Discrete Modeling of Granular Media: A NURBS-based Approach

This dissertation is concerned with the development of a new discrete element method (DEM) based on Non-Uniform Rational Basis Splines (NURBS). With NURBS, the new DEM is able to capture sphericity and angularity, the two particle morphological measures used in characterizing real grain geometries. By taking advantage of the parametric nature of NURBS, the Lipschitzian dividing rectangle (DIRECT) global optimization procedure is employed as a solution procedure to the closest-point projection problem, which enables the contact treatment of non-convex particles. A contact dynamics (CD) approach to the NURBS-based discrete method is also formulated. By combining particle shape flexibility, properties of implicit time-integration, and non-penetrating constraints, we target applications in which the classical DEM either performs poorly or simply fails, i.e., in granular systems composed of rigid or highly stiff angular particles and subjected to quasistatic or dynamic flow conditions. The CD implementation is made simple by adopting a variational framework, which enables the resulting discrete problem to be readily solved using off-the-shelf mathematical programming solvers. The capabilities of the NURBS-based DEM are demonstrated through 2D numerical examples that highlight the effects of particle morphology on the macroscopic response of granular assemblies under quasistatic and dynamic flow conditions, and a 3D characterization of material response in the shear band of a real triaxial specimen.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.

On distribution of pollen in lakes. [Translation from: Palynology in geological research in the Baltic Soviet Republics p113-119. Vilnius, 1973.]

Systematic investigations of the distribution of pollen in stationary water bodies until now have hardly been conducted. For clarification of the problem of how the pollen of different plants which falls into a lake is deposited in relation to its physical properties, the character of the lake, wind currents and other factors, pollen analyses were carried out of surface samples of the bottom sediments of 13 Lithuanian lakes. Lakes were selected of different sizes (areas from 2333 ha. to 8 ha.) and different depths, not uniformly overgrown, situated in different physico-geographic regions of Lithuania. As a result of the investigation, it was established that in the surface layer of the sediments of the lakes of Lithuania pollen of woody species predominates.

Essays in Behavioral Decision Theory

This thesis studies decision making under uncertainty and how economic agents respond to information. The classic model of subjective expected utility and Bayesian updating is often at odds with empirical and experimental results; people exhibit systematic biases in information processing and often exhibit aversion to ambiguity. The aim of this work is to develop simple models that capture observed biases and study their economic implications.

In the first chapter I present an axiomatic model of cognitive dissonance, in which an agent's response to information explicitly depends upon past actions. I introduce novel behavioral axioms and derive a representation in which beliefs are directionally updated. The agent twists the information and overweights states in which his past actions provide a higher payoff. I then characterize two special cases of the representation. In the first case, the agent distorts the likelihood ratio of two states by a function of the utility values of the previous action in those states. In the second case, the agent's posterior beliefs are a convex combination of the Bayesian belief and the one which maximizes the conditional value of the previous action. Within the second case a unique parameter captures the agent's sensitivity to dissonance, and I characterize a way to compare sensitivity to dissonance between individuals. Lastly, I develop several simple applications and show that cognitive dissonance contributes to the equity premium and price volatility, asymmetric reaction to news, and belief polarization.

The second chapter characterizes a decision maker with sticky beliefs. That is, a decision maker who does not update enough in response to information, where enough means as a Bayesian decision maker would. This chapter provides axiomatic foundations for sticky beliefs by weakening the standard axioms of dynamic consistency and consequentialism. I derive a representation in which updated beliefs are a convex combination of the prior and the Bayesian posterior. A unique parameter captures the weight on the prior and is interpreted as the agent's measure of belief stickiness or conservatism bias. This parameter is endogenously identified from preferences and is easily elicited from experimental data.

The third chapter deals with updating in the face of ambiguity, using the framework of Gilboa and Schmeidler. There is no consensus on the correct way way to update a set of priors. Current methods either do not allow a decision maker to make an inference about her priors or require an extreme level of inference. In this chapter I propose and axiomatize a general model of updating a set of priors. A decision maker who updates her beliefs in accordance with the model can be thought of as one that chooses a threshold that is used to determine whether a prior is plausible, given some observation. She retains the plausible priors and applies Bayes' rule. This model includes generalized Bayesian updating and maximum likelihood updating as special cases.

