CN1121658C - Computer numerical control curve path speed control method and device - Google Patents

Abstract

In CNC machining of parametric curves, the feed rate is a determining factor for the quality of the machining; the invention provides a parameterized curve interpolator with rate control, which develops the operation modes of (1) a fixed rate interpolator, and (2) an acceleration and deceleration interpolator, etc.; the operation mode of the constant-speed interpolator can ensure that the parameterized curve keeps a constant speed value in the interpolation realization process, and further, smoother acceleration and deceleration motion is obtained by utilizing the acceleration and deceleration design of the parametric curve interpolator.

Description

Computer numerical control curve path speed control method and device

The present invention relates to a computer numerical control (CNC) processing machine, in particular to a computer numerical control processing machine designed with a speed-controlled parametric curve interpolator.

Traditional CAD/CNC system operators first design the required workpieces through computer-aided design (CAD), such as: 3-D curve representation of molds, turbine blades, aircraft models, etc., where the representation of curves is generally parametric in CAD systems The format of the curve, and since the general CNC system only provides interpolators for straight lines and circles, CAD must still segment the designed workpiece surface curve and transmit it to the CNC for execution. When the linear interpolator of the CNC system receives the segmented information of the workpiece surface curve, some algorithms are used to generate the interpolation points of the tool movement, and these interpolation points are the position input commands of the servo drive.

Although the traditional curve processing method is relatively simple, there are still some disadvantages in application: (1) In order to enable a more accurate representation of the curve, the curve must be segmented into a large number of segments and transmitted between CAD and CNC, However, it is easy to cause errors when transmitting a large number of signals, such as: data loss and noise interference; (2) The discontinuity of the cutting section will cause the surface accuracy of the processed workpiece to deteriorate. (3) Due to the linearization of the curve section, the speed does not change smoothly in each straight line section, especially after considering the automatic acceleration and deceleration characteristics of the CNC interpolator.

The traditional way of acceleration and deceleration is to pass the position command through a low-pass filter to generate a smoother command input. However, this method will cause radial and chord height errors in the command path with large changes, such as near the starting point and end point of the processing curve. Although Kim (D.I.Kim) proposed to use the acceleration and deceleration method of parameters to achieve the purpose of parameterized curve acceleration and deceleration, at the same time, because the operation of acceleration and deceleration is performed on parameters, the interpolation point has a small radial error, which can make the interpolation The generation of points has the function of acceleration and deceleration, but the purpose of speed acceleration and deceleration cannot be really achieved on some curves, because the parameter space and the curve are not uniformly corresponding. In addition, when the curvature radius of the curve is too small so that the feed rate command is too large, it will cause obvious position command error.

In view of the known parametric curve interpolation process, the speed accuracy of the curve is not easy to control, and the feed rate is the decisive factor of the processing quality, so the present invention proposes the design of a parametric curve interpolator with speed control.

The object of the present invention is to provide an operation mode of an interpolator with a constant rate, which can maintain a constant rate during the interpolation process of the parameterized curve to ensure the processing quality of the computer numerical control curve path rate control method and device.

Another object of the present invention is to provide a computer numerical control curve path rate control method and device with acceleration and deceleration interpolator operation mode, using the acceleration and deceleration design of the parametric curve interpolator to obtain smoother acceleration and deceleration motion.

The computer numerical control (CNC) curve path rate control method of the present invention is to process the curve path with constant speed or acceleration and deceleration control operation, comprising the following steps:

(1) After receiving the information of the curve interpreted by computer-aided design (CAD), the parametric curve interpolator generates continuous interpolation points by the parameter iteration method (Iteration) of the parametric curve;

(2) Only the influence of the chord height error (chord height error) of the position command error generated by the interpolator is considered, and the radial error (radial error) is not considered;

(3) Deriving an approximate parameter iteration rule of the parametric curve formulation;

(4) Design a parameter iteration rule with rate control, propose a parameter compensation based on an approximate parameter iteration rule of the expression, and obtain a higher-precision curve rate control through the correlation between the compensation amount and the curve;

(5) Select the compensation value near the zero point to obtain a more reliable parameter iteration result; and

(6) Send the continuous interpolation points obtained from the compensation value to the control system to move the tool center along the parameterized curve.

The parameterized curve used in the computer numerical control (CNC) curve path speed control method of the present invention is non-equidistant B nebula (NURBS), and the parameter iteration method used by the parameterized curve interpolator is based on Chou (J.J.Chou ) and Yang (D.C.H.Yang) proposed a generalized parameter iteration rule based on the feed rate control parameter iteration method.

The computer numerical control curve path rate control device of the present invention comprises:

Computer, for aided design (CAD) work, which includes making parametric curve formats and segmented curves;

Computer Numerical Control (CNC) mechanisms, including:

The parametric curve interpolator, after receiving the parametric curve information interpreted by the computer, generates continuous interpolation points in the way of generating points of the parametric curve;

The controller receives the continuous interpolation points generated by the parametric curve interpolator, and sends out control processing signals;

The tool receives the control processing signal sent by the controller, and the center moves along the parameterized curve; and

A man-machine interface device that allows the operator to have a man-machine dialogue with the parametric curve interpolator;

Wherein the parameterized curve interpolator can perform constant rate interpolation operation mode and acceleration/deceleration interpolation operation mode for the parameterized curve.

The parametric curve interpolator device of the computer numerical control curve path rate control of the present invention comprises: a constant rate interpolator, which keeps the parametric curve path at a constant rate during the interpolation realization process; an acceleration and deceleration interpolator, which makes the processing curve path Get smoother acceleration and deceleration motion.

Fig. 1 is the processing system block diagram of the parametric curve of the embodiment of the present invention;

Fig. 2 is a schematic diagram of the position command error generated by the interpolator when the tool moves in a straight line between two interpolation points;

与参数u_i、u_i+1′、u_i+1间的几何关系；Figure 3 is the compensation amount ε _1,2 (u _i ) formula, the vector curve differential vector

Geometric relationship with parameters u _i , u _i+1 ′, u _i+1 ;

Fig. 4 is the situation of forming the same real value or conjugate imaginary value compensation ε _1,2 (u _i );

Fig. 5 is the command path of the bow-tie curve applied in this embodiment;

Fig. 6 is a first order approximation (1st order approximation) interpolation rate command result curve rate change situation;

Fig. 7 is the situation of the rate change of the curve rate of the second order approximation (2nd order approximation) interpolation rate command result;

Fig. 8 is the rate change situation of the interpolation rate command result curve in the fixed rate operation mode of this embodiment;

Fig. 9 is the calculation result of the parameter compensation value ε _1,2 (u _i ) in the constant rate operation mode;

Fig. 10 is the curve situation of the first-order (1st) approximation of the speed difference value change of quadratic curve acceleration and deceleration mode;

Fig. 11 is the curve situation of the second-order (2nd) approximation of the rate difference change of the quadratic curve acceleration and deceleration mode;

Fig. 12 is the curve situation of the speed difference change constant speed operation mode of the quadratic curve acceleration and deceleration mode.

