3) σ σ ith some diligence, we can show that er Z i e Z = i or i = and Zi π Zi = or i = and. σ (33) It ollows that we have i= β er Zi Z = i Z i = e + e. Z Z πσ (34) ence, Equation (3) becomes ( µ ) σ ( +µ ) σ Z Z. (35) () = e + e e + e d 4πσ σ Since, thereore we can introduce a new variable θ π/ deined by Thereore, we have sin θ. (36) 9

10 d cosθdθ = sin θ = cos θ. (37) By substituting Equations (36) and (37) into (35), we obtain () π/ ( sinθ µ ) σ ( sinθ+µ ) σ ( cosθ µ ) σ ( cosθ+µ ) σ (38) = e + e e + e d θ. 4πσ σ This is the expression or computing the pd Chi squared distribution or the miss distance, since = x e by Equation (9). It is noted that the upper limit o integration is π/. The random variable appears only in the integrand. Even though the RS o Equation (38) involves (and not ), this is not an obstacle because is an explicit unction o. I we wish to plot the pd o the Chi squared distribution, note that the horizontal axis is and not. Next, i we wish to obtain the pd o the Chi distribution, we may invoke Equation (4) and dierentiate the RS wrt. That is, we use the ollowing analog o Equation (3) df =. (39) d owever, we can circumvent this process by using the relation (6), since we already have the pd o the Chi squared distribution given by Equation (38). Thus, we have π/ ( sinθ µ ) σ ( sinθ+µ ) σ ( cosθ µ ) σ ( cosθ+µ ) σ (4) = e + e e + e d θ. πσ σ Note that the RS o Equation (4) conveniently involves (and not ) and this greatly simpliies the display o the results. I we wish to plot the pd o the Chi distribution, note that the horizontal axis is and not. It may be shown that or the special case o zero means and equal standard deviations, we obtain the Rayleigh pd rom Equation (4). Similarly, or the case o non-zero means and equal standard deviations, we obtain the Rician pd. Appendix A gives a reormulation o Equation (38) in an alternative orm in which the expression involves and not. DISCUSSION Thus ar, we have strictly considered how to reormulate the pd o the miss distance in terms o the non-central chi distribution, but we have not justiied to how this yields the answer obtained rom the Monte Carlo simulations mentioned in the Introduction. There, we stated that when we

11 are in the encounter region [3], the velocities v and v o the two objects can each be treated as constant so that their motions are rectilinear and their relative motion is also rectilinear. ence, the relative velocity vector v = v v is constant in direction. Next, consider pairs o seeds or the Monte Carlo simulations chosen according to the Gaussian nature o the three-dimensional individual covariances o the two objects at ingress into the encounter region. Since the encounter plane or each pair o seeds is perpendicular to this normal vector v, it ollows that the encounter planes or all such pairs o seeds are parallel to the encounter plane or the nominal case. Moreover, because the transit time through the encounter region is very short (o the order o seconds or low orbiting objects (LEOs), the covariances o the two objects while in the encounter region can also be considered as constant. Thus, the combined covariance (being the sum o the two covariances) is constant in size, shape and orientation. For the nominal case described in the Introduction, one o the two orbiting objects is at the origin o this combined covariance or the uncertainties o the relative position vector o one object with respect to the other. That is, we have (See Figure ) + σ σ (,) = e. (4) πσ σ Let the encounter plane obtained or each pair o seeds be displaced rom the nominal one by a distance N along the direction o the normal vector v. In the (,, N) coordinate system, let the combined covariance C describing the relative position uncertainties be written as σ ρ σ σ ρ σ σ C = ρ σ σ σ ρ σ σ ρ σ σ ρ σ σ σ N N N N N N N N N. (4) From this, we obtain the inverse matrix C - and the determinant C. Then, we obtain the ollowing pd ( ρn ) ( ρn ) ( ρ )N + + σ σ σn 3(,, N) = Aexp R ( ρ ρnρn )(43) ( ρn ρρn )N ( ρn ρρn )N σσ σσn σσn where A and R are given by A = ( π) 3 Rσ σ σ N ( N N N N ) R = + ρ ρ ρ ρ ρ ρ. (44) Equation (43) gives the pd o the uncertainty o Object considering that Object is at its nominal position. Thereore, Equation (43) also gives the uncertainty o the seed or Object assuming that the seed or Object is at its relative position. Since, at the instant o closest

