Singular Perturbations and Time Scales in Guidance and Control of Aerospace Systems: A Survey

The physical phenomenon of near equilibrium glide in entry flight of a lifting vehicle has long been a very useful utility in flight mechanics analysis and entry guidance designs. But no rigorous mathematical characterization of this quasi-equilibrium flight condition exists that can help provide a deeper understanding of when and how such a flight condition takes place and what is the precise relationship between exact and quasi-equilibrium glide. We present a formal analysis to quasi-equilibrium entry flight in this paper. We embed the quasi-equilibrium gliding flight problem in a class of regular perturbation problems and obtain solutions in asymptotic expansions. We show that the zeroth-order solution is an exact equilibrium glide solution. We provide the complete solution structure and convergence analysis. Our analysis yields quantitative accuracy estimates to the quasi-equilibrium glide condition that can guide how this approximation can be best used. The solution structure we obtain naturally leads to what can be done to improve the accuracy of the quasi-equilibrium glide condition.

Asymptotic Analysis of Quasi-Equilibrium Glide in Lifting Entry Flight

Research and design for lifting reentry

An improved technique for matching the asymptotic solutions of nonlinear differential equations is presented and successfully applied to the three-dimensional atmospheric skip trajectories. The classic method of matched asymptotic expansions is generally applied to two-boundary value problems. When we apply the classic method to initial valueproblems, becauseoferrorpropagation, theresulting accuracy usually depends on physical problems. In the proposed technique, second-order solutions are obtained by e rst generating a set of equations for the small perturbations,whicharethediscrepanciesbetweentheuniformly valide rst-ordersolutionsandtheexactsolutions. 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 second-order matching. In addition, in this improved technique the endpoint boundaries are artie cially extended or constructed to strengthen the physical assumptions on the outer and inner expansions for matching. Compared with the solutions obtained by numerical integration over a wide rangeof entry conditions, the second-order solutions obtained by this improved techniquearevery accurate.Thetrajectory elements at the lowestaltitude and at exit, aswell astheir accuracy, are evaluated.

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Improved matched asymptotic solutions for three-dimensional atmospheric skip trajectories

Optimal aerodynamic control by matched asymptotic expansions

The problem of determining the trajectories, partially or wholly contained in the atmosphere of a spherical, nonrotating planet, is considered. The exact equations of motion for three-dimensional, aerodynamically affected flight are derived. Modified Chapman variables are introduced and the equations are transformed into a set suitable for analytic integration using asymptotic expansions. The trajectory is solved in two regions: the outer region, where the force may be considered a gravitational field with aerodynamic perturbations, and the inner region, where the force is predominantly aerodynamic, with gravity as a perturbation. The two solutions are matched directly. A composite solution, valid everywhere, is constructed by additive composition. This approach of directly matched asymptotic expansions applied to the exact equations of motion couched in terms of modified Chapman variables yields an analytical solution which should prove to be a powerful tool for aerodynamic orbit calculations.

R. D. Culp

Papers

Solution of the exact equations for three-dimensional atmospheric entry using directly matched asymptotic expansions