CN108415374A - Generating tool axis vector method for fairing based on lathe swivel feeding axis kinematics characteristic - Google Patents

Abstract

The invention relates to a tool axis vector smoothing method based on the kinematics characteristics of the rotary feed axis of a machine tool, belonging to the technical field of high-precision and high-efficiency milling of complex curved surface parts, and relates to a tool axis vector smoothing method based on the kinematics characteristics of the machine tool rotary feed axis. According to the geometric features of the surface, the method generates the machining tool path with the constraint of equal residual height. The coordinate transformation relationship between the workpiece coordinate system and the machine tool coordinate system is established, the tool axis vector sequence corresponding to the tool contact is converted into the rotation angle sequence of the machine tool's rotary feed axis, and the interval to be optimized is determined according to the angle change of the machine tool's rotary feed axis. Based on the quaternion method, calculate the coordinates of the tool axis vector corresponding to the tool contact in the interval to be optimized, and use the least squares fitting method to smooth the angle curve of the rotary feed axis of the machine tool after optimization, and to optimize the tool axis vector Perform interference check and adjustment. The method effectively reduces the abrupt movement of the rotary feed axis of the machine tool during the machining process, realizes smooth machining, and improves the surface machining quality.

Description

Tool axis vector smoothing method based on kinematics characteristics of machine tool rotary feed axis

technical field

The invention belongs to the technical field of high-precision and high-efficiency milling of complex curved surface parts, and relates to a tool axis vector smoothing method based on the kinematics characteristics of a rotary feed axis of a machine tool.

Background technique

Complex curved surface parts are widely used in various fields such as aerospace, automobiles, and ships. How to realize high-quality and efficient processing of complex curved surface parts is a hot and difficult point of current research. Compared with three-axis CNC machining, five-axis CNC machining adds two rotation axes. The contact state between the tool and the machined surface can be controlled by adjusting the included angle of the tool relative to the local coordinate system to ensure the processing quality and efficiency of the part, and at the same time avoid the gap between the tool and the workpiece. local and global interference. At present, the direction of the tool axis vector in five-axis machining is determined according to the local geometric information of the complex surface, but with the needs of the current industrial development, the shape of the curved surface is more complex, and the local geometric information has the characteristics of rapid change. The tool axis vector determined according to the local geometric information of the surface easily leads to a large change of the tool axis vector, which seriously restricts the improvement of the surface processing quality. Therefore, obtaining a smooth tool axis vector is the key link to improve the surface processing quality. At present, scholars at home and abroad have carried out a lot of research work on the optimization of the tool axis vector. There are two main ideas: one is to use UG and other processing software to generate the tool position and then adjust it; the other is to calculate the position of the tool contact point. The feasible tool axis space of , is then optimized under certain constraints and checked for interference.

In Wang Jing et al.’s paper “Overall Optimization Method of Tool Axis for Five-axis Machining of Complex Surface Parts”, Acta Aeronautica Sinica, 2013, 34(06): 1452-1462, first calculated the feasible tool axis space at all tangent points, and obtained On the basis of the critical tool axis vector, planar mapping is performed on it to establish the initial feasible region of the tool axis swing; secondly, a new feasible region of the tool axis swing is constructed by uniformly discretizing the initial feasible region; finally, the current The tool axis vector optimization model with no interference in the cutting line and the smallest change of adjacent tool axes realizes smooth control of the tool axis vector without interference in five-axis machining of free-form surfaces. However, this method has a large amount of calculation and does not consider the kinematics of the rotary feed axis of the machine tool during the optimization process, which has great limitations. The literature by Zhou Bo et al. "Research on Tool Axis Vector Optimization Method for Five-axis CNC Machining of Complex Surfaces", Chinese Journal of Mechanical Engineering, 2013, 49(7), 184-192, by inserting tool contacts in the non-interference area, in the interference area The improved C-space method is used to generate a smooth tool axis vector. This method can handle global interference and local interference, but inserting a tangential point reduces the actual feed rate and affects processing efficiency. It is not suitable for high-precision and high-efficiency processing. .