12 approach, these two seeds lie on their own encounter plane (by deinition), thereore Equation (43) also gives the uncertainty in the position o this encounter plane. Thereore, the probability dp o inding an encounter plane or the pair o seeds at a distance between N and (N + dn) rom the nominal encounter plane is given by dp = (,, N) dn. (45) 3 Consequently, i we choose Gaussian random numbers or the seeds individually according to their own covariances at ingress, the pd or the relative position vector is obtained by integrating Equation (43) with respect to N rom to +. Thus, in eect, we are obtaining the general marginal two-dimensional pd. This has already been done in Reerence [3] and the result is + ρ + ( ρ ) σ σ σ σ (,) 3(,, N)dN = e. (46) πσ σ ρ In our case, by choosing and to be along the principal directions, the correlation coeicient ρ is zero. Thereore, we obtain the pd given in Equation (4). ence, in the limit o a large number o runs, the pd o the uncertainty o the relative position vector obtained rom the Monte Carlo simulations is precisely the same as that in the nominal encounter plane as given by Equation (4). Thereore, or a given set o mean values (µ, µ ) and unequal variances (σ, σ ), the pd or the miss distance is given by the non-central chi distribution we have obtained previously. Note that not all the conjunctions o the pairs o seeds occur at the same place and at the same time. The assumptions o constant velocities and time-independent combined covariance have been crucial in obtaining the apparently obvious result above. owever, the reasoning leading to this result is rather subtle. It would have been allacious to use the ollowing argument: For each pair o seeds, we obtain an encounter plane with the pd given by Equation (4) because we have chosen the center o the combined pd to be at one o the two orbiting objects. Thereore, the pd o the relative position vector obtained rom the Monte Carlo simulations is the same as that in the nominal encounter plane as given by Equation (4). There is an error in logic in this argument: It is implicitly assumed that each seed has a covariance centered on itsel; and this is not so. In retrospect, one could ask the ollowing question: e were dealing with the three-dimensional (,, N)-space, why didn t we consider the random variable chi χ ( + + N ) deined as the square root o the sum o the squares o three Gaussian variables, and N? The reason was that we were not dealing with the pd o the unrestricted radial distance rom the origin (Generalize Figure to three dimensions.) The problem o concern was to obtain the distribution o the separation between two objects at the point o closest approach, i we perorm a large number o simulation runs using random seed inputs chosen in accordance with two threedimensional Gaussian pds, one or each seed. e established that and showed how this distribution o miss distance is obtained rom the non-central chi distribution with unequal variances involving two (not three) degrees o reedom.

13 NUMERICAL RESULTS Most o our attention has been directed to obtaining Equation (4) or the pd o the random variable χ ( + ), because that was what we sought or the distribution o the miss distance. An example o a pd Chi distribution is shown in Figure 3 or the sample case o σ =.5 km, σ =. km, µ =.5 km, µ =. km. Figure 3 Example o Probability Density Function o Miss Distance It turns out, however, that the more useul representation or our application is actually the Cumulative Distribution Function o the Chi random variable χ ( + ), because it gives the probability the miss distance does not exceed some value. Furthermore, the inormation contained in this CDF o χ is exactly the same as that contained in the CDF o the Chi Squared random variable χ +, because the latter gives the probability the miss distance squared does not exceed the square o that value. Figure 4 shows the CDF o χ or the same sample case as that in Figure 3. An example application is the determination o the probability or which the miss distance does not exceed a certain speciied value, say α. For this, we need to ind the area under the curve in Figure 3 rom to α. owever, we can obtain the same inormation by merely obtaining the probability corresponding to α in the curve in Figure 4. Using this approach, we can ind the lower and upper values o the miss distance, excluding the probability below.5 and the probability above.975, so as to give us a 95% conidence interval. Figure 5 depicts these limits or the nominal miss distance obtained rom Two Line Element sets or the ive days preceding the collision o Iridium 33 and osmos 5 on February, 9. 3

