CN105094047A - Extended Fourier amplitude based extraction method of machine tool important geometric error source - Google Patents

Abstract

The invention discloses an extended Fourier amplitude based extraction method of a machine tool important geometric error source. Based on error measuring data, by use of an index matrix form of a screw theory, on the basis of the topology structure of a machine tool, an integral space error model of the machine tool is established, a high-order term of the error model is eliminated, and a basis equation of the error model is obtained. According to an EFAST global sensitivity analysis method, through selecting a proper conversion function, the error model is converted into a one-dimensional function from an eighteen-dimensional function, Fourier series expansion is carried out on the one-dimensional function, and a model caused by each parameter and a total variance of model output can be obtained. The EFAST method can simultaneously examine the influences exerted by change of multiple geometric errors on the result of a spinor error model and can also analyze the direct and indirect influences exerted by change of each geometric error on the model result, and the method provided by the invention can be applied to extraction of key geometric error terms having quite large influences on processing precision of the machine tool.

Description

A kind of extracting method of the important geometric error source of lathe based on expansion Fourier amplitude

Technical field

The invention provides a kind of extracting method of the important geometric error source of lathe based on expansion Fourier amplitude, belong to Precision of NC Machine Tool design field.

Background technology

High-precision numerical control machine is usually used in modern production, be used in particular among high-level efficiency and complex-curved part, and this is also the important component part of processing and manufacturing and high-performance equipment manufacturing.Lathe space error is the most important part affecting machining precision, and geometric error accounts for about 40% of all errors, especially under precision and ultraprecise processing situation.In machining, machine finish is finally to be fixed a cutting tool by lathe and relative displacement between workpiece determines.Lathe is fixed a cutting tool and relative displacement available motion modeling technique between workpiece calculates.

The main manufacturing accuracy deriving from its functional part of the geometric error of lathe also has installation accuracy equal error.In order to better improve the precision of numerically-controlled machine, the foundation of error model is also very important, and sane accurate error model is also the first step of error identification.Domestic and international experts and scholars are setting up numerically-controlled machine spatial error model field always and are carrying out unremitting exploration and research, have carried out many-sided work.The margin of error that actual measurement is arrived, the reliable measurement mechanism of main dependence, efficiently measuring method etc.Domestic and international many scholars are studied geometrical error modeling method, propose many effective modeling methods.Conventional modeling method has: such as triangle relation modeling, the error moments tactical deployment of troops, secondary relational model method, theory of mechanisms modeling, rigid body kinematics method, Multi-system modeling method etc.

The geometric error of numerically-controlled machine intercouples, and such as, the geometric error of three axle lathes is 21, and they influence each other.Therefore, finding out the crucial geometric error affecting machine finish is a very difficult job.Fortunately, susceptibility assays can be used for identifying that the uncertainty change of geometric error is on the impact of machining precision and the geometric error item identifying most critical.According to the scope of sensitivity analysis, Local sensitivity analysis and Global sensitivity analysis can be divided into.Local sensitivity analysis only checks that single geometric error item is on the impact of machining precision, and Global sensitivity analysis then considers the impact of the effect of intercoupling on machining precision of multiple geometric error item, and geometric error item coupling effect is outwardness.Therefore, global sensitivity analysis method, compared with Local sensitivity analysis method, tallies with the actual situation more, has the advantage of its uniqueness.

The precision sensitivity analysis of relevant lathe or other mechanisms has been carried out in many research work.EFAST method is a kind of effective global sensitivity analysis method.Up to now, EFAST method is successfully applied to power industry and biological industry.In the present invention, in order to extract the crucial geometric error affecting machining precision, on spinor theory basis, a kind of extracting method of the important geometric error source of lathe based on expansion Fourier amplitude is suggested.

At present, Local sensitivity analysis method and global sensitivity analysis method all have this well to apply in respective evolution, but are considering the sensitivity analysis problem under lathe geometric error coupling condition, the but solution of neither one comparison system.Patent of the present invention, based on this kind of starting point, proposes a kind of lathe crucial geometric error sensitivity recognition methods based on expansion Fourier amplitude.The advantages such as the sample number that the method has steadily and surely, calculating efficient and need is lower; EFAST method passes through the decomposition to model output variance, each rank sensitivity indices of each geometric error of acquisition that can be quantitative and total responsive number index.That is EFAST method not only can check the change of multinomial geometric error on the impact of spinor error model result simultaneously, can also analyze the change of each geometric error to the direct of model result and remote effect, the method can be used to extract the crucial geometric error item larger on machine finish impact.Shown by the result of example calculation, method proposed by the invention is effective.

So set up accurate error model and just seem very important, the susceptibility how utilizing error model to analyze geometric error item accurately is originally one of the key issue in invention.

Summary of the invention

Object of the present invention provides a kind of extracting method of the important geometric error source of lathe based on expansion Fourier amplitude.Based on error measurement data, utilize the exponential matrix form of spinor theory, on the basis of the topological structure of lathe, set up the spatial error model of lathe entirety, the high-order term of error model is cut down, obtains the fundamental equation of error model.This error model has that computing velocity is fast, simple operation and other advantages; According to EFAST Global sensitivity analysis method, by choosing suitable transfer function, error model being converted to one-dimensional functions by ten octuple functions, fourier progression expanding method is carried out to this one-dimensional functions, the model side V that each parameter causes can be obtained _iand the population variance V that model exports.The population variance exported due to model is obtained jointly by coupling between each parameter and parameter, and EFAST method passes through the decomposition to model output variance, the responsive number index in each rank of each geometric error of acquisition that can be quantitative and total responsive number index.That is EFAST method not only can check the change of multinomial geometric error on the impact of spinor error model result simultaneously, the change of each geometric error can also be analyzed to the direct of model result and remote effect, finally have identified the critical error affecting machine finish, achieve Error Tracing &, the design for precise numerical control machine provides important theoretical foundation.The advantages such as the sample number that EFAST method has steadily and surely, calculating efficient and need is lower.

For achieving the above object, the technical solution used in the present invention is the crucial geometric error recognition methods of the modeling of a kind of numerically-controlled machine space error and lathe, for solving the technical matters in the crucial geometric error identifying of numerically-controlled machine.The implementation procedure of the method is as follows, Figure 1 shows that the concrete implementation step of this method.

Step one sets up the spatial synthesis error model of lathe according to spinor theory

According to the exponential matrix form of spinor theory, by abstract for each motion parts of lathe be the vector form of 6 × 1; By forms of motion and composition error modularized processing, and with the statement of exponential matrix form, set up the spatial synthesis error model of lathe according to the topological structure of lathe;

The exponential matrix form of step 1.1 spinor theory

The motion of any rigid body can be broken down into two parts: translation motion vertically and the rotary motion around axle; That is, all parts is regarded as spinor; Unit spinor becomes as follows at Pl ü cker coordinate:

Ψ＝[k ^Tu ^T] ^T＝[k ₁,k ₂,k ₃,u ₁,u ₂,u ₃] ^T(1)

state rigid body arbitrary motion form spatially, then have:

$\widehat{\mathrm{\Ψ}}=\left[\begin{array}{cc}\hat{k}& u\\ 0& 0\end{array}\right]---\left(2\right)$

Wherein, u=[u ₁u ₂u ₃] ^t, make difficulties title matrix, if k=is [k ₁k ₂k ₃] ^t, then be expressed as:

$\hat{k}=\left[\begin{array}{ccc}0& -k_3& k_2\\ k_3& 0& -k_1\\ -k_2& k_1& 0\end{array}\right]---\left(3\right)$

Rigid motion generally all comprises translation and rotation, and vectorial h is identical at rigid body coordinate system and reference frame; Then the homogeneous transform matrix of rigid body is:

$T=\left[\begin{array}{cc}R& h\\ 0& 1\end{array}\right]---\left(4\right)$

The secondly transformation matrix that the exponential form of spinor is corresponding can be written as: as k=0, rigid body only has translation motion, then homogeneous transform matrix can be written as:

$T=e^{\widehat{\mathrm{\Ψ}}t}=\left[\begin{array}{cc}{I}_{3\×3}& ut\\ 0& 1\end{array}\right]---\left(5\right)$

When k ≠ 0, for rigid body, also there is rotary motion, now exponential matrix is:

$e^{\widehat{\mathrm{\Ψ}}t}=\left[\begin{array}{cc}{e}^{\hat{k}t}& \left(I-e^{\frac{\hat{k}t}{\left|\right|k\left|\right|}\mathrm{\θ}}\right)\frac{\stackrel{\→}{k}\×\stackrel{\→}{u}}{{\mathrm{\θ}}^2}+\frac{\stackrel{\→}{k}{\stackrel{\→}{k}}^T\stackrel{\→}{u}}{{\mathrm{\θ}}^2}\\ 0& 1\end{array}\right]---\left(6\right)$

Wherein trigonometric series expansion be expressed as:

$e^{\hat{k}t}=I+\frac{\hat{k}}{\left|\right|k\left|\right|}\sin \mathrm{\θ}+\frac{\widehat{k^2}}{\left|\right|k|{|}^2}\left(1-\cos \mathrm{\θ}\right)---\left(7\right)$

To sum up, then the exponential matrix for rigid body arbitrary motion form is in space had to be expressed as:

$T=e^{\widehat{\mathrm{\Ψ}}\mathrm{\θ}}=\left\{\begin{array}{cc}\left[\begin{array}{cc}{I}_{3\×3}& \stackrel{\→}{u}\mathrm{\θ}\\ 0& 1\end{array}\right]& if\left|\right|k\left|\right|=0\\ \left[\begin{array}{cc}{e}^{\widehat{\mathrm{\Ψ}}\mathrm{\θ}}& \left(I-e^{\hat{k}\mathrm{\θ}}\right)\left(\stackrel{\→}{k}\×\stackrel{\→}{u}\right)+\stackrel{\→}{k}{\stackrel{\→}{k}}^T\stackrel{\→}{u}\\ 0& 1\end{array}\right]& if\left|\right|k\left|\right|\≠0\end{array}\right.---\left(8\right)$

When ψ is unit spinor, || during k|| ≠ 0, the rotation angle at mechanical position is expressed as || during k||=0, the distance of translation is expressed as the representation of point in different coordinates is different, and the difference transformation matrix between them states its relation; Spinor is also interpreted as a point in coordinate system, also different at the form of presentation of different coordinates, therefore also needs the form of transformation matrix to state the relation of spinor in different coordinates, is referred to as adjoint matrix; If the motion spinor θ ψ of rigid body, the exponential matrix of its variation can be expressed as:

$e^{\widehat{\mathrm{\Ψ}}\mathrm{\θ}}=\left[\begin{array}{cc}R& h\\ 0& 1\end{array}\right]---\left(9\right)$

Adjoint exponential matrix form then under its this coordinate system:

$Adj\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)=\left[\begin{array}{cc}R& 0\\ \hat{h}R& R\end{array}\right]---\left(10\right)$

Following character is met with exponential matrix:

${\stackrel{\→}{\mathrm{\Ψ}}}_1=Adj\left(e^{\widehat{\mathrm{\θ}}\widehat{\mathrm{\Ψ}}}\right)S_2=e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}{\widehat{\mathrm{\Ψ}}}_2{\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)}^{-1}---\left(11\right)$

