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

NURBS (Non-Uniform Rational B-Spline) curve adaptive interpolation control method based on de Boor algorithm Download PDF

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CN102608956A
CN102608956A CN 201210055422 CN201210055422A CN102608956A CN 102608956 A CN102608956 A CN 102608956A CN 201210055422 CN201210055422 CN 201210055422 CN 201210055422 A CN201210055422 A CN 201210055422A CN 102608956 A CN102608956 A CN 102608956A
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curve
interpolation
nurbs
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南余荣
吴攀峰
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浙江工业大学
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Abstract

The invention discloses a NURBS (Non-Uniform Rational B-Spline) curve adaptive interpolation control method based on a de Boor algorithm. An upper computer calculates an interpolation point and transfers the calculated coordinate value list to a lower computer; and the lower computer converts the coordinate value into corresponding pulse number and sends a pulse to a motor driver which controls a motor to rotate and drives a mechanical structure to act. The NURBS curve adaptive interpolation control method comprises the following steps: a parameter value in a curve definition domain is set to calculate a corresponding point p(u) on the B-spline curve; and a NURBS curve adaptive interpolation module based on the de Boor algorithm applies the de Boor algorithm to the NURBS curve interpolation by use of a recursion formula of the de Boor algorithm. The invention provides the NURBS curve adaptive interpolation control method based on the de Boor algorithm, which reduces the complexity and has good real-time performance and relatively high interpolation efficiency.

Description

—种基于de Boor算法的NURBS曲线自适应插补控制方法 - Adaptive kinds of NURBS curve interpolation method for controlling de Boor algorithm

技术领域 FIELD

[0001] 本发明涉及一种NURBS曲线自适应插补控制方法。 [0001] The present invention relates to an adaptive NURBS curve interpolation control method.

背景技术 Background technique

[0002] 在现代计算机辅助设计(CAD)/计算机辅助制造(CAM)系统中,复杂形状零件,如模具、飞机机翼和汽车模型等通常用参数方程表示,而传统计算机数控(CNC)机床只提供线性和圆弧插补,CAD/CAM系统不得不按照要求的精度将参数曲线离散成大量的微小线段后传到CNC中进行零件加工,但这样的处理方式会带来加工精度降低、数控加工进给速度不稳定、零件生产效率降低等缺点。 [0002] In modern computer-aided design (CAD) / Computer Aided Manufacturing (CAM) system, complex shapes, such as molds, aircraft wings, and automotive models commonly used parametric equations, the conventional computer numerical control (CNC) machines only after providing the linear and circular interpolation, CAD / CAM systems have required accuracy in accordance with the parameter curve of minute line segments into a large number of discrete passes for the CNC machining parts, but this approach may impair the accuracy of machining, NC feed speed is unstable, reducing the productivity of parts and other shortcomings. 为了克服这些缺点,现代数控系统开始应用参数曲线插补,参数曲线插补可以直接将曲线传到CNC中,不必将曲线分解成微小线段,从而使CAD/ CAM和CNC之间的信息流连续。 To overcome these drawbacks, the system began to use modern numerical parameter curve interpolation, curve interpolation parameter curve may be directly transmitted to the CNC, the curve need not be decomposed into minute line segments, so that the flow of information between the CAD / CAM and the CNC continuous. 目前常用的是NURBS曲线插补,它综合了曲线曲面造型中隐式表达式与参数多项式的优点,可以统一地表达曲线曲面和解析曲线曲面。 The most commonly used is the NURBS curve interpolation, which combines the advantages of curve and surface modeling in polynomial expressions and implicit parameters, it can be unified expression of curves and surfaces and analytic curves and surfaces. NURBS曲线其泰勒展开式I阶近似算法基本可以实现进给速度均匀,而泰勒展开式2阶近似算法可以进一步减少速度波动,然而所有这些算法都致力于取得恒定的进给速度而没有考虑精度。 NURBS curve with a Taylor expansion approximation algorithm for Stage I substantially uniform rate of feed can be achieved, while the second order Taylor series approximation algorithm speed fluctuation can be further reduced, however, all of these algorithms are committed to achieve a constant feed rate without regard to precision. 对于NURBS曲线插补,进给速度的控制是其插补精度的保证,所以实现数控加工中进给速度的平滑过渡、减少速度急剧变化时对机床的冲击是插补时所要解决的问题。 For NURBS curve interpolation, the feed rate is controlled to ensure that interpolation accuracy, so that a smooth transition NC machining feedrate, the impact on the machine speed decreases abruptly changes is an interpolation problem to be solved. 因此,NURBS曲线插补过程中进给速度的控制和插补的轮廓精度对插补的精度都起决定性作用。 Thus, NURBS curve interpolation process feeding speed control and contour interpolation accuracy of interpolation accuracy are decisive.

