CN106950916A - Generating tool axis vector method for fairing is processed based on AB type five-axle number control machine tools endless knife - Google Patents

Abstract

Generating tool axis vector method for fairing is processed based on AB type five-axle number control machine tools endless knife the invention discloses one kind, comprised the following steps：The relation equation set up between endless knife generating tool axis vector and cutter spacing design variable；The motion transform equation set up between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B；The relation equation set up between endless knife cutter spacing design variable and five-axle number control machine tool gyroaxis A and B；The design variable, object function and constraints of endless knife generating tool axis vector fairing are determined, sets up and generating tool axis vector fairing Mathematical Modeling is processed based on AB type five-axle number control machine tools endless knife；Determine the method for solving of above-mentioned generating tool axis vector fairing Mathematical Modeling.This method can avoid the drastically change of machine tool rotary axle, make the motion of machine tool rotary axle more steady and smooth, the angular speed and angular acceleration of machine tool rotary axle is greatly reduced, so that the crudy and processing efficiency of curved surface are improved, with stronger actual application value.

Description

Generating tool axis vector method for fairing is processed based on AB type five-axle number control machine tools endless knife

Technical field

The present invention relates to a kind of five-axle number control machine tool generating tool axis vector method for fairing, more particularly to based on AB type five shafts numerical controlled machines Bed endless knife processing generating tool axis vector method for fairing, belongs to five-shaft numerical control processing technique field.

Background technology

When using endless knife five-axis robot complex-curved, because surface geometry property is poor, such as the normal vector of curved surface, Principal direction, curvature etc., are easily caused generated endless knife generating tool axis vector and undergo mutation and fluctuation.Even if using most simple Cutter positioning method (such as Sturz methods) above-mentioned curved surface area of five-axis robot, can also cause the acute variation of generating tool axis vector, so that shadow Ring the nonlinearity erron in the stationarity, the servo ability beyond machine tool feed axle and increase process of five-axis machine tool motion Deng.Therefore obtaining the smooth generating tool axis vector of endless knife in five-shaft numerical control processing turns into the important research direction of Machining of Curved Surface technology.For Smooth generating tool axis vector is obtained, domestic and foreign scholars have carried out numerous studies work in terms of optimal tool orientation, and propose many Five-axis robot optimal tool orientation method, focuses primarily upon two aspects：One is that geometrical constraint is only considered in workpiece coordinate system Optimal tool orientation method；Two be the optimal tool orientation side that geometrical constraint and kinematical constraint are considered in workpiece coordinate system Method.

Prior art one, (cycle, Zhao Jibin and Liu Weijun, complex-curved five-shaft numerical control process optimal tool orientation to document Technique study mechanical engineering journals, 2013 (07):184-192) propose that a kind of complex-curved five-shaft numerical control processing generating tool axis vector is excellent Change method.The processing stand position that effectively insertion is limited first in non-interfering domain, it is ensured that the global optimization of generating tool axis vector；Simultaneously dry Relate to and improved C-Space methods are used in domain, generate generating tool axis vector fairing feasible zone.

Prior art two, document (Wang Jing etc., complex curved surface parts five-axis robot cutter shaft global optimization method aviation journals, 2013(06):1452-1462) propose a kind of five axle generating tool axis vector global optimization methods based on critical constraint.Construct first There is feasible pendulum knife plane at given point of contact, and critical generating tool axis vector is calculated according to critical constraint in pendulum knife plane, Obtain on the basis of critical generating tool axis vector, Planar Mapping is carried out to it, establish the initial feasible zone of cutter shaft swing；Secondly, lead to Cross uniform to the progress of initial feasible zone discrete, adjacency matrix is constructed according to relative position relation between discrete point, and combine most short Path search algorithm obtains initial reference cutter shaft, so that constructing new cutter shaft swings feasible zone；Finally, current cutting is set up Without interference and the minimum optimal tool orientation model of adjacent cutter shaft change in row, realize free form surface five-axis robot without interference cutter shaft The smooth control of vector.Above-mentioned prior art at least has the following disadvantages：