Generally, the curve generated by the CAD system is expressed as a parametric curve, and the CAD system also provides the relevant parameter data transmission format of the parametric curve, so the parameter information of the parametric curve can be directly used in the design of the CAD/CNC system as the CAD/CNC system. Processing information transmission between CNC systems and CNC interpolation. The present invention discusses the processing system of parametric curve as shown in Figure 1, and among the figure 1 is computer, and it is done the work of aided design (CAD), comprises making parametric curve format and segmented curve, and 2 is computer numerical control (CNC) mechanism , wherein the parameterized curve interpolator 3, after receiving the parameterized curve information 7 interpreted by the computer 1, generates continuous interpolation points in the way of generating points of the parameterized curve and sends them to the controller 5 to make the center of the tool 6 along the parameterized Curve movement, 4 of which are human-machine interface devices, which allow the operator to communicate with the CNC. Common parametric curves are: Bezier curve, B spline (B Spline), and non-equidistant B spline (NURBS); the present invention uses NURBS parametric curve.

The process realized by the parametric curve interpolator 3 is mainly to calculate the parametric curve points. Generally, calculation algorithms with higher curve position and speed accuracy require more complex calculation methods and a larger amount of calculation, which is disadvantageous in the environment where real-time commands are generated. Therefore, the choice and design of the calculation method must consider the trade-off between curve accuracy and calculation time. The most commonly used calculation method format is the following parameter iteration method:

u _i+1 ＝u _i +Δ(u _i )

Wherein, u _i represents a parameter of the current time series, u _i+1 represents a parameter of the next time series, and Δ(u _i ) represents a calculation amount related to the current time series. Under the parameterized curve, u _i represents the parameter of the curve, so the parameter must be substituted into the mathematical expression of the curve to obtain the point at this time series. When designing a parametric curve interpolator, for the application of any parametric curve mathematical model, since the relationship between parameters and curves or geometric paths in the parametric curve is not obvious, it is not easy to perform inverse operations, that is, it is not easy to use known The corresponding parameter value is deduced from the curve point, so it is not easy to have a good control of the curve rate in the iterative method of the parameter. In actual processing, the parametric curve interpolator will cause an error in the interpolation result curve speed due to inappropriate parameter iteration rules. The curve speed error refers to the difference between the moving speed between adjacent interpolation points and the feed rate of the interpolator. Difference, when the error of the curve rate is larger, it means that the movement between the interpolation points cannot be operated as the feed rate, which often results in excessive position error, long processing time and poor surface of the processed workpiece. Therefore, in the design of the parametric curve interpolator, the parameter iteration method must accurately control the curve rate to reduce the error. In addition to the curve rate error, since the interpolator generates the interpolation point when the tool moves, and the tool moves in a straight line between the two interpolation points, the position command error generated by the interpolator generally includes: radial error (radial error) and chord High error (chord height error). The relationship between the radial error, the chord height error and the curve can be shown in Figure 2, where 8 is the workpiece contour curve, 9 is the tool movement path, 10 is the radial error, 11 is the height error, and 12 is the interpolation point.

The radial error is the shortest distance between the position point and the curve, while the chord height error is the secant line formed by the two position points The maximum distance between CD and cutting arc. In the design of a parametric curve interpolator, the cause of the radial error is generally the rounding error of the floating point number, and the cause of the chord height error is caused by an incorrect feed rate. Because in the representation of the parametric curve, the circular tangent error can be reduced by precise computer operation, and the chord height error is often much larger than the radial error, so in the design of the interpolator, the present invention only considers the influence of the chord height error, And the chord height error should be controlled within one basic length unit (BLU).

In the study of the parameter iteration method of the parametric curve, in the methods used by Bedi (S.Bedi), Ali (I.Ali), and Quan (N.Quan), Δ(u _i ) is a constant value, and at this time This iteration method is called uniform parameter iteration. Although this method is simple, the equidistant selection of parameters cannot guarantee the boundary range of the curve position command chord height error and the change of the curve speed, because the relationship between the parameters and the curve points is not a consistent correspondence. According to the dynamic characteristics of the cutting tool path and motion system, Chou and Yang proposed an accurate parametric curve parameter iteration method to simultaneously control the position, speed and acceleration rate of the system. The reason for the more precision is that the method of Chou and Yang added parameters and Consider the dynamic relationship between time. Kim discusses the parameter iteration method of parameter acceleration and deceleration in the parameterized curve. Through the parameter acceleration and deceleration method, high-precision position and smooth speed acceleration and deceleration results are obtained, but because the relationship between the parameters and the curve is not consistent (uniform) correspondence , so the speed may not have the expected acceleration and deceleration behavior.

Because the position command and curve speed accuracy generated by the interpolator will affect the quality of the processed workpiece; the inaccurate position command will cause the processing result with insufficient precision, and the curve speed error will cause the prolongation of working hours and the fluctuation of the speed will cause vibration. Affect the surface accuracy of the processed workpiece. Therefore, in the process of realizing any parametric curve, how to design the iterative process of parameters to produce position and curve speed accuracy can meet the design requirements, that is, to achieve low position command chord height error and low curve speed error, which is parametric curve interpolation An important key to the implementation of the device.

In the method of parameter iteration, although the parameter iteration method proposed by Chou and Yang has the dynamic relationship between parameters and time, the situation of curve rate changes is improved, but the degree of improvement is still limited by the approximation degree of Taylor expansion. At the same time, it is impossible to accurately control the speed of the curve. Therefore, the present invention proposes a parameter iteration method of feed rate control based on the generalized parameter iteration rule proposed by Chou and Yang to accurately control the curve rate generated by the interpolator. The parameter iteration method with rate control is to add a compensation amount to the original first-order approximate parameter iteration method of Chou and Yang. Since the compensation amount considers the approximate relationship between the parameter and the curve rate, the fluctuation of the curve rate can be greatly reduced. Based on the control of the feed rate, the interpolation point generation of the parametric curve interpolator can be operated in the constant speed operation mode and the acceleration and deceleration operation mode according to the requirements of the processing conditions. The constant rate operation mode is to make the interpolator maintain the same curve rate during the curve generation process to reduce the processing error caused by the rate change. In the acceleration and deceleration operation mode, since the parameter iteration method of the rate control can obtain high-precision curve rate control, in the design of the acceleration and deceleration, the acceleration and deceleration can be performed according to the feed rate parameter so that the curve rate can be carried out in the same acceleration and deceleration mode and Get real acceleration and deceleration rate results, and also fully understand the rate change during acceleration and deceleration.

