CN104950821B - A kind of digital control system speed planning method based on fractional calculus - Google Patents

Abstract

A kind of digital control system speed planning method based on fractional calculus, the present invention relates to digital control system speed planning method.Causing lathe to produce the present invention is to solve existing speed planning method, high vibration, amount of calculation are huge, and programming is complicated and sudden change of acceleration occurs, the problem of producing rigid shock.The present invention is, by 1, using continuous function y=f (t), to derive Fractional Derivative；2nd, t is tried to achieve₁、t₂And t₃Corresponding nominal acceleration is respectively a₁、a₂And a₃；3rd, required accelerating sections speed planning curve is obtained；4th, total displacement increment is calculated；5th, displacement is tried to achieve for D (m) according to formula (3)；6th, controlled device is not up to maximal rate V_maxThen reduced speed now from the step of deceleration point i-th until speed is 0；7th, it is 0 to be reduced speed now from deceleration point jth step to speed；8th, controlled device carries out decelerating to what speed was realized for the step such as 0 according to class S types curve.The present invention is applied to digital control system speed planning field.

Description

Numerical control system speed planning method based on fractional calculus

Technical Field

The invention relates to a speed planning method of a numerical control system, in particular to a speed planning method of a numerical control system based on fractional calculus.

Background

Speed planning is the core technology of numerical control systems. Common speed planning methods include a trapezoidal speed planning method, an S-shaped curve speed planning method, an exponential speed planning method, and the like. Although the trapezoidal speed planning method has small calculated amount and simple programming, the phenomenon of acceleration sudden change exists in the acceleration and deceleration stage, so that the machine tool generates severe vibration and is not suitable for high-speed and high-precision machining. The conventional S-shaped curve speed planning method divides the whole speed planning into 7 stages by utilizing a polynomial representation method, and then discusses in each stage, so that not only is the boundary judgment required to be carried out between adjacent stages in the implementation process, but also several stages are judged to exist in the actual motion, for example, the motion distance is very small (several millimeters), one or more stages such as a constant speed does not exist, and therefore, the calculation amount is huge, and the programming is complex. The same exponential speed planning method can generate acceleration sudden change at the moment of acceleration and deceleration switching, generates rigid impact, and is not suitable for high-speed and high-precision occasions.

Disclosure of Invention

The invention aims to solve the problems that the existing speed planning method causes severe vibration, huge calculated amount, complex programming, sudden acceleration change and rigid impact of a machine tool, and provides a numerical control system speed planning method based on fractional calculus.

The above-mentioned invention purpose is realized through the following technical scheme:

step one, according to the controlled object motion time t, by using a continuous function y ═ f (t), a fractional order derivative is derived:

wherein h is the time step, j is the index of the continuous addition operation sigma, and the frequency of the fractional order integration is alpha; c is the calculus onset time, typically c ═ 0; when alpha is greater than 0, the fractional order differential operation is carried out, and when alpha is less than 0, the fractional order integral operation is carried out;

step two, dividing the movement time T of the controlled object acceleration section into three sections which are respectively T₁、t₂And t₃And t is₁、t₂And t₃Respectively corresponding nominal accelerations are respectively a₁、a₂And a₃(ii) a Wherein, a₁、a₂And a₃Is constrained by₂>a₁>0,a₃<0；

Step three, order α<0, nominal acceleration a using equation (2)₁,a₂,a₃Performing fractional order integration to obtain a required speed planning curve of the acceleration section, wherein the curve is an S-like curve;

step four, the speed of the ith movement period of the controlled object is v (i), and the movement period, namely the time step length is h; if the displacement increment is d (i), ending the Nth period, namely the total displacement increment;

step five, t₁Reaches the maximum speed V at any moment_maxThe number of steps taken in the Nth cycle is m ═ t₁Obtaining the displacement D (m) by using the formula (3) when the m is equal to N;

step six, the controlled object moves according to the S-shaped curve, if the stroke S is<D (m), the controlled object does not reach the maximum speed V_maxStarting to decelerate from the ith deceleration point until the speed is 0, and satisfying s/2 ═ D (i);

step seven, the controlled object moves according to the S-shaped curve; if the stroke s>D (m), the controlled object reaches the maximum speed V_maxThen, the mixture passes through a constant speed V_maxAfter the process, the speed is reduced from the jth step of the speed reduction point to the speed of 0, and the speed reduction point meets s-D (j) ═ D (m);

step eight, when the stroke s is equal to D (m), the controlled object reaches the maximum speed V_maxThen, the controlled object is decelerated to the speed of 0 according to the S-shaped curve; thus, the numerical control system speed planning method based on fractional calculus is completed.