$e^{{\widehat{\mathrm{\Ψ}}}_1}=e^{e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}{\widehat{\mathrm{\Ψ}}}_2{\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)}^{-1}}=e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}{e}^{\mathrm{\θ}{\widehat{\mathrm{\Ψ}}}_2}{\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)}^{-1}---\left(12\right)$

For the physical construction that lathe is such, stating its structure with exponential matrix then has:

$T=e^{{\widehat{\mathrm{\Ψ}}}_1{\mathrm{\θ}}_1}\·e^{{\widehat{\mathrm{\ψ}}}_1{\mathrm{\θ}}_2}\mathrm{...}\·e^{{\widehat{\mathrm{\Ψ}}}_n{\mathrm{\θ}}_n}\·T\left(0\right)---\left(13\right)$

T (0) represents its original transform matrix, and formula (13) can be applicable to the error modeling of lathe;

The space error modeling of step 1.2 lathe

General, each axial motion can have 6 direction degree of freedom, can produce error and 3 errors of rotating of 3 translations simultaneously; Utilize spinor theory, define error module m _eΨ _e;

m _eΨ _e＝[β _x,β _y,β _z,α _x,α _y,α _z] ^T(14)

With X to geometric error consist of example, be divided into three parts; Part I Ψ _xxcomprise positioning error and rolling pendulum error alpha in the direction _xx, β _xx; Part II Ψ _yxlinear error and the top pendulum error alpha of surface level _yxβ _yx; Part III Ψ _zxlinear error and the Run-out error α of vertical plane _zx, β _zx;

Ψ _xx＝[β _xx,0,0,α _xx,0,0] ^T(15)

Ψ _yx＝[0,β _yx,0,0,α _yx,0] ^T(16)

Ψ _zx＝[0,0,β _zx,0,0,α _zx] ^T(17)

The space error of X-axis can be expressed as:

$e^{{\widehat{\mathrm{\Ψ}}}_{xe}}=e^{{\widehat{\mathrm{\Ψ}}}_{xx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zx}}---\left(18\right)$

The error model exponential matrix form of X-axis, is expressed as:

$T_a^x=e^{x{\widehat{\mathrm{\Ψ}}}_x}\·e^{{\widehat{\mathrm{\Ψ}}}_{xe}}=e^{x{\widehat{\mathrm{\Ψ}}}_x}\·e^{{\widehat{\mathrm{\Ψ}}}_{xx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zx}}---\left(19\right)$

In like manner can obtain the space error module of other axles and the error model of exponential matrix;

Step 1.3 is about the exponential matrix form of verticality and parallelism error

Axle and axle ideally due to reality are difference to some extent, and two adjacent axles are not absolute 90 °; That is there is the error of perpendicularity; For three translation shaft, definition Y-axis is desired axis, there is not the error of perpendicularity; The error of perpendicularity then between X-axis and Y-axis is γ _xy, the error of perpendicularity between Y-axis and Z axis is γ _yz, the verticality between X and Z is γ _xz; In the plane that the X axis of Y-axis and actual installation forms, only γ is existed for X-axis _xy, in like manner there are other two error of perpendicularitys at actual Z axis; Because actual axis direction inevitably will depart from the position of desired axis, therefore should consider to add the error of perpendicularity in coordinate transform, for ideal coordinates transformation of axis form:

For X to, X ideally to unit spinor representation is:

Ψ _xi＝[0,0,0,1,0,0] ^T(20)

Add the error of perpendicularity under actual state, then X to the unit spinor representation of reality is:

Ψ _xs＝[0,0,0,cos(γ _xy),-sin(γ _xy),0] ^T(21)

Corresponding exponential matrix is expressed as:

$e^{x{\widehat{\mathrm{\Ψ}}}_{xs}}=\left[\begin{array}{cccc}1& 0& 0& x\cos \left({\mathrm{\γ}}_{xy}\right)\\ 0& 1& 0& -x\sin \left({\mathrm{\γ}}_{xy}\right)\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]---\left(22\right)$

Another kind of literary style, utilizes the form of adjoint matrix to rotate-γ around Z axis desirable X-axis _xyangle reaches the effect that X-axis and Y-axis are 90 °, that is:

${\mathrm{\Ψ}}_{xs}=Adj\left(e^{-{\mathrm{\γ}}_{xy}{\widehat{\mathrm{\Ψ}}}_{zr}}\right)\·{\mathrm{\Ψ}}_{xi}---\left(23\right)$

Ψ _zr＝[0,0,1,0,0,0] ^T

Under actual conditions, the unit spinor representation of Z-direction is:

Ψ _zi＝[0,0,0,0,0,1] ^T

Two error of perpendicularitys taken into account, the positional representation of actual Z axis is:

$e^{z{\widehat{\mathrm{\Ψ}}}_{zs}}=\left[\begin{array}{cccc}1& 0& 0& -z\sin \left({\mathrm{\γ}}_{xz}\right)\\ 0& 1& 0& -z\sin \left({\mathrm{\γ}}_{yz}\right)\cos \left({\mathrm{\γ}}_{xz}\right)\\ 0& 0& 1& z\cos \left({\mathrm{\γ}}_{yz}\right)\cos \left({\mathrm{\γ}}_{xz}\right)\\ 0& 0& 0& 1\end{array}\right]---\left(24\right)$

Wherein, Ψ _zs=[0,0,0 ,-sin (γ _xz) ,-sin (γ _yz) cos (γ _xz), cos (γ _yz) cos (γ _xz)] ^t

Step 1.4 is based on the foundation of the direction of principal axis error model of three under topological structure

Theory of multi body system provides the very detailed topology controlment about lathe, can apply too in exponential matrix, this method choose the topological structure of three axle lathes as shown in Figure 2.

Ideally, lathe there is not error; Matrixing equation ideally can use T _widealrepresent:

$T_{wideal}=e^{-x{\widehat{\mathrm{\Ψ}}}_{xi}}\·e^{-y{\widehat{\mathrm{\Ψ}}}_{yi}}\·e^{z{\widehat{\mathrm{\Ψ}}}_{zi}}---\left(25\right)$

Under actual conditions, due to the error of position between the error of machine tool component self and parts, global facility error spinor is joined in spinor module; Use T _wactualrepresent:

$T_{wactual}=e^{-{\widehat{\mathrm{\Ψ}}}_{xe}}\·e^{-x{\widehat{\mathrm{\Ψ}}}_{xs}}\·e^{-{\widehat{\mathrm{\Ψ}}}_{ye}}\·e^{-y{\widehat{\mathrm{\Ψ}}}_{ys}}\·e^{{\widehat{\mathrm{\Ψ}}}_F}\·e^{z{\widehat{\mathrm{\Ψ}}}_{zs}}\·e^{{\widehat{\mathrm{\Ψ}}}_{ze}}---\left(26\right)$

In above formula, S _f=[0,0,0,0,0,0], represents ground spinor.

In workpiece coordinate system, according to actual and ideally matrixing equation, obtain the spatial error model of multi-axis NC Machine Tools:

$E=T_{wideal}^{-1}{T}_{wactual}---\left(27\right)$

Corresponding space error component e in three axial directions _x, e _y, e _zbe expressed as:

[e _x,e _y,e _z,1] ^T＝E·[0,0,0,1] ^T(28)

Omit high-order term more than second order and second order in formula, just obtain the fundamental equation of space error;

Step 2 is based on the global sensitivity analysis method of EFAST method

EFAST method is a kind of global sensitivity analysis method proposed in conjunction with the advantage of Fourier modulus sensitivity assays method.The method is sane, calculating is efficient and sample number that is that need is lower.It has employed the thought of variance analysis, thinks that the variance that model exports is caused by the interaction between each input parameter and parameter, can reflect that model exports the susceptibility to input parameter.Therefore, the contribution proportion of the coupling between parameters and parameter to population variance can be obtained by the decomposition of model variance, be parameter sensitivity sex index.Application the method should know geometry distribution and the span of each geometric error item to be analyzed in advance.

Now definition has model Y=f (x), and input parameter is X (x ₁, x ₂..., x _n).Each parameter has certain variation range and distribution form (Fig. 1), constitutes a multi-C parameter space.Based on the method parameter of analytic model susceptibility process as shown in Figure 1.

In the method, in order to analyze the size degree that every geometric error affects machining precision, first the error of perpendicularity defined between three axles is known, and get work space x=300, y=300, z=200 place is analysis site.Then error model can be expressed as E=[e _x, e _y, e _z, 0] ^t, wherein

E=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅, x ₁₆, x ₁₇, x ₁₈] ^tfor the function of 18 geometric errors of Water demand, if

α _xx,α _yx,α _zx,β _xx,β _yx,β _zx,α _xy,α _yy,α _zy,β _yy,β _zy,β _xy,α _xz,α _yz,α _zz,β _xz,β _zz,β _yz＝

x ₁,x ₂,x ₃,x ₄,x ₅,x ₆,x ₇,x ₈,x ₉,x ₁₀,x ₁₁,x ₁₂,x ₁₃,x ₁₄,x ₁₅,x ₁₆,x ₁₇,x ₁₈

First be the parameter X of Water demand _i(i=1,2 ..., n) choose a sampling function x _i:

Wherein, for a random phase between [0,2 π], get

In formula, n is the total number of uncertain parameters; w _ifor the characteristic frequency of parameter; S is the common independent variable of all parameters, gets N at equal intervals in (-π, π) interval _s=2Mw _max+ 1 value, constant M here gets 4 or 6, w usually _maxfor sequence { w _iin maximal value.Sequence { w _ishould nonlinear correlation be met, namely

$\underset{i=1}{\overset{n}{\Σ}}{r}_i\·w_i\≠0$

In formula, r _ifor integer.

Then y=f (x ₁, x ₂..., x ₁₈) y=f (s) can be converted into namely

f(s)＝f[X ₁(sinw ₁s),X ₂(sinw ₂s),...,X ₁₈(sinw ₁₈s)]s∈(-π,π)(30)

Carry out fourier progression expanding method to f (s) to obtain:

$y=f\left(s\right)\≈\underset{-\frac{N_s-1}{2}}{\overset{\frac{N_s-1}{2}}{\Σ}}\{A_j\·cos(j\·s)+B_j\·\sin (j\·s)\}---\left(31\right)$

Wherein:

$A_j=\frac{1}{N_s}\underset{k=1}{\overset{N_s}{\Σ}}f\left(s_k\right)\·cos(j\·s_k),B_j=\frac{1}{N_s}\underset{k=1}{\overset{N_s}{\Σ}}f\left(s_k\right)\·sin(j\·s_k)---\left(32\right)$

Wherein, N _sfor number of samples, $j\∈Z=\{-\frac{N_s-1}{2},...,-2,-1,0,1,2,...,+\frac{N_s-1}{2}\}$

s _k＝π/N _s(2k-N _s-1),k＝1,2,...,N _s

$z^0=z-\left\{0\right\}=\{-\frac{N_s-1}{2},...,-2,-1,1,2,...,+\frac{N_s-1}{2}\}$

Then Fourier spectrum curve definitions is: $\mathrm{\Λ}p=A_j^2+B_j^2---\left(33\right)$ A _-j=A _j, B _-j=B _j, Λ _-j=Λ _jthen by parameter x _imodel result variance V caused by input change _iparameter can be expressed as

W _i1,2,3 ... the spectrum curve sum at M frequency multiplication place, that is:

$V_i=\underset{p\∈Z^0}{\Σ}{\mathrm{\Λ}}_{{\mathrm{pw}}_i}=2\·\underset{p=1}{\overset{M}{\Σ}}\left(A_{p\·w_i}^2+B_{p\·w_i}^2\right)---\left(34\right)$

Wherein: z ⁰=z-{0}, ω _ifor parameter x _ithe integer frequency defined, wherein M generally get 4 or 6, p be integer.

