The ever-growing resident space object population poses a continual threat in that a hyper velocity impact is likely to be catastrophic to an active satellite. To avoid these scenarios, space operators compute a probability of collision metric for each potential conjunction. Uncertainty trends are studied in the conjunction plane and operational decisions to mitigate any high-risk situations are made based off this information. There are many methods of uncertainty propagation and probability of collision formulations and knowledge of their realism is required to maintain a sustainable space environment. Thus, this research studies the effect of Chan, Alfano, Foster, Gaussian mixture, and Monte Carlo probability of collision calculations and their correlation to uncertainty realism metrics. The linear, unscented transform, entropy-based, and Monte Carlo propagation techniques are utilized alongside the collision calculations and it is shown that there are important correlations any space operator should be aware of to support maintenance of a healthy spacecraft.

1. 1
A Study of Non-Gaussian Error Volumes and Non-Linear
Uncertainty Propagation for Use in Conjunction Assessment
Justin A. Spurbeck1
The University of Texas at Austin, Austin, TX, 78705, United States
The ever-growing resident space object population poses a continual threat in that a hyper
velocity impact is likely to be catastrophic to an active satellite. To avoid these scenarios, space
operators compute a probability of collision metric for each potential conjunction.
Uncertainty trends are studied in the conjunction plane and operational decisions to mitigate
any high-risk situations are made based off this information. There are many methods of
uncertainty propagation and probability of collision formulations and knowledge of their
realism is required to maintain a sustainable space environment. Thus, this research studies
the effect of Chan, Alfano, Foster, Gaussian mixture, and Monte Carlo probability of collision
calculations and their correlation to uncertainty realism metrics. The linear, unscented
transform, entropy-based, and Monte Carlo propagation techniques are utilized alongside the
collision calculations and it is shown that there are important correlations any space operator
should be aware of to support maintenance of a healthy spacecraft.
I. Introduction
One of the core topics taught in the Orbital Debris course this semester was the phenomenon of a resident space
object’s (RSO’s) uncertainty growing non-Gaussian as it is propagated throughout an orbit due to non-linearities. The
ability to accurately represent an RSO’s uncertainty is particularly important in conjunction assessment, among other
things. This paper serves to study the effects of non-Gaussian error volumes in conjunction assessment and probability
of collision (Pc) calculations as well as uncertainty realism trends for various propagation techniques.
II. Conjunction Simulation
To accomplish the desired study, a realistic simulation of a conjunction must be produced. Having experience as
a space operator in industry, the author chose to model a conjunction off of actual Conjunction Data Messages (CDMs)
provided by the Combined Space Operations Center (CSpOC). The state values used for both the primary and
secondary were produced from a slightly perturbed Cartesian Orbital Element (COE) set derived partially from a real
CDM. The secondary approach angle used was 155.0 deg. and was precisely the true value. The primary object’s
velocity was rotated by the approach angle about the primary’s radial direction to produce the secondary velocity’s
orientation. Some minor rotation about the secondary’s cross track unit vector was included to slightly perturb the
eccentricity.
The secondary’s state was generated by adding the given Radial-Intrack-Crosstrack (RIC) miss distance at Time
of Closet Approach (TCA) to the primary’s state. In general, it was assumed the primary RSO was a large satellite in
a sun-synchronous orbit that is well tracked via GPS measurements and thus demonstrated a consistently small
covariance. A range of covariance sizes were studied for the secondary and will be presented along with each
simulation as relevant. Other basic spacecraft and secondary assumptions are listed in Table 1 below.
A visualization of the conjunction produced in MATLAB is included in Fig. 1. Unsurprisingly, the conjunction is
in a polar region – a high traffic area for sun-synchronous orbits. The primary satellite is denoted by the blue trajectory
while the red represents the secondary debris with the circle and ‘x’ markers representing their respective positions at
TCA. Only ninety percent of each orbital period is plotted.
1
Graduate Research Assistant, Aerospace Engineering and Engineering Mechanics Department.

