Solutions to Rapid Collision-Avoidance Maneuvers Constrained by Mission Performance Requirements

What is it about?

For any collision avoidance scenario, the goal is to reach a desired separation distance between the two objects at the predicted time of collision by performing a fuel-optimal maneuver. One main factor driving the planning of the maneuver is the starting and desired orientation of the primary object. Depending on the current operating conditions, the spacecraft may be pointed in a certain direction to best use its solar panels, scientific equipment, or other directional hardware. Time must be allotted to reorient the spacecraft so that the main thrusters are pointed in the desired direction to perform the maneuver. Additional time will be allotted to ensure that the thrust is applied at the most optimal point. With these in mind, a maneuver is planned (assuming an impulsive burn) to reach the desired separation distance at the predicted time of collision with the least amount of fuel. These maneuvers can take hours, if not days, to plan and optimize. Unfortunately, with the uncertainty in the position of any two objects and the inability to track objects smaller than 10 cm in diameter, not all collisions can be avoided. Debris from such a collision or other debris-creating events could put other satellites on collision courses with the newly created debris with minimal time until the collision (possibly even less than half an orbit in duration). Currently, minimal previous work exists in avoiding collisions with such a small notification time. When planning avoidance maneuvers for collisions between two cataloged objects, covariance data are often known for both objects. However, when the secondary object is newly created, less opportunity exists to determine similar characteristics. Therefore, in scenarios with minimal time to collision, the secondary object must be considered as only a point mass, and the probability must be determined with the information known about the primary object (its covariance assumed to be oriented in the direction of the velocity vector). In addition, it is assumed that no time is available to rotate the spacecraft to a more desirable orientation, and so the thrust is assumed to be applied in the original orientation, and the thrust magnitude must be chosen as such. It is the analysis of the optimal burn direction that led to the discovery of a semi-analytical solution to the problem. Assuming that the spacecraft of interest is in low Earth orbit (LEO), a semianalytical solution to the fuel-optimal collision-avoidance maneuver is solved. This is accomplished by minimizing the finite burn time to reach the minimum separation distance at the time of closest approach for a given spacecraft location, thrust magnitude (assumed to be constant), thrust direction, and remaining time until the collision (time to collision). Once a relationship between the optimal burn time and the desired separation distance at the time of closest approach is found, the separation distance can be converted into a desired collision probability threshold, resulting in a single relationship between the spacecraft parameters, the required burn time, and the desired collision probability threshold. Varying the desired collision probability, the resulting changes in the required thrust duration time (and, thus, fuel use) can be observed, providing options for trading the fuel use and likelihood of a collision. Additionally, both the probability of collision and the fuel use contribute directly to the ability of the spacecraft to adhere to the desired mission specifications. As the collision probability and required burn time increase, the mission performance decreases with it. The level of robustness necessary in the mission specifications can then be used to limit the desired collision probability. This is accomplished by determining the time and fuel required to perform the collision-avoidance maneuver to the desired probability level and analyzing the effect of the time spent away from the mission orbit and the quantity of fuel required to perform the maneuver on the mission performance. It was found that, for notification times less than around 20 min, it is best to decrease the collision probability as much as the available fuel will allow without regard for the time duration of the maneuver. As the notification time increases past 20 min, more emphasis can be placed on the time required to perform the entire maneuver, and it was found that simultaneously minimizing the maneuver time and collision probability outweighed the slight extra fuel required for such a maneuver. These observations allow us to determine an optimal collision probability (typically a subjective variable) for rapid collision scenarios.

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Why is it important?

Collision-avoidance maneuvers with less than half an orbit to both plan and execute involve additional complexities not found in typical advanced notification avoidance maneuvers due to the limited time available. For instance, finite burn analysis must be applied instead of assuming an impulsive burn maneuver. The limited time available also leads to complexity in choosing both an optimal burn direction and burn location, which will both be covered in detail later. Working under these assumptions, a numerical solution can be determined for a fuel-optimal avoidance maneuver.

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