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Merging Lie perturbation theory and Taylor Differential algebra to address space debris challenges

Final Report Summary - HOPT (Merging Lie perturbation theory and Taylor Differential algebra to address space debris challenges)

In an increasingly saturated space about the Earth, aerospace engineers confront the mathematical problem of accurately predicting the position of Earth’s artificial satellites. This is required not only for the correct operation of satellites, but also for preserving the integrity of space assets and the services they provide to citizens. Operational satellites are threatened by the possibility of a collision with a defunct satellite, but most probably by the impact with other uncontrolled man-made space objects—all of them commonly called space debris.

The present international concern in space situational awareness (SSA) has produced a renewed interest in analytical and semi-analytical (SA) theories for the fast and efficient propagation of catalogs of data. Within this framework, it is widely accepted by experts that perturbation theory based on Lie transforms (LT) is the most accurate and efficient method to derive semi-analytical propagators. In a semi-analytical approach, the highest frequencies of the motion are filtered analytically via averaging procedures, allowing the numerical integration of the averaged system to proceed with very long step sizes. Then, the short-period terms can be recovered from their analytical expression.

Another fundamental need in SSA is the efficient management of uncertainties that characterize the motion of orbiting objects. To this aim Taylor differential algebraic (DA) and Taylor model (TM) techniques have been transferred in the last decade from beam physics field to astrodynamics. These techniques, by allowing high order expansions of the flow of the dynamics and rigorous estimate of the associated approximation errors, have shown to be a powerful tool for managing uncertainties both in initial conditions and model parameters.

The objective of this project was to merge SA theory based on the LT with DA, and to apply the resulting techniques to SSA problems. The main results achieved in the project are

1. The efficiency of SA propagators was improved by using DA techniques to deal with non-conservative forces. The high order Taylor expansions enabled by DA are used to reduce the computational time of numerical quadratures, which represent a bottle-neck for SA propagators. This new approach is particularly useful for long-term propagations, e.g. those required for re-entry date predictions.
2. In 2013 the researcher and his collaborators proposed a new technique called high-order transfer map (HOTM) method for the fast and accurate propagation of space debris. The technique is based on the automatic high-order expansion of the solution of ordinary differential equations by DA. A HOTM is generated via a numerical integration of a single orbital revolution in DA arithmetic. It is then followed by the repeated analytical evaluation of the HOTM to advance the orbital propagation by several orbital periods. Within this project the HOTM technique has been extended to include non-autonomous perturbations (e.g. luni-solar perturbation) and formulated in new variables to increase its accuracy. The developed method has been proved to be a powerful tool for short-term propagations of debris clouds and has potentials to be used for mission design (e.g. formation flying and repeat ground-track missions).
3. A set of tools for initial orbit determination and data association of space debris have been implemented. A general method for mapping probability density function through nonlinear transformations has been implemented and applied to space debris orbit determination for the case of both radar and optical observation. DA techniques have been used to accurately describe the uncertainty region associated with initial orbit determination enabling a powerful tool for data association and initialization of particle filters.
4. A new arbitrary dynamics Lambert’s solver has been implemented. This solver is based on an analytical solution of the J2 problem obtained by means of LT. DA techniques are then used to implement a continuation techniques that allows finding a solution in an arbitrary dynamical model without the need of iterations. The method allows dealing with transfers with hundred revolutions and it was successfully applied to the design of multiple space debris removal missions.

The activity performed within this project is expected to contribute towards the successful implementation of the European space surveillance and tracking system and therefore the sustainable use of space.