2. 2
Table 1. Primary and secondary RSO simulation parameters at TCA.
Primary Secondary
Semi-major axis 6,984.50 km 6,984.88 km
Inclination 97.88 deg 82.35 deg
Eccentricity 0.0008 0.0007742
Argument of Latitude 268.80 deg 267.57 deg
Node (RAAN) 177.07 deg 22.30 deg
Period (Tp) 96.819 min 96.827 min
Mass 2,000 kg 3 kg
Area / Radar Cross Section (RCS) 15 m2
0.0136 m2
Type Active satellite Fragmentation debris
Hard-body radius (incl. safety margin) 15 m 0.0583 m
(a) (b)
Figure 1. (a) Equatorial view of simulated conjunction. (b) View from South Pole.
A priori covariance matrices were selected based on given CDM values and general knowledge of satellite tracking
capabilities. Many satellite operators artificially inflate their covariance as a safety margin, and such a process was
also included in this analysis. Diagonal variance values were defined in the RIC frame and then rotated to ECI before
propagating for three orbital periods. The force model used was the same as in a majority of the course homework
assignments – two-body, J2, and J3 gravity along with an exponential drag model. Table 2 includes the uncertainty
values along the diagonals at one orbital period before TCA in the form of a single standard deviation (not variance).
Table 2. A priori covariance diagonal standard deviations.
Primary Secondary
Radial position [m] 15 50
In-track position [m] 300 1500
Cross-track position [m] 20 72
Radial velocity [mm/s] 1 9
In-track velocity [mm/s] 2 15
Cross-track velocity [mm/s] 1 8
Utilizing the probability of collision calculations by Alfano and Chan discussed in lecture produced the values
listed in Table 3. Other common parameters used to analyze a conjunction such as Mahalanobis distance and miss

3. 3
distance were also generated. It is reported that CSpOC uses a probability of collision calculation quite similar to
Foster [1]. Thus, to add another method to compare against for the simulated conjunction, the general Foster form was
implemented as per [2] in Eq. (1) which also matches the lecture slides.
𝑃𝑐 =
1
2𝜋𝜎𝑥 𝜎 𝑦
∫ ∫ 𝑒
(−
1
2
[(
𝑥−𝑥̅
𝜎 𝑥
)
2
+(
𝑦−𝑦̅
𝜎 𝑦
)
2
])
√𝑂𝐵𝐽2−𝑥2
−√𝑂𝐵𝐽2−𝑥2
𝑂𝐵𝐽
−𝑂𝐵𝐽
𝑑𝑦𝑑𝑥 (1)
The symbol definitions for Eq. (1) match the form given in the lecture slides and [2]. The double integral was
evaluated numerically using MATLAB’s dblquad function with a default tolerance of 1E-06. Per the Table 3 results,
all three methods for calculating probability of collision agree quite nicely. A visualization of the three-dimensional
covariance ellipsoids is shown in Fig. 2 to get a sense of the shape, scale, and orientation of the conjunction event.
Table 3. Conjunction simulation results.
Value
Pc Alfano 1.144446e-04
Pc Chan 1.144741e-04
Pc Foster 1.137170e-04
Mahalanobis distance 0.955013
Miss distance [m] 203.2237
Approach angle [deg] 155.0
Figure 2. Visualization of the primary (smaller of the two) and secondary 3σ covariance ellipsoids at TCA.
III. Conjunction Plane Analysis
A more common domain used by satellite operators to visually analyze a conjunction is the conjunction plane. Per
the author’s experience in industry, the particular conjunction plane used is the Near Vertical Near Horizontal (NVNH)
frame. To inject some operational realism into generation of such a plane, the primary and secondary states,
covariances, and miss distance will match the format given in a CDM. Per [3], the states are given in True of Date
Rotating (TDR), also called Earth Fixed Greenwich (EFG), or more commonly Earth Centered Earth Fixed (ECEF),
coordinates. The relative vector provided for miss distance is given in asset-centered RIC (UVW) coordinates and
covariances are also given in their respective RIC frames. Thus, to match this format, the following rotation sequences
were employed,

