CN115562183A - Motion control system high-precision circular interpolation method based on direct function method - Google Patents
Motion control system high-precision circular interpolation method based on direct function method Download PDFInfo
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Abstract
The invention discloses a motion control system high-precision circular interpolation method based on a direct function method, which is used for a foam cutting system. By combining double-S-shaped speed planning, S-shaped acceleration and deceleration from a current interpolation point to a next interpolation point in the circular arc track machining process is realized, the feeding speed of the current interpolation point and the period from the current interpolation point to the next interpolation point are obtained, the coordinate and the feeding speed of the next interpolation point are further obtained, and the machining precision and the machining efficiency of the circular arc track are improved in the whole interpolation process.
Description
Technical Field
The invention belongs to the field of motion control, and particularly relates to a high-precision circular interpolation method of a motion control system based on a direct function method.
Background
In a foam cutting system, machining is generally performed by a cutting machine, and a rich and efficient interpolation function is one of important indexes for measuring system performance. The interpolation mode only comprises linear interpolation and circular interpolation at present, the traditional method adopts the steps that firstly, the track is subjected to rough interpolation and then is dispersed through software to form continuous small line segments connected end to end, and then, a controller is used for carrying out linear interpolation on each small line segment. The direct function method of the circular interpolation adopts approximate calculation, is easy to cause larger errors, does not meet the requirements of the current high-precision cutting machine, and provides a high-precision circular interpolation method of a motion control system based on the direct function method by combining a double-S-shaped speed planning model in order to improve the processing performance of the cutting machine and ensure continuous and smooth operation of the feeding speed.
Disclosure of Invention
The invention aims to provide a motion control system high-precision circular interpolation method based on a direct function method aiming at the problems.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the invention provides a motion control system high-precision circular interpolation method based on a direct function method, which is used for a foam cutting system, and comprises the following steps:
obtaining the circular arc track to be processed, knowing the coordinates of the circle center and the coordinates of the current interpolation point, setting the coordinates of the next interpolation point with unknown number, the first feeding speed of the current interpolation point, the period from the current interpolation point to the next interpolation point, the corresponding feeding step length between the current interpolation point and the next interpolation point and the step angle corresponding to the feeding step length to be represented by letters, and obtaining a first expression of the step angle on the radius, the first feeding speed and the period according to the geometric relationship and the Taylor formula.
And obtaining a first maximum feeding speed allowed by the arc track according to a second maximum feeding speed allowed by the foam cutting system, a third maximum feeding speed of the arc track limited by the arch height difference of the arc track and a speed of the arc track limited by the centripetal acceleration of the arc track, and then obtaining a period and a first feeding speed by combining double-S-shaped speed planning so as to obtain a step angle.
And obtaining the displacement increment of the next interpolation point relative to the current interpolation point about the step angle and the current coordinate according to the geometric relation, further obtaining the coordinate of the next interpolation point, obtaining the speed increment of the next interpolation point relative to the current interpolation point through the period and the displacement increment, further obtaining the second feeding speed of the next interpolation point, and finishing the interpolation.
Preferably, acquiring the circular arc track to be processed, knowing the coordinates of the circle center and the coordinates of the current interpolation point, includes:
the known center coordinate is O (0,0) and the coordinate of the current interpolation point is A (X) i ，Y i ) The radius of the circular arc trajectory is expressed as
Setting the coordinates of the next interpolation point with unknown number, the first feeding speed of the current interpolation point, the period from the current interpolation point to the next interpolation point, the corresponding feeding step length between the current interpolation point and the next interpolation point and the step angle corresponding to the feeding step length to be represented by letters, and comprising the following steps:
let the coordinate of the next interpolation point be B (X) i+1 ,Y i+1 ) The first feed speed of the current interpolation point is denoted as V m The period from the point A to the point B is represented as T, the corresponding feeding step length between the point A and the point B is represented as f, the feeding step length f is a chord AB, the step angle corresponding to the feeding step length f is represented as theta, the rotating angle of the step angle theta is represented as a step pulse signal sent by a foam cutting system, namely the positive and negative of the pulse direction represent the interpolation direction;