Effects of the invention

Aiming at the defects in the prior art, the invention provides a novel method for planning the speed curve of the numerical control system, which realizes the stable acceleration and deceleration of the controlled object. The technique used is a fractional calculus technique. The fractional calculus adopted by the invention is a generalized popularization of the integral fractional calculus. Its study is almost simultaneous with the traditional newton calculus system, which has historically been studied by many large mathematicians. Various definitions have been given in the respective fields, among which the definition Gr ü nwald-Letnikov (G-L) is the most suitable for our study.

The fractional calculus adopted by the invention has memorability and non-locality, and can well smooth data and refine the data. The acceleration is integrated by adopting a fractional calculus technology, so that a smooth acceleration and deceleration speed curve can be obtained, and the requirement of stable speed planning is met. The acceleration and deceleration curve under the long stroke as shown in fig. 3c has a complete acceleration stage, a constant speed stage and a deceleration stage.

Drawings

FIG. 1 is a schematic diagram of nominal acceleration according to one embodiment;

FIG. 2 is a schematic diagram of a velocity profile according to an embodiment;

FIG. 3a is a schematic view of an acceleration/deceleration curve in different ranges according to an embodiment;

FIG. 3b is a schematic view of an acceleration/deceleration curve in different ranges according to an embodiment;

fig. 3c is a schematic view of an acceleration/deceleration curve in different ranges according to an embodiment.

Detailed Description

The first embodiment is as follows: the numerical control system speed planning method based on fractional calculus of the embodiment is specifically prepared according to the following steps:

a numerical control system speed planning method based on fractional calculus utilizes the memory characteristic of fractional calculus to carry out one-time fractional calculus on predefined nominal acceleration, and a smooth acceleration and deceleration speed planning curve can be obtained

Step one, according to the controlled object motion time t, by using a continuous function y ═ f (t), a fractional order derivative is derived:

wherein h is the time step, j is the index of the continuous addition operation sigma, and the frequency of the fractional order integration is alpha; c is the calculus onset time, typically c ═ 0; when alpha is greater than 0, the fractional order differential operation is carried out, and when alpha is less than 0, the fractional order integral operation is carried out; the invention uses formula (2) to carry out fractional order integration on the integrand to obtain the expected acceleration and deceleration curve;

step two, dividing the movement time T of the controlled object acceleration section into three sections which are respectively T₁、t₂And t₃And t is₁、t₂And t₃Respectively corresponding nominal accelerations are respectively a₁、a₂And a₃(ii) a Wherein, a₁、a₂And a₃Is constrained by₂>a₁>0,a₃<0；

Step three, order α<0, nominal acceleration a using equation (2)₁,a₂,a₃Performing fractional order integration to obtain a required acceleration section speed planning curve which is an S-like curve, and finally obtaining the acceleration section speed planning curve in the whole process as shown in figure 2;

step four, the speed planning is divided into three stages: an acceleration stage, a constant speed stage and a deceleration stage; the deceleration stage and the acceleration stage are mirror images, and the constant speed stage is at a maximum speed V_maxRunning at a constant speed until a deceleration point is reached; thus, the whole speed planning is the planning of the acceleration stage, and the speed planning method of the acceleration stage is introduced below; the acceleration phase speed plan has two performance indexes: first, acceleration time T, second, maximum speed V_maxThe two parameters are known parameters as input by a user; then a nominal acceleration defining an acceleration phase, as shown in fig. 1; in the actual movement process, according to the size of the stroke, the speed of the ith movement period of the controlled object is v (i), and the movement period, namely the time step length is h; the displacement increment is d (i) (the displacement increment is the displacement of each step, so the total displacement increment is the sum of the displacement increments), and the Nth period is ended, namely the total displacement increment;

step five, shown in FIG. 2, t₁Reaches the maximum speed V at any moment_maxThe number of steps taken in the Nth cycle is m ═ t₁Obtaining the displacement D (m) by using the formula (3) when the m is equal to N;

step six, the controlled object moves according to the S-shaped curve, if the stroke S is<D (m), the controlled object does not reach the maximum speed V_maxStarting to decelerate from the ith deceleration point until the speed is 0, and satisfying s/2 ═ D (i); FIG. 3a shows the acceleration and deceleration curve for a short stroke, when deceleration begins without reaching a maximum speed;

step seven, the controlled object moves according to the S-shaped curve; if the stroke s>D (m), the controlled object reaches the maximum speed V_maxThen, the mixture passes through a constant speed V_maxAfter the process, the speed is reduced from the jth step of the speed reduction point to the speed of 0, and the speed reduction point meets the condition that the speed reduction section and the speed reduction section are symmetrical mirror images according to a speed planning curve; as shown in fig. 3c, the acceleration/deceleration curve under a long stroke has a complete acceleration stage, a constant speed stage and a deceleration stage;

step eight, when the stroke s is equal to D (m), the controlled object reaches the maximum speed V_maxThen, the controlled object is decelerated to the speed of 0 according to the S-shaped curve, and the controlled object just reaches the maximum speed as shown in fig. 3b, but does not have a uniform speed stage, and directly enters a deceleration stage; thus, the numerical control system speed planning method based on fractional calculus is completed.

