A virtual inspection framework for precision manufacturing of aerofoil components

The finite element method plays an extremely important role in forging process design as it provides a valid means to quantify forging errors and thereby govern die shape modification to improve the dimensional accuracy of the component. However, this dependency on process simulation could raise significant problems and present a major drawback if the finite element simulation results were inaccurate. This paper presents a novel approach to assess the dimensional accuracy and shape quality of aeroengine blades formed from finite element hot-forging simulation. The proposed virtual inspection system uses conventional algorithms adopted by modern coordinate measurement processes as well as the latest free-form surface evaluation techniques to provide a robust framework for virtual forging error assessment. Established techniques for the physical registration of real components have been adapted to localise virtual models in relation to a nominal design model. Blades are then automatically analysed using a series of intelligent routines to generate measurement data and compute dimensional errors. The results of a comparison study indicate that the virtual inspection results and actual coordinate measurement data are highly comparable and the procedures for registration and virtual inspection are computationally efficient, validating the approach as an effective and accurate means to quantify forging error in a virtual environment. Consequently, this provides adequate justification for the implementation of the virtual inspection system in the virtual process design, modelling and validation of forged aeroengine blades in industry. Highlights? The paper presents a virtual inspection framework to assess the accuracy of aero-engine. ? Both the 3-2-1 approach and the ICP method are developed for part registration. ? The ICP method achieves a better solution than the 3-2-1 approach in registration. ? A case study produces reveals a good correlation exists between the virtual inspection data and the actual measurement data.


Introduction
As a result of the increasing demands to maximise the performance and quality of manufactured components within the aerospace industry, parts are inspected to ensure their features adhere to the geometrical and dimensional specifications. In particular, the inspection of complex parts comprising free-form geometry, such as forged aeroengine compressor blades, is becoming ever-more important due to requirements for higher precision and efficiency. The premise behind most inspection processes involves determining the extent to which a component deviates from a given set of specifications by comparing its actual shape to a nominal model. For quality assurance purposes, high precision dimensional measurement techniques are employed to evaluate the dimensional tolerance of forged aerofoil blades. Generally, these inspection processes may be categorised into two main groups, utilising either contact or non-contact measurement. The later acquires surface information without physically contacting the part using sensing devices such as laser/optical scanners, X-rays and CT scans [1]. The main limitation with this approach is that the measurement data may be affected by factors such as part colour, surface roughness, viewpoint and lighting [2]. Conversely, the contact inspection process of coordinate measurement is an effective measurement technique, providing both high accuracy and repeatability. The process employs a computer controlled coordinate measurement machine (CMM) to inspect the part automatically by moving a tactile probe along the workpiece surface, to measure the coordinates of individual contact points. In terms of forged blade inspection, CMM is by far the most commonly used tool owing to its ease of use, automation and measurement precision.
Compressor blades for aeroengine applications are normally manufactured using the closed die hotforging process. Forging of the work material at elevated temperatures creates distortion due to thermal contraction and spring-back, and shape errors due to the elastic deflection of the die and forging press. In industry, due to the complexity of the hot-forging process, forging process design is often dependent upon forging trials requiring prolonged lead times and increased costs. This iterative process involves modifying the die shape by a fraction of the measured forging error until the blade dimensions are within the specified tolerance and the aerofoil errors are sufficiently reduced. Therefore, dimensional inspection forms an integral part of this process by providing the necessary feedback to control the entire design and manufacturing process and achieve the desired results.
In blade forging design and manufacturing, finite element analysis has been widely used to simulate the material flow, stress/strain-rate distribution, thermal behaviour and forming load/energy requirements in blade forging [3,4]. By removing the need to conduct expensive forging trials, finite element simulation may also be used as a design tool to quantify forging errors and thereby govern die shape modification for improved dimensional accuracy [5]. However, in order to assess the accuracy of the forging errors on blade models generated from a forging simulation, it is necessary to verify the dimensional and shape accuracies via comparison to actual measurement data. Little research has been reported in this area.
The aim of this research is to develop a generic virtual inspection system to assess the dimensional accuracy of forged aerofoil blades in a virtual environment thereby allowing a fast, automated correlation between virtual forging design and actual forging production. Similar to the conventional procedure for measuring a physical blade, the inspection process for the virtual blade model comprises three main stages: part localisation, aerofoil section inspection and parameter analysis.
Both the classical 3-2-1 approach [2] and the iterative closest point (ICP) method [1] are implemented in part localisation. Aerofoil profile tolerances, aerofoil thickness and angular deviations from the nominal shape at three sections along the blade are evaluated by the aerofoil section inspection and parameter analysis modules. As a means of validation, a case study is presented to compare actual measurement data with the virtual inspection results. The results indicate that although the 3-2-1 approach is computationally efficient, the ICP method provides a much better solution to the part localisation of the blade model. A statistical analysis revealed a strong correlation between the virtual inspection results and the coordinate measurement data.
The remainder of this paper is organised as follows: Section 2 provides a comprehensive review of recent research in the area of dimensional inspection with specific emphasis on aeroengine blades. Section 3 describes the main aspects of the virtual inspection system. A more detailed description of the relevant theories and methodologies for part localisation and blade inspection is reported in section 4. An overview of the inspection systems software framework is provided in Section 5. The results of a case study detailing a comparison between virtual inspection data and actual measurement data are presented and discussed in Section 6. Concluding remarks are reported in section 7.