deriving a first expression for the step angle with respect to radius, first feed speed, and period from the geometric relationship and the taylor equation, comprising:
by geometric relationships:
the above formula is developed by Taylor formula and taken to be second order to obtain:
and according to the known formula:
f＝V m T；
the first expression is then expressed as:
preferably, the first maximum feeding speed allowed by the circular arc track is obtained according to the second maximum feeding speed allowed by the foam cutting system, the third maximum feeding speed of the circular arc track limited by the arch height difference of the circular arc track and the speed of the circular arc track limited by the centripetal acceleration of the circular arc track, and the method comprises the following steps:
let the foam cutting system allow a second maximum feed speed denoted as F max The first maximum acceleration is represented as A max The maximum jerk is denoted as J max Maximum centripetal acceleration is denoted by U circ And the bow height error is expressed as
And (3) expanding the bow height error by using a Taylor formula:
combining the formula (1):
the third maximum feed speed v that can be reached by the circular arc trajectory is limited according to the bow height error err Expressed as:
then the speed v of the circular arc track is limited by the centripetal acceleration a Expressed as:
the first maximum feed speed v allowed by the circular arc trajectory circ_max Is taken to be the second maximum feed speed denoted as F max Third maximum feed speed v err And centripetal acceleration limits the speed v of the circular arc trajectory a The minimum value in (b) is expressed as:
v circ_max ＝min(F max ，v err ，v a ) (6)；
then combine two S type speed plans to try out cycle, first feed speed again, and then try out the step angle, include:
performing double-S-shaped speed planning from the point A to the point B;
given the known boundary conditions:
initial velocity v 0 Termination velocity v 1 ；
Initial acceleration of a 0 The terminal acceleration is a 1 And are all 0;
when the feed step f is too small, the first feed speed cannot be selected from v 0 Change to v 1 Then only the plus degree segment v exists 0 <v 1 Or deceleration section v 0 >v 1 First, it is checked whether an acceleration section and a deceleration section exist simultaneously, as follows:
let the acceleration time T of a single pulse * Expressed as:
if it is
Then the acceleration reaches a maximum, the jerk is 0, and it is necessary to satisfy:
if the above formula (8) is satisfied, an acceleration section and a deceleration section exist at the same time, and the parameter first feeding speed can be obtained through calculation, otherwise, the first feeding speed cannot be obtained from v 0 Change to v 1 And the actual fourth maximum feeding speed allowed by limiting the arc track by the arch height difference of the arc track and the centripetal acceleration of the arc track arranged on the foam cutting system is represented as v lim The actual second maximum acceleration is denoted as a lim And there are two cases as follows:
v lim ＝v circ_max ；
v lim <v circ_max ；
in the double S-shaped speed planning, an acceleration section and a deceleration section are both S-shaped, and the acceleration time is defined as T 1 (ii) a Uniform acceleration time of T 2 (ii) a Decreasing acceleration time to T 3 (ii) a At uniform speed of T 4 (ii) a Acceleration and deceleration time is T 5 (ii) a Uniform deceleration time of T 6 (ii) a Reducing the deceleration time to T 7 (ii) a Acceleration period of time T a (ii) a The deceleration section has a time T d (ii) a The total time is the period from point a to point B, T = T a +T 4 +T d ；
When the first maximum feed speed satisfies v circ_max ＝v lim If the constant speed section T4 exists;
calculating an acceleration period time T a ：
When A is max Satisfy the requirement of
When a is lim Is less than A max The maximum value cannot be reached, there is no uniform acceleration segment, and the time of the acceleration segment is expressed as:
T a ＝T 1 +T 3 (9)；
and the number of the first and second electrodes,
if not, a lim Is equal to A max Can reach the maximum value, and the time T of the acceleration section a Comprises the following steps:
and the number of the first and second electrodes,
calculating the time T of the deceleration section d ：
When A is max Satisfy the requirements of
When a is lim Is less than A max The maximum value cannot be reached, there is no uniform deceleration section, and the time of the deceleration section is expressed as:
T d ＝T 5 +T 7 (13)；
and the number of the first and second groups is,
if not, a lim Is equal to A max Can reach the maximum value, the time T of the deceleration section d Comprises the following steps:
and the number of the first and second electrodes,
the uniform velocity segment time T can be obtained according to the above equations (9) - (16) 4 Comprises the following steps:
if uniform velocity section T 4 ≥0，a lim Is equal to A max Can reach the maximum value, and directly uses the formula (17) to calculate T 4 ；
If uniform velocity section T 4 <0, then v circ_max <v lim Constant velocity time T 4 =0, the maximum acceleration A of the system in the acceleration section and the deceleration section is set max If all the acceleration period time, the deceleration period time, the uniform acceleration period time and the uniform deceleration period time are present, the acceleration period time, the deceleration period time, the uniform acceleration period time and the uniform deceleration period time are as follows:
wherein, for the sake of brevity, Λ represents a string of formulas;
according to the following formula without the feeding step length of the constant speed section:
wherein the content of the first and second substances,
respectively represent the terminal speed of the acceleration section and the initial speed of the deceleration section, and
the maximum speeds of the acceleration section and the deceleration section, respectively.
