CN106774145A - A kind of projection algorithm for generating the five-axis robot track without interference - Google Patents
A kind of projection algorithm for generating the five-axis robot track without interference Download PDFInfo
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- G05B19/00—Programme-control systems
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- 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 belongs to Milling Process correlative technology field, it discloses a kind of projection algorithm for generating the five-axis robot track without interference, it is comprised the following steps：(1) workpiece surface is separated into triangle model；(2) AABB box are set up along projecting direction to cutter；(3) triangle model is screened；(4) cutter is projected to single triangular plate：Projection equation is obtained according to constraints, the calculating of cutter-contact point is concentrated on into tool coordinate system；When cutter rings body is projected to the side of triangular plate, when projecting direction is along cutter axis orientation, unary biquadratic equation is derived according to projection equation and analytic solutions are tried to achieve；When projecting direction is any direction, Eight equation of element one is derived according to the projection equation, and based on Bezier intercept methods subdivided interval solving subpoint；(5) the corresponding subpoint of most short projector distance is exported.The present invention provides cutter and projects to implementing for triangular plate along any direction.
Description
Technical field
The invention belongs to Milling Process correlative technology field, more particularly, to a kind of for generating five axles without interference
The projection algorithm of machining locus.
Background technology
Computer numerical control (CNC) processing occupies critical positions in manufacturing industry.Compared to three axis machining, five-axis robot increases
Two frees degree of rotary shaft, are capable of achieving stickiness more preferable between cutter and surface to be machined, reduce process time, raising
Surface processing accuracy.Either three axis machining or five-axis robot are directed to the calculating of process tool track, at present conventional knife
Tool trajectory calculation strategy is substantially divided into three kinds：(1) CL-surface strategies；(2) Contact-driven strategies；(3)
Projection-based strategies, as shown in Figure 1.For CL-surface strategies, knife during by cutter and just tangent workpiece
The set in site (CL) is defined as CL surface, and cutter path directly can be calculated by the CL surface for determining, such as
Document《Mesh-based tool path generation for constant scallop-height machining》It is public
Open and asked friendship to determine CL tracks the discrete triangle model of drive surface and screw rotor, the method can calculate CL tracks
During detect and eliminate interference, be applicable to all of APT (Automatically Programmed Tools) cutter three
Axle is processed and ball head knife five-axis robot, but is not applied for flat-bottomed cutter, endless knife and general APT cutter five-axis robots；
In Contact-driven strategies, CC tracks are calculated first, then determine cutter shaft, then calculate CL tracks, such as document《Adaptive
iso-planar tool path generation for machining of free-form surfaces》And《Five-
axis tool path generation based on machine-dependent potential field》In have Jie
Continue, the track that these methods are ultimately generated there may be cuts, and causes machining accuracy not high, and machined surface quality is poor；For
Projection-based strategies, it is first determined drive surface, planning drives track, then using projection algorithm generation processing rail
Mark.Such as document《Fixed-axis tool positioning with built-in global interference
checking for NC path generation》Disclose the triangle to Surface tessellation using the cutter based on driving track
The algorithm generation tool sharpening track of piece projection, patent CN201410597489.5 describes a kind of for five-shaft numerical control processing
The generation method of constant scallop-height cutter-contact point trace, for the five-axis robot of complex mesh model, proposes that the projection of self adaptation is inclined
Put method to generate the residual cutter-contact point trace such as five axles, the algorithm can be directly generated without the machining locus cut excessively, it is adaptable to any
APT cutters, but trajectory calculation is less efficient.
The projection algorithm of current cutter path is also primarily present following main limitation：1st, most projection algorithm is on anti-
Cross and cut the research for the treatment of and be only applicable to three axis machining；2nd, projection algorithm is used to generate machining locus, and machining locus computational efficiency is past
It is past relatively low, it is necessary to solve nonlinear high-order equation, equation root is subpoint position, if equation has multiple roots, is easily omitted
Some roots, that is, can not find optimal solution, so as to cause projection algorithm to fail, cannot get correct cutter-contact point so that machining locus lack
Lose, have a strong impact on work piece surface quality；3rd, because five-axis robot increased two rotary shaft frees degree, each knife is caused
The cutter axis orientation of contact is all arbitrary, and current projection algorithm is not given and projects to the specific of triangular plate along any direction
Implementation, limits use of the projection algorithm in five-axis robot so that machining locus do not have rule relative to driving track
And it is uneven.
