Variational Lambert’s Problem with Uncertain Dynamics

Lambert’s problem is a widely-known problem in astrodynamics that addresses the need of finding a trajectory given two position vectors and the time of flight between them. It is widely used in mission design and in on-line guidance algorithm in order to predict the needed maneuvers or the spacecraft state on the computed trajectory. Previous work has investigated the influence of uncertainty in the positions vectors and linearization of the classical Lambert’s problem for spacecraft autonomous applications. These approaches allow the uncertainty quantification, maneuver correction and orbit determination to be performed with respect to a nominal trajectory in a perfectly-known environment. Unfortunately, the increase number of missions to partially-known bodies of the Solar System, such as asteroids, comets and dwarf planets, requires to abandon the hypothesis of a deterministic dynamical environment as the forces acting on the spacecraft are accurately quantified only when the geophysical property of the body are known, thus when orbiting around it. This leads to the need of considering a stochastic dynamics to take into account uncertainties and errors introduced during mission design. This paper presents the variational Lambert’s problem with uncertain dynamics around a nominal trajectory and gather the formulas to characterize the probability density function and covariances of position, velocities and dynamical parameters. Then numerical simulations are presented by considering several dynamics effects, such as the spherical harmonics gravity, in order to validate the developed approach by comparison with Monte Carlo simulations. Results show good agreement between the two obtained solution. Finally, an operational simulation is presented to show an on-board autonomous application of the developed algorithm. In this scenario the spacecraft estimates on-board the new dynamics and corrects the guidance maneuvers by using the output of the variational Lambert’s problem and the navigation data. The corrected trajectory shows a decrease of the error with respect to the nominal trajectory that implies the effectiveness of the applied corrections.