Orbital Guidance Using Higher-Order State Transition Tensors

This paper derives the equations necessary to use state transition tensors (STTs) in a spacecraft guidance problem. The derivation includes all necessary equations up to the fourth-order expansions in the dynamics and the controls. These are first derived for the general case for a finite burn beginning at the epoch time; simplifications are made for when the target is the spacecraft state at a desired final time, and the controls at the epoch time are impulsive Delta v's. An application is presented comparing the STT method with a linearized numerical predictor-corrector for computing an impulsive station-keeping maneuver in an unstable halo orbit around the Earth-moon L1 Lagrange point. The STT method is shown to perform significantly faster, while also converging on accurate solutions for long-horizon targets where the predictor-corrector method fails. The formulation is able to accommodate any number of perturbations in the dynamics, rendering it suitable for onboard operational usage. This approach provides promise for fast and efficient determination of the desired controls to achieve a nonlinear target constraint surface in a highly nonlinear dynamic environment.