CN102608956A - NURBS (Non-Uniform Rational B-Spline) curve adaptive interpolation control method based on de Boor algorithm - Google Patents

Abstract

A kind of nurbs curve adaptive interpolation control method based on de Boor algorithm, host computer calculates interpolated point, calculated coordinate value list is transmitted to slave computer, coordinate value is converted into corresponding pulses number by slave computer, and pulse is issued to motor driver, control motor rotation and drive mechanism movement; The nurbs curve adaptive interpolation control method includes following procedure: setting provides a parameter value in curve definitions domain

It is intended to calculate on the B-spline curves a corresponding point p (u), using the recurrence formula of De Boor algorithm, de Boor algorithm is applied in nurbs curve interpolation by the nurbs curve adaptive interpolation module based on de Boor algorithm. The present invention provides that a kind of reduction complexity, real-time be good, the higher nurbs curve adaptive interpolation control method based on de Boor algorithm of interpolation efficiency.

Description

A kind of nurbs curve adaptive interpolation control method based on de Boor algorithm

Technical field

The present invention relates to a kind of nurbs curve adaptive interpolation control method.

Background technology

In computer-aided design (CAD) in modern times (CAD)/computer-aided manufacturing (CAM) system; The complicated shape part; Usually represent with parametric equation like mould, aircraft wing and car model etc.; And traditional computer numerical control (CNC) lathe only provides linearity and circular interpolation; Have to as requested precision of CAD/CAM system is separated into parametric line to pass to behind a large amount of mini line segments and carries out part processing among the CNC, but such processing mode can be brought shortcomings such as machining precision reduction, digital control processing speed of feed instability, the reduction of part production efficiency.In order to overcome these shortcomings, modern digital control system begins the application parameter curve interpolating, and the parametric line interpolation can directly be passed to curve among the CNC, will not resolve into mini line segment by curve, thereby makes the information flow between CAD/CAM and the CNC continuous.At present commonly used is the nurbs curve interpolation, and it combines the advantage of implied expression formula and parametric polynomial in the curve and surface moulding, can express curve and surface and parsing curve and surface uniformly.Its Taylor expansion 1 rank approximate data of nurbs curve can realize that basically speed of feed is even; And Taylor expansion 2 rank approximate datas can further reduce velocity perturbation, yet all these algorithms all are devoted to obtain constant speed of feed and are not considered precision.For the nurbs curve interpolation, the control of speed of feed is the assurance of its interpolation precision, so realize the seamlessly transitting of speed of feed in the digital control processing, problem to be solved when when minimizing speed sharply changes the impact of lathe being interpolation.Therefore, the contour accuracy of the control of speed of feed and interpolation all plays a decisive role to the precision of interpolation in the nurbs curve interpolation process.

Summary of the invention

For the deficiency that complexity is high, real-time is poor, interpolation efficient is lower that overcomes existing nurbs curve interpolating method, the present invention provides a kind of complexity, good, higher nurbs curve adaptive interpolation control method based on de Boor algorithm of interpolation efficient of real-time of reducing.

The technical solution adopted for the present invention to solve the technical problems is:

A kind of nurbs curve adaptive interpolation control method based on de Boor algorithm; Host computer calculates interpolated point; Pass to slave computer to the coordinate figure tabulation that calculates; Slave computer converts coordinate figure to the corresponding pulses number, and sends pulse to motor driver, and the control motor rotates and the action of driving device structure; Said nurbs curve adaptive interpolation control method comprises following process:

Setting provides a parameter value in the curve definitions territory desire and calculates corresponding 1 p (u) on these B-spline curves, adopts the recursion formula of moral Boolean algorithm:

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_j{N}_{j,k}\left(u\right)=\underset{j=i-k}{\overset{i-l}{\mathrm{\Σ}}}{d}_j^l{N}_{j,k-1}\left(u\right)=...=d_{i-k}^k,---\left(2\right)$

$u\∈[u_i,u_{i+1}]\⋐[u_k,u_{n+1}]$

$d_j^l=\left\{\begin{array}{cc}{d}_j,& l=0\\ (1-a_j^l)d_j^{l-1}+a_j^l{d}_{j+1}^{l-1},& j=i-k,i-k+1,...,i-l;l=\mathrm{1,2},...,k\end{array}\right.---\left(3\right)$