Eggs, Larvae and Young of The Striped Bass, Roccus saxatilis

Detailed descriptions of the early development of the striped bass, Roccus saxitilis (Walbaum), with emphasis on variation in size and morphology, sequence of fin formation, changes in body form, and attainment of the full complement of maristic numbers, are presented and illustrated for the first time. The egg is spherical, transparent, non-adhesive and relatively large. It is pelagic and buoyant, although it sinks in quiet fresh water. When unfertilized, it averages 1.3 mm, in diameter, but is 3.4 mm. when fertilized and water-hardened. The granular yolk sac, green when alive and whitish-yellow when preserved, averages 1.2 mm., and the single amber-colored oil globule is about 0.6 mm. in diameter. Newly hatched striped bass prolarvae, which range from 2.9-3.7 mm. in total length, are relatively undeveloped and nearly transparent, with no mouth opening, unpigmented eyes, and a greatly enlarged yolk sac with the large oil globule projecting beyond the head. When 5-6 mm. long the yolk sac and oil globule are assimilated and the postlarvae I show advanced development of the internal anatomy. Although the fish is still transparent, scattered melanophores are found on the head and body and chromatophores in the eyes and the ventro-posterior edge of the body. Postlarvae transform to young between 7 and 10 mm. in length when the finfolds are lost except in the dorsal, anal and caudal regions. The largest fish in this group possess a well-formed skeleton with a full complement of 25 vertebrae. Between 10 and 20 mm. in length all fish are fully transformed, muscular tissue renders most of the internal structure obscure, and the myotomes, which generally correspond in number with the vertebrae, are no longer visible. At fish lengths of 20-30 mm. scales are found on all specimens, and with the exception of the pectoral fin-rays, a full complement of meristic structures is present in all other fins. At this stage the body is pigmented uniformly with small spots. Linear regressions between several dependent variables and the , independent variable of standard length indicate that the rate of development of head, eye. and snout to anus lengths is proportional to the length of the larvae and young. Body depth and standard length are non-linear among newly-hatched larvae. Hatchery-reared striped bass demonstrated a slow rate of growth, and were regarded as "stunted," when compared to growth rates observed in another study and field collections. Observations were also made on abnormal eggs and teratological larvae and young. Blue-sac disease is tentatively identified and described for the first time in larvae and pugnosed larvae and young are also described for the first time in striped bass.

透镜自重变形引起波像差的有限元分析

在用大口径、长焦距平行光管模拟激光远场特性时，其光组透镜在重力作用下的变形不能忽略，为了分析对出射光束质量的影响，采用有限元分析软件“ANSYS”建立了平行光管光组中声400mm平凸透镜的有限元模型，给出一种分析透镜轴向变形引起的波像差的方法，在不同工况下，计算了平凸透镜在重力作用下轴向变形的峰谷值和均方根值，对轴向变形量均方根值最小工况画出了透镜表面变形的等值线图，计算了声350mm通光口径内的波像差峰谷值和均方根值，对平行光管光组的波像差做出估计，验证了设计的合理性。

Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set P of all paranormal operators in B(H), the set of all bounded endomorphisms on the complex Hilbert space H. T ϵ B(H) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L, the set of all T ϵ B(H) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N≤P≤L.

If the uniform operator (norm) topology is placed on B(H), then the relative topological properties of N, P, L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B(H) when dim H ≥ 2, 3) N = P when dimH ˂ ∞ , 4) N is a nowhere dense subset of P when dimH ˂ ∞ , 5) P is not a nowhere dense subset of L when dimH ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dimH ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.

A Direct Approach to Robustness Optimization

This dissertation reformulates and streamlines the core tools of robustness analysis for linear time invariant systems using now-standard methods in convex optimization. In particular, robust performance analysis can be formulated as a primal convex optimization in the form of a semidefinite program using a semidefinite representation of a set of Gramians. The same approach with semidefinite programming duality is applied to develop a linear matrix inequality test for well-connectedness analysis, and many existing results such as the Kalman-Yakubovich--Popov lemma and various scaled small gain tests are derived in an elegant fashion. More importantly, unlike the classical approach, a decision variable in this novel optimization framework contains all inner products of signals in a system, and an algorithm for constructing an input and state pair of a system corresponding to the optimal solution of robustness optimization is presented based on this information. This insight may open up new research directions, and as one such example, this dissertation proposes a semidefinite programming relaxation of a cardinality constrained variant of the H ∞ norm, which we term sparse H ∞ analysis, where an adversarial disturbance can use only a limited number of channels. Finally, sparse H ∞ analysis is applied to the linearized swing dynamics in order to detect potential vulnerable spots in power networks.

Effects of density differences on lateral mixing in open-channel flows

This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source.