Contents of the invention

Aiming at the defects of the prior art, the present invention invents a tool axis vector smoothing method for five-axis numerical control machining of complex curved surfaces based on the kinematics characteristics of the rotary feed axis of the machine tool. The proposed method can effectively reduce the sudden change in the motion of the rotary feed axis of the machine tool during the machining process, achieve a stable machining process, improve the surface machining quality, and provide theoretical and technical support for high-precision and efficient machining of complex curved surfaces.

The technical scheme of the present invention is a tool axis vector smoothing method based on the kinematics characteristics of the rotary feed axis of the machine tool, which is characterized in that, according to the geometric characteristics of the curved surface, the processing tool path is generated with the equal residual height as a constraint; the workpiece coordinate system is established with the machine tool The conversion relationship between the coordinate systems, the tool axis vector corresponding to the tool contact point is converted into the rotation angle of the machine tool's rotary feed axis, and the interval to be optimized is determined according to the relationship between the machine tool's rotary feed axis angle and the cumulative arc length of the tool path; The boundary tool axis vector is constrained, and the tool axis homogenization method based on quaternion is used to optimize the tool axis vector, and the least square fitting principle is used to adjust the maximum curvature of the rotation angle of the rotary feed axis and the cumulative arc length of the tool path after optimization. The tool axis vector; finally carry out the interference check. The specific steps of the method are as follows:

Step 1: Generate the contact position of the tool track based on the equal residual height method

Take the complex surface as S(u,v), in the CAM software, use the ball-end cutter as the processing tool, use the method of equal residual height to generate the tool path r(ξ), and obtain the tool position file {P,V}, where P is Tool tip coordinates, V is the unit tool axis vector, according to the tool tip coordinates in the tool location file, the tool contact coordinates P _C can be obtained as:

P _C ＝P+R·VR·N (1)

In the formula, R represents the radius of the ball-end cutter; N represents the unit normal vector of the curved surface corresponding to the cutter contact.

Calculate geometric information such as unit normal vector N and unit tangent vector T on the surface S(u,v) according to formula (2):

In the formula, S _u (u,v) and S _v (u,v) are the first-order partial derivatives of the surface, T is the unit tangent vector of the surface along the direction of the tool path, r′(ξ) is one of the specific tool paths of the surface The order derivative, K is the cross product of the surface unit normal vector and the unit tangent vector.

Take the knife contact P _C as the origin, and take T, K, N as the three coordinate axes respectively, establish the local coordinate system P _C TKN, and obtain the distance between the ball-end knife-knife axis vector and the control angle in the local coordinate system from the coordinate transformation relationship The relationship equation:

In the formula, Indicates the tool axis vector in the local coordinate system, α indicates the forward inclination angle of the ball-end cutter around the K axis in the local coordinate system, β indicates the roll angle of the ball-end cutter around the axis N in the local coordinate system, Rot(K,α) and Rot (N, β) are the rotation matrices with K and N as the rotation axes and the rotation angles α and β respectively, expressed as follows:

Step 2: Calculation of the angle of the rotary feed axis in the machine tool coordinate system

Taking the AC double turntable five-axis CNC machine tool as an example, the rotation axis of the rotary table A is parallel to the X axis of the machine tool coordinate system, and the rotation axis of the rotary table C is parallel to the Z axis of the machine tool coordinate system. According to the configuration of the feed axis of the machine tool, the transformation matrix E from the workpiece base coordinate system to the tool coordinate system is:

In the formula, Rot(X,-θ _A ) and Rot(Z,-θ _C ) represent the rotation matrix of the rotary table with the X and Z axes as the rotation axes and the rotation angles of θ _A and θ _C respectively; Trans(x, y, z) represents the translation matrix. Specifically:

Note that the unit tool axis vector V in the workpiece coordinate system = [i ^* , j ^* , k ^* , 0], take the initial direction vector V _base of the tool axis to point to the positive direction of the Z axis of the machine tool coordinate system, that is, V _base = [0,0, 1,0], then:

V＝E·V _base (5)

From formulas (4) and (5), we can get:

In the formula, i ^* , j ^* , k ^* represent the three components of the unit tool axis vector in the workpiece coordinate system.