$a_j^l=\frac{u-u_{j+l}}{u_{j+k+1}-u_{j+l}}$

Regulation $\frac{0}{0}=0,$

To k nurbs curve p (u), order $c\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{d}_i{N}_{i,k}\left(u\right),$ $w\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{N}_{i,k}\left(u\right),$ Find the solution c (u) and w (u) respectively application formula (2)～(3), promptly tries to achieve nurbs curve:

p(u)＝c(u)/w(u)(7)。

Further, the bow high level error that utilize to limit is carried out self-adaptation to the speed of feed of interpolation and is regulated, and detailed process is following: obtain on the nurbs curve of space the curvature k of any arbitrarily according to formula (19) _i, and then try to achieve radius-of-curvature ρ _i=1/k _i

$k_i=\frac{\frac{{\mathrm{dC}}_x\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_y\left(u\right)}{{\mathrm{du}}^2}-\frac{{\mathrm{dC}}_y\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_x\left(u\right)}{{\mathrm{du}}^2}{|}_{u=u_i}}{{\left|\right|\frac{\mathrm{dC}\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^3}---\left(19\right)$

Wherein, P (u _i) and P (u _I+1) be respectively u=u on the approximate circular arc _iAnd u=u _I+1The interpolated point at place; C (u _i) and C (u _I+1) be respectively u=u on the nurbs curve _iAnd u=u _I+1The interpolated point at place is because C (u _i)=C (u _I+1), make L _i=|| P (u _I+1)-P (u _i) ||, speed of feed V (u then _i) be expressed as approx

$V\left(u_i\right)=\frac{L_i}{T_s}---\left(20\right)$

Bow high level error ER is the deviation of equivalent chord length and actual SPL, is expressed as

$\mathrm{ER}={\mathrm{\ρ}}_i-\sqrt{{\mathrm{\ρ}}_i^2-{\left(\frac{L}{2}\right)}^2}---\left(21\right)$

If limit the size of bow high level error ER, then corresponding speed of feed V (u _i) do

$V\left(u_i\right)=\frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}---\left(22\right)$

Generally, ρ _iSo＞＞ER is V (u in the formula (22) _i) the result be a real number value

Formula (22) shows speed of feed V (u _i) should be with ER and ρ _iThe adaptive adjustment of variation, regulation rule is following:

$V\left(u_i\right)=\left\{\begin{array}{cc}F,& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}>F\\ \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2},& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}\≤F\end{array}\right.---\left(23\right)$

Wherein, F is the speed of feed command value; If the radius-of-curvature of current point is enough little on the space curve; Then bend high level error and possibly surpass limit value, at this moment interpolation algorithm with speed of feed F is reduced to

to satisfy the requirement of the bow high level error ER limit; Otherwise the speed of feed F with given proceeds interpolation.

Technical conceive of the present invention is: the contour accuracy with speed of feed control and interpolation in the realization nurbs curve interpolation process is a target, makes nurbs curve when high-speed interpolation, can keep keeping again the adjustment of speed the progress of interpolation.The technical matters that solves: the one, de Boor algorithm application in the nurbs curve interpolation, is reduced the complicacy of interpolation algorithm, thereby improve the real-time and the interpolation efficient of CNC system; The 2nd, in nurbs curve interpolation process, limit the bow high level error speed of feed of interpolation realized that self-adaptation regulates, realize seamlessly transitting of speed of feed in the numerical control process, when minimizing speed sharply changes to the impact of lathe.

Beneficial effect of the present invention mainly shows: 1) the present invention with de Boor algorithm application in the nurbs curve interpolation; Avoided the iterative solution process of B spline base function; The greatly lower complicacy of algorithm, thus the real-time and the interpolation efficient of CNC system improved.2) the present invention carries out self-adaptation to the speed of feed of interpolation and regulates with limiting the bow high level error, has realized seamlessly transitting of speed of feed in the digital control processing, has reduced when speed sharply changes the impact to lathe.

Description of drawings

Fig. 1 is the estimation synoptic diagram to next interpolated point.

Embodiment

Below in conjunction with accompanying drawing the present invention is further described.

