CN110516388A - Surface tessellation point cloud model ring cutting knife rail generating method based on reconciliation mapping - Google Patents
Surface tessellation point cloud model ring cutting knife rail generating method based on reconciliation mapping Download PDFInfo
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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
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- 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
Abstract
The present invention is based on the Surface tessellation point cloud model ring cutting knife rail generating methods of reconciliation mapping to belong to Free-form Surface Parts precision processing technology field, is related to a kind of Surface tessellation point cloud model ring cutting knife rail generating method based on reconciliation mapping.This method carries out triangular mesh generation using indirect subdivision method combination Delaunay algorithm is parameterized, to spatial spreading point cloud model.Based on reconciliation mapping method, three-dimensional space grid is flattened to two-dimensional surface with energy deflection minimum constraint, while the deflection that each tri patch generates in reconciliation mapping process is measured using mapping drawing coefficient.Endless knife rail is generated by calculating Triangular meshes method resultant curvature and cutting step length and line-spacing in the two-dimensional surface being converted to.Finally, three-dimensional space is returned into knife rail inverse mapping and generates machining locus.This method generates the circular cutter rail of Surface tessellation point cloud model by the method for reconciliation mapping, improves processing quality and processing efficiency, and this method is suitable for free form surface part Precision Machining.
Description
Technical field
The invention belongs to Free-form Surface Parts precision processing technology fields, are related to a kind of Surface tessellation based on reconciliation mapping
Point cloud model ring cutting knife rail generating method.
Background technique
With the fast development of digital measuring technique, reverse-engineering has been widely used for the numerical control processing of free form surface
In.But since free form surface class workpiece can not be expressed by specific parametric equation, and usually there is complicated geometrical characteristic,
Knife rail directly cannot be generated by traditional numerical control modeling method, therefore the discrete point cloud based on free form surface realizes high-precision high-efficiency
Tool-path planning be still the hot spot studied in computer-aided manufacturing.Cutter path rule are carried out to free form surface discrete point cloud
The research drawn is concentrated mainly on three aspects: first is that carrying out proper treatment to point cloud model directly generates cutter path, but phase
The geometry of pass calculates extremely complex；Second is that cutter path is planned based on reconstructed surface, however the process for putting cloud reconstructed surface needs
A series of processing such as feature extraction, area dividing, surface fitting and cutting, and reconstructed surface and original point cloud are carried out to cloud
There is a certain error；Third is that converting triangle mesh curved surface for point cloud model, and through dimension-reduction treatment in two-dimensional parameter domain
Plan knife rail, this method avoid complexity in three dimensions, and friendship to be asked to calculate, but some problems are still unresolved at present, such as plan
Cutter path there are wedge angle, mapped boundaries, there are deformation distortions etc..Therefore have studied it is a kind of based on reconcile mapping curved surface from
Scatterplot cloud model ring cutting knife rail generating method considers the stretcher strain generated in reconciliation mapping process, by free form surface discrete point
Cloud data generate high-precision processing knife rail, realize the Precision Machining of free form surface class workpiece.
Feng Yixiong et al. patent " grid free form surface annular cutter trajectory planning based on improved B utterfly subdivision
Method ", patent publication No. CN105739432B.The patent passes through the contour parallel milling model of building triangle gridding free form surface, with
Model boundary inwardly biases for initial tool track and forms screw type cutter path, however this method still needs to the space knife to biasing
Rail carries out that friendship is asked to calculate.Technical literature " Iso-parametric tool-path planning for point clouds ",
Zou Qiang etc., Computer-Aided Design, 2013,45 (11): 1459-1468, the document is used is reflected based on conformal
The mesh parameterization method penetrated, however there are conformal distortion for the cutter path of this method generation.
Summary of the invention
The present invention in view of the drawbacks of the prior art, has invented a kind of Surface tessellation point cloud model ring cutting based on reconciliation mapping
Knife rail generating method.This method first carries out spatial point cloud model using the indirect subdivision method combination Delaunay algorithm of parametrization
Triangular mesh generation realizes space lattice and parameter field by reconciling the method mapped for processing curve dimensionality reduction to two-dimensional surface
The mapping of net boundary vertex and internal vertex.Then, machining locus is planned in parameter field, finally by machining locus inverse mapping
Three-dimensional space is returned, the numerical control machining knife rail of free form surface is generated.The method of dimension-reduction treatment avoids complexity in three dimensions
The deformation asked friendship to calculate, substantially increase computational efficiency, and be introduced into mapping drawing coefficient analysis reconciliation mapping, reduces net
Distortion in lattice parameterized procedure, the high-precision knife track gauge for realizing free form surface are drawn, and processing quality and processing efficiency are improved.
