Improved Matched Asymptotic Solutions for Three-Dimensional Atmospheric Skip Trajectories

Abstract

An improved technique for matching the asymptotic solutions of non-linear differential equations is presented and successfully applied to the three-dimensional atmospheric skip trajectories. The classical method of matched asymptotic expansions (MAE) is generally applied to two-point-boundary value problems. When we apply the MAE method to initial value problems, due to error propagation, the resulting accuracy usually depends on the physical problems. In the proposed technique, the second-order solutions are obtained by first generating a set of equations for the small perturbations which are the discrepancies between the uniformly valid first-order solutions and the exact solutions. Then, the equations of the small perturbations are integrated separately near the outer and inner boundaries to obtain the perturbed outer and inner expansion solutions, respectively, for a secondorder matching. In addition, in this improved technique the end-point boundaries are artificially extended or constructed to strengthen the physical assumptions on the outer and inner expansions for the matching while in the evaluation of the constants of integration in the uniformly valid first-order solutions, the prescribed end-points are effectively enforced. In this paper, to show the applicability of the improved technique, we first apply it to the rectilinear restricted three-body problem. We men consider the threedimensional skip trajectory. Compared to the solutions obtained by numerical integration over a wide range of entry conditions, the second-order solutions obtained by this improved technique are very accurate. The trajectory elements at the lowest altitude and at exit as well as their accuracy are evaluated.

Skip Entry Trajectory Planning and Guidance

“…For entry vehicles with relatively low lift-to-drag (L/D) ratios, a known strategy since the Apollo era for achieving long downrange is to allow the vehicle to skip out of the atmosphere [4,5,6,7,8,9,10,11]. Figure 1.1 shows an illustration of this concept with the first entry, skip, then reentry of the atmosphere and final entry phase to the landing site.…”

“…Modern computer capabilities permit generation of a feasible trajectory onboard the vehicle and use of a similar Shuttle-like tracking technique [27]. Analytic approaches using asymptotic expansions and optimal control techniques have been attempted for skip entry and aeroassist maneuvers [6,8,19,20]. With the advent of flight computers, onboard calculation of a trajectory has become viable through the use of numerical predictorcorrector methods [9,10,11,20,21,22,23,24,25,26].…”

Analytic Development of a Reference Trajectory for Skip Entry

“…Note that a complete analytic solution to the skip-entry problem should determine the conditions at exit (range to go, velocity V, and flightpath angle ) that render the entry conditions for the final phase. The central difficulty in analytically determining the exit conditions lies in the limitations that various approximations and linearization assumptions have been found to have when estimating the range flown at low-drag altitudes [2,[17][18][19]. The analytic determination of the exit conditions remains an elusive problem whose resolution is not intended in this Note.…”

Asymptotic Solutions of the Restricted Three-Body Problem by Use of Perturbation Methods.

“…On the other hand, in the improved MAE method (iMAE), 14,15) instead of a straightforward matching of the outer and inner expansions to higher-order terms, the higherorder solutions are derived first by generating perturbations between the lower-order composite solutions and the exact solutions. The perturbation equations are then integrated in the outer and inner regions, and a higher-order matching is performed for the expansions of the perturbations.…”