Since the first orbital launch in 1957, the number of artificial objects in Earth orbit has been steadily increasing. This

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1 SpaceOps Conferences 28 May - 1 June 2018, 2018, Marseille, France 2018 SpaceOps Conference / Collision probability through time integration Implementation and operational results Vincent Schaeffer a, Sophie Laurens a, Pierre Seimandi b, and Florian Delmas a a Centre National d Etudes Spatiales (CNES), Toulouse, France b GMV Innovating Solutions, Toulouse, France In 2012, V.T. Coppola proposed a new formulation for the computation of the probability of collision between two satellites based on a time-integration of the probability density flux entering the surface the hard body radius sphere. This formulation takes into account the evolution of the covariance over time, as well as the velocity uncertainty. It should therefore give access to a more realistic probability of collision, especially for low-velocity encounters. This paper describes the work currently done at CNES on the method, which was tested on more than generated cases, in addition to a selection of past collision risks. The paper highlights several issues encountered during the implementation of the method and focuses on a few conjunctions of interest. I. Introduction Since the first orbital launch in 1957, the number of artificial objects in Earth orbit has been steadily increasing. This has led to a corresponding increase in the number of threats to active satellites from hyper-velocity collision, putting in jeopardy crucial services that benefit human society. To mitigate such risks, one of the approaches commonly adopted consists in the planning and the execution of collision avoidance maneuvers for operational satellites. This relies on the analysis of the probability of collision between the satellite of interest and any space object entering a safety volume around it. An accurate estimation of this probability of collision is thus primordial for the management of the spacecraft population. An under-estimated probability could lead to serious risks being ignored, while an over-estimated one could trigger unnecessary maneuvers, leading to mission perturbations and reducing of the operational lifespan of the satellite. To estimate this probability, CNES currently relies on a simple model based on the linearized states of the objects at time of closest approach (TCA) [1, 2]. Assuming that the encounter is of short-term nature, the relative trajectory is locally approximated as rectilinear, the positional uncertainty is considered as time-independent and the velocity uncertainty is neglected. The probability of collision can then be computed by a two-dimensional integration in the encounter plane, making this method easy to implement and fast to compute (an important feature given the number of alerts treated daily). This 2D-model is however not applicable to every encounter. By construction, it is not adapted to low-velocity encounters, where the two objects evolve close to one another for long periods of time. During these encounters, the relative trajectory and the states uncertainties can change significantly: the assumptions described above are simply not valid anymore. More importantly, significant discrepancies have also been observed for several high-velocity alerts. While the 2D-probability computed for these alerts was negligible, the probability returned by a Monte-Carlo (MC) algorithm was sufficiently high to warrant an evasion maneuver. This is problematic operationally since it implies that a serious collision risk was missed. For these reasons, alternative methods for the computation of the probability of collision were investigated. Among them, the method proposed by Coppola [3] in 2012 is of particular interest. This method, referred to here as the 3D-probability method, is based on the integration of the flux of a time-dependent PDF entering the surface the hard body radius (HBR) sphere over a given time-span. Compared to the 2D-probability method currently used operationally, it takes into account the velocity uncertainty and the evolution of the covariance over time. It should therefore give access to a more realistic probability of collision, especially for low-speed encounters, and provide an alternative to expensive Monte-Carlo computations. More precisely, the method defines the probability density flux penetrating the HBR sphere at each time t. The probability of collision is then computed by integrating this flux over the time-span considered. As an example, figure 1 represents the evolution of the flux during a high-velocity encounter. The bell-shaped flux seen Corresponding author: vincent.schaeffer@cnes.fr Corresponding author: sophie.laurens@cnes.fr 1 Copyright 2018 by CNES. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2 here is typical of this kind of encounters, for which the main contribution to the probability of collision occurs over only a few seconds. Fig. 1 Typical evolution of the probability density flux during a high-velocity encounter (TCA is at t = 0) Coppolas formulation relies on the following hypotheses: The uncertainty on the position and velocity of each object remains Gaussian at each time: For our applications, the initial covariance matrices are usually propagated linearly, through a transition matrix. This assumption is thus met by construction. One should however be careful when propagating over long periods of time. Although a covariance matrix could still be computed, there is no guarantee that it would reflect the real uncertainty (which may not even be Gaussian anymore). Here, we will assume that this type of propagation is valid for the time-spans considered; The dynamic model and the probability distribution function of each object are independent: This is a valid assumption for most encounters. However, it would not be appropriate for objects flying in formation, with trajectory correction based upon the location of other objects; Each collision trajectory must cross the combined HBR sphere instantly: This assumption means no collision trajectory must linger on the surface of the HBR sphere for any amount of time. While it is difficult to verify this assumption, we expect that it will be met in most cases; Each collision trajectory crosses the combined HBR sphere only once: Although we expect this assumption to be met for most encounters, it can be disputable for objects staying close to each other for a large period of time, such as objects flying in formation or for some geostationary conjunctions; While these assumptions seem reasonable in most cases, it is difficult to assess beforehand whether they are actually met. Failing that, the method was tested against more than generated collision risks, as well as on a selection of past collision risks. The goal of this first validation was to apply Coppola s method on a large variety of cases in order to identify implementation issues and abnormal behaviors. After a short description of the cases tested in section II, this paper focuses on a few points of interest highlighted by our study. More specifically, section III discusses the formulation for the estimation of the encounter duration proposed by Coppola in a second paper [4]. Section IV then discusses the use of Lebedev s quadrature [5] for the integration over the surface of the HBR sphere and highlight a possible lack of convergence for some configurations. Finally, section V presents two high-velocity cases resulting in a very under-estimated 2D-probability, despite being significant collision risks. II. Test cases description As said in section I, Coppola s method have been tested on more than generated collision risks (28512 exactly). Each case defines the combined HBR, the mean states of the objects at TCA and the corresponding position and velocity uncertainties. These collision risks correspond to 44 unique encounters with respect to the orbital parameters. For each one of these encounters, three combined HBR values (1 m, 10 m or 100 m) are then considered, as well as various initial uncertainties. For the secondary object, the 1-sigma volume of the covariance ranges from 10 m 3 to m 3, every intermediate order of magnitude being covered. The aspect ratio of this volume (i.e. the degree of stretching in a specific direction) is between 1 to 1000 for the most part, but it can reach about for a few cases. The covariance of the primary object is negligible. 2

3 A few key-characteristics on these encounters are listed in table 1. This table shows that, although only 44 unique encounters are considered, a wide-variety of conjunctions are actually represented by these cases. Note that the collision risks tested do not aim to be always realistic (in particular with regard to the covariance used). The main goal of these cases was to test the limits of the method to identify abnormal behaviors or implementation issues. In addition to these generated cases, a selection of past collision risks were also studied. When discussed in this paper, these cases are indicated as "real collision risk". All the results presented here use a basic Keplerian propagation for both the mean orbits and the covariance states (only the central gravitational force is considered). Table 1 Generated cases characteristics Field Values Combined HBR 1 m, 10 m and 100 m Miss distance 1 m, 10 m, 100 m and 2000 m Relative speed 0.79 m s 1 to m s 1 Orbit classification LEO-1/2/3, GEO, HEO Altitude 340 km to km Eccentricity to 0.67 Inclination 0.11 to The probabilities computed for these generated cases are compared figure 2, with the 2D-probability on the x-axis and the 3D-probability on the y-axis. The solid black line indicates a 2D-probability equal to the 3D-probability and the dashed black lines mark a difference of one order of magnitude between the two probabilities. The dotted gray lines represent the 10 8 threshold for both probabilities (below this value, we consider the probability as negligible). High-velocity encounters are plotted in orange and low-velocity encounters in blue, with a threshold of 50 m s 1 to discriminate the two kinds of encounter. In addition to these two graphs, table 2 presents the breakdown of the cases according to the relative error between the 2D-probability and the 3D-probability. As a note, % of these cases are high-velocity encounters and % are the low-velocity encounters. As expected, the two probabilities are in good agreement for most high-velocity encounters. Only 2.08 % yield a difference of more than one order of magnitude and most of these cases return a negligible probability with both methods. However, a few high-velocity cases yield a negligible 2D-probability despite a relatively high 3D-probability. A group in particular returns a 3D-probability above 10 5, which is almost high enough to warrant an evasion maneuver. These cases will be further analyzed in section V, along with a past collision risk exhibiting the same problem. For low-velocity encounters, results are still in relatively good agreement % of the cases result in less than 10% difference and % result in a difference lower than one order of magnitude. However, a notable number of cases yield a negligible 2D-probability and a significant 3D-probability, or vice-versa. These differences can be explained by the nature of these encounters. The 2D-probability model is not adapted and the probability of collision is often poorly evaluated. Table 2 Breakdown of the generated cases according to the relative error on the probability of collision Relative error High-speed Low-speed Less than 1 % % % Between 1 % and 10 % 3.05 % % More than 10 % and less than one order of magnitude 1.79 % % More than one order of magnitude 2.08 % 8.87 % More than one order of magnitude and any P c > % 4.08 % 3

4 (a) All cases (b) Both 2D and 3D probabilities above Fig. 2 Comparison of the 2D and 3D-probability of collision for the generated cases. High-velocity encounters (> 50 m s 1 ) are plotted in orange and low-velocity encounters ( 50 m s 1 ) in blue. III. Encounter duration As described in section I, the formulation proposed by Coppola consists of a time-integration of the probability density flux. To estimate the probability of collision, the time-span to integrate must thus be defined. Coppola addresses this particular problem in a second paper [4], in which he proposes a formulation that characterizes the duration of the encounter using the short-term hypotheses. This formulation yields two parameters, τ 0 and τ 1, that define the duration of the encounter relatively to the TCA. These parameters are: τ 0 = 2α c σ v + q 0 δ max and τ 1 = + 2αc σ v + q 0 δ min, v 0 v 0 where v 0 is the relative speed at TCA and σ v is the relative position uncertainty normal to the encounter plane. The terms q 0, δ min and δ max depend on the relative position in the encounter plan, the combined HBR and the initial uncertainty. The terms δ min and δ max are often negligible compared to σ v and the parameter α c controls the length of the time span returned, while the term q 0 defines its shift relatively to the TCA. Note that the parameter α c must be high enough to ensure a low approximation error under the short-term encounter hypotheses. Coppola indicates in his paper that a value of is enough to yield an error below Here, we chose to use a higher value: n σ = 2α c = 15 (i.e. α c = 10.61). This formulation works well for the vast majority of the cases studied. Figure 1 is representative of the result obtained when it is applied to high-velocity encounters. The time span returned is shifted relatively to the TCA (located at t = 0) and centered on the bell-shaped flux. The shift of the interval computed by the method is however incorrect in a few cases, resulting in an under-evaluated probability induced by a flux either truncated or completely missed. Figure 3 gives an example of a past collision risk impacted by this problem. It is a high-velocity encounter with a relative speed around 14.7 km s 1. While still under investigation, a few things can be considered to palliate the issue. The flux at the beginning and at the end of the integration interval could for example be compared to the maximum value computed in order to check whether the whole flux was actually integrated. If not, the integration interval could be extended and the probability recomputed. Another possible solution could be to automatically include the TCA (i.e. t = 0) in the interval since a lot of cases impacted by the problem (but not all) exhibit a shift too important and the encounter occurs somewhere between the time span computed and the TCA. Although Coppola s formulation is designed for short-term encounters, it can also be useful for low-velocity encounters. Figure 4 shows the probability density flux for a typical low-velocity encounter of a real GEO satellite. During this conjunction, the two objects evolve close to each other for a long period of time with a relative speed of 2 m s 1. The time span returned by Coppola s formulation is still correctly shifted (by about 1800 s) and the flux keeps a similar shape as the one observed for high-velocity encounters, the only difference being the duration of the encounter 4

