CN114879602B - Design method for five-axis side milling machining single-cutter-position envelope characteristic line of rotary cutter - Google Patents
Design method for five-axis side milling machining single-cutter-position envelope characteristic line of rotary cutter Download PDFInfo
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Abstract
The invention belongs to the field of numerical control machining, and discloses a design method of a five-axis side milling machining single-cutter-position envelope characteristic line of a rotary cutter, which comprises the following steps: firstly, discrete sampling is carried out on a cutter shaft of a selected cutter position, and the distribution state of an ideal envelope characteristic line on a cutter curved surface is determined along a normal projection vector from a sampling point to a design curved surface; then, based on an envelope theory, taking the motion curve tangent vector of a fixed point on a cutter shaft as a variable, establishing an envelope characteristic point control equation, and approximating an ideal envelope characteristic line; on the basis, carrying out parametric expression on the current tool position, calculating the initial value of the tangent vector of the tool position, and obtaining an equivalent substitution model of the envelope characteristic line approximation problem; and finally, solving the model to obtain an envelope characteristic line of the current tool position, and calculating a machining error. The method can construct the single-cutter-position envelope characteristic line under the condition of no cutter path, solves the problem that cutter envelope information cannot be calculated in the cutter-position discrete generation process, and can be used for planning high-precision side milling cutter positions.
Description
Technical Field
The invention relates to a design method of a five-axis side milling machining single-cutter-position enveloping characteristic line of a rotary cutter, belonging to the technical field of numerical control machining.
Background
In five-axis side milling, the machining error is an important basis for planning the pose of the cutter and evaluating the machining quality. According to the numerical control machining forming theory, the motion envelope surface of the cutter can not be separated through accurate calculation of machining errors. Patent 1 "a five-axis machining cutter envelope surface calculation method based on motion vector: CN 112464399A [ 2 ], [ P ].2021.03.09 ] and patent 2 "a method and system for analyzing and calculating the rigid motion envelope surface of a rotary cutter: CN 112487657A (P), 2021.03.12 (CN 112487657A) all propose concise and efficient tool five-axis machining motion envelope modeling methods, but these methods can only determine the tool envelope under the condition that the tool track or the tool bit sequence is known, and in the process of tool bit point-by-point planning, because of lacking the tool motion information necessary for envelope modeling, the tool bit envelope characteristic line cannot be calculated, so that the machining error of a single tool bit is difficult to accurately obtain. In order to solve this problem, researchers often approximate the machining error by the geometric deviation between the tool and the curved surface, and use the machining error for positioning the tool (document 1, connecting-predicting tool path generation for minimizing machining errors in five-axes CNC planar milling of drilled surfaces [ J ]. Journal of Manufacturing Systems,2020,55 171-178 "), but from the perspective of improving machining accuracy, the error evaluation based on tool envelope information is undoubtedly more advantageous, and therefore, how to obtain the envelope feature line in the tool position planning becomes an urgent technical problem to be solved. Patent 3 "a method for calculating envelope characteristic lines of a rotary tool based on envelope theory and meridian division: CN 104866655 [ A ]. 2015.08.26' replaces the moving direction of the cutter with the feeding direction preset on the design curved surface, and then calculates the envelope characteristic line of the discrete cutter position by the envelope condition. However, due to the complexity of five-axis machining motion, the real motion direction of the tool deviates from the preset direction of the curved surface, and the influence of the deviation on the calculation of the tool envelope characteristic line is difficult to quantitatively estimate. In addition, the method provided by patent 1 is only suitable for envelope modeling of a local area of the tool, and it is difficult to determine the overall envelope characteristic line of the current tool position.
The method starts from an envelope theory, based on the relation between the envelope characteristic point and the cutter shaft motion tangent vector, and by optimizing the tangent vector of the cutter shaft fixed position motion curve, the approximation of the single cutter position envelope characteristic line to an ideal envelope characteristic line is realized, so that the difficult problem that the single cutter position envelope characteristic line cannot be calculated can be effectively solved, and the method has great significance for planning the high-precision side milling cutter position.
Disclosure of Invention
The invention aims to achieve the purpose that the enveloping characteristic points of the cutter approach to ideal scattered points by controlling the cutting vector at a fixed position on the cutter shaft by utilizing the relation between the enveloping characteristic points of the rotary cutter and the motion cutting vector of the cutter shaft, thereby realizing the single-cutter-position enveloping design.
