Collision Probability

Report
Determining & Evaluating High
Risk Conjunction Events
Improving Space Operations
Workshop
Boulder, CO
5 – 6 April 2011
Introduction
•
•
•
•
•
•
•
Conjunction Assessment Process
Collision Probability
Sigma Level Analysis
Alignment of Radial Vectors
Collision Probability Sensitivity
Maximum Collision Probability
High Risk Events
Conjunction Assessment Process (1/2)
• Conjunction assessment is performed based on state and
state uncertainty data generated and disseminated by the
JSpOC.
• Close approach predictions are generated based on a 5-day
screening span.
• Results are sent out daily, or more frequently for high risk
conjunction events.
• Predictions are made using data from the JSpOC highaccuracy space object catalog.
Conjunction Assessment Process (2/2)
• The typical conjunction assessment process for a satellite
program is to receive Orbital Conjunction Messages (OCMs)
for secondary objects that are predicted to violate a
designated screening volume around the primary satellite.
• OCMs contain sufficient data to calculate a Pc value; i.e., state
vector and state uncertainty data.
• The screening volume must be large enough to capture close
approaches with objects with a wide range of covariance
sizes—this can result in large amounts of data and many
conjunction events that are not a threat.
Collision Probability (1/2)
• Collision probability (Pc) is a measure of the overlap of the
error distribution of the two objects, where the error
distribution is given by the covariance matrix.
• When the covariances of the objects are combined, Pc can be
thought of as the relationship of the miss vector to the
combined covariance.
Primary & Secondary object with
covariance ellipsoids
Combined covariance with keep-out
region positioned by miss vector
Collision Probability (2/2)
• Pc is used as the primary measure of risk since it captures the
miss distance, the relative geometry, and the associated
uncertainty of the close approach.
• A conjunction assessment process based on miss distance
alone does not account for the uncertainty inherent in the
problem.
• Miss distance is used as the basis of the screening process,
leading to the receipt of many OCMs for conjunction events
with zero Pc values.
• This approach can lead to large amounts of data; especially
for satellites in LEO.
Sigma Level Analysis (1/2)
• The miss vector and the state uncertainty of the two objects
can be used for a streamlined screening process.
• The sigma level can be calculated using the Radial, Intrack,
and Crosstrack (RIC) directions of the primary.
   =

 + 
 −    =
where  ≡  

2
+ 
 + 
2

=
 + 
and  ≡   
• OCMs are not required for these calculations.
Sigma Level Analysis (2/2)
• The two semi-major axes equal the largest possible in-plane
uncertainty component,  +  ; and the semi-minor axis
equals the sum of the radial uncertainties,  +  .
• A high sigma level from either calculation should result in a
near zero Pc value, but a low sigma level does not necessarily
lead to a high Pc value.
• A sigma level > 4 is recommended.








Sigma Level Analysis Results
• Sample report of data required:
Predicted Miss Distances
Primary Error at TCA
Secondary Error at TCA
Total
Radial
Intrack
Crosstrack
Radial
Intrack
Crosstrack
Radial
Intrack
Crosstrack










• Sample results with corresponding collision probability:
Case
Radial
Sigma Level
In-plane
Sigma Level
Collision
Probability
1
2.29
0.35
7.78e-5
2
1.13
3.42
5.18e-9
3
-9.29
0.05
0
4
0.84
0.36
1.18e-3
5
4.17
0.09
4.74e-012
Alignment of Radial Vectors
 =

2
+ 
2
Since  ≫  for tangible conjunction
events,  is small and the two Radial unit
vectors can be considered collinear for the
purposes of this analysis.




≡     & 
  

 = arc

≡     


• The Radial unit vectors of the two

objects nearly align for conjunction
events, therefore, the Radial

direction can be decoupled from
the Intrack and Crosstrack directions.


Earth
center



Collision Probability Sensitivity
log Pc
• Covariance size can be scaled to determine the sensitivity of
the Pc value.
• Since covariance is propagated from OD epoch to Time of
Close Approach (TCA), a
Collision Probability Sensitivity
-4
reduction in covariance size
gives an indication of how
-5
the Pc will evolve.
-6
• This calculation is
-7
performed with a static
-8
miss distance.
-9
-10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Covariance Size
0.9
1
Maximum Collision Probability
Normalized Covariance Size
12
10
8
6
4
2
0
4
Sigma Level
log Pc
• The Pc sensitivity curve allows evaluation of the Pc max
condition.
• Pc max tends to occur when the miss vector lies on the
1-sigma uncertainty ellipsoid.
Collision Probability Sensitivity
-2
• Most conjunction events
-3
evolve to the left of the
-4
Pc max condition.
Pc
.
-5
Radial Sigma Level
• Those that don’t tend to
-6
In-plane Sigma Level
be of concern …
-7
• Notice that in this example -8
the covariance was enlarged -9
to show Pc max.
-10
0
0.5
1
1.5
2
2.5
3
3.5
log Pc
• Some conjunction events show little change in Pc as the
covariance is contracted.
• This occurs when the miss vector is within the 1 or 2-sigma of
the combined covariance
Collision Probability Sensitivity
-2.5
2.5
ellipsoid.
2
• This condition can be a sign
Pc
.
Radial Sigma Level
that the conjunction event -3
In-plane Sigma Level 1.5
will remain a threat as
the TCA approaches.
1
-3.5
0.5
-4
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Covariance Size
0.9
0
1
Sigma Level
High Risk Events

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