Then model population variance V is:

$V=\underset{p\∈Z^0}{\Σ}{\mathrm{\Λ}}_p=2\·\underset{j=1}{\overset{\frac{N_s-1}{2}}{\Σ}}{\mathrm{\Λ}}_j---\left(35\right)$

By A _pand B _pand parameter x _icorresponding frequencies omega _ithrough type (34), (35) can obtain the model side V that each parameter causes _iand the population variance V that model exports.The population variance exported due to model is obtained jointly by coupling between each parameter and parameter, can by as follows for its exploded representation:

$V=\underset{i}{\Σ}{V}_i+\underset{i\≠j}{\Σ}{V}_{ij}+\underset{i\≠j\≠m}{\Σ}{V}_{ijm}+\mathrm{...}+V_{\mathrm{12...}k}---\left(36\right)$

Wherein: V _ijfor parameter x _iby parameter x _jthe variance (coupling variance) of contribution; V _ijmfor parameter x _iby parameter x _jand x _mthe variance of contribution; V _12...kfor parameter x _iby parameter x ₁, x ₂..., x _kthe variance of contribution.Therefore, by normalized, parameter x _isingle order sensitivity indices S _ican be defined as follows:

$S_i=\frac{V_i}{V}---\left(37\right)$

What this sensitivity indices reflected is that parameter exports the direct contribution ratio of population variance to model.In like manner, parameter x _isecond order and three rank sensitivity indices may be defined as:

$S_{ij}=\frac{V_{ij}}{V},S_{ijm}=\frac{V_{ijm}}{V}$

For a Multi-parameter coupling model, parameter x _ioverall sensitivity indices be each rank sensitivity indices sum, can be expressed as:

S _Ti＝S _i+S _ij+S _ijm+...+S _12...i...k

$S_{T_i}=\frac{V-V_{-i}}{V}---\left(38\right)$

Total sensitivity indices is reflected parameter direct contribution ratio and indirectly model is exported to the contribution sum of population variance by the coupling interaction effect with other parameters, and between parameter without coupling time, S _ijand S _ijm0, S is Deng item _ti=S _i, EFAST analyzes and is equal to local sensitivity analysis.EFAST method, by the decomposition to model output variance, can obtain each rank of each parameter and total susceptibility quantitatively, and distinguish the possible influence degree of parameter to analog result according to susceptibility.This just makes EFAST method that the change of multiple parameter not only can be checked the impact of error model result, also can analyze each Parameters variation to the direct of analog result and remote effect.

The circular of total sensitivity is as follows: first distribute to parameter X _ia larger frequency w _i, and be the different integer frequency { w that remaining other setting parameter one group is less _i, and should relation w be met _i>=2Mmax{w ' _i.So just Dividing in frequency domain is become two parts [1, Mmax (w ' _i)] and [Mmax (w ' _i)+1, (N _s-1)/2]

Finally can calculate parameter x by formula (39) _itotal sensitivity value:

${\mathrm{ST}}_i=\frac{\underset{j=M\·\max \left(\left\{w_i^{\′}\right\}\right)+1}{\overset{\frac{N_s-1}{2}}{\Σ}}\left(A_j^2+B_j^2\right)}{\underset{j=1}{\overset{\frac{N_s-1}{2}}{\Σ}}\left(A_j^2+B_j^2\right)}---\left(39\right)$

Accompanying drawing explanation

Fig. 1 .EFAST method carries out the process flow diagram of global sensitivity analysis.

Fig. 2. the topology diagram of three weeks vertical machining centres.

Fig. 3. the line frame graph of three shaft vertical machining centers.

Fig. 4. at Chosen Point one order broken line graph in X direction.

Fig. 5. at Chosen Point global sensitivity broken line graph in X direction.

Fig. 6. at the one order broken line graph of Chosen Point along Y-direction.

Fig. 7. at the global sensitivity broken line graph of Chosen Point along Y-direction.

Fig. 8. at the one order broken line graph of Chosen Point along Z-direction.

Fig. 9. at the global sensitivity broken line graph of Chosen Point along Z-direction.

Figure 10. the distribution schematic diagram of test point.

Figure 11. at the broken line graph of whole work space one order in X direction.

Figure 12. at the broken line graph of whole work space one order in X direction.

Figure 13. at the broken line graph of whole work space one order in X direction.

Figure 14. at the broken line graph of whole work space one order in X direction.

Figure 15. at the broken line graph of whole work space one order in X direction.

Figure 16. at the broken line graph of whole work space one order in X direction.

Embodiment

Example: for three-shaft linkage numerical control machine tool (Fig. 3)

Step one sets up the spatial synthesis error model of lathe according to spinor theory

According to the exponential matrix form of spinor theory, by abstract for each motion parts of lathe be the vector form of 6 × 1.By forms of motion and error module process, and with the statement of exponential matrix form, set up the spatial error model of lathe according to the topological structure (Fig. 2) of lathe.

The exponential matrix form of step 1.1 spinor theory

The motion of any rigid body can be broken down into two parts: translation motion vertically and the rotary motion around axle.Can all parts be regarded as spinor.Unit spinor is into as follows at Pl ü cker coordinate:

Ψ＝[k ^Tu ^T] ^T＝[k ₁,k ₂,k ₃,u ₁,u ₂,u ₃] ^T(40)

state rigid body arbitrary motion form spatially, then have:

$\widehat{\mathrm{\Ψ}}=\left[\begin{array}{cc}\hat{k}& u\\ 0& 0\end{array}\right]---\left(41\right)$

Wherein, u=[u ₁u ₂u ₃] ^t, make difficulties title matrix, if k=is [k ₁k ₂k ₃] ^t, then can be expressed as:

$\hat{k}=\left[\begin{array}{ccc}0& -k_3& k_2\\ k_3& 0& -k_1\\ -k_2& k_1& 0\end{array}\right]---\left(42\right)$

Rigid motion generally all comprises translation and rotation, and vectorial h is identical at rigid body coordinate system and reference frame.Then the homogeneous transform matrix of rigid body is:

$T=\left[\begin{array}{cc}R& h\\ 0& 1\end{array}\right]---\left(43\right)$

The secondly transformation matrix that the exponential form of spinor is corresponding can be written as: as k=0, rigid body only has translation motion, then homogeneous transform matrix is:

$T=e^{\widehat{\mathrm{\Ψ}}t}=\left[\begin{array}{cc}{I}_{3\×3}& ut\\ 0& 1\end{array}\right]---\left(44\right)$

When k ≠ 0, also there is rotary motion for rigid body, now exponential matrix is:

$e^{\widehat{\mathrm{\Ψ}}t}=\left[\begin{array}{cc}{e}^{\hat{k}t}& \left(I-e^{\frac{\hat{k}t}{\left|\right|k\left|\right|}\mathrm{\θ}}\right)\frac{\stackrel{\→}{k}\×\stackrel{\→}{u}}{{\mathrm{\θ}}^2}+\frac{\stackrel{\→}{k}{\stackrel{\→}{k}}^T\stackrel{\→}{u}}{{\mathrm{\θ}}^2}\\ 0& 1\end{array}\right]---\left(45\right)$

Wherein trigonometric series expansion be expressed as:

$e^{\hat{k}t}=I+\frac{\hat{k}}{\left|\right|k\left|\right|}\sin \mathrm{\θ}+\frac{\widehat{k^2}}{\left|\right|k|{|}^2}\left(1-\cos \mathrm{\θ}\right)---\left(46\right)$

To sum up, then the exponential matrix for rigid body arbitrary motion form is in space had to be expressed as:

$T=e^{\widehat{\mathrm{\Ψ}}\mathrm{\θ}}=\left\{\begin{array}{cc}\left[\begin{array}{cc}{I}_{3\×3}& \stackrel{\→}{u}\mathrm{\θ}\\ 0& 1\end{array}\right]& if\left|\right|k\left|\right|=0\\ \left[\begin{array}{cc}{e}^{\widehat{\mathrm{\Ψ}}\mathrm{\θ}}& \left(I-e^{\hat{k}\mathrm{\θ}}\right)\left(\stackrel{\→}{k}\×\stackrel{\→}{u}\right)+\stackrel{\→}{k}{\stackrel{\→}{k}}^T\stackrel{\→}{u}\\ 0& 1\end{array}\right]& if\left|\right|k\left|\right|\≠0\end{array}\right.---\left(47\right)$

When ψ is unit spinor, || during k|| ≠ 0, the rotation angle at mechanical position is expressed as || during k||=0, the distance of translation is expressed as the representation of point in different coordinates is different, and the difference transformation matrix between them states its relation; Spinor is also interpreted as a point in coordinate system, also different at the form of presentation of different coordinates, therefore also needs the form of transformation matrix to state the relation of spinor in different coordinates, is referred to as adjoint matrix; If the motion spinor θ ψ of rigid body, the exponential matrix of its variation can be expressed as:

$e^{\widehat{\mathrm{\Ψ}}\mathrm{\θ}}=\left[\begin{array}{cc}R& h\\ 0& 1\end{array}\right]---\left(48\right)$

Adjoint exponential matrix form then under its this coordinate system:

$Adj\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)=\left[\begin{array}{cc}R& 0\\ \hat{h}R& R\end{array}\right]---\left(49\right)$

Following character is met with exponential matrix:

${\widehat{\mathrm{\Ψ}}}_1=Adj\left(e^{\widehat{\mathrm{\θ}}\widehat{\mathrm{\Ψ}}}\right)S_2=e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}{\widehat{\mathrm{\Ψ}}}_2{\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)}^{-1}---\left(50\right)$

$e^{{\widehat{\mathrm{\Ψ}}}_1}=e^{e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}{\widehat{\mathrm{\Ψ}}}_2{\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)}^{-1}}=e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}{e}^{{\widehat{\mathrm{\Ψ}}}_2}{\left(e^{\mathrm{\θ}\widehat{\mathrm{\Ψ}}}\right)}^{-1}---\left(51\right)$

For the physical construction that lathe is such, stating its structure with exponential matrix then has:

$T=e^{{\widehat{\mathrm{\Ψ}}}_1{\mathrm{\θ}}_1}\·e^{{\widehat{\mathrm{\ψ}}}_1{\mathrm{\θ}}_2}\mathrm{...}\·e^{{\widehat{\mathrm{\Ψ}}}_n{\mathrm{\θ}}_n}\·T\left(0\right)---\left(52\right)$

T (0) represents its original transform matrix, can be applied to the error modeling of lathe.

