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CN109669395B - Variable-radius circular interpolation method for axisymmetric aspheric surface - Google Patents

Variable-radius circular interpolation method for axisymmetric aspheric surface Download PDF

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CN109669395B
CN109669395B CN201811471921.0A CN201811471921A CN109669395B CN 109669395 B CN109669395 B CN 109669395B CN 201811471921 A CN201811471921 A CN 201811471921A CN 109669395 B CN109669395 B CN 109669395B
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interpolation
radius
aspheric surface
interval
axisymmetric aspheric
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CN109669395A (en
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王朋
张昊
杨坤
李伟皓
薛栋柏
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Tianjin Jinhang Institute of Technical Physics
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/41—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by interpolation, e.g. the computation of intermediate points between programmed end points to define the path to be followed and the rate of travel along that path
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/34—Director, elements to supervisory
    • G05B2219/34083—Interpolation general

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Abstract

The invention discloses a variable radius circular interpolation method of an axisymmetric aspheric surface, which comprises the following steps: a. determining an interpolated diameter D; b. determining an interpolation interval dx; c. calculating an interpolation radius ri of any interpolation interval; d. and d, calculating the interpolation radius value of each interpolation interval according to the established method in the steps a, b and c, and finishing the interpolation. On the basis of the existing interpolation mode, the invention improves the mode of adopting variable-radius circular interpolation, and effectively reduces the influence of interpolation errors on the surface shape precision of the axisymmetric aspheric surface; the method has simple calculation, can directly carry out parameterized optimization after programming through programming software, does not need to supplement hardware, can improve the processing precision, has high efficiency and low cost, and is a quick and effective axisymmetric aspheric surface programming interpolation method.

Description

Variable-radius circular interpolation method for axisymmetric aspheric surface
Technical Field
The invention belongs to the technical field of ultra-precision machining, and relates to a variable-radius circular interpolation method for an axisymmetric aspheric surface.
Background
With the continuous improvement of the requirements of optical systems, the application of the axisymmetric aspheric surface is more and more extensive, wherein ultra-precision milling and single-point diamond turning are one of two main processing methods for processing the axisymmetric aspheric surface. The common point of the two is that the cutter and the processed surface are in point-to-point contact mode in the processing process, and the processing precision is influenced by various factors such as cutter profile precision, machine tool precision, tool setting precision, clamping stress, numerical control interpolation precision and the like in the contact mode. Among the above influencing factors, numerical control interpolation accuracy is the only soft influencing factor, i.e., one of the ways of improving the machining accuracy by optimizing the interpolation program.
At present, the interpolation mode commonly used in the process of the axisymmetric aspheric surface interpolation is mainly a linear interpolation mode, and in commercial software, two interpolation point selection modes are provided, one is an equal linear quantity interval interpolation point selection method, and the other is an equal arc length interval selection method. However, in any selection method, in the linear interpolation method, a straight line connection method is adopted between two adjacent points, and a rise difference, that is, a surface shape accuracy error, is generated between the connection straight line and the aspheric surface curve between the two points in the optical axis direction. Therefore, it is one of the directions of research to change the interpolation method between two points after the interpolation point is determined to reduce the surface shape accuracy error.
Disclosure of Invention
Objects of the invention
The purpose of the invention is: the variable-radius circular interpolation method for the axisymmetric aspheric surface adopts a variable-radius circular interpolation mode, so that the adjustable-radius circular interpolation is adopted between two adjacent interpolation points, the vector height difference is reduced, and the fitting precision of the axisymmetric aspheric surface is improved.
(II) technical scheme
In order to solve the above technical problem, the present invention provides a variable radius circular interpolation method for an axisymmetric aspheric surface, which includes the following steps:
a. determining an interpolated diameter D;
b. determining an interpolation interval dx;
c. calculating an interpolation radius ri of any interpolation interval;
d. and d, calculating the interpolation radius value of each interpolation interval according to the established method in the steps a, b and c, and finishing the interpolation.
(III) advantageous effects
The variable-radius circular interpolation method for the axisymmetric aspheric surface provided by the technical scheme improves the mode of adopting variable-radius circular interpolation on the basis of the conventional interpolation mode, and effectively reduces the influence of interpolation errors on the surface shape precision of the axisymmetric aspheric surface; the method has simple calculation, can directly carry out parameterized optimization after programming through programming software, does not need to supplement hardware, can improve the processing precision, has high efficiency and low cost, and is a quick and effective axisymmetric aspheric surface programming interpolation method.
Drawings
Fig. 1 is a schematic diagram of an axisymmetric aspheric surface and a circular interpolation.
FIG. 2 is a schematic diagram of the process of the present invention.
Fig. 3 is a view of the arc interpolation according to the present invention.
Fig. 4 is a schematic view of an aspheric surface.
Detailed Description
In order to make the objects, contents, and advantages of the present invention clearer, the following detailed description of the embodiments of the present invention will be made in conjunction with the accompanying drawings and examples.
The invention relates to a variable radius circular interpolation method of an axisymmetric aspheric surface, which comprises the following steps:
a. determining an interpolated diameter D
And determining the interpolation diameter D according to the axial symmetry aspheric surface equation z ═ f (x), drawing requirement indexes, the processed material, the blank and other factors.
The interpolation diameter D should be larger than the design diameter D0 of the axisymmetric aspheric surface that needs to be processed actually.
Typically, D is 0.5mm to 2mm greater than D0.
b. The interpolation interval dx is determined.
And determining an interpolation interval dx according to parameters such as an aspheric equation z ═ f (x), drawing requirement indexes, an interpolation diameter D and the like.
Further, the interpolation interval dx is determined to be equal in the x-axis direction, i.e., dx is constant.
Further, D/(2dx) should be an integer, i.e., the patch point must pass through the aspheric vertex.
Further, dx is selected as: when interpolation is carried out according to the straight line, the maximum value of the difference value of the straight line and the axial symmetry aspheric surface along the Z direction does not exceed the PV value required in the technical index, and the PV value is the difference between the maximum value and the minimum value in a comparison curve of the actual surface and the theoretical surface.
c. An interpolation radius ri of an arbitrary interpolation interval is calculated.
And (3) regarding the aspheric surface between any intervals as an off-axis aspheric surface, and determining the best fitting spherical surface according to the principle of minimum height difference along the z-axis. The sphere at this time is a circle passing through the two interpolation points and having a center on the z-axis, and the radius of the circle is the interpolation radius riThe specific calculation method is as follows:
suppose that two interpolation points at an arbitrary interpolation interval are Pi(xi，zi) And Pi+1(xi+1，zi+1) The midpoint of the line between the two points is Pm(xm，zm) Then, then
xi+1-xi＝dx (2)
According to the attached figure 1, let
Figure BDA0001891169490000031
Figure BDA0001891169490000032
Then the process of the first step is carried out,
zm＝k(xm-xi)+zi (5)
suppose OiHas a coordinate value of (0, Z)oi) Then, then
Figure BDA0001891169490000033
From the trigonometric relationship, r can be obtainediIs expressed as
Figure BDA0001891169490000041
Further, the joint type (3) to (7) can obtain the value of the arc interpolation curvature radius required for the interval.
d. And d, calculating the interpolation radius value of each interpolation interval according to the established method in the steps a, b and c, and finishing the interpolation.
Examples
With reference to the schematic interpolation diagrams of aspheric surfaces shown in fig. 1 and fig. 2 and the flowchart of the present invention, the following steps are described for determining the specific implementation of the method of the present invention, taking the aspheric surface shown in fig. 4 as an example, with reference to other drawings:
a. an interpolated radius D is determined. In the aspheric drawing shown in fig. 4, D0 requires 11mm, parameters of the aspheric surface are given, and the interpolation radius D is 12mm according to the requirements of D0.
b. The interpolation interval dx is determined. According to the determination principle that passes through the origin of the aspheric surface and that D/(2dx) should be an integer, in this example, dx is determined as:
dx＝1mm
c. an interpolation radius of an arbitrary interpolation interval is calculated. Because of the axisymmetric aspheric surface, the aspheric surface is completely consistent in the areas of-6-0 and 0-6 in the x-axis direction. Therefore, only the directions of 0 to 6 are considered as xi2 and xi+1Two points corresponding to 3 are calculated as an interval.
X is to bei2 and xi+1Substituting the aspheric equations with the parameters shown in fig. 4 into 3, the coordinate between the two adjacent points can be obtained as Pi(2, 0.2338) and Pi+1(3，0.537)
The coordinates of the two points obtained are substituted for equations (3) to (7), and the radius of the interpolated arc at that time can be determined to be ri＝8.6306mm。
d. According to the above calculation method, when dx is 1, the corresponding ri value is:
r1 r2 r3 r4 r5 r6
8.6859mm 8.6689mm 8.6306mm 8.5596mm 8.4436mm 8.3mm
by simulation, it can be seen that the interpolation curve is shown in fig. 3.
Since the present invention is directed to an axisymmetric aspherical surface, only the x-axis positive portion may be considered in the calculation.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (1)

