Dynamic Systems Approach to the Lander Descent Problem

Abstract

T HE gravity turn landing scheme was originally developed for the 1966–1968 Lunar Surveyor landing mission [1]. Since then it has been applied in many landing missions such as the Viking Lander and Mars Polar Lander [2]. The descent method has the advantage of being near fuel optimal while guaranteeing a vertical landing [3,4]. However, themethod requires a control system that can apply the thrust antiparallel to the instantaneous velocity vector for the entire descent. This Note focuses on the dynamic aspects of the landing scheme. The immediate problem here is to solve for the control that takes the lander from a circular, or near-circular, initial parking orbit to the surface of the planet with zero final relative velocity and zero flightpath angle. In the past, this problem has been studied primarily through various assumptions. The studies can generally be divided into two groups corresponding to two different assumptions: 1) descent from low altitude and 2) aflat planet. In [3,5] both assumptions are considered and compared for the case of a constant thrustto-weight ratio. Using the first assumption, the authors argue for the validity of approximating the full nonlinear gravity with a constant gravity while maintaining, or at least an approximation of, the Coriolis force. This way, one of the equations decouple from the remaining two equations, which can then be solved by quadrature. Using the second assumption (see also [4,6]), both the gravity gradient and the Coriolis forces are neglected. This truncation can be solved analytically. In particular, the authors in [6] obtain analytical solutions with an inclusion of a quadratic air drag. It is shown that the effect of the drag is to increase the effective thrust–weight ratio so that the descent is completed with a lower thrust compared with the vacuum case. In this Note, the problem is considered without any assumptions on the gravity and the Coriolis forces. The approach undertaken here is also more geometric and includes an initial qualitative analysis. Furthermore, a closed-form solution is derived for the case of a controller that depends upon the flight-path angle. Finally, a numerical analysis is provided for the full problem with constant thrust to mass ratio. Through an appropriate choice of scaling the numerical solution of the required control is presented through a single planar plot, regardless of the parameters of the problem. Model