CN111240275B - Feed rate planning method based on logarithmic probability function under motion and error limitation - Google Patents

Feed rate planning method based on logarithmic probability function under motion and error limitation Download PDF

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CN111240275B
CN111240275B CN202010044905.4A CN202010044905A CN111240275B CN 111240275 B CN111240275 B CN 111240275B CN 202010044905 A CN202010044905 A CN 202010044905A CN 111240275 B CN111240275 B CN 111240275B
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feed rate
parameter
acceleration
curve
value
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CN111240275A (en
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姜鑫
李禾雄
霍冠英
苏澄
叶丹蕾
陆泽宏
胡一飞
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Beihang University
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    • G05B19/00—Programme-control systems
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    • G05B19/416—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by control of velocity, acceleration or deceleration
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Abstract

The invention discloses a feed rate planning method for numerical control machining based on a logarithmic probability function, which defines a feed rate adjustment factor as a half power of the ratio of error limit to calculation error, takes an equally spaced parameter sequence, and obtains a final feed rate by continuously utilizing the adjustment factor to adjust each parameter according to the bow height error limit; defining a discrimination value as an absolute value of the difference between the front difference and the rear difference, calculating the discrimination value at a known parameter, wherein the discrimination value is a calculated inflection point when being larger, the inflection point, the starting point and the end point divide a curve into a plurality of sections, and each section is an acceleration, deceleration or uniform speed process; utilizing a Sigmoid function to control the feed rate in the acceleration and deceleration process, and solving a feed rate expression suitable for the process according to the requirements of acceleration, acceleration and smoothness; the acceleration and deceleration processes are spliced by using a forward scanning process and a backward scanning process. The method can reduce the processing error on the premise of ensuring the acceleration and the jerk, and realize continuous feed rate planning.

Description

Feed rate planning method based on logarithmic probability function under motion and error limitation
Technical Field
The invention relates to the technical field of numerical control machining and manufacturing, in particular to curve geometric feature extraction, function transformation and application, kinematic analysis and modeling, and particularly relates to a feed rate planning method for numerical control machining based on a logarithm probability function.
Background
In the digital processing process under the guidance of the numerical control system, the numerical control system receives input processing information, the processing information mainly comprises processing track, feed rate, cutter position offset and the like, then the processing information enters an interpolation module, and interpolation points in the actual processing process are calculated in the interpolation module according to the processing information. Before performing interpolation calculation, the feed rate during machining needs to be planned in order to obtain a better machining effect.
In a given machining trajectory, the feed rate of the trajectory is typically a constant feed rate. For a more complex processing track, the constant feed rate cannot adapt to the change of a curve during actual processing, frequent stopping and starting can be caused, limitation of bow height error, acceleration and jerk cannot be guaranteed, and the processing effect is poor.
In order to improve the quality of processing a complex curve, i.e. a non-basic configuration curve, the reasonable and effective feed rate planning for the complex curve becomes a research hotspot. With continuous progress of CAD technology, parameterization technology of complex curves matures day by day, processing track curves are feasible by using parameter expression, and feed rate planning based on the parameter expression curves is generated. The basic idea of feed rate planning is to divide a machining track into a plurality of sections and design an acceleration and deceleration process in each section, and during design, acceleration and jerk are guaranteed to be within a limit range. The existing acceleration and deceleration design has three basic modes: linear acceleration and deceleration control, exponential acceleration and deceleration control and S-shaped acceleration and deceleration control. The above three modes have respective advantages and disadvantages: the linear acceleration and deceleration control design is simple and easy to realize, but the continuity cannot be ensured; the design of exponential acceleration and deceleration control is complex and difficult to realize, and the smoothness cannot be ensured; the S-type acceleration and deceleration control considers the acceleration and deceleration process in a segmented manner, and the adjustment is complicated.
Disclosure of Invention
In view of the above, the present invention provides a feed rate planning method for numerical control machining based on a logarithmic probability function, which is used to reduce a machining error and realize continuous feed rate planning on the premise of ensuring an acceleration and a jerk.
Therefore, the invention provides a feed rate planning method for numerical control machining based on a logarithmic probability function, which comprises the following steps of:
s1: according to a parameter curve of a numerical control machining track, obtaining an equidistant parameter sequence, and according to the limitation of bow height errors, obtaining the feed rate numerical values of all parameters in the equidistant parameter sequence to obtain a series of discrete number pairs;
s2: obtaining a feed rate curve by using the obtained series of discrete number pairs, searching inflection points in the series of discrete number pairs, and dividing the feed rate curve into a plurality of acceleration processes, a plurality of deceleration processes and a plurality of uniform speed processes by using the inflection points;
s3: performing feed rate design on each acceleration process and each deceleration process by using a logarithmic probability function;
s4: and sequentially utilizing the forward feed rate adjustment process and the backward feed rate adjustment process to adjust the feed rate, so as to realize the splicing of each acceleration process, each deceleration process and each uniform speed process.