1. A variable radius circular interpolation method of an axisymmetric aspheric surface is characterized by comprising the following steps:
a. determining an interpolated diameter D;
b. determining an interpolation interval dx;
c. calculating an interpolation radius r of an arbitrary interpolation intervali；
d. According to the method established in the steps a, b and c, calculating the interpolation radius value of each interpolation interval to finish the interpolation;
in the step a, the interpolation diameter D is larger than the design diameter D0 of the axisymmetric aspheric surface which needs to be processed actually;
in the step a, D is 0.5mm-2mm larger than D0;
in the step b, the interpolation interval dx is determined as equal interval along the x axis, that is, dx is a constant;
in the step b, D/(2dx) is an integer;
in the step c, the aspheric surface between any intervals is regarded as an off-axis aspheric surface, and the best fitting spherical surface is determined according to the principle of minimum height difference along the z-axis; the sphere at this time is a circle passing through the two interpolation points and having a center on the z axis, and the radius of the circle is the interpolation radius ri;
the interpolated radius riThe calculation method comprises the following steps:
let P be the two interpolation points at any interpolation intervali(xi，zi) And Pi+1(xi+1，zi+1) The midpoint of the line between the two points is Pm(xm，zm) Then, then
xi+1-xi＝dx (2)
Order:
Figure FDA0003064469450000011
Figure FDA0003064469450000012
then:
zm＝k(xm-xi)+zi (5)
suppose OiHas a coordinate value of (0, Z)oi) And then:
Figure FDA0003064469450000021
from the trigonometric relationship, r is obtainediIs expressed as
Figure FDA0003064469450000022
Further, the joint type (3) to (7) can obtain the value of the arc interpolation curvature radius required for the interval.
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