With reference to Fig. 1; A kind of nurbs curve adaptive interpolation control method based on de Boor algorithm; Host computer calculates interpolated point, passes to slave computer to the coordinate figure tabulation that calculates, and slave computer converts coordinate figure to the corresponding pulses number; And sending pulse to motor driver, the control motor rotates and the action of driving device structure; Said nurbs curve adaptive interpolation control method comprises following process:

Setting provides a parameter value in the curve definitions territory desire and calculates corresponding 1 p (u) on these B-spline curves, adopts the recursion formula of moral Boolean algorithm:

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_j{N}_{j,k}\left(u\right)=\underset{j=i-k}{\overset{i-l}{\mathrm{\Σ}}}{d}_j^l{N}_{j,k-1}\left(u\right)=...=d_{i-k}^k,---\left(2\right)$

$u\∈[u_i,u_{i+1}]\⋐[u_k,u_{n+1}]$

$d_j^l=\left\{\begin{array}{cc}{d}_j,& l=0\\ (1-a_j^l)d_j^{l-1}+a_j^l{d}_{j+1}^{l-1},& j=i-k,i-k+1,...,i-l;l=\mathrm{1,2},...,k\end{array}\right.---\left(3\right)$

$a_j^l=\frac{u-u_{j+l}}{u_{j+k+1}-u_{j+l}}$

Regulation $\frac{0}{0}=0,$

To k nurbs curve p (u), order $c\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{d}_i{N}_{i,k}\left(u\right),$ $w\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{N}_{i,k}\left(u\right),$ Find the solution c (u) and w (u) respectively application formula (2)～(3), promptly tries to achieve nurbs curve:

p(u)＝c(u)/w(u) (7)。

Further, the bow high level error that utilize to limit is carried out self-adaptation to the speed of feed of interpolation and is regulated, and detailed process is following: obtain on the nurbs curve of space the curvature k of any arbitrarily according to formula (19) _i, and then try to achieve radius-of-curvature ρ _i=1/k _i

$k_i=\frac{\frac{{\mathrm{dC}}_x\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_y\left(u\right)}{{\mathrm{du}}^2}-\frac{{\mathrm{dC}}_y\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_x\left(u\right)}{{\mathrm{du}}^2}{|}_{u=u_i}}{{\left|\right|\frac{\mathrm{dC}\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^3}---\left(19\right)$

Wherein, P (u _i) and P (u _I+1) be respectively u=u on the approximate circular arc _iAnd u=u _I+1The interpolated point at place; C (u _i) and C (u _I+1) be respectively u=u on the nurbs curve _iAnd u=u _I+1The interpolated point at place is because C (u _i)=C (u _I+1), make L _i=|| P (u _I+1)-P (u _i) ||, speed of feed V (u then _i) be expressed as approx

$V\left(u_i\right)=\frac{L_i}{T_s}---\left(20\right)$

Bow high level error ER is the deviation of equivalent chord length and actual SPL, is expressed as

$\mathrm{ER}={\mathrm{\ρ}}_i-\sqrt{{\mathrm{\ρ}}_i^2-{\left(\frac{L}{2}\right)}^2}---\left(21\right)$

If limit the size of bow high level error ER, then corresponding speed of feed V (u _i) do

$V\left(u_i\right)=\frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}---\left(22\right)$

Generally, ρ _iSo＞＞ER is V (u in the formula (22) _i) the result be a real number value

Formula (22) shows speed of feed V (u _i) should be with ER and ρ _iThe adaptive adjustment of variation, regulation rule is following:

$V\left(u_i\right)=\left\{\begin{array}{cc}F,& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}>F\\ \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2},& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}\≤F\end{array}\right.---\left(23\right)$

Wherein, F is the speed of feed command value; If the radius-of-curvature of current point is enough little on the space curve; Then bend high level error and possibly surpass limit value, at this moment interpolation algorithm with speed of feed F is reduced to

to satisfy the requirement of the bow high level error ER limit; Otherwise the speed of feed F with given proceeds interpolation.

