Анализ геометрической разрешимости при сборке сложных изделий как задача принятия решений

Computer aided assembly planning (CAAP) of complex products is an important and urgent problem of state-of-the-art information technologies. A configuration of the technical system imposes fundamental restrictions on the design solutions of the assembly process. The CAAP studies offer various methods for modelling geometric constraints. The most accurate results are obtained from the studies of geometric obstacles, which prohibit the part movement to the appropriate position in the product, by the collision analysis methods. An assembly of complex technical systems by this approach requires very high computational costs, since the analysis should be performed for each part and in several directions.The article describes a method for minimizing the number of direct checks for geometric obstacle avoidance. Introduces a concept of the geometric situation to formalize such fragments of a structure, which require checking for geometric obstacle avoidance. Formulates two statements about geometric heredity during the assembly. Poses the task of minimizing the number of direct checks as a non-antagonistic two-person game on two-colour painting of an ordered set. Presents the main decision criteria under uncertainty. To determine the best criterion, a computational experiment was carried out on painting the ordered sets with radically different structural properties. All the connected ordered sets are divided into 13 subclasses, each of which includes structurally similar instances. To implement the experiment, a special program has been developed that creates a random ordered set in the selected subclass, implements a game session on its coloration, and also collects and processes statistical data on a group of the homogeneous experiments.The computational experiment has shown that in all subclasses of the partial orders, the Hurwitz criterion with a confidence coefficient of 2/3 and that of Bayes-Laplace demonstrate the best results. The Wald and Savage criteria have demonstrated the worst results. In the experiment, a difference between the best and worst results reached 70%. With increasing height (maximum number of levels) of an ordered set, this difference tends to grow rapidly. In the subclass of pseudo-chains, all criteria showed approximately equal results.The game model of geometric obstacles avoidance allows formalizing data on geometric heredity and obtaining data on the composition and the minimum number of configurations, the test of which objectifies all existing-in-the-product geometric constraints on the movements of parts during assembly.Автоматизация проектирования процессов сборки сложных изделий – это важная и актуальная проблема современной информационной технологии. Фундаментальные ограничения на проектные решения сборочного передела накладывает геометрия технической системы. В исследованиях по CAAP предложены различные методы моделирования геометрических связей. Самые точные результаты дает исследования геометрических препятствии, которые запрещают движение детали в служебное положение в изделии, методами анализа столкновений. Для сборки сложных технических систем данный подход требует очень высоких вычислительных затрат, поскольку анализ следует выполнить для каждой детали и в нескольких направлениях.В статье описан метод минимизации числа прямых проверок на геометрическую разрешимость. Введено понятие геометрической ситуации, которое формализует такие фрагменты конструкции, для которых требуется проверка геометрическую разрешимость. Сформулированы два утверждения о геометрической наследственности при сборке. Задача минимизации числа прямых проверок поставлена как неантагонистиченская игра двух лиц по окрашиванию упорядоченного множества в два цвета. Приведены основные критерии принятия решений в условиях неопределенности. Для определения лучшего критерия проведен вычислительный эксперимент по окрашиванию упорядоченных множеств с радикально различными структурными свойствами. Все связные упорядоченные множества разбиты на 13 подклассов, в каждый из которых входят структурно подобные экземпляры. Для реализации эксперимента создана специальная программа, которая создает случайное упорядоченное множество в выбранном подклассе, реализует игровой сеанс по его окрашиванию, а также собирает и обрабатывает статистические данные по группе однородных экспериментов.Вычислительный эксперимент показал, что во всех подклассах частичных порядков лучшие результаты у критериев Гурвица с коэффициентом доверия 2/3 и Байеса-Лапласа. Худшие результаты продемонстрировали критерии Вальда и Севиджа. Разница между лучшими и худшими результатами достигала в эксперименте 70%. Эта разница имеет тенденцию к быстрому росту с увеличением высоты (максимального числа уровней) упорядоченного множества. В подклассе псевдоцепей все критерии показали примерно равные результаты.Игровая модель геометрической разрешимости позволяет формализовать данные о геометрической наследственности и получить данные о составе и минимальном числе конфигураций, проверка которых объективирует все геометрические ограничения на движения деталей при сборке, существующие в изделии

On reconfiguration of disks in the plane and related problems

We revisit two natural reconfiguration models for systems of disjoint objects in the plane: translation and sliding. Consider a set of n pairwise interior-disjoint objects in the plane that need to be brought from a given start (initial) configuration S into a desired goal (target) configuration T, without causing collisions. In the translation model, in one move an object is translated along a fixed direction to another position in the plane. In the sliding model, one move is sliding an object to another location in the plane by means of an arbitrarily complex continuous motion (that could involve rotations). We obtain various combinatorial and computational results for these two models: (I) For systems of n congruent disks in the translation model, Abellanas et al. showed that 2n − 1 moves always suffice and ⌊8n/5 ⌋ moves are sometimes necessary for transforming the start configuration into the target configuration. Here we further improve the lower bound to ⌊5n/3 ⌋ − 1, and thereby give a partial answer to one of their open problems. (II) We show that the reconfiguration problem with congruent disks in the translation model is NPhard, in both the labeled and unlabeled variants. This answers another open problem of Abellanas et al. (III) We also show that the reconfiguration problem with congruent disks in the sliding model is NP-hard, in both the labeled and unlabeled variants. (IV) For the reconfiguration with translations of n arbitrary convex bodies in the plane, 2n moves are always sufficient and sometimes necessary

Translation separability of sets of polygons

We consider the problem of separating a set of polygons by a sequence of translations (one such collision-free translation motion for each polygon). If all translations are performed in a common direction the separability problem so obtained has been referred to as the uni-directional separability problem; for different translation directions, the more general multi-directional separability problem arises. The class of such separability problems has been studied previously and arises e.g. in computer graphics and robotics. Existing solutions to the uni-directional problem typically assume the objects to have a certain predetermined shape (e.g., rectangular or convex objects), or to have a direction of separation already available. Here we show how to compute all directions of unidirectional separability for sets of arbitrary simple polygons. The problem of determining whether a set of polygons is multi-directionally separable had been posed by G.T. Toussaint. Here we present an algorithm for solving this problem which, in addition to detecting whether or not the given set is multidirectionally separable, also provides an ordering in which to separate the polygons. In case that the entire set is not multi-directionally separable, the algorithm will find the largest separable subset