# Browsing by Subject "Trajectory optimization"

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Item Analysis and synthesis of collaborative opportunistic navigation systems(2014-05) Kassas, Zaher; Humphreys, Todd Edwin; Arapostathis, Ari, 1954-Show more Navigation is an invisible utility that is often taken for granted with considerable societal and economic impacts. Not only is navigation essential to our modern life, but the more it advances, the more possibilities are created. Navigation is at the heart of three emerging fields: autonomous vehicles, location-based services, and intelligent transportation systems. Global navigation satellite systems (GNSS) are insufficient for reliable anytime, anywhere navigation, particularly indoors, in deep urban canyons, and in environments under malicious attacks (e.g., jamming and spoofing). The conventional approach to overcome the limitations of GNSS-based navigation is to couple GNSS receivers with dead reckoning sensors. A new paradigm, termed opportunistic navigation (OpNav), is emerging. OpNav is analogous to how living creatures naturally navigate: by learning their environment. OpNav aims to exploit the plenitude of ambient radio frequency signals of opportunity (SOPs) in the environment. OpNav radio receivers, which may be handheld or vehicle-mounted, continuously search for opportune signals from which to draw position and timing information, employing on-the-fly signal characterization as necessary. In collaborative opportunistic navigation (COpNav), multiple receivers share information to construct and continuously refine a global signal landscape. For the sake of motivation, consider the following problem. A number of receivers with no a priori knowledge about their own states are dropped in an environment comprising multiple unknown terrestrial SOPs. The receivers draw pseudorange observations from the SOPs. The receivers' objective is to build a high-fidelity signal landscape map of the environment within which they localize themselves in space and time. We then ask: (i) Under what conditions is the environment fully observable? (ii) In cases where the environment is not fully observable, what are the observable states? (iii) How would receiver-controlled maneuvers affect observability? (iv) What is the degree of observability of the various states in the environment? (v) What motion planning strategy should the receivers employ for optimal information gathering? (vi) How effective are receding horizon strategies over greedy for receiver trajectory optimization, and what are their limitations? (vii) What level of collaboration between the receivers achieves a minimal price of anarchy? This dissertation addresses these fundamental questions and validates the theoretical conclusions numerically and experimentally.Show more Item Application of automatic differentiation to trajectory optimization via direct multiple shooting(2003) Garza, David Marcelo; Fowler, Wallace T.Show more Automatic differentiation, also called computational differentiation and algorithmic differentiation, is the process of computing the derivatives or Taylor series of functions from the computer source code implementing the functions. To date, general-purpose trajectory optimization codes have relied on finite-differencing to compute the gradients needed by the nonlinear programming (NLP) algorithms within the codes. These codes typically support the selection of an arbitrary objective and constraint set from a library of a few hundred output variables. The use of automatic differentiation in these trajectory optimization programs can provide objective and constraint gradients to the same precision as the underlying functions without requiring the generation of hundreds of analytic derivative expressions by hand or via symbolic algebra packages. This work combines automatic differentiation with a direct multiple shooting method and uses the resulting method to solve a pair of example problems. The first is the well-known lunar launch problem, while the second is a launch vehicle ascent problem similar in complexity to that which would be computed by a program such as the Program to Optimize Simulated Trajectories (POST) for use in vehicle design studies. Results include comparisons of convergence behavior of the NLP problem and solution accuracy. Tests comparing the use of Euler angles versus quaternion elements as control variables demonstrate the versatility of automatic differentiation. For loose convergence levels automatic differentiation provided faster convergence than finite differencing on the launcher ascent problem. For tight accuracy requirements, automatic differentiation resulted in fewer major iterations on the lunar launch problem.Show more Item Low-Earth Orbit trajectory optimization in the presence of atmospheric uncertainty(2022-05-03) Brown, Aaron Jay; Jones, Brandon A.; Bettadpur, Srinivas V; Akella, Maruthi R; Hull, David G; Abusali, P.A.M.Show more The previous 20 to 