An investigation of fuel optimal terminal descent
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Current renewed interest in exploration of the moon, Mars, and other planetary objects is driving technology development in many fields of space system design. In particular, there is a desire to land both robotic and human missions on the moon and elsewhere. The core of a successful landing is a robust guidance, navigation, and control system (GN&C). In particular, the landing guidance system must be able to deliver the vehicle from an orbit above the planet to a desired soft landing, while meeting several constraints necessary for the safety of the vehicle. In addition, due to the performance limitations of current launch vehicles, it is desired to minimize the amount of propellant used during the landing. To make matters even more complicated, the landing site may change in real-time in order to avoid previously undetected hazards which become apparent during the landing maneuver. The Apollo program relied heavily on the eyes of the astronauts to avoid such hazards through manual control. However, for missions to the lunar polar regions, poor lighting conditions will make this much more difficult; for robotic missions, this is not an option. It is desired to find a solution to the landing problem such that the fuel used is minimized while meeting constraints on the initial state, final state, bounded thrust acceleration magnitude, and bounded pitch attitude. With the assumptions of constant gravity and negligible atmosphere, the form of the optimal steering law is found, and the equations of motion are integrated analytically, resulting in a system of five equations in five unknowns. When the pitch over constraint is ignored, it is shown that this system of equations can be reduced analytically to two equations in two unknowns. In addition, when an assumption of a constant thrust acceleration magnitude is made, this system can be reduced further to one equation in one unknown. It is shown that these unknowns can be bounded analytically. An algorithm is developed to quickly and reliably solve the resulting one-dimensional bounded search. The algorithm is used as a real-time guidance and is applied to lunar and Mars landing test cases.