发明内容 SUMMARY

[0003] 为了克服已有NURBS曲线插补方法的复杂度高、实时性差、插补效率较低的不足, 本发明提供一种降低复杂度、实时性良好、插补效率较高的基于de Boor算法的NURBS曲线自适应插补控制方法。 [0003] In order to overcome the existing high complexity NURBS interpolation method, real-time difference, the lower the efficiency of interpolation is insufficient, the present invention provides a method of reducing complexity, good real-time, high efficiency interpolation based on de Boor NURBS curve interpolation algorithm for adaptive control method.

[0004] 本发明解决其技术问题所采用的技术方案是: [0004] aspect of the present invention to solve the technical problem are:

[0005] —种基于de Boor算法的NURBS曲线自适应插补控制方法,上位机计算出插补点, 把计算出的坐标值列表传给下位机,下位机把坐标值转换成相应脉冲数,并发出脉冲到电机驱动器,控制电机转动并驱动机械结构动作;所述NURBS曲线自适应插补控制方法包括以下过程: [0005] - kinds of NURBS curve interpolation de Boor algorithm for adaptive control method, the host computer calculates the interpolation points, the calculated coordinate values ​​to the list of the next crew, the next crew to convert the coordinate values ​​corresponding to the number of pulses, and emits a pulse to the motor driver, the motor rotates and drives the control operation of the mechanical structure; the NURBS curve interpolation adaptive control method comprising the following procedures:

[0006] 设定给出曲线定义域内一参数值M £[%,%.+1]=[义,?/„+1],欲计算该B样条曲线上对应一点P (u),采用德布尔算法的递推公式: [0006] curve definition given in the art to set a parameter value M £ [%,%. + 1] = [Yi,? / '+ 1], is calculated to be the corresponding point P (u) on the B-spline curves, using De Boor's algorithm is recursive formula:

n il n il

[0007] p(u) = YjdjNhk(u) = Z d)Nhk_x(u) = --- = dlk, [0007] p (u) = YjdjNhk (u) = Z d) Nhk_x (u) = --- = dlk,

j=o j=ik j = o j = ik

[0008] //e[u+1]c[&,//M+1] [0008] // e [u + 1] c [&, // M + 1]

[0009] ] I (I —a;)-1 +«:J, )= /• — 々,/• — 々+ 1,...,/• — /;/= 1,2,...,众CN 102608956 A [0009]] I (I -a;) - 1 + «: J,) = / • - 々, / • - 々 + 1, ..., / • - /; / = 1,2, ... , all the CN 102608956 A

book

Bright

[0011]规定 [0011] provisions

Figure CN102608956AD00051

[0012]对 k 次NURBS 曲线P (U),令 [0012] The k-th NURBS curve P (U), so that

Figure CN102608956AD00052

应用式⑵〜 Application type ⑵~

i=0 i=0 i = 0 i = 0

(3)分别对c(u)和w(u)进行求解,即求得NURBS曲线: (3) respectively, c (u) and w (u) is solved, i.e. NURBS curves obtained:

[0013] p(u) = c(u)/w(u) (7) o [0013] p (u) = c (u) / w (u) (7) o

[0014] 进一步,利用限定的弓高误差对插补的进给速度实行自适应调节,具体过程如下根据式(19)求出空间NURBS曲线上任意一点的曲率匕,进而求得曲率半径P i = IAi ; [0014] Further, with a defined chord error imposed on the feed rate of the adaptive interpolation procedure is as follows according to the NURBS curve calculated on any point of the space (19) of curvature dagger, then obtain the curvature radius P i = IAi;

Figure CN102608956AD00053

[0016] 其中,P (Ui)和P(ui+1)分别是近似圆弧上u = Ui和u = ui+1处的插补点;C (Ui)和C(ui+1)分别是NURBS曲线上u = Ui和u = ui+1处的插补点,由于C(Ui) = C(ui+1),令Li = [0016] wherein, P (Ui) and P (ui + 1) are approximately circular arc and u = u = Ui at the interpolation point ui + 1; C (Ui) and C (ui + 1) are u = Ui on the NURBS curve and u = ui + 1 at the interpolation point, since C (Ui) = C (ui + 1), so that Li =