The relationship between the rotation angle of the machine tool's rotary feed axis and the tool axis vector V in the workpiece coordinate system can be obtained from formula (6):

In order to obtain the rotation angle value of the rotary feed axis of the machine tool, according to formula (7), the rotation angle sequence S{S _A , S _C } of the machine tool rotary feed axis is obtained from the tool axis vector V corresponding to the tool contact P _C in the workpiece coordinate system.

Step 3: Establishment of the mathematical model for calculating the angular velocity and angular acceleration of the rotary feed axis of the machine tool

Denote the processed surface as S(u,v), and the angular velocity ω and angular acceleration a of the rotating feed axis in five-axis NC machining of complex curved surfaces are:

In the formula, θ represents the angular position of the rotary feed axis of the machine tool, θ _ξ and θ _ξξ represent the first and second derivatives of the rotation angle variable of the rotary feed axis of the five-axis machining machine tool to the parameter ξ of the machining trajectory curve, respectively, and Respectively represent the first-order and second-order derivatives of the processing trajectory curve parameter ξ to the processing time t.

Since the tool contact is a discrete point in the actual process, the discrete method is used to solve the angular velocity and angular acceleration of the rotary feed axis of the machine tool. Taking the mth tool path as an example, the calculation is as follows:

Assuming that there are n tool contacts on the m-th tool track, the angular velocity of the rotary feed axes A and C of the machine tool corresponding to the i-th tool contact P _Ci is:

In the formula, and Respectively represent the rotation angle values of the machine tool A and C rotary feed axes corresponding to the i-th tool contact, and Respectively represent the angular velocity of the rotary feed axis of the machine tool A and C corresponding to the i-th tool contact, L _i represents the distance from the tool contact P _Ci to the adjacent tool contact P _Ci+1 on the curved surface, and v represents the constant progress during machining give speed.

From this, the angular velocity of the rotary feed axis of the machine tool can be obtained as:

In the formula, _ωi represents the angular velocity of the rotary feed axis of the machine tool corresponding to the i-th tool contact.

The angular acceleration of the rotary feed axis of machine tool A and C corresponding to the i-th tool contact point P _Ci is:

In the formula, a _Ai and a _Ci represent the angular acceleration of the rotary feed axes of machine tools A and C corresponding to the i-th tool contact, respectively.

From this, the angular acceleration value of the rotary feed axis of the machine tool can be obtained as:

In the formula, a _i represents the angular acceleration of the rotary feed axis of the machine tool corresponding to the i-th tool contact.

Step 4: Selection of the interval to be optimized and smoothing of the tool axis vector

The optimization interval is selected according to the included angle of the vector formed by adjacent tool contacts in the relationship curve between the rotation angle of the machine tool's rotary feed axis and the cumulative arc length of the tool path. Firstly, the rotation angle value sequence S{S _A , S _C } of the machine tool rotation feed axis corresponding to each tool contact point is obtained by formula (7), and the arc length is represented by the distance L _i between adjacent tool contacts, and the rotation feed axis of the machine tool is obtained For the angle curve of the axis A, C rotation angle with respect to the cumulative arc length of the tool path, the following vectors are defined by the adjacent tool contacts P _Ci-1 , P _Ci and P _Ci+1 on the curve:

P _Ci P _Ci-1 ＝(b _i ,c _i ),P _Ci P _Ci+1 ＝(b _i+1 ,c _i+1 )

In the formula, b _i ＝L _i means the arc length between two adjacent tool contacts; c _i ＝θ _i -θ _i-1 means the difference in rotation angle between two adjacent tool contacts corresponding to the rotary feed axis of the machine tool value.

Calculate the vector angle formed by the ith knife contact on the curve and the adjacent discrete knife contact as:

In the formula, _m θ _i represents the vector angle formed by the i-th tool contact of the m-th tool track and the adjacent tool contacts.

When i=2,...,n-1, get the vector angle formed by each knife contact and adjacent knife contacts, and then calculate the average value of all knife contact position vector angles for:

By comparing the vector angle value and the average value formed by each knife contact and the adjacent knife contact, the optimization interval is selected:

In the formula, e represents the initial tool contact number of the section to be optimized on the tool track, and f indicates the end tool contact number of the section to be optimized on the tool track.

