A Systematic Approach for Geometrical and Dimensional Tolerancing in Reverse Engineering

In the fields of mechanical engineering and industrial manufacturing the term Reverse Engineering (RE) refers to the process of creating engineering design data from existing parts and/or assemblies. While conventional engineering transforms engineering concepts and models into real parts, in the reverse engineering approach real parts are transformed into engineering models and concepts (Várady et al., 1997). However, apart from its application in mechanical components, RE is a very common practice in a broad range of diverse fields such as software engineering, animation/entertainment industry, microchips, chemicals, electronics, and pharmaceutical products. Despite this diversity, the reasons for using RE appear to be common in all these application fields, e.g. the original design data and documentation of a product are either not updated, not accessible, do not exist or even have never existed. Focusing on the mechanical engineering domain, through the application of RE techniques an existing part is recreated by acquiring its’ surface and/ or geometrical features’ data using contact or non contact scanning or measuring devices (Wego, 2011). The creation of an RE component computer model takes advantage of the extensive use of CAD/CAM/CAE systems and apparently provides enormous gains in improving the quality and efficiency of RE design, manufacture and analysis. Therefore, RE is now considered one of the technologies that provide substantial business benefits in shortening the product development cycle (Raja & Fernandes, 2008).


Introduction
In the fields of mechanical engineering and industrial manufacturing the term Reverse Engineering (RE) refers to the process of creating engineering design data from existing parts and/or assemblies.While conventional engineering transforms engineering concepts and models into real parts, in the reverse engineering approach real parts are transformed into engineering models and concepts (Várady et al., 1997).However, apart from its application in mechanical components, RE is a very common practice in a broad range of diverse fields such as software engineering, animation/entertainment industry, microchips, chemicals, electronics, and pharmaceutical products.Despite this diversity, the reasons for using RE appear to be common in all these application fields, e.g. the original design data and documentation of a product are either not updated, not accessible, do not exist or even have never existed.Focusing on the mechanical engineering domain, through the application of RE techniques an existing part is recreated by acquiring its' surface and/ or geometrical features' data using contact or non contact scanning or measuring devices (Wego, 2011).The creation of an RE component computer model takes advantage of the extensive use of CAD/CAM/CAE systems and apparently provides enormous gains in improving the quality and efficiency of RE design, manufacture and analysis.Therefore, RE is now considered one of the technologies that provide substantial business benefits in shortening the product development cycle (Raja & Fernandes, 2008).
Tolerance assignment is fundamental for successful mechanical assembly, conformance with functional requirements and component interchangeability, since manufacturing with perfect geometry is virtually unrealistic.In engineering drawings Geometric Dimensioning and Tolerancing (GD&T) correlates size, form, orientation and location of the geometric elements of the design model with the design intent, therefore it has a profound impact on the manufacturability, ease of assembly, performance and ultimate cost of the component.High geometrical and dimensional accuracy leads to high quality; however, tight tolerances lead to an exponential increase of the manufacturing cost.Though the importance of mechanical components and assemblies that have been inevitably modified during several stages of their life cycle, e.g.surface modifications of automotive components during prototype functionality testing (Chant et al., 1998;Yau, 1997), mapping of sheet metal forming deviations on free form surface parts (Yuan et al., 2001), monitoring on the geometrical stability during test runs of mock-up turbine blades used in nuclear power generators (Chen & Lin, 2000), repair time compression by efficient RE modeling of the broken area and subsequent rapid spare part manufacturing (Zheng et al., 2004), recording of die distortions due to thermal effects for the design optimization of fan blades used in aero engines (Mavromihales et al., 2003).
The principles and applications of tolerancing in modern industrial engineering can also be found in several reference publications, e.g.(Drake, 1999;Fischer, 2004).An extensive and systematic review of the conducted research and the state of the art in the field of dimensioning and tolerancing techniques is provided by several recent review papers, e.g.(Singh et al. 2009a(Singh et al. , 2009b) ) and need not be reiterated here.In the large number of research articles on various tolerancing issues in design for manufacturing that have been published over the last years, the designation of geometrical tolerances has been adequately studied under various aspects including tolerance analysis and synthesis, composite positional tolerancing, geometric tolerance propagation, datum establishment, virtual manufacturing, inspection and verification methods for GD&T specifications, e.g.(Anselmetti and Louati, 2005;Diplaris & Sfantsikopoulos, 2006;Martin et al., 2011).
Although RE-tolerancing is a very important and frequently met industrial problem, the need for the development of a systematic approach to extract appropriate design specifications that concern the geometric accuracy of a reconstructed component has been only recently pointed out, (Borja et al., 2001;Thompson et al., 1999;VREPI, 2003).In one of the earliest research works that systematically addresses the RE-tolerancing problem (Kaisarlis et al., 2000), presents the preliminary concept of a knowledge-based system that aims to the allocation of standard tolerances as per ISO-286.The issue of datum identification in RE geometric tolerancing is approached in a systematic way by (Kaisarlis et al, 2004) in a later publication.Recently, (Kaisarlis et al, 2007(Kaisarlis et al, , 2008) ) have further extend the research on this area by focusing on the RE assignment of position tolerances in the case of fixed and floating fasteners respectively.The methodology that is presented in this Chapter further develops the approach that is proposed on these last two publications.The novel contribution reported here deals with (i) the systematic assignment of both geometrical and dimensional tolerances in RE and their possible interrelation through the application of material modifiers on both the RE features and datums and (ii) the consideration of costeffective, competent tolerance designation in RE in a systematic way.