Substituting the formula (21) into the above formulas (18), (19) and (20) to obtain:
if it satisfies
The above is true, and the period T is calculated;
if not, at least one section of the acceleration section and the deceleration section can not reach the maximum acceleration, and the condition needs to adopt a proportional attenuation method for iterative operation on the acceleration until T is obtained a <0 or T d <0 and maximum acceleration a in the foam cutting System lim I.e. corresponding T a =0 or T d ＝0；
The update rate plan parameters are as follows:
v circ_max ＝v lim ；
A max ＝a lim ；
from the time parameters, the velocity and acceleration of the current interpolation point can be calculated as follows:
v m ＝v lim ＝v 0 +(T a -T 1 )a lima ＝v 1 -(T d -T 5 )a limd ； (23)；
a lima ＝J max T 1 ； (24)；
a limd ＝J max T 5 ； (25)；
wherein, a lima Acceleration expressed as acceleration segment, a limd Acceleration expressed as a deceleration segment;
substituting the formula (23) into the formula (1) to obtain a step angle theta which accords with the S-shaped speed plan in the interpolation period T and satisfies the following conditions:
preferably, the obtaining a second expression of the horizontal and vertical coordinates of the next interpolation point expressed by the current interpolation point coordinates and the step angle according to the geometric relationship, obtaining the displacement increment of the horizontal and vertical coordinates of the next interpolation point relative to the current interpolation point through the second expression, obtaining the feed speed increment of the next interpolation point relative to the current interpolation point through the interpolation period, further obtaining the feed speed of the next interpolation point, and further completing the interpolation comprises:
in a triangular AOB, the geometric relationship yields:
α＝φi+θ/2 (27)；
wherein alpha is an angle formed by the chord AB and a horizontal axis, phi i is an angle formed by the radius of the point A on the circular arc and a vertical axis, the step angle theta is an angle of the angle AOB, and the increment of the next interpolation point B relative to the current interpolation point A in the horizontal and vertical axis directions is delta X and delta Y in sequence, so that the coordinate of the point B satisfies X i+1 ＝X i +ΔX，Y i+1 ＝Y i + Δ Y, derived from the geometric relationship:
wherein the content of the first and second substances,
substituting the formulas (27) and (28) to obtain:
expanding the relation of the coordinate of the next interpolation point to the displacement increment through a Taylor formula, calculating to obtain the displacement increment, and further calculating the coordinate of the next interpolation point, wherein the method comprises the following steps:
and (3) expanding the trigonometric functions in the formulas (4) and (5) by a Taylor formula to obtain the second order of the trigonometric functions:
and sequentially substituting the expressions (32) and (33) into the corresponding expressions (30) and (31), and calculating to obtain the coordinate satisfaction of the next interpolation point B:
available according to equation (26):
then:
and the speed increment of the next interpolation point B along the direction of the horizontal axis relative to the feeding speed of the current interpolation point A
Velocity increment along longitudinal axis
Obtaining:
further obtaining the coordinate of the point B and a second feeding speed;
and converting the speed unit of each shaft into pulse frequency, outputting the displacement increment which is the pulse increment to a servo motor corresponding to each shaft, and finishing the interpolation.
Compared with the prior art, the invention has the following beneficial effects: by combining double-S-shaped speed planning, S-shaped acceleration and deceleration from the current interpolation point to the next interpolation point in the arc track machining process is realized, the feeding speed of the current interpolation point and the period from the current interpolation point to the next interpolation point are obtained, the coordinate and the feeding speed of the next interpolation point are further obtained, the machining precision and the machining efficiency of the arc track are improved in the whole interpolation process, meanwhile, the feeding speed is continuous and smooth, and errors are reduced.
Drawings
FIG. 1 is a schematic flow chart of a high-precision circular interpolation method of a motion control system based on a direct function method according to the present invention;
FIG. 2 is a schematic diagram of a circular arc trajectory to be processed according to the present invention;
FIG. 3 is a schematic diagram of an S-shaped velocity profile of the present invention.
Detailed Description
The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
It will be understood that when an element is referred to as being "connected" to another element, it can be directly connected to the other element or intervening elements may also be present. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used in the description of the present application herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the application.
