Collision Probability with Gaussian Mixture Orbit ...

Collision Probability with Gaussian Mixture Orbit Uncertainty Kyle J. DeMars∗ Missouri University of Science and Technology, Rolla, MO 65409

Yang Cheng† Mississippi State University, Mississippi State, MS 39762

Moriba K. Jah‡ Air Force Research Laboratory, Kirtland AFB, NM 87117

I.

Introduction

In view of the high value of space assets and a growing space debris population increasingly caused by random collisions1 in a congested space environment, collision analysis is of great significance. Collisions between space objects (spacecraft and space debris) can at best be determined in a probabilistic manner because of the lack of perfect knowledge about the parameters and motions of the space objects. Collision analysis based on the nominal orbital parameters without taking into account the uncertainty in those parameters or in the orbit motions is inaccurate. The probability of collision between two space objects provides a quantitative measure of the likelihood that the space objects will collide with each other. Collision probability is closely related to close approaches,2, 3 collision detection,4–6 and conflict probability7 and has been derived for low Earth orbits (including the International Space Station),8–13 ∗

Assistant Professor, Department of Mechanical and Aerospace Engineering. Email: [email protected]. Senior Member AIAA. † Assistant Professor, Department of Aerospace Engineering. Email: [email protected]. Associate Fellow AIAA. ‡ Senior Research Engineer. Associate Fellow AIAA. 1 of 17

geosynchronous orbits,14, 15 spacecraft formations,16, 17 and other orbiting objects.18 Except for the early work based on the kinetic theory of gases and the Poisson model,18, 19 collision probabilities are derived from probabilistic models of space object relative motion parameters.8–17, 20–36 The derivation of collision probability is based on many assumptions. For example, space objects are usually modeled as spheres,20 although other shape models have also been used.2, 26, 29, 30, 34 The spherical model does not require the space object orientation information, which is unavailable in many cases, and therefore greatly simplifies the derivation. A large body of work on collision probability is focused on short-term encounters,8, 9, 11, 22, 24, 25, 27, 31–33 in which linear relative motion, constant Gaussian relative position uncertainty, and zero relative velocity uncertainty are assumed. These assumptions can be justified by the high relative speed and short encounter time in short-term encounters. For scenarios with low relative velocities or long-term encounters, the importance of including nonlinear relative motion and non-zero relative velocity uncertainty has been shown.16, 17, 21, 23, 28, 34 Monte Carlo simulations have been used to compute the collision probability.17, 25, 30, 35 The Monte Carlo analysis is reliable and applicable to both short-term and long-term encounters; it is computationally expensive, however, especially when the collision probability is small and accurate values of the collision probability are needed. In addition to the Monte Carlo method, two deterministic methods have been used to derive the collision probability formulas. For the sake of simplicity, the brief review of the two methods below assumes that the two space objects are spheres and that there is no relative velocity uncertainty. A collision between two spherical objects occurs when the distance between the centers of the spheres is less than the sum of the radii of the spheres. The collision probability is determined by analysis of the relationship between the hard ball attached to one object and the probability distribution of the relative position centered at the other object, where the hard ball is a sphere of radius equal to the sum of the radii of the two spheres. The first method is based on the volume swept out by the hard ball, whose dynamics relative to the other object are governed by the relative motion model.8, 13, 14, 20–22, 24, 26, 27, 29, 32–34, 36 The collision probability is given by the expectation of the swept-out volume with respect to the relative position probability distribution. For short-term encounters with linear relative motion, the swept-out volume is approximately a cylinder. The collision probability can then be more conveniently determined as a two-dimensional integral after the swept-out volume of the hard ball and the relative position probability distribution (visualized as a three-dimensional ellipsoid for Gaussian relative position probability distribution) are projected at the time of closest approach onto the collision plane, which is perpendicular to the relative velocity vector. The second method, also known as the direct method, is based on the influx of the 2 of 17

