Dissertations / Theses on the topic 'Trajectory optimization. Launch vehicles (Astronautics)'

Trajectory optimization is an important part of launch vehicle design and operation. With the high costs of launching payload into orbit, every pound that can be saved increases affordability. One way to save weight in launch vehicle design and operation is by optimizing the ascent trajectory. Launch vehicle trajectory optimization is a field that has been studied since the 1950’s. Originally, analytic solutions were sought because computers were slow and inefficient. With the advent of computers, however, different algorithms were developed for the purpose of trajectory optimization. Computer resources were still limited, and as such the algorithms were limited to local optimization methods, which can get stuck in specific regions of the design space. Local methods for trajectory optimization have been well studied and developed. Computer technology continues to advance, and in recent years global optimization has become available for application to a wide variety of problems, including trajectory optimization. The aim of this thesis is to create a methodology that applies global optimization to the trajectory optimization problem. Using information from a global search, the optimization design space can be reduced and a much smaller design space can be analyzed using already existing local methods. This allows for areas of interest in the design space to be identified and further studied and helps overcome the fact that many local methods can get stuck in local optima. The design space included in trajectory optimization is also considered in this thesis. The typical optimization variables are initial conditions and flight control variables. For direct optimization methods, the trajectory phase structure is currently chosen a priori. Including trajectory phase structure variables in the optimization process can yield better solutions. The methodology and phase structure optimization is demonstrated using an earth-to-orbit trajectory of a Delta IV Medium launch vehicle. Different methods of performing the global search and reducing the design space are compared. Local optimization is performed using the industry standard trajectory optimization tool POST. Finally, methods for varying the trajectory phase structure are presented and the results are compared.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2001.
Includes bibliographical references (p. 167-168).
Onboard trajectory generation dispenses with the pre-defined trajectories used. in Reusable Launch Vehicle (RLV) guidance since the early days of the Shuttle era. This shift, enabled by a new breed of algorithms harnessing modern computer power, offers improvements in performance, robustness, operational cost, and safety. This thesis develops a set of algorithms providing onboard trajectory generation for low lift-over-drag gliding RLVs in subsonic flight below 40,000 ft. The NASA/Orbital Sciences X-34 is used as a representative model for which feasible trajectories are designed over a range of initial conditions without human intervention. In addition to being autonomous, the guidance output of the onboard trajectory generator differs from current Shuttle-based approaches, providing a realistic "future history" in a propagated plan, rather than output commands reacting to perceived instantaneous vehicle needs. Hence, this approach serves an enabling role in a larger research effort to develop a next generation guidance system using an integrated control function. To assess feasibility, the onboard trajectory generator is benchmarked against traditional X-34 guidance for a drop test scenario. The results match in basic form, with differences showcasing the autonomous algorithms' preference for maximum robustness. The true strength of the onboard trajectory generator lies in its ability to handle off-nominal conditions. A series of test cases highlight the ability of the algorithms to effectively cope with anomalous initial drop conditions, reach the desired terminal states, and provide maximum late-trajectory robustness. Computation time is sufficiently brief to suggest a real-time application, after straightforward improvements are made.
by André R. Girerd.
S.M.

Thesis (Ph.D. in Astronautical Engineering)--Naval Postgraduate School, December 2006.
Dissertation supervisor(s): I. Michael Ross. "December 2006." Includes bibliographical references (p. 395-411). Also available in print.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.
Includes bibliographical references (p. 189-197).
Small unmanned aerial vehicles (UAVs), equipped with navigation systems and video capability, are currently being deployed for intelligence, reconnaissance and surveillance missions. One particular mission of interest involves computing location estimates for targets detected by onboard sensors. Combining UAV state estimates with information gathered by the imaging sensors leads to bearing measurements of the target that can be used to determine the target's location. This 3-D bearings-only estimation problem is nonlinear and traditional filtering methods produce biased and uncertain estimates, occasionally leading to filter instabilities. Careful selection of the measurement locations greatly enhances filter performance, motivating the development of UAV trajectories that minimize target location estimation error and improve filter convergence. The objective of this work is to develop guidance algorithms that enable the UAV to fly trajectories that increase the amount of information provided by the measurements and improve overall estimation observability, resulting in proper target tracking and an accurate target location estimate. The performance of the target estimation is dependent upon the positions from which measurements are taken relative to the target and to previous measurements. Past research has provided methods to quantify the information content of a set of measurements using the Fisher Information Matrix (FIM). Forming objective functions based on the FIM and using numerical optimization methods produce UAV trajectories that locally maximize the information content for a given number of measurements. In this project, trajectory optimization leads to the development of UAV flight paths that provide the highest amount of information about the target, while considering sensor restrictions, vehicle dynamics and operation constraints.
(cont.) The UAV trajectory optimization is performed for stationary targets, dynamic targets and multiple targets, for many different scenarios of vehicle motion constraints. The resulting trajectories show spiral paths taken by the UAV, which focus on increasing the angular separation between measurements and reducing the relative range to the target, thus maximizing the information provided by each measurement and improving the performance of the estimation. The main drawback of information based trajectory design is the dependence of the Fisher Information Matrix on the true target location. This issue is addressed in this project by executing simultaneous target location estimation and UAV trajectory optimization. Two estimation algorithms, the Extended Kalman Filter and the Particle Filter are considered, and the trajectory optimization is performed using the mean value of the target estimation in lieu of the true target location. The estimation and optimization algorithms run in sequence and are updated in real-time. The results show spiral UAV trajectories that increase filter convergence and overall estimation accuracy, illustrating the importance of information-based trajectory design for target localization using small UAVs.
by Sameera S. Ponda.
S.M.

Proper engine selection for Reusable Launch Vehicles (RLVs) is a key factor in the design of low cost reusable launch systems for routine access to space. RLVs typically use combinations of different types of engines used in sequence over the duration of the flight. Also, in order to properly choose which engines are best for an RLV design concept and mission, the optimal trajectory that maximizes or minimizes the mission objective must be found for that engine configuration. Typically this is done by the designer iteratively choosing engine combinations based on his/her judgment and running each individual combination through a full trajectory optimization to find out how well the engine configuration performed on board the desired RLV design. This thesis presents a new method to reliably predict the optimal engine configuration and optimal trajectory for a fixed design of a conceptual RLV in an automated manner. This method is accomplished using the original code Steele-Flight. This code uses a combination of a Genetic Algorithm (GA) and a Non-Linear Programming (NLP) based trajectory optimizer known as GPOPS II to simultaneously find the optimal engine configuration from a user provided selection pool of engine models and the matching optimal trajectory. This method allows the user to explore a broad range of possible engine configurations that they wouldn't have time to consider and do so in less time than if they attempted to manually select and analyze each possible engine combination. This method was validated in two separate ways. The codes ability to optimize trajectories was compared to the German trajectory optimizer suite known as ASTOS where only minimal differences in the output trajectory were noticed. Afterwards another test was performed to verify the method used by Steele-Flight for engine selection. In this test, Steele-Flight was provided a vehicle model based on the German Saenger TSTO RLV concept and models of turbofans, turbojets, ramjets, scramjets and rockets. Steele-Flight explored the design space through the use of a Genetic Algorithm to find the optimal engine combination to maximize payload. The results output by Steele-Flight were verified by a study in which the designer manually chose the engine combinations one at a time, running each through the trajectory optimization routine to determine the best engine combination. For the most part, these methods yielded the same optimal engine configurations with only minor variation. The code itself provides RLV researchers with a new tool to perform conceptual level engine selection from a gathering of user provided conceptual engine data models and RLV structural designs and trajectory optimization for fixed RLV designs and fixed mission requirement.
Master of Science

