WO2018046948A1

WO2018046948A1 - Three-dimensional shape error detection

Info

Publication number: WO2018046948A1
Application number: PCT/GB2017/052639
Authority: WO
Inventors: Abhishek Das; Pasquale FRANCIOSA; Darek CEGLAREK
Original assignee: The University Of Warwick
Priority date: 2016-09-09
Filing date: 2017-09-08
Publication date: 2018-03-15
Also published as: GB201615376D0

Abstract

There is disclosed a method and system for identifying product form errors for use in the control of product manufacture. The method involves scanning multiple products produced at a common stage of a manufacturing process so as to output a scanned profile for each product (18) and comparing each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile. Form errors are grouped into a plurality of discrete form error modes and each form error mode is parameterized such that each form error mode can be characterised by one or more parameter value. Form error modes that are common to a plurality of said multiple products are identified and a statistical model for each common form error mode is output. A manufacturing process may accommodate discrete form error modes applied to the nominal product profile.. A product inspection system using the method is disclosed.

Description

Three-Dimensional Shape Error Detection

This disclosure concerns systems and methods for identifying shape errors in

manufactured products so as to be able to account for such errors in subsequent product handling, processing and/or assembly steps.

Within the field of product manufacture, it is widely recognised that products are produced within certain tolerance specifications, with more stringent tolerance requirements generally being placed upon precision products. However for assemblies, shape errors incurred in the manufacture of different components can be compounded so as to cause prominent product defects.

Furthermore, certain manufacturing processes require highly accurate understanding of the shape of a product in order to perform the relevant operation. One example of such a process is the emerging technology of remote laser welding, which is highly sensitive to part-to-part fit up. That is to say, if the components to be welded are not held in the correct proximity, i.e. if the gap between the components is too great, the remote laser welding process may produce an ineffective joint. Shape error It is generally known to work from a nominal Computer Aided Design (CAD) model of a product in assessing the shape of a manufactured product. Some existing CAD packages offer tools to model shape variation in a deterministic manner so as to analyse tolerances associated with a product and generate a mathematical/geometric definition for one or more tolerance zone.

Aside from an entirely theoretical approach, it has also been proposed in the art to measure the shape of a non-ideal product produced by the manufacturing method and to compare the measured shape to the nominal CAD model. Decomposition of the shape may allow shape errors to be identified and quantified for a single part.

One example of a previously attempted technique to model shape error comprises a Constructive Solid Geometry-based (CSG) approach in which form errors are embedded with the nominal geometry using control points. With relatively few control points available, principally simple geometry can be controlled with a few form error types. However such techniques are not suited to complex three-dimensional geometric profiles because the complexity causes a significant increase in form errors and the limited number control points and error types are insufficient to represent accurately the form errors that can arise.

Other examples of techniques for representation of form errors are available in the prior art but those techniques are either limited to modeling a specific type of surface (e.g. planar surfaces) or else fail to adequately account for the random nature of errors found in highly three-dimensional surfaces.

However whilst previous techniques provide a potential solution for individual, or a small number, of products and the associated shape errors, there exist problems in trying to apply such techniques to real production lines in which large volumes of products are produced, particularly for products having complex geometries.

For example, the accurate scanning of products creates a significant volume of data which cannot effectively be handled in real time. Therefore the current applications for such scanning processes are limited to reverse engineering and inspection at single part level rather than process control. Furthermore, existing modelling techniques can assess individual error types well but are not suited to handling the random nature of errors that can occur over the shape of a complex three-dimensional part. The application of existing techniques to cover all such pseudo-random shape errors would vastly increase the computational burden and complexity to the point that such techniques are not practical to implement.

It is an aim of the invention to provide a shape error identification system which mitigates one or more of the above, or other, problems associated with the prior art.

It may be considered an aim of the present invention to provide a shape error identification and/or classification technique that is better suited to batch or volume production of three- dimensional and/or complex product shapes. It may be considered an additional or alternative aim of the invention to provide a shape error identification technique that can allow improved automated control of a subsequent product processing or assembly step.

According to a first aspect of the present invention there is provided a method of identifying product form errors for use in the control of product manufacture, the method comprising scanning multiple products produced at a common stage of a manufacturing process so as to output a scanned profile for each product, comparing each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile, grouping form errors into a plurality of discrete form error modes and parameterizing each form error mode such that each form error mode can be characterised by one or more parameter value, identifying form error modes that are common to a plurality of said multiple products and outputting a statistical model for each common form error mode.

The product may be a three-dimensional product, such as for example profiled sheet material products and/or machined products. The statistical model for each form error mode, or at least one or more thereof, may be advantageously used in the control of one or manufacturing process in which the product is made/used. Accordingly, in a second aspect of the invention, there is provided a method of controlling the manufacture of goods comprising said products, in which the method of the first aspect is used.

According to a third aspect of the invention, there is provided a product form error inspection system comprising a product scanner arranged to scan multiple products produced at a common stage of a manufacturing process so as to output a scanned surface profile for each product, a surface comparator arranged to receive the scanned surface profiles and to compare each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile, and an error processor arranged to group the form errors output by the comparator into a plurality of discrete form error modes and parameterize each form error mode such that each form error mode can be characterised by one or more parameter value, wherein the error processor comprises a statistical model generator arranged to identify and log form error modes that are common to a plurality of said multiple products and output a statistical model for each common form error mode.

According to a further aspect of the invention, there is provided a data carrier comprising machine readable instructions for the operation of one or more processor to: access a scanned profile for each of multiple products produced at a common stage of a manufacturing process; compare each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile; group form errors into a plurality of discrete form error modes; parameterize each form error mode such that each form error mode can be characterised by one or more parameter value; identifying form error modes that are common to a plurality of said multiple products; and output a statistical model for each common form error mode.

