CN108073138A - Suitable for the elliptic arc smooth compression interpolation algorithm of high speed and high precision processing - Google Patents
Suitable for the elliptic arc smooth compression interpolation algorithm of high speed and high precision processing Download PDFInfo
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- G—PHYSICS
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- 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 present invention relates to a kind of elliptic arc smooth compression interpolation algorithms suitable for high speed and high precision processing, comprise the following steps：Identify continuous mini line segment machining area；Data point is chosen in continuous mini line segment machining area and it is fitted to obtain quadratic rational Bézier；Elliptic arc is identified according to curvilinear characteristic, and geometric format conversion is carried out to it；Same elliptical adjacent ellipses arc will be belonged to merge, obtain interpolation curve；Elliptic arc interpolation is carried out on interpolation curve.The method of the present invention utilizes the feature of spline curve, identifies the elliptic arc in machining path, interpolation is carried out with its geometric format, and the corresponding interpolation parameters of arc length can be precisely calculated, and reduces calculation amount and processes the frequent fluctuation of medium velocity, realizes high speed and high precision processing.
Description
Technical field
The fitting of parametric spline curve and the identification of oval segmental arc belong to several with merging in being processed the present invention relates to high quality
Control processing technique field.
Background technology
Ellipse is one kind in conic section, relatively conventional in work pieces process program.Since the calculating of oval arc length relates to
And to ellptic integral problem, and its original function cannot be represented with finite form, so can not accurate interpolation directly be carried out to ellipse.
Again because it is with stronger engineering practicability, scholar both domestic and external conducts extensive research this.
Such as reflection method, by Elliptic Mappings to a certain plane, it is made to be projected as justifying, by carrying out interpolation to circle, then utilized
Coordinate transform obtains the interpolated point on ellipse.Eccentric angle method of addition, in the case where arc length increment relative elliptical perimeter is very small,
Interpolation is carried out by next interpolated point and current interpolated point and the relation for centrifuging angle increment.Though above method simple possible, all
There are problems that profile errors and velocity perturbation are unable to control.Then equal error method is proposed, although improving elliptical add
Work precision, but still velocity perturbation can not be efficiently controlled.Therefore a large amount of scholars study numerical method, by more smart
Really derive the relation of interpolated point and arc length, to be effectively reduced velocity perturbation, but these methods are because being related to successive ignition
With complicated numerical computations, real-time is poor, so can not be applied to digital control system.
With the development of design and fabrication technology, more and more people using CAD (CAD) system come into
The design of row complex parts.But since most of digital control system does not support the transmission of Parameter Spline data, usually utilize
Computer-aided manufacturing (CAM) system is by the free curve or curved surface of CAD design, with a series of foldings in specific range of tolerable variance
Line goes to cover, so as to generate by largely instructing the nc program a little formed.
By carrying out digital control processing to small straightway, can not have to consider complicated oval arc length computational problem.But by
In the frequent variation of acceleration and acceleration, the vibrations of lathe can be caused, reduce the efficiency and quality of processing.And now for
The research of mini line segment High-speed machining is broadly divided into two methods.A kind of is corner's insertion transition sample in adjacent mini line segment
Curve.For example, by way of being inserted into B-spline Curve or Bézier curve around the corner, corner is improved
Speed, so as to improve processing efficiency, but since cycling of the interpolated point on straightway and transition spline curve section occurs so that insert
It is inconsistent to mend step-length, so as to cause the fluctuation of process velocity, if instruction point is more intensive, the fluctuation of speed will be more frequent.Separately
It is a kind of, it is to be fitted to smooth machining path interpolation or by discrete instruction point by way of approaching.It for example, will be by even
The machining path that continuous mini line segment is specified is converted into the machining path represented by quadratic B é zier curves, by Bézier curve
The high speed and high precision processing to free curve is realized in interpolation.Although this mode can preferably approach former design curve, due to
Matched curve is more complicated, can not accurately calculate the corresponding interpolation parameters of Interpolation step-length, process velocity is caused to fluctuate, reduce processing
Precision.