In order to achieve the aim, the invention discloses a design method of a five-axis side milling machining single-cutter-position envelope characteristic line of a rotary cutter, which comprises the following steps:
(1) Discrete sampling is carried out on the cutter shaft of the selected cutter position, and the distribution state of an ideal envelope characteristic line on the curved surface of the cutter is determined along the normal projection vector from the sampling point to the designed curved surface;
(2) Based on an envelope theory, taking a motion curve tangent vector of a fixed point on a cutter shaft as a variable, establishing an envelope characteristic point control equation, and approximating an ideal envelope characteristic line;
(3) Carrying out parametric expression on the current tool position, calculating an initial value of a tangent vector of the current tool position, and obtaining an equivalent substitution model of an envelope characteristic line approximation problem;
(4) And (4) solving the model in the step (3) to obtain an envelope characteristic line of the current tool position, and calculating a machining error.
The step 1 comprises the following steps:
aiming at the selected cutter position, inputting the length of a cutter to be L, uniformly sampling the L along the cutter shaft direction, and calculating the normal projection from a sampling point to a designed curved surface S; setting the sampling point with the normal projection point as P i And the number is n, let P i The corresponding shadow point is M i (ii) a Along P i M i Making ray and intersecting with the curved surface of the tool to obtain an intersection point Q i Point set
The distribution state of the ideal envelope characteristic line of the current tool position is described.
The step 2 comprises the following steps:
get P B And P T Respectively the knife point of the current knife position and the point on the knife shaft which is L away from the knife point, and setting the corresponding motion track tangent vectors as X and Y, respectively, then the point on the knife shaft and the point P are B The tangent vector T corresponding to a point with a distance h is:
according to the envelope theory, if one point E on the curved surface of the cutter is an envelope characteristic point, the E satisfies the following condition:
D t ·(D-E)＝0
wherein D is a point on the cutter shaft, D t Is the tangent vector corresponding to the point; meanwhile, the following relationship exists between E and D: and D is the intersection point of the normal vector of the position of the E and the cutter shaft.
On the basis of this, use D i Represents Q i The intersection point of the normal vector and the cutter shaft is recorded as D i And Q i The difference is: n is a radical of i ＝D i -Q i (ii) a When is Q i For an ideal envelope point, a homogeneous linear system of equations with X, Y as the unknown variable can be obtained as:
wherein h is i Is D i From the nose point P B The distance of (c).
The step 3 comprises the following steps:
analyzing the space distribution state of the existing knife location point, and calculating the corresponding parameter expression according to the position of the current knife location as follows:
wherein j represents the ordinal number of the current tool position in the existing tool position sequence, P B,k And P T,k The cutter point respectively represents the kth cutter position and a point on the cutter shaft with the distance L from the cutter point. When j is taken as a variable, P B,j 、P T,j And P B,k 、P T,k The meaning is the same.
Reconstruction of passing points P separately using local splines B,j-3 ～P B,j And P T,j-3 ～P T,j The obtained curve control point matrix is:
DB j ＝Ω[P B，j-3 ，P B，j-2， P B，j-1 ，P B，j ] T
DT j ＝Ω[P T，j-3 ，P T，j-2， P T，j-1 ，P T，j ] T
where Ω is a coefficient matrix whose constituent elements u j-3 To u j Parameter values corresponding to the j-3 th to j-th tool positions:
at this time, the initial value of the tangent vector corresponding to the jth cutter position is:
non-zero X and Y pairs of ideal envelope points Q i Approximation deviation of (e) i Can be approximated by:
let ε = max { ε i I is more than or equal to 1 and less than or equal to n, and the solution problem of the homogeneous linear equation set can be equivalently converted into the following optimization model:
min.d T var
s.t.Avar≤b
[X ini ，Y ini ，-1]-Δ≤var≤[X ini ，Y ini ，1]+Δ
where var represents a vector of X, Y and ε: var = [ X, Y, ε ]]；d＝[0，0，0，0，0，0，1] T (ii) a Delta is a 7-membered row vector, the constituent elements of which are all [0.05,0.1 ]]A constant within a range; b is a column vector containing 2n zeros; a is a 2n × 7 type coefficient matrix whose row vectors
Is expressed in the form of:
the step 4 comprises the following steps:
solving the tangent vector optimization model by using an interior point method to obtain the optimal solution [ X ] of the tangent vector of the current tool position opt ,Y opt ]And calculating an envelope characteristic line on the rotary cutter according to the envelope characteristic line. By discretizing the characteristic line into m characteristic points E i I is not less than 1 and not more than m, and E is calculated i And obtaining the processing error distribution of the current cutter position by the distance from the designed curved surface S.