The plurality of products may comprise a batch or sample of the products, for example comprising in excess of 10, 50 or 100 products. Scanning such a batch of products may be used to infer the statistical model for error modes found therein so as to allow said statistical models to be used to predict form errors associated with ongoing manufacture of the same type of product by way of the same process. The statistical model may be based upon the one or more parameter value used to characterise each error mode. The statistical model for a form error mode may be based upon the frequency of occurrence of said form error mode in the scanned products and/or magnitude of one or more form error determined within said form error mode (e.g. the maximum deviation from the nominal profile).

The invention may comprise discretisation of the nominal profile and/or the scanned profiles. The invention may comprise applying a mesh to the nominal profile and/or the scanned profiles. The invention may comprise defining a plurality of voxels over the nominal profile and/or scanned profiles. The invention may comprise a discretisation module or tool.

The invention may comprise determination and/or logging of a form error for a plurality of discrete portions, e.g. voxels, of the scanned profile. A discrete form error smoothing technique may be used, e.g. to infer a gradual change in error between adjacent discrete portions in the event that the discretisation process cause step changes in form error. A Laplace smoothing technique may be applied.

The discretisation and/or grouping of form errors advantageously allow decomposition of the nominal profile and/or shape error data, e.g. scanned point data, so as to be able to be able to quantify the form error modes.

A mathematical transform of shape errors and/or points on the scanned profile may be applied. A sinusoid transform, e.g. a Fourier or Fourier-related transform, may be used. A Discrete Cosine Transform may be used. A mathematical transform may be used to derive one or more defining parameter for each, or a group, of form errors. An amplitude and/or frequency parameter may be used. Thus the form errors or groups thereof may be parameterised in determining/identifying the form error modes, e.g. in addition to the parameterisation of the form error modes once identified.

A form error mode prioritisation and/or selection process may be used. A subset of all the identified form error modes may be selected. One or more form error truncation criterion may or may not be implemented. An energy compaction assessment may be used to identify and/or select a subset of form error modes from the identified form error modes.

One or more form error correction criterion/process may be used, for example on a selected subset of the identified form error modes. A form error magnitude/amplitude correction criterion may be applied.

A form error correlation may be performed between the discretised form errors and form errors determined from the scanned profile, e.g. prior to discretisation. Form error modes may be selected from the identified form error modes based on said correlation, e.g. with the selection criterion comprising a threshold correlation value or ratio.

A weighting may be applied to each, or each selected, form error mode. A least squares approach may be used.

The surface error profile may be reconstructed from the discretised surface error profile using the identified/selected form error modes. The reconstructed surface error profile may be compared with a surface error profile determined from the scanned profile data. An error mode parameter set may be determined. The error mode parameter set may comprise an error mode vector/matrix, which may be used in statistical modelling.

Statistical model generation for each form error mode may comprise fitting a statistical distribution, e.g. a predetermined statistical distribution, to the occurrence of the form error mode within scanned profile of the multiple products. Statistical model distribution generation may comprise probability density estimation, e.g. to determine/characterise a suitable statistical distribution to be applied. Probability density function estimation may be performed using a data-driven and/or non-parametric approach, e.g. by fitting an empirical distribution to a form error data set. Kernel Density Estimation may be used. A bandwidth or smoothing parameter may be determined. One or more composite product profile may be determined, comprising each of the selected/identified form error modes. The one or more composite product profile may comprise a virtual/model profile used to amass the form error modes. The composite product profile may be advantageous in determining suitable control of a manufacturing process for the product. A clustering process may be used to define clusters of form error modes to be included in a single composite product profile, e.g. in the event that a plurality of composite product profiles are generated to encompass all the different form error modes identified. Statistical model generation for the form error modes may allow generation of a plurality or batch of variational product profile models. A selection of one or more form error mode and/or an amplitude of said one or more form error mode may be made for each variational product profile model. The selection of form error mode or amplitude thereof may differ between each variational product profile model generated.

The invention may allow use of the scanned profile data for statistical manufacturing process control. Existing control charts are not capable to cope with the high volume of captured surface data and/or to transform the captured data into useful and interpretable information for automated process control. The invention thus allows advantageous characterisation and use of the shape errors associated with a batch of parts/products.

The batch form error model processing associated with the present invention allows simulation of shape errors associated with batch of products, which may be used to identify the risk of shape errors causing a defect in a manufacturing process for the product or goods/assemblies containing the product. This may allow modification of a manufacturing process to accommodate such risks, for example by modifying the manner in which a product is held, e.g. in a fixture, for processing, or a processing parameter itself, such as a joining/assembly/machining process parameter. In some examples, the characterisation of form error modes may allow simplified inspection of further products produced by the manufacturing method such that common form error defects can be easily identified and quantified without requiring a fully detailed product scan. Subsequent handling/processing of the product may thus be modified automatically to accommodate the determined form error modes and/or amplitude thereof. In other examples of the invention, the form error modes, particularly the statistical models thereof, may be fed back into the product design such that modifications to the design can be implemented to avoid problematic form error modes during manufacture. Practicable embodiments of the invention are described in further detail below with reference to the accompanying drawings, of which:

Fig. 1 shows a high level overview of the process implemented by an example of the invention;

Fig. 2 shows an example of form error determination for a cloud of measured surface data points from a scanned surface;

Fig. 3 shows a flow diagram of an example of a process for geometric modal analysis of form errors in products used by an example of the invention;

Fig. 4 shows an example of computational product surface decomposition for scanned and nominal surfaces; Figs 5a- 5d show examples of product models for a common product surface taken through different stages of a form error identification process according to an example of the invention;

Fig. 6 shows examples of different form error modes identified for products manufactured according to the nominal product model associated with the product of Fig. 5;

Fig.7 shows a flow diagram of a statistical modelling process for form error modes according to an example of the invention; and Fig. 8 shows a chart of modelled statistical variations for a number of identified form error modes.