4. 4
𝑟𝐸𝐶𝐸𝐹 = [𝑊(𝑡) 𝑇][𝑅(𝑡) 𝑇][𝑁(𝑡) 𝑇][𝑃(𝑡) 𝑇]𝑟𝐸𝐶𝐼 (2)
𝑣 𝐸𝐶𝐸𝐹 = [𝑊(𝑡) 𝑇][[𝑅(𝑡) 𝑇][𝑁(𝑡) 𝑇][𝑃(𝑡) 𝑇]𝑣 𝐸𝐶𝐼 − 𝜔 𝐸 × [𝑊(𝑡)]𝑟𝐸𝐶𝐼] (3)
where the preceding rotation matrices are defined in [4]. It’s noted that the inclusion of polar motion, precession, and
nutation are likely a bit overkill for the accuracy requirements in this study. To get the miss distance in asset-centered
RIC,
𝑟 𝑚𝑖𝑠𝑠,𝑅𝐼𝐶 = 𝛽 𝑅𝐼𝐶
𝐸𝐶𝐼
𝑟 𝑚𝑖𝑠𝑠,𝐸𝐶𝐼 (4)
where the beta rotation matrix is defined as,
𝛽 𝑅𝐼𝐶
𝐸𝐶𝐼
= [
𝑅 𝑥 𝑅 𝑦 𝑅 𝑧
𝐼𝑥 𝐼 𝑦 𝐼𝑧
𝐶 𝑥 𝐶 𝑦 𝐶𝑧
]
(5)
and each component of the beta matrix in Eq. (5) is the corresponding ECI direction cosine component of the radial,
in-track, and cross-track unit normal vectors respectively [5]. To get the RIC covariance matrices, the following
sequence is performed,
𝑃𝑅𝐼𝐶 = 𝛽 𝑅𝐼𝐶
𝐸𝐶𝐼
𝑃𝐸𝐶𝐼 𝛽 𝑅𝐼𝐶
𝐸𝐶𝐼 𝑇 (6)
which can be used for both primary and secondary RSOs. Now that we have a starting point as it would be in receiving
a CDM, more rotations are to be performed. Depending on the formulation and preferred coordinate frames, there are
multiple ways to produce a conjunction plane. To get the NVNH frame, ECEF coordinates used. Thus, after rotating
both covariances to ECEF we can add the RIC miss vector to the primary to get the secondary state in the ECEF frame
of the primary, of course after rotating the RIC miss vector to ECEF again. All that is left now is to transform the
secondary covariance into the primary ECEF frame. Using the unit vectors for the primary in ECEF coordinates to
form another beta matrix allows us to rotate the secondary’s covariance into the proper relative frame. From here we
take the following projection,
𝑟̂𝑟𝑒𝑙 = ‖𝑟𝑝𝑟𝑖𝑚𝑎𝑟𝑦,𝐸𝐶𝐸𝐹‖ (7)
𝑣̂ 𝑟𝑒𝑙 = ‖𝑣 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦,𝐸𝐶𝐸𝐹 − 𝑣 𝑝𝑟𝑖𝑚𝑎𝑟𝑦,𝐸𝐶𝐸𝐹‖ (8)
𝑛̂ 𝑟𝑒𝑙 = ‖𝑟̂𝑟𝑒𝑙 × 𝑣̂ 𝑟𝑒𝑙‖ (9)
𝑇 = [
𝑛̂ 𝑟𝑒𝑙
𝑇
𝑟̂𝑟𝑒𝑙
𝑇 ] (10)
𝑟𝑟𝑒𝑙,𝑁𝑉𝑁𝐻 = 𝑇(𝑟𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦,𝑟𝑒𝑙,𝐸𝐶𝐸𝐹 − 𝑟𝑝𝑟𝑖𝑚𝑎𝑟𝑦,𝐸𝐶𝐸𝐹) (11)
𝑃𝑝𝑟𝑖𝑚𝑎𝑟𝑦,𝑁𝑉𝑁𝐻 = 𝑇𝑃𝑝𝑟𝑖𝑚𝑎𝑟𝑦,𝐸𝐶𝐸𝐹 𝑇 𝑇
(12)
𝑃𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦,𝑁𝑉𝑁𝐻 = 𝑇𝑃𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦,𝐸𝐶𝐸𝐹 𝑇 𝑇 (13)
where 𝑟𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦,𝑟𝑒𝑙,𝐸𝐶𝐸𝐹 is the relative position of the secondary in the ECEF frame of the primary. Putting this all
together and plotting the primary and secondary yields the results in Fig. 3. As both orbits are near circular, it makes
intuitive sense that they display a symmetry about their axes. As expected, the secondary object has a larger overall
covariance. The predicted miss distance is visualized by the secondary mean’s distance from the origin, which is the
location of the primary RSO. In this case, both covariances were propagated with the standard “EKF-like” method
using the state transition matrix (STM) as defined in Eq. (14) below,

