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Collision Risk Estimation

Contents

  1. Introduction
  2. Methods for Estimating the Collision Probability
    1. Collision Risk Using the Short-term Encounter Model
    2. Collision Risk from Differential Algebra
    3. Collision Risk from Fragment Clouds
  3. References

Introduction

As the trajectory of an object has an associated uncertainty, the covariance, the exact conjunction geometry cannot be perfectly known and therefore only a collision probability, $P_{c}$, can be determined. Several methods exist for calculating this collision probability.

Methods for Estimating the Collision Probability

Collision Risk Using the Short-term Encounter Model

The geometry of the approach can be used to derive the probability of collision. For scenarios where the approach can be considered as high velocity, the short-term encounter model can be used. Although valid for most conjunctions in the LEO and GEO regimes, the assumptions of this approach are violated in cases of formation flying, or GEO colocation, where the objects have low relative velocities, resulting in more involved calculations. This family of geometric methods, however, suffer from the effects of risk dilution, where increasing the covariance or uncertainty on an object can reduce the risk of collision.

Derivation

Consider a scenario in which two LEO objects, a target object (primary, $p$, e.g. an operational satellite) and a chaser (secondary, $s$, e.g. debris) pose a mutual collision threat. The state vector (position and velocity), or orbit, of each object is not precisely known due to both measurement, and orbit determination and propagation errors. This uncertainty in each orbit is described by a covariance metric. As a result of this uncertainty, the exact conjunction geometry cannot be known, and thus only a collision probability can be determined.

Establishing this probability, essential for determining if either object should feasibly perform a collision avoidance manoeuvre, can be determined using simplied geometric considerations. Similar approaches to this problem, such as the short-term encounter or near-miss conjunction models, discussed by [1], [2] and [3], which consider the scenario of two spherical objects under Gaussian distributed uncertainty, share the same assumptions. For both target and chaser, about the time of closest approach (TCA),

  1. The geometrical shape of each object is assumed to be spherical.
  2. The target and chaser objects move along straight lines with constant velocity during the encounter.
  3. Uncertainty in the velocities are considered to be negligible, and is therefore neglected.
  4. Uncertainty in the positions of the objects may be modelled by a 3D Gaussian distribution.
  5. Position uncertainties of both objects are considered to be uncorrelated, and thus the Gaussian distributions may be treated as independent.
  6. The position uncertainties during the encounter are constant, and thus correspond to the covariances at TCA.

Here, we consider the collision, or hit, condition to be if the distance between the centres of the two spheres (at TCA) is smaller than the sum of their radii. This critical radius, $R_{c}$ is called the hard body radius (HBR), and defines the collision area and volume,

\[R_{c} = R_{p} + R_{s}\] \[A_{c} = \pi R_{c}^{2}\] \[V_{c} = \frac{4\pi}{3} R_{c}^{3}\]

To determine if this criteria is met, we require the relative position vector ($\Delta\vec{r}$) of the two bodies at TCA. As the uncertainty distribution of the positions of both objects individually may be considered to be Gaussian distributed, this difference vector is also a Gaussian. Neglecting the velocity components of the covariances, and assuming that the remaining elements are uncorrelated between both objects, we may combine both positional covariances into one defining both objects, $C$. The probability density function in $\Delta\vec{r}$, in the vicinity of the point of closest approach, may then be described by,

\[p(\Delta\vec{r})= \frac{1}{\sqrt{(2\pi)^{3} \det(C)}} \exp\left( -\frac{1}{2} \Delta\vec{r}^{T} C^{-1} \Delta\vec{r}\right).\]

The collision probability, $P_{c}$, can then be interpreted as a volume integral of the 3D probability density function over the spherical region $V_{c}$, centred on the chaser object position (all uncertainty $C$ given to chaser),

\[P_{c} = \int_{V_{c}} p(\Delta\vec{r}) dV\]

It is intuitive to consider this scenario geometrically using an encounter plane (or B-plane). The origin of this plane is the target trajectory crossing point at TCA, with the x-axis defined as the direction to the chaser crossing point. The distance between the objects along this axis is therefore the miss distance. In this set-up, the relative position vector, $\Delta\vec{r} = \vec{r_{s}} - \vec{r_{p}}$, is parallel to the encounter plane at TCA. This is perpendicular to the relative velocity vector, $\Delta\vec{v} = \vec{v_{s}} - \vec{r_{p}}$. Using this B-plane geometry allows us to reduce the collision risk volume integral to a 2D Gaussian surface integral over a disc by mapping the position error covariance ellipsoid onto elliptical contours of constant probability on the B-plane [3].

\[P_{c}(R_{c}, C_{B}, x_{B}, y_{B}) = \frac{1}{\sqrt{(2\pi)^{2} \det(C_{B})}} \int_{-R_{c}}^{+R_{c}} \int_{-\sqrt{R_{c}^{2} - x_{B}^{2}}}^{+\sqrt{R_{c}^{2} - x_{B}^{2}}} \exp\left( -\frac{1}{2} \Delta\vec{r}_{B}^{T} C_{B}^{-1} \Delta\vec{r}_{B}\right) dy_{B} dx_{B}\]

The new integrand is the probability density function of the relative coordinates in the encounter plane. The integration domain is the size of the combined collision cross-section $A_{c}$.

