CN108073138B - Elliptical arc smooth compression interpolation algorithm suitable for high-speed high-precision machining - Google Patents
Elliptical arc smooth compression interpolation algorithm suitable for high-speed high-precision machining Download PDFInfo
- Publication number
- CN108073138B CN108073138B CN201610978793.3A CN201610978793A CN108073138B CN 108073138 B CN108073138 B CN 108073138B CN 201610978793 A CN201610978793 A CN 201610978793A CN 108073138 B CN108073138 B CN 108073138B
- Authority
- CN
- China
- Prior art keywords
- point
- curve
- arc
- interpolation
- elliptical arc
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Classifications
-
- 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
- G05B19/4103—Digital interpolation
Abstract
The invention relates to an elliptic arc smooth compression interpolation algorithm suitable for high-speed high-precision machining, which comprises the following steps: identifying a continuous tiny line segment processing area; selecting model value points in the continuous micro line segment processing area and fitting the model value points to obtain a secondary rational Bezier curve; identifying an elliptical arc according to the curve characteristics, and carrying out geometric form conversion on the elliptical arc; merging adjacent elliptic arcs belonging to the same ellipse to obtain an interpolation curve; and performing elliptic arc interpolation on the interpolation curve. The method identifies the elliptical arc in the processing path by using the characteristics of the spline curve, performs interpolation by using the geometric form of the elliptical arc, can accurately calculate the interpolation parameter corresponding to the arc length, reduces the calculated amount and frequent fluctuation of the speed in the processing, and realizes high-speed and high-precision processing.
Description
Technical Field
The invention relates to fitting of a parameter spline curve and identification and combination of elliptical arc sections in high-quality processing, and belongs to the technical field of numerical control processing.
Background
An ellipse is one of the conic sections, which is common in workpiece machining procedures. Since the calculation of the arc length of an ellipse involves the problem of ellipse integration, and the primitive function cannot be represented in a finite form, the ellipse cannot be directly and accurately interpolated. And because of its strong engineering practicability, the study of scholars at home and abroad has been extensively conducted.
For example, in the mapping method, an ellipse is mapped to a certain plane, projected as a circle, and the circle is interpolated, and then, an interpolation point on the ellipse is obtained by coordinate transformation. And the centrifugal angle increment method performs interpolation according to the relationship between the next interpolation point and the current interpolation point and the centrifugal angle increment under the condition that the arc length increment is very small relative to the ellipse perimeter. Although the methods are simple and feasible, the problems of contour error and uncontrollable speed fluctuation exist. Accordingly, an equal error method has been proposed, and although the accuracy of processing an ellipse is improved, the speed fluctuation cannot be effectively controlled. Therefore, a large number of scholars study numerical methods, and the relationship between the interpolation point and the arc length is accurately deduced to effectively reduce the speed fluctuation, but the methods are poor in instantaneity because of involving multiple iterations and complex numerical calculation, so that the methods cannot be applied to a numerical control system.
As design and manufacturing techniques have evolved, more and more people use Computer Aided Design (CAD) systems to design complex parts. However, since most numerical control systems do not support the transmission of parametric spline data, a Computer Aided Manufacturing (CAM) system is usually used to overlay a free curve or curved surface of a CAD design with a series of broken lines within a specific tolerance range, thereby generating a numerical control machining program consisting of a large number of command points.
By carrying out numerical control processing on the tiny straight line segments, the problem of complicated ellipse arc length calculation can be avoided. However, frequent changes in acceleration and jerk may cause vibrations in the machine tool, reducing the efficiency and quality of the machining. The current research on the high-speed processing of the tiny line segments is mainly divided into two methods. One is to insert transition splines at the corners of adjacent tiny line segments. For example, the machining efficiency is improved by inserting cubic B-spline curves or quadratic bezier curves at the corners to increase the speed at the corners, but the interpolation step length is inconsistent due to the cyclic occurrence of interpolation points on the straight line segment and the transition spline curve segment, so that the machining speed fluctuates more frequently if the instruction points are denser. The other is to fit discrete command points to a smooth machining path by interpolation or approximation. For example, a machining path specified by a continuous minute line segment is converted into a machining path represented by a quadratic bezier curve, and high-speed and high-precision machining of a free curve is realized by interpolating the bezier curve. Although the method can better approximate the original design curve, the fitting curve is complex, and the interpolation parameters corresponding to the interpolation step length cannot be accurately calculated, so that the machining speed fluctuates, and the machining precision is reduced.
Disclosure of Invention
In order to overcome the defects that a direct interpolation algorithm and the existing small-line-segment interpolation algorithm cannot carry out high-precision machining and have speed fluctuation, the invention aims to provide a parameter spline fitting method.