IP (ui+1)-P (Ui) I |,则进给速度V(Ui)近似地表示为 IP (ui + 1) -P (Ui) it |, then the feed speed V (Ui) is approximately expressed as

[0017] [0017]

[0018] [0018]

[0019] [0019]

[0020] [0020]

[0021] [0021]

[0022] [0022]

[0023] [0023]

V(U1) = I (20) V (U1) = I (20)

弓高误差ER是等效弦长与实际样条曲线的偏差,表示为 Chord error ER is equivalent to the deviation from the actual chord spline curve, expressed as

ER = P1-^-(-)2 (21) ER = P1 - ^ - (-) 2 (21)

如果限定弓高误差ER的大小,则相应的进给速度V(Ui)为V(U1) = -(Pi-ERf (22) If the chord error ER defined size, the corresponding feed speed V (Ui) to V (U1) = - (Pi-ERf (22)

通常情况下,P i >> ER,所以式(22)中V(Ui)的结果是一个实数值式(22)表明进给速度V(Ui)应随ER和Pi的变化自适应的调整,调整规则如下 Typically, P i >> ER, and therefore equation (22) results in V (Ui) is to show a real value of formula (22) feed rate V (Ui) to be adjusted with the adaptive variations ER and Pi, adjust the rules are as follows

[0025] 其中,F是进给速度指令值,如果空间曲线上当前点的曲率半径足够小,则弓高误差可能超过限定值,这时插补算法将进给速度由F减小到2 -(P, -ER)2,以满足限定的弓高误差ER的要求;否则以给定的进给速度F继续进行插补。 [0025] where, F is the feed speed command value, if the current point on the radius of curvature sufficiently small space curve, the chord error may exceed the limit value, then the interpolation algorithm F from the feed speed is reduced to 2 - (P, -ER) 2, in order to meet the requirements of the chord error ER defined; or a given feed rate F continues to be interpolated.

[0026] 本发明的技术构思为:以实现NURBS曲线插补过程中进给速度控制和插补的轮廓精度为目标,使NURBS曲线在高速插补时既能保持对速度的调整又能保持插补的进度。 [0026] The technical idea of ​​the present invention are: to achieve a NURBS interpolation process feed speed control and contour accuracy of interpolation for the target, so that adjustments can maintain the NURBS curve at high speed while maintaining interpolation interpolation make progress. 要解决的技术问题:一是将de Boor算法应用到NURBS曲线插补中,降低插补算法的复杂性, 从而提高CNC系统的实时性和插补效率;二是在NURBS曲线插补过程中限定弓高误差对插补的进给速度实现自适应调节,实现数控过程中进给速度的平滑过渡,减少速度急剧变化时对机床的冲击。 To solve the technical problems: First, the de Boor algorithm to NURBS interpolation, reducing the complexity of interpolation algorithm to improve timeliness and efficiency of CNC interpolation system; the second is limited to NURBS interpolation process chord error adaptive adjustment of the feed rate interpolation, NC smooth transition during the feed rate, to reduce the impact when the machine speed changes abruptly.

[0027] 本发明的有益效果主要表现在:1)本发明将de Boor算法应用到NURBS曲线插补中,避免了B样条基函数的迭代解过程,极大地较低了算法的复杂性,从而提高了CNC系统的实时性和插补效率。 [0027] Advantageous effects of the present invention are mainly: 1) of the present invention will be de Boor algorithm to NURBS curve interpolation, avoids the iterative solution procedure B-spline, which greatly lower the complexity of the algorithm, thereby improving the timeliness and efficiency of CNC interpolation system. 2)本发明用限定弓高误差对插补的进给速度实行自适应调节,实现了数控加工中进给速度的平滑过渡,减少了速度急剧变化时对机床的冲击。 2) The present invention is defined by chord error implement adaptive interpolation of feed rate, to achieve the NC machining feed speed of the smooth transition, reducing the impact when the machine speed changes abruptly.

附图说明 BRIEF DESCRIPTION

[0028] 图I是对下一个插补点的估计示意图。 [0028] FIG. I is a schematic view of a next estimation of the interpolation point.

具体实施方式 detailed description

[0029] 下面结合附图对本发明作进一步描述。 [0029] The following drawings in conjunction with the present invention will be further described.

