Collision avoidance maneuver design based on multi-objective optimization

The possibility of having collision between a satellite and a space debris or another satellite is becoming frequent. The amount of propellant is directly related to a satellite’s operational lifetime and revenue. Thus, collision avoidance maneuvers should be performed in the most efficient and effective manner possible. In this work the problem is formulated as a multi-objective optimization. The first objective is the Δv, whereas the second and third one are the collision probability and relative distance between the satellite and the threatening object in a given time window after the maneuver. This is to take into account that multiple conjunctions might occur in the short-term. This is particularly true for the GEO regime, where close conjunction between a pair of object can occur approximately every 12h for a few days. Thus, a CAM can in principle reduce the collision probability for one event, but significantly increase it for others. Another objective function is then added to manage mission constraint. To evaluate the objective function, the TLE are propagated with SGP4/SDP4 to the current time of the maneuver, then the Δv is applied. This allow to compute the corresponding “modified” TLE after the maneuver and identify (in a given time window after the CAM) all the relative minima of the squared distance between the spacecraft and the approaching object, by solving a global optimization problem rigorously by means of the verified global optimizer COSY-GO. Finally the collision probability for the sieved encounters can be computed. A Multi-Objective Particle Swarm Optimizer is used to compute the set of Pareto optimal solutions.The method has been applied to two test cases, one that considers a conjunction in GEO and another in LEO. Results show that, in particular for the GEO case, considering all the possible conjunctions after one week of the execution of a CAM can prevent the occurrence of new close encounters in the short-term

An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $C^1$-Lohner algorithm proposed by Zgliczy\'nski and it provides sharper bounds. As an application of the algorithm, we give a computer-assisted proof of the existence of an attractor set in the R\"ossler system, and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, that is, conjugated to subshift of finite type with positive topological entropy.Comment: 33 pages, 11 figure

On a computer-aided approach to the computation of Abelian integrals

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four