Advances in Satellite Conjunction Analysis with OR.A.SI

Advances in Satellite
Conjunction Analysis with
OR.A.SI ©
Antonios Arkas
Flight Dynamics Engineer

Table of Contents
Preface
1. Close Approach Analytic Simulator (CASI)
2. Examples of Simulated Close Approach Events Produced from CASI
3. Detection and Visualization of Probability Dilution in the Kp, Ks Scale Factor
Space
4. Range Determination for the K Scale Factor – Covariance Realism
5. Selection of the Optimal Evasive Maneuver Based on the Scaled Probability of
Collision

Preface
OR.A.SI Flight Dynamics software has been endowed with satellite conjunction analysis, middle man
features [4] and collision avoidance manoeuvre calculations capabilities since 2012. The assessment of
probability of collision, based on S.Alfano’s numerical method [6], was validated with an integrated
Monte Carlo close approach simulator which produces Gaussian multivariate random state vectors for
both primary and secondary objects and calculates the close approach statistics based on a large number of
iterations. This simulator, despite being very useful for the validation purposes for which it was
specifically developed, is characterized by very long run times to reach realistic probabilistic results. In
order to remedy this drawback, an analytic close approach simulator (CASI) has been developed which
can produce on demand close approach events, tailored with the desired characteristics (miss distance and
individual covariances on TCA) for short-term encounters [1] under the assumption of Gaussian positional
uncertainty for each spacecraft. The advantage of this new module is that it allows the study of the impact
of the various close approach characteristics, in a controllable fashion, on the probability of collision and
the shape of the probability dilution area in the scale factor Kp and Ks space, without the need of any CDM
(Conjunction Data Message).
The presentation of the characteristics and capabilities of this newly developed analytic simulator, will be
followed by results concerning the detection and graphical representation of the probability dilution
occurrence in the two dimensional space of the Kp, Ks scale factors, which determine the minimum
accuracy requirements for a meaningful probability assessment [7].
Finally the very interesting and critical issue of covariance realism will be addressed along with the code
developed for the determination of the realistic scale factors K interval, which avoids over estimation of
the collision risk, based on Kolmogorov-Smirnov hypothesis testing [5] [3].
The code used for the production of all the presented results, is part of the legacy Flight Dynamics
software OR.A.SI, developed with C++ Borland Builder 6 IDE.

1. Close Approach Analytic
Simulator (CASI)

1.1 Close Approach Analytic Simulator (CASI) Characteristics
CASI is an analytic close approach simulator whose basic purpose is the production of two state vectors
and two covariance matrices corresponding to the geometry and positional uncertainty which define the
desired close approach scenario. This module has the following characteristics:
 Production of close approach events for whatever altitude on TCA and whatever type of orbit
for the primary object (LEO, MEO or GEO).
 Automatic calculation of the state vectors for the primary and the secondary objects, based on:
1. the altitude of the close approach
2. the characteristics of the primary object’s orbit (major semi axis, eccentricity, inclination)
3. the desired relative position of the secondary with respect to the primary, referred to the RTN local
orbital frame of the primary
4. the direction of the secondary velocity with respect to the one of the primary
 Calculation of the covariance matrices by defining the desired size of each error ellipsoid
(lengths of the three semi axes) and its orientation with respect to the RTN local orbital frame
of the corresponding object. The error ellipsoid for each object is initially assumed to be
aligned with its longest dimension along the in-track direction of the RTN frame and with its
shortest one aligned with the radial direction. The final orientation of the error ellipsoid, with
respect to the aforementioned RTN frame, is defined by three successive intrinsic Euler
rotations (z-x-z convention) [3]. Even thought the rotation of each error ellipsoid imbues the
corresponding covariance matrix with correlations, the two matrices are assumed not to be
cross-correlated.
 Assessment of the rectilinear approximation validity necessary to guarantee the legitimacy of
passing from a three dimensional probability of collision integral, to a two dimensional one [1].

