Efficient n-to-n Collision Detection for Space Debris using 4D AABB Trees (Extended Report)
Collision detection algorithms are used in aerospace, swarm robotics, automotive, video gaming, dynamics simulation and other domains. As many applications of collision detection run online, timing requirements are imposed on the algorithm runtime: algorithms must, at a minimum, keep up with the passage of time. In practice, this places a limit on the number of objects, n, that can be tracked at the same time. In this paper, we improve the scalability of collision detection, effectively raising the limit n for online object tracking. 
The key to our approach is the use of a four-dimensional axis-aligned bounding box (AABB) tree, which stores each object's three-dimensional occupancy region in space during a one-dimensional interval of time. This improves efficiency by permitting per-object variable times steps. Further, we describe partitioning strategies that can decompose the 4D AABB tree search into several smaller-dimensional problems that can be solved in parallel. We formalize the collision detection problem and prove our algorithm's correctness. We demonstrate the feasibility of online collision detection for an orbital space debris application, using publicly available data on the full catalog of n=16848 objects provided by www.space-track.org.