[0030] 参照图I, 一种基于de Boor算法的NURBS曲线自适应插补控制方法,上位机计算出插补点,把计算出的坐标值列表传给下位机,下位机把坐标值转换成相应脉冲数,并发出脉冲到电机驱动器,控制电机转动并驱动机械结构动作;所述NURBS曲线自适应插补控制方法包括以下过程: [0030] Referring to FIG I, one kind of adaptive NURBS curve interpolation method for controlling de Boor algorithm, the host computer calculates the interpolation points, the calculated coordinate values ​​to the list of the next crew, the next crew converted into coordinate values the corresponding number of pulses, and emits a pulse to the motor driver, the motor rotates and drives the control operation of the mechanical structure; the NURBS curve interpolation adaptive control method comprising the following procedures:

[0031] 设定给出曲线定义域内一参数值MG |^.,%.+1][[义,?/„+1],欲计算该B样条曲线上对应一点P (U),采用德布尔算法的递推公式: [0031] curve definition given in the art to set a parameter value MG | ^,% + 1] [[Yi, / '1 +?], Calculated to be the corresponding point P (U) on the B-spline curve employed. De Boor's algorithm is recursive formula:

[0032] [0032]

[0033] [0033]

Figure CN102608956AD00061

[0036] [0036]

[0037]对 k 次NURBS 曲线p (u),令咖、=Z ^dlNl k (u) ,w(u) = ^ W1Nljc (u),应用式⑵〜 [0037] The k-th NURBS curve p (u), to make coffee, = Z ^ dlNl k (u), w (u) = ^ W1Nljc (u), Formula application ⑵~

i=0 i=0 i = 0 i = 0

(3)分别对c(u)和w(u)进行求解,即求得NURBS曲线: (3) respectively, c (u) and w (u) is solved, i.e. NURBS curves obtained:

[0038] p(u) = c (u)/w (U) (7)。 [0038] p (u) = c (u) / w (U) (7).

[0039] 进一步,利用限定的弓高误差对插补的进给速度实行自适应调节,具体过程如下根据式(19)求出空间NURBS曲线上任意一点的曲率匕,进而求得曲率半径P i = IAi ; [0039] Further, with a defined chord error imposed on the feed rate of the adaptive interpolation procedure is as follows according to the NURBS curve calculated on any point of the space (19) of curvature dagger, then obtain the curvature radius P i = IAi;

Figure CN102608956AD00062

[0041] 其中,P (Ui)和P(ui+1)分别是近似圆弧上u = Ui和u = ui+1处的插补点;C (Ui)和C(ui+1)分别是NURBS曲线上u = Ui和u = ui+1处的插补点,由于C(Ui) = C(ui+1),令Li = [0041] wherein, P (Ui) and P (ui + 1) are approximately circular arc and u = u = Ui at the interpolation point ui + 1; C (Ui) and C (ui + 1) are u = Ui on the NURBS curve and u = ui + 1 at the interpolation point, since C (Ui) = C (ui + 1), so that Li =

IP (ui+1)-P (Ui) I |,则进给速度V(Ui)近似地表示为 IP (ui + 1) -P (Ui) it |, then the feed speed V (Ui) is approximately expressed as

Figure CN102608956AD00063

[0043] 弓高误差ER是等效弦长与实际样条曲线的偏差,表示为 [0043] ER chord error is equivalent to the deviation from the actual chord spline curve, expressed as

[0044] [0044]

Figure CN102608956AD00071

[0045] 如果限定弓高误差ER的大小,则相应的进给速度V(Ui)为 [0045] If the size of the chord error ER is defined, the corresponding feed speed V (Ui) of

[0046] [0046]

Figure CN102608956AD00072

[0047] 通常情况下,P i >> ER,所以式(22)中V(Ui)的结果是一个实数值 [0047] Typically, P i >> ER, and therefore equation (22) results in V (Ui) is a real-valued

[0048] 式(22)表明进给速度V(Ui)应随ER和P ,的变化自适应的调整,调整规则如下: [0048] Formula (22) shows that the feed rate V (Ui) should be adjusted adaptively with the change of P and ER, adjusted rules are as follows:

[0049] [0049]

Figure CN102608956AD00073

[0050] 其中,F是进给速度指令值,如果空间曲线上当前点的曲率半径足够小,则弓高误差可能超过限定值,这时插补算法将进给速度由F减小到 [0050] where, F is the feed speed command value, if the current point on the radius of curvature sufficiently small space curve, the chord error may exceed the limit value, then the interpolation algorithm of the feed speed is reduced to F

Figure CN102608956AD00074

,以满足限定的 To meet defined

弓高误差ER的要求;否则以给定的进给速度F继续进行插补。 Chord error ER required; otherwise, at a given feed rate F continues interpolation.

