Studies of the equation of state and elasticity of mantle minerals

Because the Earth’s upper mantle is inaccessible to us, in order to understand the chemical and physical processes that occur in the Earth’s interior we must rely on both experimental work and computational modeling. This thesis addresses both of these geochemical methods. In the first chapter, I develop an internally consistent comprehensive molar volume model for spinels in the oxide system FeO-MgO-Fe₂O₃-Cr₂O₃-Al₂O₃-TiO₂. The model is compared to the current MELTS spinel model with a demonstration of the impact of the model difference on the estimated spinel-garnet lherzolite transition pressure. In the second chapter, I calibrate a molar volume model for cubic garnets in the system SiO₂-Al₂O₃-TiO₂-Fe₂O₃-Cr₂O₃-FeO-MnO-MgO-CaO-Na₂O. I use the method of singular value analysis to calibrate excess volume of mixing parameters for the garnet model. The implications the model has for the density of the lithospheric mantle are explored. In the third chapter, I discuss the nuclear inelastic X-ray scattering (NRIXS) method, and present analysis of three orthopyroxene samples with different Fe contents. Longitudinal and shear wave velocities, elastic parameters, and other thermodynamic information are extracted from the raw NRIXS data.

Kinematics and Local Motion Planning for Quasi-static Whole-body Mobile Manipulation

This thesis studies mobile robotic manipulators, where one or more robot manipulator arms are integrated with a mobile robotic base. The base could be a wheeled or tracked vehicle, or it might be a multi-limbed locomotor. As robots are increasingly deployed in complex and unstructured environments, the need for mobile manipulation increases. Mobile robotic assistants have the potential to revolutionize human lives in a large variety of settings including home, industrial and outdoor environments.

Mobile Manipulation is the use or study of such mobile robots as they interact with physical objects in their environment. As compared to fixed base manipulators, mobile manipulators can take advantage of the base mechanism’s added degrees of freedom in the task planning and execution process. But their use also poses new problems in the analysis and control of base system stability, and the planning of coordinated base and arm motions. For mobile manipulators to be successfully and efficiently used, a thorough understanding of their kinematics, stability, and capabilities is required. Moreover, because mobile manipulators typically possess a large number of actuators, new and efficient methods to coordinate their large numbers of degrees of freedom are needed to make them practically deployable. This thesis develops new kinematic and stability analyses of mobile manipulation, and new algorithms to efficiently plan their motions.

I first develop detailed and novel descriptions of the kinematics governing the operation of multi- limbed legged robots working in the presence of gravity, and whose limbs may also be simultaneously used for manipulation. The fundamental stance constraint that arises from simple assumptions about friction and the ground contact and feasible motions is derived. Thereafter, a local relationship between joint motions and motions of the robot abdomen and reaching limbs is developed. Baseeon these relationships, one can define and analyze local kinematic qualities including limberness, wrench resistance and local dexterity. While previous researchers have noted the similarity between multi- fingered grasping and quasi-static manipulation, this thesis makes explicit connections between these two problems.

The kinematic expressions form the basis for a local motion planning problem that that determines the joint motions to achieve several simultaneous objectives while maintaining stance stability in the presence of gravity. This problem is translated into a convex quadratic program entitled the balanced priority solution, whose existence and uniqueness properties are developed. This problem is related in spirit to the classical redundancy resoxlution and task-priority approaches. With some simple modifications, this local planning and optimization problem can be extended to handle a large variety of goals and constraints that arise in mobile-manipulation. This local planning problem applies readily to other mobile bases including wheeled and articulated bases. This thesis describes the use of the local planning techniques to generate global plans, as well as for use within a feedback loop. The work in this thesis is motivated in part by many practical tasks involving the Surrogate and RoboSimian robots at NASA/JPL, and a large number of examples involving the two robots, both real and simulated, are provided.