Note that the interval conforming to formula (15) is the interval to be optimized R=[e,f]. Based on the principle of uniform change of the tool axis vector and considering the constraint of the boundary tool axis vector, the tool axis homogenization method based on quaternion is used to optimize the The tool axis vector of the interval tool contact position. The tool axis homogenization method based on quaternions first selects the initial tool axis vector V ₁ and the end tool axis vector V _n according to the boundary of the interval to be optimized, and takes V ₁ and V _n as the boundary, according to the formula (16 ) to optimize the tool axis vector of the tool contact position within the interval [e, f] to be optimized.

In the formula, V _i represents the optimized tool axis vector in the area to be optimized [e, f], V ₁ and V _n represent the initial knife contact P _Ce and the end knife contact P _Cf of the area to be optimized [e, f] respectively The corresponding tool axis vector, θ _Q =arccos(V ₁ ·V _n ), represents the angle formed by the tool axis vectors V ₁ and V _n .

Thus, the optimized tool axis vector of the corresponding tool contact position in the interval to be optimized is obtained, and the angle value of the machine tool rotary feed axis at the corresponding tool contact position is obtained according to formula (7), and then the optimized rotary feed axis is obtained The corner sequence is

In the formula, and Respectively represent the rotation angle sequences of the A-axis and C-axis of the machine tool after optimization.

In order to smooth the curve of the relationship between the rotation angle of the rotary feed axis of the machine tool and the cumulative arc length of the tool path, and avoid sudden changes in the angular acceleration of the rotary feed axis, analyze the optimized curve of the relationship between the rotation angle of the rotary feed axis of the machine tool and the cumulative arc length of the tool path, and select the curve The knife contact P _Cε with the largest curvature, that is, the position near the tip point, adjusts the shape of the curve near the tip point on the curve by the least square fitting principle, so that the curve F satisfies the following constraint equation:

In the formula, ω _i represents the weight coefficient of each data point, θ ^* (L _i ) represents the rotation angle value of the machine tool rotation feed axis after adjusting the tool contact P _Ci , and L _i and θ _i represent the horizontal and vertical coordinates of the curve, respectively.

According to the formula (18), the rotation angle of the rotary feed axis and the cumulative arc length curve of the tool path can be smoothly transitioned, and the re-optimized sequence of the rotary feed axis is obtained as

In the formula, and Respectively represent the rotation angle sequences of the A-axis and C-axis of the machine tool after the final optimization.

The rotation angle sequence of the rotary feed axis can be transformed into the tool axis vector sequence _k V = { _k V ₁ ,…, _k V _n } expressed in the workpiece coordinate system by the formula (6), and the optimized tool axis vector in the workpiece coordinate system can be obtained The coordinate value below.

Calculate the rake angle and roll angle of the tool axis vector in the corresponding workpiece local coordinate system at this time, compare with the feasible region of the tool axis vector, and compare according to the local curvature radius of the surface to be processed at the tool contact point and the effective cutting radius of the tool , to judge whether there is interference, if there is interference, adjust the direction of the tool axis vector to avoid collision interference, and thus the optimized tool axis vector can be obtained.

The notable effect and benefit of the present invention is that a new principle of tool axis vector optimization interval selection is proposed, which can satisfy situations such as sudden changes in the geometric characteristics of complex surfaces; The tool axis vector smoothing method solves the problem that the existing methods are difficult to ensure the smooth movement of the machine tool's rotary feed axis in the five-axis CNC machining of complex surfaces, and improves the surface processing quality; The kinematic characteristics of the rotary feed axis and the processing of the vector interference of the tool axis are more comprehensive. This method is based on the tool axis vector smoothing method based on the kinematics characteristics of the rotary feed axis of the machine tool. It has strong applicability in five-axis CNC machining of complex curved surfaces, and is suitable for five-axis precise and efficient machining of various complex characteristic curved surfaces. It is very important for improving the processing quality of curved surfaces. It is of great practical significance to improve the machining quality of the workpiece surface by giving full play to the kinematics performance of the machine tool and improving the efficiency.

Description of drawings

Figure 1—The overall flowchart of the tool axis vector optimization method.

Figure 2—Schematic diagram of the direction of the tool axis vector in the local coordinate system.