L iterature Review
A critique of the latest work and technological developments in the area of aeroengine blade inspection is provided in this section. Relevant topics such as localisation part registration, methods and procedures for measurement data acquisition, as well as tolerance evaluation algorithms and techniques for quantifying geometrical and dimensional discrepancies are discussed in detail.

Part Localisation T echniques
The process of part localisation, also known as registration, mathematically locates the part prior to inspection by determining a rigid body 3D coordinate transformation between the design coordinate system (DCS) and the measurement coordinate system (MCS). Traditionally, the design coordinate system is located using the six point principle or 3-2-1 approach [1,2,6]. For parts with regular features such as planar surfaces or cylindrical features, coordinate systems may be easily established.
However, as aerofoil blades are largely composed of freeform surfaces, it is difficult to locate enough planar surfaces to act as datum planes [7]. Consequently, various localisation techniques have been developed for free-form surface inspection.
Based on a concept of finding the closest point set between two free-from surfaces, the iterative closest point (ICP) approach may be used to establish a 3D transformation matrix by aligning the two surfaces through an iterative process [8]. Huang et al described an ICP approach which minimises the sum of the squared distances between the measured points and their closest points (corresponding points) on the nominal surface, also known a priori [9]. A transformation matrix generated by the approach comprises six parameters which define the position and orientation of the coordinate frame.
Using the pseudo-inverse method, the sum of the squared distances may be minimised iteratively between the respective point sets. Menq et al [10] proposed an optimal match algorithm to determine a rigid body transformation of one surface related to another, also based on least-squares minimisation. Ainsworth et al [11,12] presented a localisation approach which required an initial manual input to gain an approximate alignment of the part. Subsequently, an ICP algorithm was applied for more accurate registration. Lai et al [13] proposed an algorithm for the registration of irregular shapes using the coordinate measurement process. The part localisation process comprised of a rough and fine alignment procedure to match the part coordinate of the CMM with the model coordinate of the CAD model.
The main drawback of the aforementioned localisation techniques when applied to the registration of aerofoil blades is that they provide only an approximate solution to a set of measured data. The solution varies when the number or location of points varies and therefore does not guarantee that the design coordinate system can be regenerated for inspection [14]. Also, the ICP approach only ensures registration when the measurement surface and nominal model are close enough in both 3D orientation and position, necessitating initial manual alignment in some cases [1]. Conversely, the process of free-form surface localisation with reference to design datums locates the measured surface data with respect to the design model using known datum references instead of the free-form surface itself. The main advantage of this approach is that it does not require the specification of the closest points on a model, thereby providing a simpler yet more robust registration procedure which can be easily implemented [12,14].
Datum reference frames may also be defined by using elementary datum reference features such as planes and cylinders [14]. Coordinate systems may be established using the normal of a planar datum or the axis of a cylindrical surface. In terms of the measured part, datums are constructed by fitting the measurement points of a feature according to a least squares principle [1]. Li et al described an approach for localisation of sculptured surfaces with datums using the concept of Datum Direction Frame (DDF) [7]. The localisation process proposed by Hsu et al for aerofoil blade inspection used an iterative algorithm incorporating CMM measurement and a coordinate upgrading procedure [6,14].
However, the process is prone to error if local deformations are present [15]. For example, if an aberration in the surface occurs at a datum point, the coordinate system will be incorrectly aligned.

