CN107450472B - The method for realizing cutter path parameter arc length based on three bezier curve interpolation - Google Patents
The method for realizing cutter path parameter arc length based on three bezier curve interpolation 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
- G05B19/19—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 positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path
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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
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Abstract
The invention discloses a kind of methods for realizing cutter path parameter arc length based on three bezier curve interpolation, comprising the following steps: (1) is fitted in advance based on three bezier curve；(2) interpolation curve is converted into a whole B-spline curves, whole B-spline curves is carried out to wait parameter samplings；(3) arc length parameters for calculating sampled point, are established using arc length parameters as the objective function of variable；(4) objective function is solved using ELSPIA algorithm, obtains the B-spline of approximation parameters arc length, and B-spline meets the processing request of chord error constraint, guarantor's type constraint and less control point simultaneously；The present invention improves the efficiency of cutter path B-spline fitting algorithm；Parameter arc length is realized, and initial spline curve meets chord error constraint and the constraint of guarantor's type, reduces velocity perturbation present in processing, is effectively improved the shape defect and the ungratified phenomenon of error of cutter path.
Description
Technical field
The invention belongs to computer-aided manufacturing and Computerized Numerical Control processing technology fields, more particularly, to one kind based on three times
The method of Bezier curve interpolation realization cutter path parameter arc length.
Background technique
Currently, the cutter path of numerical control processing is usually expressed with small line segment, small line segment is by CAM software according to mismachining tolerance
By the discrete acquisition of original surface.The cutter path that small line segment indicates has the disadvantage in that (1) small line segment enormous amount, data are deposited
Storage and transmission quantity are big；(2) cutter path only has G0 continuity, and G1, G2 are discontinuous, and system continually acceleration and deceleration easily cause
Machine vibration reduces the machining accuracy and surface quality of part；(3) tool-path smoothing is poor, parts profile machining accuracy and
Surface quality is poor.The cutter path that small line segment indicates is difficult to meet the high-precision process requirements of High-speed Computer number control, therefore, actual processing
In, it will usually the cutter path that small line segment indicates is fitted using the better parameter curve of geometric continuity.Geometric continuity
Is defined as: 1. G0 is continuous, i.e., two sections of curves are connected to same point；2. G1 is continuous, i.e. tangential direction of the two sections of curves in junction
It is identical；3. G2 is continuous, i.e. buckling vector of the two sections of curves in junction is identical.
The characteristic that B-spline curves are realized with its versatility and easily is widely used in cutter path fitting.B-spline by
Control point and knot vector can define, and indicate that cutter path can simplify G code with it, reduce data storage capacity.Furthermore B sample
Item itself has higher continuity, and there is better fairness in the track after fitting, and without legacy data point, has preferable
Noise suppression effect；Furthermore the track that relatively primitive small line segment indicates, the track after fitting is elongated, is suitble to High-speed machining, such as
Siemens 840D realizes the real-time interpolation of spline curve, effectively simplifies machining code, improves processing quality.
It is non-linear relation between curve arc long and parameter when spline interpolation.This non-linear relation makes real-time interpolation
In be difficult to efficiently calculate next interpolation parameters, to generate velocity perturbation, influence processing efficiency.If spline curve is with arc length
For parameter, parameter arc length is realized, then quickly can accurately obtain next interpolation according to linear relationship in real-time interpolation
Parameter avoids velocity perturbation, improves real-time interpolation efficiency.
The parameter and arc length of spline curve do not have accurate analytical expression, currently used interpolation parameters calculation method packet
Include the Taylor method of development, the numerical method of the differential equation, iterative approximation and parameter arc length fitting process.
The main solution of parameter arc length is as follows: the cutter path of given nonparametric arc length is converted into parameter
The B-spline cutter path of arc length；The B-spline cutter path of a nonparametric arc length is resolved into several Bezier first
Then line segment samples these Bezier line segments and calculates the arc length of sampled point, is finally fitted these using least square method and adopts
Sampling point obtains the B-spline curves of a parameter arc length, but this method does not consider the action error of spline curve
(chord error)。
In addition, still an alternative is that constructing several Bezier in every two adjacent data points using local interpolation algorithm
Then curve is converted into the controlling curve of a parameter arc length, finally use least square approximation method construct one
New B-spline (approximating curve) approaches controlling curve.This method considers the approximate error of approximating curve and controlling curve, but
Due to not considering the error of controlling curve and raw data points, do not ensure that the error of approximating curve and legacy data point meets
Processing request.