A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

The instability of fluids with time dependent heating

The stability of a fluid having a non-uniform temperature stratification is examined analytically for the response of infinitesimal disturbances. The growth rates of disturbances have been established for a semi-infinite fluid for Rayleigh numbers of 10³, 10⁴, and 10⁵ and for Prandtl numbers of 7.0 and 0.7.

The critical Rayleigh number for a semi-infinite fluid, based on the effective fluid depth, is found to be 32, while it is shown that for a finite fluid layer the critical Rayleigh number depends on the rate of heating. The minimum critical Rayleigh number, based on the depth of a fluid layer, is found to be 1340.

The stability of a finite fluid layer is examined for two special forms of heating. The first is constant flux heating, while in the second, the temperature of the lower surface is increased uniformly in time. In both cases, it is shown that for moderate rates of heating the critical Rayleigh number is reduced, over the value for very slow heating, while for very rapid heating the critical Rayleigh number is greatly increased. These results agree with published experimental observations.

The question of steady, non-cellular convection is given qualitative consideration. It is concluded that, although the motion may originate from infinitesimal disturbances during non-uniform heating, the final flow field is intrinsically non-linear.

An investigation of the stresses and deflections of swept plates

The problem in this investigation was to determine the stress and deflection patterns of a thick cantilever plate at various angles of sweepback.

The plate was tested at angles of sweepback of zero, twenty, forty, and sixty degrees under uniform shear load at the tip, uniformly distributed load and torsional loading.

For all angles of sweep and for all types of loading the area of critical stress is near the intersection of the root and trailing edge. Stresses near the leading edge at the root decreased rapidly with increase in angle of sweep for all types of loading. In the outer portion of the plate near the trailing edge the stresses due to the uniform shear and the uniformly distributed load did not vary for angles of sweep up to forty degrees. For the uniform shear and the uniformly distributed loads for all angles of sweep the area in which end effect is pronounced extends from the root to approximately three quarters of a chord length outboard of a line perpendicular to the axis of the plate through the trailing edge root. In case of uniform shear and uniformly distributed loads the deflections near the edge at seventy-five per cent semi-span decreased with increase in angle of sweep. Deflections near the trailing edge under the same loading conditions increased with increase in angle of sweep for small angles and then decreased at the higher angles of sweep. The maximum deflection due to torsional loading increased with increase in angle of sweep.

Morfologia dentofacial e características oclusais dos índios Arara: revisitando o papel da hereditariedade e da dieta na etiologia da má oclusão

A influência da dieta e da hereditariedade nas características dentofaciais foi avaliada através do exame de duas populações indígenas amazônicas divididas por um processo de fissão linear. Os indígenas que constituem a aldeia Arara-Iriri são descendentes de um único casal expulso da aldeia Arara-Laranjal. O crescimento da aldeia Iriri ocorreu pelo acasalamento de parentes próximos, ratificado por um alto coeficiente de consanguinidade (F=0,25, p<0,001). A epidemiologia da má oclusão e das características da face foi analisada nos indivíduos entre dois e 22 anos, das aldeias Iriri (n=46) e Laranjal (n=130). A biometria da dentição e da face foi obtida em 55 indígenas em dentição permanente sem perdas dentárias, através da fotogrametria facial e dos modelos de gesso. O desgaste dentário foi examinado em 126 indivíduos através da análise de regressão múltipla. Os resultados revelaram uma determinação significativa da idade no desgaste dos dentes (R2=87,6, p<0,0001), que se mostrou semelhante entre as aldeias (R2=0,027, p=0,0935). Por outro lado, diferenças marcantes foram observadas nas características dentofaciais. Revelou-se uma face mais vertical (dolicofacial) entre os índios Iriri e o predomínio do tipo braquifacial nos indígenas da aldeia original, corroborado pela fotogrametria. Uma face sagitalmente normal foi observada em 97,7% da aldeia Laranjal, enquanto faces convexas (26,1%, RR-16,96) e côncavas (15,2%, RR=19,78) eram mais prevalentes na aldeia Iriri (p<0,001). A biprotrusão, com consequente redução do ângulo nasolabial, era uma característica comum entre os Arara, porém com maior prevalência no grupo Iriri (RP=1,52, p=0,0002). A prevalência da má oclusão foi significativamente mais alta na aldeia Iriri (RP= 1,75, p=0,0007). A maioria da população da aldeia original (83,8%) apresentou uma relação normal entre os arcos dentários, contudo, na aldeia resultante (Iriri), 34,6% dos indivíduos era Classe III (RP=6,01, p<0,001) e 21,7% era Classe II (RP=2,02, p=0,05). Enquanto nenhum caso de apinhamento e de sobremordida foi observado na aldeia Iriri, a razão da prevalência era 2,64 vezes maior para a mordida aberta anterior (p=0,003), 2,83 vezes (p<0,001) para a mordida cruzada anterior, 3,93 (p=0,03) para a sobressaliência aumentada, e de 4,71 (p=0,02) para a mordida cruzada posterior. Observou-se uma alta prevalência das perdas dentárias, sem diferença entre as aldeias (RP=1,46, p=0,11). O exame dos modelos revelou uma tendência de incisivos maiores e pré-molares e caninos menores na aldeia Iriri, delineando uma semelhança na massa dentária total entre as aldeias, que, aliada a arcadas dentárias maiores, justificaram o menor índice de irregularidade dos incisivos entre esses indígenas. Esses resultados minimizam a influência do desgaste dentário, uma evidência direta de como um indivíduo se alimentou no passado, no desenvolvimento dentofacial e enfatizam o predomínio da hereditariedade, através da endogamia, na etiologia da variação anormal da oclusão dentária e da morfologia da face.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Two and Three Finger Caging of Polygons and Polyhedra

