Differential corrections provide a method to improve the real-time accuracy and precision of GPS, but there are several sources of differential corrections and each have an associated accuracy and precision. In dynamic applications, the speed and heading of the rover may also have an effect on the accuracy and precision reported by the GPS receiver. These factors may have more of an effect on one differential correction method than another. An experiment was designed to test the differential correction methods under dynamic conditions. No corrections, OmniStar HP corrections, and RT2 corrections from a local base station were tested at several speeds and headings. The experiment was designed to determine what relationship, if any, exists between these factors and positional accuracy and precision of the differential correction sources. The results of the experiment will help designers choose the most effective solution for their positioning needs. The experiment showed that local RT2 corrections offered the most precision under dynamic conditions. The precision of OmniStar HP was close to that of RT2 corrections. The system with no corrections was the least precise of the three tested. The speed and direction of the vehicle were not observed to have a significant affect on the precision of the systems tested. The type of differential corrections used was not seen to have any influence on relative accuracy. The speed and direction of the vehicle did have an influence on the relative accuracy of the systems. / Master of Science

<div>Applications of electric propulsion to spaceflight in multi-body environments require a targeting algorithm to produce suitable trajectories on the ground and on board spacecraft. The two-level targeter with low thrust (TLT-LT) provides a framework to implement differential corrections in computationally-limited autonomous spacecraft applications as well as the larger design space of pre-mission planning. Extending existing two-level corrections algorithms, applications of the TLT-LT to spacecraft with a range of propulsive capabilities, from nearly-impulsive to low-thrust, are explored. The process of determining partial derivatives is generalized, allowing reduced logical complexity and increased flexibility in designing sequences of thrusting and ballistic segments. Various implementation strategies are explored to enforce constraints on time and other design variables as well as to improve convergence behavior through the use of dynamical systems theory and attenuation factors. The TLT-LT is applied to both nearly-impulsive and low-thrust spacecraft applications in the circular restricted three-body problem to demonstrate the flexibility of the framework to correct trajectories across the spectrum of thrust magnitude. Finally, parameter continuation is employed to extend a family of trajectories from a solution with nearly-impulsive thrust events to the low-thrust regime, and the characteristics of this transition are investigated.</div>

<p>Increasingly, space flight missions are planned to traverse regions of space with complex dynamical environments influenced by multiple gravitational bodies. The nature of these systems produces motion and regions of sensitivity that are, at times, unintuitive,</p> <p>and the accumulation of trajectory dispersions from a variety of sources guarantees that spacecraft will deviate from their pre-planned trajectories in this complex environment, necessitating the use of a targeting process to generate a new feasible reference path. To ensure mission success and a robust path planning process, trajectory designers require insight into the interaction between the targeting process, the baseline trajectory, and the dynamical environment. In this investigation, the convergence behavior of these targeting processes is examined. This work summarizes a framework for characterizing and predicting the convergence behavior of perturbed targeting problems, consisting of a set of constraints, design variables, perturbation variables, and a reference solution within a dynamical system. First, this work identifies the typical features of a convergence basin and identifies a measure of worst-case performance. In the absence of an analytical method, efficient numerical discretization procedures are proposed based on the evaluation of partial derivatives at the reference solution to the perturbed targeting problem. A method is also proposed for approximating the tradespace of position and velocity perturbations that achieve reliable</p> <p>convergence toward the baseline solution. Additionally, evaluated scalar quantities are introduced to serve as predictors of the simulation-measured worst-case convergence behavior based on the local rate of growth in the constraints as well as the local relative change in the targeting-employed partial derivatives with respect to perturbations.</p> <p><br></p> <p>A variety of applications in different dynamical regions and force models are introduced to evaluate the improved discretization techniques and their correlation to the predictive metrics of convergence behavior. Segments of periodic orbits and transfer trajectories from past and planned missions are employed to evaluate the relative convergence performance across sets of candidate solutions. In the circular restricted three-body problem (CRTBP), perturbed targeting problems are formulated along a distant retrograde orbit and a near-rectilinear halo orbit (NRHO) in the Earth-Moon system. To investigate the persistence of results from the CRTBP in an ephemeris force model, a targeting problem applied to an NRHO is analyzed in both force models. Next, an L1 -to-L2 transit trajectory in the Sun-Earth system is studied to explore the effect of moving a maneuver downstream along</p> <p>a trajectory and altering the orientations of the gravitational bodies. Finally, a trans-lunar return trajectory is explored, and the convergence behavior is analyzed as the final maneuver time is varied.</p>

