Total differential methods based universal post processing algorithm considering geometric error for multi-axis NC machine tool

Total differential methods based universal post processing algorithm considering geometric error for multi-axis NC machine tool

Author's Accepted Manuscript Total Differential Methods Based Universal Post Processing Algorithm Considering Geometric Error for Multi-Axis NC Machi...

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Author's Accepted Manuscript

Total Differential Methods Based Universal Post Processing Algorithm Considering Geometric Error for Multi-Axis NC Machine Tool F.Y. Peng, J.Y. Ma, W. Wang, X.Y. Duan, P.P. Sun, R. Yan

www.elsevier.com/locate/ijmactool

PII: DOI: Reference:

S0890-6955(13)00017-5 http://dx.doi.org/10.1016/j.ijmachtools.2013.02.001 MTM2835

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International Journal of Machine Tools & Manufacture

Received date: Revised date: Accepted date:

19 June 2012 1 February 2013 1 February 2013

Cite this article as: F.Y. Peng, J.Y. Ma, W. Wang, X.Y. Duan, P.P. Sun, R. Yan, Total Differential Methods Based Universal Post Processing Algorithm Considering Geometric Error for Multi-Axis NC Machine Tool, International Journal of Machine Tools & Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2013.02.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Total Differential Methods Based Universal Post Processing Algorithm Considering Geometric Error for Multi-Axis NC Machine Tool F.Y. Peng 1,* J.Y. Ma 2, W. Wang 2, X.Y. Duan 2, P.P. Sun 2, R. Yan 2 1

State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;

2

National NC System Engineering Research Center, School of Mechanical Science and Engineering, Huazhong

University of Science and Technology, Wuhan 430074, China *Corresponding author. Tel.: +86 27 87542513; Cel.: +86 13986168308; fax: +86 27 87540024. E-mail address: [email protected] (F.Y. Peng). Permanent address: B418, Advanced manufacturing building, 1037 Luoyu Road, Huazhong Univ. of Sci. & Tech., Wuhan, China

Abstract:Multi-axis numerical control machining for free-form surfaces needs CAD/CAM system for the cutter location and orientation data. Since these data are defined with respect to the coordinate of workpiece, they need converting for machine control commands in machine coordinate system, through a processing procedure called post processing. In this work, a new universal post processing algorithm considering geometric error for multi-axis machine tool with arbitrary configuration. Firstly, ideal kinematic model and real kinematic model of the multi-axis NC machine tool are built respectively. Difference between the two kinematic models is only whether to consider the machine tool’s geometric error or not. Secondly, a universal generalized post processing algorithm containing forward and inverse kinematics solution is designed to solve kinematic models of multi-axis machine tool. Specially, the inverse kinematics solution is used for the ideal kinematic model, while the forward kinematics solution is used for the real kinematic model. Then, a total differential algorithm is applied to improve the calculation speed and reduce the difficulty of inverse kinematics solution. Realization principle of the total differential algorithm is to transform the inverse kinematics solution problem into that one of solving linear equations based on spatial relationship of adjacent cutter locations. Thirdly, to reduce the complexity of geometric error calibration experiment, effect weight of geometric error components is determined by the sensitivity analysis based on orthogonal method, and then the real kinematic model considering geometric error is established. Finally, the universal post processing algorithm based on total differential methods is implemented and demonstrated experimentally in a five-axis machine tool. The results show that the maximum error value can be decreased to one-fifth using the proposed method in this paper. Key words: Total Differential Methods, Multi-axis machine tool, Geometric error, Universal post processing, Arbitrary configuration

NOMENCLATURE CL NC CNC TDA UPPA IKM RKM

cutter location numerical control computerized numerical control total differential algorithm universal post processing algorithm ideal kinematic model real kinematic model