The technical scheme adopted by the invention for solving the technical problems is as follows: an elliptic arc smooth compression interpolation algorithm suitable for high-speed high-precision machining comprises the following steps:
identifying a continuous tiny line segment processing area;
selecting model value points in the continuous micro line segment processing area and fitting the model value points to obtain a secondary rational Bezier curve;
identifying an elliptical arc and the conversion of the geometric form of the elliptical arc according to the curve characteristics;
merging adjacent elliptic arcs belonging to the same ellipse to obtain an interpolation curve;
and performing elliptic arc interpolation on the interpolation curve.
The fitting comprises the following steps:
2-1) obtaining the weight of the secondary rational Bezier curve;
2-2) obtaining a fitting curve of the quadratic Bezier by calculating the average value according to the weight.
The weights are obtained by:
wherein, w1As a weight, P0Is a first type value point, P2Is a last-type value point, P1Is a control point,P is an instruction point, u is P0Q and QP2Q is P1A straight line segment [ P ] as a projection center0P2]And projecting to the projection point of the quadratic Bezier fitting curve.
The method for obtaining the fitting curve of the quadratic rational Bezier by solving the average value according to the weight comprises the following steps:
4-1) obtaining a weight w corresponding to the quadratic Bezier curve according to a weight solving formulakAnd corresponding shoulder point coordinates sk＝wk/(1+wk),k＝i+1,...,j-1；
4-2) passing pairs skAveraging to obtain an average shoulder point coordinate s and an average weight w which is s/(1-s);
4-3) according to P0，P1，P2And w determines the type value point QiAnd QjA fitted curve of quadratic Bezier between them.
The identification of the elliptical arc and the conversion of the geometric form thereof according to the curve characteristics comprises the following steps:
5-1) when | P0P1|！＝|P1P2L and 0 < w1If the curve is less than 1, the curve is an elliptic arc;
5-2) obtaining geometric information of the elliptical arc by the following steps:
Ci(u) is a standard quadratic rational Bezier curve for a section of the machining path, which corresponds to an elliptical arc section according to Ci(u) obtaining a start coordinate p _ start and an end coordinate p _ end of the elliptical arc segment; the central coordinates of the elliptical arc segment are,
P1+(S+T)
the long radius and the short radius of the elliptic arc section are respectively,
let λ be2＞λ1Is greater than 0 and is the root of the following quadratic equation,
2λ2-(kη+4β)λ+2(k-1)＝0
＝|S×T|2,η＝|S-T|2，β＝S·T
two points on the long axis of the elliptical arc section are,
Q1＝P1+(+r1x0)S+(+r1y0)T
Q2＝P1+(-r1x0)S+(-r1y0)T
according to Q1、Q2Obtaining the slope kl of the long shaft and the included angle d _ kl between the long shaft and the positive half shaft of the x shaft; establishing a local coordinate system by using the major axis as the x 'axis and the minor axis as the y' axis, and using Q1、Q2Obtaining a starting angle d _ start, a terminal angle d _ end and an elliptical arc direction turn of the elliptical arc, wherein the starting point of the elliptical arc is connected with the center, and the terminal point of the elliptical arc is connected with the center, and the starting angle d _ start and the terminal angle d _ end of the elliptical arc are respectively relative to a positive half shaft of a local coordinate system x';
when the elliptical arc and the geometric form thereof are identified according to the curve characteristics, an elliptical arc segment group Ellipse _ Arcs [ ] is obtained, and the structure of each data in the group is as follows:
the invention has the following beneficial effects and advantages:
1. the method is simple to control, can effectively reduce the frequent fluctuation of the elliptical arc processing speed, and realizes the high-quality elliptical arc processing.
2. The compression amount is large and the smoothness is high. The method can identify the elliptical arc in the numerical control machining program and can represent the elliptical arc in a geometric form, so that the number of program segments is greatly reduced, and the smoothness of the machining path is improved by the geometric representation form of the elliptical arc.
3. The processing precision and the processing efficiency are high. The method uses the elliptic arc in the geometric form for interpolation, can accurately calculate the interpolation parameter corresponding to the arc length, reduces the calculation complexity and the frequent fluctuation of the speed in the processing, and improves the processing quality and the processing efficiency.
Drawings
FIG. 1 is a flow chart of the method of the present invention
FIG. 2 is a schematic view of the identification of a continuous micro-line machining area;
FIG. 3 is a schematic illustration of the identification of local curvature maxima;
FIG. 4 is a schematic diagram of a quadratic Bezier curve;
FIG. 5 is a schematic diagram of the calculation of the section point tangent vector;
FIG. 6 is a schematic diagram of the calculation of elliptical arc geometry information;
FIG. 7 is a diagram illustrating the division of regions and variables.
Detailed Description
The present invention will be described in further detail with reference to examples.