In the present embodiment, with de Boor algorithm application in nurbs curve: given k B-spline curves:

$p\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{d}_i{N}_{i,k}\left(u\right)---\left(1\right)$

Control vertex d wherein _i(i=0,1 ..., n), knot vector U=[u ₀, u ₁..., u _N+k+1].Calculate corresponding 1 p (u) on these B-spline curves if provide a parameter value in the curve definitions territory

desire, can adopt the recursion formula of moral Boolean algorithm:

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_j{N}_{j,k}\left(u\right)=\underset{j=i-k}{\overset{i-l}{\mathrm{\Σ}}}{d}_j^l{N}_{j,k-1}\left(u\right)=...=d_{i-k}^k,---\left(2\right)$

$u\∈[u_i,u_{i+1}]\⋐[u_k,u_{n+1}]$

$d_j^l=\left\{\begin{array}{cc}{d}_j,& l=0\\ (1-a_j^l)d_j^{l-1}+a_j^l{d}_{j+1}^{l-1},& j=i-k,i-k+1,...,i-l;l=\mathrm{1,2},...,k\end{array}\right.---\left(3\right)$

$a_j^l=\frac{u-u_{j+l}}{u_{j+k+1}-u_{j+l}}$

Regulation $\frac{0}{0}=0.$

Also can ask the arrow of leading of B-spline curves with the moral Boolean algorithm; As ask k some r rank at place on the B-spline curves to lead and vow p (r) (u), can calculate by following recursion formula:

$p^{\left(r\right)}\left(u\right)=\frac{d^r}{{\mathrm{du}}^r}\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_j{N}_{j,k}\left(u\right)=\underset{j=i-k}{\overset{i-r}{\mathrm{\Σ}}}{d}_j^r{N}_{j,k-r}\left(u\right)---\left(4\right)$

Wherein new reference mark does

$d_j^l=\left\{\begin{array}{cc}{d}_j,& l=0\\ (k-l+1)\frac{d_{j+1}^{l-1}-d_j^{l-1}}{u_{j+k+1}-u_{j+l}},& j=i-k,i-k+1,...,i-r;l=\mathrm{1,2},...,r\end{array}\right.---\left(5\right)$

Knot vector does

U ^r＝[u ₀ ^(r)，u ₁ ^(r)，…，u ^(r) _n+k?2r+1]＝[0，…，0，u _k+1，…，u _n，1，…，1](6)

Wherein 0 and 1 all is k+1-r.

Can know that through analyzing the molecule denominator of nurbs curve is B-spline curves, therefore can use de Boor algorithm to find the solution respectively to the molecule denominator, the ratio of the two be institute and asks.To k nurbs curve p (u), order $c\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{d}_i{N}_{i,k}\left(u\right),$ $w\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{N}_{i,k}\left(u\right),$ Application formula (2)～(3) can be found the solution c (u) and w (u) respectively, promptly try to achieve nurbs curve:

p(u)＝c(u)/w(u)(7)

The single order that differentiate can get p (u) to formula (7) is led and is vowed and do

$p^{\′}\left(u\right)=\frac{w\left(u\right)c^{\′}\left(u\right)-w^{\′}\left(u\right)c\left(u\right)}{{\left[w\left(u\right)\right]}^2}---\left(8\right)$

The second order that differentiate can get p (u) to formula (8) is led and is vowed and do

$p^{\′\′}\left(u\right)=\frac{c^{\′\′}\left(u\right){\left[w\left(u\right)\right]}^2-c\left(u\right)w^{\′\′}\left(u\right)w\left(u\right)-2c^{\′}\left(u\right)w\left(u\right)w^{\′}\left(u\right)+2c\left(u\right){\left[w\left(u\right)\right]}^2}{{\left[w\left(u\right)\right]}^3}---\left(9\right)$

Wherein, c ' (u), w ' (u), c " (u), w " (u) can try to achieve by the de Boor algorithm of formula (2)～(6).

In real time de Boor algorithm application is in the nurbs curve interpolation: from seeing in essence; Make the numerical control device of drive unit with servomotor; Its digital control system is a discrete numerical control device, and its digital control system is a discrete sampled-data system, so adopt the discrete data sampling interpolation here.During work, in each sampling period, digital control system goes out the cutter position that will arrive of following one-period according to speed of feed V (t) real-time interpolation of setting, and the motion of Control Servo System thus.When this parametric line of processing nurbs curve, because position and parameter value on the curve are one to one, so the just continuous recursion calculating parameter u of interpolation process _iProcess.

