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1.School of Mechanical and Materials Engineering，North China University of Technology，Beijing 100144，P.R.China；2.AVIC Manufacturing Technology Institute，Beijing 100024，P.R.China；3.School of Mechanical Engineering and Automation，Beihang University，Beijing 100191，P.R.China

Abstract:As a result of the recently increasing demands on high-performance aero-engine，the machining accuracy of blade profile is becoming more stringent.However，in the current profile，precision milling，grinding or near-netshape technology has to undergo a tedious iterative error compensation.Thus，if the profile error area and boundary can be determined automatically and quickly，it will help to improve the efficiency of subsequent re-machining correction process.To this end，an error boundary intersection approach is presented aiming at the error area determination of complex profile，including the phaseⅠof cross sectional non-rigid registration based on the minimum error area and the phaseⅡof boundary identification based on triangular meshes intersection.Some practical cases are given to demonstrate the effectiveness and superiority of the proposed approach.

Key words：aero-engine blade；error area；boundary identification；non-rigid registration；NC re-machining

0 Introduction

Aero-engine blade possesses complex profile and strict design tolerance，which has always been regarded as the difficulty in aeronautical manufactur?ing.During the finishing process of blade profile be?fore mass production，because of the effect of elastic and permanent deformation fractions，the iterative correction process of error compensation is always necessary but tedious［1］.The most common method used by technologists is to modify the NC program to obtain a more accuracy profile.

The main manufacturing errors of blade profile are from geometric and thermo-mechanical error of machine tool，deflection errors of cutting tool，workpiece and machine tool structure，dynamic er?rors of motion control method and software［2］.There are a lot of approaches to compensate these error sources［3-4］，especially the machining-relat?ed［5］，which is always compensated by online remachining technology［6］.However，the necessary error area and boundary for blade profile are consid?ered by manual operations with non-automation and poor-efficiency.The current digital determination focusses on the extraction and classification of scan?ning data geometric attributes.Two main principles are found in Refs.［7-10］：（1）Curvature method，（2）normal vector method.These researches are carried out for extreme points of principal curvature and abrupt points of normal vectors in the main di?rection of the scanned data.However，the curva?ture and normal vectors of the blade profile possess complex variation，especially in the case of small machining allowance，it is difficult to determinate the geometric attributes by the aforementioned prin?ciples.In order to achieve the numerical control and automation of the blade profile error compensation，the paper carries out a boundary intersection meth?od.

1 Overview and Terminology

In order to achieve high efficiency for re-ma?chining technology［11］，the proposed approach in?cludes two phases：the phase Ⅰof cross sectional non-rigid registration based on the minimum error area and the phaseⅡof boundary identification based on triangular meshes intersection，where the mini?mum area objective function is used to reduce the subsequent re-machining time.Because design toler?ance constraints are introduced in solving the objec?tive function，the profile deformation distribution is approximated by the transformation and rotation of several typical cross sections under the assumption of continuous distribution of blade profile deforma?tion［12］，reducing the computational complexity of solving constrained non-linear optimization problem.Subsequently，the triangle-triangle intersection is carried out between the measured data and the toler?ance surface of the optimum blade profile.Then the closed and open loops of error boundaries are identi?fied by the intersection point sorting algorithms.The flowchart of the proposed approach is shown in Fig.1.

Fig.1 Flowchart of the proposed approach

The proposed approach follows the basic princi?ple of blade design from simplicity to complexity，namely from cross sections to skinning profile.Fig.2 shows the cross-sectional form-position tolerance，which consists of displacement（?），orientation（φ），and profile tolerance（υrorυs）［13-14］.Measured data can be rigidly transformed to the optimal posi?tion through translation and rotation matrixes in the Section 2.1，which is composed of two translations（Tx，Ty）and one rotation（Rz），and the relational inequalities are given by Eq.（1）for rigid transforma?tion purpose.

Fig.2 Cross-sectional sketch of aero-engine blade

The measured data are conducted with two different devices，CMM CENTURY 977（AVIC BPEI，Beijing，China；Renishaw TP7M touch trigger probe）and GOM ATOS Ⅱ-400（GOM mbH，Braunschweig，Germany）.Our previous re?search［15］is helpful to improve the accuracy and ef?ficiency of measured data，and the slicing process is adopted on each cross section，as shown in Fig.3.

