Coevolving Defender Strategies Within Adversarial Ground Station Transit Time Games via Competitive Coevolution

Abstract

Emerging Proliferated Low Earth Orbit (P-LEO) constellations may be susceptible to attacks launched by malevolent actors capable of compromising orbiting satellites. We introduce and investigate an adversarial ground station transit time game as a proxy to study the ability to rapidly detect and respond to satellite attacks. Investigation of this problem allows us to study more complex real-world dynamics involving threats to satellite systems. Unfortunately, the problem proves to be daunting to solve due to the high-dimensionality of the solution space and the difficulty of predicting action consequences in dynamic adversarial settings. For this reason, an effective method to identify successful strategies as solutions to the problem is necessary. In this work, an artificial intelligence approach called competitive coevolution is employed to solve scenarios featuring an attacker evolving strategic locations to degrade the performance of a constellation, while a defender evolves intelligent strategies to counter the attacker’s action. The proposed solution outperforms both minimal and complex strategies, while showing versatility and robustness over a variety of scenarios.

Appendix A: Heatmap Rotation

The set of GP primitives for the defender includes several functions to “rotate” a heatmap and produce a new one. This can be conceptualized as rotating a spherical globe with the heatmap’s data on its surface, while fixing the grid of the heatmap in place without rotation. The output maps each cell of the heatmap to the new data in that location after the rotation. Points on the heatmap are mapped to latitude, longitude coordinates, so these rotations can be performed by converting to Cartesian coordinates and multiplying by an appropriate rotation matrix. However, heatmaps are made up of discrete data cells, which generally won’t line up after a rotation, so these need to be resampled from the original heatmap.

For this to work, the heatmap must be accessible by a continuous function on the sphere. Let Hcell(i, j) be the accessor function for the heatmap array at row i and column j. When mapped to the sphere, for a given grid resolution res, the centers of these grid cells are located at $-\frac{\pi }{2}+i \cdot res$ latitude and $-\pi +j \cdot res$ longitude. We will define Hmap(lat, lon) as a function mapping a latitude, longitude point to a heatmap value, which gives the value of the heatmap cell with the nearest center to that point:

$$\begin{aligned} Hmap(lat, lon) = Hcell\left(round\left( \frac{lat + \pi /2}{res}\right) , round\left( \frac{(lon+\pi ) \mathbin {mod} 2\pi }{res}\right) \right) \end{aligned}$$

where round rounds to the nearest integer, and $\mathbin {mod}$ is the Euclidean modulo operator.

Let Rot(lat, lon) be the rotation of a point (lat, lon) following a given transformation, with $Rot^{-1}(lat, lon)$ as the inverse transformation. Then the value of the rotated heatmap at (lat, lon) is Hmap(Rot(lat, lon)). In order to define the cells of the rotated heatmap, we take this value at the four corners of each cell, and average them to get the value of that cell. The inner boundaries of the cell have the coordinates:

$$\begin{aligned} \begin{aligned} top&= min\left( lat + \frac{res}{2} - \epsilon , \frac{\pi }{2}\right) \\ bottom&= max\left( lat - \frac{res}{2} + \epsilon , -\frac{\pi }{2}\right) \\ right&= lon + \frac{res}{2} - \epsilon \\ left&= lon - \frac{res}{2} + \epsilon \end{aligned} \end{aligned}$$

where $\epsilon$ is a value much smaller than the resolution to ensure that the ties are broken in the direction of the current cell, so that the identity rotation does not modify the heatmap. From these, the values of the rotated heatmap are:

$$\begin{aligned} \begin{aligned} Hmap_{rot}(lat, lon)&= (Hmap(Rot^{-1}(top, right))\\&\quad + Hmap(Rot^{-1}(top, left))\\&\quad + Hmap(Rot^{-1}(bottom, right))\\&\quad + Hmap(Rot^{-1}(bottom, left))) / 4\\ Hcell_{rot}(i, j)&= Hmap_{rot}\left( -\frac{\pi }{2}+i \cdot res, -\pi +j \cdot res\right) \end{aligned} \end{aligned}$$