CN109828535B - NURBS curve interpolation method based on fourth-order Runge-Kutta algorithm - Google Patents
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Abstract
The invention discloses a NURBS curve interpolation method based on a fourth-order Runge-Kutta algorithm, which specifically comprises the following steps: defining a NURBS curve, solving a node vector increment delta u by a fourth-order Runge-Kutta algorithm, calculating an interpolation track of the NURBS curve, and calculating a maximum bow height error constraint step size, a maximum acceleration constraint step size and a step size relative deviation. The method solves the node vector increment delta u by adopting a fourth-order Runge-Kutta algorithm, so that the curve of the node vector increment delta u interpolates the local truncation error gamma of the track calculation result in real timen+1＝Ο(T5) A backward difference method is adopted to replace a complex derivation process, so that the interpolation efficiency can be effectively improved; the method can enable the interpolation error to be in a controllable range by adopting an error control method of a maximum bow height error constraint step length and a maximum acceleration constraint step length. In conclusion, the curve interpolation method provided by the invention can better meet the curve processing requirements of high speed and high precision of the current numerical control machine tool.
Description
Technical Field
The invention relates to the technical field of NURBS curve interpolation calculation, in particular to a NURBS curve interpolation method based on a fourth-order Runge-Kutta algorithm.
Background
With the rapid development of intelligent manufacturing industry in China, the traditional linear and circular interpolation technology of the numerical control machine tool cannot meet the processing requirements of modern high-precision industrial products, and the interpolation technology is gradually replaced by parameterized curve interpolation.
The NURBS curve is a non-uniform rational B-spline free curve that can be accurately represented by corresponding parameters, and is often used in parametric curve interpolation because it can accurately represent free curves and free surfaces in product design and manufacture.
Compared with the traditional interpolation method, the NURBS curve interpolation technology has the advantages of relatively simple interpolation program, high machining efficiency, good part machining surface quality and the like, so that the research on the NURBS curve interpolation is very meaningful in the field of high-precision machining of numerical control machines.
At present, Taylor first-order or second-order expansion approximation algorithm and nonlinear equation method are mostly adopted for NURBS curve interpolation to solve node vector increment, and because the derivation calculation process of the method is relatively complex under the high-precision requirement, and the processing precision of the surface of a part and the interpolation efficiency cannot be well guaranteed.
Disclosure of Invention
The invention aims to provide a NURBS curve interpolation method based on a fourth-order Runge-Kutta algorithm, which can effectively improve the interpolation precision and the interpolation efficiency of a NURBS curve.
The NURBS curve can be accurately represented by corresponding parameters, so that the NURBS curve can accurately represent a free curve in product design and manufacture; the fourth-order Runge-Kutta algorithm is a very common numerical method in solving nonlinear differential equations, and has the advantages of high calculation precision and simple and convenient calculation process; therefore, aiming at the densification process of the parameter points, a classic fourth-order Runge-Kutta algorithm is introduced, the fourth-order Runge-Kutta algorithm is used for solving the value of the node vector increment delta u, and the parameterized curve is interpolated.
Specifically, the fourth order longge-kutta equation is:
wherein:
k1＝f(tn,yn) Formula (2)
k4＝f(tn+h,yn+hk3) Formula (5)
Order node vector u of NURBS curvei＝ynAnd obtaining a node vector u based on a fourth-order Runge-Kutta algorithm by taking the interpolation period T of the NURBS curve as hiThe recurrence equation of (c):
wherein:
in the formula uiA node vector of the ith node on the NURBS curve; t is NURBS curve interpolation period, the specific value is set artificially, and T is Ti+1-tiInterpolating periods for NURBS curves;
next, the above equations (7) to (10) are substituted into the equation (6), and the recurrence equation of the increment Δ u of the node vector is further organized as:
wherein,
+ and
the determination may be made by a higher order backward difference calculation, specifically,
further, the formula (10) may be further organized as:
the equation (16) is the final fourth-order Runge-Kutta algorithm with respect to the node vector uiThe recurrence equation of (c).