The invention disclosed herein is concerned with the generation of a system by which shape errors of a manufactured surface profile can be assessed and numerically characterised in manner that allows the output of the process to be applied to the ongoing manufacture of corresponding products. The invention is concerned particularly with the manufacture of products by forming a profiled surface in the product. The description below proceeds in relation to sheet products, for example sheet metal products, which may be formed by pressing of the sheet material to form the desired surface profile. However it will be appreciated that there are a significant number of products and associated manufacturing techniques to which the invention could be applied and in which a profiled surface is produced by a manufacturing process in order to achieve a desired/nominal surface profile. Such examples could comprise other methods of deforming thin-walled/sheet material (e.g. including composite product layups or pressing/stamping of other materials) or other profile forming methods, including material removal/machining, deposition or moulding processes. The invention is particularly well suited to processes in which accurate understanding of the formed surface profile is needed in order to guide subsequent manufacturing processes, such as joining/assembly with an adjacent product/component or subsequent surface profiling processes. Accordingly, certain aspects of the invention may encompass a manufacturing method or a method of manufacture setup comprising the invention.

However the invention may find application in other areas. Other aspects of the invention may comprise such methods and/or systems for inspection of products. In examples of the invention, accurate understanding of the surface profile may be used accept or reject products, or inspection may be performed for used products in which the deviation from a nominal profile may be used to assess the degradation of the product in use.

An overview of the process 10 described hereinbelow is shown in Fig. 1 . The process involves the creation of a Computer Aided Design (CAD) model 12 for the product being manufactured. This is typically achieved using conventional CAD modelling tools, which will not be described herein for simplicity since such tools will be well known to the person skilled in the art. Such tools allow creation of a mathematical/geometric definition of a three- dimensional surface of a product, herein referred to as a CAD model or nominal profile 14.

In order to determine shape errors between the nominal model 14 and the manufactured products, the real, manufactured products are measured by a surface measuring process 16 in order to generate a measured surface profile 18. The measured surface profile 18 is compared to the nominal profile 14 in order to determine the form errors in the manufactured product surface. In order to facilitate use of the invention for complex/three-dimensional surface profiles, free form shape errors are extracted from measured part data, e.g. to simulate geometric tolerance requirements.

Surface measuring/scanning is achieved using conventional 3D non-contact metrology sensors that sense reflections off the surface so as to produce point data for points identified at the surface. The points may be defined as locations in a suitable coordinate system, e.g. a 3D Cartesian coordinate system, which can be correlated/aligned with the nominal model. Suitable surface measuring equipment may comprise 3D laser scanners or 3D white-light scanners.

The stage 16 of measuring the product profile is repeated for a batch of products intended to be representative of the range of product forms producible by the manufacturing process. Accordingly the selection of the batch size is a careful consideration and may vary depending on the production line under consideration but will typically include tens or hundreds of products. The larger the batch size, the greater the certainty in the form error results but the larger the computational effort. In certain examples of the invention, an initial batch size may be selected for generation of initial form error models but ongoing product samples may be scanned, e.g. as part of an in-line process, so as to check product profiles against the existing form error models and/or update the models in an ongoing manner.

The amassed form errors for the multiple products are processed at statistical modelling stage 20 in which identified modal analysis, or 'decomposition', of the form errors is undertaken so as to identify common modes of form error creation. Each mode of form error creation may thus represent a 'family' of errors according to a parametric definition such that common types of form error can easily be identified and categorised by reference to its corresponding error mode. By categorising errors in this manner, statistical models for each form error mode can be produced. Such statistical models are particularly useful since they allow quantification of error types that can be used to control manufacturing process parameters.

A process for implementing the general example of Fig. 1 is shown in Fig. 3. In generating the process of Fig. 3, two hypotheses are introduced to simplify the modelling process in a practical/computational sense:

(i) Smoothness assumption: shape error field signal has sufficient smoothness such that the high spatial frequency components (short wavelength error such as surface roughness and waviness) are small and can be ignored. This assumption implies that shape error is highly spatially correlated.

(ii) Normal deviation assumption: the shape errors of a real part surface can be represented as a normal deviation function f(x,y,z) defined in 3D domain. Normal deviation calculation has limitation around the curved features.

Fig. 2 shows an example of normal deviation calculations from nominal profile features for a cloud of points (CoP) from scanned surface data. The form error field is defined as the differences between the actual measured surfaces 22 and nominal surfaces 24. The measured surface 22 may be determined where the CoP is generally aligned to form a coherent surface, e.g. discarding potentially erroneous outlying points. Two example points 26 and 28 are shown, for which the actual direction of deviation from the nominal surface 24 is given by dashed line 30, whereas the normal direction of deviation relative to the surface is shown at 32.

The form error field may be given as: f(x,y,z) = Factual - F ^'nominal , where Fnominal = Fn (x,y,z) denotes the nominal position of the data point and Factual = Fa (x,y,z) denotes the actual position of the data point. In general, part surface form error field is sampled as discrete space signals. The sampled error data set f(x,y,z) = f (ΙΔχ, mAy, ηΔζ) where /, n and m represent the sample size of the in three dimensional axes. For typical smooth error field a large proportion of signal energy is expected to be concentrated in a small number of the modes (transform coefficients). In general, for three-dimensional signal (sampled data), with number of sample points equals to N3 (or LxMxN, if L≠M≠N), the forward and inverse transforms (models generation and reconstruction, respectively) are given as follows:

where T(u,v,w) are independent transformation parameters representing contribution of the error modes with space frequency of u, ι/ and w are in three axes x, y and z respectively. The g(x,y,z,u,v,w) and h(x,y,z,u,v,w) are called the forward and inverse transformation kernels. In this study, 3D Discrete Cosine Transform has been used as the main kernel for decomposition of form errors. This has been found to be particularly apt for modelling the random nature of the shape errors under consideration. Transform coefficients have been selected for form error characterisation as will be described in further detail below.

Turning back to Fig. 3, this example of a methodology for performing Geometric Modal Analysis (GMA) comprises of three major steps:

(i) Data pre-processing 34 which includes generation of a mesh model at 36 from nominal CAD model 14. Corresponding steps for the measured part data (CoP) 18 may be performed by post-processing the CoP data to determine the measured surface 22 so as to be able to obtain shape error deviations.

