# Sensitivity and prediction analysis of an Apollo-type earth orbital reentry trajectory

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This study investigates the suitability of using the sensitivity coefficients provided by the fundamental guidance equation to analyze and compare, in an engineering sense, the results and effects of state and control variable perturbations on an optimal Apollo-type reentry trajectory. The general formulation of the nonlinear two-point boundary value optimization problem and the formulation of the fundamental guidance equation are presented. The perturbation equations used in the development of the fundamental guidance equation are usually available to the optimizer who uses the more common optimization techniques. The four suboptimal control profiles used to approximate the optimal control history are: (1) the Curve Fit Approximation; (2) the Straight Line Approximation; (3) the Flat Step Approximation; and (4) the Constant Control Profile Approximation. The sensitivity coefficients of the fundamental guidance equation are used to compare the four control profiles. These coefficients are an indication of the state variable sensitivity to initial state and constant bias control deviations. Moreover, these coefficients are used to linearly predict terminal state deviations which may be a result of initial state and constant control bias deviations from the selected nominal control history. The linearly predicted data are compared to actual integrated error data to obtain some indication of the limiting accuracy of the fundamental guidance equation for this problem. Note that more than one nominal control history could be used in referencing the trajectory deviation predictions. Real-time deviation predictions could he facilitated by referencing deviations to the control history which most nearly approximates the real-world case. The fundamental guidance equations were found to provide an abundance of information concerning state and control variable influence upon one another. Linear predictions for 2-percent initial state variable and 10-percent constant control variable deviations were found to agree well with the integrated error data which were used as the base line. As would be expected, the linear prediction method was found to be significantly faster than numerically integrating families of error trajectories.