Through the technical scheme of the invention, the following beneficial effects can be achieved:
(1) The calculation of the characteristic points strictly follows the envelope condition, and the result is accurate and reliable; meanwhile, the scheme is suitable for the overall envelope design of various side milling cutters such as cylindrical cutters, conical cutters, drum cutters and the like.
(2) In the process of selecting cutter shaft vector optimization, the spatial distribution state of the existing cutter positions is considered, so that the optimization result of the vector can not only meet the calculation requirement of the current cutter position planning on the machining error, but also lay a foundation for the subsequent generation of cutter paths integrating the vector information.
Drawings
FIG. 1 is a flow chart of a design method of a five-axis side milling machining single-cutter-location envelope characteristic line of a rotary cutter.
Fig. 2 is a schematic diagram of calculation of ideal envelope feature points.
Fig. 3 is a diagram showing a relationship between the envelope characteristic point and a tangent at a selected position of the knife shaft.
Fig. 4 is a schematic diagram of calculation of a single-tool machining error.
Detailed Description
Reference will now be made in detail to specific embodiments of the present invention, examples of which are illustrated in the accompanying drawings, which will assist those skilled in the art in further understanding the invention, and are not intended to limit the invention in any way. It should be noted that, for those skilled in the art, modifications or equivalent substitutions can be made without departing from the spirit of the present invention, and these are within the scope of the present invention.
As shown in the flowchart of fig. 1, the method for designing the five-axis side milling machining single-tool-location envelope characteristic line of the rotary tool according to the present invention comprises the following steps:
(1) Discrete sampling is carried out on the cutter shaft of the selected cutter position, and the distribution state of an ideal envelope characteristic line on the curved surface of the cutter is determined along the normal projection vector from the sampling point to the designed curved surface;
(2) Based on an envelope theory, taking a motion curve tangent vector of a fixed point on a cutter shaft as a variable, establishing an envelope characteristic point control equation, and approximating an ideal envelope characteristic line;
(3) Carrying out parametric expression on the current tool position, calculating an initial value of a tangent vector of the current tool position, and obtaining an equivalent substitution model of the envelope characteristic line approximation problem;
(4) And (4) solving the model in the step (3) to obtain an envelope characteristic line of the current tool position, and calculating a machining error.
As shown in FIG. 2, for a taper milling cutter as an example, the selected cutting position is P B And P T Is represented by the formula, wherein P B And P T Respectively a tool point of the current tool position and a point on the tool shaft which is L away from the tool point. At P B And P T And uniformly sampling, and calculating the normal projection from the sampling point to the design curved surface S. Let P be the sampling point of the normal projection i Projection result is M i ，P i (M i ) N, for these points, along P i M i Making ray and intersecting with the curved surface of the tool to obtain an intersection point Q i 。
Let Q i The section intersection circle of the plane perpendicular to the cutter shaft and the curved surface of the rotary cutter is
To Q i In other words, Q i The normal projection distance to the designed curved surface S corresponds to->
The distance from S is extreme and can be inferred based on the geometrical characteristics when Q is used i As->
At the envelope point of (4), is greater than or equal to>
The machining error caused is minimal. Therefore, the point column distributed along the cutter shaft direction->
Is considered as an ideal envelope feature point. Starting from the envelope theory, if a point E on the curved surface of the cutter is an envelope characteristic point, the E satisfies the following condition:
D t .(D-E)＝0 (1)
wherein D is a point on the cutter shaft, D t Is the pointA corresponding tangent vector; meanwhile, the following relationship exists between E and D: and D is the intersection point of the normal vector of the curved surface position of the cutter where E is located and the cutter shaft.