[0051] 本实施例中,将de Boor算法应用于NURBS曲线:给定一条k次B样条曲线: [0051] In this embodiment, the de Boor algorithm to NURBS curve: k given a B-spline curve:

[0052] [0052]

Figure CN102608956AD00075

[0053] 其中控制顶点屯(i = 0,1, “^n),节点矢量U = [uQ,Up如若给出曲线定义域内一参数值^[%,%+1][[乂,?/„+1],欲计算该B样条曲线上对应一点p(u),可采用德布尔算法的递推公式: [0053] wherein the control vertex Tun (i = 0,1, "^ n), a knot vector U = [uQ, Up art should the curve definition is given a parameter value ^ [%% + 1] [[qe,? / "+ 1], B to be calculated corresponding to one o'clock p (u) on the spline, the recurrence formula can be de Boer algorithm:

[0054] [0054]

Figure CN102608956AD00076

[0058]规定| [0058] provisions |

Figure CN102608956AD00077

[0059] 用德布尔算法也可求B样条曲线的导矢,如求k次B样条曲线上的一点处的r阶导矢p(r) (11),〃£[%,%.+1][[义,?/„+1],可按如下递推公式计算: [0059] The algorithm may also be evaluated by de Boer B-spline curve derivative vector, such as seeking order derivative vector r P (r) at the k-th B-spline curve point (11), 〃 £ [%,%. +1] [? [Yi, / "+ 1], according to the following recursion formula:

[0060] [0060]

Figure CN102608956AD00078

[0061] 其中新的控制点为 [0061] wherein the new control points

Figure CN102608956AD00079

[0063] 节点矢量为 [0063] node vector

[0064] Ur = [u0(r), u1(r),…,u(r)n+k 2r+1] = [0, ...,0,uk+1, ...,un,l, ...,1](6) [0064] Ur = [u0 (r), u1 (r), ..., u (r) n + k 2r + 1] = [0, ..., 0, uk + 1, ..., un, l , ..., 1] (6)

[0065] 其中0和1都为k+lr个。 [0065] 0 and 1 wherein k + lr are th.

[0066] 经过分析可知NURBS曲线的分子分母均为B样条曲线,因此可对分子分母分别使用de Boor算法进行求解,二者的比值即为所求。 [0066] After analysis shows that the numerator and denominator are NURBS curve B-spline curve, and therefore can be used separately de Boor algorithm by the numerator and denominator of the ratio of the two is also desired. 对k次NURBS曲线p (u),令 NURBS curve of the k p (u), so that

Figure CN102608956AD00081

应用式⑵〜(3)可分别对c(u)和w(u)进行求解, Application of the formula ⑵~ (3) respectively of c (u) and w (u) is solved,

即求得NURBS曲线: I.e. NURBS curves obtained:

[0067] p (u) = c(u)/w(u) (7) [0067] p (u) = c (u) / w (u) (7)

[0068] 对式(7)求导可得p(u)的一阶导矢为 Derivative [0068] The formula (7) can be obtained p (u) is a derivative of vector

[0069] [0069]

Figure CN102608956AD00082

[0070] 对式(8)求导可得p(u)的二阶导矢为 Derivative [0070] The formula (8) can be obtained p (u) is the second derivative vector

「00711 "00711

Figure CN102608956AD00083

[0072]其中,c' (u)、w' (u)、c〃(u)、w〃 (u)可由式⑵〜(6)的de Boor 算法求得。 [0072] wherein, c '(u), w' (u), c〃 (u), w〃 (u) by the formula ⑵~ (6) is obtained by de Boor algorithm.

[0073] 实时de Boor算法应用于NURBS曲线插补:从本质上看,用伺服电机作驱动装置的数控设备,其数控系统是一个离散的数控设备,其数控系统是一个离散的数据采样系统,所以这里采用离散数据采样插补。 [0073] Real de Boor NURBS interpolation algorithm is applied: In essence, the servomotor driving device for numerical control equipment, numerical control system which is a discrete numerical control apparatus, a numerical control system is discrete data sampling system, so here discrete data sampling interpolation. 工作时,在每个采样周期内,数控系统根据设定的进给速度V(t)实时插补出下一周期刀具要到达的位置,并由此控制伺服系统的运动。 In operation, at each sampling period, the CNC according to the set feed speed V (t) in real time the position of the next interpolation cycle the tool to be reached, and thereby control movement of the servo system. 在加工NURBS 曲线这种参数曲线时,由于曲线上的位置与参数值是一一对应的,所以插补过程也就是连续递推计算参数^的过程。 When the process of this processing parameter curve NURBS curve, since the position on the curve and the parameter value is one to one, so that the interpolation process is a continuous ^ recursive calculation parameters.

