Optimal phase-angle design for the powered-descent phase of lunar lander

In the optimal trajectory design problem for the lunar powered-descent phase, the periapsis of the de-orbit burn phase is usually chosen as a starting point or an initial state. The resultant trajectory in these cases shows that the altitude of the lander increases to gain more sufficient time to reduce large initial horizontal velocity. However, the periapsis of the de-orbit burn phase is not the optimal choice. In this study, the optimal initial phase angle can be found by applying the modified trajectory-optimization problem, where the initial state is considered as a free variable. In this problem, any additional assumption and change in hardware compared with the traditional optimal lunar-landing problem are not imposed except for the initial phase angle. Using the proposed numerical approach, we show that the optimal phase angle is not always equivalent to the periapsis, and fuel consumption can be reduced by changing the starting point of the powered descent phase.