The content of the invention
For the disadvantages described above or Improvement requirement of prior art, the invention provides a kind of for generating five axles without interference
The projection algorithm of machining locus, its be based on Projection-based strategy in projection algorithm the characteristics of, for for generating
The projection algorithm of machining locus is designed.The projection algorithm selects the corresponding cutter of most short projector distance and triangular plate mould
Used as cutter-contact point, it is applicable to three axles and five axles, the processing of any workpiece surface and any APT cutters to the contact point of type, and
Can ensure that the track of generation is cut without mistake；The projection algorithm gives the principle that cutter-face and cutter are projected to the side of triangular plate,
The calculating of cutter-contact point is concentrated under tool coordinate system CCS, is then transformed under workpiece coordinate system WCS, improve projection efficiency；
Along the projection of cutter axis orientation be converted into unitary four times cutter for the projection of cutter rings body-side by the projection algorithm
The direct solution of equation, it is to avoid use iterative numerical, improves computational efficiency；The projection algorithm for cutter rings body-
Side projects, it is proposed that cutter is converted into the method blocked using Bezier along any direction projection and is tried to achieve only comprising a real root
Solution Interval Set merge all real roots of the equation of n th order n of unitary 8 tried to achieve using dichotomy, subpoint is further calculated, to be processed
Track, improves the flexibility of searching algorithm so that projection algorithm be applied to five-axis robot, it is ensured that machining locus it is uniform
It is regular.
To achieve the above object, the invention provides a kind of projection calculation for generating the five-axis robot track without interference
Method, it is comprised the following steps：
(1) workpiece surface is separated into triangle model；
(2) to each cutter initial position, AABB box are set up to cutter along projecting direction；
(3) triangle models of the AABB box along the projection covering of projecting direction is selected；
(4) cutter is projected to single triangular plate：First, cutter is projected to the face of triangular plate, if cutter is to the face of triangular plate
Project successfully, then export subpoint and the corresponding projector distance of the subpoint, otherwise carry out cutter and projected to the side of triangular plate,
Wherein, cutter is projected to triangular plate when projection includes cutter rings body to triangular plate, and cutter rings body is to triangular plate
When side projects, a projection equation is obtained according to constraints, when projecting direction is along cutter axis orientation, the projection equation is turned
Unary biquadratic equation is turned to, the unitary biquadratic is tried to achieve according to Abel-Ruffini theories and Ferrari Lodovico methods
The analytic solutions of journey；When projecting direction is any direction, the projection equation is converted into Eight equation of element one, using improved
Bezier method for cutting subdivided interval simultaneously solves all real roots of the Eight equation of element one using dichotomy, and then try to achieve projection
Point, if cutter exports subpoint and corresponding projector distance to the projection success on the side of triangular plate, otherwise carries out cutter to three
The summit projection of gusset plate, if cutter is projected successfully to the summit of triangular plate, exports subpoint and corresponding projector distance, otherwise
Cutter is projected to triangular plate and failed；Record projects successful subpoint and corresponding projector distance every time, and calculate cutter-contact point and
Cutter location, the process that circulation projects cutter to single triangular plate, until cycle-index is equal to the quantity of the triangular plate of screening；
(5) minimum projection is found out apart from corresponding subpoint as final cutter-contact point, to each cutter initial position,
Corresponding cutter-contact point and cutter location are calculated, the machining locus of curve surface of workpiece are finally obtained.
Further, cutter is projected respectively according to the method on perspective plane to the face projection and cutter of triangular plate to the side of triangular plate
Arrow NFThe subpoint under tool coordinate system CCS is calculated with Projection Constraint condition, the subpoint for obtaining then will be calculated and is transformed into work
Under part coordinate system WCS, cutter-contact point is exported.
Further, cutter rings body is projected to the side of triangular plate, edge-vector and projection side of the projection plane by triangular plate
To decision, the constraints is：(1) method arrow of the subpoint on torus is vertical with the edge-vector of triangular plate；(2) subpoint
On a projection plane.
Further, the unary biquadratic equation is：
(S1 2-1)(d(-UzS1)+Q)2+R2((UxVx+UyVy+UzVz)S1)2=0
In formula, S1=cos (θ) Vy-sin(θ)Vx(d, h) is the seat of the center of circle in tool coordinate system of cutter rings body portion
Scale value, R is the radius of the circular arc portion of cutter rings body, V (Vx, Vy, Vz) it is the edge-vector of triangular plate, PV(Ux, Uy, Uz) it is throwing
Shadow direction, θ is the angle of latitude of the surface of revolution of cutter rings body, Q=hUxVy-hUyVx-PxUyVz+PxUzVy+PyUxVz-PyUzVx-
PzUxVy+PzUyV1P1, (Px, Py, Pz) on the side of triangular plate any one summit.
Further, the Eight equation of element one is：
F (x)=a8x8+a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0=0
In formula, ai(i=0,1,2 ... 8) is true according to the parameter of cutter rings body, the edge-vector of triangular plate and projecting direction
Fixed coefficient,θ is the angle of latitude of the surface of revolution of cutter rings body.
