Fuel-optimal acquisition and control of a cartwheel formation in Earth displaced heliocentric orbit

An optimization approach for cartwheel formation acquisition and maintenance in an Earth Displaced heliocentric orbit is presented. This work considers non-gravitational perturbations such as solar radiation pressure, thus extending the studies previously performed for the mission LISA. The problem is tackled as a Nonlinear Programming problem using a multiple shooting method. The optimization process is performed in two steps: first, the orbital elements of each satellite in heliocentric orbit are optimized to guarantee the stability during the science phase hence easing maintenance of the cartwheel formation in presence of orbital perturbations. Then, the obtained initial states are propagated to obtain a set of target orbital states which become the final target of a second optimization covering the transfer phase from Earth. For the science phase optimization presents two alternative cost functions are introduced, one based on the arm-length evolution and one on the arm-length-rate evolution. The performance of each cost function is analysed for different initial displacement angles: for target arm-lengths below 2.5 million kilometers the arm-length cost function provides the best results while no significant difference between the two optimized solutions is observed above this value. The transfer phase optimization presents two different approaches, one considering an injection on a trajectory more favourable for one of the three spacecraft and one considering an injection on an intermediate trajectory which minimizes the overall acquisition cost of all spacecraft. The proposed optimization approach performance are studied on a set of test cases covering both transfer and science phase, showing that stable configuration conditions can be found even in presence of orbital perturbations and that the multiple injection transfer is capable of providing a more homogeneous fuel consumption among the three spacecraft.