25 years have seen a tremendous increase in space exploration, and with that an increase in the level of logistics planning needed to ensure mission success. For spacecraft that are designed to be periodically re-supplied, a key logistics consumable is propellant, as it constitutes the greatest up-mass on re-supply vehicles. A trajectory design strategy is therefore desired that minimizes propellant usage in order to ease the demand for propellant re-supply missions. This thesis develops such a strategy in three stages, and uses the International Space Station (ISS) as its testbed, as no other LEO spacecraft is more challenging from a space logistics standpoint. First, the ISS trajectory planning problem is formulated as a constrained burn optimization problem assuming a deterministic atmosphere. The cost function is total ∆v, with constraints imposed on longitude of ascending node (LAN) and semi-major axis (SMA) altitude. Analytic derivatives are constructed for both the cost and constraints, which are necessary given the 6-week to 2-year time frames being considered. A gradient-based optimizer is then utilized to find locally-optimal solutions to real-world ISS trajectory planning problems. Second, atmospheric uncertainty is addressed by constructing a probabilistic model of space weather data using Gaussian Processes (GPs). Bayesian inference is performed using the GP model to generate mean and covariance estimates for space weather predictions, whose pedigree is assessed against test data. The predictions are then mapped into atmospheric density via the analytic Jacchia-Roberts density model, and the effect of space weather uncertainty on orbital lifetime is examined. Third, an ISS burn execution uncertainty model is developed. This model, along with the space weather uncertainty model, are deployed in a linear covariance analysis to ascertain their combined effect on LAN and SMA altitude dispersions. The deterministic constraints from the original problem are re-formulated as stochastic constraints, where now the constraint uncertainty interval is required to fall within specified bounds. An updated optimization framework is constructed using the original ∆v cost function along with the stochastic constraints to solve the trajectory optimization problem under atmospheric uncertainty. Finally, the complete architecture is summarized for deployment in an operational setting.Show more Item Numerical analysis of complex-step differentiation in spacecraft trajectory optimization problems(2011-05) Campbell, Alan Robert; Hull, David G.; Ocampo, CesarShow more An analysis of the use of complex-step differentiation (CSD) in optimization problems is presented. Complex-step differentiation is a numerical approximation of the derivative of a function valid for any real-valued analytic function. The primary benefit of this method is that the approximation does not depend on a difference term; therefore round-off error is reduced to the machine word-length. A suitably small choice of the perturbation length, h, then results in the virtual elimination of truncation error in the series approximation. The theoretical basis for this method is derived highlighting its merits and limitations. The Lunar Ascent Problem is used to compare CSD to traditional forward differencing in applications useful to the solution of optimization problems. Complex-step derivatives are shown to sufficiently apply in various interpolation and integration methods, and, in fact, consistently outperform traditional methods. Further, the Optimal Orbit Transfer Problem is used to test the accuracy, robustness, and runtime of CSD in comparison to central differencing. It is shown that CSD is a considerably more accurate derivative approximation which results in an increased robustness and decreased optimization time. Also, it is shown that each approximation is computed in less time using CSD than central differences. Overall, complex-step derivatives are shown to be a fast, accurate, and easy to implement differentiation method ideally suited for most optimization problems.Show more Item Optimal lunar orbit insertion from a free return trajectory(2012-05) Jesick, Mark Christopher; Ocampo, Cesar; Fowler, Wallace; Hull, David; Marchand, Belinda; Russell, RyanShow more With the discovery of water ice at the moon's south pole, future human lunar exploration will likely occur at polar sites and, therefore, require high inclination orbits. Also of importance for human missions is the capability to abort if unfavorable circumstances arise. This dissertation addresses both of these concerns by creating an automated, systematic architecture for constructing minimum propellant lunar orbit insertion sequences while ensuring crew safety by maintaining a ballistic Earth return trajectory. To ensure a maneuver-free abort option, the spacecraft is required to depart Earth on a free return trajectory, which is a ballistic Earth-moon-Earth segment that requires no propulsive maneuvers