The invention relates to an elliptic arc smooth compression interpolation algorithm suitable for high-speed high-precision machining, which divides a machining path into a discontinuous micro line segment machining area and a continuous micro line segment machining area according to double-arch-height error limitation. And for the non-continuous micro line segment processing area, interpolation calculation is directly carried out on the straight line segment formed by the adjacent instruction points so as to ensure the processing precision. For a continuous tiny line segment processing area, fitting a curvature extreme point and a curvature point according to a curvature value of a discrete instruction point, and converting a broken line processing path into a smooth secondary rational Bezier curve processing path; then, identifying an elliptical arc by utilizing the characteristics of the secondary rational Bezier curve, and converting the elliptical arc into a geometric form; and finally, combining adjacent elliptical arc sections, and performing interpolation calculation.
The invention provides an elliptic arc smooth compression interpolation algorithm suitable for high-speed high-precision machining, which comprises the following steps:
1. and (4) identifying the machining area, namely, according to the double-arch-height error judgment condition, calling two adjacent points which do not meet the condition and a command point between the two adjacent points as a continuous tiny line segment machining area.
2. Selecting a model value point, calculating the curvature value of the instruction point in the continuous micro line segment machining area through a discrete point curvature calculation formula, finding out the local curvature maximum value point and the inflection point in the machining path according to the curvature value and the judgment condition of the adjacent instruction point, and marking the two end points, the local curvature maximum value point and the inflection point of the continuous micro line segment machining area as the model value point.
3. And fitting the type value points, and converting the broken line processing path specified by the instruction points into a smooth secondary rational Bezier curve processing path according to the coordinate values and the unit tangent vectors of the type value points under the condition of ensuring the processing precision.
4. And identifying the elliptical arc, identifying an elliptical arc segment according to the characteristics of the secondary rational Bezier curve, and converting the secondary rational Bezier form of the elliptical arc into a geometric form.
5. And merging the elliptical arc sections, namely identifying whether the adjacent elliptical arc sections belong to the same ellipse or not according to the geometric information of the adjacent elliptical arc sections, and merging the adjacent elliptical arc sections belonging to the same ellipse.
6. And (4) interpolation of the elliptic arc, namely performing interpolation calculation on the elliptic arc in a geometric form to realize high-speed and high-precision machining of the elliptic arc.
As shown in figure 1, the invention provides an elliptic arc smooth compression interpolation algorithm suitable for high-speed high-precision machining, which solves the problem of small line segment interpolation, and the method consists of 6 parts, namely identification of a machining area, selection of a type value point, fitting of the type value point, identification of an elliptic arc, conversion of a geometric form, combination of elliptic arc segments and interpolation of the elliptic arc, so that the machining quality and efficiency are improved.
According to the double-arch-height error judgment condition, two adjacent points which do not meet the condition and a command point between the two adjacent points are called as continuous micro-line segment processing areas.
And calculating the curvature value of the instruction point in the continuous micro-segment machining area through a discrete point curvature calculation formula, finding out the local curvature maximum value point and the inflection point in the machining path according to the curvature values and judgment conditions of adjacent instruction points, and marking the two end points, the local curvature maximum value point and the inflection point of the continuous micro-segment machining area as the model value points.
And under the condition of ensuring the machining precision, converting the broken line machining path specified by the instruction point into a smooth secondary rational Bezier curve machining path according to the coordinate value and the unit tangent vector of the type value point.
And identifying an elliptical arc segment according to the characteristics of the secondary rational Bezier curve, and converting the secondary rational Bezier form of the elliptical arc into a geometric form.
And identifying whether the adjacent elliptic arc sections belong to the same ellipse or not according to the geometric information of the adjacent elliptic arc sections, and combining the adjacent elliptic arc sections belonging to the same ellipse.
The interpolation of the elliptic arc is realized by performing interpolation calculation on the elliptic arc in a geometric form.
The method comprises the following specific steps:
1. identification of continuous micro-line segment processing area
As shown in FIG. 2, Pi-1、PiAnd Pi+1Three instruction points adjacent in sequence, l1、l2Is the segment length of the small line segment, theta is the corner between the small line segments, the double arch height error judgment condition is as follows,
wherein the content of the first and second substances,1、2respectively a small line segment Pi-1PiAnd PiPi+1Bow height error of phi1Is OPi-1And OPiHalf of the included angle. Phi is a2Is OPiAnd OPi+1Half of the included angle.
If it is not1Or2Greater than a set maximum bow height error valuemaxThen P isiIs a breakpoint; if there is a command point between two adjacent break points, the two break points and the command point between them form a continuous tiny line segment processing area.
2. Selection of model point
In order to reduce the fitting times of continuous tiny line segment processing areas and increase the compression amount of program segments, the model value points are selected through the following three steps.
