Mesh refinement in direct transcription methods for optimal control

Betts, John T.; Huffman, William P.

doi:10.1002/(sici)1099-1514(199801/02)19:1<1::aid-oca616>3.0.co;2-q

Cited by 127 publications

(83 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main idea behind all these adaptive grid methods is to use a high resolution (dense) grid only in the vicinity of control switches, constraint boundaries etc, and a coarse grid elsewhere. Examples of such adaptive gridding techniques for the solution of optimal control problems are [10,13,70,12,43,35].…”

Section: Optimal Trajectory Generationmentioning

confidence: 99%

Mathematical Models for Aircraft Trajectory Design: A Survey

Delahaye

Puechmorel

Tsiotras

et al. 2014

Lecture Notes in Electrical Engineering

View full text Add to dashboard Cite

Air traffic management ensures the safety of flight by optimizing flows and maintaining separation between aircraft. After giving some definitions, some typical feature of aircraft trajectories are presented. Trajectories are objects belonging to spaces with infinite dimensions. The naive way to address such problem is to sample trajectories at some regular points and to create a big vector of positions (and or speeds). In order to manipulate such objects with algorithms, one must reduce the dimension of the search space by using more efficient representations. Some dimension reduction tricks are then presented for which advantages and drawbacks are presented. Then, front propagation approaches are introduced with a focus on Fast Marching Algorithms and Ordered upwind algorithms. An example of application of such algorithm to a real instance of air traffic control problem is also given. When aircraft dynamics have to be included in the model, optimal control approaches are really efficient. We present also some application to aircraft trajectory design. Finally, we introduce some path planning techniques via natural language processing and mathematical programming.

show abstract

Section: Optimal Trajectory Generationmentioning

confidence: 99%

Mathematical Models for Aircraft Trajectory Design: A Survey

Delahaye

Puechmorel

Tsiotras

et al. 2014

Lecture Notes in Electrical Engineering

View full text Add to dashboard Cite

show abstract

“…For example, Betts and Huffman [43] describe an automated process for designing the time grid such that discretization errors in the solution are reduced. Moore et al [44] compare Betts and Huffman's approach [43] with a similar method that aims to reduce discretization errors in the energy evolution and also examines the use of time adaptive variational integrators to generate an initial guess with continuous, variable step size profile.…”

Section: Creation Of Initial Guessmentioning

confidence: 99%

Trajectory Design Combining Invariant Manifolds with Discrete Mechanics and Optimal Control

Moore

Ober‐Blöbaum

Marsden

2012

Journal of Guidance, Control, and Dynamics

View full text Add to dashboard Cite

A mission design technique that combines invariant manifold techniques, discrete mechanics, and optimal control produces locally optimal low-energy trajectories. Previously, invariant manifolds of the planar circular restricted three-body problem have been used to design trajectories with relatively small midcourse change in velocity V. A different method of using invariant manifolds is explored to design trajectories directly in the four-body problem. Then, using the local optimal control method DMOC (Discrete Mechanics and Optimal Control), it is possible to reduce the midcourse V to zero. The influence of different boundary conditions on the optimal trajectory is also demonstrated. These methods are tested on a trajectory that begins in Earth orbit and ends in ballistic capture at the moon. Impulsive DMOC trajectories require up to 19% less V than trajectories using a Hohmann transfer. When applied to low-thrust trajectories, DMOC produces an improvement of up to 59% in the mass fraction and 22% in travel time when compared with results from shooting methods. = mass consumption, kg P = spacecraft Q = configuration space q = configuration variable q k = discrete configuration variable r a , r p = radius of elliptical orbit at apoapsis and perigee, km r M H , a H = perigee radius, semimajor axis of Hohmann transfer ellipse, km T = low-thrust magnitude, N T k = discrete thrust magnitude applied at node k, N T max = maximum allowable thrust, N TQ = state space T t = normalized time denoting transition from low thrust to no thrust T x;k = discrete thrust at node k in x direction, N T y;k = discrete thrust at node k in y direction, N t k = discrete time variable u = continuous control parameter u k = discrete control variable u x = control force in x direction u y = control force in y direction v r = normalized radial velocity v x;k = normalized discrete velocity at node k in x direction v y;k = normalized discrete velocity at node k in y direction x i , y i = x, y position of body i for i E; S; M V = scales nondimensional velocity to meters per second V = magnitude of change in velocity, m=s or km=s V C = V necessary to enter elliptical capture orbit at moon, km=s V H = V necessary to leave Earth orbit using a Hohmann transfer, km=s V i = V for circular orbit insertion at body i, i E; M, m=s or km=s V traj = V applied throughout trajectory, m=s t = discrete time grid t = refined discrete time grid = variational operator q k = discrete configuration variation i = angle of body i with respect to x axis of coordinate system for i M; S, rad = mass parameter of planar circular restricted three-body problem k = discrete thrust optimization variable, N = angle of semimajor axis of elliptical orbit at the moon with respect to the Earth-moon x axis, deg = phase of Earth-moon frame with respect to sun-Earth frame, deg = effective potential in rotating frame ! i = normalized rotation rate of body i for i M; S

