CN112486094B - Cutter optimization and processing track generation method for complex curved surface side milling - Google Patents

Cutter optimization and processing track generation method for complex curved surface side milling Download PDF

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CN112486094B
CN112486094B CN202011377249.6A CN202011377249A CN112486094B CN 112486094 B CN112486094 B CN 112486094B CN 202011377249 A CN202011377249 A CN 202011377249A CN 112486094 B CN112486094 B CN 112486094B
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CN112486094A (en
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伯彭波
张晓苗
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Harbin Institute of Technology Weihai
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Abstract

The invention discloses a cutter optimization and processing track generation method for complex curved surface side milling, which comprises the following steps: step one, interactively designing initial curves A (t) and B (t) to generate an initial tool machining path; step two, generating an initial cutter shape; step three, optimizing a processing path and a cutter shape; step four, error calculation; and step five, calculating the curved surface of the cutter. The main contribution of the invention is to provide a calculation framework which can directly calculate the continuous parametric surface representation of the shape of the cutter for the side milling processing of the complex curved surface and has higher precision and higher calculation efficiency. Different from the existing method, the method of the invention directly optimizes a B-spline-expressed cutter curved surface to obtain a continuously expressed cutter shape, and the method can process over-cut control and cutter jitter control.

Description

Cutter optimization and processing track generation method for complex curved surface side milling
Technical Field
The invention relates to a cutter shape optimization method facing side milling, in particular to a cutter optimization method for complex curved surface high-precision side milling and a processing track generation method corresponding to a generated cutter.
Background
In the machining of complex curved surfaces such as impeller blades of aircraft engines, end milling machining and side milling machining are commonly used as machining techniques. The end milling technology has the problems of easy interference generation, low processing efficiency and the like. The side milling belongs to line contact processing and has wider processing line spacing, so the processing efficiency is far higher than that of end milling. The side milling has higher processing precision and is generally applied to the finish machining stage. The side milling requires that the tool and the workpiece curved surface are tangent to a curve at each moment, so that the path planning of the side milling is difficult due to the strict geometric constraint, and the side milling is mainly used for processing a ruled surface at present.
For a non-developable ruled surface or a free-form surface, a high-precision machining result is difficult to obtain by adopting a standard cutter, a proper cutter curved surface shape and a corresponding machining track need to be calculated aiming at a designed curved surface, and then the cutter is customized for actual machining. The existing method can obtain the representation of the discrete cutter curved surface, the representation precision is not high enough, and the existing side milling cutter shape optimization method does not consider the control of motion shake and over-cut.
Disclosure of Invention
The invention aims to provide a tool optimization method for high-precision side milling of a complex curved surface and a method for generating a processing track corresponding to a generated tool. Different from the existing method, the method of the invention directly optimizes a cutter curved surface represented by a B-spline to obtain the cutter shape represented by a continuous parameter curved surface and simultaneously obtain the continuous representation of the movement path of the cutter, and the method can control over-cutting and cutter shaking.