Introduction 5-axis machining offers better shaping capability and higher productivity than 3-axis machining with two more degrees of freedom in positioning the cutting tool relative to the surface. And the accessibility using 5-axis machining meets the needs of manufacturing of sculptured surface such as marine propeller, compressor impeller and so forth. Machining accuracy of those core parts with complex curved surfaces is key to improve performance of relevant products. But various errors during processing, especially on large machine tool, limit the improvement of machining accuracy. Geometric error, thermal distortion effects, loads deformation, dynamic force fluctuation, location errors, motion control delay, and tool wear are all error sources affecting the machining accuracy finished workpiece. So it is quite necessary to study processing method of improving machining accuracy. Before numerical control (NC) machining process, a CAM module calculates and generates tool path of cutter including cutter tip position and the tool orientation from CAD model. Significant researches are conducted on tool path planning to improve machining efficiency and surface quality currently. Ye[1-2] presents a novel tool path planning method considering machining kinematics in multi-axis machining. Zhu[3-4] proposes an approach for third-order reconstruction of the cutter envelope surface and design a tool positioning strategy for efficiently machining free-form surfaces with non-ball-end cutters. Lei[5] proposes to generate new paths through evaluating position deviation equation using fundamental equation of NURBS path. Guo [6] proposes a tool path planning method to reduce the geometric errors and cutting force fluctuations aroused by cutter runout. Most of these researches focused on methods of tool path planning to improve machining accuracy directly. However, the final accuracy of processed surfaces depends not only on tool path planning but also on post processing and controlling over each axis of NC system. Post processing is to covert the tool path, output as a machine controller independent Cutter Location Data file (CLDATA), to NC data. Postprocessor is used to perform the task. The main solutions of postprocessor include derivation method, geometric method and numerical method. Multifarious studies utilize homogeneous transformation matrix to describe spatial transformation and developed postprocessor for three typical types of five-axis machine tools [7-11]. She [12-14] derives homogeneous transformation matrix of five-axis machine tools with orthogonal or non-orthogonal structure. Based on it, relevant postprocessor is developed. Yun [15] adopts a geometric method to derive post processing on large-scale propeller machining machine tool with

three rotational axes and two linear axes, in which the problem of nonlinearity and non-derivation of kinematic functions is solved. There are also some investigations on post processing method considering geometric error which leads to inaccuracy of finished workpiece measured through error calibration experiments. Mahbubur [16] points out that manufacture error, assembling error and etc. lead to the cutter tip deviating from the ideal position and machine tool is sensitive to angular error among all geometric error items. Through interpolation of CL data and post processing, the online compensation to geometric error can be achieved. Xiong [17] proposed an error elimination approach providing an effective measure of quality control for multi-axis CNC machining and robot manipulation. Much research has focused on error compensation method implemented in NC system and gotten effective achievements. Hsu [18] suggests that tool orientation error is only related to rotational axes and develops a new algorithm considering geometric error on rotational axes and linear axes severally. Schwenke [19] presents a technique of digital error compensation to compensate geometric error. Uddin [20] proposes a simulation method of geometric error, which identifies geometric error of five-axis machine tool with an inclined turntable and compensates kinematic error. Nevertheless, previous methods mentioned above considers geometric error not in post processing stage, or only apply to machine tool of certain type. Universal post processing algorithm considering geometric error is not further investigated. Therefore, we propose a universal post processing algorithm considering geometric error of multi-axis NC machine tool with arbitrary configuration. The generated NC data can eliminate inaccuracy caused by geometric errors. By generating more precise NC data in post processing stage, we can obtain higher accuracy in subsequent processing. The paper is organized as follows. Section 1 gives the ideal and real kinematic models of the multi-axis NC machine tool respectively. Section 2 proposes a universal post processing algorithm (UPPA) for multi-axis NC machine tool with arbitrary configuration. The UPPA is devoted to generate the NC code considering the geometric error. Then a total differential algorithm (TDA) is utilized to deal with the complexity of inverse kinematics solution. At last the effectiveness of UPPA is proofed by proving the convergence of iterative algorithm. In Section 3 the algorithm of TDA and UPPA through machining experiments on a five-axis mill-turn machine tool is verified. Then the sensitivity analysis is carried out by orthogonal method to simplify the geometric error calibration experiment in section 4. In section 5, the effectiveness of the proposed kinematics solution algorithm considering geometric error is verified through machining experiments on a five-axis mill-turn machine tool.