" Arc-length parameterized spline curves for real-time simulation " text
The segmentation arc length for calculating the cubic B-spline of input is disclosed, several is then looked for equally distributed to adopt on batten according to arc length
Sampling point, and by the parameter of dichotomy calculating sampled point, it finally constructs an interpolation curve and passes through these sampled points；But this method
The error between input curve and interpolation curve is not considered.
Summary of the invention
Aiming at the above defects or improvement requirements of the prior art, the present invention provides one kind to be based on three bezier curve
The method that interpolation realizes cutter path parameter arc length, its object is to obtain the B-spline of an approximation parameters arc length, and B
Batten meets the processing request of chord error constraint, guarantor's type constraint and less control point simultaneously.
To achieve the above object, according to one aspect of the present invention, it provides a kind of based on cubic Bezier curve interpolation
The method for realizing cutter path parameter arc length, which comprises the steps of:
(1) local interpolation is carried out to adjacent data point using three bezier curve according to discrete cutter path, if obtaining
Dry G2 is continuous and meets the cubic Bezier curve of chord error constraint and the constraint of guarantor's type；Interpolation curve is each consecutive number
The set of Bezier curve between strong point；
(2) interpolation curve is converted into a whole B-spline curves, whole B-spline curves is carried out to wait parameter samplings:
(3) arc length parameters for calculating sampled point, are established using arc length parameters as the objective function of variable；
(4) objective function is solved using ELSPIA algorithm, obtains the B-spline of approximation parameters arc length, and B-spline is full simultaneously
Sufficient chord error constraint, the processing request of guarantor's type constraint and less control point；
Preferably, step (1) includes following sub-step:
S11, the two adjacent data point Q that interpolation will be participated in0,Q1As the first and last control point of three bezier curve interpolation, lead to
It crosses Renner method and obtains data point Q0,Q1The unit tangent vector T at place0,T1；Wherein, first control point b0=Q0, last control point b3=Q1；
S12, pass through the continuous condition of G1 at the parameter expression and endpoint of three bezier curve, acquisition control point { b0,
b1,b2,b3, data point Q0,Q1, unit tangent vector T0,T1Relationship it is as follows:
Wherein, l0,l1Refer to that the mould of endpoint tangent vector is long；
S13, the long l of mould for determining endpoint tangent vector is constrained according to chord error constraint, the constraint of guarantor's type, fairness0,l1；
S14, according to the long l of mould0,l1Obtain control point b1,b2, establish data point Q0,Q1Between cubic Bezier
Curve P (t), and obtain the Bezier curve between all consecutive number strong points.
Preferably, step (2) includes following sub-step:
S21, the B-spline that interpolation curve is converted to an entirety；For data point Q0,Q1Between cubic Bezier it is bent
Line P (t), enabling the knot vector of B-spline is U=[0,0,0,0,1,1,1,1], control point b0,b1,b2,b3, by Q0,Q1Between
Bezier curve P (t) be converted to B-spline, and all Piecewise Bezier Curves can be converted to segmentation cubic B-spline；
Using data point parameter as node, and the multiplicity of interior nodes is 3, obtains an interpolation in the first of all data points
Beginning B-spline c (t), the control point of initial B-spline are b0,b1,b2,b3…,bn, data point are as follows:Wherein b3k=Qk, k
=0,1,2,3 ... m；N=3m；
S22, to parameter samplings such as whole B-splines；
For node interval [ts,te], sampled point parameter
Wherein, M is that the number of sampled point subtracts 1, and M=3* (m+1), m are that the number of data point subtracts 1；
If not having sampled point parameter in node interval, the median of node interval is inserted into as new sampled point parameter；
If the parameter of sampled point isBy the defined formula of B-spline, sampled point is obtained
Preferably, step (3) includes following sub-step:
S31, the arc length parameters for calculating sampled point；
Two neighboring sampled point CiAnd Ci+1Between arc length
Numerical integration method based on Bool formula calculates arc length:
x0=ti, x4=ti+1,xi+1=xi+ h ..., f (x)=| c ' (t) |, fi=f (xi), i=
0,1,..4；
Total arc length of total arc length of initial spline curveThe arc length parameters of sampled point s0=0；
S32, it establishes objective function and solves fitting B-spline c (ss), so that ssj
Indicate data point arc length parameters, j=1,2,3 ... ..m.