An advanced ascent guidance algorithm for rocket-powered launch vehicles is developed. The ascent guidance function is responsible for commanding attitude, throttle and setting during the powered ascent phase of flight so that the vehicle attains target cutoff conditions in a near-optimal manner while satisfying path constraints such as maximum allowed bending moment and maximum allowed axial acceleration. This algorithm cyclically solves the calculus-of-variations two-point boundary-value problem starting at vertical rise completion through orbit insertion. This is different from traditional ascent guidance algorithms which operate in an open-loop mode until the high dynamic pressure portion of the trajectory is over, at which time there is a switch to a closed loop guidance mode that operates under the assumption of negligible aerodynamic forces. The main contribution of this research is an algorithm of the predictor-corrector type wherein the state/costate system is propagated with known (navigated) initial state and guessed initial costate to predict the state/costate at engine cutoff. The initial costate guess is corrected, using a multi-dimensional Newtons method, based on errors in the terminal state constraints and the transversality conditions. Path constraints are enforced within the propagation process. A modified multiple shooting method is shown to be a very effective numerical technique for this application. Results for a single stage to orbit launch vehicle are given. In addition, the formulation for the free final time multi-arc trajectory optimization problem is given. Results for a two-stage launch vehicle burn-coast-burn ascent to orbit in a closed-loop guidance mode are shown. An abort to landing site formulation of the algorithm and numerical results are presented. A technique for numerically treating the transversality conditions is discussed that eliminates part of the analytical and coding burden associated with optimal control theory.

This dissertation presents the development of a new concept and technology exploration methodology for aerospace architectures. The methodology is based on modeling the design space by a graph, and optimizing the graph using Ant Colony Optimization. The results show that the proposed design methodology can explore more efficiently the concept and technology space of a launch vehicle architecture than traditional optimization approaches such as Genetic Algorithm and Simulated Annealing. The purpose of the method is to introduce quantitative and simultaneous exploration of concept and technology alternatives during the early phases of conceptual design. To achieve this goal, technical challenges such as expanding the size of the design space, exploring more efficiently the design options, and simultaneously considering technologies and concepts are overcome. The total number of design alternatives grows factorially with the number of concepts in the design space. Under these circumstances, the design space is difficult to explore in its totality. Considering more alternatives has been the focus of several researchers, using Genetic Algorithms and Simulated Annealing. The large number of incompatibilities between alternatives, however, limits these optimization algorithms and reduces the number of concepts or technologies that can be considered. To address these problems, a concept and technology selection methodology is developed. The methodology proposes a way to automatically generate aerospace architectures, and to model concept and technology incompatibilities by means of a graph. In conjunction with this new modeling approach, a graph-based stochastic optimization algorithm is used to efficiently explore the design space. This design methodology is applied to the simultaneous concept and technology exploration of an expendable launch vehicle architecture. This study demonstrates that the consideration of more design alternatives can help design engineers to make more informed decisions during the concept and technology selection process. Moreover, the simultaneous exploration of concepts and technologies has the potential to identify a different set of solutions than the standard approach where the technologies are explored after the concepts have initially been selected.

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.

A satellite launch vehicle not subjected to any perturbations, external or internal, could be guided along a trajectory by following a stored, pre-computed steering program. In practice, perturbations do occur, and in order to take account of them and to achieve an accurate injection, a closed loop guidance system is required. Guidance algorithm is developed by solving the optimal control problem. Closed form solution is difficult because the necessary conditions are in the form of Two Point Boundary Value Problems (TBVP) or Multi Point Boundary Value Problems (MPBVP). Development of non-iterative guidance algorithm is taken as a prime objective of this thesis to ensure reliable on-board implementation. If non-iterative algorithms are required, the usual practice is to approximate the system equations to derive closed form solutions. In the present work, approximations cannot be used because the algorithm has to cater to a wide variety of vehicles and missions. Present development adopts an alternate approach by splitting the reconfigurable algorithm development in to smaller sub-problems such that each sub-problem has closed form solution. The splitting is done in such a way that the solution of the sub-problems can be used as building blocks to construct the final solution. By adding or removing the building blocks, the algorithm can be configured to suit specific requirements. Chapter 1 discusses the motivation and objectives of the thesis and gives a literature survey. In chapter 2, Classical Flat Earth (CFE) guidance algorithm is discussed. The assumptions and the nature of solution are closely analyzed because CFE guidance is used as the baseline for further developments. New contribution in chapter 2 is the extension of CFE guidance for a generalized propulsion system in which liquid and solid engines are present. In chapter 3, CFE guidance is applied for a mission with large pitch steering angles. The result shows loss of optimality and performance. An algorithm based on regular perturbation is developed to compensate for the small angle approximation. The new contribution in chapter 3 is the development of Regular Perturbation based FE (RPFE) guidance as an extension of CFE guidance. RPFE guidance can be configured as CFE guidance and FEGP. Algorithms presented up to chapter 3 are developed to inject a satellite in to orbits with unspecified inertial orientation. Communication satellite missions demand injection in to an orbit with a specific inertial orientation defined by argument of perigee. This problem is formulated using Calculus of Variations in chapter 4. A non-iterative closed form solution (Predicted target Flat Earth or PFE guidance) is derived for this problem. In chapter 5, PFE guidance is extended to a multi-stage vehicle with a constraint on the impact point of spent lower stage. Since the problem is not analytically solvable, the original problem is split in to three sub-problems and solved. Chapter 6 has two parts. First part gives theoretical analysis of the sub-optimal strategies with special emphasis to guidance. Behavior of predicted terminal error and control commands in presence of plant approximations are theoretically analyzed for a class of optimal control problems and the results are presented as six theorems. Chapter 7 presents the conclusions and future works.

