Numerical control interpolation method for controlling bow height error

Abstract

The embodiment of the invention discloses a numerical control interpolation method for controlling a bow height error, and the method comprises the following steps in a preprocessing stage: 1, pre-interpolation calculation: analyzing a cubic NURBS curve instruction, constructing a model, calculating a curvature radius, a maximum allowable feeding speed, a next interpolation point parameter and a bow height error, adjusting a hyper-bound error by using a dichotomy, and generating an information vector; 2, curve segmentation: checking a speed extreme point, and segmenting a curve by combining starting and ending points; 3, information matrix generation: generating an ascending-order information matrix according to the segmentation points and the speed; 4, pre-scanning processing: forward scanning the matrix, searching a specific structure, checking feasibility, and deleting columns which do not meet the constraint; in the feeding speed planning stage, a bidirectional scanning strategy is adopted to process different block structures, A and A-U structures are processed in the forward direction, D and U-D structures are processed in the reverse direction, residual structures are finally scanned and processed, and corresponding planning methods are adopted for different structures to process the situation that errors exceed boundaries or are not feasible.

Description

Technical Field

[0001] The present application relates to the technical field of numerical control machining, and in particular to a numerical control interpolation method for controlling bow height error. Background Art

[0002] As a core of computer numerical control machining, interpolation algorithm is crucial to industrial production and an indispensable part of modern industrial manufacturing. Motion planning is a key issue in interpolation and has therefore received widespread attention.

[0003] There are various methods for high precision and high efficiency. The methods can be divided into two categories: construction methods and optimization methods. Generally speaking, construction methods use certain simplifications or strong assumptions to achieve fast or even real-time calculations, at the cost of relatively low processing efficiency or violation of constraints. In contrast, optimization methods usually establish complex global optimization models to obtain high processing efficiency, at the cost of longer calculation time.

[0004] In CNC machining, tool paths are often represented by smooth curves or G01 polyline segments. For the former, speed planning can be performed directly. For the latter, a smooth path can be obtained by using a fillet transition algorithm or a global fitting method.

[0005] The acceleration / deceleration control (ACC / DEC) method is widely used due to its simplicity and the fact that the axis motion can be controlled by the tangential motion and the normal motion. This method can explicitly construct the feed rate curve. In order to make the feed rate curve smooth, it needs to have global C^2 continuity. In order to avoid machine vibration, the magnitude of the acceleration needs to be controlled. In order to obtain higher machining efficiency, the movement of the tool needs to be bang-bang controlled, that is, there is always an axis motion or tangential motion that reaches the upper limit. The 7-segment S-shaped feed rate curve meets the above requirements and is widely used.

[0006] The 7-segment polynomial feed rate curve cannot guarantee the global continuity of the jerk, which will lead to machine vibration and poor machining results. To address this problem, a method to limit the size of the jump (the derivative of the jerk with respect to time) is proposed.

[0007] In addition to polynomial functions, trigonometric functions can also be used to construct feed rate curves. The latter can construct feed rate curves with continuous acceleration, more stable force and higher processing quality.

[0008] However, all the above methods have the defect of not being able to strictly control the bow height error. Bow height error is the distance between the line segment connecting two adjacent interpolation points and the curved path during the machining process. Because the tool cuts along the broken line segment connecting the interpolation points, the bow height error describes the geometric appearance error of the workpiece, which directly affects the accuracy and quality of the machining. There are two reasons why the previous acceleration and deceleration methods will cause the bow height error to exceed the limit: 1. The bow height error between two points on the path is estimated based on the curvature circle, not the actual value; 2. In speed planning, only the feed speed of the segment point takes into account the bow height error limit. Summary of the invention

[0009] The embodiments of this specification provide a CNC interpolation method for controlling bow height error to ensure that the bow height error during the processing is always strictly controlled within the set error limit, avoiding the problem of bow height error exceeding the limit in traditional methods.

