The method of singular-perturbation analysis is applied to the determination of range-fuel-time optimal aircraft trajectories. The problem is shown to break down into three sub-problems which are studied separately. In particular, the inner layer containing the altitude path-angle dynamics is analyzed in detail. The outer solutions are discussed in an earlier work. As a step forward in solving the ensuing nonlinear two-point boundary-value problem, linearization of the equations is suggested. Conditions for the stability of the linearized boundary-layer equations are discussed. Also, the question of parameter selection to fit the solution to the split boundary conditions is resolved. Generation of feedback laws for the angle-of-attack from the linear analysis is discussed. Finally, the techniques discussed are applied to a numerical example of a missile. The linearized feedback solution is compared to the exact solution obtained using a multiple shooting method. / Master of Science

Singular subarcs arise in quite a few problems of flight dynamics. The present study is devoted to the specific problem of ascent and acceleration of a vehicle in atmospheric flight in which a variable-thrust arc forms a part of the optimal trajectory. A two-parameter family of singular arcs was generated for time-range-fuel problems of an ascending rocket, using the modelling of Zlatskiy and Kiforenko. The short-term optimality of the singular subarcs has been checked in terms of certain necessary conditions: the classical Clebsch condition, the Kelley condition or the Generalized Legendre-Clebsch condition and the Goh condition. All these are found to be satisfied computationally for all the candidates. The calculations were repeated for simplified thrust-along-the-path modelling and similar results on optimality obtained. / Master of Science

In aircraft research, there is keen interest in the procedure of determining the set of controls required to perform a maneuver from a definition of the trajectory. This is called the inverse problem. It has been proposed that if a complete set of states and state time derivatives can be derived from a trajectory then a model-following solution can allocate the controls necessary for the maneuver. This paper explores the problem of finding the complete state definition and provides a solution that requires numerical differentiation, fixed point iteration and a Newton's method solution to nonlinear equations. It considers trajectories that are smooth, piecewise smooth, and noise ridden. The resulting formulation was coded into a FORTRAN program. When tested against simple smooth maneuvers, the program output was very successful but demonstrated the limitations imposed by the assumptions and approximations in the development. / Master of Science

The capability to perform rendezvous and proximity operations about space objects is central to the next generation of space situational awareness. The ability to diagnose and respond to spacecraft anomalies is often hampered by the lack of capability to perform inspection or testing on the target vehicle in flight. While some limited ability to perform inspection can be provided by an extensible boom, such as the robotic arms deployed on the space shuttle and space station, a free-flying companion vehicle provides maximum flexibility of movement about the target. Safe and efficient utilization of a companion vehicle requires trajectories capable of minimizing spacecraft resources, e.g., time or fuel, while adhering to complex path and state constraints. This paper develops an efficient solution method capable of handling complex constraints based on a grid search A* algorithm and compares solution results against a state-of-the-art nonlinear optimization method. Trajectories are investigated that include nonlinear constraints, such as complex keep-out-regions and thruster plume impingement, that may be required for inspection of a specific target area in a complex environment. This work is widely applicable and can be expanded to apply to a variety of satellite relative motion trajectory planning problems. / The capability to perform rendezvous and proximity operations about space objects is central to the next generation of space situational awareness. The ability to diagnose and respond to spacecraft anomalies is often hampered by the lack of capability to perform inspection or testing on the target vehicle in flight. While some limited ability to perform inspection can be provided by an extensible boom, such as the robotic arms deployed on the space shuttle and space station, a free-flying companion vehicle provides maximum flexibility of movement about the target. Safe and efficient utilization of a companion vehicle requires trajectories capable of minimizing spacecraft resources, e.g., time or fuel, while adhering to complex path and state constraints. This paper develops an efficient solution method capable of handling complex constraints based on a grid search A* algorithm and compares solution results against a state-of-the-art nonlinear optimization method. Trajectories are investigated that include complex nonlinear constraints, such as complex keep-out-regions and thruster plume impingement, that may be required for inspection of a specific target area in a complex environment. This work is widely applicable and can be expanded to apply to a variety of satellite relative motion trajectory planning problems. / Master of Science / The ability of one satellite to perform actions near a second space satellite or other space object is important for understanding the space environment and accomplishing space mission goals. The development of a method to plan the path that one satellite takes near a second satellite such that fuel usage is minimized and other constraints satisfied is important for accomplishing mission goals. This thesis focuses on developing a fast solution method capable of handling complex constraints that can be applied to plan paths satellite relative motion operations. The solution method developed in this thesis is then compared to an existing solution method to determine the efficiency and accuracy of the method.

