Effective Sensitivity Analyses of Radar Systems in Formation Flying using Differential Algebra

Spacecraft Formation Flying is a technology that has been increasingly used in recent years. Distributed formations of low-cost SmallSats, both deployable and free-flying, can provide mission ca- pability comparable to or greater than large monolithic spacecraft, but with greatly improved flexibility (adaptability, scalability, evolvability and maintainability) and robustness (reliability, survivability and fault tolerance). These advantages are accompanied by an increase in complexity in maintaining and managing the system. Especially in the case of systems whose purpose is the creation of Synthetic Aper- ture Radar (SAR), relative positions and relative velocities must be finely controlled so that the acquired data are not partially or totally corrupted. Several studies deal with the active and passive control of satellite formations with the aim of minimizing the fuel needed to carry out orbital correction maneuvers or to lengthen the life of the mission. Such works often take place in a deterministic context. Accurate simulations must take into account the uncertainties affecting the quantities involved. In reality, the initial state of the system may not be known precisely, so it should be treated as a random variable. In this context, uncertainty propagation helps to know how the uncertainty on the initial state influences the uncertainty on the final state when the system is subject to complex nonlinear dynamics. The Monte- Carlo procedure (MC) is the best known and most intuitive method for uncertainty propagation. The problem is that in case of large state vectors (as in the case of control of satellite formations with 6-DoF dynamics control) or of propagation of the system for multiple orbits, MC is difficult to implement due to the high computational cost. In this paper, a different methodology based on Differential Algebra (DA) is presented. DA is able to obtain the same information as the classical MC method in a shorter amount of time, at the cost of an acceptable loss of accuracy. In particular, we present simulation studies of Uncertainty Propagation and Sensitivity Analysis applied to Formation Flying by using DA, and discuss the advantages and disadvantages with respect to the MC method.