[0010] To solve the above technical problems, the embodiments of this specification are implemented as follows:

[0011] The present invention provides a numerical control interpolation method for controlling bow height error, which is applied to a numerical control machining system for a cylinder block of an automobile engine, comprising:

[0012] S1. Path reception and analysis, including:

[0013] During the CNC machining process of the automobile engine cylinder block, the CNC system receives a cylinder block machining path instruction represented by a cubic non-uniform rational B-spline NURBS curve; parses the control point coordinates, node vectors and weight factors in the cylinder block machining path instruction, and converts them into a data format that can be recognized and processed by the CNC machining system, thereby constructing a mathematical model of the cylinder block machining path;

[0014] S2, pre-interpolation processing, specifically including:

[0015] S21, initialization operation, including: setting the initial index i=1, setting the curve parameter u i Set to the left edge of the path definition domain U 1 ;

[0016] S22, calculation of curvature radius, including: based on the mathematical characteristics of the cubic NURBS curve, the CNC system calculates u based on the curve parameter equation i The radius of curvature at i , the radius of curvature ρ i Used to reflect the bending degree of the cylinder body processing path at the corresponding point;

[0017] S23, determining the maximum allowable feed speed, including: according to the calculated curvature radius ρ i , and the bow height error limit δ and feed speed upper limit V pre-set by the CNC systemm , Normal acceleration upper limit A n And the upper limit of jerk J n , first use the formula Preliminary estimate of speed value, and then through Determine i The maximum permissible feed speed v at i ;

[0018] S24, calculation of the next interpolation point parameters, including: using the second-order Taylor expansion Calculate the curve parameter u of the next interpolation point i+1 ; where the symbol T s Indicates the interpolation cycle set by the numerical control machining system;

[0019] S25. Calculation and judgment of bow height error, including: calculation of C(u i ) and C(u i+1 ) i ; If δ i ≤δandu i+1 In the path definition domain, the index i is incremented by 1 and the above calculation steps are repeated; if δ i ≤δ but u i+1 Not within the path definition domain, terminate pre-interpolation; if δ i >δ, using the dichotomy method in the speed range [0,v i ] Adjust v i After the pre-interpolation is completed, multiple information vectors containing curve parameters u and the corresponding maximum allowable feed speed v are generated, and the second element of the first and last information vectors is set to 0;

[0020] S26, curve segmentation, including: analyzing the cylinder machining curve after pre-interpolation, starting from i=2, checking u i Is the velocity at the extreme point, that is, whether it satisfies v i-1 ≤(<)v i ,v i >(≥)v i+1 , or v i-1 ≥(>)v i ,v i <(≤)v i+1 If the condition is met, record the point;

[0021] when i <n pre -1, continue to check the next point; if i ≥ n pre -1, the check is terminated; the curve is segmented according to the recorded extreme points and the starting and ending points of the path, and the segmentation points are determined, where the symbol n pre Indicates the number of pre-interpolation points;

[0022] S26, generating an information matrix, including: generating an information matrix of the cylinder body processing path according to the segmentation points obtained by segmenting the curve and the corresponding maximum allowable feed speed, wherein each column of the matrix represents a segmentation point, the first row element is the curve parameter u corresponding to the segmentation point, the second row element is the maximum allowable feed speed v corresponding to the segmentation point, and the matrix is ​​arranged in ascending order of the curve parameter u;

[0023] S26, pre-scanning processing, including: forward pre-scanning the generated information matrix to find the AD and AUD structures therein; feasibility testing the found structures according to the tangential kinematic constraints and bow height error limits set by the numerical control system; if the structure does not meet the constraints, deleting the corresponding columns in the information matrix to ensure that the information matrix meets the requirements of subsequent cylinder body processing;

[0024] S27, bidirectional scanning and feeding speed planning, specifically including the following steps:

[0025] S271, forward scanning, including: processing the A structure and the AU structure in the information matrix, determining the reasonable feed speed of each point, so as to improve the efficiency and accuracy of cylinder body processing;

[0026] S272, reverse scanning, including: processing D structure and UD structure, further optimizing feed speed planning, making the processing process more stable;

[0027] S273, final scanning, including: performing forward scanning again, processing the residual AD and AUD structures, and generating a complete feed speed planning of the cylinder body machining tool motion trajectory; in the final scanning process, the parallel computing resources of the numerical control system are utilized to improve the computing efficiency;

[0028] S274. Planning and processing of different block structures, including: for A structure, AU structure, AD structure and AUD structure, different planning strategies are respectively adopted to further optimize the feed speed and control the bow height error to ensure the processing quality of the automobile engine cylinder block.

[0029] In an optional embodiment, in step S1, when the CNC system performs coordinate transformation on the control point coordinates, it transforms them from the coordinate system of the design software to the working coordinate system of the CNC machining system, and at the same time performs data integrity and validity checks on the node vectors and weight factors to ensure compliance with the mathematical definition of the cubic NURBS curve.