(ii) GMA decomposition 38 involves voxelisation of mesh model, Laplace interpolation, and 3D DCT decomposition, and

(iii) GMA mode identification which involves mode selection criteria and mode magnitude correction to achieve desired model accuracy. A. Data Pre-processing

The nominal features of the part (i.e. according to the CAD model) are composed of B- spline or NURBS surfaces which are not sufficient to embed the free shape errors.

A mesh model of the nominal features helps to easily integrate part shape errors with the nominal part which leads to several benefits, such as normal vector of the mesh nodes can be utilized to compute the shape deviation; and variational pattern can be easily incorporated. Conventional meshing tools will be known to the skilled person as they are used in other computer aided engineering applications such as Finite Element Analysis and Computational Fluid Dynamics. Meshing tools allow a three-dimensional grid/mesh network to be applied over a structure or domain wherein nodes are represented as points at which the mesh structure intersects, i.e. at internal and external vertices of the mesh structure.

In the application of the present invention, the surface profile of the nominal/scanned product is meshed. A quad mesh is used, for example a regular quad mesh.

The part measurement data captured through 3D non-contact scanner in terms of CoP is used to calculate deviation at each mesh node. In this proposed method, alignment of CoP with nominal CAD model is highly significant for model accuracy. Let N_n be the number of mesh node and D_n is set of calculated deviation at N_n. B. GMA Based Shape Decomposition

Building a unified functional shape error model is not trivial as it involves (i) transforming the 3D irregular surface model (such as 3D sheet metal parts with complex geometries, curvatures, holes and slots) to uniform 3D volumes structure to facilitate shape error decomposition into orthogonal shape error patters/modes, (ii) truncation and selection of most significant shape error modes with engineering importance, and (iii) accurately emulate real part shape errors with fewer selected modes.

The GMA decomposition involves three major steps: (i) Voxelisation of mesh nodes to envelope 3D complex shape which creates non-uniform scattered voxel structure, (ii) Laplace interpolation to smooth the non-uniform scattered voxel structure, and (iii) 3D DCT decomposition to obtain the shape error modes. The use of such combined processes is believed to be novel in itself and may provide a further aspect of the invention, either alone, or in combination with any other aspect defined herein. An example of the voxelisation of the mesh structure 40 is shown in Fig. 4, in which the space about the product surface is divided into adjoining volumes - i.e. voxels 42. The voxels 42 are typically hexahedral, e.g. comprising regular, right-angled hexahedra, such that every portion of the surface profile is captured in a voxel. The error field decomposition using 3D DCT can be applied on the uniform grid data. Therefore, a complex shaped part does not need to be used directly for decomposition but instead it can be discretized into a set of 3D uniform grid points. A structure containing scattered deviation has been achieved through voxelisation of mesh nodes 44. For this purpose, a voxel grid of length, heidh and width diesnions L x M x N is used as shown in Fig. 4 where each mesh node 44 position of the nominal part is described as point coordinate,

N_k = {x y

, *V[l, 2,3, ....,n]

where k is the node number and {x y z}_k represents the Cartesian coordinate of a mesh node k. For constant mapping of mesh node 44 coordinates to voxel space, a bounding box is computed enveloping the mesh model 40. The voxels 42 containing the mesh nodes 44 have been identified by linear mapping of the node coordinates to voxel space. All the voxel elements 42 containing the mesh nodes 44 are identified and calculated deviations at mesh nodes are allocated to the corresponding voxel elements. Relying on the chosen L M N voxel grid size, more than one mesh node may belong to same voxel in few cases and the allocated deviations of those voxels are computed as average of belonging node deviations. Therefore, error field in voxel space differs from the original shape error field due to averaging. Optimal voxel grid size can be chosen by minimizing this difference.

Calculation of deviation of the scanned product profile from the nominal profile may be performed at mesh nodes 44.

Voxel smoothing may be used as a result if the dsicretisation process. The nominal mesh nodes 44 are enveloped with the voxel grid to enable 3D DCT transformation on the voxel structure. After the voxelisation process, many voxel elements 42 in the voxel grid will not contain mesh node deviation and remain as empty. This implies a non-continuous voxel deviation field will result from the deviation determination process, i.e., a non-uniform scattered voxel structure. Since DCT attempts to fit a set of continuous cosine function to the given data field, as soon as discontinuities are detected, a large number of undesirable fitting modes could be generated. This result is potentially detrimental because the main shape error patterns may not be readily distinguishable from the others. In order to smooth the voxel model and make a continuous data field, a Laplacian smoothing technique is applied to assign a meaningful deviation value in at least some empty voxel elements 42, wherein the original deviation represents an internal boundary constraint.

In the voxel grid space of L χ M χ N, any voxel element deviation can be defined as f(i,j,k), where /=[1 ,2, .../.], y=[1 ,2,...M\, and k=\\ ,2,...N\. From the voxelisation process, the voxel elements 42 containing the original mesh node 44 deviations have been identified. 3D La lace definition

can be generalized to: f {i -l, j,k) - 2f {i, j,k) + f {i + 1, j, k )

Ax²

f {i, j -\,k) - 2f (i, j,k) + f {i, j + \,k)

Ay²

f (i, j, k -\) - 2f {i, j,k) + f (i, j,k) _

+ Az² to calculate deviation at each empty voxel element. Ax , Ay , and Az represent the voxel element length in L, M and N voxel directions respectively.

For 3D form error decomposition, the 3D DCT transformation given below is applied on the Laplace interpolated voxel structure in order to decompose the shape error field into significant error modes.

^'πν(2] + ϊ) w(2k + \)

cos cos f(i, j,k)

2M 2N

1

where, «( ) ^:

1 ΐ/, ξ≠0

In the above transform, modes C(u,v,w) represent the 3D DCT transformed coefficients which are a class of orthogonal transformations and u,v,w represent the modal position in voxel space. C. GMA Mode Identification

It is desirable to include only a few modes or transform coefficients in the model without losing much information on the shape error field to keep the shape error model tractable. However, the model should meet the desired accuracy of acceptable limit defined by the user. Thus it is proposed to identify the most prominent form error modes as part of the form error mode identification/characterization process. In order to retain dominant shape errors that have engineering importance, two criteria have been imposed as mode truncation criteria and selected modes amplitude correction criteria.

