JP3628672B2 - Curve interpolation by arc - Google Patents

Curve interpolation by arc Download PDF

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JP3628672B2
JP3628672B2 JP2002170070A JP2002170070A JP3628672B2 JP 3628672 B2 JP3628672 B2 JP 3628672B2 JP 2002170070 A JP2002170070 A JP 2002170070A JP 2002170070 A JP2002170070 A JP 2002170070A JP 3628672 B2 JP3628672 B2 JP 3628672B2
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Prior art keywords
curve
arc
interpolation
approximate
tolerance
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JP2004013797A (en
Inventor
栄一 八木
茂雄 安達
雅幸 掃部
昇 目黒
周平 高原
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川崎重工業株式会社
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Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a curve interpolation method capable of realizing efficient curve interpolation with less error in a control device for machine tools, robots, and the like, and also in a design support device for designing machining dimensions and shapes. .
[0002]
[Prior art]
The curve of the tooth surface of the gear is represented by a complicated mathematical expression consisting of a trigonometric function, as represented by an involute curve, a cycloid curve, a trochoid curve, a hypoid curve, and the like. Also, mathematical formulas are used to express the shape of aircraft blades, turbine fins, and products that emphasize design mystery and aesthetics.
Currently, there are few machine tools and robots that are widely used in the market that have the ability to input arbitrary mathematical formulas directly into the control unit. It is necessary to interpolate the curve represented by the equation by some method and to input the interpolation point and interpolation parameter data to the control device. In general, the most frequently used method and the method of generating the interpolation point relatively easily is interpolation by line segment. At present, it can be said that no matter how inexpensive a machine tool or robot is, it has the function of moving the tool along a specified line segment between two points. However, for example, gear machining generally requires high-precision machining, and therefore interpolation by line segments requires many interpolation points, resulting in high design costs and machining costs.
[0003]
As means for solving this problem, Japanese Patent Application Laid-Open No. 2000-3212 discloses a method of generating an approximate line segment that makes full use of the tolerance of tolerance, but means for greatly reducing the number of interpolation points. I have n’t gone. In addition, in a mechanism that transmits an angle and torque, such as a gear, not only the tolerance between the theoretical value and the machining value should be as small as possible, but also the approximate ripples for interpolation must be smoothly connected to each other. Cause deterioration of the performance of the machine itself that uses the gear as a component.
[0004]
As a method for compensating for the drawback of such line segment interpolation, for example, an interpolation method using a spline curve as described in JP-A-10-240329, and further described in JP-A-10-228306. There are many interpolation methods using such NURBS curves, interpolation methods using Bezier curves as described in JP-A-6-162184, and interpolation methods using clothoid curves as described in JP-A-6-259567. Proposed. These interpolation methods are currently becoming common in the field of machining, and there are many machine tools having these interpolation functions. However, they are a relatively expensive class of existing machine tools and are not effective for any machine tool. Interpolation using these curves is unfamiliar to non-designers who specialize in it, and works smoothly at the stage of trial and error during design and at the stage of inspection by a third party after design completion. This leads to delays in the design process and inspection errors.
[0005]
[Problems to be solved by the invention]
Now, when a general machine designer designs a curve such as a tooth profile, interpolation is performed using both arcs and line segments. The arc is more familiar to general designers than the previous spline curve and the like, and the design work can be carried out smoothly for the designer himself / herself and for the third party who performs the inspection. The interpolation features are approximated by arcs in areas where the unevenness of the curve is deep, and approximated by line segments in areas where the unevenness is shallow, or in the inflection points that change from concave to convex, or from convex to concave. Interpolation points that smoothly connect the approximate lines are generated. However, there is no regularity in the generation method of these approximate lines, and in general, an approximate line is generated based on the experience of the designer or trial and error, and then it is verified whether the approximate line is within the tolerance range. If it is not within the range, a backward design is performed such as correcting the interpolation points of the approximate line or increasing the number of interpolation points. In such a method, not only tolerance tolerance is not efficiently demonstrated, but it is not easy to generate an interpolation point that smoothly connects a group of arcs and a line segment. Is required.