In one embodiment, as shown in fig. 1 to 3, a motion control system high-precision circular interpolation method based on a direct function method is used for a foam cutting system, and the motion control system high-precision circular interpolation method based on the direct function method comprises the following steps:
s1, obtaining a circular arc track to be processed, knowing a circle center coordinate and a current interpolation point coordinate, setting the coordinate of a next interpolation point with unknown number, a first feeding speed of the current interpolation point, a period from the current interpolation point to the next interpolation point, a corresponding feeding step length between the current interpolation point and the next interpolation point and a step angle corresponding to the feeding step length to be represented by letters, and solving a first expression of the step angle relative to a radius, a first feeding speed and the period according to a geometric relation and a Taylor formula.
Specifically, the feeding speed of the foam cutting system has a close relation with the processing precision, the surface roughness and the production efficiency, and the direct function circular arc interpolation method is a process of approximating a given section of circular arc track to be processed by a group of continuous tiny line segments through formula derivation. The most important is to calculate the increment value of each axis in each interpolation period and then perform fine interpolation calculation on the increment value. Firstly, calculating an interpolation period of an interpolation point, a feeding step length in the interpolation period and a feeding speed of the interpolation point, wherein the feeding step length is equal to the interpolation period multiplied by the feeding speed.
Acquiring a circular arc track to be processed, knowing a circle center coordinate and a current interpolation point coordinate, and comprising the following steps of:
as shown in fig. 2, the circular arc locus P to be processed s P e The coordinate of the known circle center is O (0,0), and the coordinate of the current interpolation point is A (X) i ，Y i ) The radius of the circular arc trajectory is expressed as
Setting the coordinates of the next interpolation point with unknown number, the first feeding speed of the current interpolation point, the period from the current interpolation point to the next interpolation point, the corresponding feeding step length between the current interpolation point and the next interpolation point and the step angle corresponding to the feeding step length to be represented by letters, and comprising the following steps:
let the coordinate of the next interpolation point be B (X) i+1 ,Y i+1 ) The first feed speed of the current interpolation point is denoted as V m The period from point a to point B is represented as T, the corresponding feed step between point a and point B is represented as f, the feed step f is chord AB, the step angle corresponding to the feed step f is represented as θ, and the angle of rotation of the step angle θ is represented as the step pulse signal emitted by the foam cutting system, i.e., the positive and negative of the pulse direction represent the interpolation direction, usually the counterclockwise interpolation time is defined as positive, and the clockwise interpolation time is defined as negative.
Deriving a first expression for the step angle with respect to radius, first feed speed, and period from the geometric relationship and the taylor equation, comprising:
as shown in fig. 2, in the triangular AOB, M is the midpoint of AB, and the angle AOB is twice the angle AOM, which is obtained by the geometrical relationship:
the above formula is developed by Taylor formula and taken to be second order to obtain:
and according to the known formula:
f＝V m T；
the first expression is then expressed as:
s2, obtaining a first maximum feeding speed allowed by the arc track according to a second maximum feeding speed allowed by the foam cutting system, a third maximum feeding speed of the arc track limited by the arch height difference of the arc track and a speed of the arc track limited by the centripetal acceleration of the arc track, and then obtaining a cycle and a first feeding speed by combining double-S-shaped speed planning, so as to obtain a step angle.
Specifically, first, a first maximum feed speed allowed by the circular arc trajectory is obtained as follows:
let the foam cutting system allow a second maximum feed speed denoted as F max The first maximum acceleration is represented as A max The maximum jerk is denoted as J max Maximum centripetal acceleration is denoted by U circ And the bow height error is expressed as
And (3) expanding the bow height error by using a Taylor formula:
combining the formula (1):
the third maximum feed speed v that can be reached by the circular arc trajectory is limited according to the bow height error err Expressed as:
then the speed v of the circular arc track is limited by the centripetal acceleration a Expressed as:
the first maximum feed speed v allowed by the circular arc trajectory circ_max Is taken to be the second maximum feed speed denoted as F max Third maximum feed speed v err And the centripetal acceleration limits the speed v of the circular arc trajectory a The minimum value in (a) is expressed as:
v circ_max ＝min(F max ，v err ，v a ) (6)。
the period and the first feeding speed are obtained by combining double-S-shaped speed planning, and then the step angle is obtained, as follows:
performing double-S-shaped speed planning from the point A to the point B;
given the known boundary conditions:
initial velocity v 0 Termination velocity v 1 ；
Initial acceleration of a 0 The terminal acceleration is a 1 And are all 0;
when the feed step f is too small, the first feed speed cannot be selected from v 0 Change to v 1 Then only the plus degree segment v exists 0 <v 1 Or a deceleration section v 0 >v 1 First, it is checked whether there are acceleration and deceleration sections at the same time, as follows:
let the acceleration time T of a single pulse * Expressed as:
if it is
Then the acceleration reaches a maximum, the jerk is 0, and it is necessary to satisfy:
if the above formula (8) is satisfied, an acceleration section and a deceleration section exist at the same time, and the parameter first feeding speed can be obtained through calculation, otherwise, the first feeding speed cannot be obtained from v 0 Change to v 1 . The actual fourth maximum feed speed which is allowed by the arch height difference of the arc track and the centripetal acceleration limit arc track arranged on the foam cutting system and the arc track is represented as v lim The actual second maximum acceleration is denoted as a lim And there are two cases:
v lim ＝v circ_max ；
v lim <v circ_max ；
as shown in FIG. 3, in the dual S-shaped speed plan, the acceleration section and the deceleration section are both S-shaped, and the acceleration time is defined as T 1 (ii) a Uniform acceleration time of T 2 (ii) a Reducing acceleration time to T 3 (ii) a At uniform speed of T 4 (ii) a Acceleration and deceleration time is T 5 (ii) a Uniform deceleration time of T 6 (ii) a Reducing the deceleration time to T 7 (ii) a Acceleration period of time T a (ii) a The deceleration section has a time T d (ii) a The total time is the period from point a to point B, T = T a +T 4 +T d ；
When the first maximum feed speed satisfies v circ_max ＝v lim If the constant speed section T4 exists;
calculating an acceleration period time T a ：
When A is max Satisfy the requirement of
When a is lim Is less than A max The maximum value cannot be reached, there is no uniform acceleration segment, and the time of the acceleration segment is expressed as:
T a ＝T 1 +T 3 (9)；
and the number of the first and second electrodes,
if not, a lim Is equal to A max Can reach the maximum value, and the time T of the acceleration section a Comprises the following steps:
and the number of the first and second electrodes,
calculating the time T of the deceleration section d ：
When A is max Satisfy the requirement of
When a is lim Is less than A max The maximum value cannot be reached, there is no uniform deceleration section, and the time of the deceleration section is expressed as:
T d ＝T 5 +T 7 (13)；
and the number of the first and second electrodes,
if not, a lim Is equal to A max Can reach the maximum value, and the time T of the deceleration section d Comprises the following steps:
and the number of the first and second electrodes,
the uniform speed period time T can be obtained according to the above equations (9) - (16) 4 Comprises the following steps:
if uniform velocity section T 4 ≥0，a lim Is equal to A max Can reach the maximum value, and directly uses the formula (17) to calculate T 4 ；
If uniform velocity section T 4 <0, then v circ_max <v lim And time of uniform velocity T 4 =0, the maximum acceleration A of the system in the acceleration section and the deceleration section is set max If all the acceleration section time, the deceleration section time, the uniform acceleration section time and the uniform deceleration section time are present, the following steps are performed:
wherein, for the sake of brevity, Λ represents a string of formulas;
according to the following formula without the feeding step length of the uniform speed section:
wherein the content of the first and second substances,
respectively represent the terminal speed of the acceleration section and the initial speed of the deceleration section, and
the maximum speeds of the acceleration section and the deceleration section, respectively.
Substituting the formula (21) into the above formulas (18), (19) and (20) to obtain:
if it satisfies
The above is true, and the period T is calculated;
if not, at least one section of the acceleration section and the deceleration section can not reach the maximum acceleration, and the condition needs to adopt a proportional attenuation method for iterative operation on the acceleration until T is obtained a <0 or T d <0 and maximum acceleration a in the foam cutting system lim I.e. corresponding T a =0 or T d ＝0；
The update rate plan parameters are as follows:
v circ_max ＝v lim ；
A max ＝a lim ；
from the time parameters, the velocity and acceleration of the current interpolation point can be calculated as follows:
v m ＝v lim ＝v 0 +(T a -T 1 )a lima ＝v 1 -(T d -T 5 )a limd ； (23)；
a lima ＝J max T 1 ； (24)；
a limd ＝J max T 5 ； (25)；
wherein, a lima Expressed as acceleration of the acceleration segment, a limd Acceleration expressed as a deceleration segment;
substituting the formula (23) into the formula (1) to obtain a step angle theta which accords with the S-shaped speed plan in the interpolation period T and satisfies the following conditions:
and S3, obtaining the displacement increment of the next interpolation point relative to the current interpolation point about the step angle and the current coordinate according to the geometric relation, further obtaining the coordinate of the next interpolation point, obtaining the speed increment of the next interpolation point relative to the current interpolation point according to the period and the displacement increment, further obtaining the second feeding speed of the next interpolation point, and finishing interpolation.