(A) Derivation of Parametric Curve Formulation:

In the derivation process of the parameter iteration method, it is assumed that C(u) is the expression of the parametric curve point, where u is the parameter representation and is a function of time t, ie: u=u(t). definition

u(t _i )=u _i ; u(t _i+1 )=u _i+1

From the Taylor expansion, we know that: $u_{i+1}=u_i+\frac{\mathrm{du}}{\mathrm{dt}}{|}_{t=t_i}\&Center\; Dot;(t_{i+1}-t_i)+\frac{1}{2}\&Center\; Dot;\frac{d^2\mathrm{<figure-callout\; id="2"\; label="u\; dt"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">u</figure-callout>}}{{\mathrm{dt}}^{}}{|}_{t=t_i}\&CenterDot;{(t_{i+1}-t_i)}^2+h.o.T----\left(1\right)$

Since the curve velocity V(u _i ) can be expressed as $V\left(u_i\right)={\left|\right|\frac{c\left(u\right)}{\mathrm{dt}}\left|\right|}_{u=u_i}={\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}\&Center\; Dot;\frac{\mathrm{du}}{\mathrm{dt}}{|}_{t=t_i}$

so $\frac{\mathrm{du}}{\mathrm{dt}}{|}_{t=t_i}=\frac{V\left(u_i\right)}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}}----\left(2\right)$

Differentiate Equation (2) again to get, $\frac{d^2\mathrm{<figure-callout\; id="2"\; label="u\; dt"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">u</figure-callout>}}{{\mathrm{dt}}^{}}{|}_{t=t_i}=\frac{-1}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^2}[V\left(u_i\right)\&Center\; Dot;\frac{d\left(\right|\left|\frac{c\left(u\right)}{\mathrm{du}}\right|\left|\right){|}_{u=u_i}}{\mathrm{dt}}]----\left(3\right)$

By simple differential calculus, we know that $\frac{d\left(\right|\left|\frac{c\left(u\right)}{\mathrm{du}}\right|\left|\right){|}_{u=u_i}}{\mathrm{dt}}=\frac{d\left(\right|\left|\frac{c\left(u\right)}{\mathrm{du}}\right|\left|\right){|}_{u=u_i}}{\mathrm{du}}\&Center\; Dot;\frac{\mathrm{du}}{\mathrm{dt}}{|}_{t=t_i}=\frac{d\left(\right|\left|\frac{c\left(u\right)}{\mathrm{du}}\right|\left|\right){|}_{u=u_i}}{\mathrm{du}}\&Center\; Dot;\frac{V\left(u_i\right)}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}}----\left(4\right)$

Substituting formula (4) into formula (3), we can get, $\frac{d^2\mathrm{<figure-callout\; id="2"\; label="u\; dt"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">u</figure-callout>}}{{\mathrm{dt}}^{}}{|}_{t=t_i}=-\frac{V^2\left(u_i\right)}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u={\mathrm{<figure-callout\; id="3"\; label="u\; i"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">u</figure-callout>}}_i}^3}\&Center\; Dot;\frac{d\left(\right|\left|\frac{c\left(u\right)}{\mathrm{du}}\right|\left|\right){|}_{u=u_i}}{\mathrm{du}}----\left(5\right)$

Also by simple differential calculus, we can know that $\frac{d\left(\right|\left|\frac{c\left(u\right)}{\mathrm{du}}\right|\left|\right){|}_{u=u_i}}{\mathrm{du}}=\frac{\frac{c\left(u\right)}{\mathrm{du}}\&Center\; Dot;\frac{d^{2}C\left(u\right)}{{\mathrm{du}}^2}}{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|{|}_{u=u_i}}----\left(6\right)$

Substituting formula (6) into formula (5), the quadratic differential formula of parameter u can be obtained as, $\frac{d^2\mathrm{<figure-callout\; id="2"\; label="u\; dt"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">u</figure-callout>}}{{\mathrm{dt}}^{}}{|}_{t=t_i}=-\frac{V^2\left(u_i\right)\&CenterDot;(\frac{c\left(u\right)}{\mathrm{du}}\&CenterDot;\frac{d^{2}C\left(u\right)}{{\mathrm{du}}^2})}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^4}----\left(7\right)$

If the generation interval of the curve points is a sampling time T _s , then

t _i+1 -t _i =T _s

Substituting equations (2) and (7) into equation (1) and omitting higher-order terms, the parameter iteration rules for primary and quadratic approximations can be obtained respectively as follows:

An approximate parameter iteration rule: $u_{i+1}=u_i+\frac{V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}}}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}}----\left(8\right)$

Parameter iteration rule for quadratic approximation: $u_{i+1}=u_i+\frac{V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}}}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}}-\frac{V^2\left(u_i\right)\&CenterDot;T_{\mathrm{the\; <figure-callout\; id="2"\; label="s"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">s</figure-callout>}}^2\&Center\; Dot;(\frac{c\left(u\right)}{\mathrm{du}}\&Center\; Dot;\frac{d^{2}C\left(u\right)}{{\mathrm{du}}^2}){|}_{u={\mathrm{<figure-callout\; id="2"\; label="u\; i"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">u</figure-callout>}}_i}}{2\&Center\; Dot;{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}^4}----\left(9\right)$

Wherein, since the desired curve rate is the set value of the feed rate, V(u _i ) can be designated as the set value of the feed rate when the parameter u=u _i in the implementation of the parametric curve interpolator. In the plane coordinate system, however, due to $\frac{c\left(u\right)}{\mathrm{du}}=\left[\begin{array}{c}\frac{c_x\left(u\right)}{\mathrm{du}}\\ \frac{c_{\mathrm{the\; y}}\left(u\right)}{\mathrm{du}}\end{array}\right];\frac{d^{2}C\left(u\right)}{{\mathrm{du}}^2}=\left[\begin{array}{c}\frac{d^2{C}_x\left(u\right)}{{\mathrm{du}}^2}\\ \frac{d^2{C}_{\mathrm{the\; y}}\left(u\right)}{{\mathrm{du}}^2}\end{array}\right]$

so ${\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}=\sqrt{{\left(\frac{c_x\left(u\right)}{\mathrm{du}}\right)}^2+{\left(\frac{c_{\mathrm{the\; y}}\left(u\right)}{\mathrm{du}}\right)}^2}$ $(\frac{c\left(u\right)}{\mathrm{du}}\&Center\; Dot;\frac{d^{2}C\left(u\right)}{{\mathrm{du}}^2}){|}_{u=u_i}=(\frac{c_x\left(u\right)}{\mathrm{du}}\&CenterDot;\frac{d^2{C}_x\left(u\right)}{{\mathrm{du}}^2}+\frac{c_{\mathrm{the\; y}}\left(u\right)}{\mathrm{du}}\&Center\; Dot;\frac{d^2{C}_{\mathrm{the\; y}}\left(u\right)}{{\mathrm{du}}^2}){|}_{u=u_i}$