The purpose of the invention is realized by the following technical scheme:
a cutter optimization and processing track generation method for complex curved surface side milling processing comprises the following steps:
step one, designing initial curves A (t) and B (t) alternately to generate an initial processing path:
(1) the user interactively appoints K +1 line segments, two end points of each line segment are positioned on the curved surface S, and a line segment sequence L is supposed to be obtained0＝(A0，B0)，...，LK＝(AK，BK) Wherein A isi,BiK represents two end points of a line segment, the length of each line segment is equal to L, and L is the length of the tool;
(2) the end point A of the line segment obtained in (1)iOr BiRespectively move a distance R along the normal direction of the curved surfaceAAnd RBNamely, updating as:
Ai＝Ai+N(Ai)·RA；
Bi＝Bi+N(Bi)·RB；
wherein N (P) represents the normal direction of the curved surface of the point P;
(3) based on the endpoint Ai、BiDefining the track curves at two ends of the axis of the cutter as follows:
Figure BDA0002807460290000021
Figure BDA0002807460290000022
wherein N isi,d(t) is the B-spline basis function of degree d;
the line segment connecting the same parameter points on A (t) and B (t) represents the axis of the tool, and the points on the axis define an axis locus R (t, s) as follows:
R(t,s)＝A(t)(1-s)+B(t)s t∈[0,1],s∈[0,L]；
step two, generating an initial cutter shape:
(1) taking a sampling point R on an axial trace plane R (t, s)i,j：
Ri,j＝R(ti,sj)＝A(ti)(1-sj)+B(ti)sj；
Wherein s isj∈[0,1]J is 0.. multidot.w, which is the sampling value in the parameter domain s, and W +1 is the number of sampling points; t is ti∈[0,1]J is 0,.. times, Y is a sampling value in the parameter domain t, and Y +1 is the number of sampling points;
(2) calculate each sample point Ri,jNearest distance d to design curved surface Si,jDefining data points(s) in a planar rectangular coordinate system X-Yj,di,j)；
(3) Data points(s)j,di,j) A function fit to the form:
Figure BDA0002807460290000031
wherein f isi,i＝0,...,NfIs a 1-dimensional control point, and the number of the control points is Nf+1；Ni,d(s) is a B-spline basis function of degree d;
assuming that the axis of the cutter curved surface is L(s), and r(s) defines a plane curve C, wherein the curve C rotates 360 degrees around the X axis to obtain a rotating curved surface as an initial cutter curved surface;
step three, optimizing a processing path and a cutter shape:
(1) sampling on the axial trace surface R (t, s) to obtain a data point R (t)i,sj)：
R(ti,sj),i＝0,...,N1,j＝0,...,N2；
Wherein N is1And N2Is an integer representing the number of sampling points on parameter t and parameter s, respectively;
calculating the sampling point R (t)i,sj) Foot point S (u) on design surface Si,j,vi,j) Calculating the point S (u) on the design surfacei,j,vi,j) Unit normal vector N (u) ofi,j,vi,j)；
(2) Solving an optimization problem defined by:
Figure BDA0002807460290000041
wherein, (A' (t)i)-B′(ti))·(B(ti)-A(ti) 0, i 0, N is a rigid motion constraint, meaning that the tool axis remains fixed in length while in motion; fApproxIs a distance error function taking into account over-cut control, FEnergyIs a stability control function of the movement of the cutter head;
step four, error calculation:
calculating R (t)i,sj) Foot point S (u) on design surface Si,j,vi,j) Calculating machining error
Figure BDA0002807460290000042
Where G is the number of sampling points; if the error meets the requirement, outputting a function A (t), a function B (t) for determining the machining path and a function r(s) for determining the shape of the cutter, and turning to the fifth step, otherwise, turning to the third step;
step five, calculating the curved surface of the cutter;
a rectangular coordinate system is established by taking the X axis of the coordinate system as the z axis, taking any vector perpendicular to the z as the X axis and taking zxx (X represents the cross product of the vector) as the y axis, and the parameterization of the tool curved surface under the coordinate system is defined as the following formula:
Figure BDA0002807460290000043
where s and θ are parameters of the tool curved surface.
In the present invention, the variables in the optimization problem include the control points A of the curve A (t)i,i＝0,…Control points B of, K, B (t)iI-0, …, K (K stands for the number of control points), and a control point f of a function r(s)i,i＝0,...,Nf(NfRepresenting the number of control points).
Compared with the prior art, the invention has the following advantages:
(1) the main contribution of the invention is to provide a calculation framework which can directly calculate the continuous parametric surface representation of the shape of the cutter for the side milling processing of the complex curved surface and has higher precision and higher calculation efficiency. The existing method can only calculate the discrete representation of the curved surface of the cutter, if more variables are needed to obtain higher precision, the more variables can also increase the calculated amount and reduce the calculation efficiency.