Fig.3 Measured data and slicing process

2 Algorithm and Implement

With the development of computer-aided de?sign，there are some differences in the representa?tion and properties of the blade profile.For conve?nience in exposition，F(xiàn)ig.4 shows the nominalE={Ei|i=0，…，m-1} cross-sectional curves can be written as［16］In thev-direction its spline can be written asV'j=withv∈[0，1]，where {?0，j，…，?m'，j}are control points of the cubic B-splint curve with data points {d0，j，…，dm-1，j}.Thus，the blade profile can be expressed as

Fig.4 Sketch of blade profile representation

whereNi，3(u)，Nj，3(v) are the normalized B-spline basis function of degree 3.

2.1 Phase Ⅰ

Step 1Selectr（3≤r≤m）cross sections formEand build subsetS={Si'|i'=0，…，r-1}?E，their cross-sectional curvesis theconstant iso-parameter curve ofW.

The cross sectional non-rigid registration be?tweenCi'andMmeasured points.The basic idea is similar to the classical iterative closest point（ICP）algorithm［17］.It is also iterative in nature，which means that the algorithm developed can be formulat?ed as an objective function by the least squares prin?ciple of rigid registration，and produces a sequence of non-rigid transformation matrixesHi'(Txi'，Tyi'，Rzi')as follows

wherePi'is the closest point inCi'corresponding toQi'.It should be noted that the subsetQi'has few or all elements equal to the measured points on the cross sectionSi'and satisfies Euclidean distance|Pi'Qi'|≤υs/2 in each iteration，as shown in Fig.5.In other words，if the Euclidean distance from the measuring point toCi'is more thanυs/2，the mea?sured point will not be brought into Eq.（3）.

Fig.5 Sketch of non-rigid registration for Pi′and Qi′

Fig.6 Sketch of Step 2 for U′i=fH(vi)?Ui

Due to the difficulty in achieving a sequence of fitting coefficientsB={b0，…，bt，…，br}of Eq.（4）within the limited time period of optimization，an ef?ficient approximation operation is developed to iden?tify triangular patches of measured points that are above the upper tolerance（υs/2）boundary surface ofW'(u，v).The area sum of these possible error patches is considered as error area.Fig.7 shows the minimum directional distancedgbetweenW'(u，v)and the center pointcgof triangular patchesTg，g=1，2，…，M∈N*.Agis the patch area ofTg.It is al?so iterative in nature，which means that the error-ar?ea determination can be formulated as an objective function by the minimal area principle of registration fitting and generatesB={b0，…，bt，…，br}，as Eq.（6）.The iterative process（Steps 1―3）may be continued until the objective function differ?ence ΔminF(D) between every two iterations is equal or less than specified thresholds，then the op?timum blade profileW'(u，v) after non-rigid regis?tration can be constructed，denoted byW"(u，v).

Fig.7 The minimum directional distance dg between center point cg of triangular patches Tg and W′(u,v)

It is assumed that the number of measured points isNon each cross section，then the time complexity of Eq.（3）isO(rN).In the course of solving minF(D)，the minimum distance calcula?tion is required forMmeasuring points，then the time complexity of Eq.（6）isO(M+rN).

2.2 Phase Ⅱ

After the approximation-based non-rigid regis?tration，W"(u，v) can meet the requirements of po?sition tolerance（?，φ）and the minimum error ar?ea，then the error boundary can be identified through computer graphic intersection algo?rithms［18］.Specifically，the error area can be defined as outer points of the upper tolerance（υs/2）surface ofW"(u，v)，which can be obtained by the surface offset algorithm.Considering the triangulation con?venience ofMmeasuring points and the tolerance boundary surface，the error boundary is obtained by triangle-triangle intersection algorithms.Fig.8 shows the result of intersection points.

Fig.8 Results of error boundary identification

There are two types of the error boundary：（1）Closed loops that represent these intersection points are away from the profile edges，and（2）open loops that represent endpoints reach to the profile edges.Fig.9 shows the initial sequence of intersection points.The sequence can be changed based on the distance threshold for each neighboring point，then the final sequence of intersection points is linked to form the error boundary.There is one open loop and one closed loop，inside which are identified as error area.

Fig.9 Closed/open loops and error area

3 Results and Discussion

In order to illustrate the validity and advance?ment of the proposed approach，three practical cas?es are presented.