[0074] 曲线弧长对时间的微分为: [0074] differentiating the curve arc length of time:

[0075] [0075]

Figure CN102608956AD00084

(10) (10)

[0076]所以: [0076] Therefore:

[0077] [0077]

Figure CN102608956AD00085

[0078] 可近似认为V' (t)等于设定的进给速度V(t)。 [0078] can be approximated that V '(t) is set equal to the feed speed V (t). 根据微分几何可知: According to differential geometry can be seen:

[0079] [0079]

Figure CN102608956AD00086

(12) du (12) du

[0080] 其中 [0080] in which

[0081] x = Px(u), y = Py(u), z = Pz(u), rn , dP (u) , dP (u) dP (u) [0081] x = Px (u), y = Py (u), z = Pz (u), rn, dP (u), dP (u) dP (u)

[0082] [0082]

Figure CN102608956AD00087

(13) (13)

[0083] 由式(11)、(12)可得参数u对时间t的求导: [0083] by formula (11), (11), the parameters of time t u derivation:

「00841 "00841

Figure CN102608956AD00088

114) 114)

[0085] 将上式进行二阶泰勒级数展开:[0086] [0085] will be the second-order Taylor series expansion formula: [0086]

Figure CN102608956AD00091

[0087]其中,= T,HOT是泰勒级数展开式的高阶小量,由式(14)可知 [0087] where, = T, HOT Taylor series expansion of a small amount of high order, by the formula (14) can be seen

Figure CN102608956AD00092

[0091]将式(16)、(17)代入式(15)并略去高次项,有 [0091] The formula (16), (17) into equation (15) and higher order terms are omitted, there

Figure CN102608956AD00093

[0094]其中,x' (Ui)、X" (Ui)、y' (Ui)、y" (Ui)、z' (Ui)、z" (Ui)由式⑷〜(9)的de Boor算法算出。在实际计算时,上式中V (t)是设定的进给速度,T是已知采样周期,则可通过式(18)即可由当前点位置坐标所对应的Ui值计算下一插补点所对应的Uw值。 [0094] where, x '(Ui), X "(Ui), y' (Ui), y" (Ui), z '(Ui), z "(Ui) by the formula ⑷~ (9) of de Boor calculating algorithm. in the actual calculation, the equation V (t) is the feed speed setting, T is the sampling period is known, can be by the formula (18) to the current point position coordinate corresponding to the calculated value Ui interpolation point corresponding to a value Uw.

[0095] NURBS曲线速度自适应控制:图1表示的是一段圆弧近似在区间u G [u” ui+1]内的NURBS曲线。P i是在u = Ui处的曲率半径,且P i =1/\。其中,ki是空间NURBS曲线上任意一点的曲率,可通过下式计算: [0095] NURBS curve speed adaptive control: Figure 1 shows a circular arc is approximately in the interval u G [u "ui + 1] NURBS curve within .P i u = the radius of curvature is at Ui and P i = 1 / \ where, Ki is the curvature at any point on the NURBS curve of space, can be calculated by the following formula:

[0096] [0096]

Figure CN102608956AD00094

[0097]图1中P(Ui)和P(Uw)分别是近似圆弧上u = Ui和u = ui+1处的插补点;C(Ui) 禾ロC(ui+1)分别是NURBS曲线上u = Ui和u = 处的插补点。 [0097] Figure 1 P (Ui) and P (Uw) are approximately circular arc u = Ui and u = ui + interpolation point at 1; C (Ui) Wo ro C (ui + 1) are the NURBS curve interpolation point Ui and u = u = at. 由于C(Ui) =C(ui+1),令Li =I |P(ui+1)-P(Ui) I,则进给速度V (Ui)可近似地表示为 Since C (Ui) = C (ui + 1), so that Li = I | P (ui + 1) -P (Ui) I, the feed rate V (Ui) can be approximately expressed as

[0098] [0098]

Figure CN102608956AD00095

[0099]弓高误差ER是等效弦长与实际样条曲线的偏差,可表示为 [0099] ER chord error is equivalent to the deviation from the actual chord spline curve can be expressed as

[0100] [0100]

Figure CN102608956AD00096

[0101]如果限定弓高误差ER的大小,则相应的进给速度V(Ui)为 [0101] If the size of the chord error ER is defined, the corresponding feed speed V (Ui) of