The technical solution adopted by the present invention is that a kind of Surface tessellation point cloud model ring cutting cutter orbit making based on reconciliation mapping
Method, which is characterized in that this method is first using the indirect subdivision method combination Delaunay algorithm of parametrization to spatial point cloud model
Triangular mesh generation is carried out, and has rebuild the topological relation of triangle gridding；Using the parametric method of reconciliation mapping, to three dimensional network
Lattice carry out dimension-reduction treatment, and introduce mapping drawing coefficient analysis mapping deformation；It is calculated in parameter field any in grid cell
The law vector and curvature of point, calculate cutting step length and line-spacing, generate the cutter path in parameter field；Finally, by parameter field
Knife rail inverse mapping returns three-dimensional space and generates processing knife rail；Specific step is as follows for method:
Step 1, free form surface discrete point cloud model triangular mesh generation
Triangular mesh generation is carried out to spatial point cloud model using indirect subdivision method combination Delaunay algorithm is parameterized；Its
Process are as follows: three-dimensional space point cloud is parameterized into two-dimensional surface domain first, and parametrization point cloud is carried out with boundary constraint
Delaunay triangulation network lattice subdivision；Then, the topological connection relation between plane domain intrinsic parameter point set is constructed, and it is carried out
Boundary Extraction and sequence processing；Finally, planar triangulations are mapped back three-dimensional space according to topological relation, to obtain three-dimensional
The triangular mesh generation result of spatial point cloud；
The process is related to topological relation building and point cloud boundary extraction and sort algorithm between parametrization point cloud:
1) parametrization point cloud topological relation building
The triangle topology relationship for tentatively establishing triangular mesh model first is closed based on the triangle topology tentatively established
System, is successively read grid vertex call number, and be stored in new grid vertex array；Then it is closed using the topology tentatively established
System, search include all triangular mesh units on vertex, successively find out and be stored in the grid index number group of vertex correspondence
In；Because vertex storage is random, therefore in triangular mesh unit process of the search comprising the vertex, order occur is also
Random；Right-hand rule processing is carried out to the topological relation on newly-established vertex and its neighborhood triangle number, makes vertex neighborhood
Triangle is numbered into arranged counterclockwise；
2) point cloud boundary extraction and sort algorithm
During contour parallel milling, usually select longest a line circle as initial knife rail, then grey iterative generation finally adds
Work track, therefore Boundary Extraction must be carried out to Surface tessellation point cloud model；For the triangle gridding of plane domain intrinsic parameter point cloud
For model, a triangular plate, referred to as boundary edge are only belonged to if there is certain side；If certain side belongs to two triangles
Piece, then this side is known as internal edges；The enclosure space polygon formed that joined end to end by boundary edge is known as boundary；According to above-mentioned side
All boundary edges in triangle grid model are found out in the definition on boundary side；In algorithm realization, since the side collection initially acquired is unordered
, therefore must be ranked up and organize, form end to end integral edge；Process is as follows:
Firstly, being initialized to parametrization triangle grid model boundary set array { edges }, used attribute is set as
false；Then, the line set of triangle gridding is traversed, TriCount is the quantitative attribute of triangle where side, finds one
Any side that TriCount attribute value is 1, i.e. any one boundary edge；Two endpoints on the side are respectively as Polygonal Boundary
Former point and current point find subsequent point by current point, and cyclic process is as follows:
1. traversing all of its neighbor triangle of current point, the side for meeting following condition is found:
Meet condition TriCount=1 in adjacent triangle；Used attribute value is false；With former point and current point structure
At side have and only 1 intersection point；
2. the one end not being overlapped with current point in the side is set as subsequent point, former point and current point are reset, before order
A little it is directed toward current point, current point is directed toward subsequent point, and sets true for the used attribute on the side.Step is transferred to if closure
3. being otherwise transferred to step 1.；
3. saving and exporting the boundary set of tracking result, process terminates；
Step 2, the D Triangulation dimension-reduction treatment based on reconciliation mapping
The key of reconciliation mapping method is to find an energy equation, referred to as objective function, and provides the target letter
Several boundary conditions, the extreme value for then solving objective function obtain a parametrization.Its process are as follows: for given mould to be processed
Type finds its boundary point, and boundary point is mapped to according to certain rules on the boundary that plane domain has predefined；For non-side
Boundary's point must guarantee that the elastic potential energy of the grid after mapping is minimum to guarantee that the deformation generated after model mapping can be minimum, thus
It converts mapping problems to and solves energy-minimum problem.The process be related to D Triangulation model boundary vertex mapping and it is interior
The mapping of portion vertex.
1) border vertices map
It enables space curved surface triangle gridding TM have k border vertices, is BV={ vi(i=1,2,3 ..., k), each vertex
It is to sequence sequence by its syntople.To turn to example to planar unit circle field parameter, define
V={ v1,v2,...,vmBe all vertex set, m be all vertex number, T={ t1,t2,...,tnBe
The set of all triangles, n are the number of all triangles；TM (T, V) is enabled to be mapped to plane PlOne round domain Ω: (x-x0)2
+(y-y0)2+(z-z0)2On=1, corresponding position of the grid surface boundary point on flat circle domain is determined using accumulation chord length method,
Enable i-th boundary edge L of ViTwo vertex be viAnd vi+1, then in triangular apex set V* in the mapped its corresponding two
A vertexThe central angle constituted with flat circle domain center of circle oIt should meet:
In formula, | Li| indicate side LiLength.