after translunar injection. Because of the need for global lunar access, the required spacecraft plane change at the moon may be large enough that a multi-maneuver sequence offers cost savings. The combination of this orbit insertion sequence with the free return orbit increases the likelihood of a safe Earth return for crew while not compromising the ability to achieve any lunar orbit. A procedure for free return trajectory generation in a simplified Earth-moon system is presented first. With two-body and circular restricted three-body models, the algorithm constructs an initial guess of the translunar injection state and time of flight. Once the initial trajectory is found, a square system of nonlinear equations is solved numerically to target Earth entry interface conditions leading to feasible free return trajectories. No trial and error is required to generate the initial estimate. The automated algorithm is used to generate families of free return orbits for analysis. A targeting and optimization procedure is developed to transfer a spacecraft from a free return trajectory to a closed lunar orbit through a multi-maneuver sequence in the circular restricted three-body model. The initial estimate procedure is automated, and analytical gradients are implemented to facilitate optimization. Cases are examined with minimum time, variable symmetric, and general free returns. The algorithm is then upgraded to include a more realistic solar system model with ephemeris-level dynamics. An impulsive engine model is used before conversion to a finite thrust model. Optimal control theory is applied and the results are compared with the linearly steered thrust model. Trends in the flight time and propellant for various orbit insertion sequences are analyzed.Show more Item Patched periodic orbits : a systematic strategy for low-energy trajectory and moon tour design(2018-09-13) Restrepo Gómez, Ricardo León; Russell, Ryan P., 1976-; Fowler, Wallece T.; Akella, Maruthi R.; Bettadpur, Srinivas V.; Lo, MartinShow more The implementation of low-energy trajectories has opened a new paradigm in trajectory design, where, at the cost of long transfer times, low-fuel consumption missions opened a new world of routes for space exploration. Low-cost lunar missions and the exploration of the gas giant planetary satellite systems are one of the main applications of these fuel-efficient trajectories. However, the flexibility in applications that low-energy trajectories provide comes at the cost of high complexity in the design. In this dissertation a systematic strategy to simplify the design process of low-energy trajectories is developed. The method, called patched periodic orbits (PPO) is analogous to the patched-conics model in high energy regimes, where conic segments are used as building blocks to give rise to full complex trajectories. In the PPO model, the building blocks are precomputed three-body periodic orbits that are patched together to build transfer mechanisms in multi-body environments. To support the PPO model, a broad database of axisymmetric three-body periodic orbits for planets and main planetary satellites in the Solar System have been generated and provided online. The periodic orbit search is performed over 24 pairs of bodies that are well approximated by the circular restricted three-body problem, resulting in approximately 3 million periodic solutions. The database contains a new set of periodic solutions that approximate heteroclinic connections between other pairs of periodic orbits. These connecting orbits provide free escape/capture mechanisms as well as natural transfers between libration point orbits, among others. In order to efficiently converge the highly sensitive solutions the database is generated using a multiple grid search strategy and a robust differential corrector with a full second-order trust region method. Examples of point design applications for several different challenging trajectory problems are introduced, including transfers between the Galilean moons, alternative endgames of a Europa mission, and low-cost Earth-Moon transfers with ballistic lunar captures. Additionally, a strategy to compute landing trajectories at Europa with a broad surface coverage is presented. The strategy uses lissajous segments as staging orbits that allow to decouple the landing phase with the Europa approach. Combining the PPO model and the landing strategy, an end-to-end solution that connects the landing phase, the staging orbit, and a Ganymede-Europa moon tour is presented.Show more Item Preliminary interplanetary trajectory design tools using ballistic and powered gravity assists(2015-08) Brennan, Martin James; Fowler, Wallace T.; Russell, Ryan; Bettadpur, Srinivas; Lightsey, E G; Olsen, CarrieShow more Preliminary interplanetary trajectory designs frequently use simplified two-body orbital mechanics and linked conics methodology to model the complex trajectories in multi-body systems. Incorporating gravity