This graduate project compares legacy simplified general perturbation model (SGP4) code developed by NASA Johnson Space Center, Orbital Debris Program Office, to a recent public release of SGP4 code by David Vallado. The legacy code is a subroutine in a larger program named PREDICT, which is used to predict the location of orbital debris in GEO. Direct comparison of the codes showed that the new code yields better results for GEO objects, which are more accurate by orders of magnitude (error in meters rather than kilometers). The public release of SGP4 also provides effective results for LEO and MEO objects on a short time scale. The public release code was debugged and modified to provide instant functionality to the Orbital Debris Program Office. Code is provided in an appendix to this paper along with an accompanying CD. A User’s Guide is presented in Chapter 7.

In order to support UCLA’s development of the 3 cm Miniature Xenon Ion (MiXI) thruster, Cal Poly has a 3 cm thruster under development. This version, called MiXI Cal Poly Version 1 (MiXI-CPv1), is complete and has been utilized in vacuum chamber thermal validation testing. Testing on this version was used to check the validity of heat transfer simulations modeled in SolidWorks. Investigations of the 3 cm ion thruster configuration were intended to discover the driving factors affecting the thermal behavior of the discharge chamber and surrounding design space. Numerical simulations indicate that the heating of the samarium cobalt permanent magnets can be mitigated through the implementation of two proposed modifications. The first modification is to implement a 2% thoriated tungsten filament cathode. This design exhibited maximum permanent magnet temperatures of 325°C, twenty-five degrees below the maximum upper temperature of 350°C. Since some magnetic degaussing effects have been observed at temperatures above 300°C, the aforementioned solution can be combined with a thruster design modification to achieve a reduced permanent magnet temperature of 298°C. This modification would involve increase the anode wall thickness from approximately 0.7 mm to 2 mm below the permanent magnet ring, creating a stepped anode design. Additionally, less effective solutions were proposed and modeled and are presented for completeness.

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

付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」 / 京都大学 / 新制・課程博士 / 博士(工学) / 甲第24891号 / 工博第5171号 / 新制||工||1987(附属図書館) / 京都大学大学院工学研究科電子工学専攻 / (主査)教授 白石 誠司, 准教授 掛谷 一弘, 教授 小野 輝男, 教授 森山 貴広 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