Further, side from described cutter rings body to triangular plate project, projecting direction for it is any to when, using improvement
Bezier method for cutting subdivided interval and using dichotomy solve Eight equation of element one all real roots, specifically include following step
Suddenly：
(51) range of variables in polynomial of degree n, f are askedn(x), x ∈ [a, b]；
(52) by polynomial of degree n fnX (), x ∈ [a, b] are expressed as n Bernstein multinomials p (t) t ∈ [0,1], defeated
Go out the polynomial coefficient { b of Bernsteini}；
(53) using the coefficient { b for trying to achieveiConstruct Bezier curve , wherein d (t) represents the point on Bernstein multinomials p (t) to the distance of t axles, Di
Represent the control point of Bezier curve D (t)；
(54) Minimum Convex Closure { A of Bezier curve D (t) is soughti}；
(55) controlling polygon { D of Bezier curve D (t) is soughtiAnd Minimum Convex Closure { the AiAnd t axles intersection point, obtain
Initial solution interval { [t on ti, ti+1]}；
(56) optimization initial solution is interval, it is ensured that there is unique root in each solution interval；
(57) the interval set of solution after output optimization；
(58) using dichotomy to obtain the solution in each interval, and subpoint and corresponding is calculated according to the solution for obtaining
Projector distance, finds out minimum projection apart from corresponding subpoint.
Further, optimization initial solution is interval, it is ensured that the interval process for having unique root of each solution is comprised the following steps：
S561, treats detection interval [ti, ti+1], by formula x=a+ (b-a) * t, inverse x, obtain interval [xi, xi+1]；
S562, by polynomial of degree n fn(x), x ∈ [xi, xi+1] n Bernstein multinomials p (t) is converted into, calculating it is
Number { bi}；
S563, judges the polynomial coefficient { b of BernsteiniSymbol, if coefficient symbols change number of times be more than 2, turn
S564；If coefficient symbols change number of times is 1, turn S565；If coefficient symbols change number of times is 0, turn S566；
S564, to the range optimization comprising two roots or unrooted, judges and removes the interval of unrooted, to comprising two roots
Interval be finely divided, with reference to maximum control point and the characteristic at minimum control point, priority choosing is carried out to the interval that subdivision is obtained
It is fixed, to determine the sequencing of interval search；
Polynomial coefficient { the b of S565, output BernsteiniVariation of sign number of times be 1 interval；
S566, detects to next interval.
In general, by the contemplated above technical scheme of the present invention compared with prior art, the use that the present invention is provided
In the projection algorithm of five-axis robot track of the generation without interference, its corresponding contact of cutter with curved surface of most short projector distance of selection
Used as cutter-contact point, it is applicable to three axles and five axles, the processing of any workpiece surface and any APT cutters to point, and can ensure life
Into track without cross cut；The projection algorithm gives the principle that cutter-face and cutter are projected to the side of triangular plate, by cutter-contact point
Calculating concentrate under tool coordinate system CCS, be then transformed under workpiece coordinate system WCS, improve projection efficiency；The projection
Along the projection of cutter axis orientation be converted into the straight of unary biquadratic equation cutter for the projection of cutter rings body-side by algorithm
Connect solution, it is to avoid use iterative numerical, improve computational efficiency；The projection algorithm is carried for cutter rings body-side projection
Go out cutter is converted into the method blocked using Bezier along any direction projection and tried to achieve only comprising a solution interval for real root
And all real roots of the equation of n th order n of unitary 8 are tried to achieve using dichotomy, and subpoint is further calculated, to obtain machining locus, improve
The flexibility of search so that projection algorithm is applied to five-axis robot, it is ensured that the uniform regularity of machining locus.
Brief description of the drawings
Fig. 1 is that three kinds of conventional cutter paths calculate strategy flow chart slightly.
Fig. 2 is the schematic diagram that cutter is projected to workpiece surface.
Fig. 3 is the schematic diagram that cutter is projected to single triangular plate.
Fig. 4 is being related to for generating the projection algorithm of the five-axis robot track without interference for better embodiment offer of the present invention
And model schematic.
Fig. 5 is the schematic diagram of APT cutters.
Fig. 6 is the flow chart of the projection algorithm for generating the five-axis robot track without interference in Fig. 4.
Fig. 7 is the APT cutters that are related to of projection algorithm for generating the five-axis robot track without interference of Fig. 4 to triangular plate
The schematic diagram of the face projection of model.
Fig. 8 is for generating APT cutters that the projection algorithm of the five-axis robot track without interference is related to triangle in Fig. 4
The schematic diagram of the side projection of piece model.
Fig. 9 is the annulus for generating the APT cutters that the projection algorithm of the five-axis robot track without interference is related in Fig. 4
The derivation schematic diagram that body is projected to the side of triangle model.
Figure 10 is the projection for generating the five-axis robot track without interference in utilizing Bezier method for cutting to solve Fig. 4
8 flow charts of projection equation that algorithm is related to.
Figure 11 is for generating that the projection algorithm of the five-axis robot track without interference is related to by adjacent two in Fig. 4
The method arrow of triangular plate determines the schematic diagram of search angular range.
Figure 12 is turning for generating the polynomial of degree n that the projection algorithm of the five-axis robot track without interference is related in Fig. 4
Turn to the n polynomial schematic diagram of Bernstein.
Figure 13 is bent for generating the construction Bezier that the projection algorithm of the five-axis robot track without interference is related in Fig. 4
The schematic diagram of line.