(2) The calculation framework of the invention directly optimizes the processing path of the cutter, and realizes the control of the cutter shake by introducing the function related to the cutter shake.
(3) The computing framework of the invention can control the over-cutting in the processing and can generate the processing result without the over-cutting.
Drawings
FIG. 1 is a sequence of line segments on a design surface;
FIG. 2 is a view of the sequence of line segments being elevated along the normal of the surface;
FIG. 3 is a schematic diagram illustrating the generation of an initial surface;
FIG. 4 is a shape definition of a general cutter;
FIG. 5 is an initial tool shape and motion;
FIG. 6 is a flow chart of the method of the present invention;
FIG. 7 shows a machining path (left) generated by the method of the present invention and a corresponding tool shape (right) approximating a conical curved surface;
FIG. 8 is a drawing of a machining path (left) and a corresponding tool shape (right) of a convex curved surface generated by the method of the present invention;
fig. 9 shows the machining path (left) generated by the method of the present invention and the corresponding tool shape (right) of the concave curved surface.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings, but not limited thereto, and any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.
In the five-axis numerical control machining technology, the side milling machining adopts the side surface of a cutter to remove materials. The tool bit rotating at high speed can be regarded as a rotating curved surface, so that the envelope curved surface generated by the motion of the rotating curved surface of the tool corresponding to the high-precision side milling machining is close to the machining curved surface at high precision. The tool to which the invention is applicable may be a milling tool or a grinding tool.
The shape and the processing track of the curved surface of the cutter are two closely related problems which need to be solved simultaneously. The existing tip shape methods do not directly result in a continuous tool curve. The invention provides a method for calculating a cutter curved surface and a processing track of a given design curved surface S, which specifically comprises the following steps as shown in FIG. 6:
1. solving the curved surface and the processing track of the cutter
And assuming that the design curved surface is S, and the envelope surface generated by the cutter curved surface phi along the processing motion track is E. The machining error is measured by the distance error between E and S. The parameters of both the tool curved surface phi (theta, s) and the machining trajectory R (t, s) together determine the quality of the machining result. The method of the invention simultaneously solves the parameters of the curved surface phi (theta, s) and the processing track R (t, s) of the cutter, and comprises the following steps:
step 1: interactive design curves A (t) and B (t) to generate an initial processing path;
the length L of the tool and the approximate width R of the tool are set. The step comprises the following substeps:
(1) the user interaction designates K +1 line segments, two end points of each line segment are located on the curved surface S, and the positions of the end points are designated by the user. Suppose that a sequence of line segments L is obtained0＝(A0，B0)，...，LK＝(AK，BK) Wherein A isi,BiK denotes two end points of a line segment, each having a length equal to L. As shown in fig. 1.
(2) End point A of the line segment obtained in (1)iOr BiRespectively move a distance R along the normal direction of the curved surfaceAAnd RBNamely, updating as:
Ai＝Ai+N(Ai)·RA；
Bi＝Bi+N(Bi)·RB；
wherein N (P) represents the normal direction of the curved surface at point P. As shown in fig. 2.
(3) Based on the endpoint Ai、BiDefining the track curves at two ends of the axis of the cutter as follows:
Figure BDA0002807460290000071
Figure BDA0002807460290000072
here, the definition of B-spline curves is used, where Ni,d(t) is the B-spline basis function of degree d. The line segment connecting the same parameter points on A (t) and B (t) represents the axis of the tool, and the points on the axis define an axial trace plane R (t, s) as shown in formula 1:
R(t,s)＝A(t)(1-s)+B(t)s t∈[0,1],s∈[0,L] (1)；
where L is the length of the tool axis. As shown in fig. 3.