Consider the problem of developing an algorithm that computes optimal preprogrammed evasive maneuvers for a Maneuvering Reentry Vehicle (MaRV) attacking a target defended with Anti-Ballistic Missiles (ABMs). The problem is large in terms of the number of optimization parameters, and perhaps in terms of the number of nonlinear constraints. Since both MaRV and ABM trajectories are expensive to compute, rapid convergence of the optimization algorithm is of prime concern. This paper examines a discontinuity in the cost function that degrades both the speed and the reliability of optimizer convergence. A solution is offered, proposing that the optimization algorithm be operated in a new parameter space, in which the discontinuity occurs at infinity. Effectively, the mapping prevents the optimization algorithm from crossing the discontinuity thereby improving optimizer convergence. Results comparing convergence with and without the parameter mapping demonstrate the effectiveness of the procedure. / Master of Science

<div>Optimization of atmospheric flight control has long been performed on the ground, prior to mission flight due to large computational requirements used to solve non-linear programming problems. Onboard trajectory optimization enables the creation of new reference trajectories and updates to guidance coefficients in real time. This thesis summarizes the methods involved in solving optimal control problems in real time using convexification and Sequential Convex Programming (SCP). The following investigation provided insight in assessing the use of state of the art SCP optimization architectures and convexification of the hypersonic equations of motion[ 1 ]–[ 3 ] with different control schemes for the purposes of enabling on-board trajectory optimization capabilities.</div><div>An architecture was constructed to solve convexified optimal control problems using direct population of sparse matrices in triplet form and an embedded conic solver to enable rapid turn around of optimized trajectories. The results of this show that convexified optimal control problems can be solved quickly and efficiently which holds promise in autonomous trajectory design to better overcome unexpected environments and mission parameter changes. It was observed that angle of attack control problems can be successfully convexified and solved using SCP methods. However, the use of multiple coupled controls is not guaranteed to be successful with this method when they act in the same plane as one another. The results of this thesis demonstrate that state of the art SCP methods have the capacity to enable onboard trajectory optimization with both angle of attack control and bank angle control schemes.</div><div><br></div>

Simulation of maneuvers with multibody models of rotorcraft vehicles is an important research area due to its complexity. During the maneuvering flight, some important design limitations are encountered such as maximum loads and maximum turning rates near the proximity of the flight envelope. This increases the demand on high fidelity models in order to define appropriate controls to steer the model close to the desired trajectory while staying inside the boundaries. A framework based on the hierarchical decomposition of the problem is used for this study. The system should be capable of generating the track by itself based on the given criteria and also capable of piloting the model of the vehicle along this track. The generated track must be compatible with the dynamic characteristics of the vehicle. Defining the constraints for the maneuver is of crucial importance when the vehicle is operating close to its performance boundaries. In order to make the problem computationally feasible, two models of the same vehicle are used where the reduced model captures the coarse level flight dynamics, while the fine scale comprehensive model represents the plant. The problem is defined by introducing planning layer and control layer strategies. The planning layer stands for solving the optimal control problem for a specific maneuver of a reduced vehicle model. The control layer takes the resulting optimal trajectory as an optimal reference path, then tracks it by using a non-linear model predictive formulation and accordingly steers the multibody model. Reduced models for the planning and tracking layers are adapted by using neural network approach online to optimize the predictive capabilities of planner and tracker. Optimal neural network architecture is obtained to augment the reduced model in the best way. The methodology of adaptive learning rate is experimented with different strategies. Some useful training modes and algorithms are proposed for these type of applications. It is observed that the neural network increased the predictive capabilities of the reduced model in a robust way. The proposed framework is demonstrated on a maneuvering problem by studying an obstacle avoidance example with violent pull-up and pull-down.