Figure 14 is the shape for generating the Minimum Convex Closure that the projection algorithm of the five-axis robot track without interference is related in Fig. 4
Into schematic diagram.
Figure 15 is the intersection point signal of the Minimum Convex Closure and t axles in the controlling polygon and Figure 13 of the Bezier curve in Figure 12
Figure.
Figure 16 is to be related to that initial solution is interval to be shown for generating the projection algorithm of the five-axis robot track without interference in Fig. 4
It is intended to.
Figure 17 is the Optimizing Flow figure in the initial solution interval in Figure 16.
Specific embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, below in conjunction with accompanying drawing and case study on implementation,
The present invention will be described in further detail.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention,
It is not intended to limit the present invention.Additionally, technical characteristic involved in invention described below each implementation method is only
Not constitute conflict each other can just be mutually combined.
Refer to Fig. 2, the projection for generating the five-axis robot track without interference that better embodiment of the present invention is provided
Algorithm, its make cutter be located at one without interference initial position, cutter along a certain given direction towards curved surface move while
Keep cutter-orientation constant (only translating, without rotation), until contact surface and tangent with curved surface, point of contact is cutter-contact point.
In order to solve the problems, such as that curve surface of workpiece shape matching is complicated, workpiece surface is approached using triangle model, by cutter
Cutter is reduced to workpiece surface On The Projection to be projected to triangle model.Fig. 3 is referred to, cutter is projected to triangle model and wrapped
Include following steps：(1) workpiece surface is separated into triangle model；(2) AABB box (axis are set up along projecting direction to cutter
aligned bounding box)；(3) triangular plate that may be projected to is screened；(4) cutter is projected to single triangular plate；(5) it is defeated
Go out the corresponding subpoint of most short projector distance.Fig. 4 is referred to, cutter is projected to single triangular plate, including cutter is to triangular plate
Face projection, cutter are projected to the side projection and cutter of triangular plate to the summit of triangular plate.Cutter can be divided into cutter conical sections
With cutter rings body portion.Because the mathematic(al) representation of cutter rings body is 4 power equations, and the mathematical expression of cutter cone
Formula is 2 equation of n th order n, thus cutter rings body to project to triangular plate more difficult.Wherein, cutter rings body is thrown to the face of triangular plate
Shadow can try to achieve subpoint according to the method in face arrow, and cutter rings body can be converted into straight line and be asked with torus to the summit projection of triangular plate
The problem of friendship.Cutter rings body is projected to the side of triangular plate, when projecting direction is cutter axis orientation, is tried to achieve according to constraints
Projection equation, derives unary biquadratic equation and seeks the analytic solutions of the unary biquadratic equation according to projection equation；When projection side
During to for any direction, the equation of n th order n of unitary 8 is derived according to the projection equation, using the iterative numerical blocked based on Bezier
Method subdivided interval, solves subpoint.The present invention ensures that projection algorithm is adapted to five-axis robot, and the track of generation is not done
Relate to, and then obtain preferable machined surface quality.
In present embodiment, the projection algorithm is applied to the generation of three axles and five axles without interference machining locus；Meanwhile, institute
State projection algorithm and be suitable for arbitrary APT (Automatically Programmed Tools) cutter, while improve track
Computational efficiency；Additionally, the projection algorithm gives the specific solution that cutter projects to triangular plate along any direction first,
Ensure that the uniformity and regularity of machining locus.
The center of circle of APT cutter rings body portions is (d, h), the knife bar length H of cutter, Tool in Cutting length L, wedge angle A1、
Side angle A2, diameter D, shown in circular arc portion radius R, such as Fig. 5 (a)；The effective cutting zone of cutter includes upper cone body, middle part
Torus and lower cone body, shown in such as Fig. 5 (b).
Refer to Fig. 3 to Fig. 8, better embodiment of the present invention provide for generating the five-axis robot track without interference
Projection algorithm is mainly included the following steps that：
Step one, sets up the triangle model of workpiece surface, is meeting in given error range, and triangle model can be forced
Nearly surface model；
Step 2, to each cutter initial position, AABB box is set up to cutter along projecting direction；
Step 3, screens triangular plate, i.e., the triangular plate that described AABB box are covered along the projection of projecting direction；
Step 4, cutter is projected to single triangular plate, is justified because cutter can be converted into cutter to the summit projection of triangular plate
Cone or cutter rings body portion and intersection between lines.Therefore cutter is only introduced to the face projection of triangular plate and cutter to triangular plate
The principle of side projection.
