Revisiting Optimal Guidance Solution for Vertical Landing and Take-Off

The vertical landing and take-off are crucial phases of a rocket-powered spacecraft's powered descent and ascent trajectories, during which its motion is confined to a single (vertical) axis. It is well-known that the minimum-propellant guidance solutions for vertical landing and take-off in a vacuum, consist of only one coast and one full-thrust arc and no singular arc exists in these problems. However, an exact solution for the vertical descent problem is not found in the literature. In the present work, exact guidance solutions for the optimal vertical landing and take-off are presented that are valid for arbitrary boundary conditions. The time-based and switching function-based guidance law implementations are discussed and their performance is compared using lunar descent and ascent simulations.