Optimal Control of Endoatmospheric Launch Vehicle Systems: Geometric and Computational Issues

Abstract

In this paper we develop a geometric analysis and a numerical algorithm, based on indirect methods, to solve optimal guidance of endo-atmospheric launch vehicle systems under mixed control-state constraints. Two main difficulties are addressed. First, we tackle the presence of Euler singularities by introducing a representation of the configuration manifold in appropriate local charts. In these local coordinates, not only the problem is free from Euler singularities but also it can be recast as an optimal control problem with only pure control constraints. The second issue concerns the initialization of the shooting method. We introduce a strategy which combines indirect methods with homotopies, thus providing high accuracy. We illustrate the efficiency of our approach by numerical simulations on missile interception problems under challenging scenarios.

Convex Optimization of Launch Vehicle Ascent Trajectory with Heat-Flux and Splash-Down Constraints

“…Besides the payload maximization, the optimization must take into account all mission requirements, which are transcribed as differential, boundary, and path constraints. The differential constraints are associated with the equations of motion ( 6)- (8). The boundary conditions include the initial, terminal, and linkage constraints.…”

“…The present algorithm does not require an accurate initialization, but, rather, in the authors' experience, any starting trajectory with an altitude profile always above sea level is sufficient to achieve convergence. Such trajectories can be easily generated via numerical integration of the original rocket equations of motion ( 6)- (8). To set up the forward propagation, the unknown control history, the duration of free-time arcs, and the initial mass must be prescribed.…”

“…More recent work based on the indirect method includes both pure trajectory design applications [3][4][5] and closed-loop guidance algorithms. [6][7][8] However, when dealing with real-world launch vehicle missions, the indirect method may be unappealing as it requires the derivation of the optimality conditions, which can be a burdensome task, and the solution of the resulting boundary value problem requires a meticulous initialization process to achieve convergence. In addition, if path constraints are included in the formulation, an a priori knowledge of the structure of the constrained arcs is necessary, which, in general, is hard to guess.…”

A minimal time optimal control for a drone landing problem

“…In [4,5], local geometric results are combined with numerical simulations and conjugate point arguments. More recently, [2] deals with geometric analysis and numerical algorithm, based on indirect methods, and provides high numerical precision for optimal trajectories; see also [18] that contains a refined geometric study of the extremals coming from PMP, which allows to implement efficient numerical methods solving the problem of the guidance of a rocket.…”

Sequential Convex Programming For Non-Linear Stochastic Optimal Control

“…DP provides optimal policies through the resolution of a partial differential equation, whereas the necessary conditions for optimality offered by the PMP allow to set up a two-point boundary value problem that, if solved, returns candidate locally optimal solutions. Both methods lead to analytical solutions only in very few cases, and complex hindrances may quickly appear in the numerical context (the stochastic setting is even more vicious than the deterministic one, the latter being better understood for a quite wide range of problems, see, e.g., [37,11]). This has fostered the investigation of more tractable approaches to solve NLPs, such as Monte Carlo simulation [34,16], Markov chain discretization [19,20] and deterministic (though non-equivalent) reformulation [2,4], among others.…”