(1) The starting point and the ending point of the continuous processing area, namely the breakpoint, are marked as type value points.
(2) The local curvature maximum point is marked as a typing point.
As shown in FIG. 2, the coordinates of three adjacent command points are Pi-1(xi-1,yi-1)、Pi(xi,yi) And Pi+1(xi+1,yi+1) Discrete instruction point PiThe curvature value of (a) is determined by the following formula,
wherein Δ Pi-1PiPi+1The area of the triangle that is signed is determined by the following equation,
suppose klIs PiLocal curvature minimum on the left, krIs PiLocal curvature minimum on the right, P if the following two conditions are satisfiediThe local curvature maximum point is shown in fig. 3.
1)|ki|＞|klAnd ki|＞|kr|
2)|ki|-|kl|≥fOr | ki|-|kr|≥f，fIs the set maximum curvature difference.
(3) The point where the bending direction of the processing path is changed, i.e., the inflection point, is marked as a type value point.
Using the value of the curvature of the instruction point calculated in the step (2) to judge if k isi-1ki> 0 and kiki+1＜0，ki-1、ki+1Are respectively an instruction point Pi-1And Pi+1The curvature value of (1) is PiIs an inflection point.
3. Fitting of shape points
For n type value points in the continuous tiny line segment machining area, a quadratic Bezier curve of n-1 segments can be used for fitting so as to achieve the purpose of smoothly compressing the machining path.
The quadratic bezier curve of the standard type is shown below,
wherein, P0、P1And P2As a control point, w1Is a weight value, u is a variable, and the range is [0, 1 ]]。
As can be seen from FIG. 4, when the leading end point P is given0And P2And the tangential direction T at these two points0And T2Can easily pass through the straight line [ P ]0T0]And [ P2T2]The intersection of (A) is obtained as P1Given a point P, the curve, and hence w, can be uniquely determined1。
The curve is considered as the point P0，P1And P2Projection of a determined parabola, P1Is the center of projection. As shown in FIG. 4, a straight line segment [ P ]0P2]Projected onto the desired curve, points P and Q are the corresponding projected points. Let w 10, straight line segment l (u) ═ P is obtained0P2]I.e. by
L (u) is a point P0And P2Is thus | P0Q | and | QP2The ratio of | is u2:(1-u)2Thereby pushing out
The u and the P are brought into a secondary rational Bezier curve to obtain w1Thereby obtainingThe curve is obtained.
Assumed value point QiAnd QjBy the instruction point Qi+1,Qi+2,…,Qj-2And Qj-1If the specified broken line processing path composition and the tangent vector of the type value point are known, the type value point QiAnd QjThe detailed steps of the fitting between are as follows.
(1) Is constructed with P0＝Qi，P1And P2＝QjFitting curves for the control points to interpolate Q respectivelyk(k ═ i + 1.., j-1). This will result in an intermediate weight wkAnd corresponding shoulder point coordinates sk＝wk/(1+wk),k＝i+1,...,j-1；
(2) By pairs of skThe average value is taken to obtain the shoulder point coordinates of the approximation curve, i.e.
The weight w in the middle is s/(1-s);
(3) according to P0，P1，P2And w is the determinable value point QiAnd QjThe fitted curve in between.
4. Calculation of type point tangent vector
Since the tangent vector at the instruction point is not provided in the nc machining program, the tangent vector at the type value point can be calculated by the type value point and four instruction points around it. As shown in FIG. 5, QiIs a type value point, Qi-2、Qi-1、Qi+1、Qi+2For four instruction points around it, QiThe unit tangent vector expression is as follows
Since the parameter u is not used in the calculation formula, the unit tangent vector can only be regarded as the direction of the tangent vector. T can be obtained by the following formula0，T1And Tn-1，Tn。
q0＝2q1-q2,q-1＝2q0-q1
qn+2＝2qn+1-qn,qn+1＝2qn-qn-1
5. Control of fitting accuracy
Although the curve segment passes through the value point QiAnd QjBut cannot guarantee it to QiAnd QjThe distance between all the command points satisfies the maximum fitting error. Thus, Q will beiAnd QjAll the command points in between are projected on the fitted curve, and whether the deviation is within the allowable error range or not, namely, the deviation is less than or equal to the maximum profile error set by the systemcIf the error requirement is met, performing curve fitting of the next section; otherwise, setting the instruction point with the maximum deviation as a new model value point, carrying out curve fitting again by using the new model value point, checking the fitting precision again, and repeating the process until the error requirement is met.
6. Identification of elliptical arcs and conversion of geometric forms
Because the standard type secondary rational B zier curve has only one weight factor, the expression capability is stronger than that of the B zier curve in a non-rational form, and a plurality of curves can be expressed when | P |0P1|！＝|P1P2L and 0 < w1If < 1, the curve is an elliptical arc.