Blade Measurement and E valuation
In general, the conventional methodology for part validation of turbine blades using the coordinate measurement process involves evaluating the dimensional accuracy of the component along several aerofoil cross sections. Hsu et al [2] described a blade section inspection approach where three cross sectional profiles were measured at the base, mid and tip sections perpendicular to the stacking axis.
The CMM employed a contour measurement mode, whereby each blade profile was measured at a constant height (z coordinate). Cardew-Hall et al [15] proposed a similar process for section inspection, whereby planes perpendicular to the stacking axis cut the aerofoil to generate spline profiles. The precision inspection system developed by Pahk et al obtained scanning path measurement coordinates by using coordinate data from the CAD nominal model [16].
Once the blade has been successfully localised, dimensional errors will still exist as a result of curvature change, blade thickness and twist [2]. As the section curvature and twist becomes greater along the length of the aerofoil, the discrepancy between the measurement data and the nominal sectional data becomes more exaggerated. Consequently, additional alignment procedures are required to compensate for this residual misalignment error [16]. By minimising the residual misalignment, the sectional measurement data may then be transformed and a new set of corresponding points are generated on the nominal section. The iterative process of transforming the measurement data and calculating the curve corresponding points continues until the convergence of computation.
The geometric design parameters of an aerofoil blade may be categorised into three groups: blade orientation and displacement, blade dimensions and profile tolerance [2]. The blade dimensions that are commonly inspected for comparative analysis include the chord length, the length of the leading/trailing edge to the stacking axis and the aerofoil thickness at the leading edge, centre and trailing edge, respectively. Blade orientation and displacement indicate the deviation of the position and orientation of the overall blade. Blade orientation is defined in terms of the orientation angle.
Blade displacement relates to the deviation between the actual and basic stacking points, where the stacking points are the construction points about which each section is defined.
Generally, the profile tolerance is used to identify the form error of an individual region on the blade section, including the pressure and suction surfaces. A tolerance zone is usually defined as the space between the offset boundaries of the nominal profile and thereby sets a limit for the variation of the form error. Often, the profile tolerance is quoted as a single value for each section and may be defined as the sum of the maximum errors on both sides of the nominal curve. Pahk et al [16] proposed a rigorous approach for profile tolerance evaluation based on the Tschebyscheff norm between the measurement data and the corresponding closest points data.
Statistical based methods may be used as a means of tolerance verification, whereby the standard deviation of the manufactured surface reflects how far the measured surface deviates from the nominal model. According to Huang et al [17], the deviation of a manufactured surface may be separated into deterministic error, d and random error, components. As the deterministic error is virtually removed after localisation, the deviation of the surface is dominated by the random component, which obeys a normal distribution. As the actual deviation value is unknown, it can be estimated from the sample data. Several example cases to test the approach were presented by both Huang et al [17] and Li et al [7]. In each case, after initial localisation of the part to the DCS using the -form surface was out of tolerance or the standard deviation was above the acceptable limit. However, after performing further localisation using an optimal match algorithm, based on an ICP approach, the aforementioned values were successfully reduced to within the required tolerance range.

