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

A Robust and Efficient Method for Solving Point Distance Problems by Homotopy

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user.Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it.This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan.Numerical results show that this hybrid method is efficient and robust

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

A Generalized Malfatti Problem

Abstract Malfatti's problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle's sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly

Modélisation Géométrique par Contraintes : Solveurs basés sur l’arithmétique des intervalles

Nowadays many applications of computer graphics and geometric require resolutionnonlinear systems.The calculation of the synthesized images by ray tracing requires the computation of the intersection points between rays (half-lines) and surfaces defined by one or more algebraic equations.Geometric modeling, and Computer Aided Design and Manufacturing (CAD/CAM) needcalculating the points of intersection between such surfaces. This yields systems of algebraic equations of small sizes. All geometric modelers used nowadays provide the possibilityto model geometric objects by a set of geometric constraints.The resolution of these geometric constraints requires the resolution of non linear algebraic systems. The sub-irreducible systems can be large (over ten unknowns and equations) and are solved by numerical methods such the iteration of Newton-Raphson, homotopy (or continuation), methods of Newton per interval and solvers using Bernstein bases or other geometric bases.Bernstein bases allow calculating good estimates of polynomials values over a grid and solving polynomial systems encountered in imaging, geometric modeling, and geometric constraints solving.This thesis presents two types of solvers. The first is based on the classic Bernstein basesand limited to small systems with 6 or 7 unknown at most, as it appears in image synthesis, byfor example ray tracing on parametric surfaces.The second is new, avoids this limitation by defining the Bernstein polytope and using thelinear programming. This second type of solvers is usable on arbitrary sized systems.Aujourd’hui plusieurs applications de l’informatique graphique ou géométrique nécessitent la résolution desystèmes non linéaires.Le calcul des images de synthèse par lancer de rayons nécessite le calcul des points d’intersection entre desrayons (des demi-droites) et des surfaces définies par une ou plusieurs équations algébriques.La modélisation géométrique, et la Conception et Fabrication Assistées par Ordinateur (CFAO) ont besoinde calculer les points d’intersection entre de telles surfaces. Il en résulte des systèmes d’équations algébriquesde petites tailles. Enfin, tous les modeleurs géométriques utilisés en CFAO fournissent aujourd’hui la possibilitéde modéliser des objets géométriques, ou de dimensionner des pièces, par un ensemble de contraintesgéométriques.La résolution de ces contraintes géométriques nécessite la résolution de systèmes d’équations algébriques nonlinéaires. Les sous systèmes irréductibles peuvent être de grande taille (plus d’une dizaine d’inconnueset d’équations) et sont résolus par des méthodes numériques : citons l’itération de Newton-Raphson,l’homotopie (ou continuation), les méthodes de Newton par intervalles, et les solveurs utilisant les basestensorielles de Bernstein ou d’autres bases géométriques.Les bases tensorielles de Bernstein permettent de calculer de bons encadrements des valeurs des polynômessur des pavés, et de résoudre les systèmes polynomiaux rencontrés en synthèse d’images, en modélisationgéométrique, et en résolution de contraintes géométriques.Cette thèse présente deux types de solveurs. Le premier est classique fondé sur les bases de Bernsteinet limité aux petits systèmes de 6 ou 7 inconnues au plus, comme il apparaît en synthèse d’images, parexemple pour le lancer de rayons sur des surfaces paramétriques.Le second est nouveau, il évite cette limitation en définissant le polytope de Bernstein et en recourant à laprogrammation linéaire. Ce deuxième type de solveurs est utilisable sur des systèmes de taille arbitraire

Design reuse in a CAD environment

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 09/03/1999.For many companies, design related information mainly exists as rooms of paper-based archives, typically in the form of manufacturing drawings and technical specifications. This 'static' information cannot be easily reused. The work presented in this thesis proposes a methodology to ease this problem. It defines and implements a computer-based design tool that will enable existing design families to be transformed into 'dynamic' CAD-based models for the Conceptual, Embodiment and Detailed stages of the design process. Two novel concepts are proposed here, i) the use of a Function Means Tree to store Conceptual and Embodiment design and ii) a Variant Method to represent Detailed design. In this way a definite link between the more abstract conceptual and the concrete detailed design stages is realised by linking individual detailed designs to means in the Function Means Tree. The use of the Variant Method, incorporating 'state-of-the-art' developments in Solid Modelling, Feature-Based Design and Parametric Design, allows an entire family of designs to be represented by a single Master Model. Therefore, instances of this Master Model need only be stored as a set of design parameters. This enables current design families and new design cases to be more created more efficiently. Industrial Case Studies, including a Lathe Chuck family, a Drive-End casting and a family of Filtration Systems are given to prove the methodology

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