relative position probability distribution into the hard ball,11, 28, 31 which is assumed to be stationary. If the relative position probability distribution is treated as a mass distribution with the total mass being unity, the collision probability is the amount of mass that enters the hard ball, which can be obtained from spherical integrals. The two methods can yield the same collision probability formulas for short-term encounters, but the direct method includes the relative velocity uncertainty more easily and is more appropriate for general probability distributions. Recent work in statistical inference as applied to space surveillance tracking has demonstrated that not all space object uncertainties are well represented by a Gaussian distributions. The non-Gaussian behavior has been shown to occur in problems that require long arcs of propagation,37–40 as well as problems for which insufficient data has been collected on a space object, i.e. initial orbit determination.41, 42 Of specific interest are the developments which make use of a Gaussian mixture parameterization of the space object orbit uncertainty representation, which motivates the desire to develop a method for the computation of the collision probability using a Gaussian mixture representation of uncertainty. The Gaussian mixture representation has been used in a recent extension of linear conjunction analysis that combines analytic formulae for short-term encounters with a Gaussian mixture to improve the calculation of the probability of collision.43 This Note derives the collision probability between space objects from first principles in the presence of large state uncertainties and nonlinear relative motion using the direct method. The large non-Gaussian uncertainty is modeled as a time-varying Gaussian mixture,44 and the developed method is applicable to both short-term and long-term encounters.

II.

Definition of Collision Probability

The probability of collision between two space objects is determined from the hard ball (combined object) and the joint probability density function (pdf) p(xt ) of the relative position rt and relative velocity vt , with xTt = [rTt vtT ], rt = r(t), vt = v(t), xt = x(t), where t denotes time. The hard ball is assumed to be of fixed shape, size, and orientation and located at the origin in the relative position space. Here, the hard ball is not necessarily a sphere. The evolution of the joint pdf p(xt ) is determined by the motions of the space objects, but there is no measurement update of p(xt ) to add new information on their motions. The rotational motions of the space objects may have an indirect effect on the orientation of the hard ball or p(xt ) but these do not appear in the collision probability derivation. The two space objects are said to collide at time t if rt ∈ V , where V is the volume of the hard ball. If the hard ball is a sphere with radius R, rt ∈ V becomes krt k ≤ R. The

collision probability Pc (t) at time t is the probability that a collision occurs by time t. It is 3 of 17

a non-decreasing function in time t, with Pc (−∞) = 0, Pc (∞) ≤ 1. The collision probability Pc (t) is related to the collision probability rate pc (t) by t

Z

Pc (t) =

pc (τ )dτ .

(1)

−∞

The collision probability rate pc (t) is the instantaneous increase in collision probability at t. The origin of time may be chosen such that Pc (0) = 0 for practical purposes. Then, Pc (t) =

t

Z

pc (τ )dτ .

(2)

0

Using the direct method for collision probability,11, 28, 31 pc (t) is defined as pc (t) =

Z Z S

vn ≤0

|vn | p(xt ) dvt dS ,

(3)

ˆ , S denotes the total surface area of the hard ball, dS denotes the surface where vn = vt · n ˆ denotes the unit normal vector to the surface area element dS. The area element, and n inequality vn ≤ 0 corresponds to the influx. The computation of Pc (t) from pc (t) is valid when p(xt ) flows through the surface S only once by t.28 Accounting for multiple hard-ball crossings is difficult using either the direct method or the swept-volume based method.14 From p(xt ) = p(vt |rt ) p(rt ), Eq. (3) can be written as pc (t) =

Z Z S

vn ≤0



|vn | p(vt |rt ) dvt p(rt ) dS .

(4)

Given p(xt ), the collision probability rate and collision probability can be computed from Eq. (4) and Eq. (1), respectively. In this Note, the collision probability formula is derived assuming that p(xt ) is a Gaussian mixture. Before proceeding to the next section, a quantity related to Pc (t) is briefly discussed. This related quantity is given by Pc′(t)

=

Z

p(rt ) dV ,

V

where V is the volume of the hard ball and dV is a volume element. Because the integral for Pc′ (t) does not include velocity, Pc′ (t) can be computed more easily than Pc (t). By analogy to mass conservation, that is, the mass increase equals the “mass in” minus the “mass out,”

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it follows that Pc′(t2 )

−

Pc′(t1 )

Z

=

t2

pc (t)dt −

t1

Z

t2

t1

Z Z S

vn >0



|vn | p(xt ) dvt dS dt ,

where the last term is the outflux. This equation shows that Pc (t) is not equivalent to Pc′ (t) in general and the time integration in Eq. (1) or (2) is essential. When the outflux is approximately zero, that is, t2

Z

t1

Z Z S

vn >0



|vn | p(xt ) dvt dS dt ≈ 0 ,

it is seen that Pc (t) ≈ Pc′ (t) for t ∈ [t1 , t2 ]. For example, if t1 = −∞, t2 = 0 and the outflux is approximately zero during (−∞, 0], then Pc (0) =

Z

0

pc (t)dt =

−∞

III.