(41) cutter is projected to the face of triangular plate
Specifically, first, refer to Fig. 7 and set up workpiece coordinate system WCS (Workpiece Coordinate System),
Then, set up tool coordinate system CCS (Cutter Coordinate System), wherein the X-Z plane of tool coordinate system CCS by
Cutter axis orientation TAN is sweared with triangular plate methodFDetermine, projector distance is defined as dP, the method arrow N of triangular plateFWith any point on torus
Method arrow NTOpposite number, wherein N each otherF=P1P2×P1P3,So that NF=-NT.Root
According to initial cutter location CL0, cutter axis orientation TA(0,0,1), projecting direction PV, tool-information TsAnd triangular plate information TF(triangular plate
Three summit P1, P2, P3)；Calculate and obtain cutter-contact point CC, cutter location CL and projector distance dP, basic procedure is as follows：
S411, N is sweared by the method below workpiece coordinate system WCSFIt is transformed under tool coordinate system CCS；
S412, calculates the tool offset vector O, i.e. point CC under CCS under tool coordinate system CCS0；
S413, point CC will be crossed under tool coordinate system CCS0Friendship is asked along the straight line and triangular plate of projecting direction, knife is can obtain
CC points and projector distance d under tool coordinate system CCSP；
S414, by formula CL=CL0+dPPVThe CL points under tool coordinate system CCS are calculated, and is transformed into workpiece coordinate system WCS
Under；If CC points are not inside triangular plate, rotor has the side projection to triangular plate, otherwise exports CC points, CL points and projector distance dP；
(42) cutter is projected to the side of triangular plate
Fig. 8 is referred to, cutter is as follows to the rudimentary algorithm flow that the side of triangular plate projects：
S421, calculates by projecting direction PVEdge-vector V with triangular plate determines projection plane；
S422, projection plane and cutter surface intersection, calculate the CC points and projector distance d under CCSP；
S423, by formula CL=CL0+dPPVThe CL points under tool coordinate system CCS are calculated, and is transformed into workpiece coordinate system WCS
Under；
S424, if CC points are not inside triangular plate, cutter is projected to the side of triangular plate, otherwise export CC points, CL points and
Side projects dP。
Because the mathematic(al) representation of cutter rings body is 4 power equations, and the mathematic(al) representation of cutter cone is 2 powers
Journey, thus torus to project to triangular plate more difficult.Wherein torus-face projection can seek subpoint according to the method in face arrow, circle
Ring body-summit projection can be converted into straight line and seek the problem of friendship with torus.Cutter rings body-specific derivation of side projection is as follows：
Fig. 9 is referred to, projection plane is by edge-vector V (Vx, Vy, Vz) and projecting direction PV(Ux, Uy, Uz) determine, while taking up an official post
One summit of meaning is represented by P1(Px, Py, Pz), torus can be described by the center of circle (d, h) and angle (α, θ).
Projection equation is derived according to projection condition, specific derivation process is as follows.
Any point P on the torusTCan be described as：
In formula,The π of 0≤θ ＜ 2.
The method arrow at any point is N on torusT
The method arrow N of projection planePCan be by the vectorial V on side and projection vector PVDetermine
When the side of triangular plate is tangent with torus, it is necessary to meet two Projection Constraint conditions：One is cutting on torus
The corresponding method arrow of point is vertical with edge-vector, and two is that point of contact is located in projection plane.Equation below can be obtained.
NTV=0
(PT-P1)·NP=0
Two above Projection Constraint equation is two nonlinear equations on variable α and θ, two equations of simultaneous
Obtain a nonlinear equation on angle, θ.
(S1 2-1)(d(VzS2-UzS1)+Q)2+r2((UxVx+UyVy+UzVz)S1-S2)2=0
S in formula1=cos (θ) Vy-sin(θ)Vx, S2=cos (θ) Uy-sin(θ)Ux
Q=hUxVy-hUyVx-PxUyVz+PxUzVy+PyUxVz-PyUzVx-PzUxVy
+PzUyV1
When projecting direction is cutter axis orientation, PV(Ux, Uy, Uz) it is cutter axis orientation TA(0,0,1), variable S2=0, projection side
Journey can abbreviation be one on variable S1Unary biquadratic equation, it is theoretical according to Abel-Ruffini and utilize Ferrari
Lodovico methods solve all of Real Number Roots of equation of n th order n of unitary 4, and then obtain subpoint.
(S1 2-1)(d(-UzS1)+Q)2+R2((UxVx+UyVy+UzVz)S1)2=0
When cutter is projected along any direction, using the universal formula of trigonometric function,Replacement coefficient ai(i
=0,1,2 ... 8) can obtain Eight equation of element one, try to achieve all of real root of the Eight equation of element one, then from the real root for obtaining
The middle root for choosing the most short projector distance of correspondence is used as subpoint, i.e. cutter-contact point.