2) internal vertex maps
Enabling mapping relations is ξ, ξ (vi)=(xi,yi,zi)T, wherein ξ (vi) (i=1,2,3 ..., k) it is to take border vertices
To flat circle domain mapping as a result, the elastic potential energy E (ξ) after surface mesh reconciliation mapping is indicated are as follows:
In formula, Ki,jFor vertex vi、vjThe side L constitutedi,jCoefficient of elasticity.Enable side Li,jFor 2 trianglesWithIt is common, then:
In formula,For triangleArea, Li,jIndicate vertex vi、vjThe side constituted, with such
It pushes away.
ξ(vi) (i=1,2,3 ..., be m) all vertex of space curved surface triangle gridding to flat circle domain mapping as a result, ξ
(vi) (i=1,2,3 ..., m) all in plane Pl: on ax+by+cZ+d=0, without loss of generality, c ≠ 0 is enabled, by ξ (vi)=(xi,
yi,zi)T, ξ (vj)=(xj,yj,zj)T, then have zi=-(axi+byi+ d)/c, zj=-(axj+byj+ d)/c, then energy equation (2)
It is changed into following form:
That is:
It is minimum for the deformation that occurs that the topological structure of curved surface triangle gridding TM when being mapped on flat circle domain, then energy
Equation (5) will reach minimum value, should meet:
Wherein, k is the border vertices number of space curved surface triangle gridding TM, and m is all number of vertex；
Enable N (vi) be and vertex viAdjacent vertex set, and vj∈N(vi).Inclined is asked to each internal vertex in (5) formula
It leads, obtains a system of linear equations:
Formula (7) are arranged and are write as matrix form, obtaining a size is 2 (n-k) × 2 (n-k), with linear coefficient matrix A
For the equation group of coefficient:
AX=B (8)
Wherein:
Space curved surface triangle gridding internal point is obtained with over-relaxation iterative method solution Sparse positive definite linear syfem of equations (9) to reflect
It is mapped to the corresponding coordinate in plane parameter domain.
There are very important stretcher strains during space curved surface triangle gridding reconciliation mapping parametersization, to join in plane
Accurate trajectory planning is carried out on number grid, it is necessary to consider that parametric grid is mapped to the stretcher strain amount of three-dimensional grid generation.Net
Lattice surface parameterization has the characteristics that fragment linearity, i.e. the parameterized results ξ of space lattice curved surface are to connect for each triangular plate
Continuous and linear, the affine transformation relationship of existence anduniquess between each pair of space triangular piece and parameter field triangular plate, therefore each triangle
Piece inside gradient and mapping drawing coefficient are constant；
Stretch measurement L2(T*) root mean square that characterizes all directions stretcher strain in triangular plate, using stretching measurement to close
It is estimated like isotropic mapping drawing coefficient σ, gives three vertex vs of space triangular piece Ti、vj、vkAnd its it is corresponding
Parameter field triangular plate T*Three vertexThe then gradient in this triangular plate are as follows:
In formula, S is parameter field triangular plate T*Area.
S=((u1-u0)(v2-v0)-(u2-u0)(v1-v0))/2 (11)
Jacobian matrix [ξu,ξv] minimax singular value be respectively as follows:
Coefficient a in formula1, a2, a3Are as follows:
Then triangular plate mapping drawing coefficient σ may be expressed as:
Given parameters domain triangular plate T*Interior initial track point e*, according to space triangular piece T and parameter field triangular plate T*Between
Affine transformation relationship, acquire corresponding tracing point e in T using area coordinate or secondary weighted method, need at this time calculate e point
The trajectory parameters at place simultaneously convert it to e according to mapping drawing coefficient σ*At point；
Step 3, the inverse mapping of mapping is reconciled
The liter dimension of knife rail is realized in the inverse mapping mapped that reconciles, and generates the knife rail for being used for processing space grid surface；Enable plane
Domain PlIn certain calculate cutter-contact point e*The triangular plate at place isThe triangular plate is corresponding three in grid surface TM
Cornual plate is T { vi,vj,vk, according to the principle that mapping front and back mesh triangles piece topological structure is constant, corresponding space knife can be obtained
Contact e.Affine frame is established, enabling (α, β) is point e*In affine coordinate systemUnder parameter coordinate, using two
The method of sublinear weighting calculates the coordinate of e:
E=vi+α(vj-vi)+β(vk-vi) (15)
Step 4, free form surface discrete point cloud model circular cutter track gauge is drawn
1) law vector and curvature estimation of triangle gridding
To draw convenient for subsequent knife track gauge, geometry of the calculating method resultant curvature analysis cutter-contact point on D Triangulation is special
Property.With the method for least square curved surface local fit to grid cell vertex normal vector, principal curvatures k1、k2With corresponding principal direction
d1、d2It is estimated, wherein k1For maximum principal curvatures, k2For minimum principal curvatures.Grid cell vertex is calculated further according to Euler's formula
Along the normal curvature of any direction d, calculation expression are as follows:
K (d)=k1·cos2θ+k2·sin2θ (16)
In formula, θ indicates any direction d and principal direction d1Between angle.