Spectroscopic orbital elements and/or related parameters have been determined for eight binary systems, using radial-velocity measurements that have a typical precision of about 15 ms⁻¹. The orbital periods of these systems range from about 10 days to 26 years, with a median of about 6 years. Orbital solutions were determined for the seven systems with shorter periods. The measurement of the mass ratio of the longest-period system, HD217166, demonstrates that this important astrophysical quantity can be estimated in a model-free manner with less than 10% of the orbital cycle observed spectroscopically.\\ Single-lined orbital solutions have been derived for five of the binaries. Two of these systems are astrometric binaries: β Ret and ν Oct. The other SB1 systems were 94 Aqr A, θ Ant, and the 10-day system, HD159656. The preliminary spectroscopic solution for θ Ant (P~18 years), is the first one derived for this system. The improvement to the precision achieved for the elements of the other four systems was typically between 1--2 orders of magnitude. The very high precision with which the spectroscopic solution for HD159656 has been measured should allow an investigation into possible apsidal motion in the near future. In addition to the variable radial velocity owing to its orbital motion, the K-giant, ν Oct, has been found to have an additional long-term irregular periodicity, attributed, for the time being, to the rotation of a large surface feature.\\ Double-lined solutions were obtained for HD206804 (K7V+K7V), which previously had two competing astrometric solutions but no spectroscopic solution, and a newly discovered seventh-magnitude system, HD181958 (F6V+F7V). This latter system has the distinction of having components and orbital characteristics whose study should be possible with present ground-based interferometers. All eight of the binary systems have had their mass ratio and the masses of their components estimated.\\ The following comments summarize the motivation for getting these results, and the manner in which the research was carried out. \\ The majority of stars exist in binary systems rather than singly as does the Sun. These systems provide astronomers with the most reliable and proven means to determine many of the fundamental properties of stars. One of these properties is the stellar mass, which is regarded as being the most important of all, since most other stellar characteristics are very sensitive to the mass. Therefore, empirical masses, combined with measurements of other stellar properties, such as radii and luminosities, are an excellent test for competing models of stellar structure and evolution.\\ Binary stars also provide opportunities to observe and investigate many extraordinary astrophysical processes that do not occur in isolated stars. These processes often arise as a result of direct and indirect interactions between the components, when they are sufficiently close to each other. Some of the interactions are relatively passive, such as the circularization of the mutual orbits, whilst others result from much more active processes, such as mass exchange leading to intense radiation emissions. \\ A complete understanding of a binary system's orbital characteristics, as well as the measurement of the all-important stellar masses, is almost always only achieved after the binary system has been studied using two or more complementary observing techniques. Two of the suitable techniques are astrometry and spectroscopy. In favourable circumstances, astrometry can deduce the angular dimensions of the orbit, the total mass of the system, and sometimes, its distance from us. Spectroscopy, on the other hand, can determine the linear scale of the orbit and the ratio of the stellar masses, based on the changing radial velocities of both stars. When a resolved astrometric orbital solution is also available, the velocities of both stars can allow the binary system's parallax to be determined, and the velocities of one star can provide a measure of the system mass ratio.\\ Unfortunately, relatively few binary systems are suited to these complementary studies. Underlying this difficulty are the facts that, typically, astrometrically-determined orbits favour those with periods of years or decades, whereas spectroscopic orbital solutions are more often measured for systems with periods of days to months. With the development of high-resolution astrometric and spectroscopic techniques in recent years, it is hoped that many more binary systems will be amenable to these complementary strategies.\\ Several months after this thesis began, a high-resolution spectrograph, HERCULES, commenced operations at the Mt John University Observatory, to be used in conjuction with the 1-metre McLellan telescope. For late-type stars, the anticipated velocity precision was ≲10 ms⁻¹. The primary goals of this thesis were: 1.~to assess the performance of HERCULES and the related reduction software that subsequently followed, 2.~to carry out an observational programme of 20 or so binary systems, and 3.~to determine the orbital and stellar parameters which characterize some of these systems. The particular focus was on those binaries that have resolved or unresolved astrometric orbital solutions, which therefore may be suited to complementary investigations.\\ HERCULES was used to acquire spectra of the programme stars, usually every few weeks, over a timespan of about three years. High-resolution spectra were acquired for the purpose of measuring precise radial velocities of the stars. When possible, orbital solutions were derived from these velocities, using the method of differential corrections.