Z

p(r0 ) dV . V

Gaussian Mixture Computation of Collision Probability

Let xa (t) = xa,t and xb (t) = xb,t represent the time-varying state vectors of two space objects, A and B, respectively, and x(t) = xt be the relative state between objects A and B, such that xt = xb,t − xa,t . Given the joint pdf of the states of objects A and B as p(xa,t , xb,t ) = p(xa,t , xa,t + xt ), the pdf of the relative state xt is given by marginalization over xa,t , yielding the relationship p(xt ) =

Z

p(xa,t , xb,t )dxa,t =

Z

p(xa,t , xa,t + xt )dxa,t .

(5)

Now, let the pdfs for the state of object A and the state of object B be given by Gaussian mixture pdfs: p(xa,t ) = p(xb,t ) =

La X

i=1 Lb X

(i)

(i)

(i)

(6a)

(j)

(j)

(j)

(6b)

wa,t pg (xa,t ; ma,t , Pa,t ) wb,t pg (xb,t ; mb,t , Pb,t ) ,

j=1

where the Gaussian pdf is −1/2

pg (x ; m, P) = |2πP|



1 exp − (x − m)T P−1 (x − m) 2

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.

Under the assumption that the states for object A and object B are independent, their joint pdf becomes a product of the individual state pdfs, such that p(xa,t , xb,t ) = p(xa,t )p(xb,t ) which, after substitution of Eqs. (6), may be written as p(xa,t )p(xb,t ) =

Lb La X X

(i)

(j)

(i)

(i)

(j)

(j)

wa,t wb,t pg (xa,t ; ma,t , Pa,t )pg (xa,t + xt ; mb,t , Pb,t ) ,

(7)

i=1 j=1

where xb,t = xa,t + xt has been used in the state pdf for object B. In Ref. [28], Coppola showed that Z pg (xa,t ; ma,t , Pa,t )pg (xb,t + xt ; mb,t , Pb,t)dxa,t = pg (xt ; µt , Σt ) , (8) where µt = ma,t − mb,t and Σt = Pa,t + Pb,t . Substituting Eq. (7) into Eq. (5) and applying Eq. (8) to each term within the resultant summation, it follows that when the state distributions for object A and object B are given by Gaussian mixture pdfs, the distribution of the relative state between objects A and B is p(xt ) =

Lb La X X

wa,t wb,t pg (xt ; µt , Σt ) ,

(ij)

(i)

(i)

(j)

(ij)

(ij)

(9)

i=1 j=1

(ij)

(i)

(j)

(j)

where µt = ma,t − mb,t and Σt = Pa,t + Pb,t . Equation (9) could be substituted into Eq. (3) in order to determine the collision probability rate, but since integration is performed with respect to vt , it is beneficial to first decompose each element of the Gaussian mixture in Eq. (9) into a product of a pdf in the (ij) relative position and a pdf in the conditional relative velocity. That is, by partitioning µt (ij)

and Σt

as (ij)

µt



=

(ij) µr,t (ij) µv,t



(ij)

and



Σt



=

(ij) Σr,t (ij) Σvr,t

(ij) Σrv,t (ij) Σv,t



,

it can be shown that the Gaussian component decomposes into a product as (ij)

(ij)

(ij)

(ij)

(ij)

(ij)

pg (xt ; µt , Σt ) = pg (rt ; µr,t , Σr,t )pg (vt′ ; (µv,t )′ , (Σv,t )′ ) , (ij)

(ij)

(ij)

(ij)

(ij)

(ij)

(ij)

(10) (ij)

(ij)

where vt′ = vt − Σvr,t (Σr,t )−1 rt , (µv,t )′ = µv,t − Σvr,t (Σr,t )−1 µr,t , and (Σv,t )′ = Σv,t − (ij) (ij) (ij) Σvr,t (Σr,t )−1 Σrv,t . Applying Eq. (10) to each component of the Gaussian mixture in Eq. (9),

6 of 17

it follows that p(xt ) may be written as p(xt ) =

Lb La X X

(i)

(j)

(ij)

(ij)

(ij)

(ij)

wa,t wb,t pg (rt ; µr,t , Σr,t )pg (vt′ ; (µv,t )′ , (Σv,t )′ ) .