5. 5
Pk+1 = ϕ(tk+1, tk)PkϕT(tk+1, tk) + Γ(tk+1, tk)QkΓT(tk+1, tk) (14)
where the symbols used in Eq. (14) are defined as in [6]. No process noise was used in the propagation for this study.
Figure 3. NVNH conjunction plane at TCA with primary RSO at origin.
One way to better visualize what “near horizontal” and “near vertical” represent is to think of near vertical as radial
and near horizontal as in-track when both orbits are precisely circular as in this case. The near horizontal axes will
transform into a combination of in-track and cross-track depending on the relative velocity vector. The NVNH plane
has the relative velocity coming into and out of the plane. For more eccentric orbits or secondaries with relative
velocities out of plane with the primary’s, the secondary covariance major and minor axes will begin to display an
apparent rotation relative to the NVNH axes. An example of this effect is seen in Fig. 4 with the same conditions after
one orbital period where the secondary was simulated to have a slightly higher eccentricity. A side effect of the relative
velocity change is that the in-plane secondary covariance size has increased based on the given orientation of the
ellipsoids.
In a real-world scenario, the probability of collision values listed in Table 3 would be high enough to warrant close
monitoring as TCA approaches. If the values began to trend in the wrong direction, a collision avoidance maneuver
(COLA) may be executed. To demonstrate the effectiveness of a small magnitude delta-v, a +0.30 m/s along-track
burn was modeled at two periods to TCA and the results of the maneuver are shown in Fig. 5, as well as in Fig. 6 in
three dimensions.

6. 6
Figure 4. NVNH conjunction plane at TCA where the secondary has a more noticeable eccentricity.
Figure 5. NVNH conjunction plane with secondary ellipse including COLA burn.

7. 7
Figure 6. Results of the COLA mitigation burn showing a 3σ miss in three-dimensions.
As can be seen in Table 4, the +0.30 m/s along-track COLA maneuver dropped the probability of collision,
Mahalanobis distance, and miss distance down to acceptable levels – effectively mitigating the conjunction. In reality,
an operator likely would not wait until two orbital periods to TCA to perform a maneuver as it leaves little room for
error. The further out an operator plans a maneuver, the smaller the magnitude needed. The last-minute burn was
modeled here to study how close one can get to TCA while still avoiding a collision with a relatively small delta-v
magnitude.
Table 4. COLA maneuver results.
Value
Pc Alfano 3.039374e-09
Pc Chan 2.798380e-09
Pc Foster 3.005086e-09
Mahalanobis distance 61.25
Miss distance [m] 10,304.80
Figure 5 also serves to promote caution when using a two-dimensional plane to analysis a three-dimensional event.
The slight shift in the NVNH plane does not seem like enough to mitigate the conjunction, but the orbit has phased
enough such that it shifts in the dimensions not displayed as in Fig. 6. In other scenarios or projection variations, one
may see a larger displacement of the COLA 3σ covariance offset such that it’s easier to see that the risk has been
reduced adequately from a visual standpoint. For instance, the conjunction plane defined in the lecture notes shows a
larger visual displacement (aside from orientation differences) in Fig. 7. It should be noted that the axes labels for Fig.
7 are defined in terms of the primary major and minor axes directions.
Another common practice in industry is to combine the covariances in the conjunction plane and plot them centered
at the secondary mean. This procedure will give a better view of the overall uncertainty induced by the secondary
object. To study the trends over time as TCA approaches, it is also common to plot updated 3σ error ellipses as new
measurements are taken. Typically, the color of the ellipse will change proportional to time until TCA. This gives
space operators an easy visual queue as to how the secondary RSO may or may not be drifting relative to the primary
as the solution converges. Instead of implementing the updated measurement trend, the author decided to focus on