5 Fig. 3 Truncated probability density flux (real collison risk) which last about 200 s instead of a few seconds. Despite the low-velocity, the relative trajectory near TCA is actually rectilinear, which could explain why Coppola s formulation is still applicable here. The probability density flux for another low-velocity conjunction is given figure 5. In this encounter, the secondary object orbits around the primary object for a long period of time, very closely at first and then slowly drifting away. This results in an important initial risk, followed by periodic spikes in the probability density flux, each one adding to the probability of collision computed. Defining an encounter duration is thus difficult and Coppola s formulation should not be used blindly. Because of this kind of encounter, low-velocity conjunctions should always be analyzed carefully (even if most of the risks studied so far behave as depicted figure 4). Fig. 4 Probability density flux for typical low-velocity encounter (real collision risk) Fig. 5 Probability density flux for a periodic, low-velocity encounter (generated case) 5

6 IV. Integration on the surface of the HBR sphere The formulation proposed by Coppola consists of a time-integration of the probability density flux over a chosen time-span. At each time t, this density probability flux can be computed through a simple two-dimensional integration over the surface of the combined HBR sphere. To compute this surface integral, Coppola suggests to use Lebedev s quadrature [5] with 1454 points (i.e. order 65), noting no significant difference when using higher orders. Since this quadrature uses points evenly distributed on the surface of the sphere, it requires fewer total grid points to achieve an accuracy similar to a standard (θ, φ) discretization and is thus very efficient. For our applications, this quadrature suffers however from several limitations. The determination of the grid weights is achieved by enforcing the scheme to integrate exactly all spherical harmonics up to a given order, which implies solving a set of nonlinear equations. Currently, these weights have been tabulated up to order 131 (5810 points). Although this order is more than enough for most cases, a default of convergence was observed for a few encounters. An example of such an encounter is given figure 6. It depicts the probability density flux obtained using Lebedev s quadrature with 1454 points (figure 6a) or 5810 points (figure 6b) and using a converged (θ, φ) discretization (figure 6c). For this particular case, the covariance in the plane orthogonal to the velocity is very small compared to the combined HBR (100 m) and Lebedev s quadrature is not able to reach convergence, even when using the highest order available. The various probabilities computed for this case are listed in table 3. The differences observed are relatively minor (less than 10 %) and it could be argued that Lebedev s quadrature is accurate enough. A possible default of convergence is however not acceptable operationally, as a poorly evaluated probability of collision could potentially lead to a missed collision risk. The use of an alternate integration scheme able to ensure convergence is thus recommended over Lebedev s quadrature. Even if higher orders were available, ensuring convergence with this quadrature would require either to determine the order to use beforehand (which can be difficult) or to test the available grids incrementally in order to check numerically if the wanted accuracy is reached (which is inefficient, since very few grid points can be reused). Table 3 Probabilities of collision corresponding to the case depicted figure 6 Method Quadrature Probability 2D-probability N.A D-probability (6a) Lebedev (1454 points) D-probability (6b) Lebedev (5810 points) D-probability (6c) Converged (θ, φ) discretization V. Under-estimated 2D-probability of collision Although the 2D and the 3D probabilities of collision are in good agreement for most high-velocity encounters (see section II), that is not systematically the case. In particular, a few conjunctions result in a 2D-probability largely under-estimated compared to the probability of collision returned by Coppola s method. The probability density flux corresponding to two of these cases are presented figures 7 and 8, the first being a generated case and the second being a past collision risk. In addition to these two graphs, table 4 lists the probabilities computed for these conjunctions, including the ones returned by a Monte-Carlo algorithm with 10 billions samples. For both cases, the 3D-probability and the Monte-Carlo probability are very close and sufficiently