Analytical Dimensional Reduction of a Fuel Optimal Powered Descent Subproblem

Abstract

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 landing guidance system must be able to deliver the vehicle to a desired soft landing while meeting several constraints necessary for the safety of the vehicle. Due to performance limitations of current launch vehicles, it is desired to minimize the amount of fuel used. In addition, the landing site may change in real-time in order to avoid previously undetected hazards which become apparent during the landing maneuver.This complicated maneuver can be broken into simpler subproblems that bound the full problem. One such subproblem is to find a minimum-fuel landing solution that meets constraints on the initial state, final state, and bounded thrust acceleration magnitude. With the assumptions of constant gravity and negligible atmosphere, the form of the optimal steering law is known, and the equations of motion can be integrated analytically, resulting in a system of five equations in five unknowns. It is shown that this system of equations can be reduced analytically to two equations in two unknowns. With an additional assumption of constant thrust acceleration magnitude, 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, and it is used as a real-time guidance applied to a lunar landing test case.

Fuel-efficient Descent and Landing Guidance Logic for a Safe Lunar Touchdown

“…The optimal controls are given by Eq. (22), and the corresponding optimal trajectory is very similar to that given in Fig. 7.…”

“…Fuel-optimal control problems formulated with a bounded thrust typically generate thrust profile consists of "bang-bang" type control. [17][18][19][20][21][22] In these optimal thrust solutions, the thrust magnitude "bangs" instantaneously between its maximum and minimum magnitudes. Even ignoring for the time being whether the lunar lander descent engine can be throttled between its minimum and maximum thrust magnitudes instantaneously, we also have a concern about the acceptability of these types of bang-bang control for crew safety.…”

Guidance for Autonomous Precision Landing on Atmosphereless Bodies

“…52 Rea and Bishop reduced the dimensionality of the landing problem, assuming constant thrust and a free time-of-flight. 53 Some of the guidance laws have been compared in a trade-off in Ref. 6.…”

A State-Triggered Line of Sight Constraint for 6-DoF Powered Descent Guidance Problems

“…The composition of a rotation and translation is represented using dual quaternion multiplication. As such, the unit dual quaternion that represents a translation by r followed by rotation q is given by (taking (11) into account)q…”

“…7,8 Several other authors continued to study the necessary conditions of the minimum fuel 3-DoF guidance problem using optimal control theory. [9][10][11] At the same time, Açıkmeşe and Ploen published work with an alternative viewpoint on the 3-DoF problem. [12][13][14] They took a convex programming approach, and showed that a non-convex (non-zero) lower bound on thrust magnitude can be relaxed by introducing a slack variable.…”