Dimensional tolerancing in reverse engineering
Reconstructed components must obviously mate with the other components of the mechanical assembly that they belong to, in (at least) the same way as their originals, in order the original assembly clearances to be observed, i.e. they must have appropriate manufacturing tolerances.As pointed out in Section 1, this is quite different and much more difficult to be achieved in RE than when designing from scratch where, through normal tolerance analysis/synthesis techniques and given clearances, critical tolerances are assigned, right from the beginning, to all the assembly components.Integration of geometric www.intechopen.comReverse Engineering -Recent Advances and Applications 136 accuracy constrains aimed at the reconstruction of 3D models of RE-conventional engineering objects from range data has been studied adequately, (Raja & Fernandes, 2008;Várady et al., 1997).These studies deal, however, with the mathematical accuracy of the reconstructed CAD model by fitting curves and surfaces to 3D measured data.Featurebased RE (Thompson et al., 1999;VREPI, 2003) does not address, on the other hand, until now issues related with the manufacturing tolerances which have to be assigned on the CAD drawings in order the particular object to be possible to be made as required.A methodology for the problem treatment is proposed in the following sections.
Engineering objects are here classified according to their shape either as free-form objects or as conventional engineering objects that typically have simple geometric surfaces (planes, cylinders, cones, spheres and tori) which meet in sharp edges or smooth blends.In the following, Feature-Based RE for mechanical assembly components of the latter category is mainly considered.Among features of size (ASME, 2009), cylindrical features such as holes in conjunction with pegs, pins or (screw) shafts are the most frequently used for critical functions as are the alignment of mating surfaces or the fastening of mating components in a mechanical assembly.As a result, their role is fundamental in mechanical engineering and, consequently, they should be assigned with appropriate dimensional and geometrical tolerances.In addition, the stochastic nature of the manufacturing deviations makes crucial, for the final RE outcome, the quantity of the available (same) components that serve as reference for the measurements.The more of them are available the more reliable will be the results.For the majority of the RE cases, however, their number is extremely limited and usually ranges from less than ten to only one available item.Mating parts can also be inaccessible for measurements and there is usually an apparent lack of adequate original design and/or manufacturing information.In the scope of this research work, the developed algorithms address the full range of possible scenarios, from "only one original componentno mating component available" to "two or more original pairs of components available", focusing on parts for which either an ISO 286-1 clearance fit (of either hole or shaft basis system) or ISO 2768 (general tolerances) were originally designated.
Assignment of RE dimensional tolerances is accomplished by the present method in five sequential steps.In the primary step (a) the analysis is appropriately directed to ISO fits or general tolerances.In the following steps, the candidate (Step b), suggested (Step c) and preferred (Step d) sets of RE-tolerances are produced.For the final RE tolerance selection (Step e) the cost-effective tolerancing approach, introduced in Section 5, is taken into consideration.For the economy of the chapter, the analysis is only presented for the "two or more original pairs of components available" case, focused on ISO 286 fits, as it is considered the most representative.

Direction of the analysis on ISO fits and/or general tolerances
Let Rd M_h , RF h , RRa h , U h and Rd M_s , RF s , RRa s , U s be the sets of the measured diameters, form deviations, surface roughness, and the uncertainty of CMM measurements for the REhole and the RE-shaft features respectively.The Δ max , Δ h_max , Δ s_max limits are calculated by, In this step, the analysis is directed on the assignment of either ISO 286-1 clearance fits or ISO 2768 general tolerances through the validation of the condition, where In case no member of the IT CAN_h set satisfies the condition (10) for a particular NS CAN_n , the latter is excluded from the BH SG set.In order to qualify for the BH SG set, candidate nominal sizes that validate the condition (10) are further confirmed against all members of the FD CAN set, the candidate IT grades of the IT CAN_s set and, as well as, the measured RE-shaft data, through the constraints, Figure 1 In case no member of the FD CAN set satisfies the condition (11) for a particular NS CAN_n , the latter is excluded from the BH SG set.Moreover, in case no member of the IT CAN_s set satisfies the condition (12) for a particular pair of FD CAN_q and NS CAN_n , validated by (11), they are both excluded from the BH SG set.In a similar manner, the production of the suggested Basic Shaft fits set is achieved by the following set of conditions, Fig. 1.Suggested Basic Hole fits qualification