[0006]
For example, when the tooth profile curve of a gear is interpolated, interpolation is required with only a convex arc in the convex area of the theoretical curve and only with a concave arc in the concave area, and the tolerance of the approximate line is also on the + side or − In many cases, specifications are required to be held only on one of the specified sides. These requirements that the designers have sensibly cannot be satisfied even if the interpolation method using the spline curve or clothoid curve is applied as it is, and it is difficult to construct an effective interpolation method for the tooth profile curve. Met.
[0007]
The present invention has been made in view of the above points, and the object of the present invention is to make the maximum use of the tolerance range of the tolerance, to make the unevenness of the arc constant by the unevenness of the theoretical curve, and to determine the sign of the tolerance of the approximate line. Another object of the present invention is to provide a curve interpolation method that can provide efficient interpolation with fewer interpolation points than a conventional interpolation method by providing a constant curve interpolation method on the other side.
[0008]
[Means for Solving the Problems]
For example, the following is a summary of the specifications that the designer has sensuously in interpolating the tooth profile curve of the gear.
(1) The error between the tooth profile curve and the approximate curve is within the tolerance value.
(2) In principle, the approximate curve has an error on the rotation center side of the gear compared to the tooth profile curve.
(3) The concave area of the tooth profile curve is approximated by a concave arc or straight line, and the convex area is approximated by a convex arc or straight line.
(4) Adjacent arcs or arcs and line segments are smoothly connected.
[0009]
Here, the tooth profile curve to be designed is referred to as a reference curve, and the rotation center of the gear among the two envelopment curves generated when the center of the circle whose radius is allowable tolerance follows on this reference curve. The curve on the side is called a tolerance curve. At this time, from the concave point of the reference curve to the inflection point, the inflection point of the reference curve passes through the inflection point and the concave arc group having the same inclination at the connection point between the reference curve and the tolerance curve. Approximate with a line segment equal to the slope of the curve, and the slope from the convex point of the reference curve to the inflection point passes through the inflection point and the convex arc group between the reference curve and the tolerance curve, with the same slope at the connecting point. Is approximated by a line segment equal to the slope of the inflection point of the reference curve.
If the tooth profile curve is approximated by a group of arcs and a line segment between the reference curve and the tolerance curve, the specifications (1) and (2) can be satisfied. Further, if the arc group and the line segment are approximated so as to have the same inclination at the connection point, the specification (4) can be satisfied. Furthermore, approximate from the concave point of the reference curve to the inflection point with a line segment that passes through the concave arc group and the inflection point and is equal to the inclination of the inflection point of the reference curve. The specification {circle over (3)} can be satisfied by approximating with a line segment that passes through the convex arc group and the inflection point and whose inclination is equal to the inclination of the inflection point of the reference curve.
[0010]
That is, the method of the present invention is a curve interpolation method for interpolating a given curve with an approximate arc group within a tolerance range of tolerance, and is allowed on either the inside or outside of the given curve. Find the approximate arcs and approximate line segments that fall between the curves generated to have the maximum tolerance, and approximate the curve with arc groups and line segments that are within the tolerance range. Approximate arcs or / and line segments that are connected smoothly by approximating the minutes to have the same slope at the connection point, and approximating with concave arcs or line segments in the concave area of the curve and convex in the convex area It is characterized by being approximated by an arc or a line segment.
In the above method, for example, the curve data used as the approximate arc is a part of the divided series of curve data.
The apparatus of the present invention is configured to have a function of performing the above-described curve interpolation method in a numerical control device, a design support device (CAD or the like), an analysis support device, or the like.
[0011]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, embodiments of the present invention will be described, but the present invention is not limited to the following embodiments, and can be implemented with appropriate modifications. The tooth profile reference curve is expressed as z r (q) = (x r (q), y r (q)), and the tolerance curve is expressed as z t (q) = (x t (q), y t (q)). . Here, q is a parameter corresponding to the curve displacement. In addition, the slopes of the curves at the point q are k r (q) and k t (q), respectively. Let q c be the inflection point.