The experimental pieces of evidence for the existence of Majorana states in topo- logical superconductors have so far been inconclusive despite intense research in the past two decades [Zha+20; Kay+20]. Combined with promising applications in quantum computing [Nay+08; Ali+11] and the resulting technological development of society, the elusiveness of Majorana states keeps motivating theoretical and ex- perimental research to this day. Our analytical findings and numerical explorations in new topological superconducting platforms suggest several tools and solutions for their future realisation in condensed matter systems. The planar Josephson junction (pJJ) introduced in 2017 by F. Pientka et al. [Pie+17] and M. Hell et al. [HLF17] is a versatile platform for topological superconductivity. It harnesses the tunability of the superconducting phase difference across the Josephson junction as an external control parameter that switches the pJJ between the trivial and topological phases of matter. The junction between the (trivial) superconductors is quasi-one-dimensional and hosts one new Majorana zero mode at each of its ends following each topological phase transition. However, the creation of a second Majorana zero mode on one end triggers a return to the trivial regime as both zero modes hybridize into a regular non-topological fermion. It is then crucial to identify the model parameters that lead to topological phases with a single Majorana state per end. Our main result on the pJJ establishes the general constraint on its microscopic parameters—including the phase difference and a magnetic field—to cross the topo- logical phase transitions. The identification of sectors in parameter space leading to a single Majorana mode becomes then straightforward. In some limits the pJJ develops a topological sector at small magnetic field for a phase difference close to the value p while it remains trivial at the same field near zero phase difference. Since the phase is sufficient to turn on and off the topology, we call this feature “switchable topology”. Looking for switchable topology is experimentally relevant as it makes the topology easily tunable while keeping intact the proximitized su- perconductivity otherwise jeopardized by the applied field. Concretely, we found switchable topology in three configurations: in wide junctions with a transparent interface with the superconducting regions, in fine-tuned narrow junctions weakly coupled to the superconducting regions, and in junctions with a strong Zeeman energy when they are ultranarrow and transparent. Thanks to our exact analytical results, setups interpolating between these limits can adjust the desired properties at will. The other important finding about the pJJ concerns the stability of its topological phases, by which we mean the presence of a sizable spectral gap in the topological sector. We observed that the Rashba spin-orbit coupling is responsible for strongly decreasing the gap in the relevant topological sector at low Zeeman field, but sym- metry arguments justify that wide, transparent junctions are generically immune to this effect for large enough Rashba coupling. After 2017, other platforms started to use the Josephson superconducting phase difference as a knob to trigger topological superconductivity [Liu+19; JY21]. We introduce here the stacked Josephson junction (sJJ) as a new platform for topological superconductivity, which is made of two non-centrosymmetric superconductors sandwiching a two-dimensional magnet around which chiral Majorana edge modes propagate. Unlike the Majorana zero modes in the pJJ, chiral Majorana modes can add to each other if they propagate in the same direction, as indicated by the integer Chern number of their topological phase. The bulk-edge correspondence, however, only constrains the net number of topological edge states and allows room for other non-topological states to coexist with the chiral Majorana states without interacting with them. We found that the presence of trivial chiral edge modes in the sJJ restricts access to the Majorana states themselves. The symmetry protection of the trivial modes, fortunately, disappears with an in-plane magnetic field applied through the magnet or with superconducting leads different on the top and at the bottom of the stacked junction. The theoretical investigations of topological platforms have currently outnum- bered the experiments with convincing signatures of Majorana edge states. This imbalance calls for new ways to probe the agreement between topological models and laboratory setups. The critical current of a Josephson junction acts as a link between the microscopic description and macroscopic observables. Thermoelectric measurements, which distinguish between supercurrent and quasiparticle current, modify this model-dependent connection, and would provide an electrical probe to estimate the validity of a model like that of the pJJ. We computed the contribution to the thermoelectric coefficient of the bulk states of a uniform superconductor, that has a similar environment to that of the pJJ (i.e., Rashba coupling and in-plane Zeeman field). The results were not conclusive and motivated us to suggest new analytical and numerical approaches to obtain the thermoelectric response of the pJJ, in particular by including the contribution of the Andreev bound states and non-linear effects.:Foreword — how to read this thesis 1 Preamble A popular short story: pencils and lightbulbs 5 Basics and concepts 1 Introduction to Majorana physics 13 1.1 The electrons & their