CN113515091B - Real-time track interpolation method and device for corner smoothing spline - Google Patents

Real-time track interpolation method and device for corner smoothing spline Download PDF

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CN113515091B
CN113515091B CN202110359418.1A CN202110359418A CN113515091B CN 113515091 B CN113515091 B CN 113515091B CN 202110359418 A CN202110359418 A CN 202110359418A CN 113515091 B CN113515091 B CN 113515091B
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spline
arc length
chord
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CN113515091A (en
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杨建中
周会成
张建雄
高嵩
朱万强
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Huazhong University of Science and Technology
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    • G05B19/00—Programme-control systems
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    • G05B19/18—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
    • G05B19/41—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 interpolation, e.g. the computation of intermediate points between programmed end points to define the path to be followed and the rate of travel along that path
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention discloses a real-time interpolation method and a real-time interpolation device for a corner smoothing spline, belonging to the field of numerical control machine tool machining, wherein the method comprises the following steps: s1: establishing a chord height error optimization model corresponding to the corner smoothing spline; s2: substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence; s3: establishing a least square fitting equation by utilizing the sampling parameter sequence and the sampling arc length sequence; solving a least square fitting equation in real time to establish an arc length parameter mapping model; s4: and (4) inputting the spline parameters of the spline with the smooth corner into the arc length parameter mapping model, and acquiring the spline parameters and coordinates corresponding to the next interpolation point to realize real-time track interpolation. The method provided by the invention can quickly calculate the high-precision spline parameters, and further obtain the high-precision interpolation point coordinates so as to reduce the fluctuation of the feeding speed.

Description

Real-time track interpolation method and device for corner smoothing spline
Technical Field
The invention belongs to the field of numerical control machine tool machining, and particularly relates to a method and a device for interpolating a corner smoothing spline in real time.
Background
In today's field of numerical control machining, straight lines and arcs are still widely used as tool trajectories in production environments due to direct coding and good support of CNC. Due to poor continuity between straight-line tracks, frequent acceleration and deceleration of the tool are required, which causes frequent fluctuation of the feed speed and the acceleration, and even in places with too large corners, the tool needs to be decelerated to approach zero to pass through, which has a great influence on the surface quality of the workpiece and also reduces the machining efficiency. Thus, many numerical control systems provide a smoother, more continuous transition between straight segment processing trajectories by constructing local splines at the corners of adjacent straight segments. However, due to the nonlinear relation between the arc length of the spline curve and the parameters, the problem of feed speed fluctuation exists in the real-time interpolation of the spline. For this purpose, a number of real-time interpolation methods of spline trajectories are produced.
The current real-time interpolation method of various spline curves is to establish an optimization function and find spline parameters meeting the precision through continuous iteration; the calculation amount of this method is usually large, the calculation is time-consuming, more importantly, the number of iterations is uncertain, the stability of the whole calculation process is difficult to guarantee, and in extreme cases, the number of iterations may be too many, which causes interpolation blocking, which is very dangerous for processing.
Disclosure of Invention
Aiming at the defects or improvement requirements of the prior art, the invention provides a real-time interpolation method and a real-time interpolation device for a corner smooth spline, aiming at rapidly calculating high-precision spline parameters and further obtaining high-precision interpolation point coordinates to reduce feed speed fluctuation, thereby solving the technical problems of low calculation efficiency, poor stability and low precision of the conventional spline interpolation method.
To achieve the above object, according to one aspect of the present invention, there is provided a method for real-time interpolation of a trajectory of a corner-smoothed spline, including:
s1: establishing a chord height error optimization model corresponding to the corner smoothing spline;
s2: substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence;
s3: establishing a least square fitting equation by using the sampling parameter sequence and the sampling arc length sequence; solving the least square fitting equation in real time to establish an arc length parameter mapping model;
s4: and inputting the spline parameters of the corner smoothing spline into the arc length parameter mapping model, and acquiring spline parameters and coordinates corresponding to a next interpolation point to realize real-time track interpolation.
In one embodiment, the S1 includes:
establishing the chord height error optimization model, and changing the distribution of sampling points to reduce the deviation of chord length and arc length; the chord height error optimization model is expressed as:
Figure BDA0003004904440000021
the chord height error optimization model meets the upper limit constraint of the real-time requirement, and the upper limit n of the number of sampling points is given; with the increase of the sampling cycle number i, the number t (i) of sampling points is increased continuously until the sampling is stopped when the upper limit n of the number of sampling points is reached.
In one embodiment, the S2 includes:
s21: substituting the chord height error estimation method into the chord height error optimization model to obtain the sampling parameter sequence U and the sampling point sequence P:
Figure BDA0003004904440000022
s22: obtaining a sampling arc length sequence S ═ S by the approximate arc length of the accumulated chord length0 s1 s2 ··· sn-1]Wherein, in the step (A),
Figure BDA0003004904440000023
in one embodiment, the chord height error estimation method in S2 includes:
s201: smoothing two adjacent sampling points p on the spline curvekAnd pk+1Tangent vector V ofkAnd Vk+1Expressed as:
Vk＝αktk Vk+1＝αk+1tk+1；
wherein the corner smoothing spline curve is a plane curve constructed at the corner of two straight line segments, pk＝C(uk)，pk+1＝C(uk+1)，αkAnd alphak+1Are each pkAnd pk+1Tangent vector mode length of (t)kAnd tk+1Are each pkAnd pk+1Unit tangent of thetakAnd thetak+1Respectively is tangent vector VkAnd Vk+1And chord line vector
Figure BDA0003004904440000031
The included angle of (A);
s202: corresponding parameter value u to the maximum error point of the chord height on the corner smooth spline curvechordAnd estimating to obtain:
Figure BDA0003004904440000032
uk、uk+1are each pkAnd pk+1The chord height error between the two, q is a proportionality coefficient;
s203: using the value u of the parameterchordEstimating to obtain a curve segment
Figure BDA0003004904440000033
Upper chord height error value epsilonchordAnd point p corresponding to the maximum value of the chord height errorchord；
Figure BDA0003004904440000034
Figure BDA0003004904440000035
In one embodiment, the S3 includes:
s31: normalizing the sampling arc length sequence to obtain a sequence theta, wherein theta is [ alpha ═ alpha [0 α1 α2··· αn-1]＝[0 s1/smax s2/smax ··· 1]；
S32: setting a boundary constraint of the arc length parameter mapping model to ensure the continuity of the interpolation speed at the endpoint, wherein the boundary constraint is expressed as:
Figure BDA0003004904440000041
s33: to determine A ═ a0 a1 a2 a3 a4 a5]TThe normalized parameter sequence Θ ═ α0α1α2···αn-1]Arc length parameter mapping quintic polynomial model for substituting to-be-determined coefficient as independent variable
Figure BDA0003004904440000042
In (3), a fitting parameter sequence is obtained
Figure BDA0003004904440000043
Figure BDA0003004904440000044
S34: combining the boundary constraint and the sequence of sampling parameters U ═ U0 u1 u2 ··· un-1]And obtaining the least square fitting equation:
Figure BDA0003004904440000045
s35: performing QR decomposition on coefficient matrix in the boundary constraint equation
Figure BDA0003004904440000046
Solving the least squares fit equation to yield:
Figure BDA0003004904440000047
wherein A ═ a0 a1 a2 a3 a4 a5]TIs the coefficient of a fitting polynomial and is,
Figure BDA0003004904440000048
a boundary constraint coefficient matrix; PhiQ is [. phi. ]1 Φ2]，QTA＝[B C]T；
S36: and solving a fitting parameter polynomial coefficient A by adopting a Gaussian orthogonal method to obtain the arc length parameter mapping model.
In one embodiment, the S4 includes:
s41: arc length s obtained by speed planningkThe arc length skInputting the arc length parameter mapping model to calculate spline parameter u of next interpolation pointk：
Figure BDA0003004904440000051
S42: the spline parameter u of the next interpolation point is usedkSubstituting the equation corresponding to the corner smooth spline curve to obtain the next interpolation point coordinate pk＝C(uk)。
In one embodiment, the method before S1 further includes:
performing power-base conversion on the corner smoothing splines to represent various different corner smoothing splines by adopting a uniform power base; the power-based expression is of the form:
Figure BDA0003004904440000052
wherein, U is [ 1U … U ]d]For a set of linearly independent bases of polynomial space, aiIs a polynomial coefficient, i is 0,1, …, d, u is a spline parameter, d is a spline order.
According to another aspect of the present invention, there is provided a device for real-time interpolation of a corner-smoothed spline trajectory, comprising:
the establishing module is used for establishing a chord height error optimization model corresponding to the corner smoothing spline;
the calculation module is used for substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence;
the solving module is used for establishing a least square fitting equation by utilizing the sampling parameter sequence and the sampling arc length sequence; solving the least square fitting equation in real time to establish an arc length parameter mapping model;
and the interpolation module is used for inputting the spline parameters of the corner smoothing splines into the arc length parameter mapping model and acquiring the spline parameters and coordinates corresponding to the next interpolation point so as to realize real-time interpolation of the track.
In general, compared with the prior art, the above technical solution contemplated by the present invention can achieve the following beneficial effects:
(1) the invention utilizes the chord height error optimization model to sample the spline track in real time, obtains the spline parameters and the sampling point sequence, and further calculates to obtain the arc length sequence. And fitting the arc length and the parameter sequence by adopting a polynomial, solving a fitting expression by adopting a QR decomposition-based calculation method, and considering the continuity constraint of corner smooth splines at the boundary to finally obtain a mapping model of the arc length and the parameters. And during spline interpolation, fast calculating high-precision spline parameters in real time by using the arc length and the parameter mapping model according to the input arc length, further calculating high-precision interpolation point coordinates, and reducing feed speed fluctuation.
(2) The sampling mode is based on the chord height error optimization model, and a chord height error rapid estimation method is provided, so that the sampling process is rapid and stable, and can be carried out in real time when corner smooth transition splines are constructed in a numerical control system. When the arc length parameter mapping model is constructed, the first-order continuity of the end points is considered, the continuity of the interpolation speed at the end points is ensured, and the fluctuation is avoided. And the method based on QR decomposition is adopted to solve the arc length parameter mapping model, so that the solving mode is fast and stable.
Drawings
FIG. 1 is a schematic diagram of a numerically controlled machined corner smooth transition spline curve in an embodiment of the present invention;
FIG. 2 is a schematic flow chart of a method for real-time interpolation of a corner smoothing spline track according to an embodiment of the present invention;
FIG. 3 is a schematic flow chart of a method for real-time interpolation of corner-smoothed spline traces according to another embodiment of the present invention;
FIG. 4 is a schematic diagram illustrating the arc length, chord length, and end tangent between two adjacent sampling points on a curve according to an embodiment of the present invention;
FIG. 5a is a schematic flow chart of the arc length parameter mapping model construction in FIG. 3;
fig. 5b is a schematic flow chart of the non-uniform sampling in fig. 3.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
FIG. 1 is a schematic diagram of a numerically controlled machined corner smooth transition spline curve in an embodiment of the present invention; for the corner smoothing transition spline curve in fig. 1, the present invention provides a real-time interpolation method for the trajectory of the corner smoothing spline, as shown in fig. 2 and fig. 3, including:
s1: establishing a chord height error optimization model corresponding to the corner smoothing spline;
s2: substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence;
s3: establishing a least square fitting equation by utilizing the sampling parameter sequence and the sampling arc length sequence; solving a least square fitting equation in real time to establish an arc length parameter mapping model;
s4: and (4) inputting the spline parameters of the spline with the smooth corner into the arc length parameter mapping model, and acquiring the spline parameters and coordinates corresponding to the next interpolation point to realize real-time track interpolation.
In one embodiment, S1 includes:
establishing a chord height error optimization model, and changing the distribution of sampling points to reduce the deviation of chord length and arc length; the chord height error optimization model is expressed as:
Figure BDA0003004904440000071
the chord height error optimization model meets the upper limit constraint of the real-time requirement, and the upper limit n of the number of sampling points is given; with the increase of the sampling cycle number i, the number t (i) of sampling points is increased continuously until the sampling is stopped when the upper limit n of the number of sampling points is reached.
Specifically, in order to quickly calculate the arc length in a real-time scene, a strategy of accumulating the chord length to approximate the arc length is generally adopted, the chord length to approximate the arc length is deviated, and the deviation of the curve and the chord is generally measured by a chord height error epsilon. The chord height error of the curve is the maximum distance from the curve to the connection line of the two end points of the curve. In order to reduce the deviation between the chord length and the arc length, the distribution of sampling points is changed, and a chord height error optimization model is established, namely:
Figure BDA0003004904440000072
where i is the number of cycles. By increasing sampling points in a continuous cycle, the string height error sequence epsilon between the sampling point sequencesiThe maximum chord height error in (1) is minimal. Meanwhile, in order to guarantee the real-time performance of the sampling process, the upper limit n of the number of the sampling points is given. The number t (i) of sampling points increases with the increase of the sampling cycle number i until the sampling is reachedAnd stopping sampling when the upper limit n of the number of the sampling points is reached.
In one embodiment, S2 includes:
s21: substituting the chord height error estimation method into a chord height error optimization model to obtain a sampling parameter sequence U and a sampling point sequence P:
Figure BDA0003004904440000081
s22: obtaining a sampling arc length sequence S ═ S by the approximate arc length of the accumulated chord length0 s1 s2 ··· sn-1]Wherein, in the step (A),
Figure BDA0003004904440000082
in one embodiment, the chord height error estimation method in S2 includes:
s201: as shown in fig. 4, smoothing the corner with two adjacent sampling points p on the spline curvekAnd pk+1Tangent vector V ofkAnd Vk+1Expressed as:
Vk＝αktk Vk+1＝αk+1tk+1；
wherein the corner smoothing spline curve is a plane curve constructed at the corner of two straight line segments, pk＝C(uk)，pk+1＝C(uk+1)，αkAnd alphak+1Are each pkAnd pk+1Tangent vector mode length of (t)kAnd tk+1Are each pkAnd pk+1Unit tangent of thetakAnd thetak+1Respectively is tangent vector VkAnd Vk+1And chord line vector
Figure BDA0003004904440000083
The included angle of (A);
s202: corresponding parameter value u to maximum error point of upper chord height of corner smooth spline curvechordAnd estimating to obtain:
Figure BDA0003004904440000091
uk、uk+1are each pkAnd pk+1The chord height error between the two, q is a proportionality coefficient;
s203: using the value u of the parameterchordEstimating to obtain a curve segment
Figure BDA0003004904440000092
Upper chord height error value epsilonchordAnd point p corresponding to the maximum value of the chord height errorchord；
Figure BDA0003004904440000093
Figure BDA0003004904440000094
In particular, in order to quickly find the parameter value u on the curvek、uk+1Two adjacent sampling points pk＝C(uk) And pk+1＝C(uk+1) Chord height error between and corresponding parameter value uchordAnd avoiding iterative solution, and estimating the chord height error by adopting a method. As shown in FIG. 5a, the corner-smoothing spline is a plane curve whose shape is primarily governed by the end tangent vector VkAnd Vk+1Chord line vector
Figure BDA0003004904440000095
Influence. Point pkAnd pk+1Tangent vector V ofkAnd Vk+1Can be expressed as:
Vk＝αktk Vk+1＝αk+1tk+1；
wherein alpha iskAnd alphak+1Are respectively a point pkAnd pk+1Tangent vector mode length of (t)kAnd tk+1Are respectively a point pkAnd pk+1Unit tangent of thetakAnd thetak+1Respectively is tangent vector VkAnd Vk+1And chord line vector
Figure BDA0003004904440000096
The included angle of (a).
Then the parameter value u corresponding to the maximum error point of the upper chord height of the curvechordIt can be estimated that:
uchord≈q·uk+(1-q)·uk+1；
wherein q is a proportionality coefficient:
Figure BDA0003004904440000097
then, according to the parameter value estimation, the curve segment is obtained
Figure BDA0003004904440000098
Upper chord height error value epsilonchordAnd point p corresponding to the maximum value of the chord height errorchord：
Figure BDA0003004904440000099
Figure BDA0003004904440000101
Further, the chord height error estimation method is substituted into a chord height error optimization model to obtain a sampling parameter sequence U and a sampling point sequence P:
Figure BDA0003004904440000102
further, the cumulative chord length is adopted to approximate the arc length, and a sampling arc length sequence is obtained:
S＝[s0 s1 s2 ··· sn-1]；
wherein:
Figure BDA0003004904440000103
in one embodiment, S3 includes:
s31: normalizing the sampling arc length sequence to obtain a sequence theta, wherein theta is [ alpha ═ alpha [ ]0 α1 α2 ··· αn-1]＝[0 s1/smax s2/smax ··· 1]；
S32: setting the boundary constraint of the arc length parameter mapping model to ensure the continuity of the interpolation speed at the endpoint, wherein the boundary constraint is expressed as:
Figure BDA0003004904440000104
s33: mapping a fifth order polynomial model to arc length parameters of coefficients to be determined
Figure BDA0003004904440000105
To determine the coefficient A ═ a0 a1 a2 a3 a4 a5]TThe normalized parameter sequence Θ is ═ α0 α1 α2 ··· αn-1]Taken as independent variable to obtain fitting parameter sequence
Figure BDA0003004904440000106
Figure BDA0003004904440000111
S34: combining boundary constraint and sampling parameter sequence U ═ U0 u1 u2 ··· un-1]And obtaining a least square fitting equation:
Figure BDA0003004904440000112
s35: QR decomposition of coefficient matrices in boundary constraint equations
Figure BDA0003004904440000113
Solving a least squares fit equation to obtain:
Figure BDA0003004904440000114
wherein A ═ a0 a1 a2 a3 a4 a5]TIs the coefficient of a fitting polynomial and is,
Figure BDA0003004904440000115
a boundary constraint coefficient matrix; PhiQ is [. phi. ]1 Φ2]，QTA＝[B C]T；
S36: and (4) solving a fitting parameter polynomial coefficient A by adopting a Gaussian orthogonal method to obtain an arc length parameter mapping model.
Specifically, after obtaining the sampling arc length and the parameter sequence, in order to obtain the arc length parameter mapping model
Figure BDA0003004904440000116
An arc length parameter fit is established to solve the equation, as shown in figure 5 b.
Normalizing the sampling arc length sequence (Min-Max Normalization) to obtain a sequence theta ═ alpha0α1α2···αn-1]＝[0s1/smax s2/smax···1]Conversion of original arc length parameter mapping model into
Figure BDA0003004904440000117
Secondly, setting the boundary constraint of the arc length parameter mapping model for ensuring the continuity of the interpolation speed at the end point:
Figure BDA0003004904440000121
mapping a fifth-order polynomial model to the arc length parameter of the coefficient to be determined
Figure BDA0003004904440000122
To determine the coefficient A ═ a0 a1 a2 a3 a4 a5]TThe normalized parameter sequence Θ is ═ α0 α1 α2 ··· αn-1]As independent variable to obtain fitting parameter sequence
Figure BDA0003004904440000123
Figure BDA0003004904440000124
Combining the sampling parameter sequence U ═ U0 u1 u2 ··· un-1]Considering the boundary constraint, obtaining a least squares fitting equation lsec (least Square with accuracy constraint) with constraint:
Figure BDA0003004904440000125
wherein A ═ a0 a1 a2 a3 a4 a5]TIs the coefficient of a fitting polynomial and is,
Figure BDA0003004904440000126
the coefficient matrix is constrained for the boundary.
To quickly solve the LSEC model, coefficient matrices in the boundary constraint equations are applied
Figure BDA0003004904440000127
Carrying out QR decomposition, i.e.
Figure BDA0003004904440000128
Converting the least squares fit equation into twoLinearly independent equations, to perform separate solutions:
Figure BDA0003004904440000129
wherein Φ Q ═ Φ1 Φ2]，QTA＝[B C]T；
And finally, solving a fitting parameter polynomial coefficient A by adopting a Gaussian orthogonal method to obtain a fitting polynomial coefficient, namely obtaining an arc length parameter mapping model.
In one embodiment, S4 includes:
s41: arc length s obtained by speed planningkWill be arc length skInputting the arc length parameter mapping model to calculate the spline parameter u of the next interpolation pointk：
Figure BDA0003004904440000131
S42: spline parameter u of the next interpolation pointkThe equation corresponding to the corner smoothing spline curve is substituted to obtain the next interpolation point coordinate pk＝C(uk)。
In one embodiment, the method before S1 further includes:
performing power base conversion on the corner smoothing splines to represent various different corner smoothing splines by adopting a uniform power base; the power-based expression is of the form:
Figure BDA0003004904440000132
wherein U ═ 1Ud]For a set of linearly independent bases of polynomial space, ai(i ═ 0,1, …, d) is the polynomial coefficient, u is the spline parameter, and d is the spline order.
Specifically, the numerical control system may use various splines as internal transition splines, and in order to provide a universal spline real-time interpolation algorithm, it is first required to use a universal expression form for all spline curves. The power-based expression is a general expression of a polynomial curve. Various commonly used spline curves, such as the Hermite curve, Bezier curve, Akima curve, etc., are therefore converted to the power-base expression form c (u) ═ x (u), y (u), z (u)), i.e.:
Figure BDA0003004904440000133
wherein U is [ 1U … U ]d]Is a set of linearly independent bases of polynomial space, ai(i-0, 1, …, d) is a polynomial coefficient.
According to another aspect of the present invention, there is provided a device for real-time interpolation of a corner-smoothed spline trajectory, comprising: the device comprises an establishing module, a calculating module, a solving module and an interpolating module. The device comprises an establishing module, a calculating module and a calculating module, wherein the establishing module is used for establishing a chord height error optimization model corresponding to a corner smoothing spline; the calculation module is used for substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence; the solving module is used for establishing a least square fitting equation by utilizing the sampling parameter sequence and the sampling arc length sequence; solving a least square fitting equation in real time to establish an arc length parameter mapping model; and the interpolation module is used for inputting the spline parameters of the corner smoothing splines into the arc length parameter mapping model and acquiring the spline parameters and coordinates corresponding to the next interpolation point so as to realize real-time interpolation of the track.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (5)

1. A real-time interpolation method for a corner smoothing spline track is characterized by comprising the following steps:
s1: establishing a chord height error optimization model corresponding to the corner smoothing spline;
s2: substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence;
s3: establishing a least square fitting equation by using the sampling parameter sequence and the sampling arc length sequence; solving the least square fitting equation in real time to establish an arc length parameter mapping model;
s4: inputting the spline parameters of the corner smoothing spline into the arc length parameter mapping model, and acquiring spline parameters and coordinates corresponding to a next interpolation point to realize real-time track interpolation;
the S1 includes:
establishing the chord height error optimization model, and changing the distribution of sampling points to reduce the deviation of chord length and arc length; the chord height error optimization model is expressed as:
Figure FDA0003511066030000011
wherein the chord height error optimization model satisfies an upper limit constraint of a real-time requirement, epsiloniA string height error sequence between sampling point sequences; giving an upper limit n of the number of sampling points; with the increase of the sampling cycle times i, the number t (i) of the sampling points is continuously increased until the sampling is stopped when the upper limit n of the number of the sampling points is reached;
the S2 includes:
s21: substituting the chord height error estimation method into the chord height error optimization model to obtain the sampling parameter sequence U and the sampling point sequence P:
Figure FDA0003511066030000012
s22: obtaining a sampling arc length sequence S ═ S by the approximate arc length of the accumulated chord length0 s1 s2 ··· sn-1]Wherein, in the step (A),
Figure DEST_PATH_IMAGE002
the chord height error estimation method in S2 includes:
s201: smoothing two adjacent sampling points p on the spline curvekAnd pk+1Tangent vector V ofkAnd Vk+1Expressed as:
Vk＝αktk Vk+1＝αk+1tk+1；
wherein the corner smoothing spline curve is a plane curve built at the corner of two straight line segments, pk＝C(uk)，pk+1＝C(uk+1)，αkAnd alphak+1Are each pkAnd pk+1Tangent vector mode length of (t)kAnd tk+1Are each pkAnd pk+1Unit tangent of thetakAnd thetak+1Respectively is tangent vector VkAnd Vk+1And chord line vector
Figure FDA0003511066030000021
The included angle of (A);
s202: corresponding parameter value u to the maximum error point of the chord height on the corner smooth spline curvechordAnd estimating to obtain:
Figure FDA0003511066030000022
uk、uk+1are each pkAnd pk+1The chord height error between the two, q is a proportionality coefficient;
s203: using the value u of the parameterchordEstimating to obtain a curve segment
Figure FDA0003511066030000023
Upper chord height error value epsilonchordAnd point p corresponding to the maximum value of the chord height errorchord；
Figure FDA0003511066030000024
2. The method for real-time interpolation of trajectories of corner-smoothing splines as claimed in claim 1, wherein said S3 comprises:
s31: normalizing the sampling arc length sequence to obtain a sequence theta, wherein theta is [ alpha ═ alpha [0 α1 α2 ··· αn-1]＝[0 s1/smax s2/smax ··· 1]；
S32: setting a boundary constraint of the arc length parameter mapping model to ensure the continuity of the interpolation speed at the endpoint, wherein the boundary constraint is expressed as:
Figure FDA0003511066030000031
s33: to determine A ═ a0 a1 a2 a3 a4 a5]TThe normalized parameter sequence Θ ═ α0 α1 α2 ··· αn-1]Arc length parameter mapping quintic polynomial model taken as independent variable into coefficient to be determined
Figure FDA0003511066030000032
To obtain a fitting parameter sequence
Figure FDA0003511066030000033
Figure FDA0003511066030000034
S34: combining the boundary constraint and the sequence of sampling parameters U ═ U0 u1 u2 ··· un-1]And obtaining the least square fitting equation:
Figure FDA0003511066030000035
s35: performing QR decomposition on coefficient matrix in the boundary constraint equation
Figure FDA0003511066030000036
Solving the least squares fit equation to yield:
Figure FDA0003511066030000037
wherein A ═ a0 a1 a2 a3 a4 a5]TIs the coefficient of a fitting polynomial and is,
Figure FDA0003511066030000038
a boundary constraint coefficient matrix; PhiQ is [. phi. ]1Φ2]，QTA＝[B C]T；
S36: and solving a fitting parameter polynomial coefficient A by adopting a Gaussian orthogonal method to obtain the arc length parameter mapping model.
3. The method for real-time interpolation of trajectories of corner-smoothing splines as claimed in claim 2, wherein said S4 comprises:
s41: arc length s obtained by speed planningkThe arc length skInputting the arc length parameter mapping model to calculate spline parameter u of next interpolation pointk：
Figure FDA0003511066030000041
S42: the spline parameter u of the next interpolation point is usedkSubstituting the equation corresponding to the corner smooth spline curve to obtain the next interpolation point coordinate pk＝C(uk)。
4. The method for real-time interpolation of trajectories of corner-smoothing splines as claimed in any one of claims 1 to 3, wherein the method further comprises before S1:
performing power-base conversion on the corner smoothing splines to represent various different corner smoothing splines by adopting a uniform power base; the power-based expression is of the form:
Figure FDA0003511066030000042
wherein, U is [ 1U … U ]d]For a set of linearly independent bases of polynomial space, aiIs a polynomial coefficient, i is 0,1, …, d, u is a spline parameter, d is a spline order.
5. A device for real-time interpolation of a trajectory of a corner-smoothing spline, for performing the method of real-time interpolation of a trajectory of a corner-smoothing spline of claim 1, comprising:
the building module is used for building a chord height error optimization model corresponding to the corner smoothing spline;
the calculation module is used for substituting the chord height error estimation method into the chord height error optimization model to obtain a sampling parameter sequence and a sampling point sequence, and calculating a sampling arc length sequence by using the sampling point sequence;
the solving module is used for establishing a least square fitting equation by utilizing the sampling parameter sequence and the sampling arc length sequence; solving the least square fitting equation in real time to establish an arc length parameter mapping model;
and the interpolation module is used for inputting the spline parameters of the corner smoothing splines into the arc length parameter mapping model and acquiring the spline parameters and coordinates corresponding to the next interpolation point so as to realize real-time interpolation of the track.
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