Step 1.2 utilization index matrix type carries out spatial synthesis error modeling to lathe

General, each axial motion can have 6 direction degree of freedom, can produce error and 3 errors of rotating of 3 translations simultaneously.Utilize spinor theory, definition error module m _eΨ _e.

m _eΨ _e＝[β _x,β _y,β _z,α _x,α _y,α _z] ^T

With X to geometric error consist of example, be mainly divided into three parts; Part I Ψ _xxcomprise positioning error and rolling pendulum error alpha in the direction _xx, β _xx; Part II Ψ _yxlinear error and the top pendulum error alpha of surface level _yx, β _yx; Part III Ψ _zxlinear error and the Run-out error α of vertical plane _zx, β _zx;

Ψ _xx＝[β _xx,0,0,α _xx,0,0] ^T

Ψ _yx＝[0,β _yx,0,0,α _yx,0] ^T

Ψ _zx＝[0,0,β _zx,0,0,α _zx] ^T

The space error of X-axis is expressed as:

$e^{{\widehat{\mathrm{\Ψ}}}_{xe}}=e^{{\widehat{\mathrm{\Ψ}}}_{xx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zx}}---\left(53\right)$

The error model exponential matrix form of X-axis, is expressed as:

$T_a^x=e^{x{\widehat{\mathrm{\Ψ}}}_x}\·e^{{\widehat{\mathrm{\Ψ}}}_{xe}}=e^{x{\widehat{\mathrm{\Ψ}}}_x}\·e^{{\widehat{\mathrm{\Ψ}}}_{xx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yx}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zx}}---\left(54\right)$

In like manner, the exponential matrix form of the error model of Y-axis and Z axis is expressed as follows:

Ψ _xy＝[β _xy,0,0,α _xy,0,0] ^T

Ψ _yy＝[0,β _yy,0,0,α _yy,0] ^T

Ψ _zy＝[0,0,β _zy,0,0,α _zy] ^T

$e^{{\widehat{\mathrm{\Ψ}}}_{ye}}=e^{{\widehat{\mathrm{\Ψ}}}_{xy}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yy}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zy}}---\left(55\right)$

$T_a^y=e^{y{\widehat{\mathrm{\Ψ}}}_y}\·e^{{\widehat{\mathrm{\Ψ}}}_{ye}}=e^{y{\widehat{\mathrm{\Ψ}}}_y}\·e^{{\widehat{\mathrm{\Ψ}}}_{xy}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yy}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zy}}---\left(56\right)$

Ψ _xz＝[β _xz,0,0,α _xz,0,0] ^T

Ψ _yz＝[0,β _yz,0,0,α _yz,0] ^T

Ψ _zz＝[0,0,β _zz,0,0,α _zz] ^T

$e^{{\widehat{\mathrm{\Ψ}}}_{ze}}=e^{{\widehat{\mathrm{\Ψ}}}_{xz}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yz}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zz}}---\left(57\right)$

$T_a^z=e^{z{\widehat{\mathrm{\Ψ}}}_z}\·e^{{\widehat{\mathrm{\Ψ}}}_{ze}}=e^{z{\widehat{\mathrm{\Ψ}}}_z}\·e^{{\widehat{\mathrm{\Ψ}}}_{ze}}=e^{z{\widehat{\mathrm{\Ψ}}}_z}\·e^{{\widehat{\mathrm{\Ψ}}}_{xz}}\·e^{{\widehat{\mathrm{\Ψ}}}_{yz}}\·e^{{\widehat{\mathrm{\Ψ}}}_{zz}}---\left(58\right)$

Step 1.3 is about the exponential matrix form of verticality and parallelism error

Axle and axle ideally due to reality are difference to some extent, and two adjacent axles are not absolute 90 °; That is there is the error of perpendicularity.For three translation shaft, Y-axis is desired axis, there is not the error of perpendicularity; The error of perpendicularity then between X-axis and Y-axis is S _xy, the error of perpendicularity between Y-axis and Z axis is S _yz, the verticality between X and Z is S _xz.In the plane that the X axis of Y-axis and actual installation forms, only S is existed for X-axis _xy, in like manner there are other two error of perpendicularitys at actual Z axis.

For X to, X ideally can be expressed as to unit spinor:

$ _xi＝[0,0,0,1,0,0] ^T

Add the error of perpendicularity under actual state, then X to the unit spinor representation of reality is:

$ _xs＝[0,0,0,cos(S _xy),-sin(S _xy),0] ^T

Corresponding exponential matrix is expressed as:

$e^{x{\widehat{\$}}_{xs}}=\left[\begin{array}{cccc}1& 0& 0& x\cos \left(S_{xy}\right)\\ 0& 1& 0& -x\sin \left(S_{xy}\right)\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]---\left(59\right)$

Another kind of mode, utilizes the form of adjoint matrix to take Z as axle ($ by desirable X-axis _zr) rotate to an angle to reach the effect that X-axis and Y-axis are 90 °, that is:

${\$}_{xs}=Adj\left(e^{-S_{xy}{\widehat{\$}}_{zr}}\right)\·{\$}_{xi}---\left(60\right)$

$ _zr＝[0,0,1,0,0,0] ^T

It is that axle rotates that Section 2 " r " in subscript represents with Section 1;

Similar, Z-direction unit spinor representation is:

$ _zi＝[0,0,0,0,0,1] ^T

Add the error of perpendicularity under actual state, then the unit spinor representation of Z-direction reality is:

$ _zs＝[0,0,0,-sin(S _xz),-sin(S _yz)cos(S _xz),cos(S _yz)cos(S _xz)] ^T(61)

$e^{z{\widehat{\$}}_{zs}}=\left[\begin{array}{cccc}1& 0& 0& -zsin\left(S_{xz}\right)\\ 0& 1& 0& -zcos\left(S_{xz}\right)sin\left(S_{yz}\right)\\ 0& 0& 1& z\cos \left(S_{xz}\right)\cos \left(S_{yz}\right)\\ 0& 0& 0& 1\end{array}\right]$

Another kind of literary style utilizes the form of adjoint matrix to rotate-γ around Z axis desirable X-axis _xyangle reaches the effect that X-axis and Y-axis are 90 °, that is:

${\mathrm{\Ψ}}_{xs}=Adj\left(e^{-{\mathrm{\γ}}_{xy}{\widehat{\mathrm{\Ψ}}}_{zr}}\right)\·{\mathrm{\Ψ}}_{xi}---\left(62\right)$

Ψ _zr＝[0,0,1,0,0,0] ^T

Under actual conditions, the unit spinor of Z-direction can be expressed as:

Ψ _zi＝[0,0,0,0,0,1] ^T

Two error of perpendicularitys taken into account, the position of actual Z axis can be expressed as:

$e^{z{\widehat{\mathrm{\Ψ}}}_{zs}}=\left[\begin{array}{cccc}1& 0& 0& -z\sin \left({\mathrm{\γ}}_{xz}\right)\\ 0& 1& 0& -z\sin \left({\mathrm{\γ}}_{yz}\right)\cos \left({\mathrm{\γ}}_{xz}\right)\\ 0& 0& 1& z\cos \left({\mathrm{\γ}}_{yz}\right)\cos \left({\mathrm{\γ}}_{xz}\right)\\ 0& 0& 0& 1\end{array}\right]---\left(63\right)$

Wherein, Ψ _zs=[0,0,0 ,-sin (γ _xz) ,-sin (γ _yz) cos (γ _xz), cos (γ _yz) cos (γ _xz)] ^t

Step 1.4 is set up based on the error model under topological structure

Theory of multi body system provides the very detailed topology controlment about lathe, can apply too in exponential matrix.Ideally, lathe there is not error; Matrixing equation T ideally _irepresent:

$T_{wideal}=e^{-x{\widehat{\mathrm{\Ψ}}}_{xi}}\·e^{-y{\widehat{\mathrm{\Ψ}}}_{yi}}\·e^{z{\widehat{\mathrm{\Ψ}}}_{zi}}---\left(64\right)$

The error extension matrix model of overall lathe is written as:

$T_{wactual}=e^{-{\widehat{\mathrm{\Ψ}}}_{xe}}\·e^{-x{\widehat{\mathrm{\Ψ}}}_{xe}}\·e^{-{\widehat{\mathrm{\Ψ}}}_{ye}}\·e^{-y{\widehat{\mathrm{\Ψ}}}_{ys}}\·e^{{\widehat{\mathrm{\Ψ}}}_F}\·e^{z{\widehat{\mathrm{\Ψ}}}_{zs}}\·e^{{\widehat{\mathrm{\Ψ}}}_{ze}}---\left(65\right)$

Wherein, S _f=[0,0,0,0,0,0] ^trepresent the spinor of ground;

Based on spinor exponential matrix represented by five-axis machine tool spatial error model be expressed as:

$E=T_{wideal}^{-1}{T}_{wactual}---\left(66\right)$

Corresponding space error component e in three axial directions _x, e _y, e _zbe expressed as:

[e _x,e _y,e _z,1] ^T＝E·[0,0,0,1] ^T(67)

For multiaxis NC maching lathe, the alignment error of cutter and workpiece has been left in the basket at this due to very little.So the space error of lathe component in three axial directions can be written as:

e _x＝-α _xx-α _xy+α _zxβ _yx+α _zyβ _yx-α _yyβ _xyβ _yx+α _zyβ _yy+xγ _xyβ _zx-α _yyβ _zx-α _zyβ _xyβ _zx

-(z+α _zz+α _yzβ _xz)(β _yx+β _yy-β _xyβ _zx)+(1+β _zx(β _xyβ _yy-β _zy)-β _yx(β _yy+β _xyβ _zy))

-y(β _zx+(1-β _yxβ _yy)β _zy+β _xy(β _yx+β _yyβ _zxβ _zy))

+(-zγ _yz+α _yz-α _zzβ _xz)(β _zx+(1-β _yxβ _yy)β _zy+β _xy(β _yx+β _yyβ _zxβ _zy))

e _y＝y-α _yx-α _zxβ _xx+β _xx(-α _zy+α _yyβ _xy)+(-α _xy+α _zyβ _yy)(β _xxβ _yx-β _zx)

+x(-β _xxβ _yx+β _zx)+xγ _xy(1+β _xxβ _yxβ _zx)-(α _yy+α _zyβ _xy)(1+β _xxβ _yxβ _zx)

+(z+α _zz+α _yzβ _xz)(β _xy+β _yyβ _zx+β _xx(1+β _yx(-β _yy+β _xyβ _zx)))

+(-zγ _xz+α _xz+α _zzβ _yz)(β _xxβ _yx-β _zx+(1+β _xxβ _yxβ _zx)(β _xyβ _yy-β _zy)

+β _xx(β _yy+β _xyβ _zy))-y((β _xxβ _yx-β _zx)β _zy+β _xx(-β _xy+β _yyβ _zy)

+(1+β _xxβ _yxβ _zx)(1+β _xyβ _yyβ _zy))+(-zγ _yz+α _yz-α _zzβ _xz)((β _xxβ _yx-β _zx)β _zy