Specifically, as shown in fig. 2, in the triangular AOB, the geometric relationship can be given as:
α＝φi+θ/2 (27)；
wherein alpha is an angle formed by the chord AB and a horizontal axis, phi i is an angle formed by the radius of the point A on the circular arc and a vertical axis, the step angle theta is an angle of the angle AOB, and the increment of the next interpolation point B relative to the current interpolation point A in the horizontal and vertical axis directions is delta X and delta Y in sequence, so that the coordinate of the point B satisfies X i+1 ＝X i +ΔX，Y i+1 ＝Y i + Δ Y, derived from the geometric relationship:
wherein, the first and the second end of the pipe are connected with each other,
substituting the formulas (27) and (28) to obtain:
expanding the relation of the coordinate of the next interpolation point to the displacement increment through a Taylor formula, calculating to obtain the displacement increment, and further calculating the coordinate of the next interpolation point, wherein the method comprises the following steps:
and (3) expanding the trigonometric functions in the formulas (4) and (5) by a Taylor formula to obtain the second order of the trigonometric functions:
and sequentially substituting the expressions (32) and (33) into the corresponding expressions (30) and (31), and calculating to obtain the coordinate satisfaction of the next interpolation point B:
available according to equation (26):
then:
and the velocity increment of the next interpolation point B in the direction of the horizontal axis relative to the feed velocity of the current interpolation point A
Velocity increment along longitudinal axis
Obtaining:
obtaining the feeding speed of the point B through the speed increment;
further solving the coordinate of the point B and a second feeding speed;
and converting the speed unit of each axis into pulse frequency, outputting the displacement increment which is the pulse increment to a servo motor corresponding to each axis, and finishing the interpolation.
The method combines double-S-shaped speed planning to realize S-shaped acceleration and deceleration from a current interpolation point to a next interpolation point in the circular arc track processing process, obtain the feeding speed of the current interpolation point and the period from the current interpolation point to the next interpolation point, further obtain the coordinate and the feeding speed of the next interpolation point, improve the processing precision and the processing efficiency of the circular arc track in the whole interpolation process, simultaneously ensure that the feeding speed is continuous and smooth, and reduce errors.
The technical features of the embodiments described above may be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the embodiments described above are not described, but should be considered as being within the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express the more specific and detailed embodiments described in the present application, but not be construed as limiting the claims. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the concept of the present application, which falls within the scope of protection of the present application. Therefore, the protection scope of the present patent shall be subject to the appended claims.
Claims (4)
1. A motion control system high-precision circular interpolation method based on a direct function method is used for a foam cutting system and is characterized in that: the motion control system high-precision circular interpolation method based on the direct function method comprises the following steps:
acquiring a circular arc track to be processed, knowing a circle center coordinate and a current interpolation point coordinate, setting the coordinate of a next interpolation point of an unknown number, a first feeding speed of the current interpolation point, a period from the current interpolation point to the next interpolation point, a corresponding feeding step length between the current interpolation point and the next interpolation point and a step angle corresponding to the feeding step length to be expressed by letters, and solving a first expression of the step angle on a radius, a first feeding speed and a period according to a geometric relationship and a Taylor formula;
obtaining a first maximum feeding speed allowed by the arc track according to a second maximum feeding speed allowed by a foam cutting system, a third maximum feeding speed of the arc track limited by the arch height difference of the arc track and a speed of the arc track limited by the centripetal acceleration of the arc track, and then obtaining a period and a first feeding speed by combining double-S-shaped speed planning so as to obtain a step angle;
and obtaining the displacement increment of the next interpolation point relative to the current interpolation point about the step angle and the current coordinate according to the geometrical relation, further obtaining the coordinate of the next interpolation point, obtaining the speed increment of the next interpolation point relative to the current interpolation point through the period and the displacement increment, further obtaining the second feeding speed of the next interpolation point, and finishing the interpolation.