The content of the invention
In order to overcome direct Interpolation Algorithm and existing small line segment interpolation algorithm cannot high finishing and there are velocity perturbations
Deficiency, the object of the present invention is to provide a kind of Parameter Spline approximating methods, by parametric spline curve feature, identify machining path
In oval segmental arc, by carrying out interpolation to oval segmental arc, to ensure the precision of processing, while reduce the fluctuation of process velocity.
The technical solution adopted by the present invention to solve the technical problems is：A kind of elliptic arc suitable for high speed and high precision processing
Smooth compression interpolation algorithm, comprises the following steps：
Identify continuous mini line segment machining area；
Data point is chosen in continuous mini line segment machining area and it is fitted to obtain Quadratic Rational B é zier songs
Line；
Elliptic arc and its geometric format conversion are identified according to curvilinear characteristic；
Same elliptical adjacent ellipses arc will be belonged to merge, obtain interpolation curve；
Elliptic arc interpolation is carried out on interpolation curve.
Described be fitted comprises the following steps：
2-1) ask for the weight of quadratic rational Bézier；
The matched curve of Quadratic Rational B é zier 2-2) is obtained by averaged according to weight.
The weight is obtained by following formula：
Wherein, w1For weight, P0Headed by data point, P2For last data point, P1In order to control point, P for instruction point, u P0Q with
QP2Ratio, Q be with P1For projection centre, straightway [P0 P2] project to the subpoints of Quadratic Rational B é zier matched curves.
The matched curve for obtaining Quadratic Rational B é zier by averaged according to weight comprises the following steps：
4-1) obtain asking for the corresponding weight w of quadratic rational Bézier according to the formula of asking for of weightkIt is and corresponding
Shoulder point coordinates sk=wk/(1+wk), k=i+1 ..., j-1；
4-2) by skIt averages to obtain average shoulder point coordinates s and average weight w=s/ (1-s)；
4-3) according to P0, P1, P2Data point Q is determined with wiWith QjBetween Quadratic Rational B é zier matched curve.
It is described to identify that elliptic arc and its geometric format conversion comprise the following steps according to curvilinear characteristic：
5-1) as | P0P1|！=| P1P2| and 0 ＜ w1During ＜ 1, then the curve is elliptic arc；
5-2) geological information of elliptic arc is obtained by following steps：
Ci(u) the standard type quadratic rational Bézier for being certain section in machining path, and it corresponds to an elliptic arc
Section, according to Ci(u) the starting point coordinate p_start of the ellipse segmental arc and terminal point coordinate p_end can be obtained；The center of the ellipse segmental arc
Coordinate is,
P1+ε(S+T)
The long and short radius of the ellipse segmental arc is respectively,
Assuming that λ2＞ λ1＞ 0, and be the root of following quadratic equation,
2δλ2- (+4 β of k η) λ+2 (k-1)=0
δ=| S × T |2, η=| S-T |2, β=ST
Two points on elliptic arc segment length's axis are,
Q1=P1+(ε+r1x0)S+(ε+r1y0)T
Q2=P1+(ε-r1x0)S+(ε-r1y0)T
According to Q1、Q2, the slope k l of long axis and the angle d_kl of long axis and x-axis positive axis can be obtained；Using long axis as x '
Axis, short axle establish local coordinate system for y ' axis, utilize Q1、Q2, centre coordinate, oval segmental arc starting point coordinate and elliptic arc segment endpoint
Coordinate acquires oval segmental arc starting point and is respectively relative to local coordinate system x ' with the line of centres, elliptic arc segment endpoint and the line of centres
Start angle d_start, the terminal angle d_end of positive axis and the direction turn of elliptic arc；
After elliptic arc and its geometric format conversion end is identified according to curvilinear characteristic, an elliptic arc hop count group is obtained
Ellipse_Arcs [], the structure of each data is as follows in array：
The invention has the advantages that and advantage：
1. the method for the present invention control is simple, the frequent fluctuation of elliptic arc process velocity can be effectively reduced, is realized oval
The high quality processing of arc.