Finally, this thesis provides an analysis of simultaneous force and motion control for multi- limbed legged robots. Starting with a classical linear stiffness relationship, an analysis of this problem for multiple point contacts is described. The local velocity planning problem is extended to include generation of forces, as well as to maintain stability using force-feedback. This thesis also provides a concise, novel definition of static stability, and proves some conditions under which it is satisfied.

Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Statistical and econometric analysis of the treatment of HIV/AIDS

The epidemic of HIV/AIDS in the United States is constantly changing and evolving, starting from patient zero to now an estimated 650,000 to 900,000 Americans infected. The nature and course of HIV changed dramatically with the introduction of antiretrovirals. This discourse examines many different facets of HIV from the beginning where there wasn't any treatment for HIV until the present era of highly active antiretroviral therapy (HAART). By utilizing statistical analysis of clinical data, this paper examines where we were, where we are and projections as to where treatment of HIV/AIDS is headed.

Chapter Two describes the datasets that were used for the analyses. The primary database utilized was collected by myself from an outpatient HIV clinic. The data included dates from 1984 until the present. The second database was from the Multicenter AIDS Cohort Study (MACS) public dataset. The data from the MACS cover the time between 1984 and October 1992. Comparisons are made between both datasets.

Chapter Three discusses where we were. Before the first anti-HIV drugs (called antiretrovirals) were approved, there was no treatment to slow the progression of HIV. The first generation of antiretrovirals, reverse transcriptase inhibitors such as AZT (zidovudine), DDI (didanosine), DDC (zalcitabine), and D4T (stavudine) provided the first treatment for HIV. The first clinical trials showed that these antiretrovirals had a significant impact on increasing patient survival. The trials also showed that patients on these drugs had increased CD4+ T cell counts. Chapter Three examines the distributions of CD4 T cell counts. The results show that the estimated distributions of CD4 T cell counts are distinctly non-Gaussian. Thus distributional assumptions regarding CD4 T cell counts must be taken, into account when performing analyses with this marker. The results also show the estimated CD4 T cell distributions for each disease stage: asymptomatic, symptomatic and AIDS are non-Gaussian. Interestingly, the distribution of CD4 T cell counts for the asymptomatic period is significantly below that of the CD4 T cell distribution for the uninfected population suggesting that even in patients with no outward symptoms of HIV infection, there exists high levels of immunosuppression.

Chapter Four discusses where we are at present. HIV quickly grew resistant to reverse transcriptase inhibitors which were given sequentially as mono or dual therapy. As resistance grew, the positive effects of the reverse transcriptase inhibitors on CD4 T cell counts and survival dissipated. As the old era faded a new era characterized by a new class of drugs and new technology changed the way that we treat HIV-infected patients. Viral load assays were able to quantify the levels of HIV RNA in the blood. By quantifying the viral load, one now had a faster, more direct way to test antiretroviral regimen efficacy. Protease inhibitors, which attacked a different region of HIV than reverse transcriptase inhibitors, when used in combination with other antiretroviral agents were found to dramatically and significantly reduce the HIV RNA levels in the blood. Patients also experienced significant increases in CD4 T cell counts. For the first time in the epidemic, there was hope. It was hypothesized that with HAART, viral levels could be kept so low that the immune system as measured by CD4 T cell counts would be able to recover. If these viral levels could be kept low enough, it would be possible for the immune system to eradicate the virus. The hypothesis of immune reconstitution, that is bringing CD4 T cell counts up to levels seen in uninfected patients, is tested in Chapter Four. It was found that for these patients, there was not enough of a CD4 T cell increase to be consistent with the hypothesis of immune reconstitution.

In Chapter Five, the effectiveness of long-term HAART is analyzed. Survival analysis was conducted on 213 patients on long-term HAART. The primary endpoint was presence of an AIDS defining illness. A high level of clinical failure, or progression to an endpoint, was found.

