Fuel-optimal powered descent guidance with free final-time and path constraints

This paper presents a convex optimization-based approach to efficiently obtain the numerical solution of the fuel-optimal powered descent problem with free final-time and path constraints. To avoid guessing the final-time, we propose to choose altitude, instead of time, as the independent variable in the system dynamics. This selection also brings great convenience in incorporating the glide-slope and thrust direction constraints in which the bounds can be altitude-dependent. Then, the formulated optimal control problem is converted into a convex problem via appropriate convexification techniques, such as the nonlinearity-kept & linearization approach and relaxation, etc. Relaxation is a critical technique, but analyzing its validity is generally very challenging, especially when path constraints are present. In this paper we can prove that the relaxation used is valid. Next, we discretize the convex problem and apply successive convex optimization to get the solution of the original problem. Nevertheless, in order to obtain the solution with high accuracy and low computational cost, we propose a new strategy of selecting nonuniform discretized points plus the Runge-Kutta 4th order or trapezoidal discretization method. Numerical results are provided to show the effectiveness and high efficiency of the proposed method in solving the powered descent problem and reveal an interesting structure of the optimal thrust magnitude profile when path constraints become active.