The foremost works on the tolerance modeling problem are found in (Requicha, 1983; Requicha, 1993) that introduced the mathematical definition of the tolerance’s semantic. He focused on constructing semantically correct tolerance zones and he proposed a solid offset approach for this purpose. Since then, a lot of models are proposed by the literature to perform the tolerance analysis of an assembly whose components may be considered as rigid parts (Hong & Chang, 2002).
The vector loop model uses vectors to represent relevant dimensions in an assembly (Chase et al., 1995; Chase et al., 1996; Chase et al., 1997a). Each vector represents either a component dimension or an assembly dimension. Vectors are arranged in chains or loops to reproduce the effects of those dimensions that stack together to determine the resultant assembly dimensions. Three types of variations are modelled in the vector loop model: dimensional variations, kinematic variations and geometric variations. Dimensional variations defined by dimensional tolerances are incorporated as +/- variations in the length of the vector. Kinematic variations describe the relative motions among mating parts, i. e. small adjustments that occur at assembly time in response to the dimensional and geometric variations of the components. Geometric variations capture those variations that are imputable to geometric tolerances.
The variational solid modelling approach involves applying variations to a computer model of a part or an assembly of parts (Martino & Gabriele, 1989; Boyer & Stewart, 1991; Gupta & Turner, 1993). To create an assembly, the designer identifies the relevant features of each component and assigns dimensional and geometrical tolerances to them. In real conditions (i. e. manufactured part), the feature has been characterized by a roto-translational displacement with respect to its nominal position. This displacement is modelled to summarize the complete effects of the dimensional and geometric variations affecting the part by means of a differential homogeneous transformation matrix. Once the variabilities of the parts are modelled, they must be assembled together. Another set of differential homogeneous transformation matrices is introduced to handle the roto-translational deviations introduced by each assembly mating relation.
The matrix model aims at deriving an explicit mathematical representation of the boundary of the entire spatial region that encloses all possible displacements due to one or more variability sources. In order to do that, homogenous transformation matrices are considered as the foundation of the mathematical representation. A displacement matrix is used to describe any roto-translational variation a feature may be subjected to. The matrix model is based on the positional tolerancing and the Technologically and Topologically Related Surfaces (TTRS) criteria (Clement et al., 1998); by classifying the surfaces into several classes, each characterized by some kind of invariance with respect to specific displacement kinds (e. g. a cylinder is invariant to any rotation about its axis) the resulting displacement matrix can be simplified (Clement et al., 1994).
In the terminology adopted by the jacobian model approach, any relevant surface involved in the tolerance stack-up is referred to as functional element (FE). In the tolerance chain, FEs are considered in pairs: the two paired surfaces may belong to the same part (internal pair), or to two different parts, and paired since they interact as mating elements (kinematic pair, also referred to as external pair). The parts should be in contact to be modelled by this model. Transformation matrices may be used to locate a FE of a pair with respect to the other: these matrices can be used to model the nominal displacement between the two FEs, but also additional small displacements due to the variabilities modelled by the tolerances. The main peculiar aspect of the jacobian approach is how such matrices are formulated, i. e. by means of an approach derived from the description of kinematic chains in robotics (Laperriere & Lafond, 1999; Laperriere & Kabore, 2001).
The torsor model uses screw parameters to model three dimensional tolerance zones (Chase et al., 1996). Screw parameters are a common approach adopted in kinematics to describe motion; they are used to describe a tolerance zone, since a tolerance zone is the region where a surface is allowed to move. The screw parameters are arranged in a particular mathematical operator called torsor, hence the name of the approach. To model the interactions between the parts of an assembly, three types of torsors (or Small Displacement Torsor SDT) are defined (Ballot & Bourdet, 1997): a part SDT for each part of the assembly to model the displacement of the part; a deviation SDT for each surface of each part to model the geometrical deviations from nominal; a gap SDT between two surfaces linking two parts to model the mating relation.
The Tolerance Maps model is being developed at the Arizona State University (Davidson et al., 2002; Mujezinovic et al., 2004; Ameta et al., 2007). It is based on a two-levels model: the local model, that models part variations in order to consider the interactions of the geometric controls applied to a feature of interest and the global model that interrelates all control frames on a part or assembly. A Tolerance-Map (T-Map) is a hypothetical solid of points in n-dimensions which represent all possible variations of a feature or an assembly. Overlaying the coordinates of the T-Map the stack-up equations to perform the tolerance analysis are obtained.
However, these methods are not easy to apply, especially for complex aerospace assemblies, since they were born to deal with elementary features, such as plane, hole, pin and so on. So the aid of computer is called for. In the recent years, the development of efficient and robust design tools has allowed to foresee manufacturing or assembly problems during the first steps of product modeling by adopting a concurrent engineering approach.
Efforts to deal with the tolerance analysis in aeronautic field were carried out. Sellakh proposed an assisted method for tolerance analysis of aircraft structures through assembly graphs and TTRS theory (Sellakh et al., 2003). Marguet presented a methodology to analyse and optimise the assembly sequence of simple shape assemblies (Marguet & Mathieu, 1998; Marguet et Mathieu, 1999). Ody showed a comparison among Error Budgeting techniques and 3D Tolerance Software Packages (Ody et al., 2001). Those papers present solutions of typical mechanical assemblies that involve the alignment of plane, holes and pins, but the aeronautic surfaces have a free form generally.
Some are the works found in the literature to deal with the tolerance analysis of compliant assemblies, i. e. assemblies that contain deformable parts, such as sheet-metal, plastics, composites and glass. Some of the first research related to this area was done by (Gordis & Flannely, 1994). They used frequency domain analysis to predict in-plane loads and displacements from misalignment of fastener holes in flexible components. Liu and Hu have used simple finite element models to predict assembly variation of flexible sheet metal assemblies (Liu & Hu, 1995; Liu & Hu, 1997). Their work was focused on the effect of part fixturing and order of assembly.
Merkley uses the assumptions of Francavilla and Zienkiewicz (Francavilla & Zienkiewicz, 1975) to linearize the elastic contact problem between mating flexible parts (Merkley, 1998). Merkley derived a method for predicting the mean and the variance of assembly forces and deformations due to assembling two flexible parts having surface variations. He describes the need for a covariance matrix representing the interrelation of variations at neighboring nodes in the finite element model. The interrelation is due to both surface continuity, which Merkley calls geometric covariance, and elastic coupling, which he calls material covariance. Merkley used random Bezier curves to describe surface variations and to calculate geometric covariance. Bihlmaier presents a new method for deriving the covariance matrix using spectral analysis techniques (Bihlmaier, 1999). The new method, called the Flexible Assembly Spectral Tolerance Analysis method, or FASTA, also includes the effect of surface variation wavelength on assemblies.