The above method is main not occur to cut between cutter and workpiece/lathe with global interference etc. as constraints, Minimum or smooth change is changed with generating tool axis vector in workpiece coordinate system and is turned to object function progress optimal tool orientation, so as to obtain Without interference and smooth generating tool axis vector in workpiece coordinate system.And five-axle number control machine tool is larger because of architectural difference, although workpiece coordinate The smooth change of generating tool axis vector in system, but it is difficult to ensure that each reference axis of five-axis machine tool especially gyroaxis can light in lathe coordinate system Sliding movement and without jumping phenomenon occur, so as to influence the stationarity of machine tool motion, beyond the servo ability of machine tool feed axle and increasing Nonlinearity erron during big processing etc..Therefore, it is necessary to carry out needing to consider during optimal tool orientation in workpiece coordinate system The situation of change of machine tool rotary axle in lathe coordinate system, or in lathe coordinate system directly to machine tool rotary axle carry out fairing it is excellent Change.

The content of the invention

To overcome the existing complex-curved generating tool axis vector of endless knife five-axis robot to undergo mutation and the problem of fluctuation, this hair It is bright to provide a kind of based on AB type five-axle number control machine tools endless knife processing generating tool axis vector method for fairing.

To achieve these goals, the technical solution adopted by the present invention is such：One kind is based on AB type five shafts numerical controlled machines Bed endless knife processing generating tool axis vector method for fairing, comprises the following steps：

A, the relation equation set up between endless knife generating tool axis vector and cutter spacing design variable；

B, the motion transform equation set up between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B；

C, the relation equation set up between endless knife cutter spacing design variable and five-axle number control machine tool gyroaxis A and B；

D, the design variable for determining endless knife generating tool axis vector fairing, object function and constraints, set up and are based on AB types five Shaft and NC Machining Test lathe endless knife processes generating tool axis vector fairing Mathematical Modeling；

E, the method for solving for determining generating tool axis vector fairing Mathematical Modeling in step D.

As preferred：The step A is specially：

(1) cutter local coordinate system is set up at cutter-contact point, local coordinate system O is derived_LX_LY_LZ_LMiddle endless knife cutter shaft arrow Relation equation between amount and cutter spacing design variable：

In formula, θ is endless knife in local coordinate system around Y_LThe top rake that axle is rotated, φ is annular cutter in local coordinate system Around Z_LThe side drift angle that axle is rotated,

In local coordinate system O_LX_LY_LZ_LEndless knife cutter location footpath arrow is at middle cutter-contact point：

In formula, r is the radius of circle that cuts of endless knife, and R is the ring heart radius of circle of endless knife.

(2) relation equation between endless knife generating tool axis vector and cutter spacing design variable in workpiece coordinate system is set up：

In formula, e₁=(x₁,y₁,z₁)^T, e₂=(x₂,y₂,z₂)^T, e₃=(x₃,y₃,z₃)^TRespectively local coordinate system O_LX_LY_LZ_LEach reference axis is in workpiece coordinate system O_wX_wY_wZ_wIn unit vector；

In workpiece coordinate system O_wX_wY_wZ_wEndless knife cutter location footpath arrow is at middle cutter-contact point：

As preferred：It is each according to AB type five-axle number control machine tools concrete structure, machine tool motion chain and lathe in the step B Relation between coordinate system, sets up relation equation between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B：

As preferred：In the step C, endless knife cutter spacing design variable and lathe are then derived in simultaneous formula (3) and (5) Relation equation between gyroaxis A and B：

As preferred：In the step D, machine tool rotary axle A and B is as design variable using in lathe coordinate system, with curved surface Every all cutter-contact point { P of row knife rail_i, i=1 ..., N } place be combined angular acceleration quadratic sum as object function, with machine tool rotary Axle A and B angle, angular speed and angular acceleration limitation scope are set up as constraints and are based on AB type five-axle number control machine tool rings Shape knife processes generating tool axis vector fairing Mathematical Modeling：

In formula, N is given row knife rail upper slitter number of contacts, β₁And β₂Machine tool rotary axle A and B corner, ω are represented respectively₁With ω₂Machine tool rotary axle A and B angular speed, α are represented respectively₁And α₂Machine tool rotary axle A and B acceleration is represented respectively,With Machine tool rotary angle beta is represented respectively₁And β₂Range of movement,WithMachine tool rotary angle beta is represented respectively₁And β₂Angular speed is limited Scope,WithMachine tool rotary angle beta is represented respectively₁And β₂Angular acceleration limits any cutter-contact point P on scope, curved surface_iPlace Compound angular acceleration is defined as：

In formula,For cutter-contact point P_iThe generating tool axis vector at place, t represents the time.

