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This paper describes a geometrical model to quantify minimal joint clearance in overconstrained 2D or 3D linkages when the shapes of components are subject to manufacturing deviations. This important property needs to be computed at the earlier stages of product development for engineers to design and produce reliable mechanisms in the context of mass production. The solution is based on a previous study dedicated to rigid assemblies. The major contributions of this work are the definition of a regularized closure function together with the computation of minimum joint clearance for a mobile linkage. Five examples illustrate the efficiency of the solution.

A manufactured over-constrained mechanism can’t be assembled and can’t be mobile without clearance in its joints. During the conceptual design phase, designers must specify the degree precision for the parts without proper quantitative decision-making tools. Those decisions have a great impact on the product’s quality and cost. In this paper, clearance and deviation are defined for an assembly of rigid parts. A geometric model is presented to measure those values in a given mechanism using the TTRS model. Optimum values are computed by minimizing a weighted function of both clearance and deviation.

It is well known that manufactured over-constrained mechanical systems cannot operate without clearance between parts. If the clearance value is too small, parts will not fit together. Conversely, if the clearance value is too large, parts float relative to each other and the movement is imperfect. This paper presents a tool providing decision support during the conceptual design of over-constrained mechanisms. A new key characteristic of the mechanical system, named F, is defined. F quantifies the floating phenomenon of the mechanical assembly. The smaller it is, the smaller is the free movement of parts around their nominal positions. Optimizations are used to compute the minimal clearance value to guarantee the mechanism assemblability and also the corresponding F value. These values are obtained with respect to a quality level of the manufacturing process. Then, the proposed tool plots diagrams. It shows areas where the design is possible and highlights inconsistencies. We consider that it is very useful to assist users during the conceptual design step. A case study including a planar mechanism made of 6 parts and 8 joints illustrates the tool.

A mechanism is a set of rigid links interconnected together with ideal joints and featuring at least one degree of freedom. An overconstrained mechanism is mobile provided the links' dimensions fulfill a certain relation named the “mobility condition”. The dimensions are not independent from each other and the goal is to obtain this mobility condition. Firstly, parameters are divided into two categories: dimensional parameters and positional parameters. Dimensional parameters represent links' sizes and positional parameters represent the relative positions between the links. The closure equation models the geometric problem by capturing the relationships between the two types of parameters. The principle of the paper is to generate the mobility condition by applying Groebner basis computation to the closure equation. Three methods are presented and investigated. The first one is called the univariate polynomial method (UNIPOL); the second method is called multi-order derivation method (MOD) and the third one is called the finitely separated configurations (FISECO) method. Practical implementation of these various techniques is explained by using a standard computer algebra system. The three methods are applied on a 2D overconstrained mechanism.

This paper presents a generic approach, to generate symbolic dependency relations between the variations on the dimensional parameters for a family of over-constrained structure. We call structure a set of rigid parts interconnected together with mechanical linkages. A structure is over-constrained when the size of parts are not independent of each other. To achieve our goal, we propose the following method. Firstly, parameters are divided into two categories: dimensional parameters and configuration parameters. Dimensional parameters represent the size of the parts and configuration parameters represent the relative position between the parts. Symbolic closed-loop equations model the geometric problem. They represent the dependency between two types of parameters. To generate symbolic equations under dimensional parameters, we use a generic method of elimination, based on Grobner basis computation. These symbolic relations are called “compatibility equations”, which guarantee the existence of the studied mechanical assembly. Generally, there are many more dimensional parameters than compatibility equations. In this paper, compatibility equations are regarded as implicit functions. We apply the implicit function theorem to generate symbolic differential equations which govern the variations of the clearance between the components. For that, the set of dimensional parameters is separated into two subsets: independent and dependent dimensional parameters. Maximum and minimum clearances are calculated by simulating harmonic variations of the rigid parts size. The solution of the differential equations allows a fast simulation. The presented generic approach is applied on a 2D over-constrained assembly. Symbolic and numerical results show the feasibility of this generic approach.

International audience; In mechanical engineering, the degree of freedom or mobility is a fundamental property of solid assemblies. To compute it, classical formulas fail when closed-loop overconstrained mechanisms are concerned. Another way to define the mobility is to consider the dimension of the algebraic variety representing the closure of the loop. The approach described here, consists in computing the conditions that ensure that an overconstrained mechanism is mobile.

This paper explores a numerical approach, which determines the number of mobility equations for a family of over-constrained mechanism and generates mobility equations under dimensional parameter variations. These equations can be used to check the mobility of the over-constrained mechanism. They are produced from a specified mobile system and are valid in the neighborhood of nominal mechanisms. This generic approach is applied on Goldberg 5 R linkages.

To model a geometrical part in Computer Aided Design systems, declarative modeling is a well-adapted solution to declare and specify geometric objects and constraints. In this chapter, we are interested in the representation of geometric objects and constraints using a new language of description and representation, Geometric Algebra (GA). GA is used here in association with the conformal model of Euclidean geometry (CGA) which requires two extra dimensions comparing to the usual vector space model. Topologically and Technologically Related Surfaces (TTRS) Theory is introduced here as a unified framework for geometric objects representation and geometric constraints solving. Based on TTRS, this chapter shows the capability of the CGA to represent geometric objects and geometric constraints through symbolic geometric constraints solving and algebraic classification.

