Rapid Verification Method for the Trajectory Optimization of Reentry Vehicles

Introduction T HE trajectory optimization of path-constrained nonlinear dynamic systems such as those arising in the guidanceand design of reentry vehicles has long been considereda difŽ cult problem.1i3 Of themany methods to solve such problems, the direct collocation method with sparse nonlinear programming (NLP) has proved to be quite effective. Because direct methods do not tie the resulting solutions to the Pontryaginmaximum principle4 (PMP), a two-step direct–indirect approach has been used successfully for Ž nding extremal solutions.5 One major drawback of the direct–indirect approach is that a signiŽ cant amount of labor is necessary to derive all of the necessary conditions including complex switching and junction conditions for state-constrainedproblems.Even with estimates of the adjoint arcs, solving the multipoint boundary-valueproblem is an elaborate task. Over the past few years, a new approach to solving complex trajectory optimizationproblems has been proposed.6i9 The Legendre pseudospectralmethod is a particularlyeffectivemethod because it providesspectrallyaccuratesolutionsfor the costatesand other covectorswithout the use of any analyticaldifferentialequationsfor the