(11)

i=1 j=1

Substituting Eq. (11) into Eq. (3), it is straightforward to show that the collision probability rate becomes pc (t) =

Lb La X X

(i) (j) wa,t wb,t

i=1 j=1

Z

(ij)

(ij)

pg (rt ; µr,t , Σr,t )ν(rt )dS,

S

where ν(rt ) is defined to be28 ν(rt ) = (ij)

and vn

(ij)

Z

(ij)

(ij) vn ≤0

(ij)

|vn(ij) | pg (vt′ ; (µv,t )′ , (Σv,t )′ )dvt′

(ij)

ˆ+n ˆ T Σvr,t (Σr,t )−1 rt . = vt′ · n

Now, assume that the hard ball for which the collision probability is computed is taken to be a sphere of radius R, such that rt = Rˆ n and dS = R2 cos θdθdφ with θ and φ representing spherical angles. Under this condition, the collision probability rate may be written as pc (t) = R

2

Lb La X X

(i) (j) wa,t wb,t

i=1 j=1

Z

2π

0

Z

π/2

(ij)

(ij)

pg (Rˆ n ; µr,t , Σr,t )ν(ˆ n) cos θdθdφ ,

(12)

−π/2

and ν(ˆ n) can be found to be given by28      ν0 (ˆ n) ν0 (ˆ n) σ(ˆ n) ν02 (ˆ n) √ − 1 − erf ν(ˆ n) = √ exp − 2 , 2σ (ˆ n) 2 2π σ(ˆ n) 2 where ν0 (ˆ n) and σ 2 (ˆ n) are given by i h  (ij) (ij) (ij) (ij) ˆ T µv,t + Σvr,t (Σr,t )−1 Rˆ ν0 (ˆ n) = n n − µr,t   (ij) (ij) (ij) −1 (ij) 2 T ˆ. ˆ Σv,t − Σvr,t (Σr,t ) Σrv,t n σ (ˆ n) = n Thus, integration over the relative velocity can be performed analytically. The remaining integrals over θ and φ represent the integration over the relative position under the assumption that hard ball is represented by a sphere of radius R. These integrals must be evaluated numerically for each element within the summation using a quadrature approach such as Lebedev’s method.45 Lebedev’s method is a numerical quadrature approach which approximates the surface integral of a function over a three-dimensional sphere by a finite single 7 of 17

summation. It is preferable to applying univariate quadrature over the two spherical angles as it replaces the double integral in Eq. (12) with a single summation as opposed to the double summation that results from univariate quadrature over the spherical angles. Lebedev’s ˆ k with associated weights wk , such that method generates a set of N quadrature points n pc (t) = R2

Lb La X X

(i)

(j)

wa,t wb,t

i=1 j=1

N X

(ij)

(ij)

wk pg (Rˆ nk ; µr,t , Σr,t )ν(ˆ nk ) .

(13)

k=1

Once the quadrature rule has been applied, the collision probability rate can be calculated at any given time and substituted into Eq. (1). Again, numerical integration techniques must be levied in order to compute the collision probability. By computing the collision probability rate at regular time intervals, the collision probability can be readily calculated using methods such as Newton-Cotes quadrature rules.46

IV.

Simulation Results

To demonstrate the application of the developed Gaussian mixture formulation for the computation of the probability of collision, a case involving a well-tracked object (taken to be object A) and an object that has just been observed for the first time (taken to be object B) is considered. Object A (the well-tracked object) has an uncertainty that is characterized by a Gaussian distribution, that is La = 1 in Eq. (6a). The mean of the Gaussian distribution at t = t0 is taken to be mTa,0 = [mTr,0 mTv,0 ], where 

40359.456070



   [km] mr,0 =  −10885.142726   16.372797

and



717.361085



   [m/s] mv,0 =  2980.939711   −4.322683

Furthermore, the covariance of the Gaussian distribution is taken to be diagonal with position variance values of (100 [m])2 and velocity variance values of (200 [mm/s])2 . These values for the covariance are consistent with the results obtained in Ref. [42] for ground-based tracking of objects in near-geosynchronous orbits. Object B, on the other hand, is assumed to have been observed only one time, such that it is not possible to represent its state uncertainty using a Gaussian distribution. Instead, a Gaussian mixture distribution is constructed using an uncertainty matching method42 based off of the single measurement of the line-of-sight and the line-of-sight rate (i.e. right ascension, declination, right ascension rate, and declination rate) of an object with respect ˙ to a ground-based observer. This measurement is taken at t0 , and is given by y0T = [α δ α˙ δ],