F (x)=a8x8+a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0=0
For above-mentioned 8 equation of n th order n, tried to achieve only comprising an interval set for root using improved Bezier method for cutting,
And solve all of real root of Eight equation of element one using dichotomy.Basic procedure is as follows：
A () asks range of variables in polynomial of degree n, fn(x), x ∈ [a, b]；
Figure 10 and Figure 11 is referred to, it can be seen from the order according to projection, to may proceed to side projection, by several for face projection failure
What relation is understood, if torus can be projected successfully to side, point of contact opposite direction one of corresponding method arrow on circular arc is positioned
Between two method arrows of adjacent triangular plate, α is defined1And θ1For representation swears NF1Parameter, α2And θ2For representation swears NF2's
Parameter, can obtain：
Work as NF1And NF2, it is known that angle [alpha]1, θ1, α2, θ2Can try to achieve, byCan obtain variable x scope [a,
b]。
B () refers to Figure 12, by polynomial of degree n fnX (), x ∈ [a, b] are expressed as n Bernstein multinomials p (t) t
Polynomial coefficient { the b of ∈ [0,1], output Bernsteini}；
C () is using the coefficient { b for trying to achieveiConstruct Bezier curve ,
Wherein d (t) represents the point on Bernstein multinomials p (t) to the distance of t axles, DiRepresent the control of Bezier curve D (t)
Point, refers to Figure 13；
D () seeks the Minimum Convex Closure { A of Bezier curve D (t)i}；
Figure 14 is referred to, the control point { D according to Bezier curve D (t)i, i=0,1 ..., n obtain Bezier curve D
Convex closure summit { the A of (t)i, comprise the following steps that：
S41, the value of the t at relatively more all control points selects the maximum point of t values, is designated as A0If there is the t values of multiple points equal,
It is A to take and the maximum point of d (t) values is taken in these points0, direction is designated as L with the positive consistent ray of the longitudinal axis0, such as shown in Figure 14 (a)；
S42, successively with AiIt is starting point, allows LiRotate counterclockwise, looks for a point A in all control pointsi+1So that with AiFor
Initial point and excessively Ai+1Ray and ray LiBetween angle it is minimum, remember that this ray is Li+1(i=0,1 ... n)；
S43, repeat step S42, until finding certain An=A0, at this moment A1, A2... AnIt is exactly whole summits of convex closure；
S44, by the ascending order line of index on convex closure whole summit, forms Minimum Convex Closure, shown in such as Figure 14 (b).
E () seeks the controlling polygon { D of Bezier curve D (t)iAnd Minimum Convex Closure { the AiAnd t axles intersection point, obtain
Initial solution interval { [t on ti, ti+1]}；
Figure 15 is referred to, the control point { D according to Bezier curveiAnd convex closure summit { Ai, it is interval to obtain initial solution
{[ti, ti+1], comprise the following steps that：
S51, judges every a line D of controlling polygoniDi+1Whether intersect with t axles, if intersecting, obtain the parameter of intersection point
tDi；
S52, judges every a line A of Minimum Convex ClosureiAi+1Whether intersect with t axles, if intersecting, obtain the parameter of its intersection point
tAi；
S53, the intersection point parameter that step S51 and step S52 are tried to achieve sorts by order from small to large, each two parameter shape
Into an interval, then corresponding initial interval [t is can obtain0, t1], [t1, t2]…。
S54, output initial solution interval { [ti, ti+1]}。
F () optimization initial solution is interval, it is ensured that there is unique root in each solution interval；
Figure 16 and Figure 17 is referred to, behind a series of solution interval for obtaining, Bezier curve is possible to and t in solution is interval
Axle is non-intersect, or more than one intersection point, as shown in figure 16 in interval [t0, t1] there is a root, in interval [t1, t2] in have two
Individual root, in interval [t2, t3] interior unrooted, therefore solution interval is optimized, comprise the following steps that：
S61, treats detection interval [ti, ti+1], by formula x=a+ (b-a) * t, inverse x, obtain interval [xi, xi+1]；
S62, by polynomial of degree n fn(x), x ∈ [xi, xi+1] n Bernstein multinomials p (t) is converted into, calculating it is
Number { bi}；
S63, judges the polynomial coefficient { b of BernsteiniSymbol, if coefficient symbols change number of times be more than 2, turn
S64；If coefficient symbols change number of times is 1, turn S65；If coefficient symbols change number of times is 0, turn S66.
S64, the range optimization to two roots or unrooted may be included, specifically, Bernstein coefficients { biSymbol changes
Become twice, then convex closure intersects with t axles, because the effect at control point is to haul close to control point SPL, therefore can be with root
Interval is divided into two parts according to the highest on interval [α, β] or minimum control point, if first control point b0＞ 0, then look for most
Low control point and its correspondence parameter γ；If first control point b0＜ 0, then look for highest control point and its correspondence parameter γ.By area
Between [α, β] be divided into [alpha, gamma] and [γ, β], the Bernstein coefficients { b on computation interval [alpha, gamma]i}1On interval [γ, β]
Bernstein coefficients { bi}2。
If coefficient { bi}1With coefficient { bi}2Not reversion, then solve equation in [alpha, gamma] and [γ, β] interior unrooted, therefore
Interval [α, β] unrooted；If coefficient { bi}1With coefficient { bi}2In at least one reversion once, show at least one root.If being
Number { bi}1Reversion once, coefficient { bi}2Reversion, does not have a root at interval [alpha, gamma]；If coefficient { bi}1Not reversion, coefficient { bi}2
Reversion once, has a root at interval [γ, β]；If coefficient { bi}1Reversion once, coefficient { bi}2Reversion once, then in interval
Respectively there is a root in [alpha, gamma] and interval [γ, β].If coefficient { bi}1With coefficient { bi}2In at least one reversion twice, all need
Want further subdivided interval.If coefficient { bi}1Reversion twice, coefficient { bi}2It is unsatisfactory for, then to interval [alpha, gamma] subdivision；If coefficient
{bi}1It is unsatisfactory for, coefficient { bi}2Reversion twice, is then segmented to interval [γ, β]；If coefficient { bi}1Reversion twice, coefficient { bi}2Become
Number twice, then to making a judgement in interval [alpha, gamma] and interval [γ, β], to preferentially being segmented comprising the big interval of root possibility.