Sets of preferred fits
A limited number of Preferred Fits out of the Suggested ones is proposed in Step (d) through the consideration of ISO proposed fits.Moreover, the implementation of manufacturing guidelines, such as the fact that it is useful to allocate a slightly larger tolerance to the hole than the shaft, preference of Basic Hole fits over Basic Shaft ones, preference of nominal sizes that are expressed in integers or with minimum possible decimal places etc, are additionally used to "filter" the final range of the preferred fits.The final selection, Step (e), out of the limited set of preferred fits and the method end result is reached by the consideration of the machine shop capabilities and expertise in conjunction with the application of the cost -effective RE tolerancing approach, presented in Section 5 of the chapter.

Geometrical tolerancing in reverse engineering
In order to observe interchangeability, geometrical as well as dimensional accuracy specifications of an RE component must comply with those of the mating part(-s).GD&T in RE must ensure that a reconstructed component will fit and perform well without affecting the function of the specific assembly.The methodology that is presented in this section focuses on the RE assignment of the main type of geometrical tolerance that is used in industry, due to its versatility and economic advantages, the True Position tolerance.However, the approach can be easily adapted for RE assignment of other location geometrical tolerances types, such as coaxiality or symmetry and, as well as, for run-out or profile tolerances.
Position tolerancing is standardized in current GD&T international and national standards, such as (ISO, 1998;ISO 1101ISO , 2004;;ASME, 2009).Although the ISO and the ASME tolerancing systems are not fully compatible, they both define position geometrical tolerance as the total permissible variation in the location of a feature about its exact true position.For cylindrical features such as holes or bosses the position tolerance zone is usually the diameter of the cylinder within which the axis of the feature must lie, the center of the tolerance zone being at the exact true position, Figure 2, whereas for size features such as slots or tabs, it is the total width of the tolerance zone within which the center plane of the feature must lie, the center plane of the zone being at the exact true position.The position tolerance of a feature is denoted with the size of the diameter of the cylindrical tolerance zone (or the distance between the parallel planes of the tolerance zone) in conjunction with the theoretically exact dimensions that determine the true position and their relevant datums, Figure 2. Datums are, consequently, fundamental building blocks of a positional tolerance frame in positional tolerancing.Datum features are chosen to position the toleranced feature in relation to a Cartesian system of three mutually perpendicular planes, jointly called Datum Reference Frame (DRF), and restrict its motion in relation to it.
Positional tolerances often require a three plane datum system, named as primary, secondary and tertiary datum planes.The required number of datums (1, 2 or 3) is derived by considering the degrees of freedom of the toleranced feature that need to be restricted.Change of the datums and/or their order of precedence in the DRF results to different geometrical accuracies, (Kaisarlis et al., 2008).
Fig. 2. Cylindrical tolerance zone and geometric true position tolerancing for a cylindrical feature according to ISO 1101 (Kaisarlis et al, 2008) The versatility and economic benefits of true position tolerances are particularly enhanced when they are assigned at the Maximum Material Condition (MMC).At MMC, an increase in position tolerance is allowed, equal to the departure of the feature from the maximum material condition size, (ISO, 1988;ASME, 2009).As a consequence, a feature with size beyond maximum material but within the dimensional tolerance zone and its axis lying inside the enlarged MMC cylinder is acceptable.The accuracy required by a position tolerance is thus relaxed through the MMC assignment and the reject rate reduced.Moreover, according to the current ISO and ASME standards, datum features of size that are included in the DRF of position tolerances can also apply on either MMC, Regardless of Feature Size (RFS) or Least Material Condition (LMC) basis.
Position tolerances mainly concern clearance fits.They achieve the intended function of a clearance fit by means of the relative positioning and orientation of the axis of the true geometric counterpart of the mating features with reference to one, two or three Cartesian datums.The relationship between mating features in such a clearance fit may be classified either as a fixed or a floating fastener type, (ASME, 2009; Drake, 1999), Figure 3. Floating fastener situation exists where two or more parts are assembled with fasteners such as bolts and nuts, and all parts have clearance holes for the bolts.In a fixed fastener situation one or more of the parts to be assembled have clearance holes and the mating part has restrained fasteners, such as screws in threaded holes or studs.The approach that is here presented deals with both the floating and fixed fastener cases by integrating the individual case methodologies published by (Kaisarlis et al. 2007;2008).Basic issues of the assignment of a Position Tolerance in RE are included in Table 1.Limited number of reference components that does not allow for statistical analysis, availability or not of the mating parts and the presence of wear may affect the reliability of the RE results.Moreover, datum selection and the size of the position tolerance itself should ensure, obviously, a stress-free mechanical mating.The analytic approach presented in this section deals with the full range of these issues in order to produce a reliable solution within realistic time.