Let P B And P T The corresponding motion track tangent vectors are respectively X and Y, D and the tool nose point P B Is h, then D t The following relationship is satisfied:
on the basis of this, use D i Represents Q i The intersection point of the normal vector and the cutter shaft is recorded as D i And Q i The difference is: n is a radical of i ＝D i -Q i (ii) a When Q is used, as shown in FIG. 3 i For an ideal envelope point, a homogeneous system of linear equations with X and Y as unknown variables can be obtained as:
wherein h is i Is D i From the nose point P B Of the distance of (c).
The obtained X and Y are solved to satisfy the envelope condition described by the formula (3) as much as possible, and the influence of adjacent cutter positions is also considered. For this reason, X and Y cannot be obtained by directly solving equation (3), and to satisfy the solving requirements of X and Y, the following measures can be taken: analyzing the space distribution state of the existing knife location point, and calculating the corresponding parameter expression according to the position of the current knife location:
wherein j represents the ordinal number of the current tool position in the existing tool position sequence, P B，k And P T，k Respectively representing a cutter point of the kth cutter position and a point on a cutter shaft with the distance L from the cutter point. Reconstruction of passing points P separately using local splines B，j-3 ～P B，j And P T，j-3 ～P T，j III of (2)And (3) obtaining a curve control point matrix as follows:
DB j ＝Ω[P B，j-3 ，P B，j-2， P B，j-1 ，P B，j ] T (5)
DT j ＝Ω[P T，j-3 ，P T，j-2， P T，j-1 ，P T，j ] T (6)
wherein j is a variable, P B，j 、P T，j And P B，k 、P T，k The same meaning, omega is a coefficient matrix, the constituent elements u of which j-3 To u j Parameter values corresponding to the j-3 th to j-th tool positions:
at this time, the initial value of the tangent vector corresponding to the jth cutter position is:
non-zero X and Y pairs of ideal envelope points Q i Approximation deviation of (c) i Can be approximated by:
let ε = max { ε i I is more than or equal to 1 and less than or equal to n, and the solution problem of the homogeneous linear equation set can be equivalently converted into the following optimization model:
min.d T var
s.t.Avar≤b (10)
[X ini ，Y ini ，-1]-Δ≤var≤[X ini ，Y ini ，1]+Δ
where var represents the vector of X, Y and ε: var = [ X, Y, ε ]]；d＝[0，0，0，0，0，0，1] T (ii) a Delta is a 7-membered row vector, the constituent elements of which are all [0.05,0.1 ]]A constant within a range; b is a column vector containing 2n zeros(ii) a A is a 2n x 7 type coefficient matrix, the row vector of which
Is expressed in the form of:
the formula (10) can be solved by an interior point method, and the solved result is recorded as [ X ] opt ，Y opt ]. It should be noted that X, Y is limited to X during the solution process ini And Y ini In a small neighborhood of (B), thus X opt 、Y opt And X ini 、Y ini Are very similar. Considering the parameterization method of equation (4), the parameter value of the current tool position is not a standard expression, and the standard value is
Wherein j is 0 Is the ordinal number of the last tool position in the tool position sequence; at the same time X opt 、Y opt Should also be multiplied by pick>
So as to eliminate the influence caused by parameter normalization. The above operation can be convenient for X opt And Y opt Integrated in subsequent tool path generation, but for the current tool position, X opt And Y opt The change of the amplitude in equal proportion does not affect the envelope modeling of the current cutter position, so the envelope modeling can be carried out according to X opt And Y opt And calculating an envelope characteristic line on the rotary cutter. As shown in fig. 4, by discretizing the characteristic line into m characteristic points E i I is not less than 1 and not more than m, and E is calculated i And obtaining the processing error distribution of the current cutter position by the distance from the designed curved surface S.
Although the executing process is specific to the conical cutter, the executing process can be easily popularized to other types of side milling cutters such as cylindrical cutters and drum-shaped cutters; meanwhile, the provided scheme is easy to realize in programming, the calculation of the single-tool-bit envelope characteristic line is efficient, the result is reliable, the calculation requirement of the current tool bit planning on the machining error can be met, and a foundation is laid for the subsequent generation of the tool path integrating the vector cutting information.