Polynomial coefficient { the b of S65, output BernsteiniVariation of sign number of times be 1 interval；
S66, detects to next interval.
G the solution after () output optimization is interval.Specifically, it would be possible on the interval [α, β] comprising two roots or unrooted
After interval is divided into two parts by highest or minimum control point, based on the variation of sign rule in step (5), it is considered to convex closure and t
The intersecting characteristic of axle, continuous subdivided interval is judged that final output includes an interval for the interval and unrooted of root.
(h) using dichotomy to obtain the solution in each interval, and according to the solution tried to achieve calculate corresponding subpoint and
Projector distance, and try to achieve subpoint.
Step 5, exports the corresponding subpoint of most short projector distance, gives cutter initial position, cutter one by one to it is all can
Successful triangular plate projection can be projected, and records corresponding subpoint and projector distance.Minimum projection is found out apart from corresponding throwing
Shadow point calculates cutter location as final cutter-contact point by cutter-contact point.To each cutter initial position, corresponding knife is calculated
Site, finally obtains the machining locus of curve surface of workpiece.
Projection algorithm provided by the present invention for generating the five-axis robot track without interference, selects most short projector distance pair
The contact point of the cutter answered and triangle model as cutter-contact point, its be applicable to three axles and five axles, any workpiece surface plus
Work and any APT cutters, and can ensure that the track of generation is cut without mistake；The projection algorithm gives cutter-face and cutter to three
The principle of the side projection of gusset plate, the calculating of cutter-contact point is concentrated under tool coordinate system CCS, is then transformed into workpiece coordinate system
Under WCS, projection efficiency is improve；The projection algorithm for the projection of cutter rings body-side, by cutter along cutter axis orientation projection
It is converted into a direct solution for unary biquadratic equation, it is to avoid use iterative numerical, improves computational efficiency；The projection
Algorithm is for cutter rings body-side projection, it is proposed that cutter is converted into the side blocked using Bezier along any direction projection
Method tries to achieve all real roots that the equation of n th order n of unitary 8 is only tried to achieve comprising a solution interval for real root and using dichotomy, further calculates
Subpoint, to obtain machining locus, improves the flexibility of searching algorithm so that projection algorithm is applied to five-axis robot, it is ensured that
The uniform regularity of machining locus.
As it will be easily appreciated by one skilled in the art that the foregoing is only presently preferred embodiments of the present invention, it is not used to
The limitation present invention, all any modification, equivalent and improvement made within the spirit and principles in the present invention etc., all should include
Within protection scope of the present invention.
Claims (7)
1. a kind of for generating the projection algorithm without the five-axis robot track interfered, it is comprised the following steps：
(1) workpiece surface is separated into triangle model；
(2) to each cutter initial position, AABB box are set up to cutter along projecting direction；
(3) triangle models of the AABB box along the projection covering of projecting direction is selected；
(4) cutter is projected to single triangular plate：First, cutter is projected to the face of triangular plate, if cutter is projected to the face of triangular plate
Success, then export subpoint and the corresponding projector distance of the subpoint, otherwise carries out cutter and is projected to the side of triangular plate, its
In, cutter is projected to triangular plate when projection includes cutter rings body to triangular plate, side from cutter rings bodies to triangular plate
During projection, a projection equation is obtained according to constraints, when projecting direction is along cutter axis orientation, by projection equation conversion
It is unary biquadratic equation, the unary biquadratic equation is tried to achieve according to Abel-Ruffini theories and Ferrari Lodovico methods
Analytic solutions；When projecting direction is any direction, the projection equation is converted into Eight equation of element one, using improved
Bezier method for cutting subdivided interval simultaneously solves all real roots of the Eight equation of element one using dichotomy, and then try to achieve projection
Point, if cutter exports subpoint and corresponding projector distance to the projection success on the side of triangular plate, otherwise carries out cutter to three
The summit projection of gusset plate, if cutter is projected successfully to the summit of triangular plate, exports subpoint and corresponding projector distance, otherwise
Cutter is projected to triangular plate and failed；Record projects successful subpoint and corresponding projector distance every time, and calculate cutter-contact point and
Cutter location, the process that circulation projects cutter to single triangular plate, until cycle-index is equal to the quantity of the triangular plate of screening；
(5) minimum projection is found out apart from corresponding subpoint as final cutter-contact point, to each cutter initial position, is calculated
Go out corresponding cutter-contact point and cutter location, finally obtain the machining locus of curve surface of workpiece.