As preferred：The step E is specially：

(1) cutter spacing at given surface sampling cutter-contact point is generated using the existing axle tool position optimization method of endless knife five, then Obtain the initial generating tool axis vector at above-mentioned sampling cutter-contact point；

(2) the corresponding machine tool rotary angle A and B in each sampling cutter-contact point place is calculated using formula (5), recycles cubic spline Machine tool rotary angle A and B at each sampling cutter-contact point is carried out Cubic Spline Fitting by interpolating function respectively；

(3) calculate often row knife rail knife using the cubic spline interpolation fitting function obtained by step (2) and touch each knife on curve Machine tool rotary angle A and B at contact；Endless knife cutter spacing design variable θ and φ at each cutter-contact point are calculated using formula (6)；Solve each knife Endless knife error of cutter position is distributed at contact, and calculates minimal orientation-distance δ between endless knife and curve surface of workpiece；If δ ＜ 0, i.e., Interfered between endless knife and curve surface of workpiece, then make endless knife along the cutter-contact point direction of normal translation distance | δ |；Utilize formula (4) and (3) calculate endless knife cutter location footpath resultant generating tool axis vector at each cutter-contact point, curve is touched until solving the row knife rail knife The cutter location footpath resultant generating tool axis vector of upper all cutter-contact points.The beneficial effects of the invention are as follows this method can avoid machine tool rotary Mutator shaft and not fairing, make the motion of machine tool rotary axle more steady and smooth, angular speed and the angle of machine tool rotary axle are greatly decreased Acceleration, so as to improve the crudy and processing efficiency of curved surface.

Brief description of the drawings

Fig. 1 is to process generating tool axis vector method for fairing flow chart based on AB type five-axle number control machine tools endless knife

Fig. 2 is endless knife Primary Location；

Fig. 3 is AB ' type five-axle number control machine tool structural representations；

Fig. 4 is the coordinate system in AB ' type five-axle number control machine tools；

Fig. 5 is to solve flow chart based on AB type five-axle number control machine tools endless knife processing generating tool axis vector fairing Mathematical Modeling.

Embodiment

One kind of the present invention is based on AB type five-axle number control machine tools endless knife and processes generating tool axis vector method for fairing, its basic procedure As shown in figure 1, preferably embodiment is, including：

Step A, the relation equation set up between endless knife generating tool axis vector and cutter spacing design variable.The step A is specially：

(1) relation equation in local coordinate system between endless knife generating tool axis vector and cutter spacing design variable

As shown in figure 1, setting endless knife machining curve surface of workpiece S, r (u, v), P_cc(u_cc,v_cc) it is any point, n on curved surface_cc Sweared for the per unit system of the point.R is made to cut radius of circle, O for endless knife_LFor the origin of local coordinate system, thenWhereinRespectively point O_LAnd P_ccCorresponding footpath arrow.Meanwhile, respectively with point P_ccThe unit of place's direction of feed cuts resultant per unit system arrow n_ccIt is used as e₁And e₃, and take e₂=e₃×e₁.Then, with O_LFor the origin of coordinates, vector e₁、e₂And e₃Place direction is respectively X_L、Y_L And Z_LAxle sets up point P_ccThe local coordinate system O at place_LX_LY_LZ_L.And O_TX_TY_TZ_TFor the tool coordinate system being connected with cutter, its coordinate original Point O_TThe center justified positioned at cutter ring heart.Make O_TX_TY_TZ_TEach change in coordinate axis direction and O in original state_LX_LY_LZ_LUnanimously, and haveWherein R is the ring heart radius of circle of endless knife.Now, cutter is in surface points P_ccPlace by Primary Location, but Coordinate system O_LX_LY_LZ_LIn still suffer from two frees degree：One is around Y_LThe top rake θ of axle rotation, two be around Z_LThe side drift angle of axle rotation φ, two jiaos of the above is cutter spacing design variable.Different cutter spacing can be obtained by adjusting above-mentioned two angle, office can be obtained Portion coordinate system O_LX_LY_LZ_LMiddle cutter-contact point P_ccThe generating tool axis vector at place and cutter location footpath arrow