1.2 Rectilinear Motion Approximation
Every numerical method of collision probability calculation is based on the reduction of the three
dimensional integral of the Gaussian probability density of the relative position, over the volume V swept
by the sphere of combined object radius, to the following two-dimensional integral [7]:
Two-dimensional probability equation in the encounter plane: • OBJ - Combined object radius.
• σx - Projected covariance ellipse
minor semi axis.
• σy - Projected covariance ellipse
major semi axis.
• (xm ,ym) - Projection of miss distance
on covariance frame.
This reduction is computationally valid only under the assumption of relative rectilinear motion for a
specific length of the orbit in order for the value of the two-dimensional integral to be very close to the
three-dimensional one with a desired degree of accuracy [1]. CASI, based on the maximum in-track
uncertainty of the two objects, assesses initially the qualitative conditions under which this assumption is
valid and presents them to the user so that he/she decides if the collision probability calculation for the
selected event is deemed to be accurate.
According to F.Chan [1] the path of the rectilinear motion should be 17σ for 15-digit accuracy and 6σ for
a 2-digit accuracy, where σ is the maximum in-track uncertainty.

1.3 Assessment of Rectilinear Motion Approximation Errors
The method for the assessment of the rectilinear approximation errors is due to F.Chan [1] and is based on
the computation of the in-track and radial errors as well as the deflection angle of the straight line from
the actual circular orbit, under the assumption of relative rectilinear motion of specific length (17σ or 6σ)
and for a specific angle of attack (relative velocity).
As an example the following figure presents CASI’s validity assessment for the same close approach
event (same separation and covariance characteristics) but for two different angles of attack. The first case
at the left shows the rectilinear approximation errors for 14o angle of attack while the second one, at the
right part, gives the same type of errors but for an angle of attack of 1o. From the deflection angle, the
very high encounter region transit time and the other errors it is evident that the second case, where the
relative velocity is very small, violates the rectilinear motion assumption and thus the probabilistic
calculations are not accurate. This case should be dealt with methods of nonlinear relative trajectory [13].
Figure 1.1: Rectilinear assumption approximation errors for 14o and 1o angles of attack

1.4 Simulation Results (1/2)
The results corresponding to CASI’s close approach scenario, are produced from the already existing
module of OR.A.SI, which was initially developed for probabilistic calculations based on ingestion of
CDM data [14]. Since the close approach characteristics are predefined, the separation and covariance
results serve as a verification of the simulation setup while the probabilistic calculations are the desired
output of the simulation process.
The geometric part of the results comprises the following:
 Relative position of the secondary with respect to primary referred to the RTN local orbital
frame of the primary.
 Relative velocity of the secondary with respect to primary referred to the RTN local orbital
frame of the primary.
 Angle of attack and angle of the apparent velocity of the secondary with respect to the
primary.
 Variances of the position uncertainty for each object, referred to its individual RTN local
orbital frame.
 Dimension of each error ellipsoid (SVD decomposition in the case of covariance ingestion
from CDM).
 Components of the miss distance on reference frame defined by the semi axes directions of
the projected combined error ellipse on the B-plane.
 Angle subtended from the miss distance direction and the major semi axis direction of the
the projected combined error ellipse on the B-plane.

1.4 Simulation Results (2/2)
The probabilistic part of the results comprises the following:
 Probability of collision based on S.Alfano method [6] for Kp = Ks =1.
 Worst case maximum probability of collision (alignment of miss distance direction with the
major semi axis of the projected combined error ellipse on the B-plane and attainment of the
maximum σx for the specific aspect ratio AR) [8].
 Scaled probability of collision for the selected Kp and Ks scale factor intervals [5].
The numerical results are augmented with the following plots and diagrams:
 3D close approach geometry and error ellipsoid rendering.
 3D full representation of the two orbits and of their separation distance on TCA.
 Projection of the flying paths and of the separation vector on the plane defined by the
vectors of the orbital velocities (determination of which object is temporally lagging or
advancing with respect to the point of intersection of the orbits in the vicinity of the close
approach).
 3D combined error ellipsoid centered on the secondary object.
 Projection of the combined error ellipsoid on the B-plane where the X-axis corresponds to
the miss distance direction.
 Projection of the combined error ellipsoid on the B-plane where the X-axis correspond to
the projection of the primary velocity on the B-plane.
 Heat map of the scaled probability of collision for the selected Kp and Ks scale factor
intervals.