Universal kinematics model of multi-axis machine tool Machine tool can be assumed as a set of links connected in a chain by joints. Suppose machine tool components are all rigid bodies, the kinematic coordinates relationship among different machine components can be represented by a 4 × 4 homogeneous transformation matrix[21]. According to whether the geometric errors are considered or not, two different kinematic

models are built. The kinematic model ignoring geometric error is taken as ideal kinematic model (IKM). Conversely, the kinematic model considering geometric error is called real kinematic model (RKM). Assuming coordinate systems OtXtYtZt, OmXmYmZm and OwXwYwZw are established on the cutting tool, machine tool and workpiece respectively. The transformation matrix of OtXtYtZt with respect to OmXmYmZm in IKM can be expressed as follows: M

where

M

m

m

i =1

i =1

QT = Π Qi +1,i = Π [T ( Li , λi ) ⋅ R(Wi , φi )]

i = 1,2,⋅ ⋅ ⋅m

(1)

QT represents relative transformation matrix of system OtXtYtZt with respect to system

OmXmYmZm . Li = L x i + L y j + Lz k is vector from origin Ot to Om . m is number of kinematic pairs. T and R are 4×4 homogeneous translation and rotation matrix adopting Paul’s notation[22]. Using the same method, the transformation matrix of system OmXmYmZm with respect to system OwXwYwZw in IKM is expressed as follows: W

QM =

n+ m

Π

i = m +1

Qi +1,i =

n+ m

Π [T ( Li , λi ) ⋅ R(Wi ,φi )]

i = 1,2,⋅ ⋅ ⋅n

(2)

i = m +1

Thus the transformation matrix of system OtXtYtZt with respect to system OwXwYwZw in IKM can be expressed as follows: W

QT =W Q M ⋅ M QT

(3)

Geometric error needs to be considered in RKM. Each moving axis contains six geometric error components:three angular errors ΔAi , ΔBi , ΔC i around x i -axis、 yi -axis、 z i -axis respectively and three linear errors ΔX i 、 ΔYi 、 ΔZ i along x i -axis、 yi -axis、 z i -axis respectively. Because of geometric error, Q i +1,i needs to be revised as follows:

Qi′+1,i = ( E + δQi +1,i ) ⋅ Qi +1,i where E is a 4×4 unit matrix, and

δQi +1,i

⎡ 0 ⎢ ΔC i =⎢ ⎢− ΔBi ⎢ ⎣ 0

,

(4)

δQi +1,i is an error matrix as follows: − ΔC i 0

ΔBi − ΔAi

ΔAi 0

0 0

ΔX i ⎤ ΔYi ⎥⎥ ΔZ i ⎥ ⎥ 0 ⎦

(5)

Introducing geometric error, transformation matrix

W

Q′T of system OtXtYtZt with respect to

system OmXmYmZm in RKM is described as follows: W

Q′T =W Q′M ⋅M Q′T

(6)

Universal post processing algorithm of multi-axis machine tool considering geometric error 2.1 Universal post processing algorithm Subsequently, forward mapping relationship between axis space and tool space of these two different kinematic models will be constructed. The first forward mapping relationship between axis space and tool space is for IKM. Based on Eq. (7), we construct it as follows: ⎡0 ⎢0 ⎡v ‘ u’⎤ W ⎢ P=⎢ ⎥ = Ψ ( Δ)= QT ⎢0 ⎣1 0 ⎦ ⎢ ⎣1