In terms of application, the second approximation parameter iteration rule of formula (9) has smaller curve rate error than the first approximation parameter iteration rule of formula (8).

(B) Parameter iteration rule design with rate control:

It can be seen from the above that the first-order and second-order approximation parameter iteration rules represented by equations (8) and (9) are obtained by simplifying the high-order terms in the Taylor expansion of the curve parameters versus time, but in application, due to unknown The impact of omitted items on the curve, the curve rate accuracy is often limited. In view of this, the present invention proposes a parameter compensation method to obtain higher precision curve rate control through the correlation between the compensation amount and the curve. The design method is to add the compensation amount of ε(u _i ) to the approximate parameter iteration method represented by formula (8) to correct the parameter iteration. Since the compensation is performed through the approximate parameter iteration method, the curve can be omitted Computation of quadratic differential information. The compensated parameter iterative formula is: $u_{i+1}=u_{i+1}^{\&prime;}+\mathrm{\&epsiv;}\left(u_i\right)$

in $u_{i+1}^{\&prime;}=u_i+\frac{V\left(u_i\right)\&Center\; Dot;T_{\mathrm{the\; s}}}{{\left|\right|\frac{c\left(u\right)}{\mathrm{du}}\left|\right|}_{u=u_i}}$

In order to obtain the compensation amount ε(u _i ) of the parameter iteration, the present invention uses a Taylor expansion of the position versus the parameter, and obtains the compensation amount through the requirements of the feed rate and the curve speed. Define the parametric curve expression as C(u), where, $C\left(u\right)=\left[\begin{array}{c}{C}_x\left(u\right)\\ C_{\mathrm{the\; y}}\left(u\right)\end{array}\right],$ C _x (u) and C _y (u) can represent the curve positions of the x-axis and y-axis respectively when the parameter u is used.

because, $u_{i+1}=u_{i+1}^{\&prime;}+\mathrm{\&epsiv;}\left(u_i\right)$

Through a Taylor expansion approximation, $C_x\left(u_{i+1}\right)=C_x\left(u_{i+1}^{\&prime;}\right)+\frac{c_x\left(u_{i+1}^{\&prime;}\right)}{\mathrm{du}}\&Center\; Dot;\mathrm{\&epsiv;}\left(u_i\right)$ $C_{\mathrm{the\; y}}\left(u_{i+1}\right)=C_{\mathrm{the\; y}}\left(u_{i+1}^{\&prime;}\right)+\frac{c_{\mathrm{the\; y}}\left(u_{i+1}^{\&prime;}\right)}{\mathrm{du}}\&Center\; Dot;\mathrm{\&epsiv;}\left(u_i\right)$

In order to make the speed of the path motion C(u _i )→C(u _i+1 ) meet the feed rate setting value, V(u _i ), so let $\frac{\sqrt{{[C_x\left(u_{i+1}\right)-C_x\left(u_i\right)]}^2+{[C_{\mathrm{the\; y}}\left(u_{i+1}\right)-C_{\mathrm{the\; y}}\left(u_i\right)]}^2}}{T_{\mathrm{the\; s}}}=V\left(u_i\right)$

Then the following quadratic function expression can be obtained:

U·ε ² +Z·ε+W＝0

in, $u=x^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2+Y^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2$ $Z=2[\mathrm{DX}\&Center\; Dot;x^{\&prime;}\left(u_{i+1}^{\&prime;}\right)+\mathrm{Dy}\&CenterDot;Y^{\&prime;}\left(u_{i+1}^{\&prime;}\right)]$

W＝DX ² +DY ² -(V(u _i )·T _s ) ² $\mathrm{DX}=C_x\left(u_{i+1}^{\&prime;}\right)-C_x\left(u_i\right)$ $\mathrm{Dy}=C_{\mathrm{the\; y}}\left(u_{i+1}^{\&prime;}\right)-C_{\mathrm{the\; y}}\left(u_i\right)$ $x^{\&prime;}\left(u_{i+1}^{\&prime;}\right)=\frac{c_x\left(u_{i+1}^{\&prime;}\right)}{\mathrm{du}}$ $Y^{\&prime;}\left(u_{i+1}\right)=\frac{c_{\mathrm{the\; y}}\left(u_{i+1}^{\&prime;}\right)}{\mathrm{du}}$

And the compensation amount ε(u _i ) can be solved as: ${\mathrm{\&epsiv;}}_{\mathrm{1,2}}\left(u_i\right)=\frac{-Z\&PlusMinus;\sqrt{Z^2-4\mathrm{UW}}}{2u}$ $=\frac{-[\mathrm{DX}\&CenterDot;x^{\&prime;}\left(u_{i+1}^{\&prime;}\right)+\mathrm{Dy}\&CenterDot;Y^{\&prime;}\left(u_{i+1}^{\&prime;}\right)]\&PlusMinus;\sqrt{[x^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2+Y^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2]\&Center\; Dot;{(V\left(u_i\right)\&Center\; Dot;T_{\mathrm{the\; s}})}^2-{[\mathrm{Dy}\&Center\; Dot;x^{\&prime;}\left(u_{i+1}^{\&prime;}\right)-\mathrm{DX}\&CenterDot;Y^{\&prime;}\left(u_{i+1}^{\&prime;}\right)]}^2}}{x^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2+Y^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2}----\left(10\right)$

In formula (10), since the compensation amount ε(u _i ) is the root value of the quadratic program, it is necessary to discuss the properties of its root value and select an appropriate root value as the compensation value of the rate control parameter iteration method. definition $\stackrel{\&Right\; Arrow;}{\mathrm{D.}}=\left[\begin{array}{c}\mathrm{DX}\\ \mathrm{Dy}\end{array}\right]$ ${\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}=\left[\begin{array}{c}{x}^{\&prime;}\left(u_{i+1}^{\&prime;}\right)\\ Y^{\&prime;}\left(u_{i+1}^{\&prime;}\right)\end{array}\right]$