2. Examples of Simulated Close
Approach Events Produced from
CASI

2.1 Characteristics of the First Simulation
The following example is indicative of the power of CASI to initialize a close approach event with
desired characteristics and transform it from a safe one to a critical situation by simply changing the
orientation of the secondary object’s error ellipsoid.
The scenario of this example corresponds to a close approach in the GEO region from an IGSO (Inclined
GeoSynchronous Orbit) intruder approaching the primary object from the South pole direction with a miss
distance of 1.7 km, primarily towards the in-track direction.
Figure 2.1: Simulation characteristics (geometry and covariances) for the first event

2.2 Close Up 3D View and Error Ellipsoids for the First Close Approach Event
The comparison of the RTN standard deviations with the sizes of the corresponding error ellipsoids is
indicative of the fact that this simulation has been initialized with no correlations for either one of the
covariance matrices (diagonal matrices with respect to their RTN reference frames). In the following
figure the blue color corresponds to the primary object and the red to the secondary one.
Figure 2.2: Close approach geometry with RTN separations : DR = -0.520 km , DT = -1.6 km and DN = 0.330 km (Miss
Distance = 1.711 km). Angle of attack = 14.0 deg (IGSO object)

2.2 3D Representation of the Orbits in Interest
The following figure shows the 3D representation of the two orbits and of the point of close approach.
This graphical representation serves as an extra verification of the correct initialization of CASI in
order to produce the desired close approach geometry.
Figure 2.3: Three dimensional view of the orbits of the primary (blue line) and of the secondary (red line). The dotted line
corresponds to the position of the primary on TCA

2.2 Projection of the Flying Paths on the Plane Defined by the Velocity Vectors
The projection of the flying paths on the plane defined by the vectors of the orbital velocities, is very
useful for the understanding of temporal relation of each object with respect to the point where the two
flight paths intersect.
Figure 2.4: Projection of the flying paths on the plane defined by the vectors of the orbital velocities. The miss distance is
represented by the dotted line. The entrance in the encounter region (direction of movement) is indicated by the colored
stars

2.2 Probabilistic Results for the First Close Approach Event.
Despite the short miss distance of 1.7 Km, the probability of collision is less than the critical limit
of 1e-5. This happens because, as it’s evident from Figure 2.6, the geometry is favorable since the
miss distance direction is towards the area of the projected error ellipse corresponding to smaller
uncertainty.
Figure 2.5: Collision probability results for the case of RTN covariances with no correlations

2.3 Projections on the B-Plane for the First Close Approach Event
Figure 2.6: Projection of the combined covariance ellipsoid (relative distance error ellipsoid ) on the B-plane
Figure 2.7: Projection of the combined covariance ellipsoid on the B-plane. X axis is defined by the projection of the
primary velocity on the B-plane and the Y axis is forming a right handed trihedron with the X axis and the Z axis
perpendicular to the B-plane

2.3 Probability of Collision as a Function of the Size of the Error Ellipsoids
Figure 2.8: Probability expressed as a function of RSS measurement errors (first plot) and as a function of the scaling
factors (second plot). The symbol X signifies the position and value of the collision probability for no scaling of the
covariances
The sensitivity of probability of collision from the individual covariances, is best represented from a heat
map plot where the coordinate axes correspond to the variable size of the two error ellipsoids. More
details are given in part No.3 of this presentation.

2.4 Characteristics of the Second Simulation (Induction of Correlations)
A dramatic change happens when with exactly the same geometry (separation characteristics on TCA)
we rotate the secondary error ellipsoid by inducing correlations in its RTN representation.
The correlations are revealed from the difference between the secondary RTN standard deviations and
the dimensions of its error ellipsoid, and they are induced by an initial rotation of -30 deg around the
N axis and a subsequent rotation of -24 deg around the resulting R axis.
Figure 2.9: Simulation characteristics (geometry and covariances) for the second event

Figure 2.10: Deliberate rotation of the secondary object error ellipsoid towards the direction of the miss distance
2.2 Close Up 3D View and Error Ellipsoids for the Second Close Approach Event
Polar view
Equatorial view

2.6 Probabilistic Results of the Second Close Approach Event
The dramatic impact of the increase of the probability by 2 orders of magnitude due to the induction
of correlations in the RTN covariance of the secondary object, is reflected on the following
probabilistic results.
Figure 2.11: Collision probability results for the case of secondary object covariance with RTN correlations

2.7 Projection on the B-Plane for the Second Close Approach Event
The angle between the miss distance and the major semi axis of the projected ellipse (direction of
major uncertainty) is 5.5 deg i.e. 6.7 times closer to the area of major uncertainty with respect to the
first close approach event.
Figure 2.13: Projection of the combined covariance
ellipsoid on the B-plane. X axis is defined by the
projection of the primary velocity on the B-plane
and the Y axis is forming a right handed trihedron
with the X axis and the Z axis perpendicular to the
B-plane
Figure 2.12: Projection of the
combined covariance ellipsoid
(relative distance error
ellipsoid ) on the B-plane)

2.7 Projection on the B-Plane for the Second Close Approach Event
factors (second plot). The symbol X signifies the position and value of the collision probability for no scaling of the
covariances
The structure of the heat maps shows a case where the computed probability lies in the probability
dilution region and thus can not be trusted. This case is an example of the need for more
measurements in order to guarantee a meaningful probability assessment.

