Fast Fencing

We consider very natural "fence enclosure" problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set $S$ of $n$ points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose $n$ unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most $k$ closed curves and pay no cost per curve. For the variant with at most $k$ closed curves, we present an algorithm that is polynomial in both $n$ and $k$. For the variant with unit cost per curve, or unit disks, we present a near-linear time algorithm. Capoyleas, Rote, and Woeginger solved the problem with at most $k$ curves in $n^{O(k)}$ time. Arkin, Khuller, and Mitchell used this to solve the unit cost per curve version in exponential time. At the time, they conjectured that the problem with $k$ curves is NP-hard for general $k$. Our polynomial time algorithm refutes this unless P equals NP

Computer tool for maximizing the placement of congruent polyhedra

Given multiple identical polyhedral objects and a parallelepiped container, how should one place the objects so that the largest number fits inside the container? This simple question is important in many applications, yet the answer is elusive. In fact, we know of no published solution for this very general formulation. Still, in many circumstances, further restrictions apply, resulting in a large number of variations requiring different algorithmic strategies. This paper is the continuation of [12] and focus on the fundamental concepts and tools that are used for this kind of problem, such as the no-fit polygon. We also present some of its many variations, giving in particular one that applies to the stereolithographic rapid prototyping technology

Flexible Object Manipulation

Flexible objects are a challenge to manipulate. Their motions are hard to predict, and the high number of degrees of freedom makes sensing, control, and planning difficult. Additionally, they have more complex friction and contact issues than rigid bodies, and they may stretch and compress. In this thesis, I explore two major types of flexible materials: cloth and string. For rigid bodies, one of the most basic problems in manipulation is the development of immobilizing grasps. The same problem exists for flexible objects. I have shown that a simple polygonal piece of cloth can be fully immobilized by grasping all convex vertices and no more than one third of the concave vertices. I also explored simple manipulation methods that make use of gravity to reduce the number of fingers necessary for grasping. I have built a system for folding a T-shirt using a 4 DOF arm and a fixed-length iron bar which simulates two fingers. The main goal with string manipulation has been to tie knots without the use of any sensing. I have developed single-piece fixtures capable of tying knots in fishing line, solder, and wire, along with a more complex track-based system for autonomously tying a knot in steel wire. I have also developed a series of different fixtures that use compressed air to tie knots in string. Additionally, I have designed four-piece fixtures, which demonstrate a way to fully enclose a knot during the insertion process, while guaranteeing that extraction will always succeed

Polygon packing approach to disconnected graph layout

Cataloged from PDF version of article.Graph layout has become an important area of research in Computer Science for the last couple of decades. There is a wide range of applications for graph layout including data structures, databases, software engineering, VLSI technology, electrical engineering, production planning, chemistry, and biology. Most layout algorithms assume the graph to be connected. However, most graphs are disconnected and a method for putting the disconnected graph objects together is needed. Two-dimensional packing algorithms have wide area of application such as in the steel and textile industry. In steel industry, problems frequently occur when the need to stamp polygonal figures from a rectangular board arises. In the textile industry, similar problems exist. The aim is same: to maximize the use of the contiguous remainder of the board. Recently, two-dimensional packing has also been used in disconnected graph layout yielding algorithms that ‘tile’ the disconnected graph objects, which are represented by rectangles. These algorithms are also required to respect the specified aspect ratio for the final layout. A more recent approach to disconnected graph layout has been the use of polyominoes for representing the graph objects resulting in more accurate packings at the cost of increased execution times. In this thesis, we use polygons for a more accurate representation of graph objects and present new algorithms for disconnected graph layout. Specifically, we apply the No-Fit Polygon approach in twodimensional packing to disconnected graph layout. We present and analyze the graph layouts resulting from our new approach and contrast the new approach with previous ones.Başköy, CihadM.S

Automatic tool path generation for numerically controlled machining of sculptured surfaces

This dissertation presents four new tool path generation approaches for numerically controlled machining of sculptured surfaces: TRI\sb-XYINDEX, FINISH, FIVEX\sb-INDEX, FIX\sb-AXIS\sb-INDEX. All of the above systems index the tool across the object surface in the Cartesian space so that evenly distributed tool paths are accomplished. TRI\sb-XYINDEX is a three-axis tool path generation system which uses a surface triangle set (STS) representation of the surface for tool position calculations. Surface edges are detected with local searching algorithms. Quick tool positioning is achieved by selecting candidate elements of polygons. Test results show that TRI\sb-XYINDEX is more efficient when machining surfaces which are relatively flat while the discrete point approach is faster for highly curved surfaces. FINISH was developed for generating three-axis ball-end tool paths for local surface finishing. It was based on the SPS. Given a surface with excess material represented by a set of discrete points, FINISH automatically identifies the undercut areas. Results show that FINISH provides significant improvements in machining efficiency. FIVEX\sb-INDEX is developed for generating five-axis flat-end tool paths. It uses an STS approximation. Contact points on the surface are derived from edge lists obtained from the intersections of vertical cutting planes with the polygon set. The distances between adjacent end points set an initial step-forward increment between surface contact points. To verify tool movements, some intermediate tool positions are interpolated. The key features of FIVEX\sb-INDEX are: (1) a polygon set representing an object which may be composed of multiple surfaces; (2) Surface contact point generation by cutting plane intersection; (3) simple tool incrementing and positioning algorithms; (4) minimal user interaction; (5) user controlled accuracy of resulting tool paths. FIX\sb-AXIS\sb-INDEX is a subsystem of FIVEX\sb-INDEX, generating tool paths for a tool with fixed orientations. Surface contact points are generated similar to FIVEX\sb-INDEX while tool positions are corrected with the highest point technique along the tool axis direction. Linear fitting is applied to output tool positions. FIX\sb-AXIS\sb-INDEX is preferred for machining surfaces curved in one direction, such as ruled surfaces. Test results show that FIX\sb-AXIS\sb-INDEX can serve as a three-axis tool path generation system but a five-axis machine is required to do it. (Abstract shortened by UMI.)