14 Figure 4 Example o Cumulative Distribution Function o Miss Distance Squared -3 Iridium-osmos Collision Close Approach Statistics -4 Probability Miss Distance (km) Elset (days beore conjunction) Figure 5 Miss Distance and Collision Probability or the Iridium 33/osmos 5 Collision 4

15 Finally, it remains to veriy that the miss distance distributions obtained analytically agree with those obtained rom the Monte Carlo simulations. A number o plots comparing these results showing good agreement have been obtained. Figure 6 illustrates one particular case o this comparison. Figure 6 Comparison o Statistical Miss Distance Distributions CONCLUSIONS () The analytical distributions agree very well with those obtained rom Monte Carlo simulations. By the use o the pd Chi distribution or the CDF Chi Squared distribution, we have circumvented the direct approach utilizing time-consuming high precision orbit propagation to obtain miss distances when they are very small (o less than m). () The Chi distribution applicable to our analysis is the one involving two degrees o reedom, not that involving three degrees o reedom, even though we have three-dimensional covariances associated with each o the two orbiting objects. REFERENCES [] Sheldon Ross, A First Course in Probability, Prentice all, p 3,, 55, 8 th ed (). [] T.. Anderson, An Introduction to Multivariate Statistical Analysis, John iley & Sons, pp 3-33 (3). [3] F. enneth Chan, Spacecrat Collision Probability, The Aerospace /AIAA Presses, pp 7-37, 47-6, 39 (8). [4] Philip Franklin, Methods o Advanced Calculus, McGraw-ill, pp 5-5 (944). 5

16 APPENDIX A: ALTERNATIVE FORM OF EQUATION (38) By multiplying out the squares in the exponents in the integrand in Equation (38), we may write this equation as () = 4 πσ σ π/ µ ( σ +µ σ ) e ( sin θ σ ) + cos θ σ µ sin θ σ µ sin θ σ µ cosθ σ µ cosθ σ e e e e e + + dθ. (A) µ ( σ +µ σ ) = e πσ σ π/ ( sin θ σ + cos θ σ ) µ sin θ µ cosθ e cosh cosh σ d θ σ For convenience, let us introduce the ollowing notation µ sin θ µ cosθ x = and y =. (A) σ σ Then, we have cosh x cosh y e e e e 4 x y x y x y x y ( e + e e + e = ) 4 = cosh ( x + y ) + cosh ( x y ). x x y y = ( + )( + ) (A3) Thereore, Equation (A) may be written as () = πσ σ π/ µ ( σ +µ σ ) e ( sin θ σ + cos θ σ ) e cosh( x+ y) + cosh( x y) d θ. (A4) In view o the expansions 4 6 (x + y) (x + y) (x + y) cosh( x + y) = ! 4! 6! 4 6 (x y) (x y) (x y) cosh( x y) = ,! 4! 6! (A5) 6

17 we obtain ( + ) + ( ) cosh x y cosh x y (x + y ) (x + 6x y + y ) (x + 5x y + 5x y + y ) = ! 4! 6! (A6) e note that the RS o Equation (A6) involves x and y to the even powers only. This means that occurs in integral powers only in the integrand o Equation (A4). That is, there are actually no terms with hal powers in in the pd o the Chi squared distribution () Equation (38). Equation (A4) oers an alternative way to compute the pd o the Chi squared distribution. 7