properties 13 1.1.1 Hamiltonian for the planar Josephson junction 17 1.2 The scattering matrix for bound states 19 1.3 Andreev bound states for topology 24 1.4 Topological superconductivity & Majorana edge states 28 1.5 Induced topological superconductivity 34 1.6 Summary 36 Appendices 37 1.A Microscopic dynamics 37 1.A.1 Origin of spin–orbit coupling 37 1.A.2 Bogoliubov-deGennes symmetrization 37 1.A.3 Andreev reflection below the coherence length 38 1.A.4 Proximity-induced superconductivity 40 1.A.5 From s- to p-wave superconductivity 41 1.B Scattering theory for bound states 44 1.B.1 Bound states as trapped waves 44 1.B.2 Scattering theory for an open region 45 1.B.3 Scattering theory for two open regions 46 1.B.4 Bound states recovered from an open region 47 1.B.5 Numerical scattering theory for bound states 48 2 Perspectives on electronic transport 53 2.1 Electric current in a metal 53 2.2 Quantum-mechanical current 54 2.2.1 Expression for the microscopic current 55 2.3 Thermoelectric current 57 2.3.1 The Boltzmann transport equation 61 2.4 Supercurrents and the superconducting coherence phase 64 2.4.1 Josephson currents 67 Appendices 71 2.A Electric current from a potential difference 71 2.B Scattering and current 71 2.C Hole-based current in metals 73 Introduction Introduction to the Research Projects 77 i Topological properties of Josephson junctions 3 Switchable topology in the planar Josephson junction 85 Motivation & Overview of the Study 85 3.1 The planar Josephson junction and the nanowire setup 87 3.1.1 Comparison with the nanowire setup. 89 3.2 Model 92 3.3 General formula for the phase transitions 94 3.3.1 Spin decoupling for the phase transitions 96 3.3.2 Exact reflection coefficients 97 3.3.3 Exact scattering formula and Andreev reflectivity 98 3.3.4 Andreev approximation 100 3.3.5 Dimensionless formulation 101 3.3.6 Numerical and analytical checks 103 3.4 Three regimes for switchable topology 105 3.4.1 Diamond-shape regime 108 3.4.2 V-shape regime 110 3.4.3 Nanowire regime 111 3.4.4 Summary: extent of the topological transitions 114 3.5 Avoiding regimes with a small topological gap 117 3.5.1 Gapless lines as BDI phase transitions 119 3.5.2 Opening the gap in f = p 120 3.5.3 Role of the Rashba coupling 121 3.6 Conclusion 125 Appendices 129 3.A Limiting cases of the pJJ 129 3.A.1 Andreev approximation 129 3.A.2 Small field limit 131 3.A.3 Delta-barrier junction 131 3.A.4 Semiconductor nanowire 132 3.B Normal reflection via surface impurity and surface refraction 134 3.C Symmetry-constrained gap closings 136 3.D Linear deviation of the gapless line near f = p 138 3.E Calculations for the scattering formula 141 3.E.1 Boundary conditions 141 3.E.2 Combinations of scattering coefficients 142 3.E.3 Andreev coefficients for the phase transitions 143 3.E.4 Formula for B > μ 145 4 Topological and trivial chiral states in the stacked Josephson junction 147 Motivation & Overview of the Study 147 4.1 The basics of the stacked Josephson junction 149 4.2 Continuous and lattice models 151 4.3 Topological index 155 4.3.1 Methodology for the Chern number 155 4.3.2 Interpretation of the results 156 4.4 Topological and trivial edge states 162 4.5 BDI phase transitions 167 4.5.1 Dimensional reduction 168 4.5.2 Link between topological invariants 170 4.5.3 Explaining the low-energy sector 171 4.6 Conclusion 174 Appendices 177 4.A Symmetries of the Hamiltonian 177 4.A.1 Class D 177 4.A.2 Class BDI 177 4.A.3 Gapless line in f = p 178 4.A.4 Symmetry around f = p 179 4.B The parity index in 2D TSC 180 ii Transport properties of the planar Josephson junction 5 An approach to thermoelectric effects in the planar Josephson junction 183 Motivation & Overview of the Study 183 5.1 From the Josephson junction to a homogeneous superconductor 185 5.2 Model and Phenomenology 187 5.2.1 Homogeneous superconductor 187 5.2.2 Analytical spectrum and two-surface approximation 188 5.2.3 Magnetoelectric supercurrent: phenomenology 191 5.3 Electric current in a spin–orbit coupled superconductor 194 5.3.1 Formula for the current 196 5.3.2 Zero-temperature current 198 5.3.3 Small perturbations at finite temperature 200 5.4 Thermoelectric current in a spin–orbit coupled superconductor 206 5.4.1 Distribution imbalance under temperature bias 208 5.4.2 Explicit formula for the thermoelectric current 209 5.5 Discussion and Outlook 213 Appendices 219 5.A The Boltzmann equation in temperature-biased superconductors 219 5.A.1 The linear approximation 220 5.A.2 The low-temperature approximation 220 5.A.3 Integral solution of the Boltzmann equation 223 5.B Diagonalisation of the planar superconductor 225 5.B.1 Eigenstates of spin–orbit coupled superconductor 225 5.B.2 Eigenstates with a small Zeeman field 227 Conclusion Majorana quasiparticles in Josephson junctions 233 Extra Material 6 Mathematical details of Scattering theory 241 6.1 Asymmetric quantum well 241 6.2 Scattering theory for an open region 243 6.2.1 Change in potential over a small region 243 6.2.2 Change in spin-orbit coupling over a small region 245 6.2.3 Change in mass over a small region 245 7 Numerical codes for chapter 4 247 7.1 BDI index 247 7.2 Chern number 255 7.3 Spectral gap 257 7.4 Localized edge states 258 8 Short courses 261 8.1 Two formulations of superconductivity 261 8.1.1 The BCS Hamiltonian 261 8.1.2 The Bogoliubov transformation 263 8.1.3 Bogoliubov-de Gennes symmetrization 264 8.1.4 Building the semiconductor representation 266 8.2 Topological band theory 270 8.3 Majorana physics in 1D 274 8.3.1 The SSH chain 275 8.3.2 The Kitaev chain 277 Bibliography 283