The method weighted using quadratic linear, by judging position of the cutter-contact point in grid cell, in conjunction with place grid
The method arrow of unit apex, curvature information calculate method arrow and curvature at cutter-contact point；
v1、v2And v3Respectively three vertex of spatial triangle grid cell,For corresponding unit
Law vector, then inside grid cell any cutter-contact point e law vectorIt indicates are as follows:
In formula, (u, v) indicates e in affine coordinate systemUnder parameter coordinate.When e is located at tri patch side
When upper, it is linear interpolation formula that formula (17), which is degenerated,.Similarly, method of the cutter-contact point inside triangle gridding unit along any direction
Curvature calculates in aforementioned manners；
2) contour parallel milling cutter path parameter calculates
The calculating of cutter path parameter includes the calculating of cutting step length and feed line-spacing.With step-length screening method planning feed step
It is long,；It enablesIt is processing cutter-contact point through ginsengs such as B-spline curves fittings
The intensive point that number is discretely formed, ε indicate known approximate error；
Given a starting pointThen in pointSelect a point in order laterThenWithBetween
Each point Pi(i=m1+1,...,m2- 1) for, it is calculated to line segmentDistance:
If meeting di< ε (i=m1+1,...,m2- 1), i.e.,WithThe line segment that two o'clock is constituted approaches the curve at place
Approximate error be less than given value, successively pick up at this timeNext point, even m2=m2+ 1, above step is repeated, until
If there is a point Pi(i∈m1+1,...,m2- 1) straightway is arrivedDistance meet di>ε.At this momentWithTwo o'clock structure
At line segment be the longest line segment for meeting approximation accuracy condition.RetainPoint, and deleteWithBetween whole
Point.It, must be by point for whole curveAs new starting point and repeat above step.
Feed line-spacing CL is calculated by formula (19):
In formula, RtIt is residual altitude, R for ball-end mill radius, hbFor radius of curvature, model takes positive sign when being convex surface, recessed
Negative sign is taken when curved surface.
Mapping drawing coefficient σ is introduced, mapping domain is calculated and corresponds to the track spacing CL at cutter-contact point*,
3) in space and parameter field circular cutter rail generation
In conjunction with the calculation method of ring cutting process tool trajectory parameters, illustrate the generation side of circular cutter rail in space and parameter field
Method:
1. the D Triangulation model boundary of extraction is fitted using uniform B-Spline three times, it may be assumed that
In formula, DiFor control point, Ni,3It (t) is cubic B-spline basic function.
Cutter-contact point on first knife rail boundary curve is discrete according to certain precision, being determined using step-length screening method,
It is mapped to parameter field according to reconciliation mapping ruler, obtains first knife rail cutter-contact point of parameter field.Updating the knife rail is previous knife
Rail.
2. calculating previous knife rail n1The line-spacing CL that a cutter-contact point biases on space lattice curved surface, the touching of correspondence mappings domain knife
The theoretical offset row of point is away from for CL*, take line-spacing minimum valueIt is practical line-spacing along mapping previous of domain knife rail normal bias, obtains
Obtain current knife rail.
3. reconciling the method for mapping inverse mapping based on step 3, the current knife rail inverse mapping in domain will be mapped and made the return trip empty a grid surface
On, obtain next knife rail cutter-contact point of space lattice curved surface.JudgementIf 4. result is transferred to for 1, if knot
2. fruit is transferred to for 0.
4. knife rail iteration terminates, contour parallel milling cutter-contact point trace in space and parameter field is generated.
Space lattice Machining of Curved Surface cutter-contact point obtained also cannot be used directly for processing, need to complete processing cutter-contact point to add
Connection side between the conversion and tracing point of work cutter location can be used for processing.Often using point of a knife point as cutter location in numerical control processing,
It processes cutter-contact point and processes the transformational relation between cutter location are as follows:
In formula, wmFor space cutter location group, emFor space cutter-contact point group,It is sweared for cutter-contact point method,For cutter axis orientation,
RtFor tool radius.
