Non-linear sensitivity of a CubeSat’s orbital motion to an asteroid’s rotational state uncertainties using Non-Intrusive Polynomial Chaos

Small solar system bodies have received increasing scientific attention over the past decades. Studying their primitive origins can reveal important insights on the formation of planets, as well as the general evolution of the Solar System. Also their potential threat to life on Earth upon collision and their potential to act as stepping stones for deep space exploration make these bodies interesting to explore. Especially CubeSats, being small and lightweight, are promising candidates for such missions. To facilitate the design of small body missions, this research investigates the non-linear effects of uncertainties in an asteroid’s environment on the orbital motion of a CubeSat. Uncertainties in the asteroid’s mass, irregular gravity field and solar radiation pressure have been studied prior. Rotational state uncertainties have not been researched, but do affect the CubeSat’s motion indirectly through the orientation of the irregular gravity field. These are therefore selected as the subject of this work. This aids in identifying orbits that are robust against these uncertainties, thereby minimizing the required fuel for trajectory corrections and maximizing the mass for science instruments or increasing the mission duration.The study of the non-linear effects of rotational state uncertainties requires the application of non-linear uncertainty propagation methods. Non-Intrusive Polynomial Chaos was selected for its ease of implementation, promising computational efficiency and ability to provide statistical information directly. In this method, a set of samples from the uncertain domain are propagated to a desired time according to the black-box dynamics that govern the orbital motion. Subsequently, a polynomial approximation, a so-called Polynomial Chaos Expansion, is constructed for these final states as a function of the uncertain variables. This then allows for finding the final states for all possible characterisations of these uncertain variables, in a Monte Carlo like fashion, without further numerical integrations. Above that, the Polynomial Chaos Expansion terms can be used to compute statistics, such as the mean and covariance, analytically. In this work, different initial conditions are propagated and the states at various times are approximated by PolynomialChaos Expansions. All Polynomial Chaos Expansions were verified by comparison with Monte Carlo simulations.A study on the settings of Non-Intrusive Polynomial Chaos was conducted. It showed that the required settings, such as polynomial order, number of samples and method of solving for the coefficients can vary significantly, depending on the studied case. In addition, limitations in the application of this method toKepler elements were encountered for orbits that approach the singularities in this element set. In general, the results show an increase in the trajectory dispersions and non-linearities encountered with an increase in propagation times and with a decrease in orbital altitude. However, exceptions to thistrend were encountered. In the case of a retrograde equatorial orbit, the inclination was found to reach its maximum dispersion already within 5 days and stagnates thereafter. This is a result of the accelerations exerted by the asteroid’s gravitational bulges in its equatorial plane, which varies in space along with changes in the rotation pole under these uncertainties.A comparison to uncertainties in the asteroid’s mass revealed that the effects of rotational state uncertainties are relatively small, especially considering the small uncertainty in mass that was used. However, again an exception was encountered. The right ascension of the ascending node of a polar orbit at 5 km was found more sensitive to changes in the asteroid’s rotation pole than in its mass. Thus, depending on the objective of an analysis, mission designers could be required to include rotational state uncertainties in their analyses.Finally, a broader study of different initial orbital geometries revealed that retrograde orbits are more stable against rotational state uncertainties. However, depending on the exact initial orbital geometry, in terms of inclination and right ascension of the ascending node, prograde orbits can be just as stable. Alsohere it was found, though, that the inclination is more stable for polar orbits than for inclined and equatorial orbits.In conclusion, the finding that retrograde orbits are more stable against rotational state uncertainties helps mission designers to select promising orbits for actual missions. As these orbits are also beneficial for geodetic parameter estimation, they both minimise the required fuel for trajectory corrections and maximise the scientific return. Nonetheless, a wide variety of non-linear effects due to rotational state uncertainties can be encountered in the asteroid’s environment. Therefore, their influence should always be checked in mission design studies, even though in general their effects are relatively small.Aerospace Engineering | Astrodynamics & Space Mission