The curve arc long time differential is:

$V^{\′}\left(t\right)=\frac{\mathrm{ds}}{\mathrm{dt}}=\left(\frac{\mathrm{ds}}{\mathrm{dt}}\right)\left(\frac{\mathrm{du}}{\mathrm{dt}}\right)---\left(10\right)$

So:

$\frac{\mathrm{du}}{\mathrm{dt}}=\frac{V^{\′}\left(t\right)}{\mathrm{ds}/\mathrm{du}}---\left(11\right)$

Can be similar to and think that V ' (t) equals the speed of feed V (t) that sets.Can know according to infinitesimal geometry:

$\frac{\mathrm{ds}}{\mathrm{du}}=\sqrt{{\left[x^{\′}\left(u\right)\right]}^2+{\left[y^{\′}\left(u\right)\right]}^2+{\left[z^{\′}\left(u\right)\right]}^2}---\left(12\right)$

Wherein

x＝P _x(u)，y＝P _y(u)，z＝P _z(u)，

$x^{\′}=\frac{{\mathrm{dP}}_x\left(u\right)}{\mathrm{du}},$ $y^{\′}=\frac{{\mathrm{dP}}_y\left(u\right)}{\mathrm{du}},$ $z^{\′}=\frac{{\mathrm{dP}}_z\left(u\right)}{\mathrm{du}}---\left(13\right)$

Can get the differentiate of parameters u by formula (11), (12) to time t:

$\frac{\mathrm{du}}{\mathrm{dt}}=\frac{V\left(t\right)}{\sqrt{{\left[x^{\′}\left(u\right)\right]}^2+{\left[y^{\′}\left(u\right)\right]}^2+{\left[z^{\′}\left(u\right)\right]}^2}}---\left(14\right)$

Following formula is carried out the second order Taylor series expansion:

$u_{i+1}=u_i+(t_{i+1}-t_i)\frac{\mathrm{du}}{\mathrm{dt}}{|}_{t=t_i}+\frac{1}{2}{(t_{i+1}-t_i)}^2\frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{t=t_i}+\mathrm{HOT}---\left(15\right)$

Wherein, t _I-1-t _i=T, HOT are the high-order a small amount of of taylor series expansion, can be known by formula (14)

$\frac{\mathrm{du}}{\mathrm{dt}}{|}_{t=t_i}=\frac{V\left(t_i\right)}{\sqrt{{\left[x^{\′}\left(u_i\right)\right]}^2+{\left[y^{\′}\left(u\right)\right]}^2+{\left[z^{\′}\left(u\right)\right]}^2}}---\left(16\right)$

$\frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{t=t_i}=\frac{\mathrm{du}/\mathrm{dt}}{\mathrm{dt}}{|}_{t=t_i}=\frac{\mathrm{dV}\left(t\right)/\mathrm{dt}{|}_{t=t_i}}{\sqrt{{\left[x^{\′}\left(u\right)\right]}^2+{\left[y^{\′}\left(u\right)\right]}^2+{\left[z^{\′}\left(u\right)\right]}^2}}---\left(17\right)$

$\frac{{\left[V\left(t_i\right)\right]}^2[x^{\′}\left(u_i\right)x^{\′\′}\left(u_i\right)+y^{\′}\left(u_i\right)y^{\′\′}\left(u_i\right)+z^{\′}\left(u_i\right)z^{\′\′}\left(u_i\right)]}{{\{{\left[x^{\′}\left(u\right)\right]}^2+{\left[y^{\′}\left(u\right)\right]}^2+{\left[z^{\′}\left(u\right)\right]}^2\}}^2}$

With formula (16), (17) substitution formula (15) and omit high-order term, have

$u_{i+1}=u_i+\frac{\mathrm{TV}\left(t_i\right)+\frac{T^2}{2}\frac{\mathrm{dV}\left(t\right)}{\mathrm{dt}}{|}_{t=t_i}}{\sqrt{{\left[x^{\′}\left(u\right)\right]}^2+{\left[y^{\′}\left(u\right)\right]}^2+{\left[z^{\′}\left(u\right)\right]}^2}}---\left(18\right)$