A supercavitating vehicle, a next-generation underwater vehicle capable of changing the paradigm of modern marine warfare, exploits supercavitation as a means to reduce drag and achieve extremely high submerged speeds. In supercavitating flows, a low-density gaseous cavity entirely envelops the vehicle and as a result the vehicle is in contact with liquid water only at its nose and partially over the afterbody. Hence, the vehicle experiences a substantially reduced skin drag and can achieve much higher speed than conventional vehicles. The development of a controllable and maneuvering supercavitating vehicle has been confronted with various challenging problems such as the potential instability of the vehicle, the unsteady nature of cavity dynamics, the complex and non-linear nature of the interaction between vehicle and cavity. Furthermore, major questions still need to be resolved regarding the basic configuration of the vehicle itself, including its control surfaces, the control system, and the cavity dynamics. In order to answer these fundamental questions, together with many similar ones, this dissertation develops an integrated simulation-based design tool to optimize the vehicle configuration subjected to operational design requirements, while predicting the complex coupled behavior of the vehicle for each design configuration. Particularly, this research attempts to include maneuvering flight as well as various operating trim conditions directly in the vehicle configurational optimization. This integrated approach provides significant improvement in performance in the preliminary design phase and indicates that trade-offs between various performance indexes are required due to their conflicting requirements. This dissertation also investigates trim conditions and dynamic characteristics of supercavitating vehicles through a full 6 DOF model. The influence of operating conditions, and cavity models and their memory effects on trim is analyzed and discussed. Unique characteristics are identified, e.g. the cavity memory effects introduce a favorable stabilizing effect by providing restoring fins and planing forces. Furthermore, this research investigates the flight envelope of a supercavitating vehicle, which is significantly different from that of a conventional vehicle due to different hydrodynamic coefficients as well as unique operational conditions.

This thesis addresses the challenges in the efficient and robust generation and optimization of three-dimensional landing trajectories for fixed-wing aircraft subject to prescribed boundary conditions and constraints on maneuverability and collision avoidance. In particular, this thesis focuses on the airliner emergency landing scenario and the minimization of landing time. The main contribution of the thesis is two-fold. First, it provides a hierarchical scheme for integrating the complementary strength of a variety of methods in path planning and trajectory optimization for the improvement in efficiency and robustness of the overall landing trajectory optimization algorithm. The second contribution is the development of new techniques and results in mesh refinement for numerical optimal control, optimal path tracking, and smooth path generation, which are all integrated in a hierarchical scheme and applied to the landing trajectory optimization problem. A density function based grid generation method is developed for the mesh refinement process during numerical optimal control. A numerical algorithm is developed based on this technique for solving general optimal control problems, and is used for optimizing aircraft landing trajectories. A path smoothing technique is proposed for recovering feasibility of the path and improving the tracking performance by modifying the path geometry. The optimal aircraft path tracking problem is studied and analytical results are presented for both the minimum-time, and minimum-energy tracking with fixed time of arrival. The path smoothing and optimal path tracking methods work together with the geometric path planner to provide a set of feasible initial guess to the numerical optimal control algorithm. The trajectory optimization algorithm in this thesis was tested by simulation experiments using flight data from two previous airliner accidents under emergency landing scenarios.The real-time application of the landing trajectory optimization algorithm as part of the aircraft on-board automation avionics system has the potential to provide effective guidelines to the pilots for improving the fuel consumption during normal landing process, and help enhancing flight safety under emergency landing scenarios. The proposed algorithms can also help design optimal take-off and landing trajectories and procedures for airports.