Multi-finger caging offers a rigorous and robust approach to robot grasping. This thesis provides several novel algorithms for caging polygons and polyhedra in two and three dimensions. Caging refers to a robotic grasp that does not necessarily immobilize an object, but prevents it from escaping to infinity. The first algorithm considers caging a polygon in two dimensions using two point fingers. The second algorithm extends the first to three dimensions. The third algorithm considers caging a convex polygon in two dimensions using three point fingers, and considers robustness of this cage to variations in the relative positions of the fingers.

This thesis describes an algorithm for finding all two-finger cage formations of planar polygonal objects based on a contact-space formulation. It shows that two-finger cages have several useful properties in contact space. First, the critical points of the cage representation in the hand’s configuration space appear as critical points of the inter-finger distance function in contact space. Second, these critical points can be graphically characterized directly on the object’s boundary. Third, contact space admits a natural rectangular decomposition such that all critical points lie on the rectangle boundaries, and the sublevel sets of contact space and free space are topologically equivalent. These properties lead to a caging graph that can be readily constructed in contact space. Starting from a desired immobilizing grasp of a polygonal object, the caging graph is searched for the minimal, intermediate, and maximal caging regions surrounding the immobilizing grasp. An example constructed from real-world data illustrates and validates the method.

A second algorithm is developed for finding caging formations of a 3D polyhedron for two point fingers using a lower dimensional contact-space formulation. Results from the two-dimensional algorithm are extended to three dimension. Critical points of the inter-finger distance function are shown to be identical to the critical points of the cage. A decomposition of contact space into 4D regions having useful properties is demonstrated. A geometric analysis of the critical points of the inter-finger distance function results in a catalog of grasps in which the cages change topology, leading to a simple test to classify critical points. With these properties established, the search algorithm from the two-dimensional case may be applied to the three-dimensional problem. An implemented example demonstrates the method.

This thesis also presents a study of cages of convex polygonal objects using three point fingers. It considers a three-parameter model of the relative position of the fingers, which gives complete generality for three point fingers in the plane. It analyzes robustness of caging grasps to variations in the relative position of the fingers without breaking the cage. Using a simple decomposition of free space around the polygon, we present an algorithm which gives all caging placements of the fingers and a characterization of the robustness of these cages.

A novel method to design flexible URAs

Aperture patterns play a vital role in coded aperture imaging ( CAI) applications. In recent years, many approaches were presented to design optimum or near-optimum aperture patterns. Uniformly redundant arrays (URAs) are, undoubtedly, the most successful for constant sidelobe of their periodic autocorrelation function. Unfortunately, the existing methods can only be used to design URAs with a limited number of array sizes and fixed autocorrelation sidelobe-to-peak ratios. In this paper, we present a novel method to design more flexible URAs. Our approach is based on a searching program driven by DIRECT, a global optimization algorithm. We transform the design question to a mathematical model, based on the DIRECT algorithm, which is advantageous for computer implementation. By changing determinative conditions, we obtain two kinds of types of URAs, including the filled URAs which can be constructed by existing methods and the sparse URAs which have never been mentioned by other authors as far as we know. Finally, we carry out an experiment to demonstrate the imaging performance of the sparse URAs.