2. The direct function method-based motion control system high-precision circular interpolation method of claim 1, wherein: the method for acquiring the arc track to be processed, the known circle center coordinate and the current interpolation point coordinate comprises the following steps:
the known center coordinate is O (0,0) and the coordinate of the current interpolation point is A (X) i ，Y i ) The radius of the circular arc trajectory is expressed as
The step of setting the coordinate of the next interpolation point with unknown number, the first feeding speed of the current interpolation point, the period from the current interpolation point to the next interpolation point, the corresponding feeding step length between the current interpolation point and the next interpolation point and the step angle corresponding to the feeding step length are represented by letters, and the method comprises the following steps:
let the coordinate of the next interpolation point be B (X) i+1 ,Y i+1 ) The first feed speed of the current interpolation point is denoted as V m The period from point A to point B is represented as T, the corresponding feeding step length between point A and point B is represented as f, the feeding step length f is a chord AB, the step angle corresponding to the feeding step length f is represented as theta, the rotating angle of the step angle theta is represented as a step pulse signal sent by a foam cutting system, namely, the positive and negative of the pulse direction represent the interpolation direction;
the first expression of the step angle with respect to the radius, the first feed speed and the period is obtained according to the geometrical relationship and the Taylor formula, and comprises the following steps:
by geometric relationships:
the above formula is developed by Taylor formula and taken to be second order to obtain:
and according to the known formula:
f＝V m T；
the first expression is then expressed as:
3. the direct function method-based motion control system high-precision circular interpolation method as claimed in claim 2, wherein: the method for obtaining the first maximum feeding speed allowed by the arc track according to the second maximum feeding speed allowed by the foam cutting system, the third maximum feeding speed of the arc track limited by the arch height difference of the arc track and the speed of the arc track limited by the centripetal acceleration of the arc track comprises the following steps of:
let the foam cutting system allow a second maximum feed speed denoted F max The first maximum acceleration is represented as A max The maximum jerk is represented as J max Maximum centripetal acceleration is denoted by U circ And the bow height error is expressed as
And (3) expanding the bow height error by using a Taylor formula:
combining the formula (1):
then limit the circle according to the height errorThird maximum feed speed v achievable by the arc trajectory err Expressed as:
then the speed v of the circular arc track is limited by the centripetal acceleration a Expressed as:
the first maximum feed speed v allowed by the circular arc trajectory circ_max Is taken to be the second maximum feed speed denoted as F max Third maximum feed speed v err And centripetal acceleration limits the speed v of the circular arc trajectory a The minimum value in (a) is expressed as:
v circ_max ＝min(F max ，v err ，v a ) (6)；
then combine two S type speed planning again to try out cycle, first feed speed, and then try out the step angle, include:
performing double S-shaped speed planning from the point A to the point B;
given the known boundary conditions:
initial velocity v 0 Termination velocity v 1 ；
Initial acceleration of a 0 The terminal acceleration is a 1 And are all 0;
when the feed step f is too small, the first feed speed cannot be selected from v 0 Change to v 1 Then only the plus degree segment v exists 0 <v 1 Or a deceleration section v 0 >v 1 First, it is checked whether there are acceleration and deceleration sections at the same time, as follows:
let the acceleration time T of a single pulse * Expressed as:
if it is
Then the acceleration reaches a maximum, the jerk is 0, and it is necessary to satisfy:
if the above formula (8) is satisfied, an acceleration section and a deceleration section exist at the same time, and the parameter first feeding speed can be obtained through calculation, otherwise, the first feeding speed cannot be obtained from v 0 Change to v 1 And the actual fourth maximum feeding speed allowed by limiting the arc track by the arch height difference of the arc track and the centripetal acceleration of the arc track arranged on the foam cutting system is represented as v lim The actual second maximum acceleration is denoted as a lim And there are two cases as follows:
v lim ＝v circ_max ；
v lim <v circ_max ；
in the double S-shaped speed planning, an acceleration section and a deceleration section are both S-shaped, and the acceleration time is defined as T 1 (ii) a Uniform acceleration time of T 2 (ii) a Decreasing acceleration time to T 3 (ii) a At uniform speed of T 4 (ii) a Acceleration and deceleration time is T 5 (ii) a Uniform deceleration time of T 6 (ii) a Reducing the deceleration time to T 7 (ii) a Acceleration period of time T a (ii) a The deceleration section has a time T d (ii) a The total time is the period from point a to point B, T = T a +T 4 +T d ；
When the first maximum feed speed satisfies v circ_max ＝v lim If the constant speed section T4 exists;
calculating an acceleration period time T a ：
When A is max Satisfy the requirements of
When a is lim Is less than A max The maximum value cannot be reached, there is no uniform acceleration segment, and the time of the acceleration segment is expressed as:
T a ＝T 1 +T 3 (9)；