V irtual Inspection System
This section provides an overview of the virtual inspection system. The first stage of virtual inspection involves registering or localising the part in relation to a nominal model. The system provides a conventional registration procedure using non-marginal datum points and datum features to localise the part and an alternative localisation algorithm for registering free-form surfaces based on least squares minimisation. After localisation, measurement data is generated and various blade dimensions and geometrical parameters are evaluated in the second and third phases of the process.

Model Localisation
The two forms of localisation offered by the system are the 3-2-1 approach and ICP localisation. The traditional 3-2-1 approach may be employed to establish the blades coordinate system, as shown in Figure 1. However, not all of the datum points are defined by basic datum features. In particular, the primary datum plane is determined by three points on the free-form concave surface of the blade, including P 1 and P 2 on the root section and P 3 on the tip section. The secondary datum plane is constructed using the central axis of cylindrical features at each end of the blade, defined by P 4 and P 5 . Finally, the tertiary datum plane, orthogonal to both previous datum planes is determined using the last datum point, P 6 , located on the root block. The normal vectors of the aforementioned datum planes, shown in Figure 2, are defined by Eq. 1 -Eq. 3. Following industrial CMM practice, the section thickness, t is defined as the Euclidean distance between opposing K points on the concave (C C) and convex (CV) blade surfaces. Overall, section thickness is measured between three sets of K points located at the leading edge, middle and trailing edge positions for each section. Therefore for each measured value of section thickness, the thickness error, 1 may be subsequently obtained as the difference between the actual thickness measurement, ' t and the respective nominal thickness, t . The angular displacement at the mid and tip sections, 2 , also known as twist error, is quantified as the angular variation from the nominal when measured at K 1 and K 5 . Finally, the vertical displacement of the mid section, otherwise known as bow error, 3 , is defined as the vertical deviation from the nominal when measured at K 3 . The bow error may be calculated by the difference between y coordinate of K 3 on the nominal and measured profiles, respectively. Table 1 describes how the geometric parameters are assessed.
The total deviation between the measured profile and the nominal curve, known as form or profile error is evaluated using a least squares based approach described in In the example shown in Figure 6, this condition is satisfied as the error does not fall outside the offset zone.

T heoretical Formulation and M ethodology
A mathematical description of the methods and computational procedures for part localisation, measurement data acquisition and blade parameters analysis are reported in detail in this section.

3-2-1 Registration A lgorithm
To initiate localisation, datum points were firstly identified on the blade formed from the finite element forging simulation. As the initial position of the blade was in close proximity to the nominal coordinate system due to constraints applied in the FE simulation, no preliminary transformation process was required. Consider Subsequently, after gaining an initial estimate of the three primary datum points, using the rules of orthogonality and sequence for datum frame construction [6], the secondary datum, perpendicular to the primary datum, is established next using the cylindrical datum reference features at the tip pip and root pip of the forging, Figure 8(a). Thus, in both the experimental CMM measurements and in the virtual inspection procedure outlined here, the known cylindrical form of the root and tip pips is used to localise and orient the aerofoil surface.
It should be noted that the axes of these cylindrical reference features are not designed to be co-linear.
However, by locating a point on the axis of each cylindrical feature, it is possible to define a line (or direction vector) on the secondary datum plane. The normal of the secondary datum plane can be determined according to Eq. 2. Each secondary datum point is determined by fitting the nodal coordinates on the feature surface according to a least squares principle.
As shown in Figure 8(b) for a cylinder defined by a point on its axis , a and radius r , an initial estimate of these parameters may be obtained by minimising the distance function between the cylinder to m points where: where i r is the distance of the i th point to the cylinder axis and with a . By rotating and translating the data at the start of each iteration so that the trial best fit cylinder had a vertical axis passing through the origin [19], it was possible to The vector of corrections to the cylinder parameters P is be given by Eq. 14 Thus, given initial estimates of the axis point, axis direction and radius, a Gauss-Newton strategy was implemented to find the solution for best fit of the tip and root pip cylinders. The detailed iteration procedure is given in [19] with the following main steps: (i) translate the point However, in order to fit the nodal coordinates, it is necessary to identify the nodes that are situated on the surface of the cylindrical pips. Thus, a shape recognition algorithm which uses the least squares cylinder approach is employed for this purpose. To reduce processing time and simplify the search for nodes which belong to the candidate shape a 2D boundary is defined in the approximate location of the tip pip and subsequently, a second boundary is defined for the root pip. For each exposed element face within the boundary at least ten of the closest nodes are located and fitted to the candidate cylinder. If the least squares algorithm does not converge or the calculated radius exceeds the nominal radius tolerance the node set is not included in the candidate point cloud. This process continues until all exposed elements within each boundary have been evaluated. After identifying the respective point clouds each dataset is submitted to the least squares calculation which yields the coordinates of the points ' 4 P and ' 5 P on the respective cylinder axes. An example of the cylindrical point clouds generated from the searching algorithm is shown in Figure 8