Then formula (10) can be rewritten as ${\mathrm{\&epsiv;}}_{\mathrm{1,2}}\left(u_i\right)=\frac{-(\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\&Center\; Dot;{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;})\&PlusMinus;\sqrt{{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{(V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}})}^2-{|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\&times;\stackrel{\&Right\; Arrow;}{\mathrm{D.}}|}^2}}{{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2}----\left(11\right)$

与参数u_i、u_i+1′、u_i+1间的几何关系可由图3表示，图中θ为向量 与曲线微分向量 间的夹角。在式(11)中，由于 ${\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{(V\left(u_i\right)\&CenterDot;T_s)}^2-{|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\&times;\stackrel{\&RightArrow;}{D}|}^2={\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{T_s}^2[V^2\left(u_i\right)-\frac{{|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\&times;\stackrel{\&RightArrow;}{D}|}^2}{{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{T_s}^2}]$ $={\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{T_s}^2[V^2\left(u_i\right)-{|\frac{{\stackrel{\&RightArrow;}{C}}^{\&prime;}}{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}\&times;\frac{\stackrel{\&RightArrow;}{D}}{T_s}|}^2]$ $={\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{T_s}^2[V^2\left(u_i\right)-{\left|\right|\frac{\stackrel{\&RightArrow;}{D}}{T_s}\left|\right|}^2\&CenterDot;{\sin }^2\mathrm{\&theta;}]$ Among them, the vector curve differential vector

The geometric relationship with the parameters u _i , u _i+1 ′, u _i+1 can be shown in Figure 3, where θ is a vector Differentiate vectors with curves angle between. In formula (11), since ${\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{(V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}})}^2-{|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\&times;\stackrel{\&Right\; Arrow;}{\mathrm{D.}}|}^2={\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&Center\; Dot;{T_{\mathrm{the\; s}}}^2[V^2\left(u_i\right)-\frac{{|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\&times;\stackrel{\&Right\; Arrow;}{\mathrm{D.}}|}^2}{{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&Center\; Dot;{T_{\mathrm{the\; s}}}^2}]$ $={\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&Center\; Dot;{T_{\mathrm{the\; s}}}^2[V^2\left(u_i\right)-{|\frac{{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}}{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}\&times;\frac{\stackrel{\&Right\; Arrow;}{\mathrm{D.}}}{T_{\mathrm{the\; s}}}|}^2]$ $={\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&Center\; Dot;{T_{\mathrm{the\; s}}}^2[V^2\left(u_i\right)-{\left|\right|\frac{\stackrel{\&Right\; Arrow;}{\mathrm{D.}}}{T_{\mathrm{the\; s}}}\left|\right|}^2\&Center\; Dot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}]$

and ${\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2>0,{T_{\mathrm{the\; <figure-callout\; id="2"\; label="s"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">s</figure-callout>}}}^2>0$

so $\{{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&CenterDot;{(V\left(u_i\right)\&Center\; Dot;T_{\mathrm{the\; s}})}^2-{|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\&times;\stackrel{\&Right\; Arrow;}{\mathrm{D.}}|}^2\}$ and $\left\{\right[V^2\left(u_i\right)-{\left|\right|\frac{\stackrel{\&Right\; Arrow;}{\mathrm{D.}}}{T_{\mathrm{the\; s}}}\left|\right|}^2\&Center\; Dot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}\left]\right\}$ same number

in, ${\left|\right|\frac{\stackrel{\&Right\; Arrow;}{\mathrm{D.}}}{T_{\mathrm{the\; s}}}\left|\right|}^2={\left(\frac{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}{T_{\mathrm{the\; s}}}\right)}^2$ is the square of the curve rate generated by the curve interpolator when the parameter u _i+1 ′ is used.

because $\{{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}^2\&Center\; Dot;{(V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}})}^2-{|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\&times;\stackrel{\&Right\; Arrow;}{\mathrm{D.}}|}^2\}$ and $\left\{\right[V^2\left(u_i\right)-{\left|\right|\frac{\stackrel{\&Right\; Arrow;}{\mathrm{D.}}}{T_{\mathrm{the\; s}}}\left|\right|}^2\&Center\; Dot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}\left]\right\}$ The same sign, so the compensation amount ε _1,2 (u _i ) will have the following three situations:

(1) when $[V^2\left(u_i\right)>{\left(\frac{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}{T_{\mathrm{the\; s}}}\right)}^2\&CenterDot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}]$ When , the compensation amount ε _1,2 (u _i ) has different real values.

(2) when $[V^2\left(u_i\right)={\left(\frac{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}{T_{\mathrm{the\; s}}}\right)}^2\&CenterDot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}]$ When , the compensation amount ε _1,2 (u _i ) has the same real value.

(3) when $[V^2\left(u_i\right)<{\left(\frac{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}{T_{\mathrm{the\; s}}}\right)}^2\&Center\; Dot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}]$ When , the compensation amount ε _1,2 (u _i ) is the conjugate imaginary value.

与曲线微分向量 间的夹角θ。因此可能形成相同实数值或共轭虚数值补偿量ε_1，2(u_i)的情况如图4所示。当u_i+1′落在a，b附近时，由于向量 与曲线微分向量 近乎垂直，因Curve rate produced by the comparison curve interpolator The feed rate V(u _i ) and the included angle θ can be known $\left\{\right[V^2\left(u_i\right)-{\left|\right|\frac{\stackrel{\&Right\; Arrow;}{\mathrm{D.}}}{T_{\mathrm{the\; s}}}\left|\right|}^2\&Center\; Dot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}\left]\right\}$ The sign of depends on the vector

Differentiate vectors with curves The angle θ between them. Therefore, it is possible to form the same real-valued or conjugate imaginary-valued compensation quantity ε _1,2 (u _i ) as shown in FIG. 4 . When u _i+1 ′ falls near a, b, since the vector Differentiate vectors with curves nearly vertical, because

