by Mark E. Connelly. / Bibliography: p. 21. / "April, 1981." / Air Force Office of Scientific Research Contract AFOSR 80-0229

Currently, many missile fire control systems use a three degree of freedom (3-DOF) trajectory model. The three degrees of freedom represent the linear motion of the missile in three dimensions. A 6-DOF model adds roll, pitch, and yaw, or angular motion in three dimensions to the first three degrees of freedom. Because more of the missile’s attributes are modeled, a 6-DOF model is more accurate than a 3-DOF model. For the same reason, a 3-DOF model is easier to develop and executes faster. Also, because a 3-DOF model ignores the seemingly random angular motion, the step sizes used to integrate 3-DOF models are larger. The goal of this project is to develop a 6-DOF re-entry model with the accuracy of a 6-DOF model with conventional equations of motion and computational speed at least comparable to the 3-DOF model. This can be achieved by transforming the equations that compute the effects of angular motion so that they are better conditioned. Essentially, this is done by fitting a sine wave to the oscillating state variables representing the orientation and angular rates, namely the quaternions and the angular velocity. This thesis shows the results of transforming the oscillating variables of the state vector. / Master of Science

A reduced order model describing the energy and heading angle dynamics of a scramjet missile is developed using a singular perturbation technique. The cruise analysis is briefly reviewed to determine the conditions at which the missile will cruise most efficiently. The turn and climb performance of the missile over the conditions of interest is then examined and a family of extremal trajectories is constructed which asymptotically approach the cruise at an intermediate altitude. / Master of Science

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The method of singular perturbations is applied to the determination of the optimal range-fuel-time trajectory for an air-breathing missile. This method is shown to lead to the reduced-order "cruise-dash" model, and this model is used in the optimization study. Earlier work in this area is extended by the inclusion of two not heretofore considered limits on the dynamical system. The results of the earlier work are shown to hold throughout much of the velocity regime in which the missile operates, but operation in the very high and very low velocity ranges is shown to be sharply curtailed, with the optimal operating points being changed drastically in some cases. Also, the effect of the non-zero minimum admissible throttle setting and the resultant throttle-chattering on the solution of the control problem is examined in some detail. / M.S.

In this thesis, the problem of intercepting highly manoeuvrable threats using seeker-less interceptors that operate in the command guidance mode, is addressed. These systems are more prone to estimation errors than standard seeker-based systems. Several non-linear and optimal estimation and guidance concepts are presented in this thesis for interception of randomly maneuvering targets by seeker-less interceptors. The key contributions of the thesis can be broadly categorized into six groups, namely (i) an optimal selection of bank of lters in interactive multiple model (IMM) scheme to cater to various maneuvers that are expected during the end-game, (ii) an innovative algorithm to reduce chattering phenomenon and formulate effective guidance algorithm based on 'differential game guidance law' (modi ed DGL), (iii) IMM/DGL and IMM/modified DGL based integrated estimation/guidance (IEG) strategy, (iv) sensitivity and robustness analysis of Kalman lters and ne tuning of lters in filter bank using innovation covariance, (v) Performance of tuned IMM/PN, tuned IMM/DGL and tuned IMM/modi ed DGL against various target maneuvers, (vi) Performance comparison with realistic missile model. An innovative generalized state estimation formulation has been proposed in this the-sis for accurately estimating the states of incoming high speed randomly maneuvering targets. The IMM scheme and an optimal selection of lters, to cater to various maneu-vers that are expected during the end-game, is described in detail. The key advantage of this formulation is that it is generic and can capture evasive target maneuver as well as straight moving targets in a uni ed framework without any change of target model and tuning parameters. In this thesis, a game optimal guidance law is described in detail for 2D and 3D engagements. The performance of the differential game based guidance law (DGL) is compared with conventional Proportional Navigation (PN) guidance law, especially for 3D interception scenarios. An innovative chatter removal algorithm is introduced by modifying the differential game based guidance law (modified DGL). In this algorithm, chattering is reduced to the maximum extent possible by introducing a boundary layer around the switching surface and using a continuous control within the boundary layer. The thesis presents performance of the modified DGL algorithm against PN and DGL, through a comparison of miss distances and achieved accelerations. Simulation results are also presented for varying fiight path angle errors. Apart from the guidance logic, two novel ideas have been presented following the evolving "integrated estimation and guidance" philosophy. In the rst approach, an in-tegrated estimation/guidance (IEG) algorithm that integrates IMM estimator with DGL law (IMM/DGL), is proposed for seeker-less interception. In this interception scenario, the target performs an evasive bang-bang maneuver, while the sensor has noisy measure-ments and the interceptor is subject to an acceleration bound. The guidance parameters (i.e., the lateral acceleration commands) are computed with the help of zero e ort miss distance. The thesis presents the performance of the IEG algorithm against combined IMM with PN (IMM/PN), through a comparison of miss distances. In the second ap-proach, a novel modi ed IEG algorithm composed of IMM estimator and modi ed DGL guidance law is introduced to eliminate the chattering phenomenon. Results from both of these integrated approaches are quite promising. Monte Carlo simulation results re-veal that modi ed IEG algorithm achieves better homing performance, even if the target maneuver model is unknown to the estimator. These results and their analysis o er an insight to the interception process and the proposed algorithms. The selection of lter tuning parameters puts forward a major challenge for scien-tists and engineers. Two recently developed metrics, based on innovation covariance, are incorporated for determining the filter tuning parameters. For predicting the proper combination of the lter tuning parameters, the metrics are evaluated for a 3D interception problem. A detailed sensitivity and robustness analysis is carried out for each type of Kalman lters. Optimal and tuned Kalman lters are selected in the IMM con guration to cater to various maneuvers that are expected during the end-game. In the interception scenario examined in this thesis, the target performs various types of maneuvers, while the sensor has noisy measurements and the interceptor is subject to acceleration bound. The tuned IMM serves as a basis for synthesis of e cient lters for tracking maneuvering targets and reducing estimation errors. A numerical study is provided which demonstrates the performance and viability of tuned IMM/modi ed DGL based modi ed IEG strategy. In this thesis, comparison is also performed between tuned IMM/PN, tuned IMM/DGL and tuned IMM/modi ed DGL in integrated estimation/guidance scheme. The results are illustrated by an extensive Monte Carlo simulation study in the presence of estimation errors. Simulation results are also presented for end game maneuvers and varying light path angle errors . Numerical simulations to study the aerodynamic e ects on integrated estimation/ guidance structure and its e ect on performance of guidance laws are presented. A detailed comparison is also performed between tuned IMM/PN, tuned IMM/DGL and tuned IMM/modi ed DGL in integrated estimation/guidance scheme with realistically modelled missile against various target maneuvers. Though the time taken to intercept is higher when a realistic model is considered, the integrated estimation/guidance law still performs better. The miss distance is observed to be similar to the one obtained by considering simpli ed kinematic models.