[0012]
The concave section approximation method z r (q), z t (q) is defined as {q | [q s , q f ]}, and the approximation of the section {q | [q s , q f ]} is as follows. . However, it is assumed that the absolute values of curvature of z r (q s ) and z t (q s ) are smaller than the absolute values of curvature of z r (q f ) and z t (q f ).
Step 0: Let q s be the interpolation midpoint q m .
Step 1: As the z r (q m), the slope of that point is k r (q m) equal arcs A1, and the line segment Lf slope k r (q f) through z r (q f), If the reference curve and the tolerance curve are touched ((a) in FIG. 1), the section {q | [q s , q f ]} is interpolated with one arc A1 and one line segment Lf. If not (step (b) in FIG. 1), the process proceeds to step 2.
Step 2: If an arc A1 exists between the reference curve and the tolerance curve, and an arc A2 that smoothly connects to A1 at the end of A1 touches between the line segment Lf, the reference curve, and the tolerance curve (FIG. 1). (C)) The section {q | [q s , q f ]} is interpolated with two arcs A1 and A2 and one line segment Lf. If not ((d) of FIG. 1), it will progress to step 3. Step 3: The arcs A1 and A2 exist between the reference curve and the tolerance curve, and a contact point of the reference curve with which A2 smoothly contacts is set as the next interpolation point q m ′ ((e) in FIG. 1).
Step 4: Return to Step 1 with q m = q m ′.
[0013]
The convex section approximation method z r (q), z t (q) is {q | [q s , q f ]}, and the approximation of the section {q | [q s , q f ]} is as follows. . However, it is assumed that the absolute values of curvature of z r (q s ) and z t (q s ) are smaller than the absolute values of curvature of z r (q f ) and z t (q f ).
If z r (q) and z t (q) are interchanged, the convex section can be regarded as a concave section. Hereinafter, the procedure of the interpolation method of the concave section is followed.
[0014]
Interpolation Method of Section Moving from Concave to Convex z r (q), z t (q) is a section moving from convex to concave {q | [q ccv , q cvx ]}, and section {q | [q ccv , Q cvx ]} approximation follows. However, q ccv is a point in the concave section, and q cvx is a point in the convex section.
A point where the line segment of the slope k r (q c ) passes through z r (q c ) and intersects with z t (q) is defined as p 1 , the path of z t (q c ) passes through, and the slope k t (q c ) A point where the line segment intersects z r (q) is defined as p 2, and a line segment connecting p 1 and p 2 is defined as Lf (FIG. 2).
Since the section {q | [q ccv , q c ]} is a concave section, the approximation of this section follows the approximation method of the concave section using the above Lf.
Since the section {q | [q cvx , q c ]} is a convex section, the approximation of this section follows the convex section approximation method using the above Lf.
[0015]
【Example】
Circular interpolation of the concave section {q | [0, q c ]} (concave point to inflection point) of the trochoid center curve shown in the following Equation 1 was performed.
[0016]
[Expression 1]
[0017]
The interpolation points and approximate lines when the allowable tolerances are gradually tightened to 0.02, 0.015, 0.01, and 0.0022 are shown in FIGS. 3, 4, 5, and 6, respectively. As shown in FIG. 3, when the allowable tolerance is 0.02 mm, interpolation is performed with one arc and one line segment. As shown in FIG. 4, when the allowable tolerance is 0.015 mm, two arcs are used. As shown in FIG. 5, when the tolerance is 0.01 mm, the interpolation is performed with three arcs and one line segment, as shown in FIG. When the allowable tolerance is 0.0022 mm, interpolation is performed with four arcs and one line segment. Thus, as the tolerance becomes tighter, the number of approximate arcs increases. In either case, it can be confirmed that the adjacent arcs are smoothly connected, are within the allowable tolerance range, and are smoothly connected to the reference curve.