In general, through the invention it is contemplated above technical scheme is compared with the prior art, can obtain down and show
Beneficial effect:
(1) method provided by the invention for realizing cutter path parameter arc length based on three bezier curve interpolation is right
The parameter samplings such as whole B-spline, improve the efficiency of cutter path B-spline fitting algorithm；
(2) method provided by the invention for realizing cutter path parameter arc length based on three bezier curve interpolation, builds
Linear relationship between vertical curve arc long and parameter, and initial spline curve meets chord error constraint and the constraint of guarantor's type, subtracts
Velocity perturbation present in few processing, is effectively improved the shape defect and the ungratified phenomenon of error of cutter path.
Detailed description of the invention
Fig. 1 is the method for realizing cutter path parameter arc length based on three bezier curve interpolation that embodiment provides
Flow chart；
Fig. 2 is the Bezier curve under chord error constraint；
Fig. 3, which is that tangent vector length is excessive, causes Bezier curve to form loop schematic diagram；
Fig. 4 is the upper bound schematic diagram of Bezier curve guarantor's type.
Specific embodiment
In order to make the objectives, technical solutions, and advantages of the present invention clearer, with reference to the accompanying drawings and embodiments, right
The present invention is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, and
It is not used in the restriction present invention.As long as in addition, technical characteristic involved in the various embodiments of the present invention described below
Not constituting a conflict with each other can be combined with each other.
To achieve the above object, it provides a kind of based on three bezier curve interpolation realization cutter path parameter arc length
Method, process is as shown in Figure 1；Specifically includes the following steps:
(1) the pre- fitting based on three bezier curve, specifically: it is bent using cubic Bezier according to discrete cutter path
Line carries out local interpolation to adjacent data point, and it is continuous and meet chord error constraint and the constraint of guarantor's type to obtain several G2
Three bezier curve；Interpolation curve is the set of Bezier curve between each consecutive number strong point；Including following sub-step:
S11, the two adjacent data point Q that interpolation will be participated in0,Q1As the first and last control point of three bezier curve interpolation, lead to
Renner method is crossed, Q is calculated0,Q1The unit tangent vector T at place0,T1；Wherein, first control point b0=Q0, last control point b3=Q1；
S12, by the continuous condition of G1 at the parameter expression and endpoint of three bezier curve, ask
Control point { b out0,b1,b2,b3, data point Q0,Q1, unit tangent vector T0,T1Relationship it is as follows:
Wherein, l0,l1Indicate that the mould of endpoint tangent vector is long；
S13、l0,l1Value influence three bezier curve quality, if l0,l1It is unreasonable that value is set, may make
At loop and chord error beyond undesirable curve shapes such as errors；
Particular by chord error constraint, the constraint of guarantor's type, fairness constraint, l is determined0,l1Length, including it is as follows
Sub-step:
S131, chord error constraint；As shown in Fig. 2, θ1,θ2Respectively T0,T1With Q0Q1Angle；According to three times
The property of Bezier curve convex closure, if control point is located at Q0Q1For axis, mismachining tolerance δ is in the cylinder of radius, then B-spline
Chord error meet the requirements；
To guarantee that Bezier curve meets chord error constraint, l0,l1It should meet:
S132, the constraint of guarantor's type；Formula (2) only constrains chord error, and there is no the shapes for considering batten.Certain
In the case of, even if long using the mould of endpoint tangent vector determined by (2) formula, batten is also possible to have undesirable shape, such as
Shown in Fig. 3.
