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Eliminating Assumptions Regarding Satellite Conjunction Analysis

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Abstract

This works shows several ways to address and eliminate many simplifying assumptions regarding satellite conjunction analysis. These assumptions treat the conjuncting object shapes as spheres with their relative motion considered linear for the encounter. The positional errors at the time of closest approach are assumed to be zero-mean, Gaussian, uncorrelated, and constant. A test is introduced to determine if the conjuncting pair’s relative velocity satisfies the linear motion assumption. To accomplish this, a fractional probability tolerance is defined based on a user’s accuracy requirement; this tolerance is then used to find the approximate minimum relative velocity that ensures adequacy. A coarse determination of this velocity is made by forcing the relative motion to be linear but allowing the covariance to vary with time. This is followed by a refined estimate where orbital dynamics are included for the trajectory motion. Nonlinear probability is computed by breaking the relative path into sufficiently small cubes, elongated discs, or parallelepipeds such that the sectional motion is nearly linear, computing the linear-motion probability associated with each section, and then summing. Three approaches are presented to determine the nonlinear probability. The first method creates a voxel grid in Mahalanobis space, computes the probability of each affected voxel as the combined object passes through this time-invariant probability density space, and sums. The second method considers each tube (elongated disc) to be cylindrical with its ends perpendicular to its axis; this does not account for gaps or overlaps of abutting cylinders. The third method is more complex, using bundled, rectangular parallelepipeds to reduce these gaps and overlaps by treating the junctions as compound miters while incorporating probability density variations. The objects are treated as spheres for testing, but the parallelepiped method is designed to handle any time-varying object shape by using dynamic pixel files of the object images.

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Abbreviations

axis12:

unit vector from r1 to r2

axis12r:

axis12 rotated

axis23:

unit vector from r2 to r3

axis23r:

axis23 rotated

C :

Covariance matrix

C2 :

2 × 2 positional covariance matrix in the encounter plane

C3 :

3 × 3 positional covariance matrix

dt :

time shift

dx :

off-axis x position

dy :

off-axis y position

dz :

endpoint adjustment

ECI:

Earth centered inertial frame

erf :

error function

f :

Patera’s aspect ratio

fpt :

fractional probability tolerance

F :

6 × 6 position/velocity Jacobian matrix

m :

counter upper limit

M f :

final Mahalanobis distance

M i :

initial Mahalanobis distance

n :

combined covariance ellipsoid scale factor

OBJ:

cross-sectional radius

P :

probability

P_S1 :

Chan’s analytical probability approximation

r :

radius of torus’ cross-sectional (alternatively Patera’s distance to combined object perimeter)

R :

radius of torus (alternatively Patera’s distance to combined object center)

TCA:

time of closest approach

t start :

start time

t end :

end time

V :

swept out volume of collision tube

VNC:

Velocity-Normal-Co-Normal frame

xm :

rotated x position of combined object center

x_maha :

x position of combined object center in Mahalanobis space

x p :

primary object’s earth-centered x position

ym :

rotated y position of combined object center

y_maha :

y position of combined object center in Mahalanobis space

y p :

primary object’s earth-centered y position

z p :

primary object’s earth-centered z position

α :

angle between the object’s distance vector and the covariance ellipse’s x axis

Δt :

transition time

Δt coarse :

coarse time change

Δt fine :

refined time change

Δt init :

initial estimate of time change

ε :

covariance-centric angle

μ :

gravitational parameter

ϕ :

object-centric angle in Mahalanobis space

Φ :

state transition matrix

Ρ :

combined object radius (also used as density function)

ρ x :

combined object radius in Mahalanobis x direction

ρ y :

combined object radius in Mahalanobis y direction

σ :

standard deviation

τ :

unitized parameter for time

θ :

object-centric angle

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Authors and Affiliations

  1. Center for Space Standards and Innovation, 7150 Campus Drive, Suite 260, Colorado Springs, CO, 80920-6522, USA

    Salvatore Alfano

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  1. Salvatore Alfano
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Correspondence to Salvatore Alfano.

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Alfano, S. Eliminating Assumptions Regarding Satellite Conjunction Analysis. J of Astronaut Sci 59, 676–705 (2012). https://doi.org/10.1007/s40295-014-0002-4

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  • DOI: https://doi.org/10.1007/s40295-014-0002-4

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