Solving Geometric Constraints by Homotopy

International audienceNumerous methods have been proposed in order to solve geometric constraints, all of them having their own advantages and drawbacks. In this article, we propose an enhancement of the classical numerical methods, which are, up to now the only ones that apply to the general case

Description of a robotics-oriented relational positioning methodology

This paper presents a relational positioning methodology for flexibly and intuitively specifying offline programmed robot tasks, as well as for assisting the execution of teleoperated tasks demanding precise movements. In relational positioning, the movements of an object can be restricted totally or partially by specifying its allowed positions in terms of a set of geometric constraints. These allowed positions are found by means of a 3D sequential geometric constraint solver called PMF – Positioning Mobile with respect to Fixed. PMF exploits the fact that in a set of geometric constraints, the rotational component can often be separated from the translational one and solved independently

Research in constraint-based layout, visualization, CAD, and related topics : a bibliographical survey

The present work compiles numerous papers in the area of computer-aided design, graphics, layout configuration, and user interfaces in general. There is nearly no conference on graphics, multimedia, and user interfaces that does not include a section on constraint-based graphics; on the other hand most conferences on constraint processing favour applications in graphics. This work of bibliographical pointers may serve as a basis for a detailed and comprehensive survey of this important and challenging field in the intersection of constraint processing and graphics. In order to reach this ambitious aim, and also to keep this study up-to-date, the authors appreciate any comment and update information

Direct tree decomposition of geometric constraint graphs

The evolution of constraint based geometric models is tightly tied to parametric and feature-based Computer-Aided Design (CAD) systems. Since the introduction of parametric design by Pro/Engineer in the 1980's, most major CAD systems adopted constraint based geometric models as a core technology. Constraint based geometric models allowed CAD systems to provide a more powerful data model while offering an intuitive user interface. Later on, the same models also found application to fields like linkage design, chemical modeling, computer vision and dynamic geometry. Constraint based geometric models are unevaluated models. A key problem related to constraint based geometric models is the geometric constraint based solving problem which, roughly speaking, can be stated as the problem of evaluating a constraint based model. Among the different approaches to geometric constraint solving, we are interested in graph-based Decomposition-Recombination solvers. In the graph-based constructive approach, the geometric problem is first translated into a graph whose vertices represent the set of geometric elements and whose edges are the constraints. Then the constraint problem is solved by decomposing the graph into a set of sub-problems, each sub-problem is recursively divided until reaching basic problems which are solved by a dedicated equational solver. The solution to the initial problem is computed by merging the solutions to the sub-problems. The approach used by DR-solvers has been particularly successful when the decomposition into subproblems and subsequent recombination of solutions to these subproblems can be described by a plan generated a priori, that is, a plan generated as a preprocessing step without actually solving the subsystems. The plan output by the DR-planner remains unchanged as numerical values of parameters change. Such a plan is known as a DR-plan and the unit in the solver that generates it is the DR-planner. In this setting, the DR-plan is then used to drive the actual solving process, that is, computing specific coordinates that properly place geometric objects with respect to each other. In this thesis we develop a new DR-planner algorithm for graph-constructive two dimensional DR-solvers. This DR-planner is based on the tree-decomposition of a graph. The triangle- or tree-decomposition of a graph decomposes a graph into three subgraphs such that subgraphs pairwise share one vertex. Shared vertices are called hinges. The tree-decomposition of a geometric constraint graph is in some sense the construction plan that solves the corresponding problem. The DR-planner algorithm first transforms the input graph into a simpler, planar graph. After that, an specific planar embedding is computed for the transformed graph where hinges, if any, can be straightly found. In the work we proof the soundness of the new algorithm. We also show that the worst case time performance of the resthe number of vertices of the input graph. The resulting algorithm is easy to implement and is as efficient as other known solving algorithms.L'evolució de models geomètrics basats en restriccions està fortament lligada al sistemes de Disseny Assistit per Computador (CAD) paramètrics i als basats en el paradigma de disseny per mitjà de característiques. Des de la introducció del disseny paramètric per part de Pro/Engineer en els anys 80, la major part de sistemes CAD utilitzaren com a tecnologia de base els models geomètrics basats en restriccions. Els models geomètrics basats en restriccions permeteren als sistemes CAD proporcionar un model d'informació més ampli i alhora oferir una interfície d'usuari intuitiva. Posteriorment, els mateixos models s'aplicaren en camps com el disseny de mecanismes, el modelatge químic, la visió per computador i la geometria dinàmica. Els models geomètrics basats en restriccions són models no avaluats. Un problema clau relacionat amb el models de restriccions geomètriques és el problema de la resolució de restriccions geomètriques, que es resumeix com el problema d'avaluar un model basat en restriccions. Entre els diferents enfocs de resolució de restriccions geomètriques, tractem els solvers de Descomposició-Recombinació (DR-solvers) basats en graphs. En l'enfoc constructiu basat en grafs, el problema geomètric es trasllada en un pas inicial a un graf, on els vèrtexs del graf representen el conjunt d'elements geomètrics i on les arestes corresponen a les restriccions geomètriques entre els elements. A continuació el problema de restriccions es resol descomposant el graf en un conjunt de subproblemes, cadascun dels quals es divideix recursivament fins a obtenir problemes bàsics, que sovint són operacions geomètriques realitzables, per exemple, amb regle i compàs, i que es resolen per mitjà d'un solver numèric específic. Finalment, la solució del problema inicial s'obté recombinant les solucions dels subproblemes. L'enfoc utilitzat pels DR-solvers ha esdevingut especialment interessant quan la descomposició en subproblemes i la posterior recombinació de solucions d'aquests subproblemes es pot descriure com un pla de construcció generat a priori, és a dir, un pla generat com a pas de pre-procés sense necessitat de resoldre realment els subsistemes. El pla generat pel DR-planner esdevé inalterable encara que els valors numèrics dels paràmetres canviin. Aquest pla es coneix com a DR-plan i la unitat en el solver que el genera és l'anomenat DR-planner. En aquest context, el DR-plan s'utilitza com a eina del procés de resolució en curs, és a dir, permet calcular les coordenades específiques que correctament posicionen els elements geomètrics uns respecte els altres. En aquesta tesi desenvolupem un nou algoritme que és la base del DR-planner per a DR-solvers constructius basats en grafs en l'espai bidimensional. Aquest DR-planner es basa en la descomposició en arbre d'un graf. La descomposició en triangles o arbre de descomposició d'un graf es basa en descomposar un graf en tres subgrafs tals que comparteixen un vèrtex 2 a 2. El conjunt de vèrtexs compartits s'anomenen \emph{hinges}. La descomposició en arbre d'un graf de restriccions geomètriques equival, en cert sentit, a resoldre el problema de restriccions geomètriques. L'algoritme del DR-planner en primer lloc transforma el graf proporcionat en un graf més simple i planar. A continuació, es calcula el dibuix en el pla del graf transformat, on les hinges, si n'hi ha, es calculen de manera directa. En aquest treball demostrem la correctesa del nou algoritme. Finalment, proporcionem l'estudi de la complexitat temporal de l'algoritme en cas pitjor i demostrem que és quadràtica en el nombre de vèrtexs del graf proporcionat. L'algoritme resultant és senzill d'implementar i tan eficient com altres algoritmes de resolució concret