0⎤ 0⎥⎥ 1⎥ ⎥ 0⎦

(7)

where Ψ represents the forward mapping function between the axis space and tool space of IKM. Similarly, forward mapping relationship for RKM can be constructed as follows: ⎡0 ⎢0 ⎡v ′ u′⎤ W ⎢ ′ ′ ( ) P′ = ⎢ Φ Δ Q = = T ⎥ ⎢0 ⎣1 0 ⎦ ⎢ ⎣1 where, v'= [v'x

0⎤ 0⎥⎥ 1⎥ ⎥ 0⎦

v'y v'z] is the cutter tip centre’s position vector; u'= [u'x u'y

(8)

u'z] is tool

orientation vector; Φ is forward mapping function in RKM which depends on the configuration of machine tool; Δ′ = (λ1′, λ2′ ...λt′ , φ1′, φ2′ ...φr′ ) is NC data, in which t is the number of linear axis and r is the number of rotational axis. Model calculation of Eq. (7) is relatively easy but does not conform to reality. Nevertheless, Eq. (8) which fits more to the reality is more difficult to solve directly for its big error and high order term. Therefore, the method to combine Eq. (7) with Eq. (8) is used as the following algorithm. The algorithm needs CL data as initial value, and NC data is obtained using inverse kinematics solution (IKS) and forward kinematics solution (FKS) separately. The algorithm

schematic is shown in Fig. 1, while the flow chart of the algorithm is shown in Fig. 2.

r2

r1

r3 r0

rp

rm

Fig.1. The algorithm schematic.

As shown in Fig. 2, the algorithm’s flow is described as follows: (1) Let NC data Δ0 be initial value, calculate CL data P0 using FKS; (2)

Let CL data P0 be initial value, calculate DC data Δ1 using IKS;

(3)

Let NC data Δ0 + ( Δ0 − Δ1 ) be initial value, calculate revised CL data

P1 using FKS; (4)

Repeat execution of the loop step (2) and (3) until Δi +1 − Δi ≤ δ , where δ

is the setting accuracy; (5)

The last DC data is the accuracy data after compensation.

Fig.2. Flow chart of algorithm.

The existence of geometric error makes IKS problem in Step (2) difficult to solve. Thus, the key of UPPA is to find an effective algorithm to deal with the IKS problem. Section 2.2 will elaborate the TDA to improve the calculation speed and reduce the difficulty of IKS problem. Because of embedding iterative structure in the UPPA, the effectiveness of the UPPA considering geometric error will be validated by proving the convergence of iterative algorithm in section 2.3 2.2 Total differential algorithm Generally, various solution algorithms for kinematic model deal with only those multi-axis NC machine tools with specific configuration case by case. Nevertheless, the TDA is a universal solution algorithm which is available for machine tool with arbitrary configuration. In section 2.1, comparing with FKS, IKS problem is extremely difficult as hundreds of error items are introduced into matrix equation. Therefore, a TDA is proposed to target just this problem. TDA described the spatial relationship of two adjacent CL points in a total differential form. Fig.3 shows spatial relationship

between two adjacent CL points:

Fig.3. Spatial relationship between two adjacent CL points.

The spatial relationship can be described as below: ⎡Δv x ⎢ Δv Pi +1 = Pi + ⎢ y ⎢ Δv z ⎢ ⎣ 0

Δu x ⎤ Δu y ⎥⎥ Δuz ⎥

(9)

⎥ 0 ⎦

where Δv x , Δv y and Δv z are increments between adjacent cutter tip centre’s location vectors; Δu x , Δu y and Δuz are increments between adjacent tool orientation vectors. Based on Eq.(7), taking five-axis machine tool with two rotational axes as example, the relationship of adjacent CL points can be described in total differential form as follows:

∂Ψ ( Δ) ∂Ψ ( Δ) ΔY + ΔX + ∂Y ∂X ∂Ψ ( Δ) ∂Ψ ( Δ) ∂Ψ ( Δ) ΔZ + ΔA + ΔC ∂Z ∂A ∂C

Pi +1 = Pi +

(10)