Step 2: an initial tool shape is generated. The step comprises the following substeps:
(1) taking some sampling points R on the axial trace plane R (t, s)i,j：
Ri,j＝R(ti,sj)＝A(ti)(1-sj)+B(ti)sj；
Wherein s isj∈[0,1]J is 0.. multidot.w, which is the sampling value in the parameter domain s, and W +1 is the number of sampling points; t is ti∈[0,1]J is 0.. and Y is the number of sampling points in the parameter domain t, and Y +1 is the number of sampling points.
(2) Calculating R at each sampling pointi,jClosest distance d to design surface Si,jOn the surface ofDefining data points(s) in a rectangular plane coordinate systemj,di,j)；
(3) For data point(s)j,di,j) Fitting a function of the form as equation (2):
Figure BDA0002807460290000081
wherein f isi,i＝0,...,NfIs a 1-dimensional control point, and the number of the control points is Nf+1，Ni,dAnd (t) is a B-spline basis function of d times, and the node vector adopts control points of two-end heavy nodes (the repetition degree d +1) so as to enable the curve to interpolate from head to tail.
Suppose the axis of the tool surface is L(s), where s ∈ [0, L ] is the arc length parameterization, and L is the length of the tool surface axis. The function r(s) of the formula (2) represents the shortest distance from the point l(s) on the axis to the curved surface of the tool, and is called as a radius function, and then the radius function r(s) can define the contour line of the curved surface of the tool, as shown in fig. 4. r(s) defines a plane curve C, and the curve C rotates 360 degrees around the X axis to obtain a rotating curved surface, namely a cutter curved surface.
The representation method of the cutter curved surface comprises special cutter curved surfaces, such as a conical cutter head, a cylindrical cutter head, a drum-shaped cutter head and the like. For example, for a cylindrical tool tip, the function r(s) is a constant, i.e., r(s) c; for a conic bit, the function r(s) is a linear function r(s) as + b, where the shape parameters a, b determine the shape of the cone.
And step 3: the machining path and the tool shape are optimized. The step comprises the following substeps:
(1) sampling on the axial trace surface R (t, s) to obtain a plurality of data points R (t)i,sj)：
R(ti,sj),i＝0,...,N1,j＝0,...,N2；
Wherein, N1And N2Are integers representing the number of sampling points on the parameter t and the parameter s, respectively. Calculating the sampling point R (t)i,sj) Foot point S (u) on design surface Si,j,vi,j) Calculating the point S (u) on the design surfacei,j,vi,j) Unit normal vector N (u) of (c)i,j,vi,j)。
(2) The optimization problem defined by equation (4) is solved. The optimization variables are the control points A of the curves A (t) and B (t)i,BiI ═ 0. -, K and the control point f of the radius functioni,i＝0,...,Nf。
Figure BDA0002807460290000091
Wherein, (A' (t)i)-B′(ti))·(B(ti)-A(ti) 0, i 0, N is a rigid motion constraint, meaning that the tool axis remains fixed in length while in motion, N is an integer representing the number of samples on parameter t; fApproxIs a distance error function for realizing over-cut control, FEnergyIs a stability control function of the movement of the cutter head.
Let N and M denote the number of samples over the parameters t and s, respectively, FApproxThe definition is as follows:
Figure BDA0002807460290000092
wherein:
DApprox,ij＝λDistance1D1,ij+λDistance2D2,ij (5)；
D1,ij＝||A(ti)(1-sj)+B(ti)sj-(S(uij,vij)+N(uij,vij)·(h+r(si)))||2 (6)；
D2,ij＝((A(ti)(1-sj)+B(ti)sj-(S(uij,vij)+N(uij,vij)·(h+r(si))))·N(uij,vij))2 (7)。
equations (6) and (7) define the point-point distance function and the point-plane distance function, respectively, meaning that the design is usedThe equidistant curved surface of the curved surface S is used as a target curved surface of distance constraint, the conversion distance of the equidistant curved surface is h, namely the over-cutting can be controlled by adjusting the value of h, and a processing result without over-cutting can be obtained; lambda [ alpha ]Distance1And λDistance2Respectively, the weights of the point-point distance function and the point-plane distance function.