Figure 3a) shows the simulation results of the relationship between the angular velocity of the rotary feed axis and the cumulative arc length of the tool path, the abscissa represents the cumulative arc length of the tool path, and the ordinate represents the angular velocity; Figure 3b) represents the angular acceleration of the rotary feed axis and the cumulative arc length of the tool path In the diagram of relational simulation results, the abscissa represents the accumulated arc length of the tool path, and the ordinate represents the angular acceleration.

Figure 4a) shows the simulation results of the relationship between the angular velocity of the rotary feed axis and the cumulative arc length of the tool path before and after the optimization of the tool axis vector, in which 1 and 2 respectively represent the relationship between the angular velocity and the cumulative arc length of the tool path before and after optimization; Figure 4b) represents the optimization The simulation result diagram of the relationship between the angular acceleration of the forward and backward rotary feed axis and the accumulated arc length of the tool path. Among them, 1 and 2 represent the relationship between the angular acceleration before and after optimization and the accumulated arc length of the tool path, respectively.

Figure 5a) shows the roughness of the machined surface before the tool axis vector optimization, and Figure 5b) shows the roughness of the machined surface after the tool axis vector optimization; Ra is the machined surface roughness.

Figure 6—Comparison and fitting diagram of the machined surface profile before and after optimization under the measurement of the three-coordinate measuring machine, 1 indicates the machined surface profile before optimization, and 2 indicates the machined surface profile after optimization.

Detailed ways

The specific embodiment of the present invention will be described in detail in conjunction with the technical scheme and the accompanying drawings

During the five-axis CNC machining of complex curved surfaces, sudden changes in the tool axis vector or unsmoothness can easily cause sudden changes in the angular velocity of the machine tool's rotary feed axis, resulting in machining vibration marks, which seriously affect the surface processing quality of the workpiece. To solve the problem of smoothing, a tool axis vector smoothing method for five-axis CNC machining of complex curved surfaces based on the kinematics characteristics of the rotary feed axis of the machine tool was invented. The overall process is shown in Figure 1.

Using an AC double turntable five-axis CNC machine tool, taking a concave-convex table with different curvature characteristics and a sudden change in curvature characteristics as an example, with the help of UG software and MATLAB software, the implementation process of the present invention is described.

First, use UG software to model the concave-convex table surface, choose the direction of the tool axis to lean forward 15° along the feed direction, that is, α = 15°, β = 0°, and generate the machining tool path with the constraint of equal residual height 0.002, and obtain the tool position file, and generate tool contact coordinates on the tool path curve according to formulas (1)-(3). See Figure 2 for the positional relationship of the tool axis vector relative to the workpiece surface, analyze the conversion relationship between the workpiece coordinate system and the machine tool coordinate system, and calculate the machine tool rotation feed corresponding to the tool contact point on the tool path curve according to formulas (4)-(7) Axis rotation angle value.

Secondly, use MATLAB software to calculate the angular velocity and angular acceleration of the rotary feed axis of the machine tool according to the formulas (8)-(12), and the calculation results are shown in Figure 3. Through calculation, the maximum angular velocity on this tool path is 24°/s, and the maximum angular acceleration is 127°/s ² .

Then select the interval to be optimized according to formulas (13)-(15), and the serial numbers of the knife contacts corresponding to the interval are {26-47, 69-72, 116-122}. In the interval to be optimized, according to formulas (16)-(19), the optimized tool axis vector direction is calculated. Taking the tool contact number 69-72 as an example, the tool axis vector direction before optimization is {(-0.828,0.358,0.432)(-0.81,0.398,0.430)(-0.791,0.438,0.428)(-0.786, 0.447,0.427)}, the tool axis vector direction optimized by the method of the present invention is {(-0.799,0.418,0.432)(-0.801,0.416,0.431)(-0.802,0.414,0.43)(-0.803,0.412 ,0.431)}. The rotation angle sequence of the machine tool rotation feed axis angle value corresponding to the tool contact in this interval before optimization is S{S _A , S _C }={(64.407,-66.599)(64.504,-63.854)(64.689,-60.987) (64.735,-60.385)} After being optimized by the method of the present invention, the optimized corner sequence obtained is

Finally, calculate the angular velocity and angular acceleration of the rotary feed axis of the machine tool after the tool axis vector optimization according to the formulas (8)-(12), and compare them with the calculation results without the tool axis vector optimization, see Figure 4. By comparison, the maximum angular velocity on the tool path after the tool axis vector optimization is 12°/s, which is 50% lower than the angular velocity when the tool axis vector is not optimized, and the maximum angular acceleration is 89°/s ² Angular acceleration reduced by 30%.