Iterative C losest Point (I C P) Localisation
The most important and computationally demanding step of an ICP algorithm involves finding the corresponding points on the reference surface. The previously mentioned ICP registration methods [2, 10] used a parametric surface representation of the design model to locate the closest points. This study uses point sets to represent the geometric data of the respective measured and nominal models, so a different technique is required to determine the corresponding points. One option is to conduct a linear search for the closest point in the nominal model to each measured point based on Euclidian distance, but for large point sets this approach is impractical as it leads to excessive computing times.
The computational efficiency may be improved by as much two orders of magnitude by building a kdimensional binary search tree or k-d tree from the data points in the nominal model and querying the tree for each point in the measured model [20]. A nearest neighbour algorithm is used to find the closest point to a given target point on the tree. At each stage of the search the algorithm makes an approximation of the nearest distance and subsequently terminates when the possibility of more than one nearest neighbour no longer exists [21]. Consequently, large portions of the search space can be avoided using this method.

L east Squares M inimisation
There are two general methods for least squares minimisation in 3D registration problems, i.e., quaternion based and singular value decomposition (SVD) based methods. Horn et al presented a quaternion based approach, whereby rotations were represented as quaternions to simplify problems due to orthogonal rotation matrices [22]. Alternatively, the SVD approach is computationally very efficient and easily generalised to 3D problems. Consequently, this study employs a SVD-based solution method similar to that proposed by Arun et al [23]. The transformation matrix generated from the least squares minimisation of the respective point sets will be represented by an orthogonal rotation matrix and a translation vector. Thus, for known correspondences between a measured point set,

Eq. 21
Applying these constraints, Eq. 19 may be represented as where i , i=1-6 represent Lagrangian multipliers used to enforce the orthogonality constraints and Thus, on substitution of Eq. 26 into Eq. 22, F may be defined as  Thus, once R is known, T is determined using Eq. 25.

T he I C P A lgorithm
The main limitation with the ICP method is that the presence of local minima within the parameter space explored by the algorithm may lead to non-global convergence. However, this scenario can be avoided by ensuring that the respective surfaces are located in close proximity before applying ICP, thereby providing a greater chance of convergence to the global minimum. In any case, for the purposes of this study, due to the constraints applied in the FE simulation, the respective models are sufficiently close to implement the ICP algorithm. The ICP algorithm implemented in this study consists of the following steps: (iii) Apply the transformation to the measured point set.
(iv) If the change in residual error from Eq. 25 is greater than a threshold value, go back to step (i); otherwise stop.
Convergence to a minimum is found by comparing the change in residual error between iterations with a pre-specified threshold value. Local minima are detected if a significant number of residuals are above the threshold. In this case, the ICP procedure is set to terminate when the difference in residual error is less then 1e -5 .

