Apollo lunar descent guidance
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.
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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
Cites methods from "Apollo lunar descent guidance"
...A trajectory based on a quartic polynomial in time was used during the Apollo mission [5]....
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TL;DR: It is shown that the minimum-landing-error trajectory generation problem can be posed as a convex optimization problem and solved to global optimality with known bounds on convergence time, which makes the approach amenable to onboard implementation for real-time applications.
Abstract: To increase the science return of future missions to Mars and to enable sample return missions, the accuracy with which a lander can be deliverer to the Martian surface must be improved by orders of magnitude. The prior work developed a convex-optimization-based minimum-fuel powered-descent guidance algorithm. In this paper, this convex-optimization-based approach is extended to handle the case when no feasible trajectory to the target exists. In this case, the objective is to generate the minimum-landing-error trajectory, which is the trajectory that minimizes the distance to the prescribed target while using the available fuel optimally. This problem is inherently a nonconvex optimal control problem due to a nonzero lower bound on the magnitude of the feasible thrust vector. It is first proven that an optimal solution of a convex relaxation of the problem is also optimal for the original nonconvex problem, which is referred to as a lossless convexification of the original nonconvex problem. Then it is shown that the minimum-landing-error trajectory generation problem can be posed as a convex optimization problem and solved to global optimality with known bounds on convergence time. This makes the approach amenable to onboard implementation for real-time applications.
301 citations
Cites methods from "Apollo lunar descent guidance"
...In the next sections, therefore, we extend the approach of [3] to solve the minimum-landing-error problem....
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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
Cites result from "Apollo lunar descent guidance"
...This is in contrast with other approaches that either compute a closed-form solution by ignoring the constraints of the problem [5], [14], propose solving a nonlinear optimization on board [6], [7], or solve a related problem that does not minimize fuel use [8]....
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TL;DR: In this paper, the structure of the manifold S is encapsulated by two operators, state displacement and its inverse, which provide a local vector-space view around a given state x. Generic estimation algorithms can then work on the manifold s mainly by replacing +/- with / where appropriate.
Abstract: Common estimation algorithms, such as least squares estimation or the Kalman filter, operate on a state in a state space S that is represented as a real-valued vector. However, for many quantities, most notably orientations in 3D, S is not a vector space, but a so-called manifold, i.e. it behaves like a vector space locally but has a more complex global topological structure. For integrating these quantities, several ad hoc approaches have been proposed. Here, we present a principled solution to this problem where the structure of the manifold S is encapsulated by two operators, state displacement :SxR^n->S and its inverse :SxS->R^n. These operators provide a local vector-space view @[email protected][email protected] around a given state x. Generic estimation algorithms can then work on the manifold S mainly by replacing +/- with / where appropriate. We analyze these operators axiomatically, and demonstrate their use in least-squares estimation and the Unscented Kalman Filter. Moreover, we exploit the idea of encapsulation from a software engineering perspective in the Manifold Toolkit, where the / operators mediate between a ''flat-vector'' view for the generic algorithm and a ''named-members'' view for the problem specific functions.
185 citations
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TL;DR: This work presents a principled solution to the problem of manifold encapsulation where the structure of the manifold S is encapsulated by two operators, state displacement and its inverse :SxS->R^n, and exploits the idea of encapsulation from a software engineering perspective in the Manifold Toolkit.
Abstract: Common estimation algorithms, such as least squares estimation or the Kalman filter, operate on a state in a state space S that is represented as a real-valued vector. However, for many quantities, most notably orientations in 3D, S is not a vector space, but a so-called manifold, i.e. it behaves like a vector space locally but has a more complex global topological structure. For integrating these quantities, several ad-hoc approaches have been proposed.
Here, we present a principled solution to this problem where the structure of the manifold S is encapsulated by two operators, state displacement [+]:S x R^n --> S and its inverse [-]: S x S --> R^n. These operators provide a local vector-space view \delta; --> x [+] \delta; around a given state x. Generic estimation algorithms can then work on the manifold S mainly by replacing +/- with [+]/[-] where appropriate. We analyze these operators axiomatically, and demonstrate their use in least-squares estimation and the Unscented Kalman Filter. Moreover, we exploit the idea of encapsulation from a software engineering perspective in the Manifold Toolkit, where the [+]/[-] operators mediate between a "flat-vector" view for the generic algorithm and a "named-members" view for the problem specific functions.
184 citations
Cites methods from "Apollo lunar descent guidance"
...Workarounds for this exist that try to avoid these parts of the state space, as was most prominently done in the guidance system of the Apollo Lunar Module [22], or switch between alternative orderings of the parametrization each of which exhibit singularities in different areas of the state space....
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...Workarounds for this exist that try to avoid these parts of the state space, as was most prominently done in the guidance system of the Apollo Lunar Module [22], or switch between alternative orderings of the parameterization each of which exhibit singularities in different areas of the state space....
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...If it is detected, it requires a recovery strategy, in the worst case manual intervention as in the aforementioned Apollo mission [22]....
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References
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24 Aug 1964
103 citations
01 Jun 1972
TL;DR: The premission planning, real-time situation, and post-flight analysis for the Apollo 11 lunar descent and ascent are described in this article, and a comparison between pre-mission planning and actual results is included.
Abstract: The premission planning, the real-time situation, and the postflight analysis for the Apollo 11 lunar descent and ascent are described. A comparison between premission planning and actual results is included. A navigation correction capability, developed from Apollo 11 postflight analysis was used successfully on Apollo 12 to provide the first pinpoint landing. An experience summary, which illustrates typical problems encountered by the mission planners, is also included.
59 citations
TL;DR: During the final few minutes and several miles of the Apollo lunar landing, the commander of the Lunar Module (LM) can manually retargel the automatie landing system.
Abstract: Manual steering, hybrid guidance system, in-flight computations and trajectory for automatic descent and landing system for LEM
32 citations
01 Jan 1970
TL;DR: Apollo 11 premission planning, real-time situation and post-flight analysis for lunar descent and ascent phases, providing navigation correction capability for Apollo 12 was discussed in detail in this article.
Abstract: Apollo 11 premission planning, real time situation and postflight analysis for lunar descent and ascent phases, providing navigation correction capability for Apollo 12
26 citations
TL;DR: The lunar module digital autopilot is a first generation digital control system which would have been difficult to implement with conventional analog autopilot design techniques.
Abstract: The lunar module digital autopilot is a first generation digital control system which would have been difficult to implement with conventional analog autopilot design techniques. The control synthesis problem was divided into the design of an attitude state estimator, the reaction-control-system (RCS) control laws, and the main engine thrust-vector control laws. The attitude state estimator derives the angular velocity and angular acceleration of the vehicle, based on measurements of vehicle attitude and assumed control response only. Rate gyros are not used. The RCS control laws employ parabolic switch curves in their phase plane logic to obtain fast response with a minimum of firings. The critical parameters are adapted in flight in response to varying flight conditions. An unconventional set of nonorthogonal axes is used to decouple the responses to RCS control about different spacecraft axes. A thirdorder minimum time control law is used to control the vehicle attitude by means of the thrusting descent engine. Performance in flight is presented.
26 citations