间并不允许有大曲率的曲线变化，同时在变化较为平缓的曲线应用上也不会发生相同实数值或共轭虚数值补偿量的情形。因此，补偿量ε_1，2(u_i)在实际的应用上一般为不同的实数值。由式(11)， ${\mathrm{\&epsiv;}}_{\mathrm{1,2}}\left(u_i\right)=\frac{-(\stackrel{\&RightArrow;}{D}\&CenterDot;\frac{{\stackrel{\&RightArrow;}{C}}^{\&prime;}}{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}})\&PlusMinus;\sqrt{{(V\left(u_i\right)\&CenterDot;T_s)}^2-{|\frac{{\stackrel{\&RightArrow;}{C}}^{\&prime;}}{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}\&times;\stackrel{\&RightArrow;}{D}|}^2}}{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}$ $=\frac{-\left(\right|\left|\stackrel{\&RightArrow;}{D}\right||\&CenterDot;\cos \mathrm{\&theta;})\&PlusMinus;\sqrt{{(V\left(u_i\right)\&CenterDot;T_s)}^2-{\left|\right|\stackrel{\&RightArrow;}{D}\left|\right|}^2\&CenterDot;{\sin }^2\mathrm{\&theta;}}}{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}$ $=\frac{-\left(\right|\left|\stackrel{\&RightArrow;}{D}\right||\&CenterDot;\cos \mathrm{\&theta;})\&PlusMinus;\sqrt{{(V\left(u_i\right)\&CenterDot;T_s)}^2-{\left|\right|\stackrel{\&RightArrow;}{D}\left|\right|}^2+{\left|\right|\stackrel{\&RightArrow;}{D}\left|\right|}^2{\cos }^2\mathrm{\&theta;}}}{\left|\right|{\stackrel{\&RightArrow;}{C}}^{\&prime;}\left|\right|}$ This sin ² θ≌1 simultaneously $[V^2\left(u_i\right)\&le;{\left(\frac{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}{T_{\mathrm{the\; s}}}\right)}^2\&CenterDot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}]$ is also more likely to occur. In other words, the same real-valued or conjugate imaginary-valued compensation amounts are more likely to occur near points where the curvature of the curve is greater. However, in practical applications, the same real value or conjugate imaginary value compensation will not occur, because based on the consideration of the accuracy of the position command, the curve parameters

Curve changes with large curvature are not allowed between them, and the same real value or conjugate imaginary value compensation amount will not occur in the application of relatively gentle curves. Therefore, the compensation amount ε _1,2 (u _i ) generally has different real values in practical applications. By formula (11), ${\mathrm{\&epsiv;}}_{\mathrm{1,2}}\left(u_i\right)=\frac{-(\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\&Center\; Dot;\frac{{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}}{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}})\&PlusMinus;\sqrt{{(V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}})}^2-{|\frac{{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}}{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}\&times;\stackrel{\&Right\; Arrow;}{\mathrm{D.}}|}^2}}{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}$ $=\frac{-\left(\right|\left|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\right||\&CenterDot;\cos \mathrm{\&theta;})\&PlusMinus;\sqrt{{(V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}})}^2-{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}^2\&CenterDot;{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&theta;}}}{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}$ $=\frac{-\left(\right|\left|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\right||\&CenterDot;\cos \mathrm{\&theta;})\&PlusMinus;\sqrt{{(V\left(u_i\right)\&CenterDot;T_{\mathrm{the\; s}})}^2-{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}^2+{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="C9812311100241.png,C9812311100271.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}}}{\left|\right|{\stackrel{\&Right\; Arrow;}{C}}^{\&prime;}\left|\right|}$

make ${(V\left(u_i\right)\&Center\; Dot;T_{\mathrm{the\; s}})}^2-{\left|\right|\stackrel{\&Right\; Arrow;}{\mathrm{D.}}\left|\right|}^2=\mathrm{\&mu;}$

When u _i+1 ′ adopts the first-order approximate parameter iteration method, μ is generally a relatively small value, so it can be known from the approximation of the Taylor expansion,

That is, ε _1,2 (u _i ) has a negative compensation value and a compensation value near the zero point, and the negative compensation value is far away from the zero point. Since the purpose of the compensation value here is to compensate for the insufficiency of the parameter u _i+1 ′ in the curve rate, the compensation value near the zero point is selected to obtain a more reliable parameter iteration result, namely $\mathrm{\&epsiv;}\left(u_i\right)=\frac{-[\mathrm{DX}\&Center\; Dot;x^{\&prime;}\left(u_{i+1}^{\&prime;}\right)+\mathrm{Dy}\&Center\; Dot;Y^{\&prime;}\left(u_{i+1}^{\&prime;}\right)]+\sqrt{[x^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2+Y^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2]\&CenterDot;{(V\left(u_i\right)\&Center\; Dot;T_{\mathrm{the\; s}})}^2-{[\mathrm{Dy}\&Center\; Dot;x^{\&prime;}\left(u_{i+1}^{\&prime;}\right)-\mathrm{DX}\&CenterDot;Y^{\&prime;}\left(u_{i+1}^{\&prime;}\right)]}^2}}{[x^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2+Y^{\&prime;}{\left(u_{i+1}^{\&prime;}\right)}^2]}$

In the method discussed in this section, because the compensation amount is added based on the incorrect interpolation curve rate, a higher accuracy of the curve rate can be obtained.

In the above-mentioned parameter iteration algorithm with rate control, since the curve rate of the generated curve can be precisely controlled, it can be applied to the design of a parametric curve interpolator, and can have different operating modes according to different processing conditions. Generally, the operation modes of the interpolator can be divided into (1) constant speed operation mode and (2) acceleration/deceleration operation mode according to different processing conditions. The constant rate operation mode means that in the process of parametric curve realization, the interpolator can generate interpolation points with a fixed curve rate. The acceleration and deceleration operation mode can be applied in the interpolation process between the beginning and the end of the curve motion to obtain a relatively gentle curve speed change.

(C) Apply the feed rate control parameter iteration method of this embodiment to design an interpolator and perform a comparison of processing examples:

(C.1) Application example

In the setting of the system, the system used in the present invention is to write the software interpolation program in C language and execute it on the system of the personal computer tool Pentium 200MHz CPU. In the setting of the parameter curve, the present invention uses a 2-degree (2degree) NURBS parameterized curve to simulate the command path of the bow-tie curve as shown in FIG. 5 . Its related NURBS parameters are set to:

1. The order of control points is $\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{c}-150\\ -150\end{array}\right],\left[\begin{array}{c}-150\\ 150\end{array}\right],\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{c}150\\ -150\end{array}\right],\left[\begin{array}{c}150\\ 150\end{array}\right],\left[\begin{array}{c}0\\ 0\end{array}\right]\mathrm{mm}.$

2. The weight vector is: W＝[1 0.85 0.85 1 0.85 0.85 1].

3. The node vector is: $u=\left[\begin{array}{cccccccccc}0& 0& 0& \frac{1}{4}& \frac{1}{2}& \frac{1}{2}& \frac{3}{4}& 1& 1& 1\end{array}\right].$

And in the following parameter iteration process, the relevant parameters are set as:

1. Sampling time: T _s =0.01 sec.

2. Feed rate: F=12m/min=200mm/sec.

(C.2) Constant rate operation mode:

Since the design of the parametric curve interpolator with speed control can precisely control the curve speed of the interpolation result, the given fixed feed rate can be used to obtain the interpolation result of the fixed curve speed during the curve interpolation process.