and the number of the first and second electrodes,
if not, then a lim Is equal to A max Can reach the maximum value, and the time T of the acceleration section a Comprises the following steps:
and the number of the first and second electrodes,
calculating the time T of the deceleration section d ：
When A is max Satisfy the requirements of
When a is lim Is less than A max The maximum value cannot be reached, there is no uniform deceleration section, and the time of the deceleration section is expressed as:
T d ＝T 5 +T 7 (13)；
and the number of the first and second electrodes,
if not, a lim Is equal to A max Can reach the maximum value, the time T of the deceleration section d Comprises the following steps:
and the number of the first and second groups is,
the uniform speed period time T can be obtained according to the above equations (9) - (16) 4 Comprises the following steps:
if uniform velocity section T 4 ≥0，a lim Is equal to A max Can reach the maximum value, and directly uses the formula (17) to calculate T 4 ；
If uniform velocity section T 4 <0, then v circ_max <v lim Constant velocity time T 4 =0, the maximum acceleration A of the system in the acceleration section and the deceleration section is set max If all the acceleration section time, the deceleration section time, the uniform acceleration section time and the uniform deceleration section time are present, the following steps are performed:
wherein, for the sake of brevity, Λ represents a string of formulas;
according to the following formula without the feeding step length of the uniform speed section:
wherein the content of the first and second substances,
respectively represent the terminal speed of the acceleration section and the initial speed of the deceleration section, and
the maximum speeds of the acceleration section and the deceleration section are respectively;
substituting the formula (21) into the above formulas (18), (19) and (20) to obtain:
if it satisfies
The above is true, and the period T is calculated;
if not, at least one section of the acceleration section and the deceleration section can not reach the maximum acceleration, and the condition needs to adopt a proportional attenuation method for iterative operation on the acceleration until T is obtained a <0 or T d <0 and maximum acceleration a in the foam cutting System lim I.e. corresponding T a =0 or T d ＝0；
The update speed planning parameters are as follows:
v circ_max ＝v lim ；
A max ＝a lim ；
from the time parameters, the velocity and acceleration of the current interpolation point can be calculated as follows:
v m ＝v lim ＝v 0 +(T a -T 1 )a lima ＝v 1 -(T d -T 5 )a limd ； (23)；
a lima ＝J max T 1 ； (24)；
a limd ＝J max T 5 ； (25)；
wherein, a lima Acceleration expressed as acceleration segment, a limd Acceleration expressed as a deceleration segment;
substituting the formula (23) into the formula (1) to obtain a step angle theta which accords with the S-shaped speed plan in the interpolation period T and satisfies the following conditions:
4. the direct function method-based motion control system high-precision circular interpolation method of claim 3, wherein: the obtaining a second expression of the horizontal and vertical coordinates of the next interpolation point expressed by the current interpolation point coordinates and the step angle according to the geometric relationship, obtaining the displacement increment of the horizontal and vertical coordinates of the next interpolation point relative to the current interpolation point through the second expression, obtaining the feeding speed increment of the next interpolation point relative to the current interpolation point through the interpolation period, further obtaining the feeding speed of the next interpolation point, and further completing the interpolation comprises the following steps:
in a triangular AOB, the geometric relationship yields:
α＝φi+θ/2 (27)；
wherein alpha is an angle formed by the chord AB and a horizontal axis, phi i is an angle formed by the radius of the point A on the circular arc and a vertical axis, the step angle theta is an angle of the angle AOB, and the increment of the next interpolation point B relative to the current interpolation point A in the horizontal and vertical axis directions is delta X and delta Y in sequence, so that the coordinate of the point B satisfies X i+1 ＝X i +ΔX，Y i+1 ＝Y i + Δ Y, derived from the geometric relationship:
wherein the content of the first and second substances,
substituting the formulas (27) and (28) to obtain:
the expanding the relational expression of the coordinates of the next interpolation point on the displacement increment through a Taylor formula, calculating to obtain the displacement increment, and further calculating the coordinates of the next interpolation point, comprises the following steps:
and (3) expanding the trigonometric functions in the formulas (4) and (5) by a Taylor formula to obtain the second order of the trigonometric functions:
and sequentially substituting the expressions (32) and (33) into the corresponding expressions (30) and (31), and calculating to obtain the coordinate satisfaction of the next interpolation point B:
available according to equation (26):
then:
and the velocity increment of the next interpolation point B in the direction of the horizontal axis relative to the feed velocity of the current interpolation point A
Velocity increment along longitudinal axis
Obtaining:
further obtaining the coordinate of the point B and a second feeding speed;
and converting the speed unit of each shaft into pulse frequency, outputting the displacement increment which is the pulse increment to a servo motor corresponding to each shaft, and finishing the interpolation.
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