# Lossless Convex Guidance for Lunar Powered Descent

19 Jan 2023-

TL;DR: In this article , a convex programming approach was used for planetary landing guidance originally developed for Mars landings and adapted to lunar soft landings, including the addition of state and control constraints that were previously not part of the lossless convexification framework.

Abstract: This paper builds upon a convex programming approach to propellant-optimal planetary landing guidance originally developed for Mars landings and adapts it to lunar soft landings. These novel adaptations include the addition of state and control constraints that were previously not part of the lossless convexification framework: maximum tilt rate, maximum tilt acceleration, maximum thrust ramp rate, and a terminal vertical descent phase. Additionally, we have included an inverse square central gravity model and a minimum altitude constraint in the Moon-centered, Moon-fixed (MCMF) frame. These constraints are convexified and the resulting second-order cone program is solved for an Apollo-like sample case.

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01 Mar 1953

TL;DR: The problem formulated below was motivated by that of determining an interval containing the point at which a unimodal function on the unit interval possesses a maximum, without postulating regularity conditions involving continuity, derivatives, etc.

Abstract: The problem formulated below was motivated by that of determining an interval containing the point at which a unimodal function on the unit interval possesses a maximum, without postulating regularity conditions involving continuity, derivatives, etc. Our solution is to give, for every e > 0 and every specified number N of values of the argument at which the function may be observed, a procedure which is €-minimax (see (1) below) among the class of all sequential nonrandomized procedures which terminate by giving an interval containing the required point, where the payoff of the computer to nature is the length of this final interval. (The same result holds if, e.g., we consider all nonrandomized procedures and let the payoff be length of interval plus c or 0 according to whether the interval fails to cover or covers the desired point, where c^l/Uif+i, the latter being defined below.) The analogous problem where errors are present in the observations was considered in [l ], but no optimum results are yet known for that more difficult case. Search for a maximum is a "second-order" search in the sense that information is given by pairs of observations. Thus, if Xix

999 citations

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TL;DR: This work presents a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing as a finite-dimensional convex optimization problem as a second-order cone programming problem.

Abstract: We present a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing. 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 second-order cone programming problem. Second-order cone programming is a subclass of convex programming, and there are efficient second-order cone programming solvers with deterministic convergence properties. Consequently, the resulting guidance algorithm can potentially be implemented onboard a spacecraft for real-time applications.

482 citations

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TL;DR: In this article, the basic theory of proportional navigation is presented and two variations on this guidance method are treated: one in which the commanded acceleration is biased by a small value of the measured rotational rate of the line of sight between the interceptor and its target.

Abstract: Proportional navigation has proved to be a useful guidance technique in several surface-to-air and air-to-air missile systems for interception of airborne targets. In this article, which is tutorial in nature, the basic theory of proportional navigation is presented and clarified. In addition, two variations on this guidance method are treated: one in which the commanded acceleration is biased by a small value of the measured rotational rate of the line of sight between the interceptor and its target, and one in which the line-of-sight rotational rate is reduced to a prescribed value (dead space) and then maintained at this rate until intercept. The analysis is directed, by example, to the case of the exoatmospheric interception of a satellite; however, the guidance theory presented is also applicable to the intercept of a nonmaneuvering airborne target.

271 citations

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TL;DR: In this paper, interactive terminal-descent guidance enables the crew to control the essentially vertical descent rate in order to land in minimum time with safe contact speed, using concepts that make gimbal lock inherently impossible.

Abstract: Apollo Lunar-descent Guidance transfers the Lunar Module from a near-circular orbit to touchdown, traversing 17^o central angle and 15 km altitude in 11 min. A group of interactive programs in an onboard computer guide the descent, controlling altitude and the descent propulsion system throttle. A ground-based program precomputes guidance targets. This paper describes the concepts involved. Explicit and implicit guidance are discussed, guidance equations are derived, and the earlier Apollo explicit equation is shown to be an inferior special case of the later implicit equation. The paper describes interactive guidance by which the two-man crew selects a landing site in favorable terrain and directs the trajectory there. Interactive terminal-descent guidance enables the crew to control the essentially vertical descent rate in order to land in minimum time with safe contact speed. The attitude maneuver routine uses concepts that make gimbal lock inherently impossible. The throttle routine yields zero steady-state thrust-acceleration error or avoids operation within a thrust region forbidden because of hardware limitations. The ground-based program precomputes guidance targets which shape the trajectory to produce an efficient descent with adequate visibility and no transients at the final phasic interface.

229 citations

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TL;DR: A convexification of the control constraints that is proven to be lossless enables the use of interior point methods of convex optimization to obtain optimal solutions of the original nonconvex optimal control problem.

Abstract: Planetary soft landing is one of the benchmark problems of optimal control theory and is gaining renewed interest due to the increased focus on the exploration of planets in the solar system, such as Mars. The soft landing problem with all relevant constraints can be posed as a finite-horizon optimal control problem with state and control constraints. The real-time generation of fuel-optimal paths to a prescribed location on a planet's surface is a challenging problem due to the constraints on the fuel, the control inputs, and the states. The main difficulty in solving this constrained problem is the existence of nonconvex constraints on the control input, which are due to a nonzero lower bound on the control input magnitude and a nonconvex constraint on its direction. This paper introduces a convexification of the control constraints that is proven to be lossless; i.e., an optimal solution of the soft landing problem can be obtained via solution of the proposed convex relaxation of the problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain optimal solutions of the original nonconvex optimal control problem.

212 citations