Analytical landing trajectories for embedded autonomy

This paper considers an optimal guidance law for the initial braking phase of a soft landing mission on a celestial body without atmosphere in which boundary conditions on height and velocity are specifed. The optimal lander attitude for the minimum fuel landing problem is found. An analytic optimal trajectory is achieved by expanding the thrust acceleration, gravitational acceleration and the cosine of the vertical attitude angle to a high-order polynomial. Coefficients of these polynomials are obtained from the boundary conditions. A fixed gain control law and a direct adaptive control law are then developed to track the analytical reference trajectory. Finally, a mission scenario is presented to illustrate the accuracy of the analytical trajectory and validity of the control laws developed. The use of direct adaptive control for embedded autonomy will be directly contrasted against a traditional fixed gain controller, using a Lunar landing scenario. The advantage of the direct adaptive control approach is that it does not require system monitoring to detect thruster failure and can adjust its gain automatically. As such, direct adaptive control combined with the developed analytical solution enables autonomy to be embedded within the lander guidance and control system. In addition, it is shown that direct adaptive control increases the probability of lander survival through faster transient response and stability than a traditional fixed gain controller with system level failure detection and recovery.