8. 8
other unique trends to be studied in coming sections instead of repeating something that is already common practice.
A good reference and industry example of the ellipse trending to TCA is discussed by Archuleta and Nicolls in [7].
Figure 7. Conjunction plane as defined in Orbital Debris lecture. The orange ellipse includes a COLA burn.
IV. Non-Gaussian Conjunction Plane Analysis
Up to this point, all the conjunction plane analysis has been completed with the linear uncertainty propagation
technique defined in Eq. (14). It seems to be industry standard to utilize the linear approximation as well. As Eq. (14)
does not contain any information on higher order moments, it is of interest to study propagation techniques that contain
this information and project them into the NVNH conjunction plane to see how they may affect operational decision
making. Following the same steps as in Section III, the unscented transform (UT) [8], an adaptive entropy-based
method (AEGIS) with a three-component split [9], and Monte Carlo techniques were propagated with the same initial
conditions, time interval, and then projected into the NVNH plane as in Fig. 8 and Fig. 9.
The unscented transform and EKF-like linear propagation of uncertainty match almost exactly. This was somewhat
surprising as intuitively I would have expected at least a slight difference caused by the sigma point propagation
accounting for the non-linear dynamics more so than the linear STM. This will be discussed more in coming sections.
Both the Monte Carlo points and Gaussian Mixture Models (GMMs) begin to spread mainly in the along-track
direction. Each spread has components in the radial and cross-track direction as well due to a more realistic sampling
of the probability density function (PDF), i.e. the “banana” begins to show up after three orbital periods.
A key distinction to note for the Monte Carlo and GMM methods is that not all of the points projected into the
NVNH plane are at TCA anymore. Due to the out of plane distance not visualized in Fig. 8 nor Fig. 9, it is still difficult
to grasp the conjunction event as it relates to time as the fourth dimension. Again, depending on covariance size,
relative velocity vector orientation, and secondary eccentricity, the Monte Carlo and GMM methods in the NVNH
plane may prove a bit more useful than were proved to be in this circular conjunction case. As for the COLA maneuver
designed in Section III to mitigate the conjunction event, knowledge of the Monte Carlo and GMM uncertainty
representations in the NVNH plane likely would not have changed the operational decision-making process (at least
for this scenario).
Some of the difficulties encountered when trying to visualize non-Gaussian error volumes in a two-dimensional
conjunction plane can be resolved with three-dimensional probability of collision methods. Such a method will be
discussed in Section V.

9. 9
Figure 8. NVNH plane with Monte Carlo and UT uncertainty included.
Figure 9. NVNH plane with AEGIS GMMs and UT uncertainty included.