i.The number of available RE components that will be measured.The more RE parts are measured, the more reliable will be the extracted results.Typically, the number of available RE components is extremely limited, usually ranging from less than ten to a single one article.
ii. Off the shelf, worn or damaged RE components.Off the shelf RE components are obviously ideal for the job, given that the extent of wear or damage is for the majority of cases difficult to be quantified or compensated.
iii.Accessibility of the mating part (-s) Assignment of RE-position tolerance for both the fixed and the floating fastener case is accomplished by the present method in five sequential steps.The analysis is performed individually for each feature that has to be toleranced in the RE-component.At least two RE reference components, intact or with negligible wear, need to be available in order to minimize the risk of measuring a possibly defective or wrongly referenced RE component and, as it is later explained in this section, to improve the method efficiency.This does not certainly mean that the method cannot be used even when only one component is available.
Mating part availability is desirable as it makes easier the datum(s) recognition.Minimum assembly clearance and, as well as, the dimensional tolerance of the RE-feature (hole, peg, pin or screw shaft) and RE-Datums (for features of size) are taken as results from the RE dimensional tolerance analysis presented in the previous section of the chapter in conjunction with those quoted in relevant application-specific standards.
The primary step (a) of the analysis concerns the recognition of the critical features on the RE component that need to be toleranced and, as well as, their fastening situation.This step is performed interactively and further directs the analysis on either the fixed or the floating fastener option.In step (b) mathematical relationships that represent the geometric constraints of the problem are formulated.They are used for the establishment of an initial set of candidate position tolerances.The next step (c) qualifies suggested sets out of the group (b) that have to be in conformance with the measured data of the particular RE-feature.The step (d) of the analysis produces then a set of preferred position tolerances by filtering out the output of step (c) by means of knowledge-based rules and/or guidelines.The capabilities and expertise of the particular machine shop, where the new components will be produced, and the cost-tolerance relationship, are taken into consideration in the last step (e) of the analysis, where the required position tolerance is finally obtained.For every datum feature that can be considered for the position tolerance assignment of an RE-feature, the input for the analysis consists of (i) the measured form deviation of the datum feature (e.g.flatness), (ii) its measured size, in case that the datum is a feature of size (e.g.diameter of a hole) and (iii) the orientation deviation (e.g.perpendicularity) of the RE-feature axis of symmetry with respect to that datum.The orientation deviations of the latter with respect to the two other d a t u m s o f t h e s a m e D R F h a v e a l s o t o b e i n c l u d e d ( p e r p e n d i c u l a r i t y , p a r a l l e l i s m , angularity).Input data relevant with the RE-feature itself include its measured size (e.g.diameter) and coordinates, e.g.X, Y measured dimensions by a CMM, that locate its axis of symmetry.Uncertainty of the measured data should conform to the pursued accuracy level.
In that context the instrumentation used for the measured input data, e.g.ISO 10360-2 accuracy threshold for CMMs, is considered appropriate for the analysis only if its uncertainty is at six times less than the minimum assembly clearance.

Sets of candidate position tolerance sizes
The size of the total position tolerance zone is determined by the minimum clearance, minCL, of the (hole, screw-shaft) assembly.It ensures that mating features will assemble even at worst case scenario, i.e. when both parts are at MMC and located at the extreme ends of the position tolerance zone (ASME, 2009).The equations (16 -i) and (16-ii) apply for the fixed and floating fastener case respectively, For the fixed fasteners case, in industrial practice the total position tolerance T POS of equation (16-i) is distributed between shaft and hole according to the ease of manufacturing, production restrictions and other factors that influence the manufacturing cost of the mating parts.In conformance with that practice a set of 9 candidate sizes for the position tolerance of the RE-shaft, R CAN_s and/ or the RE-hole, R CAN_h , is created by the method with a (T POS /10) step, which includes the 50% -50% case, For the floating fasteners case the total position tolerance T POS of equation (16-ii) actually concerns only RE-features of the Hole type.Therefore, the R CAN_h set only contains the T POS element.The above tolerances attain, apparently, their maximum values when the RE feature own dimensional tolerance zone is added,