[0102] [0102]

Figure CN102608956AD00097

[0103]通常情况下,P i >> ER,所以式02)中V(Ui)的结果是ー个实数值 [0103] Typically, P i >> ER, and therefore equation 02) results in V (Ui) is a real-valued ー

[0104]式Q2)表明进给速度V(Ui)应随ER和P i的变化自适应的调整,调整规则如下: [0104] Formula Q2) showed that the feed rate V (Ui) should be changed adaptively with ER and P i is adjusted, the adjustment rules are as follows:

Figure CN102608956AD00101

[0106] 其中,F是进给速度指令值。 [0106] where, F is the feed speed command value. 如果空间曲线上当前点的曲率半径足够小,则弓高误差可能超过限定值,这时插补算法将进给速度由F减小到2 -(P, -ER)2,以满足限定的弓高误差ER的要求;否则以给定的进给速度F继续进行插补。 If the current point on the radius of curvature sufficiently small space curve, the chord error may exceed the limit value, then the interpolation algorithm F from the feed speed is reduced to 2 - (P, -ER) 2, in order to meet a defined bow high error ER required; otherwise, at a given feed rate F continues interpolation.

[0107] 将调整后的进给速度代入式(18),得到下一个插补点处的参数值ui+1,并将其代入式(13)求出下一个插补点的X, Y,z坐标值。 [0107] The feed rate adjusted substituted into the formula (18), the parameter value obtained at the next interpolation point ui + 1, and obtains the next interpolation is substituted into the formula (13) the point X, Y, z-coordinate value.

[0108] 实例:用MATLAB编程实现此算法。 [0108] Example: This algorithm is implemented using MATLAB programming. 为描述简单,只在平面内画二维图像,即让控制点第三轴坐标取零,参数信息如下:次数k = 3,权因子w = {1,1,1,1,1,1},控制点信息: For the simplicity of description, only the drawn two-dimensional image plane, i.e., so that the third axis coordinate of the control point to take zero, the following parameter information: the number k = 3, the weight factor w = {1,1,1,1,1,1}, control point information:

[0109] b = {(10,10,0), (60,80,0), (100,90,0),(200,20,0),(250,30,0),(300,120, 0)}节点序列u = {0,0,0,0,0. 3,0. 7,0,0,0,0},进给速度v = 50mm/s,要求的弓高误差设 [0109] b = {(10,10,0), (60,80,0), (100,90,0), (200,20,0), (250,30,0), (300,120 , 0)} with node u = {0,0,0,0,0. 3,0. 7,0,0,0,0}, feed speed v = 50mm / s, the required chord error provided

为0. 002_,计算的插补点数据庞大,只给出部分开始和结尾段域的点。 0. 002_, the calculated interpolation point data of a large, given only the start and end point portion segment domains.

[0110] [0110]

Figure CN102608956AD00102
Figure CN102608956AD00103

[0111]表I。 [0111] Table I.

Claims (2)