Remarkable result and benefit of the invention is this method for during Free-form Surface Parts actual processing, is based on a cloud
The problem that knife rail planning algorithm is complicated, machining accuracy is low reflects the processing of Surface tessellation point cloud model triangle gridding by reconciling
Complicated space knife rail planning problem is converted two-dimensional surface knife rail planning problem by the method penetrated, and minimizes mapping and become
Shape generates the processing knife rail of Free-form Surface Parts eventually by the inverse mapping of reconciliation mapping, improves processing quality and processing is imitated
Rate, this method are suitable for free form surface part Precision Machining.The method of dimension-reduction treatment avoids complexity in three dimensions and asks friendship
The deformation for calculating, substantially increasing computational efficiency, and be introduced into mapping drawing coefficient analysis reconciliation mapping, reduces grid ginseng
Distortion during numberization, the high-precision knife track gauge for realizing free form surface are drawn, and processing quality and processing efficiency are improved.
Detailed description of the invention
Fig. 1-method overall flow figure.
Fig. 2-reconciliation maps stretcher strain figure；Wherein, { o；Xyz } it is rectangular coordinate system in space, { o；Ou, ov } it is parameter
Domain coordinate system, T are space triangular dough sheet, T*For space triangular dough sheet, corresponding parameter field tri patch, ξ are after reconciling and mapping
Reconciliation mapping relations between D Triangulation and parameter field triangle gridding.
Part Surface Roughness after the knife rail processing that Fig. 3-is planned with this method.
The piece surface that Fig. 4-is processed with this method.
The Part Surface Roughness after the method processing that knife track gauge is drawn is carried out after Fig. 5-point cloud reconstruct.
Fig. 6-use point cloud reconstructing method processing piece surface.
Specific embodiment
Combination technology scheme and the attached drawing specific embodiment that the present invention will be described in detail.
It is low based on a cloud knife rail planning algorithm complexity, machining accuracy during Free-form Surface Parts actual processing
Problem has invented a kind of Surface tessellation point cloud model ring cutting knife rail generating method based on reconciliation mapping, overall flow such as attached drawing 1
It is shown.
By taking bounding box is having a size of the Surface tessellation point cloud model of 40 × 40 × 3mm as an example, the present invention will be described in detail implemented
Journey.
First by step 1, using parameterize indirect subdivision method combine more mature Delaunay algorithm to space curved surface from
Scatterplot cloud model carries out triangular mesh generation, includes 3452 triangular plates and 2093 vertex after triangle gridding.Construct point set
Between topological connection relation, using a table, triangle adjacency list indicate, to promote vertex neighborhood triangle information representing search capability,
And to triangle gridding, it carries out Boundary Extraction and sequence is handled.
Then by step 2, based on mapping method is reconciled, D Triangulation model boundary vertex and internal vertex are realized
Mapping introduces the deflection that mapping drawing coefficient analysis mapping process generates, three-dimensional space grid is flattened to two-dimensional surface, is adjusted
It is as shown in Fig. 2 with mapping stretcher strain figure.
By step 4, grid cell vertex normal vector is estimatedPrincipal curvatures k1、k2With corresponding principal direction d1、d2, according to Euler
Formula calculates grid cell vertex along the normal curvature of any direction d.And method arrow, the curvature letter of grid cell apex where combining
Breath calculates method arrow and curvature at cutter-contact point, and by the calculation method of contour parallel milling cutting step length feed line-spacing, in parameter field
Plan cutter path.Based on the inverse mapping method of step 3 reconciliation mapping, the knife rail in parameter field is mapped back into three-dimensional space, and
Processing cutter-contact point is completed to the connection between the conversion and tracing point of processing cutter location, ultimately generates three-dimensional space curved surface discrete point
Cloud model machining locus.
Knife track gauge is carried out to free form surface discrete point cloud model with this method to draw, and is drawn with knife track gauge is carried out after cloud reconstruct
Method compare, and using process time and surface roughness as the judgment criteria of processing quality quality.Test process equipment
For Mikron HSM500 high-rate wireless LAN center, the rose cutter of Φ 4 is selected to carry out machining experiment, the speed of mainshaft is set as
5000r/min, feed speed 250mm/min, residual altitude size are 20 μm.
It is fixed to survey in measurement process using the roughness of Talyrond Hobson roughness contourgraph measurement finished surface
Instrument fixture is measured, every time by adjusting probe location, guarantees that is surveyed is the roughness of machined surface same position.Every group of survey 3 times is simultaneously
It averages, wherein one group of measurement result, as shown in Fig. 3.It is thick with the piece surface after the knife rail processing of this method planning
Ra=0.9459 μm of rugosity, process time 11min42s, finished surface is shown in attached drawing 4；As shown in Fig. 5, it is reconstructed with cloud
Ra=1.2637 μm of Part Surface Roughness after carrying out the method processing that knife track gauge is drawn afterwards, process time 13min54s adds
Attached drawing 6 is seen on work surface.Compared to a cloud reconstructing method, processed with the knife rail that this method is planned, machined surface roughness reduces
25.15%, the time needed for process shortens 15.66%., it is apparent that when the complex pattern of processing variable curvature,
There is wedge angle in the contour parallel milling track of point cloud reconstructing method planning, and can be generated using this method with boundary consistency, track
Smooth continuous high quality contour parallel milling track, and processing efficiency significantly improves.