Sets of candidate DRFs and theoretical dimensions
To ensure proper RE-part interfacing and safeguard repeatability, datum features of the original part and those of the RE-part should, ideally, coincide.In order to observe this principle the original datum features and their order of precedence have to be determined.Initial recognition of datum features among the features of the RE-part is performed interactively following long established design criteria for locating or functional surfaces and the same, and taking into consideration the mating parts function.Out of all candidate recognized datums an initial set of candidate DRFs, D CAN_INIT , is produced by taking all combinations in couples and in triads between them.A valid DRF should conform with the constraints that have to do with the arrangement and the geometrical deviations of its datums.Only DRFs that arrest all degrees of freedom of the particular RE-feature and consequently have three or at least two datums are considered.DRF qualification for geometric feasibility is verified by reference to the list of the valid geometrical relationships between datums as given in (ASME, 1994).The geometric relationship for instance, for the usual case of three datum planes that construct a candidate DRF is in this way validated, i.e. the primary datum not to be parallel to the secondary and the plane used as tertiary datum not to be parallel to the line constructed by the intersection of the primary and secondary datum planes.Planar or axial datum features are only considered by the method as primary when the axis of the RE-feature is perpendicular in the first case or parallel, in the second one, to them.
The following analysis applies for both the hole and the shaft and is common for the fixed and floating fasteners case.Consequently, the indexes "h" or "s" are not used hereafter.It is here also noted that the index "i" only concerns the fixed fastener case.For the floating fastener case the index "i" has a constant value of 1.Let RF DF be the set of the measured form deviations of a candidate datum feature and RO the orientation deviations of the RE feature axis of symmetry with respect to that datum.Fitness of the members of the initial DRF set, D CAN_INIT , against the members of the R CAN set of candidate position tolerance sizes is confirmed regarding the primary datum through the following constraints, Mutual orientation deviations of the secondary and/or tertiary datums, RO DF , in a valid DRF should also conform with the position tolerance of equation ( 16), where k is a weight coefficient depending on the accuracy level of the case.A set of Candidate DRFs is thus created, D CAN (i) , that is addressed to each i member (i=1,…9) of the R CAN set.
Sets of Candidate Theoretical Dimensions, [(C CAN (ij) X, C CAN (ij) Y), i=1,2,…9, j=1,2,…,n], which locate the RE feature axis of symmetry with reference to every one of the n candidate DRF (i) j of the D CAN (i) set are generated at the next stage of the analysis.Measured, from all the available RE reference parts, axis location coordinates are at first integrated into sets, Sets of Candidate Theoretical Dimensions are then produced in successive steps starting from those calculated from the integral part of the difference between the minimum measured coordinates and the size of the position tolerance, T POSi , Following members of the C CAN (ij) X, C CAN (ij) Y sets are calculated by an incremental increase δ of the theoretical dimensions X where as upper limit is taken that of the maximum measured X (ij) M , Y (ij) M coordinates plus the position tolerance T POSi , with the populations p, q of the produced C CAN (ij) X and C CAN (ij) Y sets of candidate theoretical dimensions not necessarily equal.In the case study that is presented δ=0.05mm.Other δvalues can be used as well.

Sets of suggested position tolerances
Sets of Suggested DFRs that are produced in step (b), D SG (i) , are qualified as subgroups of the sets of Candidate DFRs, D CAN (i) , in accordance with their conformance with the measured location coordinates and the application or not of the Maximum or Least Material Conditions to the RE-feature size or to the RE-Datum size.In conjunction with equation ( 16), qualification criterion for the Suggested DFR's, DRF (i) j j=1,2,…, n, is, Figure 4 where, In case constraint ( 25) is not satisfied, a DRF (i) j can only be further considered, when Maximum or Least Material Conditions are applied to RE-feature size, Figure 4 k=1,2,…,q ; u=1,2,…,r Candidate Theoretical Dimensions that satisfy the constraints (28) but not the constraint (29) can apparently be further considered in conjunction with constraint (27) when Maximum or Least Material Conditions are used.In these cases they are respectively qualified by the conditions, e.g. for the case of RE-feature /Hole, Figure 5 When applicable, the case of MMC or LMC on a RE-Datum feature of size may be also investigated.For that purpose, the size tolerance of the datum, T S_DF , must be added on the right part of the relationships ( 27) and ( 28).In that context, the constraints ( 30) and ( 31), e.g. for the case of RE-feature /Hole -RE-Datum /Hole on MMC, are then formulated as, k=1,2,…,q ; u=1,2,…,r where d Mu_DF is the measured diameter of the datum on the u-th RE-part and MMC DF the MMC size of the RE-Datum.