2. decrement is big, smoothness is high.The method of the present invention can recognize that the elliptic arc in nc program, and can use
The geometric format of elliptic arc is indicated, and greatly reduces the quantity of program segment, while the geometric representation form of elliptic arc improves
The smoothness of machining path.
3. machining accuracy and high in machining efficiency.The method of the present invention carries out interpolation with the elliptic arc of geometric format, can be more
The corresponding interpolation parameters of arc length is precisely calculated, reduces the complexity of calculating and the frequent fluctuation of processing medium velocity, improves
Processing quality and processing efficiency.
Description of the drawings
Fig. 1 is flow chart of the method for the present invention
Fig. 2 is the identification schematic diagram of continuous mini line segment machining area；
Fig. 3 is the identification schematic diagram of local curvature's maximum；
Fig. 4 is quadratic rational Bézier schematic diagram；
Fig. 5 is the schematic diagram calculation that data point cuts arrow；
Fig. 6 is the schematic diagram calculation of elliptic arc geological information；
Fig. 7 is the division schematic diagram of region and variable.
Specific embodiment
With reference to embodiment, the present invention is described in further detail.
The present invention is a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing, according to the high error of bouble-bow
Machining path is divided into discontinuous mini line segment machining area and continuous mini line segment machining area by limitation.For discontinuous micro-
Small line segment machining area directly carries out interpolation calculating, to ensure machining accuracy on the straightway of adjacent instructions point composition.For
Continuous mini line segment machining area, according to the curvature value of discrete command point, is fitted point with extreme curvature and inflection point, by broken line
Machining path is converted into smooth quadratic rational Bézier machining path；Then, it is special using quadratic rational Bézier
Sign identifies elliptic arc, and is converted to geometric format；Finally, after adjacent ellipses segmental arc is merged, interpolation calculating is carried out.
The present invention provides a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing, including following step
Suddenly：
1. the identification of machining area, according to the high error judgment condition of bouble-bow, by the two neighboring point for being unsatisfactory for condition and it
Between instruction point be known as continuous mini line segment machining area.
2. the selection of data point for the instruction point in continuous mini line segment machining area, is calculated by discrete point curvature
Formula, calculates its curvature value, and according to the curvature value and Rule of judgment of adjacent instructions point, finds out in machining path local curvature most
Big value point and inflection point, offset is labeled as by the two-end-point of continuous mini line segment machining area, local curvature's maximum of points and inflection point
Point.
It, will 3. under conditions of machining accuracy is ensured, arrow is cut according to the coordinate value of data point and unit for the fitting of data point
The broken line machining path that instruction point is specified is converted into smooth quadratic rational Bézier machining path.
4. the identification of elliptic arc, oval segmental arc is gone out according to the feature recognition of quadratic rational Bézier, and by elliptic arc
Quadratic Rational B é zier forms be converted to geometric format.
5. the merging of oval segmental arc, to adjacent oval segmental arc, according to its geological information, identifies whether they belong to same
Ellipse is merged for belonging to same elliptical adjacent ellipses segmental arc.
6. the interpolation of elliptic arc by carrying out interpolation calculating on the elliptic arc of geometric format, realizes the high speed of elliptic arc
Height finishing.
As shown in Figure 1, the present invention proposes a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing,
Solves the problems, such as small line segment interpolation, method is by the identification of machining area, the selection of data point, the fitting of data point, elliptic arc
The converting of identification and geometric format, oval segmental arc merge and 6 parts of the interpolation of elliptic arc form, improve the matter of processing
Amount and efficiency.
According to the high error judgment condition of bouble-bow, the two neighboring point for being unsatisfactory for condition and the instruction point between them are known as
Continuous mini line segment machining area.