In order to further verify the effectiveness of the proposed method, carry out the tool axis vector optimization process and the tool axis vector optimization method comparison experiment in the CAM software, the experimental results show that the machined surface quality after the tool axis vector optimization of the present invention is obviously better than that of CAM The machining quality after the tool axis vector optimization in the software. The surface roughness of the processed workpiece was measured, and the results shown are shown in Figure 5. The measurement results show that the processing quality of the present invention is improved at the sudden change of curvature, and the surface roughness is reduced from Ra=1.6143 μm to Ra=1.1868 μm, a decrease of 26.4%. The coordinates of the middle position of the workpiece are measured by a three-coordinate measuring machine and the measured data are fitted to analyze the surface morphology after processing. The fitting results show that before the optimization of the tool axis vector, there are phenomena such as overcut and undercut during machining. After optimization This phenomenon can be improved, and the results shown are shown in Figure 6. The measurement results are in good agreement with the experimental results, indicating that the tool axis vector smoothing method based on the kinematics characteristics of the rotary feed axis of the machine tool of the present invention can make the rotary feed axis of the machine tool run smoothly, and obviously improve the processing quality of the sudden change of curvature. It provides guidance for high-quality and efficient machining of variable-curvature surface parts in actual engineering.

Claims (1)

1. a kind of generating tool axis vector method for fairing based on lathe swivel feeding axis kinematics characteristic, which is characterized in that this method root It is that constraint generates processing knife rail with equal scallop-heights according to surface geometry feature；It establishes between workpiece coordinate system and lathe coordinate system Transformational relation, the corresponding generating tool axis vector of cutter-contact point is converted to the corner of lathe swivel feeding axis, and be rotated into according to lathe Section to be optimized is determined to the relationship of Shaft angle and the cumulative arc length of knife rail；With interval border generating tool axis vector be constraint, using based on The cutter shaft homogenizing method of quaternary number optimizes generating tool axis vector, and with swivel feeding Shaft angle after least square fitting principle adjusting and optimizing With the generating tool axis vector of the cumulative arc length curve mean curvature maximum of knife rail；Finally carry out interference checking；Method is as follows：

The first step：Knife rail cutter-contact point position is generated based on Constant scallop-height

It is S (u, v) to take complex-curved, in CAM softwares, using ball head knife as process tool, is generated using constant scallop-height method Knife rail r (ξ) obtains cutter location file { P, V }, and wherein P is point of a knife point coordinates, and V is unit generating tool axis vector, according in cutter location file Point of a knife point coordinates can obtain cutter-contact point coordinate P_CFor：

P_C=P+RV-RN (1)

In formula, R indicates ball head knife tool radius；Ν indicates the corresponding curved surface per unit system arrow of cutter-contact point；

The geological informations such as per unit system arrow N, the unit tangent vector T on curved surface S (u, v) are calculated according to formula (2)：

In formula, S_u(u,v)、S_v(u, v) is the first-order partial derivative of curved surface, and T is curved surface along the unit tangent vector in knife rail direction, r ' (ξ) For the first derivative of the specific knife rail of curved surface, K is the cross product of curved surface per unit system arrow and unit tangent vector；

With cutter-contact point P_CFor origin, it is respectively three reference axis with T, K, N, establishes local coordinate system P_CTKN is converted by coordinate and is closed System obtains the relation equation under ball head knife generating tool axis vector and local coordinate system between pilot angle：

In formula,Indicate that the generating tool axis vector under local coordinate system, α indicate ball head knife leaning forward around K axis under local coordinate system Angle, β indicate that for ball head knife around the angle of heel of axis N, Rot (K, α) and Rot (N, β) are respectively rotation with K and N under local coordinate system Axis, rotation angle are the spin matrix of α and β, are indicated as follows：