(1) Mode Selection or Truncation Criteria:

To check mode significance, two criteria are proposed: (a) Energy Compaction, and (b) Pearson's Linear Correlation. (a) Energy Compaction Criteria: This criterion is derived from Parseval's theorem (energy preservation of DCT):

The ratio of the energy in a selected number of significant modes to the total energy of the signal (sampled data) can be used to characterize the energy compaction of the model. To achieve a given energy compaction (threshold) of 0 ≤ E ≤ 100%, the most significant modes/coefficients should be included in a coefficient index set Ω_θ such that:

∑∑∑C²(",v,w)

-⁰ _{Ι Μ} ⁰Γ °_Ι ^≥E (u, v, w) e Q_e

∑∑∑f²(i, j,k)

i

The above truncation criterion is based on coefficients from sampled data of an individual part. The truncation is equivalent to selection of coefficients in case the amplitude of the coefficients is monotonically decaying.

(b) Pearson's Correlation Criteria: All the energy compacted modes (Ω_β) are selected to evaluate correlation coefficients by comparing to original shape deviation, D_n. Each energy compacted coefficient has unique pattern of error distribution over the mesh node and the mesh node deviations corresponding to each coefficients are kept as τ = [τ_{ T₂ T₃ . . Γ_ω ]_ηχΩ . The mesh node deviations corresponding to each coefficient are compared with original deviations to evaluate the coefficients with higher correlation, p, which are calculated as

where, ^ = (i,2,3 _e) is the set of indices of the energy compacted modes.

A threshold value, a, has been applied for further reduction in the number of modes. Only those modes are taken to model the shape error which have correlation coefficient higher than the given threshold, a. The truncated highly correlated modes are kept in the coefficients index set Ω.₀ (p_q>a). In case where E reaches 100% and a to 0, all the decomposed modes are included in the model.

Higher energy compaction of a coefficient indicates the significance of this specific error pattern which should be considered in the model. Therefore, the energy compaction criterion should be used together with the Pearson's linear correlation criteria simultaneously for coefficients selection (truncation). For an energy compaction E and given correlation threshold a, the truncated error model must include the coefficients C(u,v,w)e Q, where Ω is an index set in which all the indices of the intersection of Ω_θ and Ω₀ are included: Ω = Ω₍ ηΩ₍

(2) Mode Magnitude Correction:

The selected modes through the mode truncation criteria are mainly to recognise the main error patterns which does not necessarily depict the correct magnitude associated with each mode. Therefore, one or more corrective measure can be applied. A least square based mode amplitude correction method is proposed and by applying 3D inverse-DCT (3D IDCT) the shape error field is recovered.

(a) Least Square Approach: To overcome the challenge associated with the magnitude of error field, least square based mode magnitude correction has been employed to obtain proper weightings to the selected coefficients. The coefficients are selected from the coefficient index set, Ω which will satisfy the following equation:

Ω

D = Y wt T

9=1

where, g=1 ,2,3,... Ω represent the number of truncated coefficients, wt_q = weightage associated with q^th coefficient, and T_q = mesh node deviations associated with q^th coefficients.

(b) Error Model using 3D inverse-DCT: Each truncated coefficient from set Ω is selected to represent the shape error model. 3D IDCT is applied, as in by reversing the above DCT process, on the selected set of truncated coefficients, Ω to recover the shape error deviation fields.

2 2 2 ^i_1 πιι{2ϊ + \)

f (»^'. ^'.*) = J - J— J—∑∑∑ a(u)a(v)a(w) cos

2L

πν{2] + \) ^w(2k + l)

cos cos C(u, v, w)

2M 2N

The function f (i,j,k) refers to the field signals (deviation) which are generated by using the truncated coefficients set, Ω = C(u, v, w) . The inverse function has been applied to obtain voxel deviations which are applied to corresponding mesh nodes to model part shape errors with few modes.

Therefore, by obtaining the weight corrected truncated coefficients and using the above IDCT function, the recovered shape error field, f (i, j,k) deviates from the original deviation field as

f (i, j, k) = f (i, j, k ) +€(i, j, k)

where, e(i, j,k) is the residual term from the original shape error deviation to recovered deviation. Results of the above described methodology are illustrated in Figs. 5 and 6 with reference to an industrial case study concerning the identification of form errors in a car hinge reinforcement part. Such a part is crucial in terms of shape error control to achieve good quality in assembly. The hinge reinforcement part has many features and varying surface curvatures such that form errors may belong to several different normal directions. If the hinge is to be remote laser welded to a corresponding car door part, the gap or clearance between the two parts is required to be 0.3 mm, i.e., the gap between the hinge reinforcement (1 .8 mm thick) and door inner panel (0.75 mm thick) should be within 0.3 mm to ensure satisfactory joining quality. Therefore, predicting part variation is crucial to ensure the gap and quality of the welding.

The captured CoP data is aligned with nominal CAD and the shape error field is calculated at nominal mesh nodes, as represented in Fig. 5(a), in normal direction to obtain the shape error field as shown in Fig. 5(b). As per the above-described voxelisation process, the mesh nodes and associated deviations are stored in the voxel structure as shown in Fig. 5(c). Fig. 5(d) represents the voxel elements that contain shape error deviation only. Thereafter, Laplace interpolation has been performed on non-uniform scattered voxel data to fill the empty voxel elements with meaningful data to make uniform structure. 3D DCT is applied on the uniform voxel grid data. The transformed coefficients are truncated based on 90% signal energy compaction and Pearson's correlation test performed to identify the most significant coefficients related to original deviations using correlation threshold, a=0.25. A total of twelve coefficients are selected to model the part shape error and by using least square approach proper weighting has been applied to the selected twelve coefficients.

A sample set of main deformation modes are plotted in Fig. 6. 3D IDCT is applied on the weight corrected coefficient set to reconstruct the part shape error field. A residue surface can be determined as the difference between the measured original surface deviation and the reconstructed surface deviation.