3. Detection and Visualization of
Probability Dilution in the Kp, Ks
Scale Factor Space
Image Source : Salvatore Alfano, 2005. Relating Position Uncertainty to Maximum
Conjunction Probability. The Journal of the Astronautical Sciences, Vol.53, No.2, April-June
2005, pp.193-205.

3.1 Dilution of Probability in Satellite Conjunction Analysis
Probability dilution occurs in satellite conjunction analysis because the positional covariances are
not expressing randomness (aleatory uncertainty) but lack of knowledge (epistemic uncertainty)
thus the best suited mathematical theory for modeling the close approach events should have been
Possibility Theory instead of Probability theory even though the two theories are closely related
[9].
As a mathematical phenomenon it can be defined as the apparent decrease of probability of
collision, past a certain point, as the variance of the position estimates increases [10]. The increase
of the size of the error ellipsoid is mathematically equivalent to dilution of the probability density
and thus decrease of the calculated probability of collision. Even though the calculation being
mathematically sound, the error is done when connecting the mathematical model to reality by the
action of, what’s called in epistemology, interpretation of the model. The interpretation of the
covariance as randomness is wrong and so does the acceptance of the meaning of the decrease of
the probability of collision in the dilution region which imbues the operator with false confidence
[11].
The dilution of probability is even more interesting when combined with the research of
covariance realism and the related issue of covariance scaling done with the determination of the
Kp and Ks intervals of the scale factors for the primary and the secondary objects. The interest lies
in the fact that scaling itself is a process that inevitably leads towards the dilution of probability.

3.2 Definition of Probability Dilution in the KP – KS Space
Following the general understanding of the meaning of probability dilution, this phenomenon can be
defined in the Kp and Ks space as the local infinitesimal decrease of probability of collision PoC
corresponding to an increase of positional uncertainty of either one, or both, of the two objects
participating in the close approach event.
Since the differential of PoC for every point in the Kp and Ks space is given as:
A point of the aforementioned space does not belong to the dilution area if and only if:
𝑑𝑃𝑜𝐶 = 𝛻 𝐾 𝑝,𝐾𝑠
𝑃𝑜𝐶 ∙ 𝑑𝐾 =
𝜕𝑃𝑜𝐶
𝜕𝐾 𝑝
𝑑𝐾 𝑝 +
𝜕𝑃𝑜𝐶
𝜕𝐾𝑠
𝑑𝐾𝑠
𝑑𝑃𝑜𝐶 > 0 ∀ 𝑑𝐾 𝑝≥ 0 𝑎𝑛𝑑 ∀ 𝑑𝐾𝑠≥ 0
If we impose this definition alternatively for the two cases:
i) dKp = 0 and dKs > 0 ii) dKp > 0 and dKs = 0
it follows that the condition so that a point (x,y) in Kp and Ks space doesn’t belong to the dilution area is
the following:
𝜕𝑃𝑜𝐶
𝜕𝐾 𝑝 𝑥
> 0 𝑎𝑛𝑑
𝜕𝑃𝑜𝐶
𝜕𝐾𝑠 𝑦
> 0

3.3 OR.A.SI Approach for the Detection of the Probability Dilution Area
Based on this observation, OR.A.SI approach for the determination of the area of probability dilution,
is to initially calculate the gradient of collision probability PoC in the aforementioned space. Having
calculated this gradient the software computes the curves delimiting the transition of either of the
components of the gradient vector, from positive to negative values. The dilution region is the area
lying below these two curves.
In the diagrams that follow, the white dotted line corresponds to the limit where the dilution is
initiated due to the primary object
𝜕𝑃𝑜𝐶
𝜕𝐾 𝑝 𝑥
> 0 and the black dotted line to where the dilution is
initiated due to the secondary object
𝜕𝑃𝑜𝐶
𝜕𝐾𝑠 𝑦
> 0 .
The main figure of interest, which is the maximum probability of collision PoC* for the selected
range of KP and KS scaling factors, is computed along with the calculation of the probability dilution
area. The area of maximum probability it is signified a with dotted yellow line.
In order to make clear the dependence of PoC from the data quality and detect significant decrease of
PoC due to possible data improvement, the probability of collision is plotted both as a function of the
scaling parameters (Kp, Ks) as well as a function of the square root of the trace of the primary and
secondary covariance matrices (RSSp, RSSs).

