Ascent Trajectories of Multistage Launch Vehicles: Numerical Optimization with Second-Order Conditions Verification

Abstract

Multistage launch vehicles are employed to place spacecraft and satellites in their operational orbits. Trajectory optimization of their ascending path is aimed at defining the maximum payload mass at orbit injection, for specified structural, propulsive, and aerodynamic data. This work describes and applies a method for optimizing the ascending path of the upper stage of a specified launch vehicle through satisfaction of the necessary conditions for optimality. The method at hand utilizes a recently introduced heuristic technique, that is, the particle swarm algorithm, to find the optimal ascent trajectory. This methodology is very intuitive and relatively easy to program. The second-order conditions, that is, the Clebsch-Legendre inequality and the conjugate point condition, are proven to hold, and their fulfillment enforces optimality of the solution. Availability of an optimal solution to the second order is an essential premise for the possible development of an efficient neighboring optimal guidance.

Multiple stage satellite launch vehicle ascent optimization using Chebyshev wavelets

“…[5][6][7][8][9][10][11] show the optimized trajectory of MSLV as computed by p the proposed strategy for k = 2, R = 6, v = 12 and M = 5. The burn duration of the final stage was obtained to be 673.1 seconds (<700 s) and the magnitude of the fuel remaining in the second-stage tank was found to be 483.59 kg.…”

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

“…Good candidates to deal with onboard processing of optimal trajectories are indirect methods (see, e.g., [11], [12], [13], [14], [15]). They leverage necessary conditions for optimality coming from the Pontryagin Maximum Principle (PMP) (see, e.g., [16], [17]) to wrap the optimal guidance problem into a two-point boundary value problem, leading to accurate and fast algorithms (see, e.g.…”

“…Indeed, in several situations, demanding performance criteria (costs C) and onerous missions (final conditions M ) force optimal trajectories to pass through points that do not lie within the domain of the local chart ϕ a (i.e., U a ), and then, by exploiting merely (GOGP) a either the optimality could be lost or, in the worst case, the numerical computations may fail. Here, the novelty consists of introducing another set of coordinates that covers the singularities (with respect to the path angle γ) of chart (U a , ϕ a ) in which the constraints c 1 and c 2 are pure control constraints, as provided by expressions (15).…”

“…Sinceq([s 1 , s 2 ]) ⊆ U a , in what follows, we merely need to consider (GOGP) when restricted to U a . Therefore, when restricted to [s 1 , s 2 ], the curve (q(·),ū(·)) is a solution for the following local version of (GOGP) in U a on the state variable q (see (15), (19)), by differentiating it w.r.t. q, one obtains ∂c ∂q (q(t),ū(t)) + ∂c ∂u (q(t),ū(t)) · ∂Φ ∂q (x(·),w(·)) = 0 .…”

Analytical Initialization of a Continuation-Based Indirect Method for Optimal Control of Endo-Atmospheric Launch Vehicle Systems

“…In [Lu et al, 2003], [Pan and Lu, 2010] the initialization problem is bypassed using finite differences algorithms and multipleshooting methods respectively. By contrast, in [Pontani and Cecchetti, 2013] the authors use second-order conditions and conjugate point theory for a multistage launch vehicle problem; the initialization of the numerical method is based on the particle swarm algorithm. However, most of these approaches remain computationally demanding and not easily applicable in view of real-time processing.…”