For the instruction point in continuous mini line segment machining area, by discrete point curvature calculation formula, its curvature is calculated
Value, and according to the curvature value and Rule of judgment of adjacent instructions point, find out local curvature's maximum of points and inflection point in machining path, incite somebody to action
Two-end-point, local curvature's maximum of points and the inflection point of continuous mini line segment machining area are labeled as data point.
Under conditions of ensureing machining accuracy, arrow is cut according to the coordinate value of data point and unit, the broken line that point will be instructed to specify
Machining path is converted into smooth quadratic rational Bézier machining path.
Oval segmental arc is gone out according to the feature recognition of quadratic rational Bézier, and by the Quadratic Rational B é zier of elliptic arc
Form is converted to geometric format.
To adjacent oval segmental arc, according to its geological information, identify whether they belong to same ellipse, it is same for belonging to
Elliptical adjacent ellipses segmental arc merges.
The interpolation of elliptic arc realizes the high speed and super precision of elliptic arc by carrying out interpolation calculating on the elliptic arc of geometric format
Processing.
The present invention is as follows：
1. the identification of continuous mini line segment machining area
As shown in Fig. 2, Pi-1、PiAnd Pi+1For three adjacent instruction points of order, l1、l2For the segment length of small line segment, θ is small
Turning between line segment, the high error judgment condition of bouble-bow is as follows,
Wherein, δ1、δ2Respectively small line segment Pi-1PiAnd PiPi+1The high error of bow, φ1For OPi-1And OPiThe half of angle.
φ2For OPiAnd OPi+1The half of angle.
If δ1Or δ2Error amount δ high more than the most longbow of settingmax, then PiFor breakpoint；If it is deposited between two adjacent breakpoints
In instruction point, then two breakpoints just constitute a continuous mini line segment machining area together with the instruction point between them.
2. the selection of data point
In order to reduce the decrement for being fitted number, increasing program segment to continuous mini line segment machining area, by as follows
Three steps choose data point.
(1) by the starting points and end point in Continuous maching region, i.e. breakpoint, labeled as data point.
(2) local curvature's maximum of points is labeled as data point.
As shown in Fig. 2, the coordinate of three adjacent instructions points is Pi-1(xi-1,yi-1)、Pi(xi,yi) and Pi+1(xi+1,yi+1),
Discrete command point PiCurvature value determined by following formula,
Wherein Δ Pi-1PiPi+1It is determined for the triangular form area of tape symbol by following formula,
Assuming that klFor PiLocal curvature's minimum value on the left side, krFor PiLocal curvature's minimum value on the right, if met following
Two conditions, then PiFor local curvature's maximum of points, as shown in Figure 3.
1)|ki| ＞ | kl| and | ki| ＞ | kr|
2)|ki|-|kl|≥δfOr | ki|-|kr|≥δf, δfFor the maximum curvature difference of setting.
(3) point for changing machining path bending direction, i.e. inflection point, labeled as data point.
The instruction point curvature value calculated using (2) step is judged, if ki-1ki＞ 0 and kiki+1＜ 0, ki-1、
ki+1Respectively instruct point Pi-1And Pi+1Curvature value, then PiFor inflection point.
3. the fitting of data point
It, can be bent with one n-1 sections of Quadratic Rational B é zier for n data point in continuous mini line segment machining area
Line is fitted, to achieve the purpose that smooth compression machining path.
The quadratic rational Bézier of standard type is as follows,
Wherein, P0、P1And P2Point in order to control, w1For weights, u is variable, and scope is [0,1].
As shown in Figure 4, as given first and last endpoint P0And P2And the tangential direction T at this 2 point0And T2When, it can hold very much
It changes places through straight line [P0 T0] and [P2 T2] intersection point obtain P1, then a point P is given with regard to the curve can be uniquely determined, thus determine
W1。
Required curve is regarded as by point P0, P1And P2Definite parabolical projection, P1For projection centre.Such as Fig. 4 institutes
Show, by straightway [P0 P2] project on the curve of requirement, then point P and Q is corresponding subpoint.Make w1=0, obtain straight line
Section L (u)=[P0 P2], i.e.,
L (u) is point P0And P2Convex combination, therefore | P0Q | and | QP2| ratio be u2:(1-u)2, so as to release
It brings u and P into quadratic rational Bézier, obtains w1, so as to obtain required curve.