If C is presenti(u) is a standard quadratic rational Bezier curve for a section of the machining path, corresponding to an elliptical arc, as shown in FIG. 6, according to Ci(u) the coordinates of the start point p _ start and the coordinates of the end point p _ end of the elliptical arc segment are obtained. The relationship between the standard quadratic rational Bezier form of the ellipse and the geometric form thereofKnowing that the central coordinates of the elliptical arc segment are,
P1+(S+T)
the long radius and the short radius of the elliptic arc section are respectively,
let λ be2＞λ1Is greater than 0 and is the root of the following quadratic equation,
2λ2-(kη+4β)λ+2(k-1)＝0
＝|S×T|2,η＝|S-T|2，β＝S·T
two points on the long axis of the elliptical arc section are,
Q1＝P1+(+r1x0)S+(+r1y0)T
Q2＝P1+(-r1x0)S+(-r1y0)T
according to Q1、Q2Obtaining the slope kl of the long shaft and the included angle d _ kl between the long shaft and the positive half shaft of the x shaft; establishing a local coordinate system by using the major axis as the x 'axis and the minor axis as the y' axis, and using Q1、Q2Obtaining a starting angle d _ start, a terminal angle d _ end and an elliptical arc direction turn of the elliptical arc, wherein the starting point of the elliptical arc is connected with the center, and the terminal point of the elliptical arc is connected with the center, and the starting angle d _ start and the terminal angle d _ end of the elliptical arc are respectively relative to a positive half shaft of a local coordinate system x'; x is the number of0、y0Is the coordinate of the starting point of the elliptical arc segment.
When the identification and geometric form conversion of the elliptical arc are completed in the continuous tiny line segment processing area, an elliptical arc segment array Ellipse _ Arcs [ ] is obtained, wherein the data structure of the elliptical arc segment is as follows:
7. merging of elliptical arcs
In order to improve the efficiency and the precision of the elliptical arc processing, the elliptical arc sections in the continuous tiny line section processing area are combined, if the adjacent elliptical arc sections have the same center, long radius r _ l, short radius r _ s, direction turn, and the included angle d _ kl between the long axis and the positive x-axis, and the terminal point coordinate p _ end of the previous elliptical arc sectioni-1And the starting point coordinate p _ start of the next elliptical arc segmentiIf they are equal, they belong to a continuous elliptical arc segment on the same elliptical arc, and the terminal angle d _ end of the last elliptical arc segment is modifiedi-1And end point coordinates p _ endi-1They are combined into an elliptical arc segment. And repeating the steps until the elliptical arc sections can not be combined.
8. Interpolation of elliptical arcs
8.1 calculation of the arc length of the ellipse
Assuming an elliptic equation of
Where a is the major semi-axis radius and b is the minor semi-axis radius, as shown in fig. 7, the ellipse is divided into four regions, and each region corresponds to different parameters as variables.
Taking the first quadrant y as an example of the variable,
the arc length between any two points in the region is,
The above equation can be converted into a product according to the Gaussian Legendre product equation,
in the case where the other quadrant x or y is variable, the arc length is derived in the same manner.
8.2 arc Length vs. variance
With the first quadrant y as a variable, the slave point E0(a,0) to Point
Taking the elliptic arc as an example, the solving of the relation between the arc length and the variable comprises the following steps:
(1) in the interval of variable y
Taking two trisection points y1And y2。
(2) Respectively calculating y according to an arc length formula of 8.1i+1＝0、y1、y2And
to the starting point y of the elliptical arciArc length of 0, s0、s1、s2、s3。
(3) Will(s)0,0)、(s1,y1)、(s2,y2) And
substituting the obtained data into a following cubic polynomial to carry out interpolation to obtain the coefficient of the polynomial, thus obtaining the relation between the arc length and the variable.
y＝a0+a1s+a2s2+a3s3，s∈[s0,s3]
Wherein, a0～a3Are coefficients. In the case that other quadrants x and y are variables, the arc length is the same as the derivation method of the relation of the variables.
8.3 interpolation of elliptical arcs
By the above process, the numerical control in the processing pathThe elliptical arc is identified and represented in geometric form. Dividing the identified elliptical arc segments according to four regions, respectively calculating the relation between the arc length and the variable in each region, and then calculating the coordinate (x) of the interpolation point of the ith interpolation period by using the relation between the arc length and the variablei′,yi') the specific procedure is as follows.