[0030] In an optional embodiment, the "pre-interpolation process" in step S2 uses a binary method to adjust v i When the speed interval [0,v i ] the middle value v mid , calculate vmid The bow height error δ is the feed speed mid ; If δ mid >δ, then update the new speed range to [0,v mid ]; if δ mid ≤δ, then update the new speed range to [v mid , v i ], repeat this process until v that meets the bow height error requirements is found i value, so as to accurately control the bow height error during the cylinder body processing.

[0031] In an optional implementation, after the feasibility check of the AD and AUD structures in the "pre-scan process", if it is found to be infeasible, the information matrix is ​​modified by deleting columns and modifying elements.

[0032] In an optional embodiment, when the forward scan of "bidirectional scanning and feed speed planning" processes structure A, if the information matrix is After conducting a feasibility analysis,

[0033] (1) If feasible, consider the next block structure;

[0034] (2) If the error is within the limit but the speed is unreachable, the CNC system uses an optimization algorithm to adjust v according to the dynamic model of the machine tool and the process requirements of the cylinder body processing. e The value of

[0035] (3) If the error exceeds the limit, first calculate v according to the reachability. e Modify it and then use the ladder construction method; specifically, set the artificial parameter β < 1 and the positive integer γ 1 , generate an arithmetic progression s = [u 1 ,u 1 +Δu,…,u 1 +(u 2 -u 1 )β], where Use s j Represents the jth element in the array;

[0036] When j = 1, the block will not be split, and the binary search method is used in the interval [v 1 ,v 2 ] to find the appropriate So that v 2 After replacing this value, the structure is feasible. If the binary search does not terminate after several iterations, let

[0037] When j>1, an extra column is inserted into the information matrix to form In the form of; first use the binary search method in the interval [v 1 ,v 2 ] to determine the appropriate v j , so that the information matrix It is feasible, and then the information matrix is The A block is processed according to the above method; by comparing the processing times corresponding to different j values, the j corresponding to the shortest processing time is selected, and the information matrix M is updated accordingly to improve the processing efficiency and accuracy of the cylinder body A structure part.

[0038] In an optional embodiment, when processing the AU structure in the forward scan of "bidirectional scanning and feed speed planning", the information matrix is ​​set to First, we determine the information matrix as (v 1 ,v 2 ) is feasible;

[0039] If feasible, no block operation is required;

[0040] If it is not feasible, then further determine the information matrix as (v 1 ,v 3 ) is feasible; if the block is feasible, use the binary search method to find the 2 u 3 ] to find the appropriate And to u 2 Modify to make the block structure feasible; if the error does not exceed the bound but is not reachable, transform M into in Calculated based on accessibility; if the bow height error still exceeds the limit even after modifying the elements based on accessibility, the step construction method is used to generate an arithmetic sequence, and the separation position of the arithmetic sequence should be limited to the interval [u 1 u 2 ]Inside;

[0041] When j = 1, the AU structure degenerates into an A block; when j > 1, the information matrix of the block structure becomes M = (v 1 ,v' 1 ,v 2 ,v 3 ); At this time, first generate a suitable v' 1 , so that the information matrix is ​​(v 1 ,v' 1 ) is feasible, and then the information matrix is ​​(v' 1 ,v 2 ,v 3 )'s AU structure is planned and processed according to the above method to optimize the processing path and speed of the cylinder body AU structure part.

[0042] In an optional embodiment, when the reverse scanning process AUD structure of "bidirectional scanning and feed speed planning" is performed, if the information matrix is

[0043] Check the matrices respectively (v 1 ,v 2 ) and (v 3 ,v 4 )’s feasibility, if all are feasible, no block operation is required;

[0044] If at least one block is infeasible, perform the following process on the infeasible block: Assume that block A is infeasible, and the detection matrix is ​​(v 1 ,v 3 ) is feasible;

[0045] If feasible, generate a feasible A block by bisection, whose matrix is make If this is not possible, a r =u 3 -u 2 ; In addition, if (v 1 ,v 2 ) is feasible, stipulate a r =0; at the same time, according to the feasibility calculation of block D a r +d r with u 3 -u 2 The relationship between the size of determines the feed speed v 2 There are two situations whether the U block can exist:

[0046] The first case is a r +d r ≤u 3 -u 2 , in which case the planning for the acceleration side and the deceleration side is independent; specifically, if a r >0, we only need to pay attention to the matrix (v 1 ,v 2 ,v' 2 ) of the AU structure, where v' 2 =(u 2 +a r ,v 2 ) T ; In this case, it is required that not only the first column of the information matrix of the AU structure should be fixed, but also the last column; when the ladder construction method is used, if the matrix is ​​(v 1 ,v' 1 ,v 2 ,v' 2) structure cannot find a feasible solution and can be degenerated into a matrix (v 1 ,v 2 ,v' 2 ) of the AU structure; for the matrix (v 3 ,v' 3 ,v 4 ) is made in the same way, where v' 3 =(u 3 -d r ,v 2 ) T ; Finally, M was replace;

[0047] The second case is a r +d r >u 3 -u 2 At this time, the block structure must first be degenerated into an AD structure, and then the feed speed planning is performed using the process for planning the AD structure.

[0048] In an optional embodiment, when the final scan of "bidirectional scanning and feed speed planning" processes the residual AD structure, the information matrix is ​​set to For the matrix A block of the feasibility analysis to find a feasible For the matrix The D block is feasibility analyzed to find a feasible

[0049] if The AD structure cannot be maintained, so the block structure will degenerate into a matrix (v 1 ,v 3 ) blocks;

[0050] If min(v 2,1 ,v 2,2 )≥max(v 1 ,v 3 ), the AD structure can be maintained, and the following processing is performed according to the bow height error of the two blocks:

[0051] (1) If both blocks can ensure that the bow height error does not exceed the limit after the accessibility analysis;

[0052] like Just v 2 Modify the value of;

[0053] like v 2 The value of is changed to This results in a decrease in the processing efficiency of block D. In order to improve the processing efficiency, the uniform speed segment generation method is used. 2 Then, add a column to the matrix to generate a uniform speed segment. Modified to

[0054] The process of the uniform speed segment generation method is as follows: First, take a trial step Where Δu is less than the predetermined value, if the block structure corresponding to this matrix is ​​feasible, use the binary search method to find the appropriate u' 2 ; Otherwise, it is considered that a high-speed uniform speed segment that improves processing efficiency cannot be generated, so no additional U blocks are generated;

[0055] When generating a uniform speed segment on the left, if v 2 The value of Then use the uniform speed segment generation method on the right side. If v 2 The value of Then use the uniform speed segment generation method on the left side;

[0056] (2) Only the first block has an out-of-bound bow height error after accessibility analysis;

[0057] Use the ladder construction method for the first block and add v 1 and v 2 Insert a new column v' 1 , for a given v' 1 , consider (v' 1 ,v 3 ) feasibility; if not feasible, degenerate to the result of j = 1, otherwise, update v as needed 2 value and use the uniform velocity segment construction method on the required side;

[0058] (3) Only the second block has an out-of-bound bow height error after accessibility analysis;

[0059] (4) If the height error of both blocks exceeds the limit after the accessibility analysis, the ladder construction method is used for both blocks to obtain two arithmetic progressions:

[0060] (c)s l =[u 1 ,u 1 +Δ 1 u,…,u 1 +(u 2 -u 1 )β],

[0061] (d)s r =[u 3 ,u3 -Δ 2 u,…,u 3 -(u 3 -u 2 )β],

[0062] Use j i Represents the serial number of the item in the i-th sequence, thus obtaining the sequence pair (j 1 ,j 2 ), using the bidirectional symmetrical ladder construction method, by specifying j 1 +j 2 =γ 1 +2, the number of sequence pairs from (γ 1 +1) 2 Reduced to γ 1 +1, thus improving computational efficiency.

[0063] One embodiment of this specification can at least achieve the following beneficial effects: The technical solution of this application ensures that the bow height error in the processing process is always strictly controlled within the set error limit through pre-interpolation calculation, accurate calculation of curvature radius and feed speed, real-time monitoring and adjustment of bow height error, etc., avoiding the problem of bow height error exceeding the limit in traditional methods, significantly improving the processing accuracy of workpieces, and is particularly suitable for aerospace, precision instruments and other fields with extremely high precision requirements. At the same time, by segmenting the curve and generating the information matrix, combined with bidirectional scanning and a variety of block structure planning methods, the tool's motion path and feed speed can be accurately planned according to the actual situation of the curve, so that the tool can always meet the kinematic limits and bow height error limits set by the user, and the degree of fit can always be guaranteed to be within the range given by the user. In addition, appropriate planning methods are used for different block structures (such as A, AU, AD, and AUD structures), such as the uniform speed segment generation method and the step construction method. Under the premise of ensuring accuracy, the feed speed is reasonably adjusted to improve the processing efficiency. The speed and acceleration are continuous, so that these two kinematic quantities will not change suddenly. At the same time, the jerk is bounded. Together, the two make the tool move smoothly, thereby avoiding damage to the machine tool or reduction in processing quality due to excessive movement. In the planning process, the tangential kinematic constraints of the CNC system (such as tangential acceleration and tangential jerk upper limit) are fully considered to ensure that the tool movement meets the performance requirements of the machine tool, avoid damage to the machine tool or reduction in processing quality due to excessive movement, and enhance the compatibility and reliability of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] In order to more clearly illustrate the embodiments of this specification or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0065] Figure 1 It is a kinematic image schematic diagram involved in a numerical control interpolation method for controlling bow height error provided by the present invention;