Second step：Swivel feeding axis angle calculation under lathe coordinate system

By taking the bis- turntable-type five-axle number control machine tools of AC as an example, the shaft of rotary table A is parallel with the X-axis of lathe coordinate system, rotation The shaft of revolving worktable C is parallel with the Z axis of lathe coordinate system；According to the configuration of machine tool feed axis, from workpiece basis coordinates system to knife Tool coordinate system transformation matrix E be：

In formula, Rot (X ,-θ_A), Rot (Z ,-θ_C) expression rotary table is using X, Z axis as rotary shaft respectively, rotation angle θ_A、θ_C Spin matrix；Trans (x, y, z) indicates translation matrix；Specially：

Remember unit generating tool axis vector V=[i under workpiece coordinate system^*,j^*,k^*, 0], take cutter shaft inceptive direction vector V_baseLathe is directed toward to sit Mark system Z axis is positive, i.e. V_base=[0,0,1,0], then have：

V=EV_base (5)

It is available by formula (4) and (5)：

In formula, i^*, j^*, k^*Indicate three components of the unit generating tool axis vector under workpiece coordinate system；

Relationship between generating tool axis vector V under lathe swivel feeding Shaft angle and workpiece coordinate system can be obtained by formula (6)：

Lathe swivel feeding shaft rotation angle value in order to obtain, according to formula (7), by cutter-contact point P under workpiece coordinate system_CCorresponding cutter shaft Vector V obtains lathe swivel feeding Shaft angle sequence S { S_A,S_C}；

Third walks：Lathe swivel feeding axis angular rate counts with angular accelerometer and learns model foundation

Note processing curve is S (u, v), and swivel feeding axis angular rate ω and angular acceleration a is in complex-curved five-shaft numerical control processing：

In formula, θ indicates the angular position of lathe swivel feeding axis, θ_ξAnd θ_ξξFive-axis robot lathe swivel feeding shaft rotation is indicated respectively Single order of the angle variable to processing trace curve parameter ξ, second dervative,Withξ pairs of processing trace curve parameter is indicated respectively The single order of process time t, second dervative；

Since in real process, cutter-contact point is discrete point, therefore discrete method is used to solve lathe swivel feeding axis angular rate and angle Acceleration；By taking the m articles knife rail as an example, calculate as follows：

It takes and shares n cutter-contact point on the m articles knife rail, then i-th of cutter-contact point P_CiCorresponding lathe A, C swivel feeding axis angular rate For：

In formula,WithCorresponding lathe A, C swivel feeding the shaft rotation angle value of i-th of cutter-contact point is indicated respectively,WithRespectively Indicate corresponding lathe A, C the swivel feeding axis angular rate of i-th of cutter-contact point, L_iIndicate cutter-contact point P on curved surface_CiTo adjacent cutter-contact point P_Ci+1Distance, v indicate processing when Constant feeding rate；

This makes it possible to obtain lathe swivel feeding axis angular rate values to be：

In formula, ω_iIndicate the corresponding lathe swivel feeding axis angular rate of i-th of cutter-contact point；

I-th of cutter-contact point P_CiCorresponding lathe A, C swivel feeding shaft angle acceleration is：

In formula, a_AiAnd a_CiCorresponding lathe A, C swivel feeding the shaft angle acceleration of i-th of cutter-contact point is indicated respectively；

This makes it possible to obtain lathe swivel feeding shaft angle acceleration values to be：

In formula, a_iIndicate the corresponding lathe swivel feeding shaft angle acceleration of i-th of cutter-contact point；

4th step：Interval selection and generating tool axis vector fairing to be optimized

According to the angle of the enough vectors of adjacent cutter-contact point in lathe swivel feeding Shaft angle and the cumulative arc length relation curve of knife rail Selection optimization section；Corresponding lathe swivel feeding Shaft angle value sequence S at each cutter-contact point is acquired by formula (7) first {S_A,S_C, with adjacent cutter-contact point distance L_iIt indicates its arc length, obtains lathe swivel feeding axis A, C corner about the cumulative arc length of knife rail Angle curve, with adjacent cutter-contact point P on curve_Ci-1、P_CiAnd P_Ci+1The following vector of definition：