show abstract

“…This idea is similar to the mesh refinement technique introduced in Ref. [29], which starts with a coarse grid (i.e. low number of discretization nodes), and if necessary, refines the discretization, and then repeats the optimization.…”

Section: Survey Of Previous Workmentioning

confidence: 99%

“…But this method can be time consuming, and it is not guaranteed to converge to a feasible answer. 2) Another method is known as the mesh refinement technique [29]. The idea is to start with a coarse grid (i.e., low number of discretization nodes) and use any guess (e.g., randomly chosen) as an initial starting point for the nonlinear solver.…”

Section: Survey Of Initialization Techniquesmentioning

confidence: 99%

“…Then, if necessary, refine the discretization (i.e., increase the number of discretization nodes), and then repeat the optimization steps using the output of the previous step as the initial guess of the current step. This process can be very time consuming, and therefore current approaches do not appear to be feasible for online planning of complex reconfiguration maneuvers consisting of several spacecraft [29,58]. 3) A third way of initializing the NLP is by using a "warm" start approach [49,63].…”

Section: Survey Of Initialization Techniquesmentioning

confidence: 99%

See 1 more Smart Citation

Two-stage path planning approach for solving multiple spacecraft reconfiguration maneuvers

Aoude

How

Garcia³

2008

J of Astronaut Sci

View full text Add to dashboard Cite

The paper presents a two-stage approach for designing optimal reconfiguration maneuvers for multiple spacecraft in close proximity. These maneuvers involve well-coordinated and highly-coupled motions of the entire fleet of spacecraft while satisfying an arbitrary number of constraints. This problem is complicated by the nonlinearity of the attitude dynamics, the non-convexity of some of the constraints, and the coupling that exists in some of the constraints between the positions and attitudes of all spacecraft. While there has been significant research to solve for the translation and/or rotation trajectories for the multiple spacecraft reconfiguration problem, the approach presented in this paper is more general and on a larger scale than the problems considered previously. The essential feature of the solution approach is the separation into two stages, the first using a simplified planning approach to obtain a feasible solution, which is then significantly improved using a smoothing stage. The first stage is solved using a bi-directional Rapidly-exploring Random Tree (RRT) planner. Then the second step optimizes the trajectories by solving an optimal control problem using the Gauss pseudospectral method (GPM). Several examples are presented to demonstrate the effectiveness of the approach for designing spacecraft reconfiguration maneuvers.

show abstract

Mesh refinement in direct transcription methods for optimal control

Cited by 127 publications

References 8 publications

Mathematical Models for Aircraft Trajectory Design: A Survey

Mathematical Models for Aircraft Trajectory Design: A Survey

Trajectory Design Combining Invariant Manifolds with Discrete Mechanics and Optimal Control

Two-stage path planning approach for solving multiple spacecraft reconfiguration maneuvers

Contact Info

Product

Resources

About