Claims (1)
1. A design method for a five-axis side milling machining single-tool-position enveloping characteristic line of a rotary cutter is characterized by comprising the following steps:
(1) Discrete sampling is carried out on the cutter shaft of the selected cutter position, and the distribution state of an ideal envelope characteristic line on the curved surface of the cutter is determined along the normal projection vector from the sampling point to the designed curved surface;
aiming at the selected cutter position, inputting the length of a cutter to be L, uniformly sampling the L along the cutter shaft direction, and calculating the normal projection from a sampling point to a designed curved surface S; setting the sampling point with the normal projection point as P i And the number is n, let P i The corresponding shadow point is M i (ii) a Along P i M i Making ray and intersecting with the curved surface of the tool to obtain an intersection point Q i Point set
Representing the distribution state of the ideal envelope characteristic line of the current cutter position;
(2) Based on an envelope theory, taking a motion curve tangent vector of a fixed point on a cutter shaft as a variable, establishing an envelope characteristic point control equation, and approximating an ideal envelope characteristic line;
get P B And P T Respectively the knife point of the current knife position and the point on the knife shaft which is L away from the knife point, and setting the corresponding motion track tangent vectors as X and Y, respectively, then the point on the knife shaft and the point P are B The tangent vector T corresponding to a point with the distance h is as follows:
according to the envelope theory, if one point E on the curved surface of the cutter is an envelope characteristic point, the E satisfies the following condition:
D t ·(D-E)＝0
wherein D is a point on the cutter shaft, D t Is the tangent vector corresponding to the point; meanwhile, the following relationship exists between E and D: the intersection point of the normal vector of the position of the E and the cutter shaft is D;
on the basis of this, D is used i Represents Q i The intersection point of the normal vector and the cutter shaft is recorded as D i And Q i The difference is: n is a radical of i ＝D i -Q i (ii) a When is Q i For an ideal envelope point, a homogeneous system of linear equations with X and Y as unknown variables can be obtained as:
wherein h is i Is D i From the nose point P B The distance of (d);
(3) Carrying out parametric expression on the current tool position, calculating an initial value of a tangent vector of the current tool position, and obtaining an equivalent substitution model of the envelope characteristic line approximation problem;
analyzing the space distribution state of the existing cutter location point, and calculating the corresponding parameter expression according to the current cutter location position as follows:
wherein j represents the ordinal number of the current tool position in the existing tool position sequence, P B，k And P T，k A cutter point representing the kth cutter position and a point on a cutter shaft with the distance L from the cutter point are respectively; when j is taken as a variable, P B，j 、P T，j And P B，k 、P T，k The meanings are the same;
reconstruction of passing points P separately using local splines B，j-3 ～P B，j And P T，j-3 ～P T，j The obtained curve control point matrix is:
D B，j ＝Ω[P B，j-3 ，P B，j-2 ，P B，j-1 ，P B，j ] T
D T，j ＝Ω[P T，j-3 ，P T，j-2 ，P T，j-1 ，P T，j ] T
wherein Ω is a coefficient matrix, and its constituent elements u j-3 To u j Parameter values corresponding to the j-3 th to j-th tool positions:
at this time, the initial value of the tangent vector corresponding to the jth cutter position is:
non-zero X and Y pairs of ideal envelope points Q i Approximation deviation of (e) i Approximated by the following formula:
let ε = max { ε i I is more than or equal to 1 and less than or equal to n, and the solution problem of the homogeneous linear equation set can be equivalently converted into the following optimization model:
min.d T var
s.t.Avar≤b
[X ini ，Y ini ，-1]-Δ≤var≤[X ini ，Y ini ，1]+Δ
where var represents the vector of X, Y and ε: var = [ X, Y, ε ]]；d＝[0，0，0，0，0，0，1] T (ii) a Delta is a 7-membered row vector, the constituent elements of which are all [0.05,0.1 ]]A constant within a range; b is a column vector containing 2n zeros; a is a 2n x 7 type coefficient matrix, the row vector of which
Is expressed in the form of:
(4) Solving the model in the step (3) to obtain an envelope characteristic line of the current tool position, and calculating a machining error;
using interior point methodSolving the tangent vector optimization model to obtain the optimal solution [ X ] of the tangent vector of the current tool position opt ,Y opt ]Calculating an envelope characteristic line on the rotary cutter according to the envelope characteristic line; by discretizing the characteristic line into m characteristic points E i I is not less than 1 and not more than m, and E is calculated i And obtaining the machining error of the current cutter position by the distance from the designed curved surface S.
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