A Powered Descent Guidance Algorithm for Mars Pinpoint Landing

Abstract

We present a powered descent guidance algorithm for Mars pinpoint landing that solves the minimum fuel trajectory optimization problem via a direct numerical method. Our main contribution is the formulation of the trajectory optimization problem, which has nonconvex control constraints, as a finite dimensional convex optimization problem, specifically as a finite dimensional semidefinite program (SDP). Since efficient SDP solvers with deterministic convergence properties exist, the resulting guidance algorithm can potentially be implemented onboard. In this paper, we present a powered descent guidance algorithm for Mars pinpoint landing. This problem is gaining importance due to the renewed interest in the manned and robotic exploration of Mars. The pinpoint landing problem can be defined as guiding a lander spacecraft to a given target on Mars’ surface with an accuracy of less than several hundred meters. This involves an entry phase through the Mars atmosphere, a phase of descent with a parachute, and then the final phase of powered descent, which is initiated with the parachute cutoff. The powered descent guidance problem for pinpoint landing is defined as finding the fuel optimal trajectory that takes a lander with a given initial state (position and velocity) to a prescribed final state in a uniform gravity field, with magnitude constraints on the available net thrust, and various state constraints. A version of this problem is also known in the optimal control literature as the soft landing problem, 1–3 and its solutions have well known characterizations. One important characterization is that the net thrust magnitude must be either at the minimum or maximum value at any given time during the maneuver. A closed form solution of the problem for the one dimensional case (with vertical motion only) is given by. 1 However, a closed form solution of the thrust profile is not available for the general three dimensional case with additional state and control constraints. Direct numerical methods for trajectory optimization are attractive because explicit consideration of the necessary conditions (adjoint equations, transversality conditions, maximum principle) are not required. 4 The infinite dimensional optimal control problem is directly converted into a finite dimensional parameter optimization problem 5–7 which is then solved via a nonlinear programming method. However, real-time onboard solution of a nonlinear program via a general iterative algorithm may not be desirable without explicit knowledge of the convergence properties of the algorithm. As Mars pinpoint landing requires realtime onboard computation of the optimal trajectory, it is essential to exploit the structure of the soft landing problem in order to design algorithms with guaranteed convergence to the global optimum. This leads us to formulate the problem in a convex optimization 8,9 framework, specifically to formulate the resulting parameter optimization problem as a semidefinite program (SDP). 10,11 SDP problems have low complexity, and they can be solved in polynomial time. There exist algorithms, such as interior-point methods, 12–14 that compute the global optimum with a deterministic stopping criteria, and with prescribed level of accuracy. Therefore, they are very well-suited for real-time onboard computations.