Assuming that the interpolation period is T, P0′(x′i-1,y′i-1) And Si-1The coordinates and the arc length of an interpolation point of an i-1 th interpolation period respectively, and the feeding speed of the i-th interpolation period is vi-1And then the increment of the arc length is Delta S ═ vi-1T, the arc length of the ith interpolation period is Si＝Si-1+ Δ S, will SiSubstituting into the relation between the arc length and the variable to obtain the corresponding variable xi' or yi'is substituted into the ellipse equation to obtain the interpolation point coordinate P' (x) of the i-th interpolation periodi′,yi′)。
Claims (5)
1. An elliptic arc smooth compression interpolation algorithm suitable for high-speed high-precision machining is characterized by comprising the following steps:
identifying a continuous tiny line segment processing area;
selecting model value points in the continuous micro line segment processing area and fitting the model value points to obtain a secondary rational Bezier curve;
selecting model value points, calculating curvature values of instruction points in a continuous micro line segment machining area through a discrete point curvature calculation formula, finding out a local curvature maximum value point and an inflection point in a machining path according to the curvature values and judgment conditions of adjacent instruction points, and marking two end points, the local curvature maximum value point and the inflection point of the continuous micro line segment machining area as the model value points;
fitting the type value points, and converting the broken line processing path specified by the instruction points into a smooth secondary rational Bezier curve processing path according to the coordinate values and unit tangent vectors of the type value points under the condition of ensuring the processing precision;
identifying an elliptical arc and the conversion of the geometric form of the elliptical arc according to the curve characteristics;
the identification of the elliptical arc and the conversion of the geometric form thereof according to the curve characteristics comprises the following steps:
5-1) when | P0P1|！＝|P1P2L and 0 < w1If the curve is less than 1, the curve is an elliptic arc;
5-2) obtaining geometric information of the elliptical arc by the following steps:
Ci(u) is a standard quadratic rational Bezier curve for a section of the machining path, which corresponds to an elliptical arc section according to Ci(u) obtaining a start coordinate p _ start and an end coordinate p _ end of the elliptical arc segment; the central coordinates of the elliptical arc segment are,
P1+(S+T)
S＝P0-P1,T＝P2-P1,
wherein, w1As a weight, P0Is a first type value point, P2Is a last-type value point, P1Is a control point;
the long radius and the short radius of the elliptic arc section are respectively,
let λ be2＞λ1Is greater than 0 and is the root of the following quadratic equation,
2λ2-(kη+4β)λ+2(k-1)＝0
＝|S×T|2,η＝|S-T|2，β＝S·T
two points on the long axis of the elliptical arc section are,
Q1＝P1+(+r1x0)S+(+r1y0)T
Q2＝P1+(-r1x0)S+(-r1y0)T
according to Q1、Q2The long axis can be obtainedThe slope kl and an included angle d _ kl between the long axis and the positive half axis of the x axis; establishing a local coordinate system by using the major axis as the x 'axis and the minor axis as the y' axis, and using Q1、Q2The starting point coordinates of the elliptic arc section and the end point coordinates of the elliptic arc section are obtained, and the starting angle d _ start and the end point angle d _ end of the starting point of the elliptic arc section and the central connecting line, the end point of the elliptic arc section and the central connecting line relative to the positive half shaft of the local coordinate system x' and the direction turn, x of the elliptic arc are obtained0、y0Coordinates of the starting point of the elliptic arc segment;
merging adjacent elliptic arcs belonging to the same ellipse to obtain an interpolation curve;
and performing elliptic arc interpolation on the interpolation curve.
2. The algorithm of claim 1, wherein the fitting comprises the following steps:
2-1) obtaining the weight of the secondary rational Bezier curve;
2-2) obtaining a fitting curve of the quadratic Bezier by calculating the average value according to the weight.
3. The algorithm of claim 2, wherein the weights are obtained by the following formula:
wherein, w1As a weight, P0First control point, P2Is the final control point, P1Is a control point, P is an instruction point, u is P0Q and QP2Q is P1A straight line segment [ P ] as a projection center0P2]And projecting to the projection point of the quadratic Bezier fitting curve.
4. The algorithm of claim 2, wherein the obtaining of the fitted curve of quadratic Bezier by averaging according to the weights comprises the following steps:
4-1) obtaining a weight w corresponding to the quadratic Bezier curve according to a weight solving formulakAnd corresponding shoulder point coordinates sk＝wk/(1+wk),k＝i+1,...,j-1；
4-2) passing pairs skAveraging to obtain an average shoulder point coordinate s and an average weight w which is s/(1-s);
4-3) according to P0，P1，P2And w determines the type value point QiAnd QjA fitted curve of quadratic Bezier between them.