Solutions to trajectory optimization problems are carried out by the direct and indirect methods. Under broad heading of these methods, numerous algorithms such as collocation, direct, indirect and multiple shooting methods have been developed and reported in the literature. Each of these algorithms has certain advantages and limitations. For example, direct shooting technique is not suitable when the number of nonlinear programming variables is large. Indirect shooting method requires analytical derivatives of the control and co-states function and a poorly guessed initial condition can result in numerical unstable values of the adjoint variable. Multiple shooting techniques can alleviate some of these difficulties by breaking down the trajectory into several segments that help in reducing the non-linearity effects of early control on later parts of the trajectory. However, multiple shooting methods then have to handle more number of variables and constraints to satisfy the defects at the segment joints. The sie of the nonlinear programming problem in the collocation method is also large and proper locations of grid points are necessary to satisfy all the path constraints. Stochastic methods such as Genetic algorithms, on the other hand, also require large number of function evaluations before convergence. To overcome some of the limitations of the conventional methods, improved solution techniques are developed. Three improved methods are proposed for the solution of trajectory optimization problems. They are • a genetic algorithm employing dominance and diploidy concept. • a collocation method using chebyshev polynomials , and • a hybrid method that combines collocation and direct shooting technique A conventional binary-coded genetic algorithm uses a haploid chromosome, where a single string contains all the variable information in the coded from. A diploid, as the name suggests, uses pair of chromosomes to store the same characteristic feature. The diploid genetic algorithm uses a dominant map for decoding genotype into a stable, consistent phenotype. In dominance, one allele takes precedence over another. Diploidy and dominance helps in retaining the previous best solution discovered and shields them from harmful selection in a changing environment. Hence, diploid and dominance affect a king of long-term memory in the genetic algorithm. They allow alternate solutions to co-exist. One solution is expressed and the other is held in abeyance. In the improved diploid genetic algorithm, dominant and recessive genes are defined based on the fitness evaluation of each string. The genotype of fittest string is declared as the dominant map. The dominant map is dynamic in nature as it is replaced with a better individual in future generations. The concept of diploidy and dominance in the improved method mimics closer to the principles used in human genetics as compared to any such algorithms reported in the literature. It is observed that the improved diploid genetic algorithm is able to locate the optima for a given trajectory optimization problem with 10% lower computational time as compared to the haploid genetic algorithm. A parameter optimization problem arising from an optimal control problem where states and control are approximated by piecewise Chebyshev polynomials is well known. These polynomials are more accurate than the interpolating segments involving equal spaced data. In the collocation method involving Chebyshev polynomials, derivatives of two neighboring polynomials are matched with the dynamics at the nodal points. This leads to a large number of equality constraints in the optimization problem. In the improved method, derivative of the polynomial is also matched with the dynamics at the center of segments. Though is appears the problem size is merely increased, the additional computations improve the accuracy of the polynomial for a larger segment. The implicit integration step size is enhanced and overall size of the problem is brought down to one-fourth of the problem size defined with a conventional collocation method using Chebyshev polynomials. Hybrid method uses both collocation and direct shooting techniques. Advantages of both the methods are combined to give more synergy. Collocation method is used in the starting phase of the hybrid method. The disadvantage of standalone collocation method is that tuning of grid points is required to satisfy the path constraints. Nevertheless, collocation method does give a good guess required for the terminal phase of the hybrid method, which uses a direct shooting approach. Results show nearly 30% reduction in computation time for the hybrid approach as compared to a method in which direct shooting alone is used, for the same initial guess of control. The solutions obtained from the three improved methods are compared with an indirect method. The indirect method requires derivations of the control and adjoint equations, which are difficult and problem specific. Due to sensitivity of the costate variables, it is often difficult to find a solution through the indirect method. Nevertheless, these methods do provide an accurate result, which defines a benchmark for comparing the solutions obtained through the improved methods. Trajectory design and optimization of a RLV(Reusable Launch Vehicle) Demonstrator mission is considered as a test problem for evaluating the performance of the improved methods. The optimization problem is difficult than a conventional launch vehicle trajectory optimization problem because of the following two reasons. • aerodynamic lift forces in the RLV add one more dimension to the already complex launch vehicle optimization problem. • as RLV performs a sub orbital flight, the ascent phase trajectory influences the re-entry trajectory. Both the ascent and re-entry optimization problem of the RLV mission is addressed. It is observed that the hybrid method gives accurate results with least computational effort, as compared with other improved techniques for the trajectory optimization problem of RLV during its ascent flight. Hybrid method is then successfully used during the re-entry phase and in designing the feasible optimal trajectories under the dispersion conditions. Analytical solutions obtained from literature are used to compare the optimized trajectory during the re-entry phase. Trajectory optimization studies are also carried out for the off-nominal performances. Being a thrusting phase, the ascent trajectory is subjected to significant deviations, mainly arising out of solid booster performance dispersions. The performance index during rhe ascent phase is modified in a novel way for handling dispersions. It minimizes the state errors in a least square sense, defined at the burnout conditions ensure possibilities of safe re-entry trajectories. The optimal trajectories under dispersion conditions serve as a benchmark for validating the closed-loop guidance algorithm that is developed for the ascent phase flight. Finally, an on-line trajectory command-reshaping algorithm is developed which meets the flight objectives under the dispersion conditions. The guidance algorithm uses a pre-computed trajectory database along with some real-time measured parameters in generating the optimal steering profiles. The flight objectives are met under the dispersion conditions and the guidance generated steering profiles matches closely with the optimal trajectories.