The haptic device recreates the sense of touch by applying forces to the user. Since the device is “rendering” forces to emulate the physical interaction, the force control is essential for haptic devices. While a dedicated force/torque sensor can close the loop of force control, the additional equipment creates extra moving mass and inertia at the tool center point (TCP). Therefore, estimating the Cartesian force at the TCP has continuously been receiving attention over the past decades. The objective of this thesis project is to develop a real-time force estimation algorithm based on the proportional current-torque relationship with the dynamic modeling of the TAU haptic device. The algorithm can be further used for the force control of the device. The research questions of the thesis are: how to design and develop an algorithm for the TAU that used for Cartesian contact force estimation, how to set up the force estimation test bench and how to evaluate the results of the force estimation algorithm. In order to achieve the force estimation algorithm, a virtual environment is built to simulate the real-time haptic physics. Then an external force/torque sensor is installed at the TCP to get the measurement of the Cartesian force at the TCP. The force estimation algorithm calculates the Cartesian force at the TCP based on the current measurement of the DC motors at the six joints. The estimation result of the Cartesian force at the TCP is then compared with the force/torque sensor measurement to determine if the estimation algorithm is sufficiently accurate. The analysis of the estimation accuracy emphasizes the feasibility of Cartesian force estimation on the TAU haptic device. / En haptikenhet gör det möjligt att förmedla en känsla av kontakt i en virtuell värld genom att skapa krafter som motverkar en rörelse . Hur denna kraft skapas och kontrolleras är av stor vikt för att få den så verklighetstrogen som möjligt. Om man har en kraftsensor kan den användas till att utforma en kraftreglering med återkoppling, men på bekostnad av en ökad massa och tröghet vid användarens hand. Detta har medfört ett ökat intresse under de senaste åren för att på olika sätt försöka uppskatta den kraft som återkopplas till användaren utan att behöva en kraftsensor. Målet för detta examensarbete är att utveckla en algoritm för att uppskatta en kontaktkraft i realtid baserat på antagandet att motormomentet är proportionellt beroende av strömmen. Algoritmen kan sedan användas för att konstruera en sluten reglerloop med kraftåterkoppling för en haptisk enhet. Forskningsfrågorna som behandlas i detta examensarbete är;  hur kan vi utforma en algoritm för estimering av kontaktkrafter för haptikenheten TAU  hur kan vi utforma en experimentell försöksuppställning för mätning av de verkliga kontaktkrafterna från TAU vid kontakt.  hur kan vi använda resultaten från experimenten för utvärdering av algoritmen För testning och utvärdering av algoritmen har en virtuell värld skapats för att efterlikna en simuleringsmiljö som haptikenheten är tänkt att användas i. En kraftsensor har monterats under det verktyg som användaren håller i när enheten används när ett typiskt ingrepp ska övas i en simulator, t.ex. borrning i en tand. Vid experimenten beräknar algoritmen den uppskattade kontaktkraften som användaren känner baserat på den uppmätta strömmen för de sex motorer som aktiveras av kontakten. Dessa beräknade värden har sedan jämförts med de från kraftsensorn uppmätta för att avgör om algoritmen är tillräckligt noggrann. Analysen visar att noggrannheten är tillräckligt bra för att vara en lovande ansats till att användas för kraftuppskattning vid reglering av kontaktkraft för haptikenheten TAU.

The field of Computational Fluid Dynamics (CFD) primarily involves the approximation of the Navier-Stokes equations. However, these equations are only valid when the flow is considered continuous such that molecular interactions are abundant and predictable. The Knudsen number, $Kn$, which is defined as the ratio of the flow's mean free path, $\lambda$, to some characteristic length, $L$, quantifies the continuity of any flow, and when this parameter is large enough, alternative methods must be employed to simulate gases. The Direct Simulation Monte Carlo (DSMC) method is one which simulates rarefied gas flows by directly simulating the particles that compose the flow and using probabilistic methods to determine their collisions and properties. This thesis discusses the development of a new DSMC simulation code, named SINATRA, which was written in object-oriented C++ and validated on Cartesian grids. The code demonstrates the ability to perform standard simulation code tasks which include reading-in a user-made input file, performing the specified simulation, and generating visualization files compatible with Tecplot 360\texttrademark, a commercial post-processing software. SINATRA strategically uses an octree data structure as a storage scheme for computational grid data and uses this a backbone for particle interactions. The discussed validation cases include comparisons of initial particle properties to theoretical data, convergence studies for the sampling of macroscopic properties, and validation of transport properties through natural diffusion and Couette flow simulations. The results show successful implementation of simple DSMC procedures, and a path for future development of the code is thoroughly discussed.