$\frac{T^2{V}^2\left(t_i\right)[x^{\′}\left(u_1\right)x^{\′\′}\left(u_i\right)+y^{\′}\left(u_i\right)y^{\′\′}\left(u_i\right)+z^{\′}\left(u_i\right)z^{\′\′}\left(u_i\right)]}{2{\{{\left[x^{\′}\left(u_i\right)\right]}^2+{\left[y^{\′}\left(u_i\right)\right]}^2+{\left[z^{\′}\left(u_i\right)\right]}^2\}}^2}$

Wherein, x ' (u _i), x " (u _i), y ' (u _i), y " (u _i), z ' (u _i), z " (u _i) calculate by the de Boor algorithm of formula (4)～(9).When actual computation, V in the following formula (t) is the speed of feed of setting, and T is the known sampling period, but then through type (18) can be by the pairing u of current point position coordinates _iValue is calculated the pairing u of next interpolated point _I+1Value.

The control of nurbs curve speed adaptive: what Fig. 1 represented is that one section circular arc is similar at interval u ∈ [u _i, u _I+1] interior nurbs curve.ρ _iBe at u=u _iThe radius-of-curvature at place, and ρ _i=1/k _iWherein, k _iBe any curvature of any on the nurbs curve of space, can pass through computes:

$k_i=\frac{\frac{{\mathrm{dC}}_x\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_y\left(u\right)}{{\mathrm{du}}^2}-\frac{{\mathrm{dC}}_y\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_x\left(u\right)}{{\mathrm{du}}^2}{|}_{u=u_i}}{{\left|\right|\frac{\mathrm{dC}\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^3}---\left(19\right)$

P (u among Fig. 1 _i) and P (u _I+1) be respectively u=u on the approximate circular arc _iAnd u=u _I+1The interpolated point at place; C (u _i) and C (u _I+1) be respectively u=u on the nurbs curve _iAnd u=u _I+1The interpolated point at place.Because C (u _i)=C (u _I+1), make L _i=|| P (u _I+1)-P (u _i) ||, speed of feed V (u then _i) can be expressed as approx

$V\left(u_i\right)=\frac{L_i}{T_s}---\left(20\right)$

Bow high level error ER is the deviation of equivalent chord length and actual SPL, can be expressed as

$\mathrm{ER}={\mathrm{\ρ}}_i-\sqrt{{\mathrm{\ρ}}_i^2-{\left(\frac{L}{2}\right)}^2}---\left(21\right)$

If limit the size of bow high level error ER, then corresponding speed of feed V (u _i) do

$V\left(u_i\right)=\frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}---\left(22\right)$

Generally, ρ _iSo＞＞ER is V (u in the formula (22) _i) the result be a real number value

Formula (22) shows speed of feed V (u _i) should be with ER and ρ _iThe adaptive adjustment of variation, regulation rule is following:

$V\left(u_i\right)=\left\{\begin{array}{cc}F,& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}>F\\ \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2},& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2\≤}F\end{array}\right.---\left(23\right)$

Wherein, F is the speed of feed command value.If the radius-of-curvature of current point is enough little on the space curve; Then bend high level error and possibly surpass limit value, at this moment interpolation algorithm with speed of feed F is reduced to

to satisfy the requirement of the bow high level error ER limit; Otherwise the speed of feed F with given proceeds interpolation.

With adjusted speed of feed substitution formula (18), obtain the parameter value u at next interpolated point place _I+1, and its substitution formula (13) obtained the x of next interpolated point, y, z coordinate figure.

Instance: realize this algorithm with the MATLAB programming.Simple for describing, only planar draw two dimensional image, promptly let the reference mark triaxial coordinate get zero, parameter information is following: number of times k=3, weight factor w={1,1,1,1,1,1}, control point information:

B={ (10,10,0), (60,80,0), (100,90; 0), (200,20,0), (250,30,0), (300; 120,0) } sequence node u={0,0,0,0,0.3,0.7,0; 0,0,0}, speed of feed v=50mm/s, the bow high level error of requirement is made as 0.002mm, and the interpolated point data of calculating are huge, only provide part and begin the point with the concluding paragraph territory.

Table 1.