Sets of preferred position tolerances
The next step of the analysis provides for three tolerance selection options and the implementation of manufacturing guidelines for datum designation in order the method to propose a limited number of Preferred Position Tolerance Sets out of the Suggested ones and hence lead the final decision to a rational end result.The first tolerance selection option is only applicable in the fixed fasteners case and focuses for a maximum tolerance size of a T POS /2.The total position tolerance T POS , whose distribution between the mating parts is unknown, will be unlikely to be exceeded in this way and therefore, even in the most unfavourable assembly conditions interference will not occur.The second selection option gives its preference to Position Tolerance Sets that are qualified regardless of the application of the Maximum or Least Material Conditions to the RE-feature size and/ or the RE-datum feature size i.e. through their conformance only with the constraint (29) and not the constraints ( 30) to (33).Moreover, guidelines for datums which are used in the above context are, (ASME 2009;Fischer, 2004): -A datum feature should be: (i) visible and easily accessible, (ii) large enough to permit location/ processing operations and (iii) geometrically accurate and offer repeatability to prevent tolerances from stacking up excessively -Physical surfaces are preferable datum features over derived and/ or virtual ones.-External datums are preferred over internal ones.-For the fixed fastener case, a preferred primary datum of the mating parts is their respective planar contact surface.-Theoretical dimensions and tolerances expressed in integers or with minimum possible decimal places are preferable.

Cost -effective RE-tolerance assignment
To assign cost optimal tolerances to the new RE-components, that have to be remanufactured, the Tolerance Element (TE) method is introduced.Accuracy cost constitutes a vital issue in production, as tight tolerances always impose additional effort and therefore higher manufacturing costs.Within the frame of further development of the CAD tools, emphasis is recently given on techniques that assign mechanical tolerances in terms not only of quality and functionality but also of minimum manufacturing cost.Cost-tolerance functions, however, are difficult to be adopted in the tolerance optimization process because their coefficients and exponents are case-driven, experimentally obtained, and they may well not be representative of the manufacturing environment where the production will take place.The TE method (Dimitrellou et al., 2007a;2007c, 2008) overcomes the mentioned inefficiencies as it automatically creates and makes use of appropriate cost-tolerance functions for the assembly chain members under consideration.The latter is accomplished through the introduction of the concept of Tolerance Elements (Dimitrellou et al., 2007b) that are geometric entities with attributes associated with the accuracy cost of the specific machining environment where the components will be manufactured.
The accuracy cost of a part dimension depends on the process and resources required for the production of this dimension within its tolerance limits.Given the workpiece material and the tolerances, the part geometrical characteristics such as shape, size, internal surfaces, feature details and/or position are taken into consideration for planning the machining operations, programming the machine tools, providing for fixtures, etc.These characteristics have thus a direct impact on the machining cost of the required accuracy and determine, indirectly, its magnitude.A Tolerance Element (TE) is defined either as a 3D form feature of particular shape, size and tolerance, or as a 3D form feature of particular position and tolerance (Dimitrellou et al., 2007a).It incorporates attributes associated with the feature shape, size, position, details and the size ratio of the principal dimensions of the part to which it belongs.For a given manufacturing environment (machine tools, inspection equipment, supporting facilities, available expertise) to each TE belongs one directly related with this environment cost-tolerance function.
TEs are classified through a five-class hierarch system, Figure 6.Class level attributes are all machining process related, generic and straightforwardly identifiable in conformance with the existing industrial understanding.In first level, TEs are classified according to the basic geometry of the part to which they belong, i.e. rotational TEs and prismatic TEs.Rotational TEs belong to rotational parts manufactured mainly by turning and boring, while prismatic TEs belong to prismatic parts mainly manufactured by milling.The contribution of the geometrical configuration of the part to the accuracy cost of a TE, is taken into account in the second level through the size ratio of the principal dimensions of the part.In this way TEs are classified as short [L/D ≤3] and long [L/D >3] TEs, following a typical way of classification.In third level TEs are classified to external and internal ones as the achievement of internal tolerances usually results to higher accuracy costs.The fourth TE classification level distinguishes between plain and complex TEs depending on the absence or presence of additional feature details (grooves, wedges, ribs, threads etc).They do not change the principal TE geometry but they indirectly contribute to the increase of the accuracy cost.In the final fifth level, the involvement of the TE size to the accuracy cost is considered.TEs are classified, to the nominal size of the chain dimension, into six groups by integrating two sequential ISO 286-1 size ranges.Based on the TE-method the actual machining accuracy capabilities and the relative cost per TE-class of a particular machine shop are recorded through an appropriately developed Database Feedback Form (DFF).The latter includes the accuracy cost for all the 96 TE-classes in the size range 3-500 mm and tolerances range IT6-IT10.DFF is filled once, at the system commissioning stage, by the expert engineers of the machine shop where the assembly components will be manufactured and can then be updated each time changes occur in the shop machines, facilities and/or expertise.The DFF data is then processed by the system through the least-square approximation and the system constructs and stores a costtolerance relationship of the power function type, per TE-class.