In formula,

It can be obtained by formula (1)

It can be obtained by formula (2)

(2) relation equation in workpiece coordinate system between annular cutter generating tool axis vector and cutter spacing design variable assumes local sit Mark system O_LX_LY_LZ_LEach reference axis is in workpiece coordinate system O_wX_wY_wZ_wIn vector be respectively e₁=(x₁,y₁,z₁)^T, e₂=(x₂,y₂, z₂)^T, e₃=(x₃,y₃,z₃)^T, then in workpiece coordinate system O_wX_wY_wZ_wMiddle generating tool axis vector and cutter location footpath arrow are

Endless knife cutter spacing design variable (i.e. top rake θ and angle of heel ψ) and generating tool axis vector can obtain by formula (5)Between Relation equation

Step B, the motion transform equation set up between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B.Institute Stating step B is specially：

Different according to gyroaxis position, AB types five-axle number control machine tool can be divided into three types again：1) AB types (A and B Axle is all located at main shaft side), 2) AB ' types (A axle positions are located at turntable side in main shaft side and B axle), 3) (A and B axle are all located at turning A ' B ' types Platform side).For simplicity, it will be hereafter illustrated using AB ' type five-axle number control machine tools as research object, its derivation formula is fitted For all AB types five-axle number control machine tools, as shown in Figure 3.Fig. 4 show in AB ' type five-axle number control machine tools between each coordinate system and closed System, general acquiescence workpiece coordinate system O_wX_wY_wZ_wWith lathe coordinate system O_mX_mY_mZ_mChange in coordinate axis direction it is consistent, digital control system can be by Both are associated.It can thus be concluded that, relation equation between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B：

In formula,

Rearrangement formula (8) can be obtained

Step C, the relation equation set up between endless knife cutter spacing design variable and five-axle number control machine tool gyroaxis A and B.Institute Stating step C is specially：

Formula (9) is substituted into formula (7) just to obtain between endless knife cutter spacing design variable and five-axle number control machine tool gyroaxis A and B Relation equation：

Step D, the design variable for determining endless knife generating tool axis vector fairing, object function and constraints, set up and are based on AB Type five-axle number control machine tool endless knife processes generating tool axis vector fairing Mathematical Modeling.The step D is specially：

(1) definition of angular acceleration is combined in five-axis robot

The concept of compound angular acceleration is introduced in lathe coordinate system, any cutter-contact point P on design curved surface_iLocate compound angle The definition of acceleration is

In formula, β₁And β₂Machine tool rotary axle A and B corner is represented respectively,For cutter-contact point P_iThe generating tool axis vector at place, t Represent time, ω₁And ω₂Machine tool rotary axle A and B angular speed, α are represented respectively₁And α₂Machine tool rotary axle A and B acceleration are represented respectively Degree.

(2) build and generating tool axis vector fairing Mathematical Modeling is processed based on AB type five-axle number control machine tools endless knife

The machine tool rotary axle A and B using in lathe coordinate system is as design variable, with the every all cutter-contact point { P of row knife rail of curved surface_i, I=1 ..., N } place is combined the quadratic sum of angular acceleration as object function, with machine tool rotary axle A and B angle, angular speed and Angular acceleration limits scope as constraints, sets up and processes generating tool axis vector fairing number based on AB type five-axle number control machine tools endless knife Learn model：

In formula,WithMachine tool rotary angle beta is represented respectively₁And β₂Range of movement,WithLathe is represented respectively Angle of revolution β₁And β₂Angular speed sets scope,WithMachine tool rotary angle beta is represented respectively₁And β₂Angular acceleration sets scope.