+β _xx(-β _xy+β _yyβ _zy)+(1+β _xxβ _yxβ _zx)(1+β _xyβ _yyβ _zy))

e _z＝-z-α _zx-α _zy+α _yxβ _xx+α _yyβ _xy-x(β _yx+β _xxβ _zx)

+(-α _xy+α _zyβ _yy)(β _yx+β _xxβ _zx)+(α _yy+α _zyβ _xy)(β _xx-β _yxβ _zx)

+xγ _xy(-β _xx+β _yxβ _zx)+(z+α _zz+α _yzβ _xz)(1+β _yx(-β _yy+β _xyβ _zx)-β _xx(β _xy+β _yyβ _zx))

+(-zγ _xz+α _xz+α _zzβ _yz)(β _yx+β _yy+β _xxβ _zx+(-β _xx+β _yxβ _zx)(β _xyβ _yy-β _zy)+β _xyβ _zy)

-y(-β _xy+β _yyβ _zy+(β _yx+β _xxβ _zx)β _zy+(-β _xx+β _yxβ _zx)(1+β _xyβ _yyβ _zy))

+(-zS _yz+α _yz-α _zzβ _xz)(-β _xy+β _yyβ _zy+(β _yx+β _xxβ _zx)β _zy

+(-β _xx+β _yxβ _zx)(1+β _xyβ _yyβ _zy))

Cancellation higher-order shear deformation item, obtaining space error equation is:

e _x＝-α _xx-α _xy-yβ _zx-yβ _zy-zβ _yx-zβ _yy

e _y＝-α _yx+xβ _zx-α _yy+zβ _xy+zβ _xx+xγ _xy-zγ _yz+α _yz(68)

e _z＝-α _zx-α _zy-xβ _yx+yβ _xy+yβ _xx+α _zz

Step 2 is based on the global sensitivity analysis method of EFAST method

EFAST method is a kind of global sensitivity analysis method proposed in conjunction with the advantage of Fourier modulus sensitivity assays method.The method is sane, calculating is efficient and sample number that is that need is lower.Adopt variance analysis, the variance that model exports is caused by the interaction between each input parameter and parameter, and reflection model exports the susceptibility to input parameter.Therefore, the contribution proportion of the coupling between parameters and parameter to population variance can be obtained by the decomposition of model variance, be parameter sensitivity sex index.Application the method should know each ginseng geometry of numbers distribution to be analyzed and span in advance.

Now definition has model Y=f (x), and input parameter is X (x ₁, x ₂..., x _n).Each parameter has certain variation range and distribution form (table 3), constitutes a multi-C parameter space.Based on the method parameter of analytic model susceptibility process as shown in Figure 1.

In the method, in order to analyze the size degree that every geometric error affects machining precision, first the error of perpendicularity defined between three axles is known, and get work space x=300, y=300, z=200 place is analysis site.Then error model can be expressed as E=[e _x, e _y, e _z, 0] ^t, wherein

E=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅, x ₁₆, x ₁₇, x ₁₈] ^tfor Water demand 18 to the function of geometric error, if

α _xx,α _yx,α _zx,β _xx,β _yx,β _zx,α _xy,α _yy,α _zy,β _yy,β _zy,β _xy,α _xz,α _yz,α _zz,β _xz,β _zz,β _yz＝

x ₁,x ₂,x ₃,x ₄,x ₅,x ₆,x ₇,x ₈,x ₉,x ₁₀,x ₁₁,x ₁₂,x ₁₃,x ₁₄,x ₁₅,x ₁₆,x ₁₇,x ₁₈

First be the parameter X of Water demand _i(i=1,2 ..., n) choose a sampling function x _i:

Wherein, for a random phase between [0,2 π], get

In formula, n is the total number of geometric error; w _ifor the characteristic frequency of parameter; S is the common independent variable of all parameters, gets N at equal intervals in (-π, π) interval _s=2Mw _max+ 1 value, constant M gets 4 or 6, w usually _maxfor sequence { w _iin maximal value.Sequence { w _ishould nonlinear correlation be met, namely

$\underset{i=1}{\overset{n}{\Σ}}{r}_i\·w_i\≠0$

In formula, r _ifor integer.

Then y=f (x ₁, x ₂..., x ₁₈) y=f (s) can be converted into namely

f(s)＝f[X ₁(sinw ₁s),X ₂(sinw ₂s),...,X ₁₈(sinw ₁₈s)]s∈(-π,π)(70)

Carry out fourier progression expanding method to f (s) to obtain:

$y=f\left(s\right)\≈\underset{-\frac{N_s-1}{2}}{\overset{\frac{N_s-1}{2}}{\Σ}}\{A_j\·cos(j\·s)+B_j\·\sin (j\·s)\}---\left(71\right)$

Wherein:

$A_j=\frac{1}{N_s}\underset{k=1}{\overset{N_s}{\Σ}}f\left(s_k\right)\·cos(j\·s_k),B_j=\frac{1}{N_s}\underset{k=1}{\overset{N_s}{\Σ}}f\left(s_k\right)\·sin(j\·s_k)---\left(72\right)$

Wherein, N _sfor number of samples, $j\∈Z=\{-\frac{N_s-1}{2},...,-2,-1,0,1,2,...,+\frac{N_s-1}{2}\}$

s _k＝π/N _s(2k-N _s-1),k＝1,2,...,N _s

$z^0=z-\left\{0\right\}=\{-\frac{N_s-1}{2},...,-2,-1,1,2,...,+\frac{N_s-1}{2}\}$

Then Fourier spectrum curve definitions is:

$\mathrm{\Λ}p=A_j^2+B_j^2---\left(73\right)$

A _-j=A _j, B _-j=B _j, Λ _-j=Λ _jthen by parameter x _imodel result variance V caused by input change _ibe expressed as

Parameter w _i1,2,3 ... the spectrum curve sum at M frequency multiplication place, that is:

$V_i=\underset{p\∈Z^0}{\Σ}{\mathrm{\Λ}}_{{\mathrm{pw}}_i}=2\·\underset{p=1}{\overset{M}{\Σ}}\left(A_{p\·w_i}^2+B_{p\·w_i}^2\right)---\left(74\right)$

Wherein: z ⁰=z-{0}, ω _ifor parameter x _ithe integer frequency defined, wherein M generally get 4 or 6, p be integer.

Then model population variance V is:

$V=\underset{p\∈Z^0}{\Σ}{\mathrm{\Λ}}_p=2\·\underset{j=1}{\overset{\frac{N_s-1}{2}}{\Σ}}{\mathrm{\Λ}}_j---\left(75\right)$

By A _pand B _pand parameter x _icorresponding frequencies omega _ithrough type (5), (6) can obtain the model side V that each parameter causes _iand the population variance V that model exports.The population variance exported due to model is obtained by coupling between each parameter and parameter, jointly by as follows for its exploded representation:

$V=\underset{i}{\Σ}{V}_i+\underset{i\≠j}{\Σ}{V}_{ij}+\underset{i\≠j\≠m}{\Σ}{V}_{ijm}+\mathrm{...}+V_{\mathrm{12...}k}---\left(76\right)$

Wherein: V _ijfor parameter x _iby parameter x _jthe variance (coupling variance) of contribution; V _ijmfor parameter x _iby parameter x _jand x _m

The variance of contribution; V _12...kfor parameter x _iby parameter x ₁, x ₂..., x _kthe variance of contribution.Therefore, by normalized, parameter x _isingle order sensitivity indices S _ican be defined as follows:

$S_i=\frac{V_i}{V}---\left(77\right)$

What this sensitivity indices reflected is that parameter exports the direct contribution ratio of population variance to model.In like manner, parameter x _isecond order and three rank sensitivity indices may be defined as:

$S_{ij}=\frac{V_j}{V},S_{ijm}=\frac{V_{ijm}}{V}---\left(78\right)$

For a Multi-parameter coupling model, parameter x _ioverall sensitivity indices be each rank sensitivity indices sum, can be expressed as:

S _Ti＝S _i+S _ij+S _ijm+...+S _12...i...k

$S_{T_i}=\frac{V-V_i}{V}$

Total sensitivity indices is reflected the direct contribution ratio of a certain geometric error item and indirectly model is exported to the contribution sum of population variance by the coupling interaction effect with other parameters, and between parameter without coupling time, S _ijand S _ijm0, S is Deng item _ti=S _i, EFAST analyzes and is equal to local sensitivity analysis.EFAST method, by the decomposition to model output variance, can obtain every each rank of geometric error and total susceptibility quantitatively, and distinguish the possible influence degree of parameter to analog result according to susceptibility.This just makes EFAST method that the change of multiple parameter not only can be checked the impact of error model result, also can analyze each Parameters variation to the direct of machining precision and remote effect.

The circular of total sensitivity is as follows: first distribute to parameter X _ia larger frequency w _i, and be the different integer frequency { w that remaining other setting parameter one group is less _i, and should relation be met

w _i≥2M·max{w′ _i}。So just Dividing in frequency domain is become two parts [1, Mmax (w ' _i)] and [Mmax (w ' _i)+1, (N _s-1)/2]

Finally can calculate parameter x by formula (79) _itotal sensitivity value:

${\mathrm{ST}}_i=\frac{\underset{j=M\·\max \left(\left\{w_i^{\′}\right\}\right)+1}{\overset{\frac{N_s-1}{2}}{\Σ}}\left(A_j^2+B_j^2\right)}{\underset{j=1}{\overset{\frac{N_s-1}{2}}{\Σ}}\left(A_j^2+B_j^2\right)}---\left(79\right)$

After considering the coupling between every geometric error as can be seen from Table 5, the susceptibility sequence of every geometric error is also little with the difference of single order sensitivity analysis acquired results.But what be worth noting that is that ratio shared by every geometric error susceptibility has had significant change.Its reason is just that the method considers the coupling between different geometric error parameter, because the interaction between different geometric error parameter is outwardness, therefore the result of Fig. 5, Fig. 7, Fig. 9 is more realistic, the advantage place of this also EFAST method just.

EFAST method is by decomposition, each rank of each geometric error of acquisition that can be quantitative and the total responsive number index to model output variance.That is EFAST method not only can check the change of multinomial geometric error on the impact of spinor error model result simultaneously, can also analyze the change of each geometric error to the direct of model result and remote effect.

At Chosen Point place, one order and global sensitivity are listed in table 5 in X-direction result of calculation.Analysis result must follow two primitive rules: (1) one order result reflects the impact of single factor test on machining precision, and (2) global sensitivity result reflects that the interaction of each factor affects machining precision.In order to analyze data more intuitively, result has shown in figures 4 and 5.

Along each body diagonal of lathe work space, equally spaced selection nine test points (totally 33 testing sites) are as Figure 10.By above-mentioned method, the result of sensitivity analysis can be obtained in each test point, then, use and ask weighted average method, obtain whole work space sensitivity analysis result.