A new suboptimal guidance law design approach for aerospace vehicles is proposed in this thesis, followed by an advanced control design for supersonic air-breathing engines. The guidance law is designed using the newly developed Generalized Model Predictive Static Programming (G-MPSP), which is based on the continuous time nonlinear optimal control framework. The key feature of this technique is one-time backward propagation of a small-dimensional weighting matrix dynamics, which is used to update the entire control history. This key feature, as well as the fact that it leads to a static optimization problem, lead to its computational efficiency. It has also been shown that the existing model predictive static programming (MPSP), which is based on the discrete time framework, is a special case of G-MPSP. The G-MPSP technique is further extended to incorporate ‘input inequality constraints’ in a limited sense using the penalty function philosophy. Next, this technique has been developed also further in a ‘flexible final time’ framework to converge rapidly to meet very stringent final conditions with limited number of iterations. Using the G-MPSP technique in a flexible final time and input inequality constrained formulation, a suboptimal guidance law for a solid motor propelled carrier launch vehicle is successfully designed for a hypersonic mission. This guidance law assures very stringent final conditions at the injection point at the end of the guidance phase for successful beginning of the hypersonic vehicle operation. It also ensures that the angle of attack and structural load bounds are not violated throughout the trajectory. A second-order autopilot has been incorporated in the simulation studies to mimic the effect of the inner-loops on the guidance performance. Simulation studies with perturbations in the thrust-time behaviour, drag coefficient and mass demonstrate that the proposed guidance can meet the stringent requirements of the hypersonic mission. The G-MPSP technique in a fixed final time and input inequality constrained formulation has also been used for optimal guidance of an aerospace vehicle propelled by supersonic air-breathing engine, where the resulting thrust can be manipulated by managing the fuel flow and nozzle area (which is not possible in solid motors). However, operation of supersonic air-breathing engines is quite complex as the thrust produced by the engine is a result of very complex nonlinear combustion dynamics inside the engine. Hence, to generate the desired thrust, accounting for a fairly detailed engine model, a dynamic inversion based nonlinear state feedback control design has been carried out. The objective of this controller is to ensure that the engine dynamically produces the thrust that tracks the commanded value of thrust generated from the guidance loop as closely as possible by regulating the fuel flow rate. Simultaneously, by manipulating throat area of the nozzle, it also manages the shock wave location in the intake for maximum pressure recovery with sufficient margin for robustness. To filter out the sensor and process noises and to estimate the states for making the control design operate based on output feedback, an extended Kalman filter (EKF) based state estimation design has also been carried out and the controller has been made to operate based on estimated states. Moreover, independent control designs have also been carried out for the actuators so that their response can be faster. In addition, this control design becomes more challenging to satisfy the imposed practical constraints like fuel-air ratio and peak combustion temperature limits. Simulation results clearly indicate that the proposed design is quite successful in assuring the desired performance of the air-breathing engine throughout the flight trajectory, i.e., both during the climb and cruise phases, while assuring adequate pressure margin for shock wave management.