Step E, the method for solving for determining generating tool axis vector fairing Mathematical Modeling in step D.The step E is specially：Propose The method of the Mathematical Modeling of generating tool axis vector fairing, solution procedure are as shown in figure 5, its detailed process is as follows in a kind of solution procedure D：

(1) cutter spacing at given surface sampling cutter-contact point is generated using the existing axle tool position optimization method of endless knife five, then Obtain the initial generating tool axis vector at above-mentioned sampling cutter-contact point.

First, by analyzing processing curve the characteristics of, the numerical value of preliminary given endless knife cutter spacing design variable, i.e. top rake θ and side drift angle φ, it is assumed that it is N to touch curve up-sampling cutter-contact point number to stationary knife rail knife, and actual cutter-contact point number is M, public to fixed step size Difference is h.Then, parameter discrete method generation sampling cutter-contact point { CC is utilized etc. on curve being touched to stationary knife rail knife_i, i=1 ..., N }； Finally, all sampling cutter-contact point CC are calculated using Sturz methods_iLocate generating tool axis vector { T_ai, i=1 ..., N } and cutter location footpath arrow { T_pi, I=1 ..., N }.

(2) each sampling cutter-contact point CC is calculated using formula (10)_iThe corresponding machine tool rotary angle A in place_iAnd B_i, recycle three Secondary spline interpolation function is by each sampling cutter-contact point CC_iThe machine tool rotary angle A at place_iAnd B_iCubic Spline Fitting is carried out, so as to obtain machine Bed angle of revolution A_iAnd B_iCubic spline interpolation fitting function F (u, A) and F (u, B), wherein u for sampling cutter-contact point parameter become Amount.

(3) for giving stationary knife rail knife to touch curve, by given M or h and the parameter discrete method such as utilize or wait action error variance Method generates actual cutter-contact point { P_Ci, i=1 ..., M }, cubic spline interpolation fitting function F (u, B) and F in recycle step (2) (u, A) calculates the row knife rail knife and touches actual cutter-contact point P on curve_CiThe corresponding machine tool rotary angle A in place_iAnd B_i；Then formula is utilized (10) actual cutter-contact point P is obtained_CiLocate machine tool rotary angle A_iAnd B_iCorresponding endless knife cutter spacing design variable θ_iAnd φ_i；Solve knife Contact P_CiLocate the distribution of endless knife error of cutter position, and calculate minimal orientation-distance δ between endless knife and curve surface of workpiece；If δ ＜ Interfered between 0, i.e. endless knife and curve surface of workpiece, then make endless knife along the cutter-contact point direction of normal translation distance | δ |；By formula And (7) obtain cutter-contact point P (6)_CiLocate the cutter location footpath arrow T of endless knife_piWith generating tool axis vector T_ai, until i=M solves the row Knife touches the cutter location and generating tool axis vector of all cutter-contact points on curve.

The foregoing is intended to be a preferred embodiment of the present invention, but protection scope of the present invention is not limited thereto, Any one skilled in the art the invention discloses technical scope in, the change or replacement that can be readily occurred in, It should all be included within the scope of the present invention.

Claims (6)

1. one kind processes generating tool axis vector method for fairing based on AB type five-axle number control machine tools endless knife, it is characterised in that including as follows Step：

A, the relation equation set up between endless knife generating tool axis vector and cutter spacing design variable；

B, the motion transform equation set up between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B；

C, the relation equation set up between endless knife cutter spacing design variable and five-axle number control machine tool gyroaxis A and B；

D, the design variable for determining endless knife generating tool axis vector fairing, object function and constraints, set up and are based on the number of axle of AB types five Control lathe endless knife processing generating tool axis vector fairing Mathematical Modeling；

E, the method for solving for determining generating tool axis vector fairing Mathematical Modeling in step D.