Blade Profile Positioning for Profile E r ror E valuation
After localisation, there remains a total deviation between the measured profile and the nominal curve. Therefore, this necessitates an algorithm to align the respective data sets in order to gain an accurate appraisal of the profile error. The profile error evaluation algorithm employed by the virtual inspection system aligns the concave and convex profiles of each section with their counterparts on the nominal at predefined locations specified by the K points close to the leading (K 1 and K 2 ) and trailing (K 5 and K 6 ) edges of the nominal model. This repositioning procedure is implemented by performing a translation, T and rotation, R in sequence. Firstly, each measured profile is translated vertically until it coincides with the relevant nominal profile at K 1 or K 2 . The second stage of the transformation involves alignment with the remaining K point at K 5 or K 6 . This is achieved by rotation about K 1 or K 2 . However, in order to find the required angle of rotation, it is necessary to use a searching algorithm to locate a direction vector ' ' 1 prof K K on the measured profile with a magnitude equal to Primarily, the input parameter file consists of nominal coordinate data for the K points and the six datum points that define design coordinate frame. Other parameters specified include tolerance values for the convergence criteria used by the various iterative algorithms employed by the system. Also, file paths are designated, defining the location of the respective input blade models and an appropriate folder to store the results of the parameters analysis module. Figure 9 shows the framework of the entire virtual inspection system.

Results and Discussion
To illustrate the effectiveness of the proposed methodology, a test case of a forged compressor blade is presented in this section. All relative geometric and dimensional tolerances of the finite element model are evaluated by the virtual inspection system and a comparison is made with inspection data from the actual component.

F inite E lement Simulation
A finite element forging simulation of an industrial case Ni-alloy blade was performed. The constitutive models defining the deformation behaviour for the billet were defined as rigid-plastic during forging and elastic-plastic during the unloading and cooling stages, whereas elastic deformation was defined for the forging dies throughout the entire simulation. Figure 10 displays the meshed 3D models of the components used in the forging simulation.
The initial temperatures specified for the billet and the dies were 1000 C and 230 C, respectively.
Other parameters defined in the pre-processing environment include the heat transfer coefficient at the interface between the workpiece and dies, h f which was set at 11 kW/m 2o C and the coefficient of friction, between the workpiece and die surface which was assumed to be 0.2 [5,24]. The forging simulation comprised of four stages including forging, unloading, cooling and trimming. Figure 11 shows the FE model including both the dies and workpiece before and after forging operations.

Profile E r ror E valuation
To assess the accuracy of the blades formed from the finite element method, a comparison is made between the actual profile error recorded by the coordinate measurement process and that generated by the virtual inspection system. A one-way analysis of variance (ANOVA) test is used to test for differences between the CMM and virtual profile error data sets. The .stl file of the designed aerofoil blade, also used as the geometry for generating the FE meshes of the forging dies, are used as blade nominal shape, whilst the nodal positions of the FE mesh from the final forged aerofoil shape excluding the flash areas are used for finding the datum features in the 3-2-1 registration method and for generating the point cloud using the ICP method. Table 2 Figure 19. This occurrence may be attributed to the inaccuracy of the FE simulation in the transition area close to the root block due to a relatively coarse mesh.

Comparison of Dimensional and G eometric Tolerances
The dimensional and geometric tolerance measurements performed by the comparative analysis module of the virtual inspection system, as well as a comparison with the corresponding CMM data are given in Tables 3-5.
In terms of the thickness data, it is evident from Table 3 that a strong correlation exists between the virtual and CMM measurements as similar values of section thickness and standard deviation are recorded. Also, relatively small discrepancies between the inspection processes are apparent for bow error measurements. Moreover, as shown in Table 4, a similar trend in the magnitude of the bow error is apparent, as in each case the maximum bow occurs at the root section. However, in terms of twist error as shown Table 5, the average value recorded from the final model is roughly three times the magnitude of that recorded by the CMM. Moreover, the CMM data indicates a change in twist angle direction between the mid and tip section, whereas the twist angle remains in the same direction for the forged finite element model. This occurrence may be related to inaccuracies in post-forging simulation.

