In the last few years, the demand for efficient and reliable algorithms for solving optimal control problems in dynamics has increased. Among all optimal control problems, time-optimal problems are probably the most challenging ones. Within the framework of this contribution, we pursue a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g. if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum.

So far, such problems have been considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this contribution is the development of an iterative, gradient-based solution strategy for solving such problems.

The proposed approach will be more robust than solving the underlying boundary value problem since the cost functional, i.e. the end time of a dynamical process, will be reduced after every iteration.

The so-called adjoint method is a promising way to compute the direction of the steepest descent, i.e. the variation of a control signal causing the largest local decrease of the cost functional.

Moreover, so-called influence differential equations are formulated to put the changes of the controls and of the final conditions into context. In order to meet the end conditions, we introduce a descent direction that, on the one hand, reduces the cost functional and, on the other hand, brings us closer to the end conditions.

As an example, a moon landing as in the Apollo program, will be considered. Time-optimal control problems have been focused beside (minimal) fuel optimal control questions in the Apollo program almost 60 years ago. In detail, we discuss the ascent, descent, rendezvous phases and abort maneuvers of the Apollo Lunar Excursion Module (LEM) to and from the moon's surface in minimum time. The LEM is described in minimal coordinates by a set of five degrees of freedom. The goal is to find the control of the thrust nozzle of the LEM to minimize a cost functional measuring the final time, where we prescribe the initial states and claim the LEM to satisfy a set of conditions at the final time. We assume that we do not know a control history, which fulfills the final conditions a priori. However, applying the adjoint-gradient method, the constrained optimum of the time-optimal control problem can be found after several updates.


The Adjoint Gradient Method for Time-Optimal Control of a Moon Landing: Ascent, Descent, and Abort

Division / Sponsors: 
Design Engineering Division and Computers & Information In Engineering Division


Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2020/volume/83907

IDETC2020 - Virtual Conference

Supporting the need among researchers worldwide for additional online and multimedia content, ASME now offers the ASME Conference Video Collection. This platform will collect and provide long-term access to conference presentation videos, encompassing all technology areas within ASME's technical conference and events program. About 30,000-40,000 engineers across all mechanical engineering and related disciplines participate in ASME conferences each year. This includes authors of approximately 6,000 to 8,000 technical papers published by ASME, the majority of whom also submit video presentations for our virtual conferences. The video collection will be available as an add-on to existing product packages or as a separate subscription.