2. according to claim 1 process generating tool axis vector method for fairing based on AB type five-axle number control machine tools endless knife, it is special Levy and be, the step A is specially：

(1) cutter local coordinate system is set up at cutter-contact point, local coordinate system O is derived_LX_LY_LZ_LMiddle endless knife generating tool axis vector with Relation equation between cutter spacing design variable：

$T_{axis}^{LCS}=R(Z_L,\mathrm{\φ})R(Y_L,\mathrm{\θ}){(0,0,1)}^T---\left(1\right)$

In formula, θ is endless knife in local coordinate system around Y_LThe top rake that axle is rotated, φ is annular cutter in local coordinate system around Z_L The side drift angle that axle is rotated,

In local coordinate system O_LX_LY_LZ_LEndless knife cutter location footpath arrow is at middle cutter-contact point：

$T_{pos}^{LCS}=R(Z_L,\mathrm{\φ})R(Y_L,\mathrm{\θ}){(-R,0,-r)}^T---\left(2\right)$

In formula, r is the radius of circle that cuts of endless knife, and R is the ring heart radius of circle of endless knife.

(2) relation equation between endless knife generating tool axis vector and cutter spacing design variable in workpiece coordinate system is set up：

$T_{axis}^{WCS}=\left(\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right)T_{axis}^{LCS}=\left(\begin{array}{c}{x}_{1}cos\mathrm{\φ}sin\mathrm{\θ}+x_{2}sin\mathrm{\φ}sin\mathrm{\θ}+x_{3}cos\mathrm{\θ}\\ y_{1}cos\mathrm{\φ}sin\mathrm{\θ}+y_{2}sin\mathrm{\φ}sin\mathrm{\θ}+y_{3}cos\mathrm{\θ}\\ z_{1}cos\mathrm{\φ}sin\mathrm{\θ}+z_{2}sin\mathrm{\φ}sin\mathrm{\θ}+z_{3}cos\mathrm{\θ}\end{array}\right)---\left(3\right)$

In formula, e₁=(x₁,y₁,z₁)^T, e₂=(x₂,y₂,z₂)^T, e₃=(x₃,y₃,z₃)^TRespectively local coordinate system O_LX_LY_LZ_LIt is each to sit Parameter is in workpiece coordinate system O_wX_wY_wZ_wIn unit vector；

In workpiece coordinate system O_wX_wY_wZ_wEndless knife cutter location footpath arrow is at middle cutter-contact point：

$T_{pos}^{WCS}=\left(\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right)T_{pos}^{LCS}---\left(4\right).$

3. according to claim 1 process generating tool axis vector method for fairing based on AB type five-axle number control machine tools endless knife, it is special Levy and be, in the step B, according between AB type five-axle number control machine tools concrete structure, machine tool motion chain and each coordinate system of lathe Relation, sets up relation equation between endless knife generating tool axis vector and five-axle number control machine tool gyroaxis A and B：

$T_{axis}^{WCS}={(\cos A\sin B,-\sin A,\cos A\cos B)}^T---\left(5\right).$

4. according to claim 1 process generating tool axis vector method for fairing based on AB type five-axle number control machine tools endless knife, it is special Levy and be, in the step C, endless knife cutter spacing design variable and machine tool rotary axle A and B are then derived in simultaneous formula (3) and (5) Between relation equation：

$\left(\begin{array}{c}\cos A\sin B\\ -\sin A\\ \cos A\cos B\end{array}\right)=\left(\begin{array}{c}{x}_{1}cos\mathrm{\φ}sin\mathrm{\θ}+x_{2}sin\mathrm{\φ}sin\mathrm{\θ}+x_{3}cos\mathrm{\θ}\\ y_{1}cos\mathrm{\φ}sin\mathrm{\θ}+y_{2}sin\mathrm{\φ}sin\mathrm{\θ}+y_{3}cos\mathrm{\θ}\\ z_{1}cos\mathrm{\φ}sin\mathrm{\θ}+z_{2}sin\mathrm{\φ}sin\mathrm{\θ}+z_{3}cos\mathrm{\θ}\end{array}\right)---\left(6\right).$

5. according to claim 1 process generating tool axis vector method for fairing based on AB type five-axle number control machine tools endless knife, it is special Levy and be, in the step D, the machine tool rotary axle A and B using in lathe coordinate system is as design variable, with the every row knife rail institute of curved surface There is cutter-contact point { P_i, i=1 ..., N } place be combined angular acceleration quadratic sum as object function, with machine tool rotary axle A and B angle Degree, angular speed and angular acceleration limitation scope are set up as constraints and process knife based on AB type five-axle number control machine tools endless knife Axial vector fairing Mathematical Modeling：