5. An elliptic arc smooth compression interpolation algorithm suitable for high-speed and high-precision machining according to claim 1 is characterized in that after the elliptic arc is identified according to curve features and the conversion of the geometrical form of the elliptic arc is finished, an elliptic arc segment array Ellipse _ Arcs [ ] is obtained, and the structure of each data in the array is as follows:
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201610978793.3A CN108073138B (en) | 2016-11-08 | 2016-11-08 | Elliptical arc smooth compression interpolation algorithm suitable for high-speed high-precision machining |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201610978793.3A CN108073138B (en) | 2016-11-08 | 2016-11-08 | Elliptical arc smooth compression interpolation algorithm suitable for high-speed high-precision machining |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN108073138A CN108073138A (en) | 2018-05-25 |
| CN108073138B true CN108073138B (en) | 2020-08-11 |
Family
ID=62153978
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201610978793.3A Active CN108073138B (en) | 2016-11-08 | 2016-11-08 | Elliptical arc smooth compression interpolation algorithm suitable for high-speed high-precision machining |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN108073138B (en) |
Families Citing this family (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108908350A (en) * | 2018-07-16 | 2018-11-30 | 苏州大学张家港工业技术研究院 | A kind of control system and its control method of interactive robot |
| CN111684374B (en) * | 2018-12-27 | 2022-05-31 | 深圳配天智能技术研究院有限公司 | Numerical control machining method, numerical control machine tool, and computer storage medium |
| CN109799792B (en) * | 2019-01-08 | 2021-06-01 | 华侨大学 | Method for smooth transition of micro line segment for high-speed numerical control machining |
| CN111552288B (en) * | 2020-04-28 | 2021-03-16 | 西南交通大学 | Mobile robot path smoothing method |
| CN111610751B (en) * | 2020-05-21 | 2023-07-28 | 天津工业大学 | Interpolation error multi-subdivision iterative calculation method for cross point set NURBS interpolation curve |
| CN113970305B (en) * | 2021-06-16 | 2023-07-28 | 广西大学 | Method for measuring axial displacement of compression bar through deflection |
| CN113381766A (en) * | 2021-06-25 | 2021-09-10 | 河北省科学院应用数学研究所 | Method, terminal and storage medium for electrocardiogram data compression |
| CN113835397B (en) * | 2021-06-30 | 2024-02-09 | 浙江大学 | Linear numerical control machining path smoothing method based on B spline curve and path integral |
Citations (10)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4243924A (en) * | 1977-06-28 | 1981-01-06 | Oki Electric Industry Co., Ltd. | System for interpolating an arc for a numerical control system |
| JPS59229618A (en) * | 1983-06-10 | 1984-12-24 | Nec Corp | Movement control device |
| CN101140460A (en) * | 2006-09-06 | 2008-03-12 | 北京数码大方科技有限公司 | Method and apparatus for driving numerically controlled machine to execute space circular arc interpolation process |
| CN101539769A (en) * | 2009-04-28 | 2009-09-23 | 中国科学院数学与系统科学研究院 | Method for fitting and interpolating G01 code based on quadratic B spline curve |
| CN101751006A (en) * | 2008-12-18 | 2010-06-23 | 陈学恭 | Interpolation method |
| CN101907876A (en) * | 2010-05-28 | 2010-12-08 | 沈阳高精数控技术有限公司 | Command point shaping, compressing and interpolating method suitable for numerical control device |
| CN102109834A (en) * | 2011-01-10 | 2011-06-29 | 浙江工业大学 | Motion control system based on non-uniform rational B-spline (NURBS) curve interpolation |
| CN102147599A (en) * | 2011-03-30 | 2011-08-10 | 华南理工大学 | Elliptic arc interpolation method |
| CN103699056A (en) * | 2013-12-02 | 2014-04-02 | 嘉兴学院 | Real-time smooth transition interpolation method of small high-speed and high-accuracy numerical control machining line segment |
| CN104597847A (en) * | 2013-10-31 | 2015-05-06 | 中国科学院沈阳计算技术研究所有限公司 | Akima spline fitting based look-ahead interpolation method |
Family Cites Families (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JPH06150013A (en) * | 1992-11-11 | 1994-05-31 | Tokyo Electric Co Ltd | Elliptic arc generator |
| JP2003331296A (en) * | 2002-05-10 | 2003-11-21 | Ricoh Co Ltd | Method of drawing circular arc |
| CN103809521B (en) * | 2012-11-14 | 2016-08-17 | 中国科学院沈阳计算技术研究所有限公司 | SPL interpolating method based on Secant Method |
| CN104268330B (en) * | 2014-09-19 | 2017-08-25 | 华南理工大学 | A kind of Bezier approximating methods of complex curve |
-
2016
- 2016-11-08 CN CN201610978793.3A patent/CN108073138B/en active Active
Patent Citations (10)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4243924A (en) * | 1977-06-28 | 1981-01-06 | Oki Electric Industry Co., Ltd. | System for interpolating an arc for a numerical control system |