Eq. (10) is indeed a 4×2 linear matrix equation, and linear Eq. (11) with variables of increments of movement along each axis can be derived from Eq.(10) as follows:

⎧a11ΔA + a12ΔC + a13 = 0 ⎪a ΔA + a ΔC + a = 0 22 23 ⎪ 21 ⎪⎪a31ΔA + a32ΔC + a33 = 0 ⎨ ⎪a41ΔA + a42ΔC + a43ΔX + a44ΔY + a45ΔZ + a46 = 0 ⎪a51ΔA + a52ΔC + a53ΔX + a54ΔY + a55ΔZ + a56 = 0 ⎪ ⎪⎩a61ΔA + a62ΔC + a63ΔX + a64ΔY + a65ΔZ + a66 = 0

(11)

where ΔX , ΔY , ΔZ , ΔA and ΔC are increments of movement along each axis. Constant coefficients

a11 , a12 , a13 … a 64 , a 65 , and a 66 in Eq.(11) can be obtained by Eq.(10).

Owing to the redundancy of Eq.(11), five of the six equations is enough for solving process, and accurate solution can be obtained. 2.3 Validity of the UPPA The universal post processing algorithm has an embedded iteration structure, so validity of the algorithm depends on the convergence of the iteration structure. The relationship between CL points and NC data can be established as below.

Pc ,i = Ψ (Δ i ) = Φ (Δ 0 + dΔ i )

i = 1,2...n

(12)

where dΔ i = Δ 0 − Δ i ; Pc ,i is revised NC data. After the first IKS, NC data are substituted into ideal kinematic model to compute intermediate CL.

Pc,1 = Ψ (Δ1 ) = Φ (Δ 0 − dΔ1 )

(13)

where Pc,1 can be expressed in total differential form ignoring high-level minim o(dΔ1 ) as follow:

Pc,1 - Pd = (X1 − X 0 ) ⋅

∂Ψ ( Δ) ∂Ψ ( Δ) ∂Ψ (Δ ) + (Z1 − Z 0 ) ⋅ + (A1 − A 0 ) ⋅ ∂X Δ =Δ0 ∂Z Δ =Δ0 ∂A Δ =Δ 0

+ (B1 − B 0 ) ⋅ = ( −dΔ1 ) ⋅

∂Ψ (Δ ) ∂Ψ (Δ) + o ( dΔ 1 ) + (C1 − C 0 ) ⋅ ∂C Δ =Δ 0 ∂B Δ =Δ 0

dΨ ( Δ) + o ( dΔ 1 ) dΔ Δ = Δ 0 (14)

Ideal CL point is Pd = Ψ (Δ 0 ) . After second IKS, NC data are substituted into RKM to compute revised CL point Pc,2 = Φ ( Δ 0 + dΔ1 ) . The magnitude of terms in error transformation matrix

δQ is very minimal. Pc ,2 is expanded in total differential form as follow:

⎡ ⎤ dΦ( Δ) Pc,2 − Pd = ⎢Pd + (−dΔ1 ) ⋅ + o(dΔ1 )⎥ dΔ Δ=Δ1 ⎢⎣ ⎥⎦ ⎡ ⎤ dΦ( Δ) + ⎢(dΔ1 ) ⋅ + o(dΔ1 )⎥ − Pd + o(dΔ1 ) dΔ Δ=Δ1 ⎢⎣ ⎥⎦ = o(dΔ1 )

(15)

It can be derived that deviation between second intermediate CL and ideal CL is high-level minim of deviation at first time. According to cutter tip vector v and tool orientation vector u of CL point P , the relationship can be expressed as follow:

⎧⎪ v c , 2 − v d = o( v c ,1 − v d ) ⎨ ⎪⎩ uc , 2 − ud = o( uc ,1 − ud )

(16)

After i-th and (i+1)-th IKS, revised CL point has the same relationship as follows:

Pc,i - Pd = (−dΔ i ) ⋅

dΨ ( Δ) + o ( dΔ i ) dΔ Δ =Δi-1

Pc,i+1 − Pd = o(dΔ i )

(17)

(18)

Functions can be derived as follows:

⎧⎪ v c ,i +1 − v d = o( v c ,i − v d ) ⎨ ⎪⎩ uc ,i −1 − ud = o( uc ,i − ud ) According to the mathematical induction, when n → ∞ , derivation of CL converges to 0. ⎧ lim (v −v ) = 0 ⎪ n →∞ c ,i +1 d ⎨ ⎪ lim ( uc ,i −1 − ud ) = 0 ⎩ n →∞

(19)

(20)

Verification of the algorithm 3.1 Model of machine tool for large propeller Verification of the universal post processing algorithm is conducted on a five-axis mill-turn machine tool, which can machine large ship propeller having up to 10 meters diameter and up to 100 tons weight. The milling configuration of the machine tool has two linear axes and three rotational axes. Tool paths generated for propeller blade processing are shown in Fig.4

Fig.4. Tool paths in machining of certain type of propeller blade. Geometric structures and kinematic chain of the machine tool are shown in Fig.5.

Zaxis

X axis

C1 axis Baxis C2 axis tool

(a) Geometric structures of machine tool

(b) Kinematic chain

Fig.5. Geometric structures of machine tool and kinematic chain. According to the configuration of machine tool, forward mapping relationship of IKM can be obtained as follows: ⎡v x ⎢v P=⎢ y ⎢vz ⎢ ⎣1

ux ⎤ u y ⎥⎥ = Ψ ( Δ) uz ⎥ ⎥ 0⎦ ⎡− l t sin B1 cos(C1 + C 2 ) + X cos C 2 ⎢ − l sin B sin(C + C ) + X sin C 1 1 2 2 =⎢ t ⎢ − l t cos B1 + Z + lc ⎢ 1 ⎣

(21) sin B1 cos(C1 + C 2 )⎤ sin B1 sin(C1 + C 2 ) ⎥⎥ ⎥ cos B1 ⎥ 0 ⎦

where l t is the length of tool; l c is the distance along Z-axis between rotational axes C1 and C 2 ; B1 , C1 , C 2 , X and Z are NC data corresponding to CL point Pi .

3.2 Verification of TDA TDA is adopted to conduct post processing on large propeller machining machine. Eq. (10) can be embodied as follows:

cos(B1 ) cos(C1 + C 2 )ΔB1 − sin(B1 ) sin(C1 + C 2 )ΔC1 − ⎧ ⎪ sin(B1 ) sin(C1 + C 2 )ΔC 2 − ΔK x = 0 ⎪ + Δ B C C B cos( ) sin( ) ⎪ 1 1 2 1 + sin(B1 ) cos(C1 + C 2 )ΔC1 + ⎪ sin(B1 ) cos(C1 + C 2 )ΔC 2 − ΔK y = 0 ⎪ − sin(B1 )ΔB1 − ΔK z = 0 ⎪ ⎨ cos(C )ΔX − l cos(B ) cos(C + C )ΔB − X sin(C ))ΔC + 2 1 1 2 1 2 2 t ⎪ ⎪l t sin(B1 ) sin(C1 + C 2 )ΔC1 + (l t sin(B1 ) sin(C1 + C 2 ) − ΔQ x = 0 ⎪ sin(C 2 )ΔX − l t cos(B1 ) sin(C1 + C 2 )ΔB1 + x cos(C 2 ))ΔC 2 − ⎪ ⎪l t sin(B1 ) cos(C1 + C 2 )ΔC1 − l t sin(B1 ) cos(C1 + C 2 ) − ΔQ y = 0 ⎪ ΔZ − l t sin(B1 )ΔB1 − ΔQz = 0 ⎩ where ΔB1 , ΔC1 , ΔC 2 , ΔX and ΔZ are increments of movement along each axis.