The effect of the embodiment is as follows:

aiming at the defects in the prior art, the embodiment provides a novel method for planning the speed curve of the numerical control system, so that the stable acceleration and deceleration of the controlled object are realized. The technique used is a fractional calculus technique. The fractional calculus employed in the present embodiment is a generalized generalization of the integral fractional calculus. Its study is almost simultaneous with the traditional newton calculus system, which has historically been studied by many large mathematicians. Various definitions have been given in the respective fields, of which the Grunnwald-Letnikov (G-L) definition is the most suitable for our study of the present embodiment.

The fractional calculus adopted by the embodiment has memorability and non-locality, and can well smooth data and achieve refinement. The acceleration is integrated by adopting a fractional calculus technology, so that a smooth acceleration and deceleration speed curve can be obtained, and the requirement of stable speed planning is met. The acceleration and deceleration curve under the long stroke as shown in fig. 3c has a complete acceleration stage, a constant speed stage and a deceleration stage.

The second embodiment is as follows: the first difference between the present embodiment and the specific embodiment is: in the first step, according to the controlled object motion time t, by using a continuous function y ═ f (t), a specific process of deriving a fractional order derivative is as follows:

(1) for the continuous function y ═ f (t), the first order derivation formula for the function y ═ f (t) is according to the classical derivation formula:

its physical meaning is understood to be the rate of change of the above function;

(2) the derivation of f' (t) obtained in step (1) into a second derivative is specifically:

its physical meaning can be understood as the curvature of a curve or the acceleration of a certain physical quantity;

(3) the derivation of f' (t) obtained according to the step (2) into a third derivative is specifically as follows:

(4) deducing the nth derivative of f (t) according to the derivation process of the steps (1) to (3) by using a mathematical induction method, wherein the nth derivative has the following form:

wherein,

from this, it is deduced that the fractional order derivative is as follows:

other steps and parameters are the same as those in the first embodiment.

The third concrete implementation mode: the present embodiment differs from the first or second embodiment in that: let α <0 in step three, the mathematical expression of the nominal acceleration defined in step two is:

and (3) performing alpha-order fractional order integration on the nominal acceleration a (t) by using a formula (2) to obtain a speed planning curve of the acceleration section. Other steps and parameters are the same as those in the first or second embodiment.

The fourth concrete implementation mode: the difference between this embodiment mode and one of the first to third embodiment modes is: in the third step, alpha is set to be adjusted between-1 and-2 so as to obtain the acceleration section speed planning curve meeting the requirement. Other steps and parameters are the same as those in one of the first to third embodiments.

Claims (4)

1. A numerical control system speed planning method based on fractional calculus is characterized in that the numerical control system speed planning method based on the fractional calculus is specifically carried out according to the following steps:

step one, according to the controlled object motion time t, by using a continuous function y ═ f (t), a fractional order derivative is derived:

$D_t^{\mathrm{\&alpha;}}{}_{c}f\left(t\right)=\underset{h\&RightArrow;0}{\lim }\frac{1}{h^{\mathrm{\&alpha;}}}\underset{j=0}{\overset{\&lsqb;(t-c)/h\&rsqb;}{\&Sigma;}}{(-1)}^j\left(\begin{array}{c}\mathrm{\&alpha;}\\ j\end{array}\right)f(t-jh)---\left(2\right);$

wherein h is the time step, j is the index of the continuous addition operation sigma, and the frequency of the fractional order integration is alpha; c is the calculus onset time, typically c ═ 0; when alpha is greater than 0, the fractional order differential operation is carried out, and when alpha is less than 0, the fractional order integral operation is carried out;

step two, dividing the movement time T of the controlled object acceleration section into three sections which are respectively T₁、t₂And t₃And t is₁、t₂And t₃Respectively corresponding nominal accelerations are respectively a₁、a₂And a₃(ii) a Wherein, a₁、a₂And a₃Is constrained by₂>a₁>0,a₃<0；

Step three, order α<0, nominal acceleration a using equation (2)₁,a₂,a₃Performing fractional order integration to obtain a required speed planning curve of the acceleration section, wherein the curve is an S-like curve;

step four, the speed of the ith movement period of the controlled object is v (i), and the movement period, namely the time step length is h; if the displacement increment is d (i), ending the Nth period, namely the total displacement increment;