8 of 17

where α = 1250795.486 [arcsec] , α˙ = 15.100466 [arcsec/s] ,

δ = −9403.750 [arcsec] , and δ˙ = 0.004452 [arcsec/s] ,

and the associated measurement noise covariance, R0 , is taken to be diagonal with angular variance values of (0.40 [arcsec])2 and angular rate variance values of (0.07 [arcsec/s])2 . Additionally, the observing station which generated the measurement has an inertial position of robs = [xobs yobs zobs ]T , where xobs = Re cos φ cos λ, yobs = Re cos φ sin λ, and zobs = Re sin φ, where Re is the radius of the Earth, φ = 15 [deg], and λ = −30 [deg]. The inertial

velocity of the observer can be computed via vobs = ω × robs where ω is the angular velocity vector of the Earth. Given the observation y0 , the region of range and range-rate values that lead to orbits with negative orbital energy is constructed (shown by the light gray region in Fig. 1(a)). This region is called the admissible region. To the admissible region, constraints on the maximum semi-major axis and the maximum eccentricity of the considered orbits are added (shown as the black and dark gray lines in Fig. 1(a), respectively). For this problem, the maximum values are taken to be amax = 45000 [km] and emax = 0.1. The intersection of the regions satisfying the constraints is called the constrained admissible region, and is shown as the darker region in Fig. 1(b). A zoomed-in version is shown in Fig. 1(c). From a probabilistic perspective, the constrained admissible region represents a bivariate uniform distribution in range and range-rate. Therefore, given the constrained admissible region, a Gaussian mixture approximation to the uniform distribution is constructed, yielding pρ,ρ˙ (ρ, ρ) ˙ =

Lb X

(j)

(j)

(j)

wρ,ρ˙ pg (ρ, ρ˙ ; mρ,ρ˙ , Pρ,ρ˙ ) ,

(14)

j=1

(j)

(j)

(j)

where wρ,ρ˙ , mρ,ρ˙ , and Pρ,ρ˙ represent the weights, means, and covariances of an Lb -component Gaussian mixture distribution that approximates the uniform distribution of the constrained admissible region shown in Figs. 1(b) and 1(c). The number of components in the Gaussian mixture of Eq. (14), Lb , is controlled by accuracy parameters σρ,max and σρ,max , which can be ˙ used to limit the size of the covariance matrices at the expense of adding components to the mixture. For this problem, the accuracy parameters are set to 5 [km] and 100 [m/s], respectively, which gives Lb = 1259. More details regarding the construction of the constrained admissible region and the Gaussian mixture distribution shown in Eq. (14) can be found in Ref. [42]. Since Eq. (14) only represents the distribution in the range/range-rate space, the observed 9 of 17

E =0 a = amax e = emax

Range-Rate [km/s]

6 4 2 0

−2

−4 −6 2

3

4

5

6

7

Range [ER]

(a) Admissible region with curves of constant semi-major axis and eccentricity 1 0.8

4

Range-Rate [km/s]

Range-Rate [km/s]

6

2 0

0.6 0.4 0.2 0

−0.2

−2

−0.4

−4

−0.6

−0.8 −6 2

3

4

5

6

−1 5.3

7

Range [ER]

5.4

5.5

5.6

5.7

5.8

5.9

Range [ER]

(b) Constrained admissible region (shown as the (c) Zoomed-in view of the constrained admissible darker region) as compared to the admissible re- region gion Figure 1. The region of admissible range/range-rate values given angle and angle-rate data. Units of [ER] represent Earth radii.

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data, y0 , and the measurement noise, R0 , are coupled with pρ,ρ˙ (ρ, ρ) ˙ under the assumption that the range and range-rate are not correlated with y0 to formulate the state distribution in range/right ascension/declination coordinates as p(˜ xb,0 ) =

Lb X

(j) (j) ˜ (j) ˜ b,0 , P wb,0 pg (˜ xb,0 ; m b,0 ) ,

(15)

j=1

where (j)

(j)

wb,0 = wρ,ρ˙ ,

(j)



˜ b,0 =  m



y0  , (j) mρ,ρ˙





˜ (j) =  R0 0  . and P b,0 (j) 0 Pρ,ρ˙

To obtain the distribution in Cartesian coordinates, the mean and covariance of each component of the Gaussian mixture in Eq. (15) is mapped from range/right ascension/declination coordinates to Cartesian coordinates using an unscented transform,47, 48 which gives the distribution in Cartesian coordinates as p(xb,0 ) =

Lb X

(j)

(j)

(j)

wb,0 pg (xb,0 ; mb,0 , Pb,0 ) .