Tolerance chain constrains
In a n-member dimensional chain the tolerances of the individual dimensions D 1 ,D 2 ,…,D n , control the variation of a critical end-dimension D 0 , according to the chain, where f(D) can be either a linear or nonlinear function.To ensure that the end-dimension will be kept within its specified tolerance zone, the worst-case constrain that provides for 100% interchangeability has to be satisfied, where t 0 is the tolerance of the end-dimension D 0 .For an (i+j)-member dimensional chain dimensions D i constitute the positive members of the chain while dimensions D j constitute its negative members.In RE tolerancing, preferred alternatives for nominal sizes and dimensional tolerances that are generated from the analysis of Section 3, for each dimension involved in the chain are further filtered out by taking into consideration the above tolerance chain constraints.

Minimum machining cost
A second sorting out is applied by taking into account the accuracy cost for each combination of alternatives that obtained in the previous stage.Cost-tolerance functions are provided by the machine shop DFF and the total accuracy cost is thus formulated as, where C(t) is the relative cost for the production of the machining tolerance ±t and A, B, k are constants.The combination of alternatives that corresponds to the minimum cost is finally selected as the optimum one.

Application examples and case studies
In order to illustrate the effectiveness of the proposed method three individual industrial case studies are presented in this section.All necessary input data measurements were performed by means of a direct computer controlled CMM (Mistral, Brown & Sharpe-DEA) with ISO 10360-2 max.permissible error 3.5 m and PC-DMIS measurement software.A Renishaw PH10M head with TP200 probe and a 10mm length tip with diameter of 2mm were used.The number and distribution of sampling points conformed with the recommendations of BS7172:1989, (Flack, 2001), (9 points for planes and 15 for cylinders).

Application example of RE geometrical tolerancing
A new optical sensor made necessary the redesign of an existing bracket that had to be fastened on the old system through the original group of 4 x M5 threaded bolts, Figure 10.The accuracy of the location and orientation of the sensor bracket was considered critical for the sensor proper function.Bracket redesign had to be based on two available old and not used reference components.In the following, the method application is focused only on the allocation of position tolerances for the four bracket mounting holes.The problem represents, apparently, a typical fixed fastener case.For the chapter economy the input data and the produced results for the allocation of the position tolerance for the through hole H1, Figure 11, are only here presented and discussed.Standard diameter of the holes is 5.3mm ± 0.1mm and minimum clearance between hole and M5 screw 0.2mm.The weight coefficient in the relationships (21) was taken k=1.  5.Although computational time difference is not significant, it is noticed that the quantity of the suggested results is strongly influenced by the number of the initially recognized possible datum features, weight coefficient k of the constraints (21), parameter δ of the equations ( 23) and the number of the available reference components.The machine shop where the part will be manufactured has an IT6 best capability and its DFF processed and the results stored.Alternatives for parts A, C and B, provided by the RE dimensional analysis of Section 3, are shown in Table 7(a) and (b) respectively.The 64 possible combinations of the part B alternatives are filtered out according to the tolerance chain constrains and, as a result, 24 combinations occur for the dimensions and tolerances D 5 ±t 5 , D 6 ±t 6 , D 7 ±t 7 as shown in Table 8.The optimum combination that corresponds to the minimum accuracy cost is the combination 64.

Conclusion
In industrial manufacturing, tolerance assignment is one of the key activities in the product creation process.However, tolerancing is much more difficult to be successfully handled in RE.In this case all or almost all of the original component design and manufacturing information is not available and the dimensional and geometric accuracy specifications for component reconstruction have to be re-established, one way or the other, practically from scratch.RE-tolerancing includes a wide range of frequently met industrial manufacturing problems and is a task that requires increased effort, cost and time, whereas the results, usually obtained by trial-and-error, may well be not the best.The proposed methodology offers a systematic solution for this problem in reasonable computing time and provides realistic and industry approved results.This research work further extends the published research on this area by focusing on type of tolerances that are widely used in industry and almost always present in reverse engineering applications.The approach, to the extent of the author's knowledge, is the first of the kind for this type of RE problems that can be directly implemented within a CAD environment.It can also be considered as a pilot for further research and development in the area of RE tolerancing.Future work is oriented towards the computational implementation of the methodology in 3D-CAD environment, the RE composite position tolerancing that concerns patterns of repetitive features, the methodology application on the whole range of GD&T types and the integration of function oriented wear simulation models in order to evaluate input data that come from RE parts that bear considerable amount of wear.