The movement of the cutter head is represented by movement curves A (t) and B (t) of two end points of the axis, and the stable movement of the cutter head can be expressed as the smoothness of the curves A (t) and B (t). The invention controls the stability of the movement and reduces the jitter through the energy functions of the curves A (t) and B (t). The smoothness of the curve is measured by the following energy function:
Figure BDA0002807460290000101
Figure BDA0002807460290000102
the jitter of the tool motion can be approximated by the following function:
Figure BDA0002807460290000103
a smooth, steady, and little jitter motion means the function E1,A、E1,B、E2,A、E2,B、E3,A、E3,BIs smaller, thus defining the total energy function:
FEnergy＝λEnergy1(E1,A+E1,B)+λEnergy2(E2,A+E2,B)+λEnergy3(E3,A+E3,B)；
wherein λ isEnergy1、λEnergy2、λEnergy3Is the weight of the energy function.
And 4, step 4: and (4) calculating an error.
Calculating R (t)i,sj) Sag on the design surface SFoot point S (u)i,j,vi,j). Calculating machining error
Figure BDA0002807460290000104
Where G is the number of sampling points. If the error meets the requirement, outputting a function A (t), B (t) for determining the machining path and a function r(s) for determining the shape of the tool, and turning to the step 5, otherwise, turning to the step 3.
And 5: and (5) calculating the curved surface of the cutter.
A rectangular coordinate system is established with the X axis of the coordinate system in fig. 4 as the z axis, an arbitrary vector perpendicular to z as the X axis, and z × X (X represents cross product of vectors) as the y axis. The parameterized representation of the tool surface in this coordinate system is defined by equation (8).
Figure BDA0002807460290000111
Wherein s and theta are parameters of the curved surface of the cutter, and L is the length of the axis of the cutter; r(s) is the tool shape definition function obtained in step 3. Fig. 5 shows the tool curvature at some locations along a machining path.
2. Results
FIGS. 7, 8 and 9 show tool shapes and corresponding side milling paths obtained by the method of the present invention, wherein the tool shape of FIG. 7 approximates a conical surface; in fig. 8, the machined curved surface is concave and the resulting tool curved surface is convex; the machined curved surface in fig. 9 is convex and the resulting tool curved surface is a concave curved surface.

Claims (3)

1. A cutter optimization and processing track generation method for complex curved surface side milling is characterized by comprising the following steps:
step one, interactive design curves A (t) and B (t) are generated, and an initial processing path is generated:
(1) the user interactively appoints K +1 line segments, two end points of each line segment are positioned on the curved surface S, and a line segment sequence L is supposed to be obtained0＝(A0，B0)，...，LK＝(AK，BK)，Wherein A isi,BiK represents two end points of a line segment, the length of each line segment is equal to L, and L is the length of the tool;
(2) the end point A of the line segment obtained in (1)iOr BiRespectively move a distance R along the normal direction of the curved surfaceAAnd RBNamely, updating as:
Ai＝Ai+N(Ai)·RA；
Bi＝Bi+N(Bi)·RB；
wherein N (P) represents the normal direction of the curved surface of the point P;
(3) based on updated endpoint Ai、BiDefining the track curves at two ends of the axis of the cutter as follows:
Figure FDA0003637218070000011
Figure FDA0003637218070000012
wherein, Ni,d(t) is the B-spline basis function of degree d;
the line segment connecting the same parameter points on A (t) and B (t) represents the axis of the tool, and the points on the axis define an axis locus R (t, s) as follows:
R(t,s)＝A(t)(1-s)+B(t)s t∈[0,1],s∈[0,L]；
step two, generating an initial cutter shape:
(1) taking a sampling point R on an axial trace plane R (t, s)i,j：
