NASA NTRS2011
The landing of a crewed lunar lander on the surface of the Moon will be the climax of any Moon mission. At touchdown, the landing mechanism must absorb the load imparted on the lander due to the vertical component of the lander's touchdown velocity. Also, a large horizontal velocity must be avoided because it could cause the lander to tip over, risking the life of the crew. To be conservative, the worst-case lander's touchdown velocity is always assumed in designing the landing mechanism, making it very heavy. Fuel-optimal guidance algorithms for soft planetary landing have been studied extensively. In most of these studies, the lander is constrained to touchdown with zero velocity. With bounds imposed on the magnitude of the engine thrust, the optimal control solutions typically have a "bang-bang" thrust profile: the thrust magnitude "bangs" instantaneously between its maximum and minimum magnitudes. But the descent engine might not be able to throttle between its extremes instantaneously. There is also a concern about the acceptability of "bang-bang" control to the crew. In our study, the optimal control of a lander is formulated with a cost function that penalizes both the touchdown velocity and the fuel cost of the descent engine. In this formulation, there is not a requirement to achieve a zero touchdown velocity. Only a touchdown velocity that is consistent with the capability of the landing gear design is required. Also, since the nominal throttle level for the terminal descent sub-phase is well below the peak engine thrust, no bound on the engine thrust is used in our formulated problem. Instead of bangbang type solution, the optimal thrust generated is a continuous function of time. With this formulation, we can easily derive analytical expressions for the optimal thrust vector, touchdown velocity components, and other system variables. These expressions provide insights into the "physics" of the optimal landing and terminal descent maneuver. These insights could help engineers to achieve a better "balance" between the conflicting needs of achieving a safe touchdown velocity, a low-weight landing mechanism, low engine fuel cost, and other design goals. In comparing the computed optimal control results with the preflight landing trajectory design of the Apollo-11 mission, we noted interesting similarities between the two missions.