Node vector u based on the fourth-order Runge-Kutta algorithmiThe step of the NURBS curve interpolation method based on the fourth-order Runge-Kutta algorithm is as follows:
s1, firstly, acquiring relevant parameters capable of describing and determining a NURBS curve according to the geometric information of the target spline curve drawn in CAD drawing software or CAM drawing software, wherein the relevant parameters comprise: node vector U ═ U0,u1,u2,…,ui,…uj,…,un]Control vertex d ═ d0,d1,d2,…,di,…dj,…,dn]And its weight factor w ═ w0,w1,w2,…,wi,…wj,…,wn]；
S2, converting the node vector uiSubstituting into a calculation formula of the node vector increment delta u based on a fourth-order Runge-Kutta algorithm:
and obtaining a node vector increment delta u, and substituting the node vector increment delta u into a formula: u. ofi+1＝ui+ΔuiIn (1), find its next node vector ui+1；
S3, the next node vector u obtained in the step S2i+1Carry over to cubic NURBS curve equation:
the coordinates of the next interpolation point are obtained:
pi+1＝p(ui+1) Or
Wherein, in the cubic NURBS curve equation, diDenotes the ith control vertex, wiRepresenting a control vertex diWeight of (1), Ni,3Represents the 3 th-order B-spline basis function, B0、B1、B2And B3Respectively, the molecular coefficients; b0、b1、b2、b3Are denominator coefficients respectively;
in the interpolation process, the arch height error of the interpolation curve segment changes along with the change of the curvature of the curve, and the arch height error h is used for restraining the feeding step length delta L in order to reduce the influence of the arch height error on the interpolation resultiBow height error h and feed step length Δ LiAnd the curve curvature rho of the interpolation section is as follows:
in the formula: rho is the curvature of the curve of the interpolation section;
the approximate calculation formula for the arch height can be obtained from the mechanism of generation of the arch height:
in the formula (20), ui+1For the next interpolation of the point node vector uiFor a known interpolation node vector, P (u)i) Is uiCoordinate of the interpolation point of (d), P (u)i+1) Is ui+1The coordinates of the interpolation points;
further, the maximum allowable bow height error is set to hmaxThen the maximum constraint step Δ L allowed by the maximum bow height errori1Comprises the following steps:
further, a maximum feed acceleration constraint a is calculatedmaxI.e. at a specified maximum feed acceleration amaxAnd maximum allowable bow height error hmaxUnder the condition of (1), calculating the maximum restraint step length Delta L determined by the feed accelerationi2：
Wherein, amaxThe specific value is related to the model of the machine used, and is determined by the fact that the machine is used for the process.
The maximum bow height error h is satisfied in the actual processing processmaxThe requirement of constraining the step length is also in accordance with the limitation of the maximum feed acceleration to the interpolation step length and the relation constraint of the period, the speed and the step length in the process of machining the numerical control machine tool, therefore, in the step S3, each time the node vector u of the estimated interpolation point is obtained by calculationi+1Then, calculating the node vector u of the estimated interpolation pointi+1Interpolation point coordinate P (u)i+1) Node vector with previous known interpolation pointuiCoordinate P (u) of interpolation pointi) The spatial distance between the two, namely the actual interpolation step length delta L, is checked to judge whether the delta L meets the condition: Δ L.ltoreq.min (Δ L)i,ΔLi1,ΔLi2) Formula (23).
If Δ L satisfies the condition: Δ L.ltoreq.min (Δ L)i,ΔLi1,ΔLi2) Then determining the coordinates P of the two adjacent interpolation pointsiAnd Pi+1The calculation of (2) meets the error requirement; if the delta L does not satisfy the above condition, selecting delta Li、ΔLi1And Δ Li2The minimum value in the interpolation step length is used as a correction value of the actual interpolation step length delta L, and then a new interpolation point coordinate P is calculatedi+1Substitute the original interpolation point position Pi+1。
Furthermore, in order to monitor Δ L of curve interpolation step length and parameter value estimation step length in real time during interpolation process
The percentage of deviation exists between the two points, the interpolation relative error result is judged by calculating and calculating the step length relative deviation delta, when the step length relative deviation delta value is larger, the larger the deviation between the theoretical step length and the actual step length is, the actual interpolation result is relatively inaccurate, and when the step length relative deviation delta falls into the allowable constraint range, whether the corresponding interpolation point is the effective interpolation point or not is judged; wherein,
the calculation formula of the step relative deviation delta is as follows:
in the formula (24), the reaction mixture is,
the step size is estimated for the parameter values,
when the delta value falls within the allowable rangeWithin a certain constraint range, the corresponding interpolation point is an effective interpolation point, and the coordinate P of the interpolation point isi+1Determining the coordinate of the next interpolation point;
when the delta value exceeds the allowable constraint range, the corresponding interpolation point is an invalid interpolation point, and the coordinates of the interpolation point need to be correspondingly corrected, wherein the specific modification method comprises the following steps:
the known interpolation point node vector u determined by the calculation processiNode vector u of the next interpolation pointi+1Delta L of curve interpolation step length and parameter value estimation step length
Into equation (26):
calculating to obtain a new node vector u after correctioni+1And then calculating to obtain the corresponding interpolation point coordinate to replace the original interpolation point coordinate Pi+1。
And S4, repeating the steps S2-S3 until the interpolation calculation of all the vector nodes U in the step S1 is completed, and further obtaining the interpolation track of the NURBS curve through all the effectively interpolated point coordinates.
The NURBS curve interpolation method based on the fourth-order Runge-Kutta algorithm has the beneficial effects that:
(1) the method is based on a fourth-order Runge-Kutta algorithm to interpolate the NURBS curve, so that the curve interpolates the local truncation error gamma of the track calculation result in real timen+1＝Ο(T5) The high-precision machining requirement is met, and compared with the traditional interpolation method, the high-precision machining method has higher precision theoretically;
(2) the method adopts a backward difference method to replace a complex derivation operation process in the interpolation process, and can effectively improve the curve interpolation efficiency;
(3) according to the method, the feeding step length is restrained through the bow height, the feeding speed and the feeding acceleration in the real-time interpolation track calculation process of the curve, so that the interpolation error falls into a controllable range.
Drawings
FIG. 1 is a flow chart of a NURBS curve interpolation method based on a fourth-order Runge-Kutta algorithm;
FIG. 2 is a NURBS curve generation diagram for a machine tool according to an embodiment of the present invention;
FIG. 3 is a NURBS curve interpolation trajectory diagram of a machine tool according to an embodiment of the present invention;
fig. 4 is a diagram illustrating relative interpolation step size deviations in a NURBS curve interpolation method according to an embodiment of the present invention.
Detailed Description
The invention will be further described with reference to the following figures and specific examples, which are not intended to limit the invention in any way.
As shown in fig. 1, a known NURBS curve is taken as an example to specifically describe a NURBS curve interpolation method based on a fourth-order longge-kutta algorithm:
according to the actual process conditions, the other conditions set in the NURBS curve generation and interpolation simulation processes in the following simulation processes include: the feed speed of the machine tool is set to 18mm/min, the interpolation period T is set to 1ms, and the upper limit h of the bow height errormax＝1μm。
Step one, determining corresponding parameters of a certain known NURBS curve according to the known NURBS curve, and comprising the following steps:
(1) the node vector U is [ 00000.48560.56530.65241111 ],
(2) the control vertex d includes: d0＝(6,6,5),d1＝(26,41,10),d2＝(51,71,17),d3＝(66,76,23),d4＝(78,74,31),d5＝(106,58,33),d6＝(133,41,42)；
(3) Controlling the corresponding weight factor w of the vertex d to be [ 11.2511.30.950.851 ];
these parameters define a spatial three-dimensional free NURBS curve as shown in fig. 2, indicated by line I in fig. 1, and the corresponding curve generated when the node vectors, control vertices and their weights are input to the machine is indicated by line II in fig. 1, so that it is necessary to perform interpolation simulation on the known NURBS curve in order to generate a curve in the machine that is consistent with the known NURBS curve;
step two, the node vector uiSubstituting into a calculation formula of the node vector increment delta u based on a fourth-order Runge-Kutta algorithm:
and obtaining a node vector increment delta u, and substituting the node vector increment delta u into a formula: u. ofi+1＝ui+ΔuiIn (1), find its next node vector ui+1；
Step three, the next node vector u obtained in the step S2i+1Carry over to cubic NURBS curve equation:
and obtaining the coordinates of the next interpolation point: p is a radical ofi+1＝p(ui+1) Or
Wherein, in the cubic NURBS curve equation, diDenotes the ith control vertex, wiRepresenting a control vertex diWeight of (1), Ni,3Representing a cubic B-spline basis function, B0、B1、B2And B3Respectively, the molecular coefficients; b0、b1、b2、b3Are denominator coefficients respectively;
in the completion step S3, the positions P of two adjacent interpolation points are determinediAnd Pi+1The spatial distance between the two, namely the actual interpolation step length delta L, is checked, and whether the delta L meets the condition is judged: Δ L.ltoreq.min (Δ L)i,ΔLi1,ΔLi2)；
Wherein, Δ LiTo allow for a feed step size with the bow height error h,
in the formula, rho is the curvature of the curve of the interpolation section;
wherein u isiFor a known interpolation node vector, ui+1Is the next interpolation point node vector, P (u)i) Is uiCoordinate of the interpolation point of (d), P (u)i+1) Is ui+1The coordinates of the interpolation points; Δ Li1To be at the maximum bow height error hmaxThe maximum constraint step size allowed under (c) is,
ΔLi2at the maximum feed acceleration amaxThe maximum constraint step size allowed under (c) is,
if Δ L satisfies the condition: Δ L.ltoreq.min (Δ L)i,ΔLi1,ΔLi2) Then determining the coordinates P of the two adjacent interpolation pointsiAnd Pi+1The calculation of (2) meets the error requirement; if the delta L does not satisfy the above condition, selecting delta Li、ΔLi1And Δ Li2The minimum value in the interpolation step length is used as a correction value of the actual interpolation step length delta L, and then a new interpolation point coordinate P is calculatedi+1Substitute the original interpolation point coordinate Pi+1。
Then, after checking the actual interpolation step length delta L, calculating the step length relative deviation delta and judging whether the step length relative deviation delta is within an allowable deviation range so as to determine whether the interpolation point is an effective interpolation point; the calculation formula of the step length relative deviation delta is as follows:
in the formula:
the step size is estimated for the parameter values,
when the delta value falls into the allowable constraint range, the corresponding interpolation point is an effective interpolation point, and the coordinate P of the interpolation pointi+1Determining the coordinate of the next interpolation point; when the delta value exceeds the allowable constraint range, the corresponding interpolation point is an invalid interpolation point, and furtherThe known interpolation point node vector u determined by the calculation processiNode vector u of the next interpolation pointi+1Delta L of curve interpolation step length and parameter value estimation step length
Substituting into the formula:
in the method, a corrected new node vector is obtained by calculation
And then calculating to obtain the corresponding interpolation point coordinate Pi+1。
And step four, repeating the steps S2-S3 until the interpolation calculation of all the vector nodes U in the step S1 is completed, and further obtaining the interpolation track of the NURBS curve through all the effectively interpolated point coordinates.
In the NURBS curve interpolation trajectory simulation of the embodiment, MATLAB is used for simulation operation and data theoretical analysis is performed on simulation result parameter values in a working area of the MATLAB, so as to further verify the feasibility and the rationality of the NURBS curve interpolation algorithm.
The partial simulation interpolation result data is shown in table 1.
Table 1:
| number of steps | u | x(mm) | y(mm) | z(mm) | ΔL(mm) |
| 1 | 0.000000 | 6.000000 | 6.000000 | 5.000000 | 0.000000 |
| 2 | 0.000000 | 6.000004 | 6.000007 | 5.000001 | 0.000008 |
| 3 | 0.000000 | 6.000032 | 6.000055 | 5.000008 | 0.000056 |
| 4 | 0.000001 | 6.000106 | 6.000186 | 5.000027 | 0.000152 |
| 5 | 0.000002 | 6.000252 | 6.000441 | 5.000063 | 0.000296 |
| 6 | 0.000003 | 6.000492 | 6.000862 | 5.000123 | 0.000488 |
| 7 | 0.000006 | 6.000851 | 6.001489 | 5.000213 | 0.000728 |
| 8 | 0.000009 | 6.001351 | 6.002364 | 5.000338 | 0.001016 |
| 9 | 0.000013 | 6.002017 | 6.003529 | 5.000504 | 0.001352 |
| 10 | 0.000019 | 6.002871 | 6.005025 | 5.000718 | 0.001736 |
| … | … | … | … | … | … |
| 750 | 0.999813 | 132.962900 | 41.023358 | 41.987638 | 0.013256 |
| 751 | 0.999863 | 132.972798 | 41.017127 | 41.990935 | 0.012152 |
| 752 | 0.999908 | 132.981835 | 41.011437 | 41.993946 | 0.011096 |
| 753 | 0.999950 | 132.990052 | 41.006264 | 41.996684 | 0.010088 |
| 754 | 0.999987 | 132.997486 | 41.001583 | 41.999162 | 0.009128 |
| 755 | 1.000000 | 133.000000 | 41.000000 | 42.000000 | 0.000000 |
The relevant parameter values for checking the interpolation step length l (mm), such as the interpolation step length l (mm), the step length error δ (%), the bow height error h (mm), and the like, generated in the real-time interpolation process are respectively calculated according to the simulation interpolation result data shown in the table 1.
The calculation results are shown in table 2 below.
Table 2:
| number of steps | ΔL,mm | h,mm | ΔLi,mm | ΔLi1,mm | ΔLi2,mm | δ,％ |
| 1 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 2 | 0.000008 | 0.000000 | 0.000008 | 0.694236 | 0.543326 | 0.000000 |
| 3 | 0.000056 | 0.000000 | 0.000056 | 0.693330 | 0.542616 | 0.000000 |
| 4 | 0.000152 | 0.000000 | 0.000152 | 0.693338 | 0.542622 | 0.000001 |
| 5 | 0.000296 | 0.000000 | 0.000296 | 0.693334 | 0.542620 | 0.000001 |
| 6 | 0.000488 | 0.000000 | 0.000488 | 0.693340 | 0.542624 | 0.000001 |
| 7 | 0.000728 | 0.000000 | 0.000728 | 0.693342 | 0.542626 | 0.000002 |
| 8 | 0.001016 | 0.000000 | 0.001016 | 0.693343 | 0.542627 | 0.000002 |
| 9 | 0.001352 | 0.000000 | 0.001352 | 0.693346 | 0.542629 | 0.000002 |
| 10 | 0.001736 | 0.000000 | 0.001736 | 0.693349 | 0.542632 | 0.000002 |
| … | … | … | … | … | … | … |
| 750 | 0.013256 | 0.000000 | 0.013256 | 0.880929 | 0.689436 | 0.000004 |
| 751 | 0.012152 | 0.000000 | 0.012152 | 0.880614 | 0.689189 | 0.000004 |
| 752 | 0.011096 | 0.000000 | 0.011096 | 0.880325 | 0.688963 | 0.000004 |
| 753 | 0.010088 | 0.000000 | 0.010088 | 0.880062 | 0.688758 | 0.000003 |
| 754 | 0.009128 | 0.000000 | 0.009128 | 0.879824 | 0.688571 | 0.000003 |
| 755 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
According to the calculation result of table 2, it is determined whether the calculation result in table 1 calculated in step S3 meets the limitation requirement of the relevant parameters in table 2, and the interpolation point coordinates are modified accordingly, and the simulated NURBS spline interpolation trajectory shown in fig. 3 is finally obtained through 755-step real-time interpolation simulation of the known NURBS curve.
As can be seen from the calculation results in tables 1 and 2 and FIG. 4, the maximum value of the step length L in the interpolation process of the NURBS curve obtained by the NURBS curve interpolation method based on the fourth-order Runge-Kutta algorithm is 0.3mm, and the minimum value is 0.8 × 10-5mm; the minimum value and the maximum value of the step error delta are respectively 0.0005 percent and 0.0972 percent, and the requirement that the maximum allowable step error delta is less than 0.1 percent is met; the maximum value of the bow height error h is 6.11 multiplied by 10-4mm, and meets the requirement that the maximum allowable bow height error h is less than 1 mu m. Therefore, the NURBS curve interpolation method based on the fourth-order Runge-Kutta algorithm can meet the requirement of machining precision and greatly improve machining efficiency, and therefore the NURBS curve interpolation method based on the fourth-order Runge-Kutta algorithm achieves the multi-aspect optimization effect on interpolation precision and interpolation efficiency.
Claims (1)
1. A NURBS curve interpolation method based on a fourth-order Runge-Kutta algorithm is characterized by comprising the following steps:
s1, determining a node vector U of the NURBS curve in the trajectory calculation formula according to the geometric information of the known spline curve [ U [ ]0,u1,u2,…,ui,…uj,…,un]Control vertex d ═ d0,d1,d2,…,di,…dj,…,dn]And its weight factor w ═ w0,w1,w2,…,wi,…wj,…,wn]；
S2, converting the node vector uiSubstituting into a calculation formula of the node vector increment delta u based on a fourth-order Runge-Kutta algorithm:
and obtaining a node vector increment delta u, and substituting the node vector increment delta u into a formula: u. ofi+1＝ui+ΔuiIn (1), find its next node vector ui+1；
S3, the next node vector u obtained in the step S2i+1Carry over to cubic NURBS curve equation:
and obtaining the coordinates of the next interpolation point: p is a radical ofi+1＝p(ui+1) Or
Wherein, in the cubic NURBS curve equation, diDenotes the ith control vertex, wiRepresenting a control vertex diWeight of (1), Ni,3Represents the 3 th-order B-spline basis function, B0、B1、B2And B3Respectively, the molecular coefficients; b0、b1、b2、b3Are denominator coefficients respectively;
and S4, repeating the steps S2-S3 until the interpolation calculation of all the vector nodes U in the step S1 is completed, and further obtaining the interpolation track of the NURBS curve through all the effective interpolation point coordinates.
In step S3, the positions P of two adjacent interpolation points are determinediAnd Pi+1The spatial distance between the two, namely the actual interpolation step length delta L, is checked, and whether the delta L meets the condition is judged: Δ L.ltoreq.min (Δ L)i,ΔLi1,ΔLi2) (ii) a Wherein, Δ LiTo allow for a feed step size with the bow height error h,
in the formula, rho is the curvature of the curve of the interpolation section;
wherein u isiFor a known interpolation node vector, ui+1Is the next interpolation point node vector, P (u)i) Is uiCoordinate of the interpolation point of (d), P (u)i+1) Is ui+1The coordinates of the interpolation points; Δ Li1To be at the maximum bow height error hmaxThe maximum constraint step size allowed under (c) is,
ΔLi2at the maximum feed acceleration amaxThe maximum constraint step size allowed under (c) is,
if Δ L satisfies the condition: Δ L.ltoreq.min (Δ L)i,ΔLi1,ΔLi2) Then determining the coordinates P of the two adjacent interpolation pointsiAnd Pi+1The calculation of (2) meets the error requirement; if the delta L does not satisfy the above condition, selecting delta Li、ΔLi1And Δ Li2The minimum value in the interpolation step length is used as a correction value of the actual interpolation step length delta L, and then a new interpolation point coordinate P is calculatedi+1Substitute the original interpolation point coordinate Pi+1；
After checking the actual interpolation step length delta L, calculating the step length relative deviation delta and judging whether the step length relative deviation delta is within an allowable deviation range so as to determine whether the interpolation point is an effective interpolation point; the calculation formula of the step length relative deviation delta is as follows:
in the formula:
the step size is estimated for the parameter values,
when the delta value falls into the allowable constraint range, the corresponding interpolation point is an effective interpolation point, and the coordinate P of the interpolation pointi+1Determining the coordinate of the next interpolation point;
when the delta value exceeds the allowable constraint range, the corresponding interpolation point is an invalid interpolation point, and the known interpolation point node vector u determined through the calculation processiNode vector u of the next interpolation pointi+1Δ L of the curve interpolation step length,And parameter value pre-estimation step length
Substituting into the formula:
in the method, a corrected new node vector is obtained by calculation
And then calculating to obtain the corresponding interpolation point coordinate Pi+1。
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