It is assumed that data point QiWith QjBetween by instruct point Qi+1,Qi+2,…,Qj-2With Qj-1The broken line machining path composition specified,
And data point cuts arrow it is known that then data point QiWith QjBetween fitting detailed step it is as described below.
(1) construction is with P0=Qi, P1And P2=QjThe matched curve put in order to control makes it distinguish interpolation in Qk(k=i+
1,...,j-1).This will generate intermediate weight wkAnd corresponding shoulder point coordinates sk=wk/(1+wk), k=i+1 ..., j-1；
(2) by skIt averages to obtain the shoulder point coordinates of approximating curve, i.e.,
Weights among then are w=s/ (1-s)；
(3) according to P0, P1, P2Data point Q can be determined with wiWith QjBetween matched curve.
4. data point cuts the calculating of arrow
Due in nc program not provide instruction point at tangent vector, but can be by data point and it around
Four instruction points calculate the tangent vector at data point.As shown in figure 5, QiFor data point, Qi-2、Qi-1、Qi+1、Qi+2For it week
The four instruction points enclosed, QiUnit cut arrow expression formula it is as follows
Since parameter u being not used in calculation formula, required unit, which cuts arrow, can only be counted as cutting the direction of arrow.Under utilization
Formula, you can obtain T0, T1And Tn-1, Tn。
q0=2q1-q2,q-1=2q0-q1
qn+2=2qn+1-qn,qn+1=2qn-qn-1
5. the control of fitting precision
Although curved section passes through data point QiWith Qj, but cannot be guaranteed that it arrives QiWith QjBetween all instructions point distance
All meet maximum error of fitting.Therefore, by QiWith QjBetween all instructions spot projection on the curve of fitting, check theirs
Whether deviation is all in the error range of permission, i.e., the largest contours error delta less than or equal to defaultcIf meet error
It is required that carry out next section of curve matching；Otherwise, the maximum instruction point of deviation is arranged to new data point, with new offset
Point re-starts curve matching, examines fitting precision again, repeats this process, until meeting error requirements.
6. the identification of elliptic arc and the conversion of geometric format
Since the quadratic rational Bézier of standard type is only there are one weight factor, ability to express is than non-rational form
Bézier curve is strong, can express a variety of curves, when | P0P1|！=| P1P2| and 0 ＜ w1During ＜ 1, then expressed curve is ellipse
Circular arc.
If Ci(u) the standard type quadratic rational Bézier for being certain section in machining path, and it is ellipse corresponding to one
Arc section, as shown in fig. 6, according to Ci(u) the starting point coordinate p_start of the ellipse segmental arc and terminal point coordinate p_end can be obtained.By
The relation of elliptical standard type Quadratic Rational B é zier forms and its geometric format understands, the centre coordinate of the ellipse segmental arc is,
P1+ε(S+T)
The long and short radius of the ellipse segmental arc is respectively,
Assuming that λ2＞ λ1＞ 0, and be the root of following quadratic equation,
2δλ2- (+4 β of k η) λ+2 (k-1)=0
δ=| S × T |2, η=| S-T |2, β=ST
Two points on elliptic arc segment length's axis are,
Q1=P1+(ε+r1x0)S+(ε+r1y0)T
Q2=P1+(ε-r1x0)S+(ε-r1y0)T
According to Q1、Q2, the slope k l of long axis and the angle d_kl of long axis and x-axis positive axis can be obtained；Using long axis as x '
Axis, short axle establish local coordinate system for y ' axis, utilize Q1、Q2, centre coordinate, oval segmental arc starting point coordinate and elliptic arc segment endpoint
Coordinate acquires oval segmental arc starting point and is respectively relative to local coordinate system x ' with the line of centres, elliptic arc segment endpoint and the line of centres
Start angle d_start, the terminal angle d_end of positive axis and the direction turn of elliptic arc；x0、y0For oval segmental arc starting point
Coordinate.