| JPS59229618A (en) * | 1983-06-10 | 1984-12-24 | Nec Corp | Movement control device |
| CN101140460A (en) * | 2006-09-06 | 2008-03-12 | 北京数码大方科技有限公司 | Method and apparatus for driving numerically controlled machine to execute space circular arc interpolation process |
| CN101751006A (en) * | 2008-12-18 | 2010-06-23 | 陈学恭 | Interpolation method |
| CN101539769A (en) * | 2009-04-28 | 2009-09-23 | 中国科学院数学与系统科学研究院 | Method for fitting and interpolating G01 code based on quadratic B spline curve |
| CN101907876A (en) * | 2010-05-28 | 2010-12-08 | 沈阳高精数控技术有限公司 | Command point shaping, compressing and interpolating method suitable for numerical control device |
| CN102109834A (en) * | 2011-01-10 | 2011-06-29 | 浙江工业大学 | Motion control system based on non-uniform rational B-spline (NURBS) curve interpolation |
| CN102147599A (en) * | 2011-03-30 | 2011-08-10 | 华南理工大学 | Elliptic arc interpolation method |
| CN104597847A (en) * | 2013-10-31 | 2015-05-06 | 中国科学院沈阳计算技术研究所有限公司 | Akima spline fitting based look-ahead interpolation method |
| CN103699056A (en) * | 2013-12-02 | 2014-04-02 | 嘉兴学院 | Real-time smooth transition interpolation method of small high-speed and high-accuracy numerical control machining line segment |
Non-Patent Citations (2)
| Title |
|---|
| 《基于二次代数样条差分插补的平面参数曲线数控加工》;单东日;《制造技术与机床》;20090228(第2期);第95-98页 * |
| 《数控加工中的平滑压缩插补算法研究》;张晓辉等;《机械工程学报》;20110331;第47卷(第5期);第156-162,169页 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN108073138A (en) | 2018-05-25 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN108073138B (en) | Elliptical arc smooth compression interpolation algorithm suitable for high-speed high-precision machining | |
| CN108062073B (en) | Circular arc smooth compression interpolation method for high-quality machining | |
| CN111061213B (en) | Processing method based on Bezier curve corner smooth transition algorithm | |
| CN109571473B (en) | Error-controllable small line segment trajectory fairing method | |
| Liu et al. | Development and implementation of a NURBS interpolator with smooth feedrate scheduling for CNC machine tools | |
| CN109976262B (en) | Global curvature continuous fairing method for micro-line segment processing | |
| Wang et al. | Design of trigonometric velocity scheduling algorithm based on pre-interpolation and look-ahead interpolation | |
| CN101539769B (en) | Method for fitting and interpolating G01 code based on quadratic B spline curve | |
| CN101907876B (en) | Command point shaping, compressing and interpolating method suitable for numerical control device | |
| Bi et al. | A general, fast and robust B-spline fitting scheme for micro-line tool path under chord error constraint | |
| CN108170094B (en) | Method for smoothly compressing cutter path | |
| CN109597359B (en) | Numerical controller | |
| Wu et al. | Implementation of CL points preprocessing methodology with NURBS curve fitting technique for high-speed machining | |
| CN113467384A (en) | Corner transition method applied to five-axis numerical control machine tool | |
| CN115202291A (en) | NURBS curve interpolation method based on elliptic arc fitting | |
| CN106844966B (en) | A kind of propeller blade face blade back Precise modeling | |
| CN104317251A (en) | Three-order NURBS curve real-time interpolation method based on Obrechkoff algorithm | |
| CN112051803B (en) | Small line segment fairing method based on space plane normal vector | |
| TW200821784A (en) | Processing path method of converting straight line to NURBS curve of computer numerical control | |
| CN109521731B (en) | G2 continuous Bezier tool path smoothing algorithm based on tolerance zone | |
| CN113504764A (en) | Continuous line segment numerical control machining path smoothing method based on position vector weighted integral | |
| CN111610751B (en) | Interpolation error multi-subdivision iterative calculation method for cross point set NURBS interpolation curve | |
| CN109991921B (en) | Smooth B spline track direct generation method | |
| CN117162492A (en) | Curved surface 3D printing path planning method based on mechanical arm | |
| CN109697272B (en) | Simple quadratic B-spline curve fitting method |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant | ||
| CP01 | Change in the name or title of a patent holder |
Address after: No. 16-2, Nanping East Road, Dongling District, Shenyang City, Liaoning Province, 110168 Patentee after: Shenyang Zhongke CNC Technology Co.,Ltd. Address before: No. 16-2, Nanping East Road, Dongling District, Shenyang City, Liaoning Province, 110168 Patentee before: Shenyang Golding Nc Intelligence Tech.co.,ltd. |
|
| CP01 | Change in the name or title of a patent holder |