10. 10
V. Probability of Collision from Non-Gaussian Error Volumes
The logical next step after projecting non-linear, non-Gaussian uncertainty methods into the conjunction plane is
to determine how these techniques may affect the probability of collision calculation. The covariance generated from
the unscented transform can be input into the same Alfano, Chan, and Foster methods as before to produce a Pc value.
The AEGIS GMM outputs can also be processed with the prior methods, but must be combined as follows from [10],
𝑃𝑐,𝐺𝑀𝑀 = ∑ 𝛼𝑖 𝑃𝑐,𝑖
𝐿
𝑖=1
(15)
which is simply the sum of each GMM probability of collision multiplied by its weight. This combination is possible
because integration is a linear operator. To produce a Pc value from the Monte Carlo distribution, a more involved
process must be followed. Ghrist and Plakalovic describe such a method that is implemented as follows from [11].
The process involves perturbing both the primary and secondary RSO from their calculated means at TCA by
sampling from their Gaussian (or non-Gaussian) distributions and then determining if the new miss distance is less
than the combined hard body radius. An important note is that after sampling from the respective distributions, the
values are not at TCA anymore and must be propagated to their new closest approach. Due to the short encounter time
spans, a simple analytical two-body propagation routine using the well-known Lagrange f and g coefficients in terms
of the universal anomaly was implemented. The particular form used is from Curtis and is reproduced below for
reference [12].
𝑟 = 𝑓𝑟0 + 𝑔𝑣0 (16)
𝑣 = 𝑓̇ 𝑟0 + 𝑔̇ 𝑣0 (17)
The f and g terms as well as their derivatives are defined as follows in Eq. (18-21),
𝑓 = 1 −
𝜒2
𝑟0
𝐶(𝛼𝜒2) (18)
𝑔 = ∆𝑡 −
1
√ 𝜇
𝜒3
𝑆(𝛼𝜒2) (19)
𝑓̇ =
√ 𝜇
𝑟𝑟0
[𝛼𝜒3
𝑆(𝛼𝜒2) − 𝜒] (20)
𝑔̇ = 1 −
𝜒2
𝑟
𝐶(𝛼𝜒2) (21)
where r and r0 are the initial and updated position, χ is the universal anomaly, α is the inverse of the semi-major axis,
and the S( ) and C( ) functions are the Stumpff functions. The newly sampled states were then propagated forward and
backwards by a designated time interval and scanned for the new TCA. Once the updated TCA was discovered, a miss
distance was calculated and checked to determined if it was less than the combined hard body radius – thus signaling
whether a collision has occurred or not. Again per [11], the probability of collision is then simply calculated as,
𝑃𝑐,𝑀𝐶 =
𝑥
𝑛
(22)
where x is the number of collisions and n is the number of Monte Carlo runs. This technique could have also been run
from the initial conditions and then propagated to TCA for a more realistic result, but the authors of [11] show this is
unnecessary and sampling at TCA is still a valid method. Also, for computational run time reasons, sampling at TCA
is highly desired.
Results of all probability of collision calculations are listed in Table 5. The initial values from the linear techniques
are included again for quick reference. The Monte Carlo technique was also reproduced by sampling from a non-
Gaussian uncertainty volume produced from the AEGIS GMM with the “waterfall” technique utilized from the Orbital

11. 11
Debris course homework six. All results use the same initial conditions, a priori covariance, and propagation interval
defined in prior sections. The Monte Carlo runs each used 50,000 samples and searched for TCA ± 1.0 seconds in
either direction from the original TCA. At LEO velocities, this corresponds to about 15 km searched along each RSO’s
trajectory. Given the relative sizes and orientations of the ellipsoids in Fig. 2, this search space seems appropriate. The
time step used for the search corresponded to roughly one-meter resolution for checking the miss distance. Using
MATLAB’s parallel processing suite allowed for 50,000 runs to complete in about eight minutes. Some effort was
spent optimizing the Lagrange coefficient and TCA search routine. A 200,000 and one million run simulation were
conducted and they both yielded results of negligible difference. It seems as if the number of runs required for Pc
convergence is approximately 50,000.
Table 5. Variations in Pc methods.
Method Value
Pc Alfano, linear 1.144446e-04
Pc Chan, linear 1.144741e-04
Pc Foster, linear 1.137170e-04
Pc Alfano, unscented 1.231718e-04
Pc Chan, unscented 1.232087e-04
Pc Foster, unscented 1.223886e-04
Pc Alfano, GMM 4.378386e-04
Pc Chan, GMM 4.436411e-04
Pc Foster, GMM 4.349804e-04
Pc Monte Carlo, Gaussian sample 8.600000e-04
Pc Monte Carlo, GMM sample 1.440000e-03
It is interesting to note the Monte Carlo and GMM methods produce larger Pc values than the other methods. The
GMM-sampled Monte Carlo run is actually a full order of magnitude higher than most techniques. It seems as if high
Pc values correlate with non-Gaussianity, at least in this case. The Monte Carlo results are shown graphically in Fig.
10 and Fig. 11. Hard body radius and thus collision threshold is indicated by the dotted line in Fig. 10. As most of the
values are on the same order of magnitude, it is difficult to say Table 5 results would sway any operational decisions.
Depending on how an operator defines a maneuver threshold, crossing into the 1e-03 range may cause for an automatic
decision to trigger a COLA mitigation burn.
Figure 10. Computed minimum miss distance versus run number during TCA search.