Claims (2)

1. nurbs curve adaptive interpolation control method based on de Boor algorithm; Host computer calculates interpolated point; Pass to slave computer to the coordinate figure tabulation that calculates; Slave computer converts coordinate figure to the corresponding pulses number, and sends pulse to motor driver, and the control motor rotates and the action of driving device structure;

It is characterized in that: said nurbs curve adaptive interpolation control method comprises following process:

Setting provides a parameter value in the curve definitions territory desire and calculates corresponding 1 p (u) on these B-spline curves, adopts the recursion formula of moral Boolean algorithm:

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_j{N}_{j,k}\left(u\right)=\underset{j=i-k}{\overset{i-l}{\mathrm{\Σ}}}{d}_j^l{N}_{j,k-1}\left(u\right)=...=d_{i-k}^k,---\left(2\right)$

$u\∈[u_i,u_{i+1}]\⋐[u_k,u_{n+1}]$

$d_j^l=\left\{\begin{array}{cc}{d}_j,& l=0\\ (1-a_j^l)d_j^{l-1}+a_j^l{d}_{j+1}^{l-1},& j=i-k,i-k+1,...,i-l;l=\mathrm{1,2},...,k\end{array}\right.---\left(3\right)$

$a_j^l=\frac{u-u_{j+l}}{u_{j+k+1}-u_{j+l}}$

Regulation $\frac{0}{0}=0,$

To k nurbs curve p (u), order $c\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{d}_i{N}_{i,k}\left(u\right),$ $w\left(u\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{w}_i{N}_{i,k}\left(u\right),$ Find the solution c (u) and w (u) respectively application formula (2)～(3), promptly tries to achieve nurbs curve:

p(u)＝c(u)/w(u)(7)。

2. a kind of nurbs curve adaptive interpolation control method as claimed in claim 1 based on de Boor algorithm; It is characterized in that: the bow high level error that utilize to limit is carried out self-adaptation to the speed of feed of interpolation and is regulated, and detailed process is following: obtain on the nurbs curve of space the curvature k of any arbitrarily according to formula (19) _i, and then try to achieve radius-of-curvature ρ _i=1/k _i

$k_i=\frac{\frac{{\mathrm{dC}}_x\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_y\left(u\right)}{{\mathrm{du}}^2}-\frac{{\mathrm{dC}}_y\left(u\right)}{\mathrm{du}}\·\frac{d^2{C}_x\left(u\right)}{{\mathrm{du}}^2}{|}_{u=u_i}}{{\left|\right|\frac{\mathrm{dC}\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^3}---\left(19\right)$

Wherein, P (u _i) and P (u _I+1) be respectively u=u on the approximate circular arc _iAnd u=u _I+1The interpolated point at place; C (u _i) and C (u _I+1) be respectively u=u on the nurbs curve _iAnd u=u _I+1The interpolated point at place is because C (u _i)=C (u _I+1), make L _i=|| P (u _I+1)-P (u _i) ||, speed of feed V (u then _i) be expressed as approx

$V\left(u_i\right)=\frac{L_i}{T_s}---\left(20\right)$

Bow high level error ER is the deviation of equivalent chord length and actual SPL, is expressed as

$\mathrm{ER}={\mathrm{\ρ}}_i-\sqrt{{\mathrm{\ρ}}_i^2-{\left(\frac{L}{2}\right)}^2}---\left(21\right)$

If limit the size of bow high level error ER, then corresponding speed of feed V (u _i) do

$V\left(u_i\right)=\frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}---\left(22\right)$

Generally, ρ _iSo＞＞ER is V (u in the formula (22) _i) the result be a real number value

Formula (22) shows speed of feed V (u _i) should be with ER and ρ _iThe adaptive adjustment of variation, regulation rule is following:

$V\left(u_i\right)=\left\{\begin{array}{cc}F,& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}>F\\ \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2},& \frac{2}{T}\sqrt{{\mathrm{\ρ}}_i^2-{({\mathrm{\ρ}}_i-\mathrm{ER})}^2}\≤F\end{array}\right.---\left(23\right)$

Wherein, F is the speed of feed command value; If the radius-of-curvature of current point is enough little on the space curve; Then bend high level error and possibly surpass limit value, at this moment interpolation algorithm with speed of feed F is reduced to

to satisfy the requirement of the bow high level error ER limit; Otherwise the speed of feed F with given proceeds interpolation.

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