Optimal Interplanetary Trajectories for Impulsive Deflection of Potentially Hazardous Asteroids under Velocity Increment Uncertainties

This paper investigates the interplanetary trajectories associated with the impulsive deflection of a potentially hazardous asteroid (PHA) considering the uncertainty of the velocity increment that a spacecraft gives to the PHA at the collision. The velocity increment is assumed to have uncertainties of magnitude and direction due to estimation errors of asteroid shape and mass distributions. The uncertainty is modeled using a convex model assuming that magnitude and direction vary independently. The effect of uncertainty is assessed by evaluating the worst (i.e. minimum) value for the closest approach distance between the PHA and Earth. The worst value of the closest approach distance can be determined analytically without searching the whole convex hull. The optimal spacecraft trajectory is designed by maximizing the worst value of the closest approach distance in terms of the Earth departure date and the asteroid arrival date of the spacecraft under C3 (Earth departure energy) constraint. Using a numerical example with a fictitious asteroid, the importance of considering the velocity increment uncertainty is demonstrated by comparing the optimal trajectory with the deterministic optimal trajectory. The uncertainty of the velocity increment direction is shown to have a significant effect on the deflection of the PHA.