(22)

Total differential algorithm and numeric algorithm [23] which combines steepest descent method and damping least square method are adopted to conduct post processing respectively. Relative errors of two methods to derivation method are shown in Fig.6.

(a) Value of B1 axis

(b) Value of B1 axis

(c) Value of C2 axis

(d) Value of C2 axis

(a) and (c) are value using total differential method; (b) and (d) are value using numeric method.

Fig.6. Relative errors of total differential method and numeric method. The results of TAD are extremely close to the results of derivation method, and calculation speed of total differential method is faster compared with numeric algorithm. Calculating 840 CL points costs only 5 seconds on computer having a 2.6 GHz frequency. The value of C1-axis in Fig.6 (b) indicates that numeric method has significant error, because small magnitude of C1-axis can lead singular of Jacobi matrix. Thus, claimed accuracy cannot be achieved and the same calculation task costs approximately 6 hours.

3.3 Verification of universal post processing algorithm

Fig.7. Verification of solving algorithm.

The universal post processing algorithm is adopted to solve RKM. Each geometric error item of the machine tool is given by the error calibration experiment in the section 4.2. To verify the correctness of the algorithm, the NC data calculated by UPPA are substituted into the RKM, and cutter tip vector and tool orientation vector are calculated by FKS. Relative error of the result to ideal CL data is shown in Fig.7. The magnitude of error between the cutter location calculated by UPPA and the ideal CL is lower than 10-4. Meanwhile the convergence speed of UPPA is quick and stable.

Geometric error sensitivity analysis and calibration experiment 4.1 Geometric error sensitivity analysis With the increase in the number of axis, geometric error of the machine tool also increases. Including the tool assembling error, the number of geometric error items of large-scale propeller machine tool is up to 37, which means that the calibration of geometric error is cumbersome and time-consuming. Therefore, orthogonal experiment design method is introduced because of the ease in allocating levels of variables by using an orthogonal array and the advantageous in reducing the number of combinations of experiments [24, 25]. Therefore, the orthogonal method is introduced. Through calculation and comparison, primary-slave relation of each error component’s impact on machining error can be determined, which is the basis of calibration experiment and foundation of RKM. Sensitivity analysis is conducted on all the motion axes of the machine tool. X-axis is taken for example to illustrate procedure. The six geometric error components of X-axis are taken as factors and each factor takes five levels: 0, ± 0.2 and ± 0.6.

RKM are established based on cases of the orthogonal experiment, then a set of 25 groups of contour error are simulated through computing with the RKM. The error distribution map and results of range analysis are shown in Fig.8 and Table 1.

Fig.8. Contour errors. Table 1 Table of analysis of range of contour error ΔZ

(mm )

Factor

ΔA (mm)

ΔB (mm)

ΔC (mm)

ΔX (mm)

ΔY (mm)

KI

77.7822

105.6045

70.2673

76.6997

53.8662

70.8804

KII

48.9320

38.8240

65.3993

54.7452

87.6474

63.3330

KIII

69.0624

36.6711

52.5295

59.4911

39.0168

63.3895

KIV

50.0501

39.0893

66.0267

56.5440

88.9498

63.5414

KV

79.1731

104.8109

70.7770

77.5198

55.5196

63.8555

Range charts R

30.2411

68.9334

18.2475

22.7746

49.9330

7.5474

It can be seen from the Fig.8 and Table 1 that all components of geometric error of X-axis have an impact on the contour error, but the angle error around Y-axis has the greatest influences on the contour error, which are up to 68.93mm. The influence of six geometric error components on the contour error decrease in order of angle error around Y-axis, linear error along Y-axis, angle error around Z-axis, linear error along Z-axis, angle error around X-axis, and linear error along X-axis. Using the same method, the orthogonal experiment of other axis is conducted. The most sensitive geometric error components of the cutter, B1 -axis, C1 -axis, Z -axis, X -axis and

C 2 -axis are linear error along Z-axis, angle error around Y-axis, linear error along X-axis, angle error around X-axis, angle error around Y-axis, angle error around Y-axis, respectively.