[0018]
Next, circular interpolation was performed in a section {q | [0, 30π / 29]} (concave point to convex point) of the trochoid parallel curve expressed by the following formula 2 from the concave to the convex. FIG. 7 shows the contact points and approximate lines when the allowable tolerance is 0.5.
[0019]
[Expression 2]
[0020]
【The invention's effect】
Since this invention is comprised as mentioned above, there exist the following effects.
(1) According to the present invention, when interpolating a given curve, an approximate arc that is smoothly connected so as to make maximum use of the allowable tolerance is generated, so that the condition of the allowable tolerance is satisfied and the interpolation is performed. The number of points can be reduced, and efficient interpolation can be realized.
(2) Since the tolerance can be maximized and efficient interpolation can be realized with a small number of interpolation points, the accuracy and processing speed of the numerical control device, the design support device, and the analysis support device can be kept high at the same time.
[Brief description of the drawings]
FIG. 1 is a diagram for explaining a curve interpolation method (an approximation method of a concave section) according to an embodiment of the present invention.
FIG. 2 is a diagram for explaining a curve interpolation method (interpolation method of a section moving from concave to convex) according to the embodiment of the present invention.
FIG. 3 is a diagram showing a result of an example of the present invention (circular interpolation of a concave section of a trochoid center curve, allowable tolerance = 0.020 [mm]);
FIG. 4 is a diagram showing a result of an example of the present invention (circular interpolation of a concave section of a trochoid center curve, allowable tolerance = 0.015 [mm]).
FIG. 5 is a diagram showing a result of an example of the present invention (circular interpolation of a concave section of a trochoid center curve, allowable tolerance = 0.010 [mm]).
FIG. 6 is a diagram showing a result of an example of the present invention (circular interpolation of a concave section of a trochoid center curve, allowable tolerance = 0.0002 [mm]).
FIG. 7 is a diagram showing the results of an example of the present invention (circular interpolation in a section where a trochoidal parallel curve moves from concave to convex, allowable tolerance = 0.5 [mm]).

Claims (2)

  1. A curve interpolation method for interpolating a given curve with an approximate arc group within a tolerance range, wherein the given curve has a maximum allowable tolerance on either the inside or outside of the curve. Approximate arcs and approximate line segments that fall between the generated curves are obtained, and the curve is approximated with arc groups and line segments that are within the tolerance range, and these arc groups and line segments have the same slope at the connection point. To generate an approximate arc or line segment that approximates and smoothly connects, and approximates with a concave arc or line segment in a concave area of a curve, and approximates with a convex arc or line segment in a convex area Curve interpolation method using characteristic arcs.
  2. The curve interpolation method according to claim 1, wherein the curve data used as the approximate arc is a part of a series of curve data .
JP2002170070A 2002-06-11 2002-06-11 Curve interpolation by arc Expired - Fee Related JP3628672B2 (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
EP1562138B1 (en) 2004-02-06 2009-08-19 Dassault Systèmes A process for drafting a curve in a computer-aided design system
SG160423A1 (en) * 2005-03-23 2010-04-29 Hurco Co Inc Method of tolerance-based trajectory planning and control
WO2006109838A1 (en) 2005-04-08 2006-10-19 Tsutomu Miyaoku Gear with cornu's spiral tooth profile
KR100745983B1 (en) 2006-07-18 2007-08-06 삼성전자주식회사 Apparatus and method for restoring using minimum value sampling
WO2008149760A1 (en) * 2007-06-01 2008-12-11 Daikin Industries, Ltd. Swing type processing device
JP4969484B2 (en) * 2008-02-25 2012-07-04 三菱重工業株式会社 Numerical controller
JP4804501B2 (en) * 2008-04-16 2011-11-02 関東自動車工業株式会社 Wire harness wiring shape display device
CN104864829B (en) * 2015-06-14 2017-06-27 吉林大学 A kind of method for fast measuring of spoon of blade

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