CN106774154A

CN106774154A - A kind of space curve interpolating method theoretical based on osculating plane

Info

Publication number: CN106774154A
Application number: CN201611258996.1A
Authority: CN
Inventors: 智红英; 闫献国; 杜娟; 张唐圣; 陈峙; 郭宏; ***; 曹启超
Original assignee: Taiyuan University of Science and Technology
Current assignee: Taiyuan University of Science and Technology
Priority date: 2016-12-30
Filing date: 2016-12-30
Publication date: 2017-05-31
Anticipated expiration: 2036-12-30
Also published as: CN106774154B

Abstract

The invention belongs to Computerized Numerical Control processing technology field, it is proposed that a kind of space curve interpolating method theoretical based on osculating plane, it is mainly used in tool-path planning operation in Multi-axis Machining.According to the local geometric shape at interpolated point on space curve, using the interpolated point of the bow error high such as equidistant surface construction satisfaction on the principal normal direction from tangent plane of each interpolated point.But the bow error high that this side up bow error high not necessarily truly, bow error function high is set up followed by analytic geometry midpoint to straight line apart from implicationThe most longbow error high calculated between two adjacent discrete points is verified.Using new method energy self-adaptative adjustment cutting step, bow error high truly is realized, be evenly distributed in the bow error high respectively approached in straightway, machining accuracy is high, and surface quality is good.

Description

A kind of space curve interpolating method theoretical based on osculating plane

Technical field：

The invention belongs to Computerized Numerical Control processing technology field, more particularly to a kind of space curve interpolation side theoretical based on osculating plane Method, is mainly used in tool-path planning operation in Machining of Curved Surface.

Background technology：

In the process industries such as aviation, aerospace industry, automobile and mould, existing largely has complex-curved part, These parts not only complex geometry, and the accuracy of manufacture and surface quality requirements are very high.With computer technology and certainly The development of dynamic control technology is more and more extensive with application of the complex-curved part in industrial products.At present, it is complex-curved The processing of part is mostly realized by Digit Control Machine Tool.Tool-path planning is the important composition of design for NC Part, its designing quality directly affects the machining accuracy and processing efficiency of part.The cutter path of Machining of Curved Surface is by one Bar cutter-contact point trace curve is constituted, and from geometrically saying, each cutter-contact point trace curve is essentially all a space curve, because This tool-path planning is converted to the interpolated point computational problem of a rule space curve.Different interpolation point calculating methods are with regard to shape Into different interpolating methods.

At present, range of application wider interpolation point calculating method such as mainly has at parameter step length method, equidistant step length, the step-length Screening method etc..

It is that parameter and curve equidistant are split Deng parameter step length method and equidistant step length, although both approaches are simple, But required in order that obtaining every section of approximate error of straight line and meeting, step-length or parameter increase can only be selected using adverse conditions the most Take, so as to cause remaining processing sequences huge, actual bow error skewness high.

Step-length screening method uses less equidistance or parameter discrete cutter rail before this using based on equal steps Mark, then checks every section of actual approximate error of straight line, finally needs to delete unnecessary discrete point, so that remaining trajectory error It is more uniform.Although step-length screening method the shortcoming of parameter step length method and equidistant step length such as overcomes, but check every section it is straight All it is approximate bow error high during bow error high in line, so as to influence the surface quality of curved surface, reduces the processing essence of workpiece Degree.

The content of the invention：

In order to solve the problems, such as conventional method in terms of Multi-axis Machining tool-path planning step size computation, the present invention is carried A kind of space curve interpolating method of the satisfaction bow high franchise theoretical based on osculating plane is gone out.It can be according to P on space curve₀Point The local geometric shape at place, is adaptively adjusted cutting step, and actual bow in every section of straight line is calculated using error function high is bent Error high so that be evenly distributed in the bow error high respectively approached in straightway, machining accuracy is high, and surface quality is good.

According to the relevant knowledge of Differential Geometry, at any on space curve, three orthogonal tangent lines, principal normals With the fundamental three-vector form that binormal constitutes curve, the osculating plane of this point is generated by them, normal plane and from tangent plane. In these three planes, osculating plane is the plane of " most pressing close to curve ", is tangent line and perpendicular to osculating plane from tangent plane.It is based on This conclusion, obtains discrete interpolated point, due at each point in the bow such as satisfaction error requirements high from the principal normal direction of tangent plane Bent curvature of a curve is different with torsion, error now bow error high not necessarily truly, so also needing to utilize The most longbow error high bent between error function calculating two-phase neighbor interpolation point high is verified.

