Efficient collision avoidance manoeuvres under multiple polynomial constraints

We propose a simple and reliable algorithm for collision avoidance manoeuvre optimisation, capable of computing impulsive, multi-impulsive, and low-thrust manoeuvres. Utilising differential algebra techniques, safety metrics such as probability of collision and miss distance are approximated by a polynomial of arbitrary order as a function of the control. Moreover, operational constraints, namely the return to the nominal trajectory and station-keeping, are employed and treated in the same way. This allows for the transcription of the collision avoidance manoeuvre problem as a polynomial programme with a quadratic (energy-optimal) objective function and high-order polynomial constraints. The polynomial constraints are successively linearised using an iterative approach, wherein the updated linear constraints are calculated from the high-order polynomial maps. This avoids the need for the recomputation of linear maps from a new expansion point. Thus, the original polynomial programme is effectively approximated by a series of quadratic programmes, enabling accurate solutions to be achieved even for highly nonlinear polynomial constraints. The method is showcased in several scenarios involving multiple short-term conjunctions and station keeping requirements. Solutions are found with run times between 0.1 s and 0.5 s, proving that the method is computationally efficient and making it potentially suitable for autonomous applications.