3.4 Probability Sensitivity, Dilution of Probability and PoC* - Case Study (1/2)
Figure 3.1: : Projection of consecutive combined covariance ellipsoids on the B-plane. X axis is defined by the projection
of the primary velocity on the B-plane and the Y axis is forming a right handed trihedron with the X axis and the Z axis
perpendicular to the B-plane
The detection of probability dilution and the connection of the probability sensitivity with the decrease of
the PoC due to improvement of data quality (decrease of covariance), is demonstrated with the aid of an
actual series of CDM’s where the data quality of the secondary object is gradually improved.
For this series, the PoC* is calculated for Kp, Ks ∈ [0.25, 4].
Table 3.1: : Close approach characteristics for a series of consecutive CDMs issued for the same event

3.4 Probability Sensitivity, Dilution of Probability and PoC* - Case Study (2/2)
Figure 3.2: Evolution of the close approach characteristics for the events of Table 3.1
From the values of Table 3.1 it is evident that the value of PoC* doesn’t get below the avoidance
action criterion of 1E-4 till CDM No.3 It will be demonstrated how the heat map of the probability
sensitivity and the dilution detection can be utilized in order to safely predict the probability trend
from the very first CDMs.

factors (second plot). The symbol X signifies the position and value of the collision probability corresponding to the
specific CDM. Dilution of probability occurs in the area lying below the black dotted line. The gradient vector field of
the PoC(Kp,Ks) is seen in the second plot
3.5 Case Study - Time to TCA 10 days (Results)

Figure 3.4: The dependence of the combined covariance matrix aspect ratio (AR) and of the elevation angle (angle
subtended by the miss distance direction and the major semi axis of the projected combined covariance on the B-plane),
from the measurement errors (RSS), justifies the identification of dilution area since the decrease of probability is not caused
by alteration of the close approach geometry
3.5 Case Study - Time to TCA 10 days (Analysis of Results)
The case of Figure 3.3 is one of severe dilution which occurs due to the very bad measurement quality of
the secondary which is almost 3 times worse than the one of primary. Since the calculated PoC*
corresponds to the lower limit of the Ks interval, it is not to be trusted because this is an indication that it
lies in the dilution area. Based to the latter observation and the fact that there is significant margin for
improvement of the secondary measurement quality (as seen from the values of the secondary RSS), it is
foreseen that both PoC and PoC* are expected to decrease significantly for the upcoming CDMs.

3.6 Case Study - Time to TCA 9 days (Results)
factors (second plot). The symbol X signifies the position and value of the collision probability corresponding to the
specific CDM. Dilution of probability occurs in the area lying below both the black and white dotted lines. The gradient
vector field of the PoC(Kp,Ks) is seen in the second plot

Figure 3.5: Probability heat maps for the CDM released 4 days before TCA
3.6 Case Study - Time to TCA 4 days (Analysis of Results)
The increase of the updated miss distance significantly altered the shape of the area where there is no
probability dilution but the value of PoC* is still above the manoeuvre avoidance criterion of 1E-4
because the increase of miss distance was not accompanied by a significant reduction of the secondary
object RSS. Despite this fact, the values of PoC in the vicinity of the lower left corner of Figure 3.5 and
the small RSS of the primary object, reinforce the anticipation for the reduction of the secondary object
RSS and the accompanying decrease of the PoC*. This is justified from the CDM which is released after
4 days. For this CDM the value of PoC* is 5.45E-5 for Kp = Ks = 4.