According to being described above, the space curve interpolating method based on the theoretical satisfaction bow franchise high of osculating plane includes following step Suddenly：

Step (1) assumes parametric equation r=r (t) of space curve Γ={ x (t), y (t), z (t) } t ∈ [a, b], When interpolation starts, the 0th interpolated point P₀Parameter be t₀=a, bends franchise ε high, the bow franchise size influence free curve high Surface quality and machining accuracy, it is better to bend the smaller surface quality of franchise high, and the bow franchise high is by manufacturer according to being processed Surface set, be known quantity；

Step (2) is in P₀At point, r (t are calculated₀),r′(t₀),r″(t₀), try to achieve equation the r " (t from tangent plane π₀)·[ρ-r (t₀)]=0, ρ be any point from tangent plane vector, to bend franchise ε high be distance construction from tangent plane π on principal normal direction Equidistant surface π₁；

Step (3) simultaneous equidistant surface π₁The equation of equation and space curve Γ, solves equidistant surface π₁With the friendship of space curve Γ The parameter t of point₁；

Step (4) calculates t₀And t₁Two corresponding interpolated point P₀And P₁Between most longbow error max δ high.Solve most longbow Error high has mid-point method, dichotomy and Fibonacci method, but is not most longbow error high truly, error degree ratio Higher, the present invention establishes P by using the knowledge of the distance at analytic geometry midpoint to straight line₀And P₁Any point between 2 points Bow error function highThe method being most worth is asked to try to achieve max δ using function；

Step (5) is if max δ<ε, then P₁As next interpolated point, satisfaction bow Gao Yun is otherwise found using line search method The point that difference is required is used as next interpolated point；

Step (6) return to step (2)；Until an end of job for space curve.

Brief description of the drawings

Fig. 1 is equidistant surface schematic diagram.

Fig. 2 is flow chart of the method for the present invention.

Fig. 3 is the interpolated point schematic diagram of space curve Γ.

Fig. 4 is the most longbow error curve diagram high between space curve Γ interpolated points.

Specific embodiment

In order to make the purpose , technical scheme and advantage of the present invention be clearer, below in conjunction with accompanying drawing 1-4 and embodiment Son, is further described in detail to the present invention.Specific implementation example described herein is only used for the explanation present invention, not For limiting the present invention.

1. assume that a parametric equation of space curve Γ is r=r (t)={ t, t²,t³T ∈ [0,100], the 0th interpolation Point P₀Parameter be t₀=0, bend franchise ε=0.001 high.

2. in P₀At point, r (t are calculated₀),r′(t₀),r″(t₀),

Then r (t₀)={ t₀,t₀ ²,t₀ ³, r, ' (t₀)={ 1,2t₀,3t₀ ², r " (t₀)={ 0,2,6t₀,

Obtain and be from tangent plane π equations：r″(t₀)·[ρ-r(t₀)]=0, i.e. y+3t₀z-t₀ ²-3t₀ ⁴=0, ρ are to take up an official post from tangent plane The vector of a bit.

To bend franchise ε high it is being distance construction from the equidistant surface π of tangent plane π from the principal normal direction of tangent plane π₁：

3. simultaneous equidistant surface π₁The equation of equation and space curve Γ, obtains intersection point P₁Parameter t₁It is equationRoot, using matlab softwares solve.

4. P is calculated₀And P₁Most longbow error max δ high between two-phase neighbor interpolation.Using distance between beeline and dot in analytic geometry Implication, establish P₀And P₁The bow error function high of any point between 2 pointsProfit The method being most worth is asked to try to achieve max δ with function；

If 5. max δ<ε, then P₁As next interpolated point, interval is otherwise progressively reduced using line search method and is found completely The point of arch of foot tolerance requirements high is used as next interpolated point；

6. return to step (2)；Until all interpolated points on a space curve have determined.

Space curve Γ interpolated points schematic diagram and most longbow error curve diagram high are as shown in figs. 34.

Claims

1. a kind of space curve interpolating method theoretical based on osculating plane, it is characterised in that comprise the following steps that：

Step (1) assumes that a parametric equation of space curve Γ is r=r (t) t ∈ [a, b], when interpolation starts, the 0th interpolation Point P₀Parameter be t₀=a, bends franchise ε high；

Step (2) is in P₀At point, r (t are calculated₀),r′(t₀),r″(t₀), try to achieve equation the r " (t from tangent plane π₀)·[ρ-r(t₀)] =0, ρ are the vector of any point from tangent plane, to bend franchise ε high are distance construction from the equidistant of tangent plane π on principal normal direction Face π₁；

Step (3) simultaneous equidistant surface π₁The equation of equation and space curve Γ, solves equidistant surface π₁With the intersection point of space curve Γ Parameter t₁；

Step (4) calculates t₀And t₁Two corresponding interpolated point P₀And P₁Between most longbow error max δ high；

Step (5) is if max δ<ε, then P₁As next interpolated point, interval is otherwise progressively reduced using line search method and is found completely The point of arch of foot tolerance requirements high is used as next interpolated point；

Step (6) return to step (2)；Until an end of job for space curve.

2. the space curve interpolating method theoretical based on osculating plane according to claim 1, it is characterised in that

Step (4) sets up P₀And P₁The bow error function high of any point between 2 points The method being most worth is asked to try to achieve max δ using function.