4.2 Geometric error calibration experiment To establish RKM of the five-axis machine tool, the geometric error needs to be calibrated. The geometric error can be obtained by measuring the direction of each motion axis, which is using a laser tracker APT T3, then geometric error can be derived and RKM can be established.

Fig.9. The measurement laser tracker. According to structure of the machine tool, the directions of the motion axis are measured successively. The most sensitive component of each moving parts given by sensitivity analysis is mainly considered. The results are shown in Table 2. Table 2 Six geometric error of each axis in machine tool ΔX

C2 axis

axis Z axis C1 axis X

B

axis

(mm)

0 0 0 0 0.0634

ΔY

(mm)

0 0 0 0.7596 0

ΔZ

(mm) 0 0 0 0 0

Δ A (°)

Δ B (°)

ΔC (°)

0 0 -0.0027 0.0099 0.0285

0 -0.0077 0.0007 0.0108 0

0 0 0 0 0

Verification experiment of kinematics solution algorithm considering geometric error Two groups of machining experiment are carried out in a five-axis machine tool. Group 1 processes a blade with NC data calculated by ideal post processing method, while Group 2 processes another blade with NC data calculated by UPPA. The other processing conditions is same for Group 1 and Group 2. The finished surface morphology of two groups are measured with a laser scanner EXAscan™, the picture of field test is shown in Fig.10.

Fig.10. Measuring experiment. Fig.11 shows triangle grid data fitting from point cloud data captured by the laser scanner.

Fig.11. Point cloud data of blades. Based on the analysis of point cloud data, machining error distribution maps of two groups are obtained.

Fig.12. Machining error distribution map of group 1.

Fig.13. Machining error distribution map of group 2. By comparing the Fig. 12 and Fig. 13, the machining error of blade reduces significantly group 2.The maximum error value decreased from 2mm to 0.4mm. The results indicate that the universal post processing algorithm is effective in decreasing geometric error of multi-axis machine tool.

Conclusions In the paper, a universal post processing algorithm considering geometric error of multi-axis NC machine tool in arbitrary configuration is proposed. Based on the ideal and real kinematic models of the multi-axis NC machine tool, a universal generalized post processing algorithm containing forward kinematics solution and inverse kinematics solution is derived to solve kinematic models. Then a total differential algorithm is applied to improve the calculation speed and reduce the difficulty of inverse kinematics solution. Based on it, inverse kinematics solution for adjacent cutter locations can be changed into linear equations about their spatial relationship. To establish kinematics model of machine tool considering geomeric error, geometric error sensitivity analysis and geometric error calibration experiments are carried out. Through orthogonal experiment, the effect weight of factors of geometric error component of each axis is obtained. Thus, distribution rule of machining error and contour error caused by geometric error is obtained. The effectiveness of the proposed kinematics solution algorithm considering geometric error is verified through machining experiments on a five-axis mill-turn machine tool. The results show that the maximum machining error value can be decreased to one-fifth using this new method. The presented algorithm can be applied to the serial multi-axis NC machine tool with arbitrary configuration. Next step would be to research universal post processing algorithm for parallel machine tool.

Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Grant No. 51075168, 50835004, 51121002).

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Highlights It proposed a universal post-processing algorithm considering geometric error for multi-axis machine tool with arbitrary configuration. Based on ideal and real kinematic models of the machine tool, a universal generalized post-processing algorithm containing forward kinematics solution and inverse kinematics solution is designed. Total differential algorithm is applied to improve the calculation speed and reduce the difficulty of inverse kinematics solution. Effect weight of factors of each axis geometric error component and distribution rules of machining error and contour error caused by geometric error are obtained through orthogonal experiment. Effectiveness of the kinematics solution algorithm considering geometric error is verified through machining experiments on a five-axis mill-turn machine tool.