Constraint-Enabled Design Information Representation for Mechanical Products Over the Internet

Global economy has made manufacturing industry become more distributed than ever before. Product design requires more involvement from various technical disciplines at different locations. In such a geographically and temporally distributed environment, efficient and effective collaboration on design is vital to maintain product quality and organizational competency. Interoperability of design information is one of major barriers for collaborative design. Current standard CAD data formats do not support design collaboration effectively in terms of design information and knowledge capturing, exchange, and integration within the design cycle. Multidisciplinary design constraints cannot be represented and transferred among different groups, and design information cannot be integrated efficiently within a distributed environment. Uncertainty of specification cannot be modeled at early design stages, while constraints for optimization are not embedded in design data. In this work, a design information model, Universal Linkage model, is developed to represent design related information for mechanical products in a distributed form. It incorporates geometric and non-geometric constraints with traditional geometry and topology elements, thus allows more design knowledge sharing in collaborative design. Segments of design data are linked and integrated into a complete product model, thus support lean design information capturing, storage, and query. The model is represented by Directed Hyper Graph and Product Markup Language to preserve extensibility and openness. Incorporating robustness consideration, an Interval Geometric Modeling scheme is presented, in which numerical parameters are represented by interval values. This scheme is able to capture uncertainty and inexactness of design and reduces the chances of conflict in constraint imposition. It provides a unified constraint representation for the process of conceptual design, detailed design, and design optimization. Corresponding interval constraint solving methods are studied

Modeling and sensitivity analysis of aircraft geometry for multidisciplinary optimization problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 415-421).A new geometry management paradigm for aircraft design utilizes Computer Aided Design (CAD) systems as the source for consistent geometry models across design phases and analysis tools. Yet various challenges inhibit the widespread application of CAD models in aircraft conceptual design because current CAD platforms are not designed for automated shape optimization. In particular, CAD models built with conventional methods can perform poorly in automated design frameworks and their associated CAD systems do not provide shape sensitivities. This thesis aims to remedy these concerns by bridging the computational geometry tools in CAD with aerospace design needs. A methodology for constructing CAD models is presented using concepts of multifidelity/multidisciplinary geometry and design motion. A formalized definition of design intent emerges from this approach that enables CAD models with parameterization flexibility, shape malleability and regeneration robustness for automated design settings. Analytic shape sensitivities are also presented to apply CAD models in gradient-based shape optimization. The parameterization and sensitivities for sketches, extrude, revolve and sweep features are given for mechanical design; shape sensitivities for B-spline curves and surfaces are also presented for airfoil and wing design. Furthermore, analytic methods modeling the sensitivity of intersection edges and nodes in a boundary representation (BRep) are given. Comparisons between analytic and finite-difference gradients show excellent agreement, however an error associated with the finite-difference gradient is found to exist if linearizing the support points of B-spline curves/surfaces and regenerating with a geometry kernel. This important outcome highlights a limitation of the finite-difference method when used on CAD models containing these entities. Finally, various example design problems are shown which highlight the application of the methods presented in the thesis. These include mechanical part design, inverse/forward design of airfoils and wings, and a multidisciplinary design space study. Gradient-based optimization is used in each design problem to compare the impact of analytic and finite-difference geometry gradients on the final designs obtained. With each of these contributions, the application of CAD-by David Sergio Lazzara.Ph.D