In order to avoid unnecessary shape defect, taking a guarantee is not in the mould of cusp and the endpoint tangent vector of loop
It is long, allow control point b1,b2It is overlapped, as shown in figure 4, obtaining l0,l1It is as follows:
S133, fairness constraint；Tensile energy EstretchWith bending energy EbendIt is the long l of mould of endpoint tangent vector0,l1
Function；
To make energy term reach minimum, tensile energy EstretchWhen minimum, l0,l1Are as follows:
Bending energy EstretchWhen minimum, l0,l1Are as follows:
Comprehensive chord error constraint, the constraint of guarantor's type and fairness requirement, according to formula (2) (3) (6) (7), obtain endpoint
The long l of the mould of tangent vector0,l1It is as follows:
S14, control point b is calculated according to formula (1)1,b2, establish data point Q0,Q1Between cubic Bezier curve P (t),
And the three bezier curve between all consecutive number strong points；Interpolation curve is the set of all Bezier curves, initial B
The control point of batten is b0,b1,b2,b3…,bn, data point are as follows:Wherein b3k=Qk, k=0,1,2,3 ... m；N=
3m；
The control point number for the curve that step (1) is obtained using interpolation method is 2~3 times of data point number, cause compared with
Big amount of storage, and only G1 is continuous at data point for interpolation curve；In order to reduce number of control points, the matter of matched curve is improved
Amount, guarantees that sampled point is uniformly distributed, interpolation curve is converted into an entirety B-spline curves c (t) first, then to whole B-spline
Curve c (t) carries out equal parameter samplings, and sampled point includes former data point.
(2) interpolation curve is converted into a whole B-spline curves, whole B-spline curves is carried out to wait parameter samplings, packet
Include following sub-step:
S21, the B-spline that interpolation curve is converted to an entirety；Interpolation curve is Bezier between each consecutive number strong point
The set of curve.
For data point Q0,Q1Between three bezier curve P (t), enable B-spline knot vector be U=[0,0,0,
0,1,1,1,1], control point b0,b1,b2,b3, then can be by Q0,Q1Between Bezier curve P (t) be converted to B-spline,
All Piecewise Bezier Curves can be similarly converted to segmentation B batten three times.
When having multiplicity in B-spline domain is the node of p, p B-spline interpolation is in corresponding control point.Due to
Bezier curve local interpolation, all data points belong to control point, therefore enable data point parameter as node, and interior nodes
Multiplicity is 3, and an available interpolation is b in the initial B-spline c (t) of all data points, the control point of initial B-spline0,
b1,b2,b3…,bn, data point are as follows:Wherein b3k=Qk, k=0,1,2,3 ... m；N=3m.
S22, to parameter samplings such as whole B-splines；
In the partial circulating of ELSPIA fitting algorithm, control point is the adjustment vector being made up of the difference vector of data point
It updates.Sampling density influences the efficiency of ELSPIA fitting algorithm, and the very few adjustment vector for control point of number of sampling points is made
With little；Number of sampling points excessively causes computationally intensive, and ELSPIA Fitting efficiency is low.
In embodiment, at 2~3 times of the number of sampled point access strong point number, ELSPIA Fitting efficiency can be obtained effectively
Improve.
For node interval [ts,te], sampled point parameter can calculate as follows:
Wherein, M is that the number of sampled point subtracts 1, and M=3* (m+1), m are that the number of data point subtracts 1；
To guarantee that sampled point includes former data point, using former data point parameter as new sampled point parameter；It is adjacent to guarantee
At least one sampled point between two nodes is inserted into the median of node interval if not having sampled point parameter in node interval
As new sampled point parameter.If the parameter of sampled point isBy the defined formula of B-spline, available sampled point
(3) arc length parameters for calculating sampled point, are established using arc length parameters as the objective function of variable:
S31, the arc length parameters for calculating sampled point；
Two neighboring sampled point CiAnd Ci+1Between arc length liIt can calculate as follows:
Wherein c ' (t) indicates the derivative of initial batten；
Without analytic solutions when calculating arc length due to three bezier curve, the numerical integration side based on Bool formula is utilized
Method calculates arc length:
x0=ti,x4=ti+1,xi+1=xi+ h ... f (x)=| c ' (t) |, fi=f (xi), i=0,
1,..4
Total arc length of total arc length of initial spline curveThe arc length parameters of sampled pointIt calculates public
Formula is as follows:
S32, objective function solution fitting B-spline c (ss) is established, so that
ssjIndicate data point arc length parameters, j=1,2,3 ... ..m.
(4) objective function is solved using ELSPIA algorithm, obtains the B-spline of approximation parameters arc length, and export.