P_CiP_Ci-1=(b_i,c_i),P_CiP_Ci+1=(b_i+1,c_i+1)

In formula, b_i=L_i, indicate the arc length between adjacent two cutter-contact point；c_i=θ_i-θ_i-1, indicate that adjacent two cutter-contact point corresponds to machine The corner difference of bed swivel feeding axis；

The vector angle that i-th of cutter-contact point is constituted with adjacent discrete cutter-contact point on calculated curve is：

In formula,_mθ_iIndicate the vector angle that the m articles knife rail, i-th of cutter-contact point is constituted with adjacent cutter-contact point；

In i=2 ..., when n-1, the vector angle that each cutter-contact point is constituted with adjacent cutter-contact point is obtained, all knives is then calculated and touches The average value of point position vector angleFor：

By comparing the size of vector angle value and average value that each cutter-contact point is constituted with adjacent cutter-contact point, selection optimization area Between：

In formula, e indicates that the initial cutter-contact point serial number in section to be optimized on knife rail, f indicate that cutter-contact point is terminated in section to be optimized on knife rail Serial number；

Remember that the section of meeting formula (15) is section R=[e, f] to be optimized, is changed uniformly for principle with generating tool axis vector, take into account boundary Generating tool axis vector constrains, and optimizes the generating tool axis vector of the section cutter-contact point position with the cutter shaft homogenizing method based on quaternary number；Based on four The cutter shaft homogenizing method of first number selectes initial generating tool axis vector V according to interval border to be optimized first₁With end generating tool axis vector V_n, with V₁And V_nTwo generating tool axis vectors are boundary, and the cutter shaft for optimizing the interior cutter-contact point position of section [e, f] to be optimized according to formula (16) is sweared Amount；

In formula, V_iIndicate the generating tool axis vector after the optimization of section to be optimized [e, f], V₁And V_nRegion to be optimized [e, f] is indicated respectively Initial cutter-contact point P_CeWith termination cutter-contact point P_CfCorresponding generating tool axis vector, θ_Q=arccos (V₁·V_n), indicate generating tool axis vector V₁And V_n The angle of composition；

Thus the generating tool axis vector after corresponding cutter-contact point position optimization in section to be optimized is obtained, and is corresponded to according to formula (7) Lathe swivel feeding shaft angle angle value at cutter-contact point position, and then the swivel feeding Shaft angle sequence after being optimized is

In formula,WithThe corner sequence of lathe A axis and C axis after optimizing is indicated respectively；

For the curve of smooth lathe swivel feeding Shaft angle and the cumulative arc length relationship of knife rail, swivel feeding shaft angle acceleration is avoided It is mutated, the curve of the lathe swivel feeding Shaft angle after analysis optimization and the cumulative arc length relationship of knife rail, trade-off curve mean curvature is most Big cutter-contact point P_Cε, i.e. tip point neighbouring position, with curved shape near the tip point on least square fitting principle adjustment curve Shape makes curve F meet following constraint equation：

In formula, ω_iIndicate the weight coefficient of each data point, θ^*(L_i) indicate cutter-contact point P after adjustment_CiLathe swivel feeding shaft rotation Angle value, L_i、θ_iCurve cross, ordinate are indicated respectively；

Swivel feeding Shaft angle and the cumulative arc length curve of knife rail can be seamlessly transitted by formula (18), obtain being rotated into after re-optimization It is to axis sequence

In formula,WithRespectively indicate final optimization pass after lathe A axis and C axis corner sequence；

Swivel feeding Shaft angle sequence can be transformed by formula (6) by the generating tool axis vector sequence indicated under workpiece coordinate system_kV= {_kV₁,…,_kV_n, coordinate value of the generating tool axis vector under workpiece coordinate system after being optimized；

Calculate corresponding workpiece local coordinate system at this time and cut the top rake and angle of heel of axial vector, with generating tool axis vector feasible zone into Row compares, and is compared according to effective radius of clean-up of the local radius of curvature of curved surface to be processed at cutter-contact point and cutter, sentences It is disconnected whether to interfere, if interfering, the adjustment of tool axis direction is carried out, it is made to avoid collision interference, this makes it possible to obtain Generating tool axis vector after optimization.