high to warrant further analysis, while the 2D-probability is under-estimated by at least one order of magnitude. The examples presented here illustrate the two kinds of behavior we observed so far. The probability density flux depicted figure 7 is comprised of two distinct bell-shaped section, located on both side of the TCA and shifted by less than a second. Conversely, the case depicted figure 8 behaves as most other high-velocity cases, characterized by a single bell-shaped flux near TCA lasting about one second. The mechanisms leading to such behaviors are not yet fully understood and must be further investigated. However, it is interesting to note that, for both cases, the covariance of the secondary object is particularly stretched in the direction of the orbital momentum. 6

7 (a) Lebedev s quadrature with 1454 points Fig. 6 (b) Lebedev s quadrature with 5810 points (c) Converged (θ, φ) discretization Probability density flux computed with: (6a) Lebedev s quadrature with 1454 points, (6b) Lebedev s quadrature with 5810 points, (6c) a converged (θ, φ) discretization (generated case) Fig. 7 Dual-bell shaped probability density flux for an encounter yielding a negligible 2D-probability (generated case) 7

8 Table 4 Probabilities of collision computed for the cases depicted figures 7 and 8 Figure 2D-Probablity 3D-Probability MC-Probability Figure Figure Fig. 8 Probability density flux for an encounter yielding negligible 2D-probability (real collision risk) VI. Conclusion & future work The formulation proposed by Coppola [3] for the computation of the probability of collision was tested on a wide-variety of cases, as well as on a few past encounters. As expected, the results for most high-velocity encounters are in good agreement with the 2D-probability of collision currently used operationally. Despite these good results, several high-velocity encounters resulted in important differences the two methods. In particular, some of these encounters yield 2D-probability significantly lower than the 3D-probability and the Monte-Carlo probability, implying that a real collision risk was potentially missed. These cases are thus very problematic operationally and must be further analyzed in order to identify the phenomena causing these differences. For low-velocity encounters, differences between the 2D-probability and the 3D-probability are more pronounced. Although it is not surprising since this kind of encounter is beyond the scope of the 2D-probability model, the results presented stress out the necessity to carefully analyze low-velocity encounters. Even if the 3D-probability seems better suited for this kind of conjunctions, properly defining the duration of the encounter can be difficult, especially for recurring risks. Moreover, the probabilities computed using Coppola s formulation should be considered with caution, as the model relies on hypotheses that might not always be satisfied. Further validations on the method are currently planned in order to increase the confidence in the probability returned. The method will be applied to a large number of recent collision risks alerts and compared to the 2D-probability of collision. The Monte-Carlo probability will then be computed for every case resulting in a strong difference between the two probabilities. A reflection on the application of the method in an operational context is also undergoing, one of the main concerns being the computational time. While relatively quick to compute, the method is still significantly slower than the 2D-probability of collision. Since the computation of the probability of collision done by CNES takes into account a possible dilatation or contraction of the covariance [6], assessing the risk for a single conjunction requires hundreds of individual evaluations. Applying the 3D-probability to every collision risk alert received would thus be inefficient. And while the relative speed at TCA can be used to decide the method to use for a specific case, the results obtained so far indicate that it is not sufficient. Additional criteria must be investigated. References [1] Patera, R., General Method for Calculating Satellite Collision Probability, Journal of Guidance, Control, and Dynamics, Vol. 24, 2001, pp doi: / [2] Alfano, S., Satellite Collision Probability Enhancements, Journal of Guidance Control and Dynamics, Vol. 29, 2006, pp

9 [3] Coppola, V., Including velocity uncertainty in the probability of collision between space objects, AAS/AIAA, Astrodynamics Specialist Conference, AAS Paper [4] Coppola, V., Evaluating the short encounter assumption of the probability of collision formula, AAS/AIAA, Astrodynamics Specialist Conference, AAS Paper [5] Lebedev, V., and Laikov, D., A quadrature formula for the sphere of the 131st algebraic order of accuracy, Russian Academy of Sciences Doklady Mathematics, Vol. 59, No. 3, 1999, pp [6] Laporte, F., JAC software, solving conjunction assessment issues, AMOS Technical Conference, Maui HI,