ELSPIA fitting algorithm includes partial circulating, middle circulation, systemic circulation and chord error refinement algorithm；
Partial circulating includes LSPIA fitting algorithm, in be cyclically updated foot point parameter, control point is inserted into systemic circulation；Three circulation controls
Data point tolerance processed meets mismachining tolerance requirement, and chord error refinement algorithm controls the chord of spline curve
Error meets mismachining tolerance requirement；
When being fitted using ELSPIA, do not update foot Point parameter, only check former data point data error and
Chord error updates control point using sampled point in partial circulating.
As it will be easily appreciated by one skilled in the art that the foregoing is merely illustrative of the preferred embodiments of the present invention, not to
The limitation present invention, any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should all include
Within protection scope of the present invention.
Claims (4)
1. a kind of method for realizing cutter path parameter arc length based on three bezier curve interpolation, which is characterized in that including
Following steps:
(1) local interpolation is carried out to adjacent data point using three bezier curve according to discrete cutter path, obtains several
G2 is continuous and meets the three bezier curve of chord error constraint and the constraint of guarantor's type；Interpolation curve is each consecutive number strong point
Between Bezier curve set；
(2) interpolation curve is converted into a whole B-spline curves, whole B-spline curves is carried out to wait parameter samplings；
(3) arc length parameters for calculating sampled point, are established using arc length parameters as the objective function of variable；
(4) objective function is solved using ELSPIA algorithm, obtains the B-spline of approximation parameters arc length, and B-spline meets simultaneously
Chord error constraint, the processing request of guarantor's type constraint and control point.
2. the method that cutter path parameter arc length is realized based on three bezier curve interpolation as described in claim 1,
It is characterized in that, step (1) includes following sub-step:
S11, the two adjacent data point Q that interpolation will be participated in0, Q1As the first and last control point of three bezier curve, pass through Renner
Method obtains data point Q0, Q1The unit tangent vector T at place0, T1；Wherein, first control point b0=Q0, last control point b3=Q1；
S12, pass through the continuous condition of G1 at the parameter expression and endpoint of three bezier curve, acquisition control point { b0, b1,
b2, b3, data point Q0, Q1, unit tangent vector T0, T1Relationship it is as follows:
Wherein, l0, l1Refer to that the mould of endpoint tangent vector is long；
S13, the long l of mould for determining endpoint tangent vector is constrained according to chord error constraint, the constraint of guarantor's type, fairness0, l1；
S14, according to the long l of mould0, l1Obtain control point b1, b2, establish data point Q0, Q1Between three bezier curve P
(t), and the three bezier curve between all consecutive number strong points is obtained.
3. the method that cutter path parameter arc length is realized based on three bezier curve interpolation as claimed in claim 2,
It is characterized in that, step (2) includes following sub-step:
S21, the B-spline that interpolation curve is converted to an entirety；For data point Q0, Q1Between three bezier curve P
(t), the knot vector for enabling B-spline is U=[0,0,0,0,1,1,1,1], control point b0, b1, b2, b3, by Q0, Q1Between
Bezier curve P (t) is converted to B-spline, and all segmentation three bezier curves can be converted to segmentation cubic B-spline；
Using data point parameter as node, and the multiplicity of interior nodes is 3, obtains an interpolation in the initial B of all data points
Batten c (t), the control point of initial B-spline are b0, b1, b2, b3..., bn, data point are as follows:Wherein b3k=Qk, k=0,
1,2,3 ... m；N=3m；
S22, to parameter samplings such as whole B-splines；
For node interval [ts, te], sampled point parameter
Wherein, M is that the number of sampled point subtracts 1, and M=3* (m+1), m are that the number of data point subtracts 1；
If not having sampled point parameter in node interval, the median of node interval is inserted into as new sampled point parameter；
If the parameter of sampled point isBy the defined formula of B-spline, sampled point is obtained
4. the method that cutter path parameter arc length is realized based on three bezier curve interpolation as claimed in claim 3,
It is characterized in that, its step (3) includes following sub-step:
S31, the arc length parameters for calculating sampled point；
Two neighboring sampled point CiAnd Ci+1Between arc length
Numerical integration method based on Bool formula calculates arc length:
F (x)=| c ' (t) |, fi=f (xi), i=0,1 ..4；
Total arc length of total arc length of initial spline curveThe arc length parameters of sampled point
S32, it establishes objective function and solves fitting B-spline c (ss), so that ssjIt indicates
Data point arc length parameters, j=1,2,3.....m.
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