12. 12
Figure 11. Instantaneous Pc calculation during a 5e4 sample Monte Carlo simulation.
VI. Gaussianity and Probability of Collision Trends
To further investigate the Section V results and to study how non-Gaussianty affects Pc results, more simulations
were run. To better assess the non-Gaussianty of a PDF, the Likelihood Agreement Measure (LAM) is introduced.
The LAM is defined by DeMars, et. al. in [9] as below,
ℒ(𝑝, 𝑞) = ∫ 𝑝(𝑥)𝑞(𝑥)𝑑𝑥 (23)
where q(x) is the defined using the Dirac delta distribution centered at each Monte Carlo point μi with weight γi equal
to 1/K as,
𝑞(𝑥) = ∑ 𝛾𝑖 𝛿(𝑥 − 𝜇𝑖)
𝐾
𝑖=1
(24)
and the p(x) distribution is defined as the GMM in Eq. (25). In this study, Eq. (25) is modified to allow use in
calculating the LAM for both the EKF-like linear covariance and unscented transform covariance. This is
accomplished by simply summing over the single covariance (i.e. a single GMM).
𝑝(𝑥) = ∑ 𝛼𝑖 𝑝 𝑔(𝑥; 𝑚𝑗, 𝑃𝑗)
𝐿
𝑖=1
(25)
Combining everything together results in the Likelihood Agreement Measure as defined in Eq. (26) below,
ℒ(𝑝, 𝑞) = ∑ ∑ 𝛾𝑖 𝛼𝑗 𝑝 𝑔(𝑥; 𝑚𝑗, 𝑃𝑗)
𝐿
𝑗=1
𝐾
𝑖=1
(26)

13. 13
Use of the LAM will generally be relative to the Monte Carlo distribution. In some parts of this study and in the
original text, the LAM is normalized to an AEGIS 3-split or 5-split method [13]. Thus, a value of unity would indicate
a PDF that is very realistic and exactly similar to the Monte Carlo distribution. The lower the value, the less realistic
the propagated PDF.
Calculating the LAM for the linear, unscented transform, GMM, and Monte Carlo methods again with all
conditions remaining as they have throughout this study yielded the results in Table 6. Unsurprisingly, nothing stayed
true to the Monte Carlo distribution due to the system’s non-linearities. The three propagation techniques still agree
well after three orbital periods.
Table 6. LAM values at Tp = 3 periods.
Method Value
Linear 0.254687
Unscented transform 0.254453
AEGIS GMM 0.254438
Monte Carlo 1.00
Two of the main factors affecting a PDF’s ability to stay true to the Monte Carlo PDF are the a priori uncertainty
and propagation interval. To stress the LAM a bit and to study where each technique begins to diverge, another
simulation was run for a range of propagation intervals. In this case, an equatorial orbit was selected for the secondary
to support easier visualization of the various PDFs. The a priori covariance was also inflated to have a 1.0 km2
variance
on all the position diagonals and a 1.0 m2
/s2
variance on the velocity diagonals. All other simulation parameters
remained the same as in prior sections. The Monte Carlo Pc calculation was not performed for the long duration
propagation as the computation time required to search for TCA increases with a growing covariance size. As the
probability of collision decreases in order of magnitude, the number of Monte Carlo runs needs to increase by an order
of magnitude as well to be able to calculate something nonzero.
As seen in Fig. 12, as the propagation interval increases, the LAM of each of the three techniques decreases. The
GMM performs the best and the unscented transform has the lowest LAM value after eight orbital periods. The
unscented transform has the lowest LAM simply due to the sigma point sensitivity to the non-linear dynamics, an
expected result. The results of Fig. 12 are reproduced in Fig. 13 but scaled to the GMM method.
Figure 12. Decreasing LAM trend as propagation interval increases.