After continuous mini line segment machining area completes the identification of elliptic arc and geometric format is converted, an ellipse can be obtained
Segmental arc array Ellipse_Arcs [], wherein the data structure of oval segmental arc is as follows：
7. the merging of elliptic arc
In order to improve the efficiency and precision of elliptic arc processing, the oval segmental arc in continuous mini line segment machining area is closed
And if adjacent oval segmental arc have identical center center, major radius r_l, short radius r_s, direction turn, long axis with
The angle d_kl of x positive axis, and the terminal point coordinate p_end of upper one oval segmental arci-1With the starting point coordinate p_ of next oval segmental arc
startiEqual, then they belong to continuous ellipse segmental arc on same elliptic arc, by the terminal angle for changing upper one oval segmental arc
Spend d_endi-1With terminal point coordinate p_endi-1They are merged into an oval segmental arc.It repeats the above steps, until oval segmental arc
It cannot remerge.
8. the interpolation of elliptic arc
The calculating of 8.1 oval arc length
Assuming that elliptic equation isWherein a is major semiaxis radius, and b is semi-minor axis radius, as shown in fig. 7, will be ellipse
Circle is divided into four regions, and each region corresponds to different parameters as variable.
By taking first quartile y is variable as an example,Then the arc length in the region between any two points is,
Wherein yi、yi+1It represents to require to appoint in the region respectively
The meaning starting point abscissa of point-to-point transmission arc length and terminal abscissa.
According to Gauss Legendre's quadrature formula, above formula can be converted to,
Other quadrant x or y are the situation of variable, and the derivation method of arc length is identical.
The relation of 8.2 arc length and variable
Using first quartile y as variable, from point E0(a, 0) arrives pointElliptic arc exemplified by, arc
Length is as follows with the relation solution procedure of variable：
(1) in the section of variable yIt is interior, take two trisection point y1And y2。
(2) y is calculated according to 8.1 arc length formula respectivelyi+1=0, y1、y2WithTo the starting point y of elliptic arci=0
Arc length, s0、s1、s2、s3。
(3) by (s0,0)、(s1,y1)、(s2,y2) andIt substitutes into following cubic polynomial into row interpolation,
Polynomial coefficient is acquired to get to the relation of arc length and variable.
Y=a0+a1s+a2s2+a3s3, s ∈ [s0,s3]
Wherein, a0~a3For coefficient.Other quadrant x or y are the derivation method of the situation of variable, arc length and variable relation
It is identical.
The interpolation of 8.3 elliptic arcs
The elliptic arc in Machining Path can be identified by the above process, and represented with the form of geometry.
For the oval segmental arc identified, it is divided according to four regions, and calculate respectively in each area arc length with
Then the relation of variable calculates the interpolation point coordinates (x of i-th of interpolation cycle using these arc length and the relation of variablei′,
yi'), it is as follows shown.
It is assumed that interpolation cycle is T, P0′(x′i-1,y′i-1) and Si-1The interpolation point coordinates of respectively (i-1)-th interpolation cycle
With arc length, the feed speed of i-th of interpolation cycle is vi-1, then arc length increment is Δ S=vi-1T, the arc length of i-th of interpolation cycle
For Si=Si-1+ Δ S, by SiIt substitutes into arc length and the relation of variable, obtains to dependent variable xi' or yi' value, then substituted into ellipse
In equation of a circle, so as to acquire the interpolation point coordinates P ' (x of i-th of interpolation cyclei′,yi′)。
Claims (6)
1. a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing, it is characterised in that comprise the steps of:
Identify continuous mini line segment machining area；
Data point is chosen in continuous mini line segment machining area and it is fitted to obtain quadratic rational Bézier；
Elliptic arc and its geometric format conversion are identified according to curvilinear characteristic；
Same elliptical adjacent ellipses arc will be belonged to merge, obtain interpolation curve；
Elliptic arc interpolation is carried out on interpolation curve.