These GMA decomposed shape error modes can be utilized for free form shape error simulation. One advantage found to be provided by the present invention when compared to other prior art techniques is that the developed GMA approach does not depend on predefined form error modes and directly decomposes the measured shape error variation. Hence the residue surface, which is not captured in an error mode region of the surface, can reach to near zero with addition of sufficient number of modes.

Further, the functional data analysis model, GMA, can be used to determine multivariate statistics for statistical process control of shape errors and 3D metrology sensor captured CoP data can be decomposed into independent shape error modes for efficient access and compact storage of real 3D parts shape information.

Statistical Modeling

With reference to Fig. 7, the outputs of the above stages of form error decomposition and selection feed into a statistical analysis tool 46, referred to herein as Statistical Geometric Modal Analysis (SGMA). Utilizing the GMA transformation, a set of representative parts of population has been decomposed. The sample batch of parts decomposed using GMA carry therein embedded information about the manufacturing process in terms of form errors. These extracted form error modes/patterns are further utilized for virtual part generation 48 and/or to synthesize one or more 'composite' part/product 50. Either/both of those processes 48, 50 are enabled by the SGMA tool 46 which commences with process 52 as described below.

Using the energy compaction and correlation test criteria as per GMA, modes are selected which explain the main process variation whilst error components that occur mainly due to uncertainly and noise in measurement data can be discarded. By using the reverse/inverse of the GMA approach, i.e. the reverse transform (IDCT) described above, the original error field deviation can be expressed, e.g. according to the equation below, where c(u, v, w) contains the truncated or preserved modes and residuals are expressed as ε . 12 / 2 1 2 ^{L~1 M~1} '^{V~1 ~} nu{2i + X)

(^l'> . = \ \ 7\ 77∑∑∑ cos

V L V M N _{u=0 v=0 w=0} 2L

*"v(2j + l)

cos cos C(w,v, w) + £

2M 2N

= Xb + e

The significant error components can be further reduced to Xb, where b is the set of energy and correlation truncated coefficient values and is composed of orthogonal shape vectors.

Suppose, q number of modes ( c , c₂ ,... C_g ) are preserved after energy truncation and correlation test expressed as

b_!

Further, considering the selected sample size m, the decomposed modal parameters can be and expressed as

The set β is composed of mainly two sets of modal parameters: (i) Common modes/patterns: present in every part in the selected sample m, and (ii) Non-common modes/patterns: appear only in few sample (<m) of the decomposed parts. Combining both common and non-common modal set, p modal parameters have been preserved for variational virtual part generation and composite part creation. These p modal parameters are again extracted from m sample parts which forms the modal parameter set β, a ρχ 1 vector, can be generalized as

Evaluation of proper amplitude of the modal parameter has been found to be important for correct evaluation. As described above for GMA, an example of error mode amplitude correction has been achieved through assigning proper weighting to p number of selected modal coefficients using least square approach. Therefore, modal parameter set β is selected for m sampled parts which creates the modal matrix for the batch and can be expressed as p m matrix.

c c

The modal matrix can be utilized for variational virtual part generation 48 and composite parts creation 50. Prior to those steps, statistical characterization has been performed by assigning proper statistical distribution to each mode of modal matrix.

Statistical Characterization of Preserved Form Error Modes

The decomposed modal matrix, β_ρχιπ , is fitted with a suitable statistical distribution for p number of modal signatures at statistical modelling stage 54. A typical assumption may be that characterized by mean modal vector and covariance matrix based on the assumption that the modal parameters are normally distributed, i.e. displaying a Gaussian distribution. However it has been found that, in many real processes, the assumption of normal distribution may be inaccurate and can produce unsatisfactory results. Therefore, use of non-parametric density estimation, such as, Kernel Density Estimation (KDE) to estimate the Probability Density Function (PDF) of the modal signatures, may overcome the problem if the modal parameters are not normally distributed.

KDE is a very powerful class of data driven techniques for non-parametric estimation of PDFs which fits an empirical distribution to a sample data sets approximating the population. Consider a kernel function K( ) and a sample set for modal parameter, [C,,c₂,...c_m] from a population distribution density F( c ), then the density estimate [25] of the sample can be written as

m

F(C,h) =—∑K

r=0 h

where, h is the window width also called as smoothing parameter or bandwidth parameter. In this approach, the kernel function is assumed to be a Gaussian kernel as the form of the kernel function is not important. On the contrary, the smoothing parameter or bandwidth determines the accuracy of the PDF. Therefore, in this example, each mode is fitted with PDF.

Fig. 8. Shows a chart of the statistical characterization of the identified form error modal signatures 56, i.e. for p modal fitting distributions. In Fig. 8, it can be seen that the statistical modeling is performed for a common form error mode coefficient or parameter, which is beneficially a quantifiable/tradable parameter determined using the methods described herein. The statistical modeling comprises determination of a form error parameter/coefficient range 58 over the batch of scanned surface profiles, as well as a mean value for the form error modal parameter/coefficient for each individual form error mode. The mean and variance/range thus allow characterization of the form error modes and a plot 62 of the form error mode distribution for each form error mode can be output as necessary.

Utilizing the statistical characterization, a number of non-ideal parts can be generated which may consist of different modal signatures and/or amplitudes relying on the different measurement data. Amplitudes of the p preserved modal signatures are drawn to generate variational virtual parts, i.e. part definitions/models comprising one or more of the identified form error modes. Using such a technique, a virtual batch of parts can be generated, e.g. corresponding to the scanned batch or else another random or pseudo-randomly generated batch using the statistical characteristics of the form error modes.