Chapter Six yields insights into where we are going. New technology such as viral genotypic testing, that looks at the genetic structure of HIV and determines where mutations have occurred, has shown that HIV is capable of producing resistance mutations that confer multiple drug resistance. This section looks at resistance issues and speculates, ceterus parabis, where the state of HIV is going. This section first addresses viral genotype and the correlates of viral load and disease progression. A second analysis looks at patients who have failed their primary attempts at HAART and subsequent salvage therapy. It was found that salvage regimens, efforts to control viral replication through the administration of different combinations of antiretrovirals, were not effective in 90 percent of the population in controlling viral replication. Thus, primary attempts at therapy offer the best change of viral suppression and delay of disease progression. Documentation of transmission of drug-resistant virus suggests that the public health crisis of HIV is far from over. Drug resistant HIV can sustain the epidemic and hamper our efforts to treat HIV infection. The data presented suggest that the decrease in the morbidity and mortality due to HIV/AIDS is transient. Deaths due to HIV will increase and public health officials must prepare for this eventuality unless new treatments become available. These results also underscore the importance of the vaccine effort.

The final chapter looks at the economic issues related to HIV. The direct and indirect costs of treating HIV/AIDS are very high. For the first time in the epidemic, there exists treatment that can actually slow disease progression. The direct costs for HAART are estimated. It is estimated that the direct lifetime costs for treating each HIV infected patient with HAART is between $353,000 to $598,000 depending on how long HAART prolongs life. If one looks at the incremental cost per year of life saved it is only $101,000. This is comparable with the incremental costs per year of life saved from coronary artery bypass surgery.

Policy makers need to be aware that although HAART can delay disease progression, it is not a cure and HIV is not over. The results presented here suggest that the decreases in the morbidity and mortality due to HIV are transient. Policymakers need to be prepared for the eventual increase in AIDS incidence and mortality. Costs associated with HIV/AIDS are also projected to increase. The cost savings seen recently have been from the dramatic decreases in the incidence of AIDS defining opportunistic infections. As patients who have been on HAART the longest start to progress to AIDS, policymakers and insurance companies will find that the cost of treating HIV/AIDS will increase.

Future of earthquake early warning : quantifying uncertainty and making fast automated decisions for applications

Earthquake early warning (EEW) systems have been rapidly developing over the past decade. Japan Meteorological Agency (JMA) has an EEW system that was operating during the 2011 M9 Tohoku earthquake in Japan, and this increased the awareness of EEW systems around the world. While longer-time earthquake prediction still faces many challenges to be practical, the availability of shorter-time EEW opens up a new door for earthquake loss mitigation. After an earthquake fault begins rupturing, an EEW system utilizes the first few seconds of recorded seismic waveform data to quickly predict the hypocenter location, magnitude, origin time and the expected shaking intensity level around the region. This early warning information is broadcast to different sites before the strong shaking arrives. The warning lead time of such a system is short, typically a few seconds to a minute or so, and the information is uncertain. These factors limit human intervention to activate mitigation actions and this must be addressed for engineering applications of EEW. This study applies a Bayesian probabilistic approach along with machine learning techniques and decision theories from economics to improve different aspects of EEW operation, including extending it to engineering applications.

Existing EEW systems are often based on a deterministic approach. Often, they assume that only a single event occurs within a short period of time, which led to many false alarms after the Tohoku earthquake in Japan. This study develops a probability-based EEW algorithm based on an existing deterministic model to extend the EEW system to the case of concurrent events, which are often observed during the aftershock sequence after a large earthquake.

To overcome the challenge of uncertain information and short lead time of EEW, this study also develops an earthquake probability-based automated decision-making (ePAD) framework to make robust decision for EEW mitigation applications. A cost-benefit model that can capture the uncertainties in EEW information and the decision process is used. This approach is called the Performance-Based Earthquake Early Warning, which is based on the PEER Performance-Based Earthquake Engineering method. Use of surrogate models is suggested to improve computational efficiency. Also, new models are proposed to add the influence of lead time into the cost-benefit analysis. For example, a value of information model is used to quantify the potential value of delaying the activation of a mitigation action for a possible reduction of the uncertainty of EEW information in the next update. Two practical examples, evacuation alert and elevator control, are studied to illustrate the ePAD framework. Potential advanced EEW applications, such as the case of multiple-action decisions and the synergy of EEW and structural health monitoring systems, are also discussed.