Simulation results of parametric curve interpolator constant rate operation mode:

Here we will compare the operation modes of parametric curve interpolators designed by different parameter iteration methods, uniform (uniform), first order approximation (1st order approximation), second order approximation (2nd order approximation), constant rate (constant feedrate) The resulting curve rate changes.

Figures 6, 7, and 8 show the speed results of the simulated interpolation curves in different interpolator operating modes, while the calculation results of parameter compensation values in the constant-rate operating mode are shown in Figure 9.

The summary of interpolation simulation results is shown in Table 1:

Table 1:

in,

1. The measurement of curve rate error refers to the difference between the curve rate generated by the interpolator and the designed feed rate.

2. The formula for calculating the rate of change of the curve rate is ${\mathrm{\&eta;}}_i=\frac{V_f-V_i}{V_f},$ Indicates the rate of change of the curve rate for each sampling time

V _i represents the curve rate generated by the interpolator per sample time

V _f is the feed rate setting

3. The parameter spacing in the uniform parameter iteration mode is the size of the sampling time.

It is obvious from the simulation results that the interpolation mode of the constant rate operation mode has the smallest curve rate error and rate change rate because the parameter iteration method of the rate control considers the relationship between the curve rate and the compensation value. At the same time, because the parameter iteration method of the first-order (1st) approximation is compensated, it can be seen that the rate accuracy of the first-order (1st) approximation after compensation is improved, and at the same time surpasses the rate provided by the parameter iteration of the second-order (2nd) approximation precision. The interpolation method of the uniform parameter operation mode has the largest curve rate error and rate change rate, because the parameter and the curve are not uniformly distributed. In the comparison of the command chord height error between the interpolation points, the first-order (1st) approximation, the second-order (2nd) approximation and the command chord height error of the constant speed operation mode are not much different, because the bowtie curve used in this simulation The speed change of the middle curve has little influence on the command chord height error between interpolation points, but the command chord height error of the uniform operation mode is significantly larger than other parameter iteration methods. In the comparison of the calculation time of each interpolation point, the time required to generate the interpolation point by the uniform parameter iteration method is only the calculation time of the curve interpolation point, so it will be the least, and the first-order (1st) approximation and the second-order In comparison with the (2nd) approximation, since the second-order (2nd) approximation requires curve quadratic differential and higher-order calculations, the calculation time will be longer than that of the first-order (1st) approximation, and the parameter iteration method of the rate control is based on a The first-order (1st) approximation is based on the second-order differential calculation of the curve, but it needs to calculate an additional curve interpolation point and some compensation value calculations, so the calculation amount will be slightly higher than the second-order (2nd) approximation. Although the parameter iteration method with rate control requires a higher amount of calculation than other parameter iteration methods, the curve rate accuracy of the interpolator is much higher than other parameter iteration methods. Based on this high-precision curve rate control, the controller Or the workpiece design engineer can grasp a clearer processing process; at the same time, compared with the sampling time, the calculation time of the interpolation point in the fixed-rate operation mode accounts for a very small proportion, so the calculation time is acceptable in the real-time realization process. To sum up, in terms of the change degree of the curve rate, the parametric curve interpolator has the smallest curve rate error in the constant rate operation mode, followed by the second-order (2nd) approximation operation mode and the first-order (1st) approximation operation mode. The parameter operating mode has the largest curve rate error.

(C.3) Acceleration and deceleration operation mode:

The traditional way of acceleration and deceleration is to pass the position command through a low-pass filter to generate a smoother command input. However, this method will cause radial and chord height errors in the command path with large changes, such as near the starting point and end point of the processing curve. In view of this, Kim proposed to use the acceleration and deceleration method of parameters to achieve the purpose of parameterized curve acceleration and deceleration. At the same time, because the acceleration and deceleration operation is performed on parameters, the interpolation point has a small radial error. Although the parameter acceleration and deceleration method proposed by Kim can make the generation of interpolation points have the function of acceleration and deceleration, it cannot really achieve the purpose of rate acceleration and deceleration on some curves, because the parameter space and the curve do not correspond uniformly. In order to obtain the real acceleration and deceleration of the curve and the change in the curve movement at an appropriate control rate, the present invention proposes the acceleration and deceleration operation mode of the parametric curve interpolator for the application of the feed rate parameter acceleration and deceleration.

Because in the parameter iteration process with rate control, the workpiece designer can accurately grasp the change of the rate, so it can be used as the realization of the acceleration and deceleration of the feed rate parameter, and can accurately grasp the acceleration and deceleration rate changes during acceleration and deceleration . The parameter iteration method with rate control is to control the iteration of parameters so that the curve rate of the interpolator can meet the set value Vu _i of the feed rate, so the feed rate parameter Vu _i can be accelerated and decelerated in the design of acceleration and deceleration operate. Then, the feed rate parameter Vu _i calculated by acceleration and deceleration is substituted into the parameter iteration formula of feed rate control to make the curve rate change as expected, so that more accurate acceleration and deceleration control can be obtained, and more accurate processing results can also be obtained.

The same application can also be used for first order (1st) and second order (2nd) approximation parameter iteration rules, or other feed rate dependent parameter iteration rules. However, in order to accurately grasp the change of acceleration rate and deceleration rate during acceleration and deceleration, the parameter iteration method must make the curve rate of the interpolator accurately reflect the feed rate setting, so it is recommended to use the parameter iteration with rate control proposed by this invention The method realizes the acceleration and deceleration operation mode of the parametric curve interpolator.

Simulation results of acceleration and deceleration operation mode of parametric curve interpolator:

In the simulation of the acceleration and deceleration operation mode of the parametric curve interpolator, since the acceleration rate and the deceleration rate are the same process, only the initial acceleration rate process of the parametric curve interpolator is compared. In the simulation in this section, three parameter iteration methods related to the feed rate will be compared: the first (1st) order approximation, the second (2nd) order approximation, and the rate control parameter iteration method applied to the curve rate during acceleration and deceleration. At the same time, in order to observe the results of applying the parameter iteration rule to different acceleration and deceleration methods, this section will compare three commonly used acceleration and deceleration methods - linear acceleration and deceleration, quadratic acceleration and deceleration (parabolic), and exponential acceleration and deceleration. Since the quadratic curve acceleration and deceleration mode is an acceleration and deceleration mode with smooth rate changes, the simulation results of the interpolation curve speed of different parameter iteration methods applied to the quadratic curve acceleration and deceleration mode are shown in Figures 10, 11, and 12. Table 2 lists the maximum curve speed difference when the parameter iteration method is applied to three different acceleration and deceleration methods.