4. Range Determination for the K
Scale Factor – Covariance Realism

4.1 Probability Sensitivity and Necessity for Confinement of the Kp and Ks Intervals
As seen from Figure 4.1, the PoC is very sensitive with respect to the covariance scale factors KP and KS.
If the calculation of PoC* is based on the same fixed range of scale factors for every conjunction and
every object, then there is a danger of overestimating the collision risk with an immediate consequence
of either frequently planning unnecessary collision avoidance maneuvers or oversizing them. Both cases
are equivalent to propellant over-consumption.
A solution for this problem is the definition of a formal process for confining the scale interval for each
object in interest, based on successive orbit determination updates. The resulting PoC* for the shrunken
intervals is called scaled probability of collision. The method to be presented and algorithmically
implemented in OR.A.SI, is due to F.Laporte from CNES [5].
Figure 4.1: Sensitivity of probability of collision from both Kp and Ks covariance scale factors for a simulated event

4.2 Mathematical Method for Shrinking the Kp and Ks Scale Intervals
The problem of shrinking the scale factor intervals is essentially a problem of Hypothesis Testing
concerning the form of cumulative distribution function. Its starting point is the observation that the
Mahalanobis distance D(𝑥, 𝜇) for a random multivariate normally distributed vector variable 𝒙 of n
dimensions from its mean 𝝁, is also a random variable with Chi-Square distribution with m degrees of
freedom [12].
𝐷 𝑥, μ = 𝑥 − 𝜇 𝑇 𝐶−1 𝑥 − 𝜇
The three dimensional random variable 𝒙 is identified with the error of the previously determined
positions of an object, referred to the latest one, and the matrix C with its corresponding covariance
matrix. Since the Least Square (LS) batch estimator is an unbiased one [12], the 𝝁 can be taken as the
result of the most recent orbit determination for the object in interest. Having defined the probability
model by the 𝝁, 𝐂 parameters and the form of the probability distribution function of the Mahalanobis
distance, the empirical estimate of the cumulative Chis-Square distribution F 𝑥 , is a random family of
functions 𝐹 𝑥 , one for each set of samples 𝒙 [2].
Shrinking the scale factor interval is equivalent to subjecting each K in the predefined interval, to a
Hypothesis Testing with null hypothesis:
𝐻 𝑜: F 𝑥 = 𝐹 𝐾 𝑥 𝑎𝑔𝑎𝑖𝑛𝑠𝑡 𝐻1: F 𝑥 ≠ 𝐹 𝐾 𝑥
where FK 𝑥 is the cumulative distribution function (c.d.f) for the scaled covariance matrix K∙C.

4.3 Hypothesis Test Design
In order to design a Hypothesis Test the first to be defined is the critical region Dc of the variable 𝒙 for
which the null hypothesis Ho is rejected. The search in a three dimensional space can alternatively be
substantially simplified to a search in a one dimensional sample space by utilizing a test statistic i.e. a
random variable q defined by a function q = g(𝒙) of the original random variable 𝒙. The decision whether
to reject of Ho is based not on the value of the vector 𝒙 but on the value of the scalar q = g(𝒙).
The statistic used for the scale factor interval shrinking is the Kolmogorov-Smirnov Distance w:
This statistic is the maximum value of the difference of the empirical cumulative distribution function
𝐹 𝑥 , computed from n samples/observations, from the theoretical cumulative distribution function
FK 𝑥 corresponding to K, for the whole range of the random variable 𝒙 .
Based on an approximation due to Kolmogorov [2], the cumulative distribution function Fw 𝑤 for the
variable w is the following:
𝑤 = max
𝑥 ∈ 0,+∞
𝐹 𝑥 − 𝐹𝑘 𝑥
𝐹𝑤 𝑤 = 1 − 2𝑒−2𝑛𝑤2
𝑓𝑜𝑟 𝑤 > 1
𝑛
The p-value (significance level) of the test concerns the probability 1 - Fw 𝑤 𝑜 i.e. the probability of the
Kolmogorov-Smirnov distance being greater than the value wo calculated from the observed samples of 𝒙.
This value is usually set to 5% but it is configurable in the implemented algorithm.
If p-value = 1 - Fw 𝑤 𝑜 < a-limit then the null hypothesis Ho and the corresponding scale factor are
rejected.