Claims (1)
1. a kind of Surface tessellation point cloud model ring cutting knife rail generating method based on reconciliation mapping, which is characterized in that this method is first
Triangular mesh generation is first carried out to spatial point cloud model using the indirect subdivision method combination Delaunay algorithm of parametrization, and is rebuild
The topological relation of triangle gridding；Using the parametric method of reconciliation mapping, dimension-reduction treatment is carried out to three-dimensional grid, and introduce and reflect
Penetrate drawing coefficient analysis mapping deformation；The law vector and curvature that any point in grid cell is calculated in parameter field, calculate feed
Step-length and line-spacing generate the cutter path in parameter field；Add finally, returning three-dimensional space and generating the knife rail inverse mapping in parameter field
Work knife rail；Specific step is as follows for method:
Step 1, free form surface discrete point cloud model triangular mesh generation
Triangular mesh generation is carried out to spatial point cloud model using indirect subdivision method combination Delaunay algorithm is parameterized；Its process
Are as follows: three-dimensional space point cloud is parameterized into two-dimensional surface domain first, and having boundary constraint is carried out to parametrization point cloud
Delaunay triangulation network lattice subdivision；Then the topological connection relation between plane domain intrinsic parameter point set is constructed, and side is carried out to it
Boundary extracts and sequence processing；Finally, planar triangulations are mapped back three-dimensional space to obtain three-dimensional space according to topological relation
Between put cloud triangular mesh generation result；
The process is related to topological relation building and point cloud boundary extraction and sort algorithm between parametrization point cloud；
1) parametrization point cloud topological relation building
The triangle topology relationship of triangular mesh model is tentatively established first；Based on the triangle topology relationship tentatively established,
It is successively read grid vertex call number, and is stored in new grid vertex array, then using the topological relation tentatively established,
Search includes all triangular mesh units on vertex, successively finds out and is stored in the grid index number group of vertex correspondence；
Because vertex storage be random, therefore search comprising the vertex triangular mesh unit process in, occur order be also with
Machine；Right-hand rule processing is carried out to the topological relation on newly-established vertex and its neighborhood triangle number, makes vertex neighborhood three
Angular number is at arranged counterclockwise；
2) point cloud boundary extracts and sorts
During contour parallel milling, usually selecting longest a line circle as initial knife rail, then grey iterative generation finally processes rail
Mark, therefore Boundary Extraction must be carried out to Surface tessellation point cloud model；For the triangle grid model of plane domain intrinsic parameter point cloud
For, a triangular plate, referred to as boundary edge are only belonged to if there is certain side；If certain side belongs to two triangular plates,
This side is known as internal edges；The enclosure space polygon formed that joined end to end by boundary edge is known as boundary；According to above-mentioned boundary edge
Definition, find out all boundary edges in triangle grid model, algorithm realization in, due to the side collection initially acquired be it is unordered,
Therefore it must be ranked up and organize, form end to end integral edge；Process is as follows:
Firstly, being initialized to parametrization triangle grid model boundary set array { edges }, used attribute is set as
false；
Then, the line set of triangle gridding is traversed, TriCount is the quantitative attribute of triangle where side, finds one
Any side that TriCount attribute value is 1, i.e. any one boundary edge, two endpoints on the side are respectively as Polygonal Boundary
Former point and current point find subsequent point by current point, and cyclic process is as follows:
1. traversing all of its neighbor triangle of current point, the side for meeting following condition is found:
Meet condition TriCount=1 in adjacent triangle；Used attribute value is false；It is constituted with former point and current point
While having and only 1 intersection point；
2. the one end not being overlapped with current point in the side is set as subsequent point, former point and current point are reset, enables former point
It is directed toward current point, current point is directed toward subsequent point, and sets true for the used attribute on the side；Step is transferred to 3. if closure,
Otherwise it is transferred to step 1.；
3. saving and exporting the boundary set of tracking result, process terminates；
Step 2, the D Triangulation dimension-reduction treatment based on reconciliation mapping
The key of reconciliation mapping method is to find an energy equation, referred to as objective function, and provides the objective function
Boundary condition, the extreme value for then solving objective function obtain a parametrization；Its process are as follows: for given model to be processed,
Its boundary point is found, boundary point is mapped to according to certain rules on the boundary that plane domain has predefined；For non-boundary