SteelConverter and Caltech VirtualShaker: Rapid Nonlinear Cloud-Based Structural Model Conversion and Analysis

STEEL, the Caltech created nonlinear large displacement analysis software, is currently used by a large number of researchers at Caltech. However, due to its complexity, lack of visualization tools (such as pre- and post-processing capabilities) rapid creation and analysis of models using this software was difficult. SteelConverter was created as a means to facilitate model creation through the use of the industry standard finite element solver ETABS. This software allows users to create models in ETABS and intelligently convert model information such as geometry, loading, releases, fixity, etc., into a format that STEEL understands. Models that would take several days to create and verify now take several hours or less. The productivity of the researcher as well as the level of confidence in the model being analyzed is greatly increased.

It has always been a major goal of Caltech to spread the knowledge created here to other universities. However, due to the complexity of STEEL it was difficult for researchers or engineers from other universities to conduct analyses. While SteelConverter did help researchers at Caltech improve their research, sending SteelConverter and its documentation to other universities was less than ideal. Issues of version control, individual computer requirements, and the difficulty of releasing updates made a more centralized solution preferred. This is where the idea for Caltech VirtualShaker was born. Through the creation of a centralized website where users could log in, submit, analyze, and process models in the cloud, all of the major concerns associated with the utilization of SteelConverter were eliminated. Caltech VirtualShaker allows users to create profiles where defaults associated with their most commonly run models are saved, and allows them to submit multiple jobs to an online virtual server to be analyzed and post-processed. The creation of this website not only allowed for more rapid distribution of this tool, but also created a means for engineers and researchers with no access to powerful computer clusters to run computationally intensive analyses without the excessive cost of building and maintaining a computer cluster.

In order to increase confidence in the use of STEEL as an analysis system, as well as verify the conversion tools, a series of comparisons were done between STEEL and ETABS. Six models of increasing complexity, ranging from a cantilever column to a twenty-story moment frame, were analyzed to determine the ability of STEEL to accurately calculate basic model properties such as elastic stiffness and damping through a free vibration analysis as well as more complex structural properties such as overall structural capacity through a pushover analysis. These analyses showed a very strong agreement between the two softwares on every aspect of each analysis. However, these analyses also showed the ability of the STEEL analysis algorithm to converge at significantly larger drifts than ETABS when using the more computationally expensive and structurally realistic fiber hinges. Following the ETABS analysis, it was decided to repeat the comparisons in a software more capable of conducting highly nonlinear analysis, called Perform. These analyses again showed a very strong agreement between the two softwares in every aspect of each analysis through instability. However, due to some limitations in Perform, free vibration analyses for the three story one bay chevron brace frame, two bay chevron brace frame, and twenty story moment frame could not be conducted. With the current trend towards ultimate capacity analysis, the ability to use fiber based models allows engineers to gain a better understanding of a building’s behavior under these extreme load scenarios.

Following this, a final study was done on Hall’s U20 structure [1] where the structure was analyzed in all three softwares and their results compared. The pushover curves from each software were compared and the differences caused by variations in software implementation explained. From this, conclusions can be drawn on the effectiveness of each analysis tool when attempting to analyze structures through the point of geometric instability. The analyses show that while ETABS was capable of accurately determining the elastic stiffness of the model, following the onset of inelastic behavior the analysis tool failed to converge. However, for the small number of time steps the ETABS analysis was converging, its results exactly matched those of STEEL, leading to the conclusion that ETABS is not an appropriate analysis package for analyzing a structure through the point of collapse when using fiber elements throughout the model. The analyses also showed that while Perform was capable of calculating the response of the structure accurately, restrictions in the material model resulted in a pushover curve that did not match that of STEEL exactly, particularly post collapse. However, such problems could be alleviated by choosing a more simplistic material model.