The generation of virtual batch using the modal signature characteristics is made as follows:

• A set of p random modal signature is drawn for N_v times to make virtual batch of N_v parts. Each modal coefficient follows the probability distribution obtained through KDE which form virtual modal matrix for N_v parts consisting of p modal signatures, i.e., β _N c c

c c c c c

where Vr = (1, 2,· · · , NJ

• From the obtained virtual modal matrix, β_ρχΝ^ , each virtual part coefficient β_ρχτ is applied with inverse transform function described above to obtain the voxel deviation considering the error term as zero. The function / ( , j,k) refers to the error field signals

(deviation) in voxel space and voxel deviations are applied to corresponding mesh nodes to generate variational virtual parts.

The statistical characterization process described above also allows generation of a hypothetical product/part comprising all the identified/selected (e.g. significant) form error modes. Depending on the type of form errors present in the measured sample set of parts, more than one composite part might be required to represent the whole population. Grouping of composite parts or form error types may be undertaken to help ensure each composite part consists of similar form errors. A clustering approach has been adopted. For the clustering, a k-means method has been applied to classify the group of parts with similar types of form errors. This clustering method helps to partition data into mutually exclusive clusters and provides the index of the parts belonging to each of the clusters. A conventional intra-cluster distance and/or inter-cluster distance based approach may be used to identify clusters and determine the number of clusters according to the spacing of points within, or between, clusters. A more quantitative way to compare the cluster solutions is to look at the average silhouette values of potential clusters.

Using the clustering process, R number of cluster has been obtained and corresponding cluster will consist of NR number of parts. Therefore, the modal matrix for NR number of arts which contain p modal signature become

where, vR = (i,2,...R) and N_R ^ N_V .

Composite parts can be obtained from the β_ρχΝι< modal matrix using different selection criteria which are mainly or entirely based on the energy compaction criteria and root sum square criteria.

Energy compaction index of a mode is the ratio of the energy of the selected mode to the total energy of all the modes (particular part). This index can be used to select modes for energy compacted composite part generation. Therefore, energy compaction index can be obtained for every element of the β _N modal matrix which can be determined as

Further, relying on the maximum and minimum energy index, a maximum and minimum set of modal coefficients can be obtained. These will create composite parts containing maximum and minimum variation of form errors respectively.

(^Ec )max = max{E- ,E~ ,·^■■∑- }

(^Ec )min = πιϊ^η{£_έ , E_£ ,— Ε_έ } Therefore, the maximum and minimum energy compaction part can be expressed as β ma = [CL , , C_{i K Λ} ,— C _Λ f

Using the above definitions, the composite parts consisting of maximum and minimum energy coefficients respectively can be obtained which can be used to represent the maximum and minimum boundary of form errors.

In addition to, or instead of the energy compaction method for composite part definition, an alternative Root Sum Square (RSS) approach may be used to determine suitable criteria for composite part: A RSS based composite part can be defined as the part from which root sum square measure to original deviations of all the parts belong in the same cluster is minimum. The original measured deviations at mesh nodes from COPs related to the NR parts are kept as £> = [£>_! D₂ ... ¾_Β ] _ΧΛ, - where n denotes the mesh node. The RSS based mesh node deviation [D_RSS ]_NXL can be obtained through the following minimization roblem:

The preserve modal parameter, p, will have the unique orthogonal shape vector of unit value and each coefficient related mesh node deviation is stored as τ =

.

Therefore, a least square approach is used to provide a solution which will minimizes the sum of squares corresponding to each part and provides the weighted coefficients, /3/._s.

The least square estimation for the weighting associated with each modal signature is computed as wt = [τ^ττγ T^TD_RSS . Therefore, the coefficient set for creating composite part using RSS criteria is given as

= [ _j x wt_x , C₂ x wt₂ , · · · C x wt ]^T

The coefficient set, jS/.s, is applied in the reverse transform process described above to obtain the error field in the voxel space and voxel deviation is applied to mesh nodes to get the composite part error field deviation.

The above SGMA techniques have been applied to different case studies such that, based on the distribution obtained from the decomposition of a batch of parts, a number of random modal signatures can be drawn from the distribution of individual modes, and a virtual modal matrix for N_v parts, β_ρχΝ consisting of p modal signatures can be obtained. According to examples of the invention, each column of the generated modal matrix represents a virtual part. Therefore, the proposed SGMA method can generalize the virtual production of parts.

It is noteable that the root sum square error based composite parts exhibit almost same deformation error patterns for all the clusters while maximum and minimum energy compaction based parts differ from cluster to cluster.

According to aspects of the invention, the identified modal signature amplitudes can be stored to form the modal matrix for generating variational virtual parts and composite parts. In examples, thereof, each modal signature may be fitted with distribution function using KDE to estimate the PDF associated with each mode.

The SGMA method quantifies the form error uncertainty associated with a batch of parts by generalizing the statistical behavior of modal signatures and synthesizing composite parts. The proposed SGMA methodology significantly enables any, or any combination of, the following areas: 1 ) dominant form error patterns identification from a batch of parts utilizing an analytical model unlike Finite Element Analysis (FEA) based case by case analysis, 2) identified error pattern parameterization allowing simulation of parts for statistical tolerance analysis based on the geometric errors, and 3) assembly process optimisation at design stage considering batch of parts error is pointing towards the robust design optimisation, such as, jig and fixture design. The virtual generation of form error patterns may additionally or alternatively be utilized to eliminate current method of offsetting the nominal surface to upper and lower boundaries and a new tolerance zone can be determined. Furthermore the invention helps to identify the relationship between Key Product Characteristics (KPCs) and Key Control Characteristics (KCCs) to evaluate conformity based process yield.

