On the Behavior of the Homogeneous Self-Dual Model for Conic Convex Optimization

There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of ε-optimal solutions, and (ii) the maximum distance of ε-optimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stopping-rule theory for HSD-based interior-point methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the ε-optimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous self-dual model that might improve the resulting solution time in practice

Redundant Sensors for Mobile Robot Navigation

Redundant sensors are needed on a mobile robot so that the accuracy with which it perceives its surroundings can be increased. Sonar and infrared sensors are used here in tandem, each compensating for deficiencies in the other. The robot combines the data from both sensors to build a representation which is more accurate than if either sensor were used alone. Another representation, the curvature primal sketch, is extracted from this perceived workspace and is used as the input to two path planning programs: one based on configuration space and one based on a generalized cone formulation of free space.

On Motion Planning with Uncertainty

Robots must successfully plan and execute tasks in the presence of uncertainty. Uncertainty arises from errors in modeling, sensing, and control. Planning in the presence of uncertainty constitutes one facet of the general motion planning problem in robotics. This problem is concerned with the automatic synthesis of motion strategies from high level task specification and geometric models of environments. In order to develop successful motion strategies, it is necessary to understand the effect of uncertainty on the geometry of object interactions. Object interactions, both static and dynamic, may be represented in geometrical terms. This thesis investigates geometrical tools for modeling and overcoming uncertainty. The thesis describes an algorithm for computing backprojections o desired task configurations. Task goals and motion states are specified in terms of a moving object's configuration space. Backprojections specify regions in configuration space from which particular motions are guaranteed to accomplish a desired task. The backprojection algorithm considers surfaces in configuration space that facilitate sliding towards the goal, while avoiding surfaces on which motions may prematurely halt. In executing a motion for a backprojection region, a plan executor must be able to recognize that a desired task has been accomplished. Since sensors are subject to uncertainty, recognition of task success is not always possible. The thesis considers the structure of backprojection regions and of task goals that ensures goal recognizability. The thesis also develops a representation of friction in configuration space, in terms of a friction cone analogous to the real space friction cone. The friction cone provides the backprojection algorithm with a geometrical tool for determining points at which motions may halt.

Image-Based View Synthesis

We present a new method for rendering novel images of flexible 3D objects from a small number of example images in correspondence. The strength of the method is the ability to synthesize images whose viewing position is significantly far away from the viewing cone of the example images ("view extrapolation"), yet without ever modeling the 3D structure of the scene. The method relies on synthesizing a chain of "trilinear tensors" that governs the warping function from the example images to the novel image, together with a multi-dimensional interpolation function that synthesizes the non-rigid motions of the viewed object from the virtual camera position. We show that two closely spaced example images alone are sufficient in practice to synthesize a significant viewing cone, thus demonstrating the ability of representing an object by a relatively small number of model images --- for the purpose of cheap and fast viewers that can run on standard hardware.

On the Fabrication of Microparticles Using Electrohydrodynamic Atomization Method

A new approach for the control of the size of particles fabricated using the Electrohydrodynamic Atomization (EHDA) method is being developed. In short, the EHDA process produces solution droplets in a controlled manner, and as the solvent evaporates from the surface of the droplets, polymeric particles are formed. By varying the voltage applied, the size of the droplets can be changed, and consequently, the size of the particles can also be controlled. By using both a nozzle electrode and a ring electrode placed axisymmetrically and slightly above the nozzle electrode, we are able to produce a Single Taylor Cone Single Jet for a wide range of voltages, contrary to just using a single nozzle electrode where the range of permissible voltage for the creation of the Single Taylor Cone Single Jet is usually very small. Phase Doppler Particle Analyzer (PDPA) test results have shown that the droplet size increases with increasing voltage applied. This trend is predicted by the electrohydrodynamic theory of the Single Taylor Cone Single Jet based on a perfect dielectric fluid model. Particles fabricated using different voltages do not show much change in the particles size, and this may be attributed to the solvent evaporation process. Nevertheless, these preliminary results do show that this method has the potential of providing us with a way of fine controlling the particles size using relatively simple method with trends predictable by existing theories.