3.1 Precision forging blade

One practical case is a precision forging blade with the size of 60 mm×90 mm，in which?=0.1 mm，υs=0.08 mm ([-0.03 mm，)+0.05 mm ]，andφ=±8'，as shown in Fig.10（a）.It is known that the machining region is the neighboring areas of leading/trailing edge（LE/TE）due to the precision forging process.Fig.10（b）shows the measured data acquired by the 3D-optical scanning system（GOM ATOS Ⅱ-400），and its error distribution illustrates that the initial machining region is pressure/suction side（PS/SS）.By reference to the phase Ⅰ，three typical cross sections（i.e.，sections ①/②/③）are conducted non-rigid registration to realize the local?ization of cross-sectional curves.

Fig.10 Precision forging blade and measured data and error distribution

The situation thatQi'has few or all elements equal to the measured data and does not satisfyυs=0.08 mm（defined in red）is not considered during the initial iterations.Figs.11（a―c）show the varia?tion of error curves（Error（mm））of the sections ①/②/③with the increase of corresponding points（｛Pi'，Qi'｝，defined in yellow）after each iteration（Iteration num（No.））.A sequence of transforma?tions and rotations are listed in Table 1.

Table 1 Non?rigid registration of three typical cross sec?tions

Fig.12 illustrates the error distribution between the optimum blade profile and measured data after phaseⅠ.Furthermore，the error boundary（black lines）are two open loops after phaseⅡ，which basical?ly tallies with the result by an experiential judgement.

Fig.11 Variation of error curves after each iteration of nonrigid registration

Fig.12 Error distribution and boundary identification for precision forging blade

3.2 High?precision grinding blade

Another practical case is a high-pressure com?pressor blade with complex profile，in whichφ=12'，?=0.1 mm，υs=0.06 mm（[-0.02 mm，+0.04 mm ]）.There are still unstable grinding error areas due to its difficult-to-machine materials and thin-wall structure［19］.Therefore，it is an essential step for error compensation to carry out the bound?ary identification on the experiment process.The GOM ATOS Ⅱ-400 is utilized to acquire measured data，as shown in Fig.13.We use five typical cross sections（i.e.，sections ①—⑤）to conduct non-rig?id registration and generate a sequence of transfor?mations and rotations in Table 2.

Fig.13 Tested blade and five typical cross sections

Table 2 Non?rigid registration of five typical cross sections

After the phaseⅠ，F(xiàn)ig.14 shows the error dis?tribution between the measured data and the opti?mum blade profile.The error area shares about 5.5% of the total profile area.Thus，the error boundaries are identified through the triangle-trian?gle intersection algorithm，including one closed loop and two open loops as shown in Fig.15.

Fig.14 Error distribution displayed by color cloud

Fig.15 Error boundary

Through modifying the cutter path based on the boundary information，the error area can be compensated automatically.Fig.16 shows the error distribution can satisfy the design requirements.

Fig.16 Error distribution after error compensation

3.3 Investment casting turbine blade

The other practical case is an investment casting turbine blade（Fig.17（a）），in whichυs=0.14 mm（［-0.07 mm，+0.07 mm］）.GOM ATOS Ⅱ-400 is utilized to acquire measured da?ta，as shown in Fig.17（b）.According to the previ?ous engineering experience，the error area is dis?tributed in the pressure side，and the error bound?ary is closed loop.Because the curvature and nor?mal vector change is not obvious in this case of small machining allowance，the geometric attribute identification method mentioned in the introduction is not suitable for determining the error area.Thus，F(xiàn)ig.17（c）illustrates the error distribution between the optimum blade profile and measured data after phaseⅠ，and four closed loops of the er?ror boundary are identified，as shown in Fig.17（d）.

Fig.17 Resluts of investment casting turbine blade

4 Conclusions

Due to the inevitable deformation of the aeroengine blade with the difficult-to-machining materi?als and complex profile，the high precision blade needs to undergo a tedious iterative correction pro?cess of error compensation.It is crucial for process engineers to determinate the profile error area and boundary automatically and quickly.To this end，we have proposed an error area determination ap?proach，which consists of two phases：The non-rig?id registration based on the minimum error area and the boundary identification based on triangular mesh?es intersection.The decoupled solution ensures the efficiency and superiority of the developed algo?rithms to preserve an optimum compromise by mini?mizing error areas and conforming design tolerance at most degree for the implementation of subsequent re-machining.

Under the framework of the proposed ap?proach，emphasis of further research activities will be conducted in solving the unfairness problem of the error boundary，which will help to a speed-gen?erating re-machining tool path and bring more auto?mation and intelligence to error compensation.