### Collision Probability

```Determining & Evaluating High
Risk Conjunction Events
Improving Space Operations
Workshop
Boulder, CO
5 – 6 April 2011
Introduction
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•
•
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Conjunction Assessment Process
Collision Probability
Sigma Level Analysis
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
Intrack
Crosstrack
Intrack
Crosstrack
Intrack
Crosstrack

• Sample results with corresponding collision probability:
Case
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
=

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

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
• 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
.
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
```