E valuation of Localisation M ethods
The Conversely, as shown in Figure 27, the ICP localised surface displays a more uniform deviation with values of similar magnitude recorded for each surface. The distribution of the data, as shown in Figure   28, is relatively symmetric. Also, the majority of data points on the normality plot shown in Figure 29 form a linear pattern, indicating that the normal distribution is a good model for this data set.  Table 6. Although better registration accuracy may be achieved using the ICP method in virtual inspection, quantitative evaluation of the effect on the dimensional and shape accuracy of the forged blade has yet to be established, which is one area for further study. In terms of the computational efficiency of the virtual inspection, less than 100 second computing time is required for the localisation computation using a normal desktop computer. As the registration algorithm was the most computationally expensive aspect of the approach, the whole virtual inspection procedure is compared more favourably to the actual CMM measurement process, which normally takes up to 5 minutes to complete the 3-2-1 registration of the actual part.
As any deviation in the shape or form of either cylindrical datum feature would significantly affect the location of the datum reference frame, it is necessary to gain an accurate appraisal of the extent to which each cylindrical surface deviates from the desired form, implied by the nominal. An indication of the cylindrical deviation may be achieved by measuring the cylindricity of each datum feature.
According to the ANSI Y14.5.1M standard [25], cylindricity is categorised as a form tolerance and is condition of a surface of revolution in which all points of the surface are equidistant A cylindricity tolerance specifies that all points of the surface must lie in some zone bounded by two coaxial cylinders whose radii differ by the specified tolerance. The cylindricity tolerance is defined according to 2 t r A P T

Eq. 43
where T is the direction vector of the cylindricity axis, A is a position vector locating the cylindricity axis, P is the position vector of a point on the surface of the datum feature, r is the radial distance from the cylindricity axis to the centre of the tolerance zone and t is the size of the cylindricity zone.
The cylindricity for both the root and tip cylindrical datum features was evaluated for each feature using the nodal positions of the finite element mesh over each features surface. The mean cylindricity calculated for the root and tip datum features was 0.10 mm and 0.04 mm respectively with a standard deviation of 0.04 mm and 0.03 mm. By using a t value similar to that of the profile error tolerance of 0.2mm, the cylindricity value for both pips meets the specified tolerance, as defined in Eq. 43.

Conclusions
In terms of the localisation process, results indicate that the classical 3-2-1 approach commonly used by the CMM in industry may not be the best approach in the application to the free-form aerofoils surface. Alternatively, the ICP approach provides a much better solution to the registration problem by considerably reducing the deviation from the nominal and generating a more uniformly localised part. Overall, the magnitude and form of the profile tolerances assessed by the system display a strong correlation with that evaluated by the CMM. Virtual inspection data obtained from FE simulation recorded for the section thickness was highly consistent with the corresponding CMM data, with a discrepancy of 6 Similarly, the results generated for the bow error were also highly comparable with the actual inspection data, as the r.m.s value generated by the system differs by only 6 relatively large deviation was apparent between the twist error values for both the mid and tip sections. This unusually large deviation in twist is likely to be attributed to the inaccuracy of the postforging simulation, in particular, the simulation of the cooling process. However, by conducting a sensitivity analysis of the effect of pre-processing conditions such as friction and mesh density on the dimensional accuracy of the component, it may be possible to obtain inspection results which are even more consistent with the CMM data.
In any case, the virtual inspection system provides a fully automated, robust procedure for the dimensional inspection of forged aerofoil blade models formed using the finite element method. The localisation process used by the system accurately registers the component with the nominal model.
The inspection and parameters analysis modules incorporate various iterative algorithms and the latest evaluation techniques to successfully quantify the forging error to within a high degree of accuracy.
The strong correlation between measurements generated from this system and actual CMM measurement data, and the normal distribution of shape deviation from the nominal model, validates the approach as an effective means to quantify the forging error using the coordinate measurement process in a virtual environment.     Table 1 Description of dimensional errors   Table 2 Profile error comparison   Table 3 Profile error comparison   Table 4 Bow error comparison Table 5 Twist error comparison Table 6 Deviation comparison between 3-2-1 and ICP localisation procedures