Defined parameters X ₁along X to, one order at w test point place be parameter X ₁along X to, global sensitivity at w test point place be therefore, for whole work space, parameter X ₁along X to one order and global sensitivity be defined as

$\begin{array}{cc}{\stackrel{x}{S}}_i=\frac{1}{33}\underset{w=1}{\overset{33}{\Σ}}\underset{w}{\overset{x}{S_i}}& \stackrel{x}{{\mathrm{ST}}_i}=\frac{1}{33}\underset{w=1}{\overset{33}{\Σ}}\underset{w}{\overset{x}{{\mathrm{ST}}_i}}\end{array}$

In whole work space, along X to single order susceptibility and global sensitivity analysis result, as shown in table 6.In order to more directly analyze data, result has been presented in Figure 11 and Figure 12.In the same way, in whole work space, along the single order susceptibility of Y-direction and Z-direction and global sensitivity analysis result as Figure 13,14,15,16.

1, as Figure 11 and 12, in the X direction, when the fluctuation of geometric error, α _xzand α _xyone order value be maximum, so the main geometric error item (space error of X-direction) affecting this direction machining precision is α _xzand α _xy.But global sensitivity analysis shows, α _xyand β _zzthe main geometric error item affecting this direction machining precision, the randomness that global sensitivity analysis has considered the geometric error of geometric error and the feature intercoupled.At overall susceptibility α on the basis analyzed _xyand β _zzbe confirmed as the geometric error affecting X-direction machining precision most critical.

2, as Figure 13 and 14, in the Y direction, when the fluctuation of geometric error, β _zyand α _yzone order value be maximum, so affecting the main geometric error item of this direction machining precision (space error in y direction) is β _zyand α _yz.But global sensitivity shows β _yyand α _yzbe the main geometric error item affecting this direction machining precision, this shows β _yyand α _yzand the interaction between other geometric errors is very large on this direction machining precision impact.As previously mentioned, in the global sensitivity randomness considering geometric error and the feature intercoupled.On this basis, β _yyand α _yzbe confirmed as the geometric error affecting Y-direction most critical.

3, as Figure 15 and 16, in z-direction, when the fluctuation of geometric error, α _zyand α _zzone order value be maximum, so affecting the main geometric error item of this direction machining precision (space error in y direction) is α _zyand α _zz.And the susceptibility in the whole world shows, α _zyand α _zzbe the main geometric error item affecting this direction machining precision, this shows α _zyand α _zzand the interaction between other geometric errors, on very large on this direction machining precision impact.Based on randomness and the feature intercoupled of geometric error, α _zyand α _zzthe geometric error affecting Y-direction most critical can be identified as.

Application and checking

Select precise vertical machining centre to be as shown in Figure 3 example, prove the analytical approach that the present invention proposes.Because the geometric error of lathe mainly causes owing to manufacturing or assembling defect, the corresponding relation between geometric error and parts precision is as follows.

The positioning error α of each linear motion axis _xx, α _yyand α _zzprimarily of the determination of the cumulative pitch error of leading screw.In other words, the impact of positioning error mainly screw rod machining precision.In vertical plane displacement of the lines error alpha _zx, α _zyand α _xzprimarily of the determination of the straightness error of guide rail in vertical plane.At water line displacement error α _yx, α _xyand α _yzdetermine primarily of the straightness error of guide rail in surface level.Rolling pendulum error β _xx, β _yyand β _zzparallelism error primarily of guide rail causes.Top pendulum error β _zx, β _xyand β _yzmainly contain the straightness error of guide rail in vertical plane to determine to determine with the length of moving component.Run-out error β _yx, β _zyand β _xzthe length of the straightness error and moving component that mainly contain guide rail in surface level determines

In conjunction with the result of Global sensitivity analysis, the geometric error item affecting the key of machine finish is

α _xy, β _zz, β _yy, α _yz, α _zyand α _zz.So take following innovative approach:

1, Y-axis and the guide rail linearity of Z axis in vertical plane is improved

2, Y-axis and the guide rail linearity of Z axis in surface level is improved

3, Z-direction guide rail adopts high-precision screw.

Table 7 is listed vertical machining centre and is transformed forward and backward positioning precision and repetitive positioning accuracy numerical value.By comparing, can reach a conclusion, the positioning precision and the repetitive positioning accuracy that carry out transforming rear lathe improve greatly.Thus, can reach a conclusion, the crucial geometric error recognition methods that this method proposes is feasible and effective, improves Precision of NC Machine Tool have great importance to the design phase.

Table 1

Table 2

Table 3

Table 4

Table 7

。

Claims (2)

1. the extracting method based on the important geometric error source of lathe of expansion Fourier amplitude, this method is based on error measurement data, utilize the exponential matrix form of spinor theory, on the basis of the topological structure of lathe, set up the spatial error model of lathe entirety, the high-order term of error model is cut down, obtains the fundamental equation of error model; According to EFAST Global sensitivity analysis method, by choosing suitable transfer function, error model being converted to one-dimensional functions by ten octuple functions, fourier progression expanding method is carried out to this one-dimensional functions, the model side V that each parameter causes can be obtained _iand the population variance V that model exports; The population variance exported due to model is obtained jointly by coupling between each parameter and parameter, and EFAST method passes through the decomposition to model output variance, the responsive number index in each rank of each geometric error of acquisition that can be quantitative and total responsive number index; That is EFAST method not only can check the change of multinomial geometric error on the impact of spinor error model result simultaneously, the change of each geometric error can also be analyzed to the direct of model result and remote effect, finally have identified the critical error affecting machine finish, achieve Error Tracing &, the design for precise numerical control machine provides important theoretical foundation;

It is characterized in that: the implementation procedure of the method is as follows,

Step one sets up the spatial synthesis error model of lathe according to spinor theory

According to the exponential matrix form of spinor theory, by abstract for each motion parts of lathe be the vector form of 6 × 1; By forms of motion and composition error modularized processing, and with the statement of exponential matrix form, set up the spatial synthesis error model of lathe according to the topological structure of lathe;

The exponential matrix form of step 1.1 spinor theory

The motion of any rigid body can be broken down into two parts: translation motion vertically and the rotary motion around axle; That is, all parts is regarded as spinor; Unit spinor becomes as follows at Pl ü cker coordinate:

Ψ＝[k ^Tu ^T] ^T＝[k ₁,k ₂,k ₃,u ₁,u ₂,u ₃] ^T(1)

state rigid body arbitrary motion form spatially, then have:

Wherein, u=[u ₁u ₂u ₃] ^t, make difficulties title matrix, if k=is [k ₁k ₂k ₃] ^t, then be expressed as:

Rigid motion generally all comprises translation and rotation, and vectorial h is identical at rigid body coordinate system and reference frame; Then the homogeneous transform matrix of rigid body is:

The secondly transformation matrix that the exponential form of spinor is corresponding can be written as: as k=0, rigid body only has translation motion, then homogeneous transform matrix can be written as:

When k ≠ 0, for rigid body, also there is rotary motion, now exponential matrix is:

Wherein trigonometric series expansion be expressed as:

To sum up, then the exponential matrix for rigid body arbitrary motion form is in space had to be expressed as:

When ψ is unit spinor, || during k|| ≠ 0, the rotation angle at mechanical position is expressed as || during k||=0, the distance of translation is expressed as the representation of point in different coordinates is different, and the difference transformation matrix between them states its relation; Spinor is also interpreted as a point in coordinate system, also different at the form of presentation of different coordinates, therefore also needs the form of transformation matrix to state the relation of spinor in different coordinates, is referred to as adjoint matrix; If the motion spinor θ ψ of rigid body, the exponential matrix of its variation can be expressed as:

Adjoint exponential matrix form then under its this coordinate system:

Following character is met with exponential matrix:

For the physical construction that lathe is such, stating its structure with exponential matrix then has:

T (0) represents its original transform matrix, and formula (13) can be applicable to the error modeling of lathe;

The space error modeling of step 1.2 lathe

General, each axial motion can have 6 direction degree of freedom, can produce error and 3 errors of rotating of 3 translations simultaneously; Utilize spinor theory, define error module m _eΨ _e;

m _eΨ _e＝[β _x,β _y,β _z,α _x,α _y,α _z] ^T(14)

With X to geometric error consist of example, be divided into three parts; Part I Ψ _xxcomprise positioning error and rolling pendulum error alpha in the direction _xx, β _xx; Part II Ψ _yxlinear error and the top pendulum error alpha of surface level _yxβ _yx; Part III Ψ _zxlinear error and the Run-out error α of vertical plane _zx, β _zx;

Ψ _xx＝[β _xx,0,0,α _xx,0,0] ^T(15)

Ψ _yx＝[0,β _yx,0,0,α _yx,0] ^T(16)

Ψ _zx＝[0,0,β _zx,0,0,α _zx] ^T(17)

The space error of X-axis can be expressed as:

The error model exponential matrix form of X-axis, is expressed as:

In like manner can obtain the space error module of other axles and the error model of exponential matrix;

Step 1.3 is about the exponential matrix form of verticality and parallelism error

Axle and axle ideally due to reality are difference to some extent, and two adjacent axles are not absolute 90 °; That is there is the error of perpendicularity; For three translation shaft, definition Y-axis is desired axis, there is not the error of perpendicularity; The error of perpendicularity then between X-axis and Y-axis is γ _xy, the error of perpendicularity between Y-axis and Z axis is γ _yz, the verticality between X and Z is γ _xz; In the plane that the X axis of Y-axis and actual installation forms, only γ is existed for X-axis _xy, in like manner there are other two error of perpendicularitys at actual Z axis; Because actual axis direction inevitably will depart from the position of desired axis, therefore should consider to add the error of perpendicularity in coordinate transform, for ideal coordinates transformation of axis form:

For X to, X ideally to unit spinor representation is:

Ψ _xi＝[0,0,0,1,0,0] ^T(20)

Add the error of perpendicularity under actual state, then X to the unit spinor representation of reality is:

Ψ _xs＝[0,0,0,cos(γ _xy),-sin(γ _xy),0] ^T(21)

Corresponding exponential matrix is expressed as:

Another kind of literary style, utilizes the form of adjoint matrix to rotate-γ around Z axis desirable X-axis _xyangle reaches the effect that X-axis and Y-axis are 90 °, that is:

Ψ _zr＝[0,0,1,0,0,0] ^T

Under actual conditions, the unit spinor representation of Z-direction is:

Ψ _zi＝[0,0,0,0,0,1] ^T

Two error of perpendicularitys taken into account, the positional representation of actual Z axis is:

Wherein, Ψ _zs=[0,0,0 ,-sin (γ _xz) ,-sin (γ _yz) cos (γ _xz), cos (γ _yz) cos (γ _xz)] ^t

Step 1.4 is based on the foundation of the direction of principal axis error model of three under topological structure

Theory of multi body system provides the very detailed topology controlment about lathe, can apply too in exponential matrix;

Ideally, lathe there is not error; Matrixing equation ideally can use T _widealrepresent:

Under actual conditions, due to the error of position between the error of machine tool component self and parts, global facility error spinor is joined in spinor module; Use T _wactualrepresent:

In above formula, S _f=[0,0,0,0,0,0], represents ground spinor;

In workpiece coordinate system, according to actual and ideally matrixing equation, obtain the spatial error model of multi-axis NC Machine Tools:

Corresponding space error component e in three axial directions _x, e _y, e _zbe expressed as:

[e _x,e _y,e _z,1] ^T＝E·[0,0,0,1] ^T(28)

Omit high-order term more than second order and second order in formula, just obtain the fundamental equation of space error;