Sodium (23Na) plays a pivotal role in cellular homeostasis throughout all life forms. In the human body it has a plethora of physiological functions including regulation of bodily fluids, muscle contraction, neuronal transmission, and connective tissue integrity. The 23Na nucleus can be visualized using magnetic resonance imaging (MRI) because it has spin 3/2 and is the 100% naturally abundant isotope of sodium. However, unlike hydrogen (1H), the nucleus of choice for MRI scanning, 23Na MR clinical applications are almost non-existent, even though it is the second most MR visible nucleus after 1H. This is primarily because 23Na requires its own customized hardware (i.e. radiofrequency RF coil) and software (i.e. pulse sequence/acquisition and image reconstruction techniques). Additionally, 23Na MR is challenging because of significantly lower signal-to-noise ratio (SNR), which translates to a requirement for lengthy scans and lower spatial resolution. Hence, development and optimization of 23Na MR acquisition techniques is an ongoing subject of research. In this dissertation development and optimization of approaches that exploit sodium as a biomarker for musculoskeletal health are described. Specifically, discussions of advanced 23Na MR techniques fall into two main categories: magnetic resonance imaging (MRI) and spectroscopy (MRS). The work first explored development of a MRI compatible electrical muscle stimulation (EMS) system that could be used as a consistent alternative to voluntary muscle contraction. A variety of MRI methods were used to confirm the similarity between stimulation and voluntary activation of muscles of the lower leg. In a follow- up study quantum filtered (QF) 23Na MRS was used to assess dynamic changes in skeletal muscle before, during and following voluntary exercise. Total (single quantum filtered, SQF) and bound intracellular sodium (triple quantum filtered, TQF) were measured in 9 healthy subjects with a 12s temporal resolution. Total sodium (SQF) significantly increased 4% (p < 0.01), while bound intracellular sodium content decreased 7% (p < 0.01) with exercise. Both returned to baseline following exercise; TQF after a few seconds and SQF after approximately 12 minutes (p < 0.05). In a preliminary study the MR-compatible EMS unit demonstrated similar QF 23Na MRS results. Sodium MR imaging, like typical 1H-MRI scanning, can be performed by rasterizing k-space. However, this approach is both temporally and SNR inefficient. There- fore newer optimized 23Na MRI pulse sequence approaches have been developed. The cost-benefit of extending the acquisition window length in terms of SNR gain and blurring in the in vivo 23Na MRI of the human knee was explored in 3 healthy subjects using a density adapted 3-dimensional projection reconstruction (DA-3DPR) sequence. Mean SNR doubled when the acquisition window was increased from 4 to 25ms. Concurrently, the FWHM, as a measure of cartilage blurring, increased by only 1±0mm across three different sections of articular cartilage. In a second imaging experiment, a pseudo-random 3D non-Cartesian k-space acquisition scheme for 23Na MRI was introduced. This scheme was highly effective in minimizing the aliasing artifacts leading to 9 times less number of shots required to cover k-space compared to a fully-sampled acquisition scheme. Hence, this resulted in a 9-fold reduction in scan time to cover k-space. Images of a resolution phantom and healthy human knee, with 3mm isotropic resolution (zero padded to 1mm isotropic resolution), were reconstructed using a non-uniform fast Fourier transform (NUFFT). / Dissertation / Doctor of Philosophy (PhD)

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In this paper, we develop a scheme to recover a single erasure when using a Cartesian code,in the context of a distributed storage system. Particularly, we develop a scheme withconsiderations to minimize the associated bandwidth and maximize the associateddimension. The problem of recovering a missing node's data exactly in a distributedstorage system is known as theexact repair problem. Previous research has studied theexact repair problem for Reed-Solomon codes. We focus on Cartesian codes, and show wecan enact the recovery using a linear exact repair scheme framework, similar to the oneoutlined by Guruswami and Wooters in 2017. / Master of Science / Distributed storage systems are systems which store a single data file over multiple storage nodes. Each storage node has a certain storage efficiency, the "space" required to store the information on that node. The value of these systems, is their ability to safely store data for extended periods of time. We want to design distributed storage systems such that if one storage node fails, we can recover it from the data in the remaining nodes. Recovering a node from the data stored in the other nodes requires the nodes to communicate data with each other. Ideally, these systems are designed to minimize the bandwidth, the inter-nodal communication required to recover a lost node, as well as maximize the storage efficiency of each node. A great mathematical framework to build these distributed storage systems on is erasure codes. In this paper, we will specifically develop distributed storage systems that use Cartesian codes. We will show that in the right setting, these systems can have a very similar bandwidth to systems build from Reed-Solomon codes, without much loss in storage efficiency.

The minimum number of colors required to color the edges of a graph so that any two distinct vertices are connected by at least one path in which no two edges are colored the same is called its rainbow connection number. This graph parameter was introduced by Chartrand et al. in 2008. The problem has garnered considerable interest and several variants of the initial version have since been introduced. The rainbow connection number of a connected graph G is denoted by rc(G). It can be shown that the rainbow connection number of a tree on n vertices is n -1. Hence |G|-1 is an upper bound for rc(G)of any non-trivial graph G. For all non-trivial, bridge-less and connected graphs G, Basavaraju etal. Showed that rc(G) can be upper-bounded by a quadratic function of its radius. In addition they also proved the tightness of the bound. It is clear that we cannot hope to get an upper-bound better than |G| - 1 in the case of graphs with bridges. An immediate and natural question is the following: Are there classes of bridge-less graphs whose rainbow connection numbers are linear functions of their radii? This question is of particular interest since the diameter is a trivial lower bound for rc(G). We answer in affirmative to the above question. In particular we studied three (graph) product operations (Cartesian, Lexicographic and Strong) and the graph powering operation. We were able to show that the rainbow connection number of the graph resulting from any of the above graph operations is upper-bounded by 2r(G)+c, where r(G) is radius of the resultant graph and c ε {0, 1, 2}.

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