In a possible implementation manner, in the feed rate planning method provided by the present invention, in step S1, the method includes obtaining feed rate values of all parameters in the equally spaced parameter sequence according to the limitation of the bow-height error by using the equally spaced parameter sequence, and obtaining a series of discrete pairs, where the method specifically includes:
assuming that the parameter curve of the processing track is expressed as P (u) ═ x (u), y (u), z (u), the parameter range is 0 ≦ u ≦ 1, and the parameter interval is Δ u, then u isi+1＝ui+Δu，u0Get an equally spaced parameter sequence as 0
Figure BDA0002369016380000021
A first order taylor's formula is adopted:
Figure BDA0002369016380000022
Figure BDA0002369016380000031
wherein x (u) represents a parametric representation of the values of the x-coordinate, y (u) represents a parametric representation of the values of the y-coordinate, z (u) represents a parametric representation of the values of the z-coordinate, uiAnd ui+1Indicating a sequence of parameters at equal intervals
Figure BDA0002369016380000032
Two adjacent parameters, T represents time, s represents arc length, v represents a preset feed rate, T represents an interpolation period, h.o.t represents high-order infinitesimal, P ' (u) represents a first derivative of a parameter curve P (u), and | P ' (u) | represents a norm of P ' (u);
from u obtainediAnd u'iCalculating two points P (u) on the parameter curvei) And P (u'i) The maximum distance between the arc segment and the line segmentIs two points P (u)i) And P (u'i) Of bow height error of (1), wherein, u'iRepresenting the parameter uiThe significance of the predicted parameter is to simulate the predicted parameter in the parameter uiThe interpolation process obtains the predicted parameter of the next interpolation point, P (u)i) Represents the parameter curve P (u) upper and the parameter uiCorresponding point, P (u'i) Representing parameter curve P (u) and estimated parameter u'iA corresponding point; regarding the arc segment as a tiny arc segment, point P (u) is usedi) The small segment of the arc of the curvature circle approaches the arc segment, the feeding step length L is regarded as the chord length of the arc, and the bow height error is as follows:
Figure BDA0002369016380000033
L＝||P(ui)-P(ui+1)||
Figure BDA0002369016380000034
where σ denotes the bow height error, ρiRepresents the parametric curve P (u) at the point P (u)i) Radius of curvature of (d), P (u)i+1) Represents the parameter curve P (u) upper and the parameter ui+1Corresponding point, P' (u)i) Represents the parametric curve P (u) at the point P (u)i) The first derivative of (d), P' (u)i) Represents the parametric curve P (u) at the point P (u)i) The second derivative of (d); the feeding step length L is known as vT according to the definition of the feeding rate; for simplification, the distance between the middle point of the arc section and the middle point of the chord length is used as the height error:
Figure BDA0002369016380000035
wherein the content of the first and second substances,
Figure BDA0002369016380000036
represents the upper and parameters of the parametric curve P (u)
Figure BDA0002369016380000037
A corresponding point; according to the relationship between the bow height error and the feeding step length and the feeding rate, the following results are obtained:
Figure BDA0002369016380000041
assuming the bow height error is limited to σεIdeal feed rate v', existing feed rate v, existing bow height error sigma and bow height error limit sigmaεThe relationship between them is:
Figure BDA0002369016380000042
traversing all parameters of the equally spaced parameter sequence to obtain each parameter uiFeed rate value v satisfying bow height error limitiTo obtain a series of discrete number pairs (u)i,vi)。
In a possible implementation manner, in the feed rate planning method provided by the present invention, in step S2, a feed rate curve is obtained by using the obtained series of discrete number pairs, an inflection point in the series of discrete number pairs is found, and the feed rate curve is divided into a plurality of acceleration processes, a plurality of deceleration processes, and a plurality of constant speed processes by using the inflection point, which specifically includes:
using the resulting series of discrete pairs (u)i,vi) Obtaining a feed rate curve;
in a series of discrete pairs, three adjacent points (u) are takeni-1,vi-1)、(ui,vi) And (u)i+1,vi+1) Calculating a discrimination value mu:
Figure BDA0002369016380000043
if the discrimination value mu is larger than the standard value xi, (u)i,vi) Is an inflection point; if the discrimination value mu is smaller than the standard value xi, (u)i,vi) Is not an inflection point;
the feed rate curve is divided into a plurality of acceleration processes, a plurality of deceleration processes and a plurality of uniform velocity processes by using inflection points.
In a possible implementation manner, in the feed rate planning method provided by the present invention, in step S3, the feed rate design is performed on each acceleration process by using a logarithmic probability function, which specifically includes the following steps:
s31: an acceleration process is represented as: block ═ vstart,vend,ustart,uendL }, wherein vstartDenotes the initial feed rate, vendIndicating the end feed rate, l the displacement of the acceleration process, ustartParameter representing a starting point, uendA parameter indicating an end point; l is obtained by integrating the calculated arc length differential ds:
Figure BDA0002369016380000051
Figure BDA0002369016380000052
wherein x ' (u) denotes the first derivative of x (u), y ' (u) denotes the first derivative of y (u), and z ' (u) denotes the first derivative of z (u); in the case of the same initial and final feed rates, according to a logarithmic probability function
Figure BDA0002369016380000053
The displacement of the feed rate movement is consistent with the displacement obtained by the linear movement, and the movement time of the acceleration process is as follows:
Figure BDA0002369016380000054
updating the acceleration block ═ vstart,vend,ustart,uend,l,t}；
S32: assuming that the log-probability function is in the original defined domainQ > 0, calculating a function value f at q1(q), two mappings may be constructed:
Figure BDA0002369016380000055
Figure BDA0002369016380000056
wherein p represents a parameter of time, g1And g2For the constructed auxiliary function, g1、g2And f1(w) substituting p into the log-probability function as a function adapted to the acceleration process
Figure BDA0002369016380000057
Let v (p) be a function after composition, where p ∈ [0, t ∈]Expressed as:
Figure BDA0002369016380000058
after derivation, the following steps are carried out:
Figure BDA0002369016380000059
and (3) solving a second derivative of the compounded function:
Figure BDA0002369016380000061
wherein v (p) represents a feed rate curve, v ' (p) represents an acceleration curve of feed rate, v ' (p) represents a jerk curve of feed rate, and g '2Represents a mapping g2First derivative of, g'1Represents a mapping g1First derivative of f1' (p) represents f1A first derivative of (p);
s33: v '(p) and v' (p) are satisfied:
Figure BDA0002369016380000062
Figure BDA0002369016380000063
wherein A ismIndicates the maximum acceleration limit in machining, JmRepresents the maximum jerk limit during machining;
when in use
Figure BDA0002369016380000064
When v' (p) has a maximum value; when in use
Figure BDA0002369016380000065
Then v "(p) has a maximum value; due to f1The value of (-q) is close to 0, so f1(-q) is regarded as 0, f1(q) is considered to be 1; then there are:
Figure BDA0002369016380000066
two solutions q are obtained1And q is2Q is q1And q is2Minimum value of (d):
q＝min{q1,q2}；
s34: calculating the acceleration a at qqA is toqComparing with the smoothness tolerance omega at the connecting point; if aqWhen omega is not more than, block is ═ vstart,vend,ustart,uendL, t, q } constitutes a feed rate design adapted to the acceleration process; if aq> omega, then decrease vstartAnd vendOf the larger value, repeating steps S33 and S34 until aqUntil omega is less than or equal to omega.
In a possible implementation manner, in the feed rate planning method provided by the present invention, in step S3, the feed rate design is performed on each deceleration process by using a logarithmic probability function, which specifically includes the following steps:
SS 31: one deceleration process is represented as: block ═ vstart,vend,ustart,uendL }, wherein vstartDenotes the initial feed rate, vendIndicating the end feed rate, l the displacement of the deceleration process, ustartParameter representing a starting point, uendA parameter indicating an end point; l is obtained by integrating the calculated arc length differential ds:
Figure BDA0002369016380000071
Figure BDA0002369016380000072
wherein x ' (u) denotes the first derivative of x (u), y ' (u) denotes the first derivative of y (u), and z ' (u) denotes the first derivative of z (u); in the case of the same initial and final feed rates, according to a logarithmic probability function
Figure BDA0002369016380000073
The displacement of the feed rate movement is consistent with the displacement obtained by the linear movement, and the movement time of the deceleration process is as follows:
Figure BDA0002369016380000074
updating the deceleration process block ═ vstart,vend,ustart,uend,l,t}；
SS 32: assuming that the value of the log-probability function in the original defined domain, q > 0, the function value at q, f (q), is computed, two mappings can be constructed:
Figure BDA0002369016380000075
Figure BDA0002369016380000076
wherein p represents a parameter of time, g1And g2For the constructed auxiliary function, g1、g2And f2(w) substituting p into the log-probability function as a function adapted to the deceleration process
Figure BDA0002369016380000077
Let v (p) be a function after composition, where p ∈ [0, t ∈]Expressed as:
Figure BDA0002369016380000078
after derivation, the following steps are carried out:
Figure BDA0002369016380000081
and (3) solving a second derivative of the compounded function:
Figure BDA0002369016380000082
wherein v (p) represents a feed rate curve, v ' (p) represents an acceleration curve of feed rate, v ' (p) represents a jerk curve of feed rate, and g '2Represents a mapping g2First derivative of, g'1Represents a mapping g1First derivative of f2' (p) represents f2A first derivative of (p);
SS 33: v '(p) and v' (p) are satisfied:
Figure BDA0002369016380000083
Figure BDA0002369016380000084
wherein A ismIndicates the maximum acceleration limit in machining, JmRepresents the maximum jerk limit during machining;
when in use
Figure BDA0002369016380000085
When v' (p) has a maximum value; when in use
Figure BDA0002369016380000086
Then v "(p) has a maximum value; due to f2The value of (-q) is close to 1, so f2(-q) is regarded as 1, f2(q) is taken as 0; then there are:
Figure BDA0002369016380000087
two solutions q are obtained1And q is2Q is q1And q is2Minimum value of (d):
q＝min{q1,q2}；
SS 34: calculating the acceleration a at qqA is toqComparing with the smoothness tolerance omega at the connecting point; if aqWhen omega is not more than, block is ═ vstart,vend,ustart,uendL, t, q } constitutes a feed rate design adapted to the deceleration process; if aq> omega, then decrease vstartAnd vendTo the larger value, repeat steps SS33 and SS34 until aqUntil omega is less than or equal to omega.
In a possible implementation manner, in the feed rate planning method provided by the present invention, in step S4, the feed rate adjustment is performed by sequentially using a forward feed rate adjustment process and a backward feed rate adjustment process, so as to implement concatenation of each acceleration process, each deceleration process, and each uniform velocity process, specifically including:
firstly, utilizing forward feed rate regulation algorithm to pair process blocksj＝{vj,start,vj,end,uj,start,uj,end,lj,tjAnd (c) the step of (c) in which, j 1, 2.. m, scanning forward from j 1, and when the process is an acceleration process, performing steps S31 to S34 while adjusting two adjacent process blocksjAnd blockj+1Value v ofj,end,tjAnd vj+1,start,tj+1；
And then utilizing a backward feeding rate adjustment algorithm to perform process block pairj＝{vj,start,vj,end,uj,start,uj,end,lj,tjAnd j is 1, 2.. m, scanning backwards from j is m, and when the process is a deceleration process, executing steps SS31 to SS34, and simultaneously adjusting two adjacent process blocksjAnd blockj+1Value v ofj,end,tjAnd vj+1,start,tj+1。
The feed rate planning method of numerical control processing based on the logarithm probability function, provided by the invention, comprises the steps of firstly defining a feed rate adjusting factor as a half power of the ratio of error limit to calculation error, taking an equally spaced parameter sequence, and adjusting each parameter according to the bow height error limit by continuously utilizing the adjusting factor to obtain the final rate; then defining a discrimination value as an absolute value of the difference between the front difference and the rear difference, calculating the discrimination value at a known parameter, wherein the discrimination value is a calculated inflection point when being larger, and the inflection point, the starting point and the ending point divide the whole curve into a plurality of sections, and each section can be regarded as an acceleration process, a deceleration process or a uniform speed process; then, the Sigmoid function is used for controlling the feed rate in the acceleration and deceleration process, and the feed rate expression suitable for the process is worked out according to the requirements of acceleration, acceleration and smoothness; and finally, splicing the acceleration and deceleration processes by using a forward scanning process and a backward scanning process. The feed rate planning method provided by the invention can reduce the processing error on the premise of ensuring the acceleration and the jerk, and realize continuous feed rate planning.
Drawings
FIG. 1 is a flow chart of a feed rate planning method for numerical control machining based on a logarithmic probability function according to the present invention;
FIG. 2 is a log probability function image used in the acceleration process;
FIG. 3 is an image of the log probability function derivative used in the acceleration process;
FIG. 4 is a log probability function image used during deceleration;
FIG. 5 is an image of log probability function derivatives used during deceleration
FIG. 6 is a graph of a processing trajectory parameter in example 1 of the present invention;
FIG. 7 is a graph of feed rate meeting the bow height error limit in example 1 of the present invention;
FIG. 8 is a schematic diagram of inflection point distribution on a feed rate curve satisfying a bow-height error limit in example 1 of the present invention;
FIG. 9 is a feed rate graph after feed rate planning according to example 1 of the present invention;
FIG. 10 is a graph comparing the bow height error for example 1 of the present invention with and without feed rate programming.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only illustrative and are not intended to limit the present invention.
The invention provides a feed rate planning method for numerical control machining based on a logarithmic probability function, which comprises the following steps as shown in figure 1:
s1: according to a parameter curve of a numerical control machining track, obtaining an equidistant parameter sequence, and according to the limitation of bow height errors, obtaining the feed rate numerical values of all parameters in the equidistant parameter sequence to obtain a series of discrete number pairs;
s2: obtaining a feed rate curve by using the obtained series of discrete number pairs, searching inflection points in the series of discrete number pairs, and dividing the feed rate curve into a plurality of acceleration processes, a plurality of deceleration processes and a plurality of uniform speed processes by using the inflection points;
s3: designing the feed rate of each acceleration process and each deceleration process by utilizing a logarithmic probability function;
s4: and the feed rate is adjusted by sequentially utilizing the forward feed rate adjustment process and the backward feed rate adjustment process, so that the splicing of each acceleration process, each deceleration process and each uniform speed process is realized.
The feed rate planning method provided by the invention considers the acceleration limit and the jerk limit and also considers the bow height error limit, designs the feed rate control process by utilizing the Sigmoid function, and has good adaptability.
In a specific implementation, in step S1 of the above feed rate planning method provided by the present invention, the feed rate values of all parameters in the equally spaced parameter sequence are obtained by using the equally spaced parameter sequence according to the limitation of the bow-height error, and a series of discrete number pairs is obtained, which may be specifically implemented by:
if the parameter curve of the machining trajectory is expressed as P (u) ═ x (u), y (u), z (u), the parameter range is 0. ltoreq. u.ltoreq.1, and the parameter interval is set to Δ u, u is calculated asi+1＝ui+Δu，u 00, for example, if the parameter interval is 0.001, it means that there are 1000 parameters;
after the parameter interval is set, an equal interval parameter sequence can be obtained
Figure BDA0002369016380000111
The bow height error is found at each parameter. The bow height error is defined as the maximum distance of the arc segment from the chord. To obtain a parameter uiCorresponding chord and arc line segments, simulating the process of interpolation calculation, and calculating the parameter u by Taylor's expansioniCorresponding interpolation point u'iWherein, u'iRepresenting the parameter uiThe significance of the predicted parameter is to simulate the predicted parameter in the parameter uiObtaining a next interpolation point prediction parameter in the interpolation process; because only analog computation is performed, the first-order taylor expansion can be adopted:
Figure BDA0002369016380000112
Figure BDA0002369016380000113
wherein x (u), y (u), z (u) represent parametric representations of respective coordinates of points on the parametric curve, the values varying with variation of the parameters, x (u) represent parametric representations of values of the x-coordinate, y (u) represent parametric representations of values of the y-coordinate, and z (u) represent parametric representations of values of the z-coordinate; u. ofiAnd ui+1Indicating a sequence of parameters at equal intervals
Figure BDA0002369016380000114
Two adjacent parameters, T represents time, s represents arc length, v represents a preset feed rate, T represents an interpolation period, h.o.t represents high-order infinitesimal, P ' (u) represents a first derivative of a parameter curve P (u), and | P ' (u) represents a norm of P ' (u);
from u obtainediAnd u'iTwo points P (u) on the parameter curve can be obtainedi) And P (u'i) The calculation of the bow height error is to calculate two points P (u)i) And P (u'i) The maximum distance between the arc segment and the line segment in between, wherein P (u)i) Represents the parameter curve P (u) upper and the parameter uiCorresponding point, P (u'i) Representing parameter curve P (u) and estimated parameter u'iA corresponding point; since the arc segment is short and can be regarded as a tiny arc segment, the point P (u) can be usedi) The small segment of the arc of the curvature circle approaches the arc segment, the feeding step length L is regarded as the chord length of the arc, and the bow height error is as follows:
Figure BDA0002369016380000121
L＝||P(ui)-P(ui+1)|| (4)
Figure BDA0002369016380000122
where σ denotes the bow height error, ρiRepresents the parametric curve P (u) at the point P (u)i) Radius of curvature of (d), P (u)i+1) Represents the parameter curve P (u) upper and the parameter ui+1Corresponding point, P' (u)i) Represents the parametric curve P (u) at the point P (u)i) The first derivative of (d), P' (u)i) Represents the parametric curve P (u) at the point P (u)i) The second derivative of (d); another calculation method of the feed step L ═ vT can be found from the definition of the feed rate; in order to simplify the operation, the distance between the middle point of the arc section and the middle point of the chord length is used as the height error to calculate:
Figure BDA0002369016380000123
wherein the content of the first and second substances,
Figure BDA0002369016380000124
represents the upper and parameters of the parametric curve P (u)
Figure BDA0002369016380000125
A corresponding point; the calculated bow height error sigma and the set bow height error limit sigma are comparedεIn contrast, when the calculated bow height error σ is smaller than the bow height error limit σεWhen the feeding rate meets the requirement, the feeding rate does not need to be adjusted; when the calculated height error sigma is larger than the height error limit sigmaεWhen the feed rate is out of the specification, the feed rate needs to be adjusted. According to the relationship between the bow height error and the feeding step length and the feeding rate, the following can be obtained:
Figure BDA0002369016380000131
as can be seen from the equation (7), since the feed rate and the second power of the bow height error are approximately in a proportional relationship, when the feed rate is adjusted, the bow height error at the corresponding position can be adjusted, and the ideal feed rate v', the existing feed rate v, the existing bow height error σ, and the bow height error limit σ can be obtainedεThe relationship between them is:
Figure BDA0002369016380000132
after the adjusted feed rate is obtained, calculating the bow height error under the existing feed rate, and if the bow height error meets the limit, namely sigma is less than or equal to sigmaεThen the adjustment process is stopped, if the bow height error still does not satisfy the constraint of σ > σεThen the adjustment process continues until satisfied;
traversing all parameters of the parameter sequence with equal intervals, and obtaining each parameter u by utilizing the processiFeed rate value v satisfying bow height error limitiSo that a series of discrete pairs (u) can be obtainedi,vi)。
A series of discrete number pairs (u) is obtained by step S1i,vi) When there are a sufficient number of these two-dimensional points, a feed rate curve can be approximated. The feed rate planning is carried out by the invention, and the aim is to obtain a continuous feed rate curve which meets the requirements of acceleration and jerk and can meet the requirement of smoothness to a certain extent on the basis of the feed rate curve.
In a specific implementation, in the step S2 of the method for planning the feed rate according to the present invention, when the feed rate curve is obtained by using the obtained series of discrete number pairs, an inflection point in the series of discrete number pairs is found, and the feed rate curve is divided into a plurality of acceleration processes, a plurality of deceleration processes, and a plurality of uniform speed processes by using the inflection point, the following method may be specifically implemented:
using the resulting series of discrete pairs (u)i,vi) Obtaining a feed rate curve;
the finding of the inflection point is realized by calculating the geometric characteristics of the inflection point. Specifically, in a series of discrete pairs, three adjacent points (u) are takeni-1,vi-1)、(ui,vi) And (u)i+1,vi+1) Looking at the intermediate point (u)i,vi) Three permutations of these three points are possible: first, three points are arranged in a monotonous way, i.e. the three points are in the same acceleration and deceleration process, i.e. the three points are accelerated at the same timeIn a process or the same deceleration process; second, the same feed rate is applied to two adjacent points among the three points, and the feed rate is different at another point, i.e., (u)i,vi) Is the turning point of the uniform speed process and the acceleration process or the turning point of the uniform speed process and the deceleration process; third, three adjacent points represent two different acceleration and deceleration processes (i.e., an acceleration process and a deceleration process), i.e., a point (u)i,vi) Is the inflection point of the change of the acceleration process and the deceleration process. The above three cases, the latter two cases, are intended to be identified by the present invention; before finding the inflection point, a premise is satisfied, a feed rate curve composed of the points is "smoother", and a discrimination value mu can be calculated by using the idea of differential calculation:
Figure BDA0002369016380000141
if the discrimination value mu is larger than the standard value xi, (u)i,vi) Is an inflection point; if the discrimination value mu is smaller than the standard value xi, (u)i,vi) Is not an inflection point;
the feed rate curve is divided into a plurality of acceleration processes, a plurality of deceleration processes and a plurality of uniform velocity processes by using inflection points, and the purpose of segmenting the feed rate curve is to decompose a complex curve into a series of acceleration, deceleration and uniform velocity processes, so that the feed rate design of each process is facilitated.
In step S2, a complex feed rate curve is divided into several acceleration, deceleration, and uniform speed steps. In the process of planning the whole feed rate curve, the splicing idea is adopted, the design of a single process is firstly completed, and then a plurality of processes are spliced to obtain a final result. The results for the split can be divided into two categories: uniform motion process and variable motion process. The uniform motion process is a trivial process and therefore does not need to be designed. For the variable speed motion process, there are only two cases: an acceleration process and a deceleration process. The following describes in detail the process of feed rate design for the acceleration process and the deceleration process, respectively.
In a specific implementation, when the step S3 of the method for planning the feed rate provided by the present invention is executed, and a logarithmic probability function is used to design the feed rate of each acceleration process, the method specifically includes the following steps:
s31: an acceleration process is represented as: block ═ vstart,vend,ustart,uendL }, wherein vstartDenotes the initial feed rate, vendIndicating the end feed rate, l the displacement of the acceleration process, ustartParameter representing a starting point, uendParameters indicating the end point, and an acceleration limit and a jerk limit are added, the maximum acceleration limit A in the machiningmMaximum jerk limit J in machiningmFurthermore, in order to be able to meet the smoothness requirements to a certain extent, a smoothness tolerance ω at the connection points is also required; l can be obtained by integrating the calculated arc length differential ds:
Figure BDA0002369016380000151
Figure BDA0002369016380000152
wherein x ' (u) denotes the first derivative of x (u), y ' (u) denotes the first derivative of y (u), and z ' (u) denotes the first derivative of z (u); in the case of the same initial and final feed rates, according to a logarithmic probability (Sigmoid) function
Figure BDA0002369016380000153
(as shown in fig. 2, the derivative of which is shown in fig. 3), the displacement of the feed rate motion coincides with the displacement obtained in a linear motion, and therefore the motion time of the acceleration process is:
Figure BDA0002369016380000154
updating the acceleration block ═ vstart,vend,ustart,uend,l,t}；
S32: assuming that the value q > 0 of the Sigmoid function in the original defined domain, the function value f at q can be calculated1(q), two mappings may be constructed:
Figure BDA0002369016380000155
Figure BDA0002369016380000156
wherein p represents a parameter of time, g1And g2The relationship between the feed rate and the time is conveniently constructed for the constructed auxiliary function; g is prepared from1、g2And f1(w) the complex is a function suitable for the acceleration process, but q is uncertain, so that finding a suitable q is the key to complete the function construction and needs to be solved according to the limiting conditions; substituting p into Sigmoid function
Figure BDA0002369016380000157
Let v (p) be a function after composition, where p ∈ [0, t ∈]It can be expressed as:
Figure BDA0002369016380000158
after derivation, the following steps are carried out:
Figure BDA0002369016380000161
and (3) solving a second derivative of the compounded function:
Figure BDA0002369016380000162
wherein v (p) represents a feed rate curve, v' (p) represents an acceleration curve of the feed rate, v ″)(p) jerk curve, g 'representing feed rate'2Represents a mapping g2First derivative of, g'1Represents a mapping g1First derivative of f1' (p) represents f1A first derivative of (p);
s33: since the feed rate control design needs to meet acceleration and jerk limit requirements, the maximum acceleration and jerk cannot exceed the limits, i.e., v' (p) and v "(p) need to meet:
Figure BDA0002369016380000163
Figure BDA0002369016380000164
wherein A ismIndicates the maximum acceleration limit in machining, JmRepresents the maximum jerk limit during machining;
when in use
Figure BDA0002369016380000165
When v' (p) has a maximum value; when in use
Figure BDA0002369016380000166
Then v "(p) has a maximum value; due to f1The value of (-q) is close to 0, so f1(-q) is regarded as 0, f1(q) is considered to be 1; then there are:
Figure BDA0002369016380000167
two solutions q are obtained1And q is2Q is q1And q is2Minimum value of (d):
q＝min{q1,q2} (20)；
s34: calculating the acceleration a at qqA is toqComparing with the smoothness tolerance omega at the connecting point; if the smoothness requirement is satisfied, i.e. aqWhen the value is less than or equal to omega, adding q into block, i.e. block is { v ═ vstart,vend,ustart,uendL, t, q }, the block can form a feed rate design suitable for the acceleration process; if the smoothness requirement is not met, i.e. aq> omega, then decrease vstartAnd vendOf the larger values, steps S33 and S34 are repeated until the smoothness requirement, i.e., a, is metqUntil omega is less than or equal to omega.
In a specific implementation, when the step S3 of the feed rate planning method provided by the present invention is executed and the logarithmic probability function is used to design the feed rate of each deceleration process, the method may specifically include the following steps:
SS 31: one deceleration process is represented as: block ═ vstart,vend,ustart,uendL }, wherein vstartDenotes the initial feed rate, vendIndicating the end feed rate, l the displacement of the deceleration process, ustartParameter representing a starting point, uendParameters indicating the end point, and an acceleration limit and a jerk limit are added, the maximum acceleration limit A in the machiningmMaximum jerk limit J in machiningmFurthermore, in order to be able to meet the smoothness requirements to a certain extent, a smoothness tolerance ω at the connection points is also required; l can be obtained by integrating the calculated arc length differential ds:
Figure BDA0002369016380000171
Figure BDA0002369016380000172
wherein x ' (u) denotes the first derivative of x (u), y ' (u) denotes the first derivative of y (u), and z ' (u) denotes the first derivative of z (u); in the case of the same initial and final feed rates, according to a logarithmic probability (Sigmoid) function
Figure BDA0002369016380000173
(as shown in fig. 4, the derivative of which is shown in fig. 5), the displacement of the feed rate motion coincides with the displacement obtained in a linear motion, and the motion time of the deceleration process is:
Figure BDA0002369016380000174
updating the deceleration process block ═ vstart,vend,ustart,uend,l,t}；
SS 32: assuming that the value q > 0 of the Sigmoid function in the original defined domain, the function value f at q can be calculated2(q), two mappings may be constructed:
Figure BDA0002369016380000175
Figure BDA0002369016380000181
wherein p represents a parameter of time, g1And g2The relationship between the feed rate and the time is conveniently constructed for the constructed auxiliary function; g is prepared from1、g2And f2(w) the function is a function suitable for the deceleration process after being compounded, but q is uncertain, so that finding a suitable q is a key for completing the function construction and needs to be solved according to a limiting condition; substituting p into a function
Figure BDA0002369016380000182
Let v (p) be a function after composition, where p ∈ [0, t ∈]It can be expressed as:
Figure BDA0002369016380000183
after derivation, the following steps are carried out:
Figure BDA0002369016380000184
and (3) solving a second derivative of the compounded function:
Figure BDA0002369016380000185
wherein v (p) represents a feed rate curve, v ' (p) represents an acceleration curve of feed rate, v ' (p) represents a jerk curve of feed rate, and g '2Represents a mapping g2First derivative of, g'1Represents a mapping g1First derivative of f2' (p) represents f2A first derivative of (p);
SS 33: since the feed rate control design needs to meet acceleration and jerk limit requirements, the maximum acceleration and jerk cannot exceed the limits, i.e., v' (p) and v "(p) need to meet:
Figure BDA0002369016380000186
Figure BDA0002369016380000187
wherein A ismIndicates the maximum acceleration limit in machining, JmRepresents the maximum jerk limit during machining;
when in use
Figure BDA0002369016380000191
When v' (p) has a maximum value; when in use
Figure BDA0002369016380000192
Then v "(p) has a maximum value; due to f2The value of (-q) is close to 1, so f2(-q) is regarded as 1, f2(q) is taken as 0; then there are:
Figure BDA0002369016380000193
two solutions q are obtained1And q is2Q is q1And q is2Minimum value of (d):
q＝min{q1,q2} (31)；
SS 34: calculating the acceleration a at qqA is toqComparing with the smoothness tolerance omega at the connecting point; if the smoothness requirement is satisfied, i.e. aqWhen the value is less than or equal to omega, adding q into block, i.e. block is { v ═ vstart,vend,ustart,uendL, t, q }, the block can form the feed rate design suitable for the deceleration process; if the smoothness requirement is not met, i.e. aq> omega, then decrease vstartAnd vendAt the larger value, steps SS33 and SS34 are repeated until the smoothness requirement, i.e., a, is metqUntil omega is less than or equal to omega.
The feed rate can be designed for each process by using the above step S3, and all processes need to be spliced after the design is completed for all processes. It should be noted that the end feed rate of the previous process and the start feed rate of the next process are originally equal, but in step S3, a feed rate adjustment process occurs, and the feed rate adjustment process makes the end feed rate of the previous process and the start feed rate of the next process different.
Based on this, in step S4, the present invention sequentially uses the two processes of forward feed rate adjustment and backward feed rate adjustment to perform feed rate adjustment, so as to realize the splicing of each acceleration process, each deceleration process, and each uniform speed process. Specifically, in the step S4 of the method for planning a feed rate according to the present invention, the feed rate is adjusted by sequentially using the forward feed rate adjustment process and the backward feed rate adjustment process, and when the splicing among the acceleration processes, the deceleration processes, and the uniform speed processes is implemented, the method may specifically be implemented as follows:
firstly, utilizing forward feed rate regulation algorithm to pair process blocksj＝{vj,start,vj,end,uj,start,uj,end,lj,tjJ-1, 2.. m, starting from j-1Starting to scan forward, and executing the step S31-step S34 when the process is an acceleration process, and simultaneously adjusting two adjacent process blocksjAnd blockj+1Value v ofj,end,tjAnd vj+1,start,tj+1；
And then utilizing a backward feeding rate adjustment algorithm to perform process block pairj＝{vj,start,vj,end,uj,start,uj,end,lj,tjAnd j is 1, 2.. m, scanning backwards from j is m, and when the process is a deceleration process, executing steps SS31 to SS34, and simultaneously adjusting two adjacent process blocksjAnd blockj+1Value v ofj,end,tjAnd vj+1,start,tj+1。
The following describes in detail a specific implementation of the above feed rate planning method provided by the present invention with a specific embodiment.
Example 1:
taking the test processing track parameter curve (as shown in fig. 6) as:
Figure BDA0002369016380000201
the definition domain of the curve is [0, 1 ]]Given a parameter interval Δ u of 0.01, a sampling period T of 0.002, a bow height error limit σε＝0.0001。
Step S1: obtaining a number pair (u, v) of parameters and feed rates satisfying the bow height error limit
To illustrate the details, u may be set to 0.5, and the first derivative of the curve is:
P′(u)＝(36u2-60u+30,15u2+60u-30,-15u2-40u+30) (2′)
substituting u-0.5 into equation (2') yields a first derivative of the curve at u-0.5:
P(0.5)′＝(9.0,3.75,6.25) (3′)
the first derivative value of the parameter u of 0.5 with respect to time was found to be 2.5904. The first derivative value is substituted for equation (1) to obtain an estimated parameter value of 0.5052.
And calculating the bow height error according to the following calculation formula:
Figure BDA0002369016380000202
and obtaining the height error caused by the estimated parameters, wherein the height error exceeds the height error limit by 0.0001.
Therefore, the original feed rate should be adjusted. The correction factor is calculated according to equation (8) when u is 0.5:
Figure BDA0002369016380000211
Figure BDA0002369016380000212
and substituting the adjusted feed rate into the formula (1) and the formula (2) to recalculate the estimated parameter value, and obtaining a new estimated parameter value of 0.5029.
The newly obtained estimated parameter value 0.5029 is substituted into equation (4'), and the bow height error is recalculated. If the height error caused by the estimated parameters is lower than the height error limit, the parameter u is 0.5 to meet the requirement and is used as the next parameter value; if the height error caused by the estimated parameters is higher than the height error limit, the parameter u is not 0.5, and the adjustment is continued:
Figure BDA0002369016380000213
estimating the parameter u to be 0.5 again by using the adjusted feed rate, calculating a new bow height error by an equation (6), obtaining a newly adjusted feed rate by substituting the newly obtained bow height error into an equation (8) until the calculated bow height error is lower than the bow height error limit, and obtaining the feed rate with the parameter u to be 0.5 at the moment
Figure BDA0002369016380000214
Feed rate point of composition
Figure BDA0002369016380000215
In the manner described above, the sequence of parameters { u } is traversediAnd obtaining the feed rate under the limitation of the bow height error corresponding to each parameter. Taking the parameter u as the abscissa and the feed rate v as the ordinate, a series of points consisting of the parameter and the feed rate can be obtained
Figure BDA0002369016380000216
From these points, a feed rate curve under the bow height error limit can be plotted as shown in fig. 7.
Step S2: finding inflection points
Three points of inflection 2, 3, 4 can be found in the feed rate curve as shown in fig. 7, and four process blocks can be formed by the three points of inflection and the starting point 1 and the ending point 51,block2,block3,block4As shown in FIG. 8, wherein block1And block4Is a uniform speed process, block2Is a deceleration process, block3Is an acceleration process. The parameters of the start point, the end point and the three inflection points and the corresponding feed rates are shown in table 1:
TABLE 1
Inflection point, start point and end point Corresponding parameter Corresponding to the feed rate
1 0.000 30.0
2 0.350 30.0
3 0.540 16.0
4 0.710 29.9
5 1.000 30.0
As is clear from table 1, the feed rates at the inflection point 4 and the termination point 5 are sufficiently close to each other, and therefore, it is a necessary principle to process data by setting the feed rate at the inflection point 4 to 30.
Calculate the displacement and time l of each segment process1,l2,l3,l4,t1,t2,t3,t4The calculation results are shown in table 2:
TABLE 2
Figure BDA0002369016380000221
Uniform motion is not required to be considered, and only accelerated motion and decelerated motion are required to be considered. Calculating block2,block3Q value of (a), given acceleration limit and jerk limit Am＝950,Jm30000. Wherein the process block216.42 and 70.59, respectively, the process block2The q value of (A) is 16.42; process block315.47 and 62.66, respectively, the process block3The q value of (2) was 15.47.
Since v '(16.42) and v' (15.47) are very close to zero, the smoothness requirement is met, and therefore corresponds to the process block2,block3The feed rate curves of (a) are expressed as:
Figure BDA0002369016380000222
Figure BDA0002369016380000223
the final finished feed rate map through the feed rate schedule is shown in fig. 9.
As shown in fig. 10, the bow height error (shown by the solid line in fig. 10) existing after the feed rate planning is compared with the bow height error (shown by the broken line in fig. 10) generated by the constant feed rate without the feed rate planning. Wherein the dashed line represents the bow height error generated at a constant feed rate, implemented as an error curve generated after feed rate planning. It can be seen that although the set bow height error limit σ is exceeded over some parameter intervalεHowever, the feed rate adjustment is most required, that is, the parameter interval in which the bow height error is the largest is limited, and the bow height error can be effectively reduced. The maximum bow height error at constant feed rate is close to 3.5 x 10-4After the feed rate planning, the error is reduced to be close to 1.0 multiplied by 10-4。
The feed rate planning method of numerical control processing based on the logarithm probability function, provided by the invention, comprises the steps of firstly defining a feed rate adjusting factor as a half power of the ratio of error limit to calculation error, taking an equally spaced parameter sequence, and adjusting each parameter according to the bow height error limit by continuously utilizing the adjusting factor to obtain the final rate; then defining a discrimination value as an absolute value of the difference between the front difference and the rear difference, calculating the discrimination value at a known parameter, wherein the discrimination value is a calculated inflection point when being larger, and the inflection point, the starting point and the ending point divide the whole curve into a plurality of sections, and each section can be regarded as an acceleration process, a deceleration process or a uniform speed process; then, the Sigmoid function is used for controlling the feed rate in the acceleration and deceleration process, and the feed rate expression suitable for the process is worked out according to the requirements of acceleration, acceleration and smoothness; and finally, splicing the acceleration and deceleration processes by using a forward scanning process and a backward scanning process. The feed rate planning method provided by the invention can reduce the processing error on the premise of ensuring the acceleration and the jerk, and realize continuous feed rate planning.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

Claims (6)

1. A feed rate planning method for numerical control machining based on a logarithmic probability function is characterized by comprising the following steps:
s1: according to a parameter curve of a numerical control machining track, obtaining an equidistant parameter sequence, and according to the limitation of bow height errors, obtaining the feed rate numerical values of all parameters in the equidistant parameter sequence to obtain a series of discrete number pairs;
s2: obtaining a feed rate curve by using the obtained series of discrete number pairs, searching inflection points in the series of discrete number pairs, and dividing the feed rate curve into a plurality of acceleration processes, a plurality of deceleration processes and a plurality of uniform speed processes by using the inflection points;
s3: performing feed rate design on each acceleration process and each deceleration process by using a logarithmic probability function;
s4: and sequentially utilizing the forward feed rate adjustment process and the backward feed rate adjustment process to adjust the feed rate, so as to realize the splicing of each acceleration process, each deceleration process and each uniform speed process.
2. The feed rate planning method of claim 1, wherein step S1, using the equally spaced parameter sequence, obtains the feed rate values of all the parameters in the equally spaced parameter sequence according to the limitation of the bow-height error, and obtains a series of discrete pairs, specifically comprising:
assuming that the parameter curve of the processing track is expressed as P (u) ═ x (u), y (u), z (u), the parameter range is 0 ≦ u ≦ 1, and the parameter interval is Δ u, then u isi+1＝ui+Δu，u0Get an equally spaced parameter sequence as 0
Figure FDA0002369016370000011
A first order taylor's formula is adopted:
Figure FDA0002369016370000012
Figure FDA0002369016370000013
wherein x (u) represents a parametric representation of the values of the x-coordinate, y (u) represents a parametric representation of the values of the y-coordinate, z (u) represents a parametric representation of the values of the z-coordinate, uiAnd ui+1Indicating a sequence of parameters at equal intervals
Figure FDA0002369016370000014
Two adjacent parameters, T represents time, s represents arc length, v represents a preset feed rate, T represents an interpolation period, h.o.t represents high-order infinitesimal, P ' (u) represents a first derivative of a parameter curve P (u), and | P ' (u) | represents a norm of P ' (u);
from u obtainediAnd u'iCalculating two points P (u) on the parameter curvei) And P (u'i) The maximum distance between the arc segment and the line segment is two points P (u)i) And P (u'i) Of bow height error of (1), wherein, u'iRepresenting the parameter uiThe significance of the predicted parameter is to simulate the predicted parameter in the parameter uiThe interpolation process obtains the predicted parameter of the next interpolation point, P (u)i) Represents the parameter curve P (u) upper and the parameter uiCorresponding point, P (u')i) To representParameter curve P (u) and estimated parameter u'iA corresponding point; regarding the arc segment as a tiny arc segment, point P (u) is usedi) The small segment of the arc of the curvature circle approaches the arc segment, the feeding step length L is regarded as the chord length of the arc, and the bow height error is as follows:
Figure FDA0002369016370000021
L＝||P(ui)-P(ui+1)||
Figure FDA0002369016370000022
where σ denotes the bow height error, ρiRepresents the parametric curve P (u) at the point P (u)i) Radius of curvature of (d), P (u)i+1) Represents the parameter curve P (u) upper and the parameter ui+1Corresponding point, P' (u)i) Represents the parametric curve P (u) at the point P (u)i) The first derivative of (d), P' (u)i) Represents the parametric curve P (u) at the point P (u)i) The second derivative of (d); the feeding step length L is known as vT according to the definition of the feeding rate; for simplification, the distance between the middle point of the arc section and the middle point of the chord length is used as the height error:
Figure FDA0002369016370000023
wherein the content of the first and second substances,
Figure FDA0002369016370000024
represents the upper and parameters of the parametric curve P (u)
Figure FDA0002369016370000025
A corresponding point; according to the relationship between the bow height error and the feeding step length and the feeding rate, the following results are obtained:
Figure FDA0002369016370000026
assuming the bow height error is limited to σεIdeal feed rate v', existing feed rate v, existing bow height error sigma and bow height error limit sigmasThe relationship between them is:
Figure FDA0002369016370000031
traversing all parameters of the equally spaced parameter sequence to obtain each parameter uiFeed rate value v satisfying bow height error limitiTo obtain a series of discrete number pairs (u)i，vi)。
3. The feed rate planning method according to claim 2, wherein step S2, obtaining a feed rate curve by using the obtained series of discrete number pairs, finding an inflection point in the series of discrete number pairs, and dividing the feed rate curve into a plurality of acceleration processes, a plurality of deceleration processes, and a plurality of constant speed processes by using the inflection point, specifically comprises:
using the resulting series of discrete pairs (u)i，vi) Obtaining a feed rate curve;
in a series of discrete pairs, three adjacent points (u) are takeni-1，vi-1)、(ui，vi) And (u)i+1，vi+1) Calculating a discrimination value mu:
Figure FDA0002369016370000032
if the discrimination value mu is larger than the standard value xi, (u)i，vi) Is an inflection point; if the discrimination value mu is smaller than the standard value xi, (u)i，vi) Is not an inflection point;
the feed rate curve is divided into a plurality of acceleration processes, a plurality of deceleration processes and a plurality of uniform velocity processes by using inflection points.
4. The feed rate planning method of claim 3, wherein the step S3 of performing feed rate design for each acceleration process by using a logarithmic probability function comprises the following steps:
s31: an acceleration process is represented as: block ═ vstart，vend，ustart，uendL }, wherein vstartDenotes the initial feed rate, vendIndicating the end feed rate, l the displacement of the acceleration process, ustartParameter representing a starting point, uendA parameter indicating an end point; l is obtained by integrating the calculated arc length differential ds:
Figure FDA0002369016370000033
Figure FDA0002369016370000034
wherein x ' (u) denotes the first derivative of x (u), y ' (u) denotes the first derivative of y (u), and z ' (u) denotes the first derivative of z (u); in the case of the same initial and final feed rates, according to a logarithmic probability function
Figure FDA0002369016370000041
The displacement of the feed rate movement is consistent with the displacement obtained by the linear movement, and the movement time of the acceleration process is as follows:
Figure FDA0002369016370000042
updating the acceleration block ═ vstart，vend，ustart，uend，l，t}；
S32: assuming that the value q > 0 of the log-probability function in the original defined domain, the function value f at q is calculated1(q), two mappings may be constructed：
Figure FDA0002369016370000043
Figure FDA0002369016370000044
Wherein p represents a parameter of time, g1And g2For the constructed auxiliary function, g1、g2And f1(w) substituting p into the log-probability function as a function adapted to the acceleration process
Figure FDA0002369016370000045
Let v (p) be a function after composition, where p ∈ [0, t ∈]Expressed as:
Figure FDA0002369016370000046
after derivation, the following steps are carried out:
Figure FDA0002369016370000047
and (3) solving a second derivative of the compounded function:
Figure FDA0002369016370000048
wherein v (p) represents a feed rate curve, v ' (p) represents an acceleration curve of the feed rate, v ' (p) represents a jerk curve of the feed rate, and g ', respectively2Represents a mapping g2First derivative of, g'1Represents a mapping g1First derivative of f1' (p) represents f1A first derivative of (p);
s33: v '(p) and v' (p) are satisfied:
Figure FDA0002369016370000051
Figure FDA0002369016370000052
wherein A ismIndicates the maximum acceleration limit in machining, JmRepresents the maximum jerk limit during machining;
when in use
Figure FDA0002369016370000053
When v' (p) has a maximum value; when in use
Figure FDA0002369016370000054
Then v "(p) has a maximum value; due to f1The value of (-q) is close to 0, so f1(-q) is regarded as 0, f1(q) is considered to be 1; then there are:
Figure FDA0002369016370000055
two solutions q are obtained1And q is2Q is q1And q is2Minimum value of (d):
q＝min{q1，q2}；
s34: calculating the acceleration a at qqA is toqComparing with the smoothness tolerance omega at the connecting point; if aqWhen omega is not more than, block is ═ vstart，vend，ustart，uendL, t, q } constitutes a feed rate design adapted to the acceleration process; if aq> omega, then decrease vstartAnd vendOf the larger value, repeating steps S33 and S34 until aqUntil omega is less than or equal to omega.
5. The feed rate planning method according to claim 3, wherein the step S3 of designing the feed rate for each deceleration process by using a logarithmic probability function specifically comprises the steps of:
SS 31: one deceleration process is represented as: block ═ vstart，vend，ustart，uendL }, wherein vstartDenotes the initial feed rate, vendIndicating the end feed rate, l the displacement of the deceleration process, ustartParameter representing a starting point, uendA parameter indicating an end point; l is obtained by integrating the calculated arc length differential ds:
Figure FDA0002369016370000056
Figure FDA0002369016370000061
wherein x ' (u) denotes the first derivative of x (u), y ' (u) denotes the first derivative of y (u), and z ' (u) denotes the first derivative of z (u); in the case of the same initial and final feed rates, according to a logarithmic probability function
Figure FDA0002369016370000062
The displacement of the feed rate movement is consistent with the displacement obtained by the linear movement, and the movement time of the deceleration process is as follows:
Figure FDA0002369016370000063
updating the deceleration process block ═ vstart，vend，ustart，uend，l，t}；
SS 32: assuming that the value q > 0 of the log-probability function in the original defined domain, the function value f at q is calculated2(q), two mappings may be constructed:
Figure FDA0002369016370000064
Figure FDA0002369016370000065
wherein p represents a parameter of time, g1And g2For the constructed auxiliary function, g1、g2And f2(w) substituting p into the log-probability function as a function adapted to the deceleration process
Figure FDA0002369016370000066
Let v (p) be a function after composition, where p ∈ [0, t ∈]Expressed as:
Figure FDA0002369016370000067
after derivation, the following steps are carried out:
Figure FDA0002369016370000068
and (3) solving a second derivative of the compounded function:
Figure FDA0002369016370000071
wherein v (p) represents a feed rate curve, v ' (p) represents an acceleration curve of the feed rate, v ' (p) represents a jerk curve of the feed rate, and g ', respectively2Represents a mapping g2First derivative of, g'1Represents a mapping g1First derivative of f2' (p) represents f2A first derivative of (p);
SS 33: v '(p) and v' (p) are satisfied:
Figure FDA0002369016370000072
Figure FDA0002369016370000073
wherein A ismIndicates the maximum acceleration limit in machining, JmRepresents the maximum jerk limit during machining;
when in use
Figure FDA0002369016370000074
When v' (p) has a maximum value; when in use
Figure FDA0002369016370000075
Then v "(p) has a maximum value; due to f2The value of (-q) is close to 1, so f2(-q) is regarded as 1, f2(q) is taken as 0; then there are:
Figure FDA0002369016370000076
two solutions q are obtained1And q is2Q is q1And q is2Minimum value of (d):
q＝min{q1，q2}；
SS 34: calculating the acceleration a at qqA is toqComparing with the smoothness tolerance omega at the connecting point; if aqWhen omega is not more than, block is ═ vstart，vend，ustart，uendL, t, q } constitutes a feed rate design adapted to the deceleration process; if aq> omega, then decrease vstartAnd vendTo the larger value, repeat steps SS33 and SS34 until aqUntil omega is less than or equal to omega.
6. The feed rate planning method according to claim 4 or 5, wherein in step S4, the feed rate adjustment is performed by sequentially using a forward feed rate adjustment process and a backward feed rate adjustment process, so as to implement the splicing among the acceleration processes, the deceleration processes, and the uniform speed processes, and specifically includes:
firstly, utilizing forward feed rate regulation algorithm to pair process blocksj＝{vj，start，vj，end，uj，start，uj，end，lj，tjAnd j is 1, 2.. m, scanning forward from j 1, and when the process is an acceleration process, performing steps S31 to S34 while adjusting two adjacent process blocksjAnd blockj+1Value v ofj，end，tjAnd vj+1，start，tj+1；
And then utilizing a backward feeding rate adjustment algorithm to perform process block pairj＝{vj，start，vj，end，uj，start，uj，end，lj，tjAnd j is 1, 2.. m, scanning backwards from j is m, and when the process is a deceleration process, executing steps SS31 to SS34, and simultaneously adjusting two adjacent process blocksjAnd blockj+1Value v ofj，end，tjAnd vj+1，start，tj+1。
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