2. a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing according to claim 1, special
Sign is that described be fitted comprises the following steps：
2-1) ask for the weight of quadratic rational Bézier；
The matched curve of Quadratic Rational B é zier 2-2) is obtained by averaged according to weight.
3. a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing according to claim 2, special
Sign is that the weight is obtained by following formula：
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Wherein, w1For weight, P0Headed by data point, P2For last data point, P1In order to control point, P for instruction point, u P0Q and QP2's
Ratio, Q are with P1For projection centre, straightway [P0 P2] project to the subpoints of Quadratic Rational B é zier matched curves.
4. a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing according to claim 2, special
Sign is that the matched curve for obtaining Quadratic Rational B é zier by averaged according to weight comprises the following steps：
4-1) obtain asking for the corresponding weight w of quadratic rational Bézier according to the formula of asking for of weightkAnd corresponding shoulder point is sat
Mark sk=wk/(1+wk), k=i+1 ..., j-1；
4-2) by skIt averages to obtain average shoulder point coordinates s and average weight w=s/ (1-s)；
4-3) according to P0, P1, P2Data point Q is determined with wiWith QjBetween Quadratic Rational B é zier matched curve.
5. a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing according to claim 1, special
Sign is that described converted according to curvilinear characteristic identification elliptic arc and its geometric format comprises the following steps：
5-1) as | P0P1|！=| P1P2| and 0 ＜ w1During ＜ 1, then the curve is elliptic arc；
5-2) geological information of elliptic arc is obtained by following steps：
Ci(u) the standard type quadratic rational Bézier for being certain section in machining path, and it corresponds to an oval segmental arc, root
According to Ci(u) the starting point coordinate p_start of the ellipse segmental arc and terminal point coordinate p_end can be obtained；The centre coordinate of the ellipse segmental arc
For,
P1+ε(S+T)
S=P0-P1, T=P2-P1,
The long and short radius of the ellipse segmental arc is respectively,
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>=</mo>
<msqrt>
<mfrac>
<mi>&epsiv;</mi>
<msub>
<mi>&lambda;</mi>
<mn>1</mn>
</msub>
</mfrac>
</msqrt>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<msqrt>
<mfrac>
<mi>&epsiv;</mi>
<msub>
<mi>&lambda;</mi>
<mn>2</mn>
</msub>
</mfrac>
</msqrt>
</mrow>
Assuming that λ2＞ λ1＞ 0, and be the root of following quadratic equation,
2δλ2- (+4 β of k η) λ+2 (k-1)=0
δ=| S × T |2, η=| S-T |2, β=ST
Two points on elliptic arc segment length's axis are,
Q1=P1+(ε+r1x0)S+(ε+r1y0)T
Q2=P1+(ε-r1x0)S+(ε-r1y0)T
According to Q1、Q2, the slope k l of long axis and the angle d_kl of long axis and x-axis positive axis can be obtained；It is short using long axis as x ' axis
Axis establishes local coordinate system for y ' axis, utilizes Q1、Q2, centre coordinate, oval segmental arc starting point coordinate and elliptic arc segment endpoint coordinate,
It acquires oval segmental arc starting point and is respectively relative to local coordinate system x ' positive axis with the line of centres, elliptic arc segment endpoint and the line of centres
Start angle d_start, terminal angle d_end and elliptic arc direction turn.
6. a kind of elliptic arc smooth compression interpolation algorithm suitable for high speed and high precision processing according to claim 5, special
Sign is after elliptic arc and its geometric format conversion end is identified according to curvilinear characteristic, to obtain an elliptic arc hop count group
Ellipse_Arcs [], the structure of each data is as follows in array：
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