International audience; This paper proposes to investigate the use of a coordinate free approach for the mapping of geometrical requirement along a product life cycle. The geometry of the studied assembly is represented using a Gram matrix that is issued from a parametric model constituted of points and vectors. This parametric model is instanced for all relevant phase of the product life cycle. The calculation of instanced parameters is based on part deformation due to changing operating conditions. This calculation is carried out thanks to existing theoretical techniques. The application presented in this paper is constituted of a simple 3D case composed of 3 articulated bars disposed as a tetrahedron and subjected to some thermal expansion.

The general objective of this paper is to analyse the dependency relations that may exist between manufacturing and design specifications of a geometric object. This object will be modelled on the basis of the TTRS concept and vector modelling synthesised by the metric tensor of a set of vectors. The first problem that we propose to solve therefore is the formal expression of these relations. After that, the dependency will be analysed and illustrated considering a 3D object.

“Good” functional tolerancing can only be obtained once the geometrical specification of the mechanism being studied has been analysed from a technological and geometrical standpoint. A geometrical analysis based on a vector approach has been developed. It enables existing tolerancing relations to be written between the design parameters relative to each part and the relative position parameters between two parts of the mechanism in question.

The goal of this paper is to present the interest to define a geometric declarative modeler according to the needs of the associated solver. The modeler presented will seem, in a first view, of a complex architecture. A more detailed description of the modeler allows the presentation of the geometric constraints management. The interest of such approach is finally illustrated by the solver presentation. This solver, divided in three modules, allows a reliable and fast resolution of the geometric problems. Indeed, each module has been defined to optimize the resolution in terms of time and robustness.

Our aim is to unify mathematically the specification and the metrological verification of a given geometrical object. The observed differences result from the fact that strictly geometrical parameters are used in CAD systems while in mechanical engineering distances are paramount. A new mathematical concept — “Near Surfaces” — will be defined. This concept enables specifications and metrological determination to be incorporated within a unified mathematical theory.

The general objective of this paper is to define the problem areas linked to the specification and overall consistency of a dimensioning diagram and to propose a solution principle to solve this problem.

The geometric models currently used in CAD-CAM modellers are chiefly based on a procedural approach. Over the past few years, the appearance of parametric and, above all, variational modellers, has opened up the way to a more declarative approach. It is true to say that parametric design is efficient, but it requires knowledge of a design chronology whose flexibility is thus diminished. On the other hand, variational design is very general and flexible but does need the use of solvers to simultaneously solve a large number of non-linear equations.

The dimensioning and the tolerancing model presented in this paper allows the functional tolerancing declaration of a class of mechanical parts independently from the geometric instantiation. That model is based on the use of the TTRS concept and on relative positioning constraints. Moreover, that model is compatible with the exchange standard ISO/CD 10303-47.

This paper presents a new approach to transform a constrained geometric object. The geometric model is based on the vectorisation of the object’s shape. Thus, elementary geometric entities and constraints are represented by a set of vectors or bivectors in order to establish the object specifications. The metric tensor is the mathematical tool used to define the vector space generated by the set of vectors and bivectors. The transformation of the initial object is realised thanks to a transformation matrix associated with a connection matrix. Based on the point of displacements of the initial object, this new method gives the final object satisfying the geometric specifications required by the designer.

The tolerancing synthesis consists in proposing a part tolerancing diagram directly obtained from the mechanism good-working conditions. However, tools that can be used to describe the functional requirements are a few and often very specific. The objective of this paper is then to propose a mean to define, from a univocal and sufficiently adaptable point of view, all the functional requirement types a design engineer is brought to specify.

To establish the specifications and to solve 2D and 3D geometric problems defined by geometric constraints, we have developed a declarative model where objects and constraints are represented by a set of vectors. These vectors are not represented by their usual Cartesian coordinates but by their respective scalar products. Then, a metric tensor characterises the geometric object. To solve the problem, we have developed a model that transforms the object from its initial state to its final state.

In preliminary project phases, designers generally prefer to express their ideas via 2D or 3D sketches. The scientific problem covered in this paper is that of the analysis of the consistency of a 3D geometric specification. Since traditional geometric Cartesian modelling does not permit easy analysis of consistency, we will present two non-Cartesian modellings: either by exclusively using points, and Cayley-Menger determinants will enable the relations between mutual distances to be written, or by using points and vectors, and Gram determinants will enable the relations between scalar products to be written. Finally, we will present the advantages of these two modellings in order to analyse the overall consistency of a specification.

This paper presents a series of methods for analyzing a geometric specification in 2 and 3 dimensional space. These are based on a new model for length and angle constraints as currently studied in variational geometry. These various tools will hopefully further the exchange of information between the designer and CAD systems.