Spectroscopic orbital elements and/or related parameters have been determined for eight binary systems, using radial-velocity measurements that have a typical precision of about 15 ms⁻¹. The orbital periods of these systems range from about 10 days to 26 years, with a median of about 6 years. Orbital solutions were determined for the seven systems with shorter periods. The measurement of the mass ratio of the longest-period system, HD217166, demonstrates that this important astrophysical quantity can be estimated in a model-free manner with less than 10% of the orbital cycle observed spectroscopically.\\ Single-lined orbital solutions have been derived for five of the binaries. Two of these systems are astrometric binaries: β Ret and ν Oct. The other SB1 systems were 94 Aqr A, θ Ant, and the 10-day system, HD159656. The preliminary spectroscopic solution for θ Ant (P~18 years), is the first one derived for this system. The improvement to the precision achieved for the elements of the other four systems was typically between 1--2 orders of magnitude. The very high precision with which the spectroscopic solution for HD159656 has been measured should allow an investigation into possible apsidal motion in the near future. In addition to the variable radial velocity owing to its orbital motion, the K-giant, ν Oct, has been found to have an additional long-term irregular periodicity, attributed, for the time being, to the rotation of a large surface feature.\\ Double-lined solutions were obtained for HD206804 (K7V+K7V), which previously had two competing astrometric solutions but no spectroscopic solution, and a newly discovered seventh-magnitude system, HD181958 (F6V+F7V). This latter system has the distinction of having components and orbital characteristics whose study should be possible with present ground-based interferometers. All eight of the binary systems have had their mass ratio and the masses of their components estimated.\\ The following comments summarize the motivation for getting these results, and the manner in which the research was carried out. \\ The majority of stars exist in binary systems rather than singly as does the Sun. These systems provide astronomers with the most reliable and proven means to determine many of the fundamental properties of stars. One of these properties is the stellar mass, which is regarded as being the most important of all, since most other stellar characteristics are very sensitive to the mass. Therefore, empirical masses, combined with measurements of other stellar properties, such as radii and luminosities, are an excellent test for competing models of stellar structure and evolution.\\ Binary stars also provide opportunities to observe and investigate many extraordinary astrophysical processes that do not occur in isolated stars. These processes often arise as a result of direct and indirect interactions between the components, when they are sufficiently close to each other. Some of the interactions are relatively passive, such as the circularization of the mutual orbits, whilst others result from much more active processes, such as mass exchange leading to intense radiation emissions. \\ A complete understanding of a binary system's orbital characteristics, as well as the measurement of the all-important stellar masses, is almost always only achieved after the binary system has been studied using two or more complementary observing techniques. Two of the suitable techniques are astrometry and spectroscopy. In favourable circumstances, astrometry can deduce the angular dimensions of the orbit, the total mass of the system, and sometimes, its distance from us. Spectroscopy, on the other hand, can determine the linear scale of the orbit and the ratio of the stellar masses, based on the changing radial velocities of both stars. When a resolved astrometric orbital solution is also available, the velocities of both stars can allow the binary system's parallax to be determined, and the velocities of one star can provide a measure of the system mass ratio.\\ Unfortunately, relatively few binary systems are suited to these complementary studies. Underlying this difficulty are the facts that, typically, astrometrically-determined orbits favour those with periods of years or decades, whereas spectroscopic orbital solutions are more often measured for systems with periods of days to months. With the development of high-resolution astrometric and spectroscopic techniques in recent years, it is hoped that many more binary systems will be amenable to these complementary strategies.\\ Several months after this thesis began, a high-resolution spectrograph, HERCULES, commenced operations at the Mt John University Observatory, to be used in conjuction with the 1-metre McLellan telescope. For late-type stars, the anticipated velocity precision was ≲10 ms⁻¹. The primary goals of this thesis were: 1.~to assess the performance of HERCULES and the related reduction software that subsequently followed, 2.~to carry out an observational programme of 20 or so binary systems, and 3.~to determine the orbital and stellar parameters which characterize some of these systems. The particular focus was on those binaries that have resolved or unresolved astrometric orbital solutions, which therefore may be suited to complementary investigations.\\ HERCULES was used to acquire spectra of the programme stars, usually every few weeks, over a timespan of about three years. High-resolution spectra were acquired for the purpose of measuring precise radial velocities of the stars. When possible, orbital solutions were derived from these velocities, using the method of differential corrections.

<div>As destinations of missions in both human and robotic spaceflight become more exotic, a foundational understanding the dynamical structures in the gravitational environments enable more informed mission trajectory designs. One particular type of structure, quasi-periodic orbits, are examined in this investigation. Specifically, efficient computation of quasi-periodic orbits and leveraging quasi-periodic orbits as trajectory design alternatives in the Earth-Moon and Sun-Earth systems. First, periodic orbits and their associated center manifold are discussed to provide the background for the existence of quasi-periodic motion on n-dimensional invariant tori, where n corresponds to the number of fundamental frequencies that define the motion. Single and multiple shooting differential corrections strategies are summarized to compute families 2-dimensional tori in the Circular Restricted Three-Body Problem (CR3BP) using a stroboscopic mapping technique, originally developed by Howell and Olikara. Three types of quasi-periodic orbit families are presented: constant energy, constant frequency ratio, and constant mapping time families. Stability of quasi-periodic orbits is summarized and characterized with a single stability index quantity. For unstable quasi-periodic orbits, hyperbolic manifolds are computed from the differential of a discretized invariant curve. The use of quasi-periodic orbits is also demonstrated for destination orbits and transfer trajectories. Quasi-DROs are examined in the CR3BP and the Sun-Earth-Moon ephemeris model to achieve constant line of sight with Earth and avoid lunar eclipsing by exploiting orbital resonance. Arcs from quasi-periodic orbits are leveraged to provide an initial guess for transfer trajectory design between a planar Lyapunov orbit and an unstable halo orbit in the Earth-Moon system. Additionally, quasi-periodic trajectory arcs are exploited for transfer trajectory initial guesses between nearly stable periodic orbits in the Earth-Moon system. Lastly, stable hyperbolic manifolds from a Sun-Earth L₁ quasi-vertical orbit are employed to design maneuver-free transfer from the LEO vicinity to a quasi-vertical orbit.</div>