4.4 Dispersion Visualization of the Object in Interest (1/)
The demonstration of the process for K interval shrinking, is done with a set of 10 successive determined
orbits and their accompanying covariances for a rocket debris.
Figure 4.2: Covariance error ellipsoid projections on the coordinate planes of the local RTN orbital frame of the last
determined orbit. The gradual decrease of position uncertainty is evident from the ellipse shrinking

4.5 Calculation of the Empirical c.d.f for K=1
Figure 4.3 presents the empirical cumulative distribution of the Mahalanobis distance of the latest orbit
with respect to the previously determined orbits, which is plotted against the theoretically expected Chi-
Square cumulative distribution with 3 degrees of freedom and scale factor K = 1.
Figure 4.3: Theoretical Chi-Square cumulative distribution and the empirical one constructed from the successive
determined orbits of the object
Empirical c.d.f
Theoretical Chi-Square c.d.f

4.6 Determination of K Range Based on K-Interval Method
An improvement of the consistency between the empirical and the theoretical c.f.d can be achieved by
the determination of the individual K coefficients for which the cumulated probability corresponding to
each successive observations, belongs to the corresponding Mahalanobis distance of the theoretical
cumulative c.d.f . The output of this process is the K-Interval = [0.10 , 0.69].
Figure 4.4: Determination of the K-Interval coefficients
CDM #1 K-Int = 0.252
CDM #2 K-Int = 0.25
CDM #3 K-Int = 0.24
CDM #4 K-Int = 0.23
CDM #5 K-Int = 0.18
CDM #6 K-Int = 0.17
CDM #7 K-Int = 0.14
CDM #8 K-Int = 0.1

4.7 Determination of K Range Based on Hypothesis Testing Method
Due to the fact that the K-Interval method is not conservative, it should be combined with the
determination of the scale factors which is based on the Kolmogorov-Smirnov distance. The software for
every value of the scale factor K, in the desired search interval, computes the probability 1 - Fw 𝑤 𝑜 of
the Kolmogorov-Smirnov distance been greater than wo and if this probability is less that the selected by
the user p-value, the K factor is rejected as non realistic. The output of this process, seen in Figure 4.5, is
the K-Kolmogorov = [0.13 , 0.69].
Figure 4.5: Determination of the K-Kolmogorov scale factors for p-value (realistic level) = 5%
Accepted K Interval

4.8 Determination of the Recommended Range for the K Scale Factor
Finally the 5% significance level recommended K range for the object in interest, is computed from the
union of the intervals computed from the two previous methods:
K – Range = K-Interval ∪ K-Kolmogorov = [0.10, 0.69]
The following figure shows the two empirical c.f.d (dotted lines) delimiting the range of the
recommended K factors, the empirical c.f.d corresponding to K = 1 and the theoretically expected one.
When this method is combined with the computation of the PoC*, presented in section 3, the resulting
maximum PoC* is the scaled collision probability which avoids the over estimation risk.
Figure 4.6: Theoretical and empirical c.f.d of the Mahalanobis distance corresponding to the p-value of 5%
K = 0.10
K = 1.0
K = 0.69

5. Selection of the Optimal Evasive Maneuver
Based on the Scaled Probability of Collision

The final part of this presentation concerns the actions taken, following the assessment of the risk of an
immanent close approach, in the form of the appropriate evasive maneuver computation. If the
probability of collision is used as the collision risk metric then this will lead to either unnecessarily
frequent execution of oversized evasive maneuver, in cases of pessimistic covariances, or planning of
undersized maneuver in cases of optimistic ones.
Due to the aforementioned dangers, the appropriateness of an evasive maneuver should be judged by the
scaled probability of collision and not simply by the probability of collision.
Past versions of OR.A.SI [14] where able to produce a series of evasive maneuvers and assess their
effectiveness from their impact on the probability of collision. The culmination of all the presented
theoretical advances is the integration of the already existing routines with the routines which assess the
scaled probability of collision so that now the effectiveness of an evasive maneuver is better measured by
its effect on the scaled probability of collision. The steps for such an assessment are the following:
1. Assessment of the orbit determination (OD) process quality from the evolution of the various OD
parameters computed from the whole batch of CDM for the specific event (Figure 5.1) .
2. Visualize the evolution of dispersions in order to get a qualitative understanding of the orbit
determination consistency i.e. depending on whether the covariances are pessimistic or optimistic,
anticipate the dynamical range of the realistic K factor interval (Figure 4.2).
3. Determine the realistic K factor interval for each object (Chapter 4 of the presentation).
4. Compute the scaled probability of collision following each evasive maneuver, for the previously
defined realistic intervals for Kp and Ks scale factors.
The presented methodology is the one implemented by CNES in JAC software [4].
5.1 Evasive Manoeuvre Computation and Scaled Probability of Collision

5.2 Assessment of OD Quality from the Evolution of the OD Parameters
Figure 5.1: Evolution of the orbit determination parameters corresponding to successive CDM
The evolution of the various OD indices found in the CDM and the compute Mahalanobis distance of the
past determined positions with respect to the last one, are quantitative measures of the of the OD quality
for each object participating in the close approach event.