Step 2 is based on the global sensitivity analysis method of EFAST method

EFAST method is a kind of global sensitivity analysis method proposed in conjunction with the advantage of Fourier modulus sensitivity assays method; The method is sane, calculating is efficient and sample number that is that need is lower; It has employed the thought of variance analysis, thinks that the variance that model exports is caused by the interaction between each input parameter and parameter, can reflect that model exports the susceptibility to input parameter; Therefore, the contribution proportion of the coupling between parameters and parameter to population variance can be obtained by the decomposition of model variance, be parameter sensitivity sex index; Application the method should know geometry distribution and the span of each geometric error item to be analyzed in advance;

Now definition has model Y=f (x), and input parameter is X (x ₁, x ₂..., x _n); Each parameter has certain variation range and distribution form, constitutes a multi-C parameter space;

In the method, in order to analyze the size degree that every geometric error affects machining precision, first the error of perpendicularity defined between three axles is known, and get work space x=300, y=300, z=200 place is analysis site; Then error model can be expressed as E=[e _x, e _y, e _z, 0] ^t, wherein

E=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅, x ₁₆, x ₁₇, x ₁₈] ^tfor the function of 18 geometric errors of Water demand, if

α _xx,α _yx,α _zx,β _xx,β _yx,β _zx,α _xy,α _yy,α _zy,β _yy,β _zy,β _xy,α _xz,α _yz,α _zz,β _xz,β _zz,β _yz＝x ₁,x ₂,x ₃,x ₄,x ₅,x ₆,x ₇,x ₈,x ₉,x ₁₀,x ₁₁,x ₁₂,x ₁₃,x ₁₄,x ₁₅,x ₁₆,x ₁₇,x ₁₈

First be the parameter X of Water demand _i(i=1,2 ..., n) choose a sampling function x _i:

Wherein, for a random phase between [0,2 π], get

In formula, n is the total number of uncertain parameters; w _ifor the characteristic frequency of parameter; S is the common independent variable of all parameters, gets N at equal intervals in (-π, π) interval _s=2Mw _max+ 1 value, constant M here gets 4 or 6, w usually _maxfor sequence { w _iin maximal value; Sequence { w _ishould nonlinear correlation be met, namely

In formula, r _ifor integer;

Then y=f (x ₁, x ₂..., x ₁₈) y=f (s) can be converted into namely

f(s)＝f[X ₁(sinw ₁s),X ₂(sinw ₂s),...,X ₁₈(sinw ₁₈s)]s∈(-π,π)(30)

Carry out fourier progression expanding method to f (s) to obtain:

Wherein:

Wherein, N _sfor number of samples,

s _k＝π/N _s(2k-N _s-1),k＝1,2,...,N _s

Then Fourier spectrum curve definitions is:

A _-j=A _j, B _-j=B _j, Λ _-j=Λ _jthen by parameter x _imodel result variance V caused by input change _iparameter can be expressed as

W _i1,2,3 ... the spectrum curve sum at M frequency multiplication place, that is:

Wherein: z ⁰=z-{0}, ω _ifor parameter x _ithe integer frequency defined, wherein M generally get 4 or 6, p be integer;

Then model population variance V is:

By A _pand B _pand parameter x _icorresponding frequencies omega _ithrough type (34), (35) can obtain the model side V that each parameter causes _iand the population variance V that model exports; The population variance exported due to model is obtained jointly by coupling between each parameter and parameter, can by as follows for its exploded representation:

Wherein: V _ijfor parameter x _iby parameter x _jthe variance (coupling variance) of contribution; V _ijmfor parameter x _iby parameter x _jand x _mthe variance of contribution; V _12...kfor parameter x _iby parameter x ₁, x ₂..., x _kthe variance of contribution; Therefore, by normalized, parameter x _isingle order sensitivity indices S _ican be defined as follows:

What this sensitivity indices reflected is that parameter exports the direct contribution ratio of population variance to model; In like manner, parameter x _isecond order and three rank sensitivity indices may be defined as:

For a Multi-parameter coupling model, parameter x _ioverall sensitivity indices be each rank sensitivity indices sum, can be expressed as:

S _Ti＝S _i+S _ij+S _ijm+...+S _12...i...k

Total sensitivity indices is reflected parameter direct contribution ratio and indirectly model is exported to the contribution sum of population variance by the coupling interaction effect with other parameters, and between parameter without coupling time, S _ijand S _ijm0, S is Deng item _ti=S _i, EFAST analyzes and is equal to local sensitivity analysis; EFAST method, by the decomposition to model output variance, can obtain each rank of each parameter and total susceptibility quantitatively, and distinguish the possible influence degree of parameter to analog result according to susceptibility; This just makes EFAST method that the change of multiple parameter not only can be checked the impact of error model result, also can analyze each Parameters variation to the direct of analog result and remote effect;

The circular of total sensitivity is as follows: first distribute to parameter X _ia larger frequency w _i, and be the different integer frequency { w that remaining other setting parameter one group is less _i, and should relation w be met _i>=2Mmax{w _i'; So just Dividing in frequency domain is become two part [1, Mmax ({ w _i')] and [Mmax ({ w _i')+1, (N _s-1)/2]

Finally can calculate parameter x by formula (39) _itotal sensitivity value:

2. the extracting method in a kind of important geometric error source of lathe based on expansion Fourier amplitude according to claim 1, is characterized in that:

Step one sets up the spatial synthesis error model of lathe according to spinor theory

According to the exponential matrix form of spinor theory, by abstract for each motion parts of lathe be the vector form of 6 × 1; By forms of motion and error module process, and with the statement of exponential matrix form, set up the spatial error model of lathe according to the topological structure of lathe;

The exponential matrix form of step 1.1 spinor theory

The motion of any rigid body can be broken down into two parts: translation motion vertically and the rotary motion around axle; Can all parts be regarded as spinor; Unit spinor is into as follows at Pl ü cker coordinate:

Ψ＝[k ^Tu ^T] ^T＝[k ₁,k ₂,k ₃,u ₁,u ₂,u ₃] ^T(40)

state rigid body arbitrary motion form spatially, then have:

Wherein, u=[u ₁u ₂u ₃] ^t, make difficulties title matrix, if k=is [k ₁k ₂k ₃] ^t, then can be expressed as:

Rigid motion generally all comprises translation and rotation, and vectorial h is identical at rigid body coordinate system and reference frame; Then the homogeneous transform matrix of rigid body is:

The secondly transformation matrix that the exponential form of spinor is corresponding can be written as: as k=0, rigid body only has translation motion, then homogeneous transform matrix is:

When k ≠ 0, also there is rotary motion for rigid body, now exponential matrix is:

Wherein trigonometric series expansion be expressed as:

To sum up, then the exponential matrix for rigid body arbitrary motion form is in space had to be expressed as:

When ψ is unit spinor, || during k|| ≠ 0, the rotation angle at mechanical position is expressed as || during k||=0, the distance of translation is expressed as the representation of point in different coordinates is different, and the difference transformation matrix between them states its relation; Spinor is also interpreted as a point in coordinate system, also different at the form of presentation of different coordinates, therefore also needs the form of transformation matrix to state the relation of spinor in different coordinates, is referred to as adjoint matrix; If the motion spinor θ ψ of rigid body, the exponential matrix of its variation can be expressed as:

Adjoint exponential matrix form then under its this coordinate system:

Following character is met with exponential matrix:

For the physical construction that lathe is such, stating its structure with exponential matrix then has:

T (0) represents its original transform matrix, can be applied to the error modeling of lathe;

Step 1.2 utilization index matrix type carries out spatial synthesis error modeling to lathe

General, each axial motion can have 6 direction degree of freedom, can produce error and 3 errors of rotating of 3 translations simultaneously; Utilize spinor theory, definition error module m _eΨ _e;

m _eΨ _e＝[β _x,β _y,β _z,α _x,α _y,α _z] ^T

With X to geometric error consist of example, be mainly divided into three parts; Part I Ψ _xxcomprise positioning error and rolling pendulum error alpha in the direction _xx, β _xx; Part II Ψ _yxlinear error and the top pendulum error alpha of surface level _yx, β _yx; Part III Ψ _zxlinear error and the Run-out error α of vertical plane _zx, β _zx;

Ψ _xx＝[β _xx,0,0,α _xx,0,0] ^T

Ψ _yx＝[0,β _yx,0,0,α _yx,0] ^T

Ψ _zx＝[0,0,β _zx,0,0,α _zx] ^T

The space error of X-axis is expressed as:

The error model exponential matrix form of X-axis, is expressed as:

In like manner, the exponential matrix form of the error model of Y-axis and Z axis is expressed as follows:

Ψ _xy＝[β _xy,0,0,α _xy,0,0] ^T

Ψ _yy＝[0,β _yy,0,0,α _yy,0] ^T

Ψ _zy＝[0,0,β _zy,0,0,α _zy] ^T

Ψ _xz＝[β _xz,0,0,α _xz,0,0] ^T

Ψ _yz＝[0,β _yz,0,0,α _yz,0] ^T

Ψ _zz＝[0,0,β _zz,0,0,α _zz] ^T

Step 1.3 is about the exponential matrix form of verticality and parallelism error

Axle and axle ideally due to reality are difference to some extent, and two adjacent axles are not absolute 90 °; That is there is the error of perpendicularity; For three translation shaft, Y-axis is desired axis, there is not the error of perpendicularity; The error of perpendicularity then between X-axis and Y-axis is S _xy, the error of perpendicularity between Y-axis and Z axis is S _yz, the verticality between X and Z is S _xz; In the plane that the X axis of Y-axis and actual installation forms, only S is existed for X-axis _xy, in like manner there are other two error of perpendicularitys at actual Z axis;

For X to, X ideally can be expressed as to unit spinor:

$ _xi＝[0,0,0,1,0,0] ^T

Add the error of perpendicularity under actual state, then X to the unit spinor representation of reality is:

$ _xs＝[0,0,0,cos(S _xy),-sin(S _xy),0] ^T

Corresponding exponential matrix is expressed as:

Another kind of mode, utilizes the form of adjoint matrix to take Z as axle ($ by desirable X-axis _zr) rotate to an angle to reach the effect that X-axis and Y-axis are 90 °, that is:

$ _zr＝[0,0,1,0,0,0] ^T

It is that axle rotates that Section 2 " r " in subscript represents with Section 1;

Similar, Z-direction unit spinor representation is:

$ _zi＝[0,0,0,0,0,1] ^T

Add the error of perpendicularity under actual state, then the unit spinor representation of Z-direction reality is:

$ _zs＝[0,0,0,-sin(S _xz),-sin(S _yz)cos(S _xz),cos(S _yz)cos(S _xz)] ^T(61)

Another kind of literary style utilizes the form of adjoint matrix to rotate-γ around Z axis desirable X-axis _xyangle reaches the effect that X-axis and Y-axis are 90 °, that is:

Ψ _zr＝[0,0,1,0,0,0] ^T

Under actual conditions, the unit spinor of Z-direction can be expressed as:

Ψ _zi＝[0,0,0,0,0,1] ^T

Two error of perpendicularitys taken into account, the position of actual Z axis can be expressed as:

Wherein, Ψ _zs=[0,0,0 ,-sin (γ _xz) ,-sin (γ _yz) cos (γ _xz), cos (γ _yz) cos (γ _xz)] ^t

Step 1.4 is set up based on the error model under topological structure

Theory of multi body system provides the very detailed topology controlment about lathe, can apply too in exponential matrix; Ideally, lathe there is not error; Matrixing equation T ideally _irepresent:

The error extension matrix model of overall lathe is written as:

Wherein, S _f=[0,0,0,0,0,0] ^trepresent the spinor of ground;

Based on spinor exponential matrix represented by five-axis machine tool spatial error model be expressed as:

Corresponding space error component e in three axial directions _x, e _y, e _zbe expressed as:

[e _x,e _y,e _z,1] ^T＝E·[0,0,0,1] ^T(67)

For multiaxis NC maching lathe, the alignment error of cutter and workpiece has been left in the basket at this due to very little; So the space error of lathe component in three axial directions can be written as:

e _x＝-α _xx-α _xy+α _zxβ _yx+α _zyβ _yx-α _yyβ _xyβ _yx+α _zyβ _yy+xγ _xyβ _zx-α _yyβ _zx-α _zyβ _xyβ _zx

-(z+α _zz+α _yzβ _xz)(β _yx+β _yy-β _xyβ _zx)+(1+β _zx(β _xyβ _yy-β _zy)-β _yx(β _yy+β _xyβ _zy))

-y(β _zx+(1-β _yxβ _yy)β _zy+β _xy(β _yx+β _yyβ _zxβ _zy))

+(-zγ _yz+α _yz-α _zzβ _xz)(β _zx+(1-β _yxβ _yy)β _zy+β _xy(β _yx+β _yyβ _zxβ _zy))

e _y＝y-α _yx-α _zxβ _xx+β _xx(-α _zy+α _yyβ _xy)+(-α _xy+α _zyβ _yy)(β _xxβ _yx-β _zx)

+x(-β _xxβ _yx+β _zx)+xγ _xy(1+β _xxβ _yxβ _zx)-(α _yy+α _zyβ _xy)(1+β _xxβ _yxβ _zx)

+(z+α _zz+α _yzβ _xz)(β _xy+β _yyβ _zx+β _xx(1+β _yx(-β _yy+β _xyβ _zx)))

+(-zγ _xz+α _xz+α _zzβ _yz)(β _xxβ _yx-β _zx+(1+β _xxβ _yxβ _zx)(β _xyβ _yy-β _zy)

+β _xx(β _yy+β _xyβ _zy))-y((β _xxβ _yx-β _zx)β _zy+β _xx(-β _xy+β _yyβ _zy)

+(1+β _xxβ _yxβ _zx)(1+β _xyβ _yyβ _zy))+(-zγ _yz+α _yz-α _zzβ _xz)((β _xxβ _yx-β _zx)β _zy

+β _xx(-β _xy+β _yyβ _zy)+(1+β _xxβ _yxβ _zx)(1+β _xyβ _yyβ _zy))

e _z＝-z-α _zx-α _zy+α _yxβ _xx+α _yyβ _xy-x(β _yx+β _xxβ _zx)

+(-α _xy+α _zyβ _yy)(β _yx+β _xxβ _zx)+(α _yy+α _zyβ _xy)(β _xx-β _yxβ _zx)

+xγ _xy(-β _xx+β _yxβ _zx)+(z+α _zz+α _yzβ _xz)(1+β _yx(-β _yy+β _xyβ _zx)-β _xx(β _xy+β _yyβ _zx))

+(-zγ _xz+α _xz+α _zzβ _yz)(β _yx+β _yy+β _xxβ _zx+(-β _xx+β _yxβ _zx)(β _xyβ _yy-β _zy)+β _xyβ _zy)

-y(-β _xy+β _yyβ _zy+(β _yx+β _xxβ _zx)β _zy+(-β _xx+β _yxβ _zx)(1+β _xyβ _yyβ _zy))

+(-zS _yz+α _yz-α _zzβ _xz)(-β _xy+β _yyβ _zy+(β _yx+β _xxβ _zx)β _zy

+(-β _xx+β _yxβ _zx)(1+β _xyβ _yyβ _zy))

Cancellation higher-order shear deformation item, obtaining space error equation is:

e _x＝-α _xx-α _xy-yβ _zx-yβ _zy-zβ _yx-zβ _yy

e _y＝-α _yx+xβ _zx-α _yy+zβ _xy+zβ _xx+xγ _xy-zγ _yz+α _yz(68)

e _z＝-α _zx-α _zy-xβ _yx+yβ _xy+yβ _xx+α _zz

Step 2 is based on the global sensitivity analysis method of EFAST method

EFAST method is a kind of global sensitivity analysis method proposed in conjunction with the advantage of Fourier modulus sensitivity assays method; The method is sane, calculating is efficient and sample number that is that need is lower; Adopt variance analysis, the variance that model exports is caused by the interaction between each input parameter and parameter, and reflection model exports the susceptibility to input parameter; Therefore, the contribution proportion of the coupling between parameters and parameter to population variance can be obtained by the decomposition of model variance, be parameter sensitivity sex index; Application the method should know each ginseng geometry of numbers distribution to be analyzed and span in advance;

Now definition has model Y=f (x), and input parameter is X (x ₁, x ₂..., x _n); Each parameter has certain variation range and distribution form (table 3), constitutes a multi-C parameter space;

In the method, in order to analyze the size degree that every geometric error affects machining precision, first the error of perpendicularity defined between three axles is known, and get work space x=300, y=300, z=200 place is analysis site; Then error model can be expressed as E=[e _x, e _y, e _z, 0] ^t, wherein

E=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅, x ₁₆, x ₁₇, x ₁₈] ^tfor Water demand 18 to the function of geometric error, if

α _xx,α _yx,α _zx,β _xx,β _yx,β _zx,α _xy,α _yy,α _zy,β _yy,β _zy,β _xy,α _xz,α _yz,α _zz,β _xz,β _zz,β _yz＝x ₁,x ₂,x ₃,x ₄,x ₅,x ₆,x ₇,x ₈,x ₉,x ₁₀,x ₁₁,x ₁₂,x ₁₃,x ₁₄,x ₁₅,x ₁₆,x ₁₇,x ₁₈

First be the parameter X of Water demand _i(i=1,2 ..., n) choose a sampling function x _i:

Wherein, for a random phase between [0,2 π], get

In formula, n is the total number of geometric error; w _ifor the characteristic frequency of parameter; S is the common independent variable of all parameters, gets N at equal intervals in (-π, π) interval _s=2Mw _max+ 1 value, constant M gets 4 or 6, w usually _maxfor sequence { w _iin maximal value; Sequence { w _ishould nonlinear correlation be met, namely

In formula, r _ifor integer;

Then y=f (x ₁, x ₂..., x ₁₈) y=f (s) can be converted into namely

f(s)＝f[X ₁(sinw ₁s),X ₂(sinw ₂s),...,X ₁₈(sinw ₁₈s)]s∈(-π,π)(70)

Carry out fourier progression expanding method to f (s) to obtain:

Wherein:

Wherein, N _sfor number of samples,

s _k＝π/N _s(2k-N _s-1),k＝1,2,...,N _s

Then Fourier spectrum curve definitions is:

A _-j=A _j, B _-j=B _j, Λ _-j=Λ _jthen by parameter x _imodel result variance V caused by input change _ibe expressed as parameter w _i1,2,3 ... the spectrum curve sum at M frequency multiplication place, that is:

Wherein: z ⁰=z-{0}, ω _ifor parameter x _ithe integer frequency defined, wherein M generally get 4 or 6, p be integer;

Then model population variance V is:

By A _pand B _pand parameter x _icorresponding frequencies omega _ithrough type (5), (6) can obtain the model side V that each parameter causes _iand the population variance V that model exports; The population variance exported due to model is obtained by coupling between each parameter and parameter, jointly by as follows for its exploded representation:

Wherein: V _ijfor parameter x _iby parameter x _jthe variance (coupling variance) of contribution; V _ijmfor parameter x _iby parameter x _jand x _m

The variance of contribution; V _12...kfor parameter x _iby parameter x ₁, x ₂..., x _kthe variance of contribution; Therefore, by normalized, parameter x _isingle order sensitivity indices S _ican be defined as follows:

What this sensitivity indices reflected is that parameter exports the direct contribution ratio of population variance to model; In like manner, parameter x _isecond order and three rank sensitivity indices may be defined as:

For a Multi-parameter coupling model, parameter x _ioverall sensitivity indices be each rank sensitivity indices sum, can be expressed as:

S _Ti＝S _i+S _ij+S _ijm+...+S _12...i...k

Total sensitivity indices is reflected the direct contribution ratio of a certain geometric error item and indirectly model is exported to the contribution sum of population variance by the coupling interaction effect with other parameters, and between parameter without coupling time, S _ijand S _ijm0, S is Deng item _ti=S _i, EFAST analyzes and is equal to local sensitivity analysis; EFAST method, by the decomposition to model output variance, can obtain every each rank of geometric error and total susceptibility quantitatively, and distinguish the possible influence degree of parameter to analog result according to susceptibility; This just makes EFAST method that the change of multiple parameter not only can be checked the impact of error model result, also can analyze each Parameters variation to the direct of machining precision and remote effect;

The circular of total sensitivity is as follows: first distribute to parameter X _ia larger frequency w _i, and be the different integer frequency { w that remaining other setting parameter one group is less _i, and should relation w be met _i>=2Mmax{w _i'; So just Dividing in frequency domain is become two part [1, Mmax ({ w _i')] and [Mmax ({ w _i')+1, (N _s-1)/2]

Finally can calculate parameter x by formula (79) _itotal sensitivity value:

After considering the coupling between every geometric error as can be seen from Table 5, the susceptibility sequence of every geometric error is also little with the difference of single order sensitivity analysis acquired results; But what be worth noting that is that ratio shared by every geometric error susceptibility has had significant change; Its reason is just that the method considers the coupling between different geometric error parameter, because the interaction between different geometric error parameter is outwardness;

EFAST method passes through the decomposition to model output variance, each rank of each geometric error of acquisition that can be quantitative and total responsive number index; That is EFAST method not only can check the change of multinomial geometric error on the impact of spinor error model result simultaneously, can also analyze the change of each geometric error to the direct of model result and remote effect;

At Chosen Point place, one order and global sensitivity are listed in table 5 in X-direction result of calculation; Analysis result must follow two primitive rules: (1) one order result reflects the impact of single factor test on machining precision, and (2) global sensitivity result reflects that the interaction of each factor affects machining precision;

Along each body diagonal of lathe work space, equally spaced selection nine test points (totally 33 testing sites); By above-mentioned method, the result of sensitivity analysis can be obtained in each test point, then, use and ask weighted average method, obtain whole work space sensitivity analysis result;

Defined parameters X ₁along X to, one order at w test point place be parameter X ₁along X to, global sensitivity at w test point place be therefore, for whole work space, parameter X ₁along X to one order and global sensitivity be defined as

In whole work space, along X to single order susceptibility and global sensitivity analysis result, as shown in table 6.

Table 1

Table 2

Table 3

Table 4

。