Table 2: Rate Error

In the acceleration rate process of the bow-tie curve, since the curvature of the curve gradually changes from small to large, the error of the curve rate will gradually increase during parameter iteration, as shown in Figures 10, 11, and 12. But when comparing different parameter iteration methods, the parameter iteration method with rate control is obviously better than other parameter iteration methods related to feed rate. In the comparison of different acceleration and deceleration methods, the exponential acceleration and deceleration method has the largest curve rate error, and the quadratic curve acceleration and deceleration method has a smoother curve rate change and the smallest rate error. Therefore, summing up the above simulation results, it can be seen that in the process of acceleration and deceleration, when the feed rate changes smoothly in the form of a quadratic curve, the parameter iteration method with rate control can make the curve rate generated by the interpolator have high accuracy The change of the quadratic curve. In other words, when the feed rate change curve of acceleration and deceleration is designed so that the interpolation result of the parametric curve interpolator has the designed acceleration rate and deceleration rate change, the parametric curve interpolation is designed through the parameter iteration method with speed control The interpolator can make the acceleration and deceleration results of the interpolator as designed, especially when the high feed rate curve is moved.

It can be seen from the above embodiment and the comparative examples that the present invention proposes a parametric curve interpolator design method with speed control, and according to the different control purposes of the curve speed, there are constant speed operation modes and acceleration and deceleration operation modes to adapt to different processing condition. The rate control parameter iteration method proposed by the present invention can obtain accurate curve rate interpolation results when applied to the design of the interpolator because the compensation amount considering the relationship between the parameter and the curve rate is added. Since the speed change will affect the surface roughness of the processed workpiece, the constant speed operation mode aims to maintain a constant value of the movement speed between adjacent interpolation points during the realization of the parameterized curve. In the discussion of the acceleration and deceleration operation mode of the parametric curve interpolator, since the present invention proposes a high-precision speed control parameter iteration method, it can be applied to the control of curve speed acceleration and deceleration, and the smoothly changing feed rate and accurate The feed rate control parameter iterates so that the acceleration and deceleration rates of the interpolated result curve rate of the parameterized curve interpolator vary as expected.

In terms of calculation time, although the parametric curve interpolation method proposed by the present invention requires more calculation process, the result can control the position and speed of the curve path more accurately, so that the machining result of the workpiece can be more accurate. Now that the speed of computers and processor chips is getting faster and faster, and the processing quality requirements are increasing, the design of the parametric curve interpolator proposed by the present invention provides good position and curve speed accuracy at the same time.

Claims (8)

1. A computer numerical control curve path speed control method is to process the constant speed control operation curve path of the interpolation point that the interpolator can produce the fixed curve speed in the process of realizing the parameterized curve, comprising the following steps: (1) After receiving the information of the curve interpreted by the computer-aided design, the parametric curve interpolator generates continuous interpolation points with the parameter iteration method of the parametric curve; (2) Only the influence of the chord height error of the position command error generated by the interpolator is considered, and the radial error is not considered; (3) derive an approximate parameter iteration rule of the parametric curve formula; (4) Design a parameter iteration rule with rate control, propose a parameter compensation based on an approximate parameter iteration rule, and obtain a higher-precision curve rate control through the correlation between the compensation amount and the curve; (5) Select the compensation value near the zero point to obtain a more reliable parameter iteration result; and (6) Send the continuous interpolation points obtained from the compensation value to the control system to move the tool center along the parameterized curve. 2. The computer numerical control curve path speed control method according to claim 1, wherein the parameterized curve used is a non-equidistant B spline. 3. computer numerical control curve path speed control method according to claim 1, is characterized in that, the parameter iteration method that parametric curve interpolator uses, is based on the generalized parameter iteration rule proposed by Chou and Yang to propose feed Parameter iteration method for rate control. 4. A computer numerical control curve path rate control method is to perform acceleration and deceleration at the feed rate parameter so that the curve rate is carried out in the same acceleration and deceleration mode and obtain the acceleration and deceleration control operation curve path processing of the real acceleration and deceleration rate results, including the following step: (1) After receiving the information of the curve interpreted by the computer-aided design, the parametric curve interpolator generates continuous interpolation points with the parameter iteration method of the parametric curve; (2) Only the influence of the chord height error of the position command error generated by the interpolator is considered, and the radial error is not considered; (3) derive an approximate parameter iteration rule of the parametric curve formula; (4) Design a parameter iteration rule with rate control, propose a parameter compensation based on an approximate parameter iteration rule, and obtain a higher-precision curve rate control through the correlation between the compensation amount and the curve; (5) Select the compensation value near the zero point to obtain a more reliable parameter iteration result; and (6) Send the continuous interpolation points obtained from the compensation value to the control system to move the tool center along the parameterized curve. 5. The computer numerical control curve path speed control method according to claim 4, characterized in that, the parameterized curve used is a non-equidistant B spline. 6. computer numerical control curve path speed control method according to claim 4 is characterized in that, the parameter iterative method that parametric curve interpolator uses is based on the generalized parameter iterative rule proposed by Chou and Yang. Parameter iteration method for rate control. 7. A computer numerical control curve path speed control device, comprising: Computer for aided design, which includes parametric curve format and segmented curves; Computer numerical control mechanism, including: The parametric curve interpolator, after receiving the parametric curve information interpreted by the computer, generates continuous interpolation points in the way of generating points of the parametric curve; The controller receives the continuous interpolation points generated by the parametric curve interpolator and sends out processing signals; The tool receives the control processing signal sent by the controller, and the center moves along the parameterized curve; A man-machine interface device that allows man-machine communication between the operator and the parametric curve interpolator; Wherein the parameterized curve interpolator can perform constant rate interpolation operation mode and acceleration/deceleration interpolation operation mode on the parameterized curve. 8. A parametric curve interpolator device for computer numerical control curve path rate control, comprising: a constant-rate interpolator that keeps the parametric curve path at a constant rate during the interpolation implementation; and The acceleration and deceleration interpolator makes the processing curve path obtain smoother acceleration and deceleration motion.

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