Point must guarantee that the elastic potential energy of the grid after mapping is minimum to guarantee that the deformation generated after model mapping can be minimum, thus will
Mapping problems, which is converted into, solves energy-minimum problem；The process is related to the mapping of D Triangulation model boundary vertex and inside
Vertex mapping；
1) border vertices map
It enables space curved surface triangle gridding TM have k border vertices, is BV={ vi(i=1,2,3 ..., k), each vertex be by
Its syntople sequences sequence；To turn to example to planar unit circle field parameter, V={ v is defined1,v2,...,vmIt is all tops
The set of point, m are the number on all vertex, T={ t1,t2,...,tnBe all triangles set, n is all triangles
Number；TM (T, V) is enabled to be mapped to plane PlOne round domain Ω: (x-x0)2+(y-y0)2+(z-z0)2On=1, using accumulation string
Regular way determines corresponding position of the grid surface boundary point on flat circle domain, enables i-th boundary edge L of ViTwo vertex be vi
And vi+1, then triangular apex set V in the mapped*In its corresponding two vertexWith flat circle domain center of circle o institute
The central angle of compositionIt should meet:
In formula, | Li| indicate side LiLength；
2) internal vertex maps
Enabling mapping relations is ξ, ξ (vi)=(xi,yi,zi)T, wherein ξ (vi) (i=1,2,3 ..., k) be to take border vertices to flat
Face circle domain mapping as a result, the surface mesh reconcile mapping after elastic potential energy E (ξ) indicate are as follows:
In formula, Ki,jFor vertex vi、vjThe side L constitutedi,jCoefficient of elasticity；Enable side Li,jFor 2 trianglesWithIt is common, then:
In formula,For triangleArea, Li,jIndicate vertex vi、vjThe side constituted, and so on；
ξ(vi) (i=1,2,3 ..., be m) all vertex of space curved surface triangle gridding to flat circle domain mapping as a result, ξ (vi)
(i=1,2,3 ..., m) all in plane Pl: on ax+by+cZ+d=0, without loss of generality, c ≠ 0 is enabled, by ξ (vi)=(xi,yi,
zi)T, ξ (vj)=(xj,yj,zj)T, then have zi=-(axi+byi+ d)/c, zj=-(axj+byj+ d)/c, then energy equation (2) turns
Become following form:
That is:
It is minimum for the deformation that occurs that the topological structure of curved surface triangle gridding TM when being mapped on flat circle domain, then energy equation
Formula (5) will reach minimum value, should meet:
Wherein, k is the border vertices number of space curved surface triangle gridding TM, and m is all number of vertex；
Enable N (vi) be and vertex viAdjacent vertex set, and vj∈N(vi)；Local derviation is asked to each internal vertex in (5) formula, is obtained
To a system of linear equations:
Formula (7) are arranged and are write as matrix form, obtaining a size is 2 (n-k) × 2 (n-k), is to be with linear coefficient matrix A
Several equation groups:
AX=B (8)
Wherein:
Space curved surface triangle gridding internal point is obtained with over-relaxation iterative method solution Sparse positive definite linear syfem of equations (9) to be mapped to
The corresponding coordinate in plane parameter domain；
There are very important stretcher strains during space curved surface triangle gridding reconciliation mapping parametersization, be in plane parameter net
Accurate trajectory planning is carried out on lattice, it is necessary to consider that parametric grid is mapped to the stretcher strain amount of three-dimensional grid generation；Grid is bent
Face parametrization has the characteristics that fragment linearity, i.e. the parameterized results ξ of space lattice curved surface for each triangular plate be it is continuous and
Linear, the affine transformation relationship of existence anduniquess between each pair of space triangular piece and parameter field triangular plate, therefore in each triangular plate
Gradient and mapping drawing coefficient are constant；
Stretch measurement L2(T*) root mean square that characterizes all directions stretcher strain in triangular plate, using stretching, measurement pairing approximation is each
It is estimated to the mapping drawing coefficient σ of the same sex, gives three vertex vs of space triangular piece Ti、vj、vkAnd its corresponding parameter
Domain triangular plate T*Three vertexThe then gradient in this triangular plate are as follows:
In formula, S is parameter field triangular plate T*Area；
S=((u1-u0)(v2-v0)-(u2-u0)(v1-v0))/2 (11)
Jacobian matrix [ξu,ξv] minimax singular value be respectively as follows:
Coefficient a in formula1, a2, a3Are as follows:
Then triangular plate mapping drawing coefficient σ is indicated are as follows:
Given parameters domain triangular plate T*Interior initial track point e*, can be according to space triangular piece T and parameter field triangular plate T*Between
Affine transformation relationship acquires corresponding tracing point e in T using area coordinate or secondary weighted method, need to calculate the rail at e point
Mark parameter simultaneously converts it to e according to mapping drawing coefficient σ*At point；
Step 3, the inverse mapping of mapping is reconciled
The liter dimension of knife rail is realized in the inverse mapping mapped that reconciles, and generates the knife rail for being used for processing space grid surface；Enable plane domain PlIn
Certain calculates cutter-contact point e*The triangular plate at place isThe triangular plate corresponding triangular plate in grid surface TM is
T{vi,vj,vk, according to the principle that mapping front and back mesh triangles piece topological structure is constant, obtain corresponding space cutter-contact point e；It builds
Vertical affine frame, enabling (α, β) is point e*In affine coordinate systemUnder parameter coordinate, using quadratic linear plus