$\begin{array}{c}\min \mathrm{\Γ}=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left|\right|{\mathrm{\α}}_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)|{|}^2\\ s.t.\left\{\begin{array}{c}{\mathrm{\β}}_1={\widehat{\mathrm{\β}}}_{1\lim it}\\ {\mathrm{\β}}_2={\widehat{\mathrm{\β}}}_{2\lim it}\\ {\mathrm{\ω}}_1={\widehat{\mathrm{\ω}}}_{1\lim it}\\ {\mathrm{\ω}}_2={\widehat{\mathrm{\ω}}}_{2\lim it}\\ {\mathrm{\α}}_1={\widehat{\mathrm{\α}}}_{1\lim it}\\ {\mathrm{\α}}_2={\widehat{\mathrm{\α}}}_{2\lim it}\end{array}\right.\end{array}---\left(7\right)$

In formula, N is given row knife rail upper slitter number of contacts, β₁And β₂Machine tool rotary axle A and B corner, ω are represented respectively₁And ω₂Point Machine tool rotary axle A and B angular speed, α are not represented₁And α₂Machine tool rotary axle A and B acceleration is represented respectively,WithGeneration respectively Table machine tool rotary angle beta₁And β₂Range of movement,WithMachine tool rotary angle beta is represented respectively₁And β₂Angular speed limits scope,WithMachine tool rotary angle beta is represented respectively₁And β₂Angular acceleration limits any cutter-contact point P on scope, curved surface_iPlace is combined Angular acceleration is defined as：

$\begin{array}{c}{\mathrm{\α}}_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)=\frac{d^2{T}_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{{\mathrm{dt}}^2}=d(\frac{\∂T_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{\mathrm{\β}}_1}{\mathrm{\ω}}_1+\frac{\∂T_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{\mathrm{\β}}_2}{\mathrm{\ω}}_2)/dt=\\ 2\frac{{\∂}^2{T}_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{\mathrm{\β}}_1{\mathrm{\β}}_2}{\mathrm{\ω}}_1{\mathrm{\ω}}_2+\frac{{\∂}^2{T}_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{{\mathrm{\β}}_1}^2}{{\mathrm{\ω}}_1}^2+\frac{{\∂}^2{T}_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{{\mathrm{\β}}_2}^2}{{\mathrm{\ω}}_2}^2+\frac{\∂T_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{\mathrm{\β}}_1}{\mathrm{\α}}_1+\frac{\∂T_{P_i}({\mathrm{\β}}_1,{\mathrm{\β}}_2)}{\∂{\mathrm{\β}}_2}{\mathrm{\α}}_2\end{array}---\left(8\right)$

In formula,For cutter-contact point P_iThe generating tool axis vector at place, t represents the time.

6. according to claim 1 process generating tool axis vector method for fairing based on AB type five-axle number control machine tools endless knife, it is special Levy and be, the step E is specially：

(1) cutter spacing at given surface sampling cutter-contact point is generated using the existing axle tool position optimization method of endless knife five, then obtained Initial generating tool axis vector at above-mentioned sampling cutter-contact point；

(2) the corresponding machine tool rotary angle A and B in each sampling cutter-contact point place is calculated using formula (5), recycles cubic spline interpolation Machine tool rotary angle A and B at each sampling cutter-contact point is carried out Cubic Spline Fitting by function respectively；

(3) calculate often row knife rail knife using the cubic spline interpolation fitting function obtained by step (2) and touch each cutter-contact point on curve Locate machine tool rotary angle A and B；Endless knife cutter spacing design variable θ and φ at each cutter-contact point are calculated using formula (6)；Solve each cutter-contact point Locate the distribution of endless knife error of cutter position, and calculate minimal orientation-distance δ between endless knife and curve surface of workpiece；It is if δ ＜ 0, i.e., annular Interfered between knife and curve surface of workpiece, then make endless knife along the cutter-contact point direction of normal translation distance | δ |；Using formula (4) and (3) endless knife cutter location footpath resultant generating tool axis vector at each cutter-contact point is calculated, institute on curve is touched until solving the row knife rail knife There is the cutter location footpath resultant generating tool axis vector of cutter-contact point.