14. 14
Figure 13. Similar plot to Fig. 12 except all LAM values are scaled to the GMM method.
Interestingly, the unscented and linear methods initially perform better than the GMM before eventually diverging
after about three orbital periods. The linear case likely has higher initial LAM because the sigma points used in the
other two methods are more sensitive to the non-linear dynamics again (drag included) and thus they “spread out”
quicker. To study how Gaussianity affects the probability of collision calculation, the LAM was plotted against Pc to
observe any correlation. Results of this simulation are shown in Fig. 14 below.
Figure 14. Study of how Gaussianty (LAM) affects various probability of collision methodologies.

15. 15
(a) (b)
(c) (d)
(e) (f)
Figure 15. Equatorial plane projection of various uncertainty methods used to produce LAM and Pc values
for the following propagation intervals: (a) Five minutes, (b) Tp = 1, (c) Tp = 3, (d) Tp = 4, (e) Tp = 6, (f) Tp = 8.
The trend in Fig. 14 indicates that as a PDF approaches the “true” Monte Carlo distribution, the probability of
collision will increase. This suggests that the higher order moments not accounted for in the EKF-like, unscented
transform, and even the GMM uncertainty representations will artificially mask the true magnitude of the collision
probability. This conclusion is reflected in the Table 5 results as well – the more moments you include, the greater the
probability of collision. This trend can also be seen visually in Fig. 15. As the non-Gaussianty of a PDF grows, it
encapsulates more of the range of potential position values that do not contain Monte Carlo points and thus when
integrating over that space, many of those values have very low probabilities.

16. 16
Using probability of collision alone to assess conjunctions is insufficient. Assuming no measurements, the
downward trend in Fig. 16 seems to indicate that as the propagation interval increases, the probability of collision will
decrease. While this is true from a mathematical standpoint, the mean values of the primary and secondary in the Fig.
16 scenario stayed to within 200 meters. The growing covariance bounds act to artificially reduce the probability of
collision, hence why the use of metrics such as miss distance and Mahalanobis distance are recommended to gain a
better assessment of a conjunction event.
To further comment on Figs. 12-16, a plot of the number of Gaussian Mixture Models required during the
propagation phase is included in Fig. 17. Unlike in the course homework, the inclusion of drag greatly increases the
number of models needed. The computation time increased so much so for the large number of GMMs that a nine-
period simulation was unable to finish in any reasonable timeframe. The maximum value required was 4,043 for the
eight-period propagation.
Figure 16. Various probability of collision method trends over an eight-period propagation interval.
Figure 17. Increasing GMM trend as propagation interval increases.

17. 17
VII. Conclusion
The results of studying non-Gaussian error volumes and non-linear propagation techniques as they relate to
conjunction assessment demonstrate that it is important for a satellite operator to be aware of their realism during
operational decision-making. It was shown that the Alfano, Chan, Foster, Gaussian mixture, and Monte Carlo
probability of collision formulations generally produce values that agree. It seems as if the probability of collision
correlates positively with higher moment uncertainty representations. It was shown that a small delta-v can be used to
mitigate high-risk conjunction events and that visual trend analysis in the NVNH (or any) conjunction plane should
be used with caution. The Likelihood Agreement Measure was generated for the various uncertainty forms over time
and cross-referenced with probability of collision values to study correlation. It was found that LAM and probability
of collision also correlated positively, supporting the results from the prior sections. It was also noted that the
probability of collision calculation by itself is not a sufficient metric to assess a conjunction and that miss distance
and Mahalanobis distance are recommended supplementary analysis tools. Ultimately, understanding the various
trends and techniques presented herein for probability of collision and uncertainty propagation support reducing
operational risk. Increased knowledge of any non-Gaussian effects in the conjunction assessment process will support
maintenance of healthy spacecraft and long-term space environment sustainability.
Acknowledgments
The author would like to thank Dr. Jones for a great semester and a very well taught, stimulating course. Also, a
big thank you is deserved of my colleagues at Maxar Technologies, Jonathan Nikkel and Adam Archuleta, for fruitful
late-night discussions on reference frames and probability of collision methods.
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