Fig. 7 .
Fig. 7. RE dimensional tolerancing case study parts and assembly

Fig. 10 .
Fig. 10.Reference RE part and mating parts assembly The CAD model of the redesigned new bracket was created in Solidworks taking into account the measured data of the reference components and, as well as, the new sensor mounting requirements.Datum and feature (hole H1) related input data are given in Table 4. Four candidate datum features A, B, C and D were considered, Figure 11.In step (a) 10 candidate DRFs (| | |C|, | |C|B|, | | |D|, | |D|B|, | |C|D|, | |D|C|, |D|A|C|, |D|A|B|, |D|B|A|, |D|C|A|) were produced by the algorithm for 7 position tolerance sizes, 0.06-0.18mmand consequently 10 sets of candidate theoretical dimensions, [(C P (ij) X, C P (ij) Y), i=3,…9, j=1,…,10].Negligible form and orientation deviations of datums A and D reduced the DRFs to the first six of them as |D|A|C|≡| |D|C|≡|D|C|A| and |D|A|B|≡| |D|B|≡|D|B|A|, having thus provided for 751 suggested tolerances in the following step (b).A representative sample of these tolerances is shown in Table5.Although computational time difference is not significant, it is noticed that the quantity of the suggested results is strongly influenced by the number of the initially recognized possible datum features, weight coefficient k of the constraints (21), parameter δ of the equations (23) and the number of the available reference components.

Fig. 11 .
Fig. 11.Case study datums (a) and the selected position tolerance of the case study (b)

3.2 Sets of candidate IT grades, fundamental deviations and nominal sizes The
|a| is the absolute value of the maximum ISO 286 clearance Fundamental Deviation (FD) for the relevant nominal sizes range (the latter is approximated by the mean value of Rd M_h , Rd M_s sets).If the above condition is not satisfied the analysis is exclusively directed on ISO 2768 general tolerances.Otherwise, the following two cases are distinguished, (i) Δ max ≤ IT 11 and (ii) IT 11< Δ max ≤ IT 18.In the first case the analysis aims only on ISO 286 fits, whereas in the second case, both ISO 286 and ISO 2768 RE tolerances are pursued.starting point for the Step (b) of the analysis is the production of the Candidate tolerance grades sets, IT CAN_h , IT CAN_s , for the hole and shaft features respectively.It is achieved by filtering the initial Candidate IT grades set, IT CAN_INIT , which includes all standardized IT grades from IT01 to IT18, by the following conditions (applied for both the h and s indexes),

IT CAN_h , IT CAN_s sets. Likewise, the set of initial Candidate Fundamental Deviations, FD CAN_INIT , that contains all the FDs applicable to clearance fits i.e. FD CAN_INIT
Other δ-values can be, obviously, used depending on the case.Since both hole and shaft have a common nominal size in ISO-286 fits, the Candidate Nominal Sizes Set, NS CAN , is then produced by the common members of

NS CAN_h , NS CAN_s , NS CAN = NS CAN_h ∩ NS CAN_s (9) 3.3 Sets of suggested fits In
Step (c) of the analysis, a combined qualification for the members of the IT CAN_h , IT CAN_s , FD CAN and NS CAN sets is performed in order to produce the two sets of suggested Basic Hole, BH SG , and Basic Shaft BS SG fits.The members of IT CAN_h and IT CAN_s sets are sorted in ascending order.For the production of the BH SG set, every candidate nominal size of the NS CAN set is initially validated against all members of the IT

Table 1 .
Issues of Geometrical Tolerance Assignment in RE . iv. Existence of repetitive features in the RE component that may have the same function (group or pattern of features).v.Type of assembly (e.g.floating or fixed fasteners).vi.The size and the form (cylindrical, circular, square, other) of the geometrical tolerance zone.vii.Candidate datums and datum reference frames.Depending on the case more possible DRFs may be considered.viii.Precedence of datum features in DRFs.www.intechopen.com

Table 2 .
Input data related to case study RE-features

Table 5 .
Representative sample of application example suggested position tolerances

Table 6 .
Set of preferred position tolerances for the application example

Table 7 .
Dimensional alternatives for parts A, C and B