Whilst the invention is defined above in relation to the output of the SGMA process, in generating the SGMA process, a novel functional-data-analysis-based GMA has been developed proposed for modeling part shape error by decomposing the error field into a series of independent shape error modes. The proposed GMA methodology has been found to beneficial in itself in the characterization of geometric errors of 3D complex shaped parts. Accordingly, in other aspects of the invention there is provided a computer implemented method for categorizing/detecting shape errors of 3D objects by determining shape errors over a measured product surface profile when compared against a nominal profile, repeating the determination stage for multiple products and decomposing the determined shape errors into common shape error modes by voxelisation of the surface profile and performing a mathematical transform of the determined shape error for each of the multiple products so as to identify common shape error modes by reference to one or more parameter value resulting from mathematical transform. A further aspect of the invention may be directed to the use of a 3D DCT operation in conjunction with a Laplace smoothing operation for a discontinuous form error profile over a measured product surface in order to decompose form errors and thereby identify quantifiable form error modes within the form errors measured in the product surface. The invention may enable shape error based Statistical Process Control (SPC) to detect the shape defects, abnormal process behavior, and unexpected faults using CoP data. Further, these shape error modes (functional data) can be utilized for efficient compaction and storage of part shape error information instead of storing CoP data (which represents non-functional data). In one example of use, the invention has been found to be beneficial to Remote Laser Welding (RLW), amongst other high tolerance joining processes, for which product/process variation is an important consideration in delivering high quality product. The joint quality is highly dependent on the part-to-part gap at the point of joining and is thus relatively intolerant to geometric variation of the kind that can be evident in stamped sheet-metal parts. This method helps fixture designer to consider a batch of non-ideal parts during the design synthesis process. Therefore, the fixture developed by this design synthesis process is not only optimised for single assembly but also for batch of assemblies which eventually represent the production population.

Additionally or alternatively, once a plurality of form error modes have been identified and characterised, inspection of subsequent manufactured parts can be tailored to identify only predetermined form error modes resulting from the processes described herein. This may allow automated handling/processing or discarding of parts according to the sensed form error modes or the amplitude thereof in subsequently manufactured parts.

Claims

Claims:

1 . A method of identifying product form errors for use in the control of product manufacture, the method comprising:

scanning multiple products produced at a common stage of a manufacturing process so as to output a scanned profile for each product;

comparing each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile;

grouping form errors into a plurality of discrete form error modes and parameterizing each form error mode such that each form error mode can be characterised by one or more parameter value;

identifying form error modes that are common to a plurality of said multiple products and outputting a statistical model for each common form error mode.

2. The method of claim 1 , wherein the products have a three-dimensional surface profile, the scanned profile being a three-dimensional profile assigned to a three- dimensional coordinate system, which is aligned with a corresponding coordinate system of the computational model of the nominal profile.

3. The method of claim 1 or 2, wherein the plurality of products comprises a batch of the products comprising tens of products or more, the statistical model being determined in accordance with the frequency with which a common form error mode is repeated over the scanned batch of the products.

4. The method of any preceding claim, wherein the statistical model is based upon the variation or mean value of the one or more parameter value used to characterise each error mode.

5. The method of any preceding claim, wherein the statistical model generation for each form error mode comprises fitting a statistical distribution to the occurrence of the form error mode within scanned profiles of the multiple products, said statistical distribution being determined by performing probability density estimation for the occurrences of the form error mode.

6. The method of any preceding claim, wherein the comparing of each scanned profile to the nominal profile comprises discretisation of the nominal profile and the scanned profiles by defining a plurality nodes on said profiles and a plurality of voxels, wherein the profiles are entirely contained within the plurality of voxels, and assigning a form error to each voxel for which a node of the scanned profile is spaced from a corresponding node of the nominal profile.

7. The method of claim 6, further comprising a Laplace smoothing process to smooth the discrepancy between form errors in adjacent voxels.

8. The method of any preceding claim, comprising applying a three-dimensional Discrete Cosine Transform of the shape errors and determining associated frequency and amplitude values for a plurality of the shape errors.

9. The method of any preceding claim, comprising applying or more form error truncation criterion so as to select a subset of the most significant form error modes from the identified form error modes, wherein a statistical model is generated for each of the selected subset of error modes.

10. The method of any preceding claim, comprising applying a weighting to each, or each selected, form error mode.

1 1 . The method of any preceding claim, comprising reconstructing the scanned profile from the nominal profile by applying the identified form error modes thereto.

12. The method of any preceding claim, comprising collating an error mode parameter set comprising an error mode matrix containing an entry for each of the identified error modes.

13. The method of any preceding claim, comprising generating one or more composite product profile, said one or more composite product profile comprising all of the selected/identified form error modes.

14. The method of claim 13, wherein a plurality of composite product profiles are generated and a clustering process is used to define clusters of form error modes to be included in each individual composite product profile.

15. The method of any preceding claim, comprising generating a plurality of variational product profile models, each model comprising one or more form error mode and an amplitude value for said one or more form error mode.

16. A method of manufacturing process control comprising:

scanning multiple products produced at a common stage of a manufacturing process so as to output a three-dimensional scanned profile for each product;

comparing each scanned profile to a three-dimensional computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile;

discretising the scanned profile by applying an array of voxels enclosing the scanned profile and applying a three-dimensional sinusoidal transform to the form errors so as to group form errors into a plurality of discrete form error modes;

parameterizing each form error mode such that each form error mode can be characterised by one or more parameter value; and

controlling the manufacturing process to accommodate said plurality of discrete form error modes applied to the nominal product profile.

17. The method of claim 16 comprising controlling the joining or assembly of the product to an adjacent part.

18. A product form error inspection system comprising:

a product scanner arranged to scan multiple products produced at a common stage of a manufacturing process so as to output a scanned surface profile for each product; a surface comparator arranged to receive the scanned surface profiles and to compare each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile; and

an error processor arranged to group the form errors output by the comparator into a plurality of discrete form error modes and parameterize each form error mode such that each form error mode can be characterised by one or more parameter value, wherein the error processor comprises a statistical model generator arranged to identify form error modes that are common to a plurality of said multiple products and output a statistical model for each common form error mode.

19. A data carrier comprising machine readable instructions for the operation of one or more processor to:

access a scanned profile for each of multiple products produced at a common stage of a manufacturing process;

compare each scanned profile to a computational model of a nominal profile for said products so as to determine form errors between the scanned profiles and the nominal profile;

group form errors into a plurality of discrete form error modes;

parameterize each form error mode such that each form error mode can be characterised by one or more parameter value;

identify form error modes that are common to a plurality of said multiple products; and output one or more composite profile comprising each common form error mode.