It is 2. as claimed in claim 1 to be used to generate the projection algorithm without the five-axis robot track interfered, it is characterised in that：Cutter
To the face projection and cutter of triangular plate N is sweared to the method that the side of triangular plate projects respectively according to perspective planeFWith Projection Constraint condition meter
The subpoint under tool coordinate system CCS is calculated, the subpoint for obtaining then will be calculated and is transformed under workpiece coordinate system WCS, export knife
Contact.
It is 3. as claimed in claim 1 to be used to generate the projection algorithm without the five-axis robot track interfered, it is characterised in that：Cutter
Torus is projected to the side of triangular plate, and projection plane is determined that the constraints is by the edge-vector and projecting direction of triangular plate：
(1) method arrow of the subpoint on torus is vertical with the edge-vector of triangular plate；(2) subpoint is on a projection plane.
It is 4. as claimed in claim 1 to be used to generate the projection algorithm without the five-axis robot track interfered, it is characterised in that：It is described
Unary biquadratic equation is：
(S1 2-1)(d(-UzS1)+Q)2+R2((UxVx+UyVy+UzVz)S1)2=0
In formula, S1=cos (θ) Vy-sin(θ)Vx, (d, h) is the coordinate of the center of circle in tool coordinate system of cutter rings body portion
Value, R is the radius of the circular arc portion of cutter rings body, V (Vx, Vy, Vz) it is the edge-vector of triangular plate, PV(Ux, Uy, Uz) it is projection
Direction, θ is the angle of latitude of the surface of revolution of cutter rings body, Q=hUxVy-hUyVx-PxUyVz+PxUzVy+PyUxVz-PyUzVx-
PzUxVyPzUyV1P1, (Px, Py, Pz) on the side of triangular plate any one summit.
It is 5. as claimed in claim 1 to be used to generate the projection algorithm without the five-axis robot track interfered, it is characterised in that：It is described
Eight equation of element one is：
F (x)=a8x8+a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0=0
In formula, ai(i=0,1,2 ... 8) be according to the parameter of cutter rings body, the edge-vector of triangular plate and projecting direction determine
Coefficient,θ is the angle of latitude of the surface of revolution of cutter rings body.
It is 6. as claimed in claim 1 to be used to generate the projection algorithm without the five-axis robot track interfered, it is characterised in that：It is described
Side from cutter rings body to triangular plate project, projecting direction for it is any to when, using improved Bezier method for cutting segment
All real roots interval and that Eight equation of element one is solved using dichotomy, specifically include following steps：
(51) range of variables in polynomial of degree n, f are askedn(x), x ∈ [a, b]；
(52) by polynomial of degree n fnX (), x ∈ [a, b] are expressed as n Bernstein multinomials p (t) t ∈ [0,1], output
Polynomial coefficient { the b of Bernsteini}；
(53) using the coefficient { b for trying to achieveiConstruct Bezier curve
Wherein d (t) represents the point on Bernstein multinomials p (t) to the distance of t axles, DiRepresent the control of Bezier curve D (t)
Point；
(54) Minimum Convex Closure { A of Bezier curve D (t) is soughti}；
(55) controlling polygon { D of Bezier curve D (t) is soughtiAnd Minimum Convex Closure { the AiAnd t axles intersection point, obtain on
Initial solution the interval { [t of ti, ti+1]}；
(56) optimization initial solution is interval, it is ensured that there is unique root in each solution interval；
(57) the interval set of solution after output optimization；
(58) using dichotomy to obtain the solution in each interval, and subpoint and corresponding projection are calculated according to the solution for obtaining
Distance, finds out minimum projection apart from corresponding subpoint.
It is 7. as claimed in claim 6 to be used to generate the projection algorithm without the five-axis robot track interfered, it is characterised in that：Optimization
Initial solution is interval, it is ensured that the interval process for having unique root of each solution is comprised the following steps：
S561, treats detection interval [ti, ti+1], by formula x=a+ (b-a) * t, inverse x, obtain interval [xi, xi+1]；
S562, by polynomial of degree n fn(x), x ∈ [xi, xi+1] n Bernstein multinomials p (t) is converted into, calculate its coefficient
{bi}；
S563, judges the polynomial coefficient { b of BernsteiniSymbol, if coefficient symbols change number of times be more than 2, turn S564；If
Coefficient symbols change number of times is 1, turns S565；If coefficient symbols change number of times is 0, turn S566；
S564, to the range optimization comprising two roots or unrooted, judges and removes the interval of unrooted, to comprising two areas of root
Between be finely divided, with reference to maximum control point and the characteristic at minimum control point, priority is carried out to the interval that subdivision is obtained and is selected, with
It is determined that the sequencing of interval search；
Polynomial coefficient { the b of S565, output BernsteiniVariation of sign number of times be 1 interval；
S566, detects to next interval.
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