Figure 5.2: Dependence of the separations on TCA, referred to the primary object RTN local orbital frame,
from the in-track DV increment and the maneuver epoch with respect to TCA
5.3 Analysis of Separation on TCA Resulting from the Execution of an Evasive Maneuver
The first criterion for the adequacy of an evasive maneuver is the comparison of the resulting separations on
TCA with the maximum uncertainty expected for the direction in interest, depending on the maximum values of
the Kp and Ks factors. In the following diagram the compact straight lines corresponds to the separation when no
maneuver is executed and the dotted lines to the worst case relative distance 1 σ uncertainty (plots correspond to
max Kp = max Ks = 2). The top series of plots correspond to West evasive maneuvers ant the bottom to East
ones. Each line corresponds to a different maneuver increment.

5.4 PoC and scaled PoC Resulting from the Execution of an Evasive Maneuver
Figure 5.3: Dependence of the PoC and the scaled PoC from the in-track DV increment and the maneuver epoch with
respect to TCA. The top series of plots correspond to West evasive maneuvers and the bottom to East ones
The final decisive criterion for the selection of the optimal evasive maneuver (smaller size), is the value of the
scaled PoC. Given that the scaled PoC is the maximum value of the PoC*, it is automatically guaranteed that
there no issue of probability dilution for this value. In the following figure, the left site plots depict the
dependence of the PoC from the maneuver increment DV and the maneuver epoch while the right ones shows the
corresponding dependence of scaled PoC. Each line corresponds to a different maneuver increment.

Figure 5.4: Detailed enlisting of the series of evasive maneuvers and their impact on PoC, scaled
PoC and RTN separations on TCA .
5.3 Separation, PoC and Scaled PoC Details for All the Computed Evasive Maneuvers
Additional to the very instructive plots, the software produces as a final product a detailed file with the RTN
separations, the PoC and the scaled PoC for all the maneuvers in the selected range of DV.

Bibliography (1/2)
1. F.Kenneth Chan, 2008, The Aerospace Corporation. Spacecraft Collision Probability.
2. Athanasios Papoulis, 1990, Prentice Hall. Probability and Statistics.
3. Goldstein, Pool & Safco, Third Edition 2002. Classical Mechanics.
4. CNES, France, Francois Laporte. JAC Software, Software, Solving Conjunction Assessment
Issues.
5. CNES, France, Francois Laporte. Operational Management of Collision Risks for LEO Satellites
at CNES
6. Salvatore Alfano, 2005. A Numerical Implementation of Spherical Object Collision Probability.
The Journal of the Astronautical Sciences, Vol.53, No.1, January-March 2005, pp.103-109.
7. Salvatore Alfano, 2005. Collision Avoidance Maneuver Planning Tool. Paper AAS 05-308
presented at the 15th AAS/AIAAAstrodynamics Specialist Conference August 7-11 , 2005
8. Salvatore Alfano, 2005. Relating Position Uncertainty to Maximum Conjunction Probability. The
Journal of the Astronautical Sciences, Vol.53, No.2, April-June 2005, pp.193-205.
9. Parul Agarwal, Dr. H.S. Nayal, Possibility Theory versus Probability Theory in Fuzzy Measure
Theory, Parul Agarwal Int. Journal of Engineering Research and Applications ISSN : 2248-9622,
Vol. 5, Issue 5, ( Part -2) May 2015, pp.37-43

Bibliography (2/2)
10. Michael Scott Balch, PhD. A Corrector for Probability Dilution in Satellite Conjunction
Analysis. 18th AIAA 2016-1445, 4-8 January 2016, San Diego, California, USA Non-
Deterministic Approaches Conference
11. Michael Scott Balch, Ryan Martin, Scott Ferson. Satellite Conjunction Analysis and the False
Confidence Theorem. arXiv:1706.08565v5 [math.ST] 24 Jul 2019
12. Maria Isabel Ribeiro, Institute for Systems and Robotics, February 2004. Gaussian
Probability Density Functions: Properties and Error Characterization.
13. David P.McKinley, AIAA. Development of Nonlinear Probability of Collision Tool for Earth
Observing System.
14. OR.A.SI – Orbit and Attitude Simulator Part 2

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