Ri,j＝R(ti,sj)＝A(ti)(1-sj)+B(ti)sj；
Wherein s isj∈[0,1]J is 0.. multidot.w, which is the sampling value in the parameter domain s, and W +1 is the number of sampling points; t is ti∈[0,1]J is 0,.. times, Y is a sampling value in the parameter domain t, and Y +1 is the number of sampling points;
(2) calculate each sample point Ri,jClosest distance d to design surface Si,jDefining data points(s) in a planar rectangular coordinate system X-Yj,di,j)；
(3) Data points(s)j,di,j) A function fit to the form:
Figure FDA0003637218070000021
wherein f isi,i＝0,...,NfIs a 1-dimensional control point, and the number of the control points is Nf+1；Ni,d(s) is a B-spline basis function of degree d;
assuming that the axis of the cutter curved surface is L(s), and r(s) defines a plane curve C, wherein the curve C rotates 360 degrees around the X axis to obtain a rotating curved surface as an initial cutter curved surface;
step three, optimizing a processing path and a cutter shape:
(1) sampling on the axial trace surface R (t, s) to obtain a data point R (t)i,sj)：
R(ti,sj),i＝0,...,N1,j＝0,...,N2；
Wherein N is1And N2Is an integer representing the number of sampling points on parameter t and parameter s, respectively;
calculating the sampling point R (t)i,sj) Foot point S (u) on design surface Si,j,vi,j) Calculating the point S (u) on the design surfacei,j,vi,j) Unit normal vector N (u) ofi,j,vi,j)；
(2) Solving an optimization problem defined by:
Figure FDA0003637218070000031
wherein, (A' (t)i)-B′(ti))·(B(ti)-A(ti) 0, i 0, N is a rigid motion constraint, meaning that the tool axis is in motionKeeping the length fixed; fApproxIs a distance error function taking into account over-cut control, FEnergyIs a stability control function of the movement of the cutter head;
step four, error calculation:
calculating R (t)i,sj) Foot point S (u) on design surface Si,j,vi,j) Calculating machining error
Figure FDA0003637218070000032
Where G is the number of sampling points; if the error meets the requirement, outputting a function A (t), a function B (t) and a function r(s) for determining the shape of the cutter, and turning to the fifth step, otherwise, turning to the third step;
step five, calculating the curved surface of the cutter;
taking the X axis of a coordinate system as a z axis, taking any vector perpendicular to the z as the X axis, and taking the zxx as a y axis to establish a rectangular coordinate system, and defining the parameterization of the tool curved surface under the coordinate system as follows:
Figure FDA0003637218070000033
where s and θ are parameters of the tool curved surface.
2. The method for tool optimization and machining trajectory generation for side milling machining of complex curved surfaces according to claim 1, wherein F is the distance between the tool and the machining trajectoryApproxThe definition is as follows:
Figure FDA0003637218070000034
wherein:
DApprox,ij＝λDistance1D1,ij+λDistance2D2,ij；
D1,ij＝||A(ti)(1-sj)+B(ti)sj-(S(ui,j,vi,j)+N(ui,j,vi,j)·(h+r(si)))||2；
D2,ij＝((A(ti)(1-sj)+B(ti)sj-(S(ui,j,vi,j)+N(ui,j,vi,j)·(h+r(si))))·N(ui,j,vi,j))2；
n and M represent the number of samples, λ, over the parameters t and s, respectivelyDistance1And λDistance2Respectively, the weights of the point-point distance function and the point-plane distance function.
3. The method for tool optimization and machining trajectory generation for side milling machining of complex curved surfaces according to claim 1, wherein F is the distance between the tool and the machining trajectoryEnergyThe definition is as follows:
FEnergy＝λEnergy1(E1,A+E1,B)+λEnergy2(E2,A+E2,B)+λEnergy3(E3,A+E3,B)；
wherein:
Figure FDA0003637218070000041
Figure FDA0003637218070000042
Figure FDA0003637218070000043
λEnergy1、λEnergy2、λEnergy3is the weight of the energy function.
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