assists provides highly efficient interplanetary trajectories, enabling otherwise infeasible spacecraft missions. Future missions may employ powered gravity assists, using a propulsive maneuver during the flyby, improving the overall trajectory performance. This dissertation provides a complete description and analysis of a new interplanetary trajectory design tool known as TRACT (TRAjectory Configuration Tool). TRACT is capable of modeling complex interplanetary trajectories, including multiple ballistic and/or powered gravity assists, deep space maneuvers, parking orbits, and other common maneuvers. TRACT utilizes an adaptable architecture of modular boundary value problem (BVP) algorithms for all trajectory segments. A bi-level optimization scheme is employed to reduce the number of optimization variables, simplifying the user provided trajectory information. The standardized optimization parameter set allows for easy use of TRACT with a variety of optimization algorithms and mission constraints. The dissertation also details new research in powered gravity assists. A review of literature on optimal powered gravity assists is presented, where many optimal solutions found are infeasible for realistic spacecraft missions. The need was identified for a mission feasible optimal powered gravity assist algorithm using only a single impulsive maneuver. The solution space was analyzed and a complete characterization was developed for solution types of the optimal single-impulse powered gravity assist. Using newfound solution space characteristics, an efficient and reliable optimal single-impulse powered gravity assist BVP algorithm was formulated. The mission constraints were strictly enforced, such as maintaining the closest approach above a minimum radius and below a maximum radius. An extension of the optimal powered gravity assist research is the development of a gravity assist BVP algorithm that utilizes an asymptote ΔV correction maneuver to produce ballistic gravity assist trajectory solutions. The efficient algorithm is tested with real interplanetary mission trajectory parameters and successfully converges upon ballistic gravity assists with improved performance compared to traditional methods. A hybrid approach is also presented, using the asymptote maneuver algorithm together with traditional gravity assist constraints to reach ballistic trajectory solutions more reliably, while improving computational performance.Show more Item Singularity-free methods for aircraft flight path optimization using Euler angles and quaternions(1982) Wuensche, Hans-Joachim; Not availableShow more Item A tabu search methodology for spacecraft tour trajectory optimization(2014-12) Johnson, Gregory Phillip; Ocampo, CesarShow more A spacecraft tour trajectory is a trajectory in which a spacecraft visits a number of objects in sequence. The target objects may consist of satellites, moons, planets or any other body in orbit, and the spacecraft may visit these in a variety of ways, for example flying by or rendezvousing with them. The key characteristic is the target object sequence which can be represented as a discrete set of decisions that must be made along the trajectory. When this sequence is free to be chosen, the result is a hybrid discrete-continuous optimization problem that combines the challenges of discrete and combinatorial optimization with continuous optimization. The problem can be viewed as a generalization of the traveling salesman problem; such problems are NP-hard and their computational complexity grows exponentially with the problem size. The focus of this dissertation is the development of a novel methodology for the solution of spacecraft tour trajectory optimization problems. A general model for spacecraft tour trajectories is first developed which defines the parameterization and decision variables for use in the rest of the work. A global search methodology based on the tabu search metaheuristic is then developed. The tabu search approach is extended to operate on a tree-based solution representation and neighborhood structure, which is shown to be especially efficient for problems with expensive solution evaluations. Concepts of tabu search including recency-based tabu memory and strategic intensification and diversification are then applied to ensure a diverse exploration of the search space. The result is an automated, adaptive and efficient search algorithm for spacecraft tour trajectory optimization problems. The algorithm is deterministic, and results in a diverse population of feasible solutions upon termination. A novel numerical search space pruning approach is then developed, based on computing upper bounds to the reachable domain of the spacecraft, to accelerate the search. Finally, the overall methodology is applied to the fourth annual Global Trajectory Optimization Competition (GTOC4), resulting in previously unknown solutions to the problem, including one exceeding the best known in the literature.Show more