The collision probability, $P_{c}$, is therefore uniquely determined by the radius of the combined spherical object, $R_{c}$, and the conjunction location within the 2D probability density distribution. The latter is defined by the miss distance, and covariance matrix of the relative position, in the encounter plane at TCA.

In one dimension, the miss distance of the two objects is uncertain and described by a Gaussian distribution defined by the summed covariance about the chaser. The risk is evaluated by integrating the distance density function in B-plane coordinates over the combined cross-section.

Collision Risk from Differential Algebra

Differential algebra (DA) is an automatic differentiation technique based on the computation of Taylor expansions. Given an intial state with a given uncertainty, $/delta$, and a sufficiently regular function, $f$, describing the dynamics (e.g. equations of motion), DA enables the propagation of the uncertainty bounds, resulting in a taylor expansion of the solution at some later time as a function of the initial uncertainty. This has significant advantages in:

DA can be easily implemented programmatically using the open-source C++ DACE repository.

This may be applied to the evaluation of collision probability by computing the Taylor expansion of the distance between the target and chaser with respect to their initial states and covariances. Differentiating to find the minimum of this gives the Taylor expansion of the distance of closest approach (DCA) including the propagated covariances. DA-based fast Monte Carlo simulations can be run on this function to compute how many samples have a target-chaser distance less than some threshold, thus giving the collision probability [4], [5].

This method has the following advantages:

Collision Risk from Fragment Clouds

When considering the collision risk from clouds of fragments (over mid to long timescales), it is impossible to propagate each member of the cloud individually due to observational and computational constraints. Typically in long-term environment models, fragment clouds are propagated using a set of representative objects [8], though this is now being extended by considering the propagated spatial density of the cloud [9], [10].

References

[1]: Serra, R., Arzelier, D., Joldes, M., Lasserre, J. B., Rondepierre, A., & Salvy, B. (2016). Fast and accurate computation of orbital collision probability for short-term encounters. Journal of Guidance, Control, and Dynamics, 1009-1021.

[2]: Alfriend, K., Akella, M., et al., (1999). Probability of Collision Error Analysis, Space Debris 1, 2135

[3]: Klinkrad, H. (2006). Space Debris: Models and Risk Analysis, Vol. 1, 1:21-35

[4]: Armellin, R., Di Lizia, P., Morselli, A., & Lavagna, M. (2012). An orbital conjunction algorithm based on Taylor models. Advances in the Astronautical Sciences.

[5]: Morselli, A., Armellin, R., Di Lizia, P., & Bernelli Zazzera, F. (2014). A high order method for orbital conjunctions analysis: Sensitivity to initial uncertainties. Advances in Space Research. https://doi.org/10.1016/j.asr.2013.11.038

[6]: Morselli, A., Armellin, R., Di Lizia, P., & Bernelli-Zazzera, F. (2012). Computing collision probability using differential algebra and advanced monte carlo methods. Proceedings of the International Astronautical Congress, IAC.

[7]: Alfano, S. (2007). Review of conjunction probability methods for short-term encounters. Advances in the Astronautical Sciences.

[8]: Frey, S., Colombo, C., Lemmens, S., & Krag, H. (2017). Evolution of fragmentation cloud in highly eccentric orbit using representative objects. Proceedings of the International Astronautical Congress, IAC.

[9]: Frey, S., Colombo, C., & Lemmens, S. (2018). Evolution of fragmentation cloud in highly eccentric earth orbits through continuum modelling. Proceedings of the International Astronautical Congress, IAC.

[10]: Frey, S., Colombo, C., & Lemmens, S. (2019). Application of density-based propagation to fragment clouds using the Starling suite. First International Orbital Debris Conference.

Bibliography

Chan, F. K. (2008). Spacecraft Collision Probability. In Spacecraft Collision Probability. https://doi.org/10.2514/4.989186

Alfano, S. (2007). Review of conjunction probability methods for short-term encounters. Advances in the Astronautical Sciences.

Greco, C., Gentile, L., Vasile, M., Minisci, E., & Bartz-Beielstein, T. (2019). Robust particle filter for space objects tracking under severe uncertainty.

Sánchez, L., Vasile, M., & Minisci, E. (2020). On the use of machine learning and evidence theory to improve collision risk management. 2nd IAA Conference in Space Situational Awareness, Arlington, United States.

Gonzalo, J. L., Colombo, C., & Di Lizia, P. (2019). Drag- and SRP-induced effects in uncertainty evolution for close approaches. 4th International Workshop on Key Topics in Orbit Propagation Applied to Space Situational Awareness (KePASSA).