The method of power calculates the coordinate of e；
E=vi+α(vj-vi)+β(vk-vi) (15)
Step 4, free form surface discrete point cloud model circular cutter track gauge is drawn
1) law vector and curvature estimation of triangle gridding
To be drawn convenient for subsequent knife track gauge, geometrical property of the calculating method resultant curvature analysis cutter-contact point on D Triangulation；
With the method for least square curved surface local fit to grid cell vertex normal vector, principal curvatures k1、k2With corresponding principal direction d1、
d2It is estimated, wherein k1For maximum principal curvatures, k2For minimum principal curvatures；Grid cell vertex edge is calculated further according to Euler's formula
The normal curvature of any direction d, calculation expression are as follows:
K (d)=k1·cos2θ+k2·sin2θ (16)
In formula, θ indicates any direction d and principal direction d1Between angle；
The method weighted using quadratic linear, by judging position of the cutter-contact point in grid cell, in conjunction with place grid cell
The method arrow of apex, curvature information calculate method arrow and curvature at cutter-contact point；
v1、v2And v3Respectively three vertex of spatial triangle grid cell,For corresponding per unit system arrow
It measures, then the law vector of any cutter-contact point e in grid cell insideIt indicates are as follows:
In formula, (u, v) indicates e in affine coordinate systemUnder parameter coordinate；When e is located on tri patch side
When, it is linear interpolation formula that formula (17), which is degenerated,；Similarly, the cutter-contact point inside triangle gridding unit is bent along the method for any direction
Rate calculates in aforementioned manners；
2) contour parallel milling cutter path parameter calculates
The calculating of cutter path parameter includes the calculating of cutting step length and feed line-spacing；Cutting step length is planned with step-length screening method,；
It enablesBe processing cutter-contact point through B-spline curves fitting etc. parameters from
The intensive point formed is dissipated, ε indicates known approximate error；
Given a starting pointThen in pointSelect a point in order laterThenWithBetween
Each point Pi(i=m1+1,...,m2- 1) for, it is calculated to line segmentDistance:
If meeting di< ε (i=m1+1,...,m2- 1), i.e.,WithThe line segment that two o'clock is constituted approaches forcing for the curve at place
Nearly error is less than given value, successively picks up at this timeNext point, even m2=m2+ 1, above step is repeated, if until
There is a point Pi(i∈m1+1,...,m2- 1) straightway is arrivedDistance meet di>ε；At this momentWithWhat two o'clock was constituted
Line segment is the longest line segment for meeting approximation accuracy condition；RetainPoint, and deleteWithBetween whole points；
It, must be by point for whole curveAs new starting point and repeat above step；
Feed line-spacing CL is calculated by formula (19):
In formula, RtIt is residual altitude, R for ball-end mill radius, hbFor radius of curvature, model takes positive sign, concave curved surface when being convex surface
When take negative sign；
Mapping drawing coefficient σ is introduced, mapping domain is calculated and corresponds to the track spacing CL at cutter-contact point*,
3) in space and parameter field circular cutter rail generation
In conjunction with the calculation method of ring cutting process tool trajectory parameters, illustrate the generation method of circular cutter rail in space and parameter field；
1. the D Triangulation model boundary of extraction is fitted using uniform B-Spline three times, it may be assumed that
In formula, DiFor control point, Ni,3It (t) is cubic B-spline basic function；
Cutter-contact point on first knife rail boundary curve is discrete according to certain precision, being determined using step-length screening method, according to
The mapping ruler that reconciles is mapped to parameter field, obtains first knife rail cutter-contact point of parameter field；Updating the knife rail is previous knife rail；
2. calculating previous knife rail n1The line-spacing CL that a cutter-contact point biases on space lattice curved surface, correspondence mappings domain cutter-contact point reason
By offset row away from for CL*, take line-spacing minimum valueIt is practical line-spacing along mapping previous of domain knife rail normal bias, is worked as
Preceding knife rail；
3. will map the current knife rail inverse mapping in domain based on the method for step 3 reconciliation mapping inverse mapping and make the return trip empty on a grid surface, obtain
Obtain next knife rail cutter-contact point of space grid surface；JudgementIf 4. result is transferred to for 1, if result is 0
It is transferred to 2.；
4. knife rail iteration terminates, contour parallel milling cutter-contact point trace in space and parameter field is generated；
Space lattice Machining of Curved Surface cutter-contact point obtained also cannot be used directly for processing, and need to complete processing cutter-contact point to processing knife
Connection side between the conversion and tracing point in site can be used for processing；Often using point of a knife point as cutter location, processing in numerical control processing
Transformational relation between cutter-contact point and processing cutter location are as follows:
In formula, wmFor space cutter location group, emFor space cutter-contact point group,It is sweared for cutter-contact point method,For cutter axis orientation, RtFor knife
Has radius.
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