(16)

j=1

The developed method for computing the probability of collision requires the time history of the Gaussian mixture distributions of object A and object B (in this example, object A is described by a 1-component Gaussian mixture distribution). The propagation stage of a Gaussian mixture unscented Kalman filter37, 42 is applied to the initial Gaussian distribution of object A and to the initial Gaussian mixture distribution of object B to determine the time history of the weights, means, and covariances for each component of the Gaussian mixture distributions representing objects A and B. For this problem, a time step of 3 seconds and a final time of 40 minutes were used. At each time step, the Gaussian mixture distributions for objects A and B are then used to compute the collision probability rate using Eq. (13) and a hard ball of radius R = 100 [m]. The time history of the collision probability rate is shown in Fig. 2, wherein it can be seen that the collision probability rate peaks at about 3 minutes past t0 and returns to near-zero at about 25 minutes past t0 . The collision probability rate is then numerically integrated with a trapezoidal method to obtain the collision probability, which is plotted against time in Fig. 3. From Fig. 3, it is seen that the probability of collision is a non-decreasing function in time that has the most increase at the point where the collision probability rate peaks. After the collision probability rate returns to zero, the probability of collision settles to a final value of Pc = 1.503 × 10−5 . 11 of 17

Collision Probability Rate, pc (t)

×10−8 7 6 5 4 3 2 1 0 0

5

10

15

20

25

30

35

40

Time Past Epoch [min]

Figure 2. The collision probability rate computed using Eq. (13) and a hard ball of radius R = 100 [m].

×10−5 1.6

Collision Probability, Pc (t)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15

20

25

30

35

40

Time Past Epoch [min]

Figure 3. The collision probability as a function of time computed using a trapezoidal method for performing the time integration in Eq. (1).

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To verify the result obtained by application of Eq. (13) coupled with a trapezoidal method for performing the time integration in Eq. (1), a Monte Carlo simulation is performed. Samples are drawn from the distributions p(xa,0 ) and p(xb,0 ) and then propagated over the collision interval. For each case where there is a time at which the position difference between the two propagated samples is smaller than the hard ball radius, a collision event is recorded. The collision probability is then computed as the total number of recorded collision events divided by the total number of Monte Carlo trials. For this study, 25 × 106 trials were conducted where each object’s position was calculated for a duration of 60 minutes at a time step of 0.1 seconds. The probability of collision as a function of the number of trials is plotted in Fig. 4, showing that probability of collision begins to converge after approximately 12 × 106 trials. Over the entire set of trials, 379 collision events were recorded, giving a

resultant probability of collision for the Monte Carlo simulation of Pc = 1.516 × 10−5 . The Gaussian mixture calculation of the collision probability only differs by 0.86% from the Monte Carlo calculation of the collision probability, which shows excellent agreement between the methods for calculating the probability of collision and provides validation for the Gaussian mixture approach. ×10−5 1.8

Collision Probability, Pc

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

10

5

15

20

25

Number (in millions) of Monte Carlo Trials

Figure 4. The collision probability as a function of the number of trials computed by Monte Carlo simulation.

V.

Conclusion

This Note introduces an extension for computing the probability of collision between two space objects when the uncertainties of the space objects are represented by Gaussian mixture distributions, rendering this method applicable to many emerging state-of-the-art

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predictors that employ Gaussian mixture representations of the state uncertainty. Several situations can arise which dictate the need for considering Gaussian mixture representations of the uncertainty. For example, space surveillance tracking problems that are characterized by sparse data necessitate long arcs of propagation which has been shown to induce nonGaussianity in orbit uncertainty. Additionally, as shown in this Note, the process of initial orbit determination induces non-Gaussian orbit uncertainty due to the lack of sufficient data to represent the orbit uncertainty by a Gaussian distribution. In each of these cases, the proposed method can be used to obtain the probability of collision between two space objects.

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