Legged robots have the potential to be a valuable technology that provides agile and adaptive locomotion over complex terrain. To realize legged locomotion's full abilities a control design must consider the nonlinear piecewise dynamics of the systems. This paper aims to develop a controller for the planar bounding of a quadrupedal robot. The bounding of the quadruped robot is characterized by a simplified hybrid model that consists of two subsystems for stance and flight phases and the switching laws between the two states. An additional model, the Multibody model, with fewer simplifications, is used concurrently to best approximate real-world behavior. The bounding gait (periodic orbit) of the robot is predicted by an optimization method based on the numerical integration of the differential equations of subsystems. To stabilize the gait, a switching controller is applied which can be split into two separate phases: stance-phase and swing-phase control. The stance phase implements reaction force control utilizing a body state feedback controller and a gait stabilizer, while the swing phase deploys position control in conjunction with a trajectory planning algorithm to ensure proper footfall. Numerical simulations are carried out for the system with/without control. The control strategy is further validated by simulations of the Simscape multibody model. The overall simulated controller results are promising and demonstrate stable bounding for four system cycles.

Analysis was done to determine fuel optimal translunar trajectories from Earth geostationary transfer orbit to a specified target lunar orbit for a small satellite navigation and communication mission. The study included the optimization of impulsive and finite burn transfers. The inclusion of finite burns was necessary due to the low thrust nature of a small satellite propulsion system. Finite burn optimization was achieved using suboptimal parameterization control theory. The orbital parameters of the initial Earth orbit as well as the target lunar orbit were varied to see how this affected the optimal transfer results. Additionally, two engine thrust levels were explored to find the impact on the fuel mass required. All optimization analyses were completed using Copernicus, a trajectory optimization software package developed at the University of Texas at Austin for the National Aeronautics and Space Administration (NASA).

Eccentric lunar frozen orbits are analyzed in this study in relation to lunar navigation and communications missions, particularly the proposed Magnolia-1 mission. An overview of the Earth/Moon system, frozen orbits, and the Magnolia-1 mission is provided. A review of existing literature is presented, and potential limitations are discussed. Both preliminary and numerical perturbation analyses are presented, and a general set of perturbations for further analysis of high-altitude lunar orbits is identified. Analysis of potential orbits for the Magnolia-1 mission is performed through calculation of a maximum deviation metric and through visualization as a function of initial orbital elements. Trends are identified within a closed search space by varying elements individually and in combination. Potential orbit designs for the Magnolia-1 mission are selected and compared to established alternatives. A method of orbit refinement is used to improve behavior, and coverage and eclipse analyses are performed to establish suitability. Conclusions are made involving general trends related to eccentric lunar frozen orbits and the specific designs proposed for the Magnolia-1 mission, and a method for the design of similar orbits is suggested. Ideas for further study are also presented.

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