  1. 1. 一种基于de Boor算法的NURBS曲线自适应插补控制方法,上位机计算出插补点,把计算出的坐标值列表传给下位机,下位机把坐标值转换成相应脉冲数,并发出脉冲到电机驱动器,控制电机转动并驱动机械结构动作;其特征在于:所述NURBS曲线自适应插补控制方法包括以下过程:设定给出曲线定义域内一参数值 1. Based on Adaptive de NURBS curve interpolation control method, PC Boor algorithm to calculate the interpolation points, the calculated coordinate values ​​to the list of the next crew, the next crew to convert the number of pulses corresponding to the coordinate value, and pulse sent to the motor driver, the motor rotates and drives the control operation of the mechanical structure; characterized in that: said adaptive NURBS curve interpolation control method comprising the following procedures: setting a parameter value curve definition given in the art
    Figure CN102608956AC00021
    ,欲计算该B样条曲线上对应一点P (u),采用德布尔算法的递推公式: , To calculate the B-spline curve corresponding to the point P (u), de Boer algorithm recursion formula:
    Figure CN102608956AC00022
    对k 次NURBS 曲线p (u),令 NURBS curve of the k p (u), so that
    Figure CN102608956AC00023
    ,应用式⑵〜⑶i=0 i=0分别对C(U)和W(U)进行求解,即求得NURBS曲线: p(u) = c (u)/w (u) (7) ο Application of the formula ⑵~⑶i = 0 i = 0 respectively C (U) and W (U) is solved, i.e. NURBS curves obtained: p (u) = c (u) / w (u) (7) ο
  2. 2.如权利要求I所述的一种基于de Boor算法的NURBS曲线自适应插补控制方法,其特征在于:利用限定的弓高误差对插补的进给速度实行自适应调节,具体过程如下:根据式(19)求出空间NURBS曲线上任意一点的曲率1^,进而求得曲率半径P i = IAi ; 2. An I according to claim NURBS-based adaptive control method of the interpolation curve de Boor algorithm, characterized in that: the use of a defined chord error implement adaptive feedrate interpolation procedure is as follows : according to the formula (19) is obtained at any point on the NURBS curve of the spatial curvature ^ 1, then obtain the curvature radius P i = IAi;
    Figure CN102608956AC00024
    其中,P(Ui)和P(ui+1)分别是近似圆弧上u = Ui和u = ui+1处的插补点! Wherein, P (Ui) and P (ui + 1) are approximately circular arc u = Ui and u = ui + 1 of the interpolation points! C(Ui)和C(ui+1)分别是NURBS曲线上u = Ui和u = ui+1处的插补点,由于C(Ui) = C(ui+1),令Li =IP (ui+1)-P (Ui) I |,则进给速度V(Ui)近似地表示为 C (Ui) and C (ui + 1) are u = Ui and u = ui + 1 at the interpolation point on the NURBS curve, since C (Ui) = C (ui + 1), so that Li = IP (ui +1) -P (Ui) I |, then the feed speed V (Ui) is approximately expressed as
    Figure CN102608956AC00025
    弓高误差ER是等效弦长与实际样条曲线的偏差,表示为 Chord error ER is equivalent to the deviation from the actual chord spline curve, expressed as
    Figure CN102608956AC00026
    如果限定弓高误差ER的大小,则相应的进给速度 If the chord error ER defined size, the corresponding feed rate
    Figure CN102608956AC00027
    通常情况下,P i >> ER,所以式(22)中V(Ui)的结果是一个实数值式(22)表明进给速度V(Ui)应随ER和Pi的变化自适应的调整,调整规则如下 Typically, P i >> ER, and therefore equation (22) results in V (Ui) is to show a real value of formula (22) feed rate V (Ui) to be adjusted with the adaptive variations ER and Pi, adjust the rules are as follows
    Figure CN102608956AC00031
    其中,F是进给速度指令值,如果空间曲线上当前点的曲率半径足够小,则弓高误差可能超过限定值,这时插补算法将进给速度由F减小到 Where, F is the feed speed command value, if the current point on the radius of curvature sufficiently small space curve, the chord error may exceed the limit value, then the interpolation algorithm of the feed speed is reduced to F
    Figure CN102608956AC00032
    ,以满足限定的弓高误差ER的要求;否则以给定的进给速度F继续进行插补。 To meet a defined chord error ER required; otherwise, at a given feed rate F continues interpolation.
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CN102880119B (en) * 2012-09-06 2015-01-14 上海交通大学 Unit arc length increment interpolation method
CN103383552B (en) * 2012-11-21 2015-08-26 深圳市智信精密仪器有限公司 A planar circular interpolation motion controller and method of controlling
CN102981456A (en) * 2012-12-04 2013-03-20 杭州电子科技大学 Non-uniform rational B-spline (NURBS) interpolation feed speed planning method aiming at embedded system
CN102981456B (en) * 2012-12-04 2014-11-26 杭州电子科技大学 Non-uniform rational B-spline (NURBS) interpolation feed speed planning method aiming at embedded system
CN104238457B (en) * 2013-06-08 2016-12-28 沈阳高精数控智能技术股份有限公司 Adaptive calculation of a complex curve interpolation method nurbs
CN105549540A (en) * 2014-10-23 2016-05-04 发那科株式会社 numerical control device
CN105785921A (en) * 2016-03-25 2016-07-20 华南理工大学 Speed planning method during NURBS curve interpolation of industrial robot
CN105785921B (en) * 2016-03-25 2018-06-22 华南理工大学 Velocity planning method when the industrial robot nurbs curve interpolation
CN106125672A (en) * 2016-08-03 2016-11-16 大连理工大学 High-efficiency processing method for complex curved surface part
CN106125672B (en) * 2016-08-03 2018-06-08 大连理工大学 Species complex surface parts efficient processing method
CN107817764A (en) * 2017-10-23 2018-03-20 山东大学 NURBS curve bi-directional adaptive interpolation algorithm based on S-curve acceleration and deceleration algorithm

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