Optimal clustering of a pair of irregular objects

Cutting and packing problems arise in many fields of applications and theory. When dealing with irregular objects, an important subproblem is the identification of the optimal clustering of two objects. Within this paper we consider a container (rectangle, circle, convex polygon) of variable sizes and two irregular objects bounded by circular arcs and/or line segments, that can be continuously translated and rotated. In addition minimal allowable distances between objects and between each object and the frontier of a container, may be imposed. The objects should be arranged within a container such that a given objective will reach its minimal value. We consider a polynomial function as the objective, which depends on the variable parameters associated with the objects and the container. The paper presents a universal mathematical model and a solution strategy which are based on the concept of phi-functions and provide new benchmark instances of finding the containing region that has either minimal area, perimeter or homothetic coefficient of a given container, as well as finding the convex polygonal hull (or its approximation) of a pair of objects

Layout problems for arc objects in convex domains

We introduce a new methodology for solving layout problems. Our objects and containers are bounded by circular arcs and line segments. We allow continuous object translations and rotations as well as minimal allowable distances between objects. For describing non-overlapping, containment and distance constraints the phi-function technique is used. We provide a general mathematical model as nonlinear programming problem with nonsmooth functions. We propose here the automatic feasible region generator, using phi-trees. The generator allows us to form ready-to-use systems of inequalities with smooth functions in order to apply efficient nonlinear optimisation procedures. We develop an efficient solution algorithm and original solver for layout problems which uses the core representation of inequlities in a sybmol form and provides exact calculation of Jacobian and Hessian matrixes. The search for local minima of NLP-problems is performed by IPOPT algorithm. An essential part of our local optimisation scheme is LORA algorithm that simplifies description of feasible region of the problem and reduces the runtime of local optimisation. It is due to this reduction our strategy can work efficiently with collections of composed objects and search for “good” local-optimal solutions for layout problems in reasonable time.Розглянуто отпимізаційну задачу упаковки довільних об'єктів, обмежених дугами кіл та відрізками прямих в опукіі області. Побудовано математичну модель у вигляді задачі недиференційованої оптимізації, множина реалізацій яко? покриває широкий клас наукових і прикладних задач геометричного проектування. Розроблено методологію розв'язання задач упаковки з урахуванням технологічних обмежень (мінімально допустимі відстані, зони заборони, можливість неперервних трансляцій та обертань об'єктів). Запропоновано генератор простору розв'язків та вирішувач (solver) для автоматичного розв'язання NLP-задач розглянутого класу.Предлагается новая методология решения оптимизационных задач компоновки произвольных объектов в контейнерах, ограниченных дугами окружностей и отрезками прямых. Строится математическая модель в виде задачи нелинейного программирования. Описывается процедура генерации области допустимых решений с применением phi-деревьев, которая позволяет формировать системы неравенств с гладкими функциями. Предлагается эффективный алгоритм поиска локально оптимальных решений.Разработан оригинальный решатель для задач негладкой оптимизации, который использует символьное представление неравенств и обеспечивает точное вычисление элементов матриц Якобиана и Гессиана. Предлагаемая методология эффективна для решения задач компоновки произвольных объектов и позволяет получать «хорошие» локально оптимальные решения за приемлемое время

Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples

Using sentinels to detect intersections of convex and nonconvex polygons

We describe finite sets of points, called sentinels, which allow us to decide if isometric copies of polygons, convex or not, intersect. As an example of the applicability of the concept of sentinel, we explain how they can be used to formulate an algorithm based on the optimization of differentiable models to pack polygons in convex sets. Mathematical subject classification: 90C53, 65K05

Part of the Computer Sciences Commons Comments Victor Milenkovic & Elisha Sacks

We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and with k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm

Efficient motion planning using generalized penetration depth computation

Motion planning is a fundamental problem in robotics and also arises in other applications including virtual prototyping, navigation, animation and computational structural biology. It has been extensively studied for more than three decades, though most practical algorithms are based on randomized sampling. In this dissertation, we address two main issues that arise with respect to these algorithms: (1) there are no good practical approaches to check for path non-existence even for low degree-of-freedom (DOF) robots; (2) the performance of sampling-based planners can degrade if the free space of a robot has narrow passages. In order to develop effective algorithms to deal with these problems, we use the concept of penetration depth (PD) computation. By quantifying the extent of the intersection between overlapping models (e.g. a robot and an obstacle), PD can provide a distance measure for the configuration space obstacle (C-obstacle). We extend the prior notion of translational PD to generalized PD, which takes into account translational as well as rotational motion to separate two overlapping models. Moreover, we formulate generalized PD computation based on appropriate model-dependent metrics and present two algorithms based on convex decomposition and local optimization. We highlight the efficiency and robustness of our PD algorithms on many complex 3D models. Based on generalized PD computation, we present the first set of practical algorithms for low DOF complete motion planning. Moreover, we use generalized PD computation to develop a retraction-based planner to effectively generate samples in narrow passages for rigid robots. The effectiveness of the resulting planner is shown by alpha puzzle benchmark and part disassembly benchmarks in virtual prototyping

Two-dimensional minkowski-sum optimization of ganged stamping blank layouts for use on precut sheet metal for convex and concave parts

Journal UnknownAs the number of parts that manufacturers need to place on a piece of material such as sheet metal increases, the need for more sophisticated algorithms for part orientation and spacing also increases. With greater part shape complexity, the ability of a skilled craftsman becomes challenged to minimize waste. Building upon the previous work of Nye, we present a Minkowski-sum method for maximizing the number of parts within gangs on a rectangular sheet of material. The example provided uses a simply shaped part to illustrate the presented method, yielding a packing efficiency of 62% that is identical to the efficiency that a skilled worker would produce without the algorithm. We also provide results for laying out a more complex part in ganged sections, demonstrating a result that would be difficult for a human to reproduce. Our work extends that of Nye by adding practical constraints such as the number of parts that can be blanked at once as well as the amount of horizontal and vertical spacing between ganged blanking sets. Additionally we add an algorithm for laying out polygons with concave geometries by separating the part into a set of convex polygons. Two examples for optimization, one of a chevron-shaped part and one of a complex shape previously used by Nye (2000) and Choi et al. (1998) are provided demonstrating the existence of a local maximum number of parts that may be stamped within a single ganged blank. Our algorithm is extendable to a program that may provide stamping manufacturers with a tool that can maximize the total number of parts stamped on stock sheet metal, or for other tiling problems

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions