CN108227630B - Free-form surface numerical control machining method adopting time parameter polynomial interpolation - Google Patents

Free-form surface numerical control machining method adopting time parameter polynomial interpolation Download PDF

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CN108227630B
CN108227630B CN201810049584.XA CN201810049584A CN108227630B CN 108227630 B CN108227630 B CN 108227630B CN 201810049584 A CN201810049584 A CN 201810049584A CN 108227630 B CN108227630 B CN 108227630B
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acceleration
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卢磊
韩江
夏链
董方方
刘海军
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Hefei University of Technology
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Abstract

The invention relates to a free-form surface numerical control machining method adopting time parameter polynomial interpolation. The method specifically comprises the following three steps: predicting prospective speed planning, generating a time parameter polynomial processing track based on B-spline and interpolating the time parameter polynomial in real time; (1) speed planning is carried out by reading a parameter curve type free-form surface processing track, and high-speed and high-precision discrete data of the free-form surface processing process are generated; 2) representing the discrete data of the high-speed and high-precision machining process after speed planning by adopting a B-spline fitting technology with given precision, and then converting the discrete data into a simple incremental time parameter polynomial surface machining track; (3) and compiling a corresponding real-time interpolation program according to the generated time parameter polynomial curved surface machining track to carry out real-time control on the numerical control machine tool so as to complete the free curved surface machining. The invention improves the contour precision and the surface quality of the processed free-form surface, obviously improves the processing efficiency of the free-form surface and reduces the data amount required by the free-form surface processing.

Description

Free-form surface numerical control machining method adopting time parameter polynomial interpolation
Technical Field
The invention belongs to the technical field of numerical control machining, and particularly relates to a free-form surface numerical control machining method, which can perform speed planning in an off-line mode according to a parameter curve machining track, generate concise curved surface machining data containing curved surface machining geometry and speed information, and develop a corresponding real-time on-line time parameter polynomial interpolation algorithm.
Background
With the development of aerospace and optics fields, the processing requirements of free-form surface parts such as aero-engine blades, optical lenses and the like are higher and higher. But free-form surfaces are generally characterized using a large number of point clouds or a large number of piecewise equations, as compared to regular surfaces such as spheres, quadrics, etc., which can be represented by equations. This greatly increases the complexity of the numerical control machining of the free-form surface. In the current production practice, the free-form surface processing still adopts a large number of small line segment discrete tracks, and then the free-form surface processing is realized through on-line fairing and speed planning. But the strict control of the online real-time calculation amount causes that the online fairing and speed planning can not realize good high-speed high-precision free-form surface processing. The disclosed small line segment real-time smooth transition interpolation method for high-speed high-precision numerical control machining adopts three times of B-spline to perform transition on a small line segment track and adopts an S acceleration and deceleration method to plan speed, but the transition precision is difficult to control, and the S acceleration and deceleration planning method is difficult to achieve the optimality of speed planning.
Therefore, in the free-form surface machining, the numerical control machining method of the parameter curve track with continuous curvature is widely applied. However, since the curve parameters and the curve arc length generally do not have a explicit function relationship, the speed planning and real-time interpolation process of the parametric curve processing is limited, so that the numerical control processing of the free-form surface of the parametric curve processing track is difficult to popularize and apply in production practice.
The prior art discloses a method for solving the problems of applying a parameter curve to the free-form surface processing, and researches an offline speed planning (Beudaert X, Lavernhe S, Tournier C (2012) Feed optimization With axis relation constraints on 5-axis NURBS and G1tool path. IntJ Mach Tools Manual 57: 73-82.) and a parameter Interpolation (Erkorkmaz K, Altintas Y (2005) Quintspling Interpolation With minimum Feed calculation. J Manual Sci Eng 127:339.) respectively by adopting the parameter curve processing track. However, the speed planning and interpolation methods are not considered cooperatively, so that the high-speed and high-precision machining process finished by the speed planning is difficult to ensure in the parameter interpolation process. And ensuring that parameter interpolation controls machine motion according to an offline speed planning process requires a large amount of redundant data.
The disclosed look-ahead interpolation system for compressing and smoothing small line segment paths is mainly characterized in that B-spline is adopted to fit small line segment tracks, an S-shaped look-ahead speed planning method and a plurality of interpolation methods are adopted. However, how to apply the result of velocity planning to the proposed interpolation methods with high accuracy is not described.
Disclosure of Invention
The invention provides a free-form surface numerical control machining method adopting time parameter polynomial interpolation, aiming at the complexity of the problem of free-form surface numerical control machining adopting a parameter curve track and the defects of the prior art. The method cooperatively considers the characteristics of online real-time calculation and offline calculation, realizes speed planning and processing data generation by adopting the offline calculation, and improves the optimality of speed planning and the simplicity of generated data. The online real-time interpolation only adopts the polynomial interpolation of time parameters, thereby reducing the calculation complexity of the real-time interpolation and improving the robustness of the real-time calculation. The application of the parametric type free-form surface processing track in production practice is promoted.
The invention is realized by the following technical scheme:
a free-form surface numerical control machining method adopting time parameter polynomial interpolation is suitable for a numerical control machine tool to machine a free-form surface, the numerical control machine tool comprises a real-time machine tool controller and a machine tool body, and the machine tool body comprises a workbench and various motion axes. The method comprises three operation steps:
(1) predictive look-ahead velocity planning
Generating high-speed and high-precision discrete data of the free-form surface processing process by reading a parameter curve type free-form surface processing track and carrying out speed planning;
(2) b-spline based time parameter polynomial machining trajectory generation
Representing the discrete data generated in the step (1) in the free-form surface processing process by adopting a B-spline fitting technology with given precision, and then converting the discrete data into a simple incremental time parameter polynomial surface processing track;
(3) time parameter polynomial real-time interpolation
According to the incremental time parameter polynomial curved surface machining track, compiling a corresponding real-time interpolation algorithm in a real-time machine tool controller, and controlling the machine tool to move to finish high-speed and high-precision machining of a free curved surface;
by the numerical control processing method for the free-form surface, the contour accuracy and the surface quality of the processed free-form surface are improved, the processing efficiency of the free-form surface is obviously improved, and the data volume required by the processing of the free-form surface is reduced.
The technical scheme for further limiting is as follows:
a free-form surface numerical control machining method adopting time parameter polynomial interpolation comprises the following specific operation steps:
(1) predictive look-ahead velocity planning
(1.1) extracting the highest-order three-time speed planning kinematic constraint according to the machining precision requirement and the operation capacity of machining equipment, reducing and simplifying the kinematic constraint by adopting a triangle inequality and a method of introducing a proportionality coefficient, and converting all the constraints into one-dimensional constraints related to curve parameters;
the speed planning kinematics constraint comprises constraint of each motion axis of the machine tool and constraint of a processing technology;
(1.2) carrying out early deceleration on track points violating one-dimensional constraints by adopting a prediction look-ahead method along a processing track, and accelerating for a time step if no track points violating one-dimensional constraints exist in the look-ahead process; according to a discrete calculation method based on a minimum value principle, calculating the jerk by using the highest-order constraint, and then obtaining discrete data of the curved surface machining process which meets the constraint and has the optimal machining efficiency through numerical integration calculation;
(2) b-spline based time parameter polynomial machining trajectory generation
(2.1) fitting by adopting B-spline of a time parameter according to discrete data of a curved surface machining process finished by speed planning, solving a B-spline node vector by adopting a node prediction method with given precision in the fitting process, and solving a control point by adopting a least square method; finally, characterizing the curved surface machining process by using the B-spline of the time parameter to obtain a B-spline machining track;
(2.2) converting the B-spline processing track into an incremental time parameter polynomial surface processing track by adopting a B-spline-polynomial conversion algorithm (converting B-spline into a polynomial);
(3) time parameter polynomial real-time interpolation algorithm
And (3) compiling a time parameter polynomial real-time interpolation program in a real-time machine controller, and controlling the motion of the numerical control machine tool through the real-time interpolation program according to the incremental time parameter polynomial curved surface machining track obtained in the step (2.2) to realize high-speed and high-precision machining of the free curved surface.
The technical scheme is further defined as follows:
in the step (1.1), the constraint of each motion axis of the machine tool is simplified by introducing a proportionality coefficient and adopting a triangle inequality method:
the machine tool motion axis constraint is summarized as follows
Figure BDA0001551984000000031
In the formula (I), the compound is shown in the specification,
Figure BDA0001551984000000032
representing the speed, acceleration and jerk of the ith axis;
Figure BDA0001551984000000033
constraints representing speed, acceleration and jerk of the ith axis;
converting multidimensional constraints of acceleration and jerk of a motion axis into one-dimensional constraints related to curve parameters by introducing a proportionality coefficient and adopting a triangle inequality method; the acceleration constraint processing formula is as follows:
Figure BDA0001551984000000034
Figure BDA0001551984000000035
the jerk constraint processing formula is as follows:
Figure BDA0001551984000000036
Figure BDA0001551984000000041
in the formula1And2is the proportional coefficient of acceleration, mu1,μ2And mu3The acceleration rate is a proportional coefficient of acceleration, and the proportional coefficient is selected according to characteristics of a machine tool and a track; in the formula
Figure BDA0001551984000000042
And
Figure BDA0001551984000000043
velocity, acceleration and jerk of the trajectory space, respectively; in the formula qs,qss,qsssThe first derivative, the second derivative and the third derivative of the curve parameter to the curve arc length are respectively.
In the step (1.2), the track points violating the constraint in the future are decelerated in advance by adopting a prediction look-ahead method, and the specific operation is as follows:
the prediction forward-looking deceleration process comprises the steps of firstly decelerating from a current point by using the allowed minimum jerk, reducing the acceleration to zero, then continuing to decrement the acceleration by using the minimum acceleration, and incrementing the acceleration to zero by using the maximum jerk every time the acceleration is decremented by one step until the speed is decremented to zero; respectively checking the speed, the acceleration and the jerk constraint in the process;
if the track point violates the speed and acceleration constraints in the stage that the initial acceleration is decreased to zero, the point can be set as a key track point violating the constraints; the processing track needs to be decelerated from an initial track point to a violation constraint key track point, and the deceleration mode is decelerated according to a prediction look-ahead search method;
in the acceleration continuous decreasing stage, if the track point violates the acceleration constraint, the jerk at the point violating the track point is recalculated according to the acceleration constraint; after the acceleration is obtained, then the deceleration algorithm is predicted to continue; if a track point violates the speed constraint, the point can be set as a key track point violating the speed constraint; the processing track needs to be decelerated from an initial track point to a key track point violating the speed constraint, and the deceleration mode is decelerated according to a predictive look-ahead search method;
in the process of acceleration increasing to zero, if the track point violates the acceleration constraint, the acceleration increasing process needs to be stopped; the deceleration prediction process decelerates to the point, the prediction look-ahead process is continuously executed from the point until the acceleration and the speed are reduced to zero, and each constraint is continuously detected;
in the process of acceleration increasing to zero, if the track point violates the speed constraint, the acceleration increasing process is stopped, and then the acceleration decreasing process is carried out; if the speed still violates the speed constraint when the point violating the speed track is reached, the process is circularly carried out; if the speed meets the constraint requirement when the point is reached, the point is the key point of the track, the processing track of the key point of the track needs to be decelerated from the initial track point to the key track point violating the constraint, and the deceleration mode decelerates according to a prediction look-ahead search method.
In the step (2.1), a node prediction method with given precision is adopted to obtain a fitting B-spline node vector, and the specific determination process is as follows:
and fitting the position data { t, p } of each motion axis along with the time for n times of B-spline, wherein the fitting error can be estimated by adopting the following formula:
Figure BDA0001551984000000051
where for the estimated n B-spline fitting errors,
Figure BDA0001551984000000052
is a vector of maximum displacement of each axis versus time derivative value of degree n,
Figure BDA0001551984000000053
forming a vector by the minimum displacement of each axis and the derivative value of time n times, wherein delta t is a node interval of time parameter n times B-spline;
according to the estimated fitting error, under the given error, the node vector is determined by the following algorithm:
step a: given node vector numberOne node value is the time T of the first position of the fitted original data1＝t1，i＝1，j＝1；
Step b: i is i +1, and judging whether i is equal to the original discrete data quantity, if so, turning to the step d; when not equal, executing step c in sequence;
step c: calculating the interval [ Tj,ti]B-spline fitting error n times, judging whether the fitting error is more than or equal to a given fitting error, and if the fitting error is more than or equal to the given fitting error, determining the next node vector as Tj+1＝tiAnd j is j + 1; turning to the step b;
step d: let node vector Tj+1And (5) ending the node vector calculation when the time is equal to the end time of the original data.
In the step (2.2), the algorithm for converting the time parameter B-spline machining track into the incremental time parameter polynomial surface machining track is as follows:
for p times the time parameter B-spline is:
Figure BDA0001551984000000054
for an incremental p-th order polynomial:
P(p)(t)＝a0+a1·t+a2·t2+a3·t3…+an·tp,0≤t≤T
A＝[a0,a1,…an]
P(p)(t)＝A·λTλ＝[1,t,t2,…tp]
for the odd time parameter B-spline, in the node interval ti,ti+1]The polynomial algorithm for converting to the incremental time parameter is as follows:
the time parameter B-spline is used for node interval ti,ti+1]Calculating function values and corresponding derivative values at corresponding end points
Figure BDA0001551984000000061
For p-th B-spline, Angstrom can be taken from the L vectorThe coefficient vector A and the time variable interval T of the p-th-order polynomial are directly obtained through the Hermite interpolation.
For the even time parameter B-spline, in the node interval ti,ti+1]1,3, 5.. the conversion to incremental time parameter polynomial algorithm is:
the time parameter B-spline is used for node interval ti,ti+1]Calculating function values and corresponding derivative values at corresponding end points
Figure BDA0001551984000000062
For the P-th B-spline, the coefficient vector A and the time variable interval T of the p-th polynomial can be directly obtained by the L vector by adopting Hermite interpolation;
for the even time parameter B-spline, in the node interval ti+1,ti+2]1,3, 5.. the conversion to incremental time parameter polynomial algorithm is:
the time parameter B-spline is used for node interval ti+1,ti+2]Calculating function values and corresponding derivative values at corresponding end points
Figure BDA0001551984000000063
For the p-degree B-spline, the coefficient vector A and the time variable interval T of the p-degree polynomial can be directly obtained from the L vector by adopting Hermite interpolation.
In the step (3), the time parameter polynomial real-time interpolation algorithm is as follows:
according to the incremental segmented time parameter polynomial curved surface processing track generated in the step (2), the coefficient vector of the ith segment of polynomial is AiSetting the total processing time at this moment as T and the polynomial time interval of each time parameter as Tj；
Then the time variable
Figure BDA0001551984000000064
Comprises the following steps:
Figure BDA0001551984000000065
variable vector lambdaiComprises the following steps:
Figure BDA0001551984000000066
the motion position calculation formula of each motion axis obtained by interpolation at the time is as follows:
P(p)＝Ai·λi T
and according to the interpolation algorithm and the total processing time T, the positions of all the motion axes of the machine tool at the moment can be obtained, and the free-form surface processing is completed by processing in sequence.
The beneficial technical effects of the invention are embodied in the following aspects:
(1) the numerical control machining method of the free-form surface with the cooperation of off-line calculation and real-time on-line is adopted, so that the calculation precision is improved, and the on-line calculation amount is reduced. The advantages of off-line calculation and real-time calculation are fully exerted.
(2) By adopting an off-line prediction prospective speed planning method, a curved surface processing process with better processing efficiency than that of an S-type acceleration and deceleration method can be obtained, and the constraint can be strictly ensured, according to the embodiment, under the same constraint, the speed loss rate of the invention is about 70%, while the speed loss rate of the invention is over 85% by adopting a commercial S-type acceleration and deceleration method;
(3) the incremental time parameter polynomial curved surface processing track is generated based on a given precision B-spline fitting algorithm, the speed planning data can be represented under the given precision, the generated representation time parameter polynomial reduces the data volume after speed planning, the data compression volume is more than 80%, and the generated incremental time parameter polynomial can effectively reduce the accumulated error;
(4) by adopting the time parameter polynomial interpolation method, a large amount of complex parameter curve geometric calculation is avoided, the real-time calculation is simple, the real-time calculation precision is higher, and the real-time calculation robustness is good.
Drawings
FIG. 1 is a flow chart of the present invention.
Fig. 2 is a schematic diagram of a dental prosthesis crown parametric type trajectory machining path in example 1.
Fig. 3a is a graph of acceleration.
Fig. 3b is a velocity profile.
FIG. 4 is a schematic diagram illustrating an acceleration incrementing process when an acceleration violates a constraint.
Fig. 5 is a graphical representation of the dental crown speed of a denture employing example 1 of the inventive arrangement.
Fig. 6X-axis velocity planning results, polynomial fitting results and fitting error schematic of denture crowns according to example 1.
Fig. 7 is a schematic diagram of the Y-axis velocity planning, polynomial fit, and fit error for a dental prosthesis crown according to example 1.
Fig. 8 is a Z-axis velocity planning result, a polynomial fitting result and a fitting error diagram of a dental prosthesis crown according to example 1.
Fig. 9 is a result graph of a free-form surface part of a dental crown of a dental prosthesis according to example 1.
Fig. 10 is a schematic diagram of a parametric type trajectory processing path of a denture root in example 2.
Fig. 11 is a schematic diagram of the tooth root velocity of a denture using example 2 of the inventive planning.
Fig. 12 is a diagram illustrating X-axis velocity planning results, polynomial fitting results and fitting errors of the denture roots in example 2.
Fig. 13 is a schematic diagram of the Y-axis velocity planning results, polynomial fitting results and fitting errors for the denture roots of example 2.
Fig. 14 is a schematic diagram of the Z-axis velocity planning results, polynomial fitting results and fitting errors of the denture roots in example 2.
Fig. 15 is a graph showing the results of processing the free-form surface part of the tooth root of the denture in example 2 using the present invention.
Detailed Description
The invention will now be further described by way of example with reference to the accompanying drawings. The following examples will assist those skilled in the art in further understanding the invention, but are not intended to limit the invention in any way. It should be noted that variations and modifications can be made by persons skilled in the art without departing from the spirit of the invention. All falling within the scope of the present invention. The following detailed description of the embodiments of the present invention is provided with reference to the accompanying drawings, but the scope of the present invention is not limited to the following embodiments.
As shown in fig. 1, the specific process of the free-form surface numerical control machining method using time parameter polynomial interpolation includes: firstly, speed planning is carried out according to a parameter type curved surface processing track designed according to a processing curved surface and a cutter shape, such as NURBS and the like. The optimal goal of speed planning is to minimize the machining time, and constraints are various constraints proposed according to the machining precision and the machining equipment capacity. After a speed planning mathematical model is built, the constraint is simplified by adopting a triangle inequality and a method of introducing a proportionality coefficient, so that the constraint is only related to a machining track parameter, and then a prediction look-ahead algorithm based on a minimum principle is adopted for speed planning. After the speed planning is finished, a large amount of discrete data is obtained, and real-time interpolation application is inconvenient. In order to solve the problem, the invention provides a time parameter polynomial processing track generation method based on B-spline. The method comprises the steps of firstly representing a speed planning result by a B-spline fitting method with given precision, and then converting the B-spline into an incremental time parameter polynomial. And finally, inputting the free-form surface processing data containing the geometric and speed information and characterized by the incremental time parameter polynomial into a real-time parameter polynomial interpolation algorithm, and controlling the machine tool to operate according to the geometric track and the planned speed through real-time interpolation. And high-speed and high-precision free-form surface machining is realized.
Example 1:
the numerical control machine tool used in the embodiment comprises a real-time machine tool controller and a machine tool body, wherein the machine tool body comprises a workbench and three motion shafts.
1. Experiments were performed using a parametric type dental crown curve machining trajectory as shown in fig. 2. Firstly, the invention is adopted to carry out speed planning on the section of track:
(1) speed planning model
Figure BDA0001551984000000081
Figure BDA0001551984000000082
Figure BDA0001551984000000083
In the formula
Figure BDA0001551984000000084
And
Figure BDA0001551984000000085
is the velocity, acceleration and jerk of the trajectory space; flim,Alim,JlimFor velocity, acceleration and jerk constraints of the trajectory space,
Figure BDA0001551984000000086
representing the speed, acceleration and jerk of the ith axis;
Figure BDA0001551984000000091
representing constraints on speed, acceleration and jerk for the ith axis.
In this embodiment, according to the machining accuracy and the equipment characteristics, the constraint conditions are selected as follows:
TABLE 1 kinematic constraints for each axis of a machine tool
Figure BDA0001551984000000092
TABLE 2 kinematic constraints along the track
Figure BDA0001551984000000093
(2) Constraint simplification
The motion axis speed constraint processing formula is as follows:
Figure BDA0001551984000000094
Figure BDA0001551984000000095
the motion axis acceleration constraint processing formula is as follows:
Figure BDA0001551984000000096
Figure BDA0001551984000000097
the constraint processing formula of the jerk of the motion axis is as follows:
Figure BDA0001551984000000098
Figure BDA0001551984000000099
in the formula1And2is the proportional coefficient of acceleration, mu1,μ2And mu3The scaling factor is selected according to the characteristics of the machine tool and the track. In the formula qs,qss,qsssThe first, second and third derivatives of the curve parameter versus the curve arc length.
TABLE 3 constrained conversion factor
Figure BDA0001551984000000101
(3) Predictive look-ahead velocity planning
After constraint simplification, only the constraint related to the trajectory parameter exists. And (4) performing speed planning on the curved surface machining process by adopting a look-ahead prediction method. The look-ahead process is shown in fig. 3a and 3b, and starts from the current point, decelerates with the minimum allowable jerk, reduces the acceleration to zero, and then continues to decrement the acceleration with the minimum acceleration, and increments the acceleration with the maximum jerk to zero every step until the speed is reduced to zero. In this process, the velocity, acceleration and jerk constraints are checked separately.
In the acceleration continue decrement phase, if a track point violates an acceleration constraint, the jerk at the violated track point is recalculated in accordance with the acceleration constraint. After the jerk is obtained, then the predictive deceleration algorithm continues. If a track point violates a speed constraint, that point can be set to violate a constraint key track point. The processing track needs to be decelerated from an initial track point to a violation constraint key track point, and the deceleration mode is decelerated according to a prediction look-ahead search method.
During acceleration ramp up to zero, if there are trace points violating the acceleration constraint, the acceleration ramp up process needs to stop as shown in FIG. 4. The deceleration prediction process decelerates to this point, from which point the prediction look-ahead process continues until both the acceleration and the velocity are reduced to zero, and each constraint is continually detected.
During the acceleration increasing process to zero, if the track points violate the speed constraint, the acceleration increasing process is stopped, and then the acceleration decreasing process is carried out. If the speed still violates the speed constraint when the point of violating the speed trajectory is reached, the process loops. If the speed meets the constraint requirement when the point is reached, the point is the key point of the track, the processing track of the key point of the track needs to be decelerated from the initial track point to the key track point violating the constraint, and the deceleration mode decelerates according to a prediction look-ahead search method.
The results of the velocity planning are shown in FIG. 5, where the average velocity was 10.99mm/s and the velocity loss rate was 70%. The continuity of the speed profile can be seen from the enlarged partial view. However, the speed planning result is a large amount of discrete data, the data amount is one hundred thousand data points, the dimensionality of each data point is four, and the time interval is 1 ms. To reduce the amount of data, the utility of speed planning is improved. The invention adopts a time parameter polynomial based on B-spline to represent the relation of the position of each shaft along with the change of time. The data size after the third-order polynomial representation is about ten thousand, and the dimensionality of each data point is seven. The amount of data compression is 80%.
2. Generating a time parameter polynomial processing track based on B-spline:
in the B-spline fitting process, preferably, a node prediction method with given precision is used to obtain a fitting B-spline node vector, and the specific determination process is as follows:
and fitting the position data { t, p } of each motion axis along with the time for n times of B-spline, wherein the fitting error can be estimated by adopting the following formula:
Figure BDA0001551984000000111
where for the estimated n B-spline fitting errors,
Figure BDA0001551984000000112
is a vector of maximum displacement of each axis versus time derivative value of degree n,
Figure BDA0001551984000000113
and (3) forming a vector of the minimum displacement of each axis to the derivative value of n times of time, wherein delta t is the B-spline node interval of n times of the time parameter.
From the estimated fitting error and the given error of 0.001mm in this example, the node vector is determined as follows:
step a: giving the first node value of the node vector as the time T of the first position of the fitted original data1＝t1，i＝1,j＝1；
Step b: i is i +1, and judging whether i is equal to the original discrete data quantity, if so, turning to the step d; when not equal, executing step c in sequence;
step c: calculating the interval [ Tj,ti]B-spline fitting error n times, judging whether the fitting error is more than or equal to a given fitting error, and if the fitting error is more than or equal to the given fitting error, determining the next node vector as Tj+1＝tiAnd j is j + 1; turning to the step b;
step d: let node vector Tj+1When the time is equal to the end point time of the original data, the node vector calculation is finished;
in order to reduce the accumulated error, the time parameter B-spline track is converted into an incremental time parameter polynomial. The algorithm for converting the time parameter B-spline processing track into the polynomial surface processing track with the segment time as the parameter comprises the following steps:
for p times the time parameter B-spline is:
Figure BDA0001551984000000114
for an incremental p-th order polynomial:
P(p)(t)＝a0+a1·t+a2·t2+a3·t3…+an·tp,0≤t≤T
A＝[a0,a1,…ap]
P(p)(t)＝A·λTλ＝[1,t,t2,…tp]
for the odd time parameter B-spline, in the node interval ti,ti+1]The polynomial algorithm for converting to the incremental time parameter is as follows:
the time parameter B-spline is used for node interval ti,ti+1]Calculating function values and corresponding derivative values at corresponding end points
Figure BDA0001551984000000121
For the p-degree B-spline, the coefficient vector A and the time variable interval T of the p-degree polynomial can be directly obtained from the L vector by adopting Hermite interpolation.
For the even time parameter B-spline, in the node interval ti,ti+1]1,3, 5.. the conversion to incremental time parameter polynomial algorithm is:
the time parameter B-spline is used for node interval ti,ti+1]Calculating function values and corresponding derivative values at corresponding end points
Figure BDA0001551984000000122
For the p-degree B-spline, the coefficient vector A and the time variable interval T of the p-degree polynomial can be directly obtained from the L vector by adopting Hermite interpolation.
For the even time parameter B-spline, in the node areaM [ t ]i+1,ti+2]1,3, 5.. the conversion to incremental time parameter polynomial algorithm is:
the time parameter B-spline is used for node interval ti+1,ti+2]Calculating function values and corresponding derivative values at corresponding end points
Figure BDA0001551984000000123
For the p-degree B-spline, the coefficient vector A and the time variable interval T of the p-degree polynomial can be directly obtained from the L vector by adopting Hermite interpolation.
Fig. 6, fig. 7, and fig. 8 are graphs of the time-varying function of the position of each axis generated by the present invention, and according to the characterization error of each axis in the graphs, it can be illustrated that the precision of the polynomial fitting method proposed by the present invention satisfies the requirement of 0.001 mm.
3. The time parameter polynomial real-time online interpolation algorithm specifically comprises the following steps:
converting a P-th B-spline processing track into a segmented time parameter polynomial track according to a time parameter, wherein the coefficient vector of the ith segment polynomial is AiSetting the total processing time at this moment as T and the polynomial time interval of each time parameter as Tj。
Then the time variable
Figure BDA0001551984000000124
Comprises the following steps:
Figure BDA0001551984000000131
variable vector lambdaiComprises the following steps:
Figure BDA0001551984000000132
the motion position calculation formula of each motion axis obtained by interpolation at the time is as follows:
P(p)＝Ai·λi T
the algorithm is compiled on a real-time controller, and the motion of a machine tool is controlled. Fig. 9 is a wax model of a crown surface of a denture machined by the numerical control machining method of the present invention, illustrating the utility of the present invention.
Example 2:
the numerically controlled machine tool used in this example was the same as that used in example 1.
To further illustrate the versatility of the present invention, a parametric denture root curve machining trajectory as shown in fig. 10 was chosen for the experiments. The constraints and conversion factors chosen are the same as in example 1. The speed planning is carried out on the section of track by adopting the method, the result of the speed planning is shown in figure 11, the average speed is 9.78mm/s, and the speed loss rate is 73%. The continuity of the speed profile can be seen from the enlarged partial view. Similarly, the utility of the speed planning result is improved in order to reduce the data amount. The invention is adopted to represent the relation of the change of the position of each axis along with the time of the speed planning result, and the data compression amount is 83 percent. According to the characterization errors of the axes in fig. 12, fig. 13 and fig. 14, the accuracy of the polynomial fitting method provided by the invention can be satisfied. Fig. 15 is a wax model of a root surface of a denture to be processed by the numerical control processing method of the present invention, illustrating the versatility of the present invention.
The foregoing description of specific embodiments of the present invention has been presented. It is to be understood that the present invention is not limited to the specific embodiments described above, and that various changes and modifications may be made by one skilled in the art within the scope of the appended claims without departing from the spirit of the invention.

Claims (1)

1. A free-form surface numerical control machining method adopting time parameter polynomial interpolation is suitable for a numerical control machine tool to machine a free-form surface, the numerical control machine tool comprises a real-time machine tool controller and a machine tool body, and the method is characterized by comprising the following three operation steps:
(1) predictive look-ahead velocity planning
Generating high-speed and high-precision discrete data of the free-form surface processing process by reading a parameter curve type free-form surface processing track and carrying out speed planning;
the specific operation is as follows:
(1.1) extracting the highest-order three-time speed planning kinematic constraint according to the machining precision requirement and the operation capacity of machining equipment, reducing and simplifying the kinematic constraint by adopting a triangle inequality and a method of introducing a proportionality coefficient, and converting all the constraints into one-dimensional constraints related to curve parameters;
specifically, the constraint of each motion axis of the machine tool is simplified by introducing a proportionality coefficient and adopting a triangle inequality method:
the machine tool motion axis constraint is summarized as follows
Figure FDA0002500205730000011
In the formula (I), the compound is shown in the specification,
Figure FDA0002500205730000012
representing the speed, acceleration and jerk of the ith axis;
Figure FDA0002500205730000013
constraints representing speed, acceleration and jerk of the ith axis;
converting multidimensional constraints of acceleration and jerk of a motion axis into one-dimensional constraints related to curve parameters by introducing a proportionality coefficient and adopting a triangle inequality method; the acceleration constraint processing formula is as follows:
Figure FDA0002500205730000014
the jerk constraint processing formula is as follows:
Figure FDA0002500205730000015
in the formula1And2is the proportional coefficient of acceleration, mu1,μ2And mu3The acceleration rate is a proportional coefficient of acceleration, and the proportional coefficient is selected according to characteristics of a machine tool and a track; in the formula
Figure FDA0002500205730000021
And
Figure FDA0002500205730000022
velocity, acceleration and jerk of the trajectory space, respectively; in the formula qs,qss,qsssThe first derivative, the second derivative and the third derivative of the curve parameter to the curve arc length are respectively;
the speed planning kinematics constraint comprises constraint of each motion axis of the machine tool and constraint of a processing technology;
(1.2) carrying out early deceleration on track points violating one-dimensional constraints by adopting a prediction look-ahead method along a processing track, and accelerating for a time step if no track points violating one-dimensional constraints exist in the look-ahead process; according to a discrete calculation method based on a minimum value principle, calculating the jerk by using the highest-order constraint, and then obtaining discrete data of the curved surface machining process which meets the constraint and has the optimal machining efficiency through numerical integration calculation;
the specific operation of adopting a prediction look-ahead method to carry out early deceleration on track points violating the constraint in the future is as follows:
the prediction forward-looking deceleration process comprises the steps of firstly decelerating from a current point by using the allowed minimum jerk, reducing the acceleration to zero, then continuing to decrement the acceleration by using the minimum acceleration, and incrementing the acceleration to zero by using the maximum jerk every time the acceleration is decremented by one step until the speed is decremented to zero; respectively checking the speed, the acceleration and the jerk constraint in the process;
if the track point violates the speed and acceleration constraints in the stage that the initial acceleration is decreased to zero, the point can be set as a key track point violating the constraints; the processing track needs to be decelerated from an initial track point to a violation constraint key track point, and the deceleration mode is decelerated according to a prediction look-ahead search method;
in the acceleration continuous decreasing stage, if the track point violates the acceleration constraint, the jerk at the point violating the track point is recalculated according to the acceleration constraint; after the acceleration is obtained, then the deceleration algorithm is predicted to continue; if a track point violates the speed constraint, the point can be set as a key track point violating the speed constraint; the processing track needs to be decelerated from an initial track point to a key track point violating the speed constraint, and the deceleration mode is decelerated according to a predictive look-ahead search method;
in the process of acceleration increasing to zero, if the track point violates the acceleration constraint, the acceleration increasing process needs to be stopped; the deceleration prediction process decelerates to the point, the prediction look-ahead process is continuously executed from the point until the acceleration and the speed are reduced to zero, and each constraint is continuously detected;
in the process of acceleration increasing to zero, if the track point violates the speed constraint, the acceleration increasing process is stopped, and then the acceleration decreasing process is carried out; if the speed still violates the speed constraint when the point violating the speed track is reached, the process is circularly carried out; if the speed meets the constraint requirement when the point is reached, the point is a key point of the track, the processing track of the key point of the track needs to be decelerated from an initial track point to a key track point violating the constraint, and the deceleration mode is decelerated according to a prediction look-ahead search method;
(2) b-spline based time parameter polynomial machining trajectory generation
Representing the discrete data generated in the step (1) in the free-form surface processing process by adopting a B-spline fitting technology with given precision, and then converting the discrete data into a simple incremental time parameter polynomial surface processing track;
the specific operation is as follows:
(2.1) fitting by adopting B-spline of a time parameter according to discrete data of a curved surface machining process finished by speed planning, solving a B-spline node vector by adopting a node prediction method with given precision in the fitting process, and solving a control point by adopting a least square method; finally, characterizing the curved surface machining process by using the B-spline of the time parameter to obtain a B-spline machining track;
the specific determination process for solving the fitting B-spline node vector by adopting the node prediction method with given precision is as follows:
for n times of B-spline fitting, the position data { t, P } of each motion axis along with the time is obtained, and the fitting error can be estimated by adopting the following formula:
Figure FDA0002500205730000031
where for the estimated n B-spline fitting errors,
Figure FDA0002500205730000032
is a vector of maximum displacement of each axis versus time derivative value of degree n,
Figure FDA0002500205730000033
forming a vector by the minimum displacement of each axis and the derivative value of time n times, wherein delta t is a node interval of time parameter n times B-spline;
according to the estimated fitting error, under the given error, the node vector is determined by the following algorithm:
step a: giving the first node value of the node vector as the time T of the first position of the fitted original data1＝t1，i＝1，j＝1；
Step b: i is i +1, and judging whether i is equal to the original discrete data quantity, if so, turning to the step d; when not equal, executing step c in sequence;
step c: calculating the interval [ Tj,ti]B-spline fitting error n times, judging whether the error is more than or equal to a given error, and if the error is more than or equal to the given error, determining the next node vector as Tj+1＝tiAnd j is j + 1; turning to the step b;
step d: let node vector Tj+1When the time is equal to the end point time of the original data, the node vector calculation is finished;
(2.2) converting the B-spline processing track into an incremental time parameter polynomial surface processing track by adopting a B-spline-polynomial conversion algorithm (converting B-spline into a polynomial);
the algorithm for converting the time parameter B-spline processing track into the incremental time parameter polynomial surface processing track is as follows:
for p times the time parameter B-spline is:
Figure FDA0002500205730000034
for an incremental p-th order polynomial:
P(p)(t)＝a0+a1·t+a2·t2+a3·t3…+an·tp,0≤t≤TΣ
A＝[a0,a1,…an]
P(p)(t)＝A·λTλ＝[1,t,t2,…tp]
for the odd time parameter B-spline, in the node interval ti,ti+1]The polynomial algorithm for converting to the incremental time parameter is as follows:
the time parameter B-spline is used for node interval ti,ti+1]Calculating function values and corresponding derivative values at corresponding end points
Figure FDA0002500205730000041
For the P-th B-spline, the coefficient vector A and the time variable interval T of the p-th polynomial can be directly obtained by the L vector by adopting Hermite interpolation;
for the even time parameter B-spline, in the node interval ti,ti+1]1,3, 5.. the conversion to incremental time parameter polynomial algorithm is:
the time parameter B-spline is used for node interval ti,ti+1]Calculating function values and corresponding derivative values at corresponding end points
Figure FDA0002500205730000042
For the P-th B-spline, the coefficient vector A and the time variable interval T of the p-th polynomial can be directly obtained by the L vector by adopting Hermite interpolation;
for the even time parameter B-spline, in the node interval ti+1,ti+2]1,3, 5.. the conversion to incremental time parameter polynomial algorithm is:
the time parameter B-spline is used for node interval ti+1,ti+2]Calculating function values and corresponding derivative values at corresponding end points
Figure FDA0002500205730000043
For the P-th B-spline, the coefficient vector A and the time variable interval T of the p-th polynomial can be directly obtained by the L vector by adopting Hermite interpolation;
(3) time parameter polynomial real-time interpolation
According to the incremental time parameter polynomial curved surface machining track, compiling a corresponding real-time interpolation algorithm in a real-time machine tool controller, and controlling the machine tool to move to finish high-speed and high-precision machining of a free curved surface;
the specific operation is as follows: compiling a time parameter polynomial real-time interpolation program in a real-time machine tool controller, and controlling the motion of a numerical control machine tool through the real-time interpolation program according to the incremental time parameter polynomial curved surface machining track obtained in the step (2.2) to realize high-speed and high-precision machining of the free curved surface;
according to the incremental segmented time parameter polynomial curved surface processing track generated in the step (2), the coefficient vector of the ith segment of polynomial is AiSetting the total processing time at this moment as T∑Each time parameter polynomial time interval is tj；
Then the time variable
Figure FDA0002500205730000051
Comprises the following steps:
Figure FDA0002500205730000052
variable vector lambdaiComprises the following steps:
Figure FDA0002500205730000053
the motion position calculation formula of each motion axis obtained by interpolation at the time is as follows:
P(p)＝Ai·λi T
according to the interpolation algorithm and the total processing time is T∑The position of each motion axis of the machine tool at the moment can be obtained, and the free-form surface machining is completed by sequentially machining;
by the numerical control processing method for the free-form surface, the contour accuracy and the surface quality of the processed free-form surface are improved, the processing efficiency of the free-form surface is obviously improved, and the data volume required by the processing of the free-form surface is reduced.
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