Continuous-thrust collision avoidance manoeuvres optimization

In the last few decades, space debris have become one of the most important hazards for space activities; preventing their formation is the best mitigation option, and avoiding an impact can be of fundamental importance for a mission’s success. This study proposes four methods to design a continuous-thrust Collision Avoidance Manoeuvre (CAM), which have been developed in the perspective of finding a fast and reliable approach, suitable for on-board implementation. The objective function to be minimized is the collision probability (Pc) at the nominal time of closest approach. The first approach consists in the conversion from an analytical fuel-optimal impulsive manoeuvre to a finite-burn arc through an indirect optimal control model. In the second approach, a new formulation of the problem is proposed, where the collision probability is considered as a terminal constraint in the minimum-Fuel and minimum-Energy Optimal Control Problem (FOP/EOP). The mathematical formulation yields a Two Boundary Value Problem, whose minimum-fuel discontinuous solution is achieved through a continuation method, and the shooting method is adopted for the resolution of the minimum-energy formulation. Both of these approaches require time consuming fully numerical iterative cycles. The last two methods are based on the linearisation of EOP, and consist in an analytical and semi-analytical approach. In the analytical approach the direction of maximum change of Pc is exploited, in order to obtain the maximum decrease of Pc with the least deviation from the nominal trajectory. In the semianalytical approach, the boundary conditions of the TPBVP associated to the minimum-energy problem are exploited, leading to a relation between the terminal cost in the cost function and the final spacecraft position. All these approaches are compared via a numerical test case set in the two-body dynamical framework, assuming circular Keplerian orbits.