$D\left(N\right)=h\underset{i=0}{\overset{N}{\&Sigma;}}v\left(i\right)---\left(3\right);$

step five, t₁Reaches the maximum speed V at any moment_maxThe number of steps taken in the Nth cycle is m ═ t₁Obtaining the displacement D (m) by using the formula (3) when the m is equal to N;

step six, the controlled object moves according to the S-shaped curve, if the stroke S is<D (m), the controlled object does not reach the maximum speed V_maxStarting to decelerate from the ith deceleration point until the speed is 0, and satisfying s/2 ═ D (i);

step seven, the controlled object moves according to the S-shaped curve; if the stroke s>D (m), the controlled object reaches the maximum speed V_maxThen, the mixture passes through a constant speed V_maxAfter the process, the speed is reduced from the jth step of the speed reduction point to the speed of 0, and the speed reduction point meets s-D (j) ═ D (m);

step eight, when the stroke s is equal to D (m), the controlled object reaches the maximum speed V_maxThen, the controlled object is decelerated to the speed of 0 according to the S-shaped curve; namely, a method based on fractional order is completedA method for planning the speed of a numerical control system of calculus.

2. The numerical control system speed planning method based on fractional calculus as claimed in claim 1, characterized in that: in the first step, according to the controlled object motion time t, by using a continuous function y ═ f (t), a specific process of deriving a fractional order derivative is as follows:

(1) for the continuous function y ═ f (t), the first order derivation formula for the function y ═ f (t) is according to the classical derivation formula:

$f^{\&prime;}\left(t\right)=\frac{df}{dt}=\underset{h\&RightArrow;0}{\lim }\frac{f\left(t\right)-f(t-h)}{h};$

(2) the derivation of f' (t) obtained in step (1) into a second derivative is specifically:

$f^{\&prime;\&prime;}\left(t\right)=\frac{d^{2}f}{{\mathrm{dt}}^2}=\underset{h\&RightArrow;0}{\lim }\frac{f^{\&prime;}\left(t\right)-f^{\&prime;}\left(t-h\right)}{h}=\underset{h\&RightArrow;0}{\lim }\frac{f\left(t\right)-2f\left(t-h\right)+f\left(t-2h\right)}{h^2};$

(3) the derivation of f' (t) obtained according to the step (2) into a third derivative is specifically as follows:

$f^{\&prime;\&prime;\&prime;}\left(t\right)=\frac{d^{3}f}{{\mathrm{dt}}^3}=\underset{h\&RightArrow;0}{\lim }\frac{f\left(t\right)-3f(t-h)+3f(t-2h)-f(t-3h)}{h^3};$

(4) deducing the nth derivative of f (t) according to the derivation process of the steps (1) to (3) by using a mathematical induction method, wherein the nth derivative has the following form:

$f^{\left(n\right)}\left(t\right)=\frac{d^{n}f}{{\mathrm{dt}}^n}=\underset{h\&RightArrow;0}{\lim }\frac{1}{h^n}\underset{r=0}{\overset{n}{\&Sigma;}}{(-1)}^r\left(\begin{array}{c}n\\ r\end{array}\right)f(t-rh)---\left(1\right)$

wherein,

$\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n(n-1)(n-2)...(n-r+1)}{r!}$

from this, it is deduced that the fractional order derivative is as follows:

$D_t^{\mathrm{\&alpha;}}{}_{c}f\left(t\right)=\underset{h\&RightArrow;0}{\lim }\frac{1}{h^{\mathrm{\&alpha;}}}\underset{j=0}{\overset{\&lsqb;(t-c)/h\&rsqb;}{\&Sigma;}}{(-1)}^j\left(\begin{array}{c}\mathrm{\&alpha;}\\ j\end{array}\right)f(t-jh)---\left(2\right).$

3. the method for numerical control system speed planning based on fractional calculus as claimed in claim 1, wherein α is obtained by the third step<0, utilizingEquation (2) for nominal acceleration a₁,a₂,a₃The mathematical expression of (a) is:

$a\left(T\right)=\left\{\begin{array}{cc}{a}_1& 0\&le;T<t_1\\ a_2& t_1\&le;T<t_2\\ a_3& t_2\&le;T\&le;t_3\end{array}\right.$

and (3) performing alpha-order fractional order integration on the nominal acceleration a (t) by using a formula (2) to obtain a speed planning curve of the acceleration section.

4. The numerical control system speed planning method based on fractional calculus as claimed in claim 1, characterized in that: in the third step, alpha is set between-1 and-2.