[0066] Figure 2 It is an algorithm flow chart of a numerical control interpolation method for controlling bow height error provided by the present invention;

[0067] Figure 3 It is a schematic diagram of a step construction method involved in a numerical control interpolation method for controlling bow height error provided by the present invention;

[0068] Figure 4 It is a schematic diagram of a uniform speed segment construction method involved in a numerical control interpolation method for controlling bow height error provided by the present invention;

[0069] Figure 5 It is a schematic diagram of a bidirectional symmetrical step construction method involved in a numerical control interpolation method for controlling bow height error provided by the present invention;

[0070] Figure 6 It is the simulation result of the trident-shaped path in the CNC interpolation method for controlling bow height error provided by the present invention; wherein, (a) (b) (c) are respectively the feed speed, tangential acceleration, and tangential jerk curves of the proposed algorithm. (d) (e) (f) are respectively the feed speed, tangential acceleration, and tangential jerk curves of Hu et al. (g) (h) (i) are respectively the feed speed, tangential acceleration, and tangential jerk curves of Wang et al.

[0071] Figure 7 It is a schematic diagram of the simulation results of the butterfly path in a CNC interpolation method for controlling bow height error provided by the present invention, wherein (a) (b) (c) are the feed speed, tangential acceleration, and tangential jerk curves of the proposed algorithm, respectively. (d) (e) (f) are the feed speed, tangential acceleration, and tangential jerk curves of Hu et al., respectively; (g) (h) (i) are the feed speed, tangential acceleration, and tangential jerk curves of Wang et al., respectively;

[0072] Figure 8The present invention provides a schematic diagram of the simulation results of the Omega-type path in a numerical control interpolation method for controlling bow height error; wherein (a) (b) (c) are respectively the feed speed, tangential acceleration, and tangential jerk curves of the proposed algorithm. (d) (e) (f) are respectively the feed speed, tangential acceleration, and tangential jerk curves of Hu et al. (g) (h) (i) are respectively the feed speed, tangential acceleration, and tangential jerk curves of Wang et al. DETAILED DESCRIPTION

[0073] In order to make the purpose, technical solutions and advantages of one or more embodiments of this specification clearer, the technical solutions of one or more embodiments of this specification will be clearly and completely described below in combination with the specific embodiments of this specification and the corresponding drawings. Obviously, the described embodiments are only part of the embodiments of this specification, not all of the embodiments. Based on the embodiments in this specification, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of one or more embodiments of this specification.

[0074] It should be understood that although the terms first, second, third, etc. may be used in this application document to describe various information, this information should not be limited to these terms. These terms are only used to distinguish the same type of information from each other.

[0075] Next, a numerical control interpolation method for controlling bow height error provided in the embodiment of the specification will be specifically described with reference to the accompanying drawings.

[0076] 1. Background

[0077] In CNC machining, tool paths are often represented by smooth curves or G01 polyline segments. For the former, speed planning can be performed directly. For the latter, a smooth path can be obtained by using a fillet transition algorithm or a global fitting method.

[0078] The Acceleration / Deceleration Control (ACC / DEC) method is widely used due to its simplicity and the fact that the axis motion can be controlled by the tangential motion and the normal motion. This method can explicitly construct the feedrate profile. In order to make the feedrate profile smooth, it needs to have a global C 2 Continuity. In order to avoid machine vibration, the magnitude of jerk needs to be controlled. In order to achieve higher machining efficiency, the tool movement needs to be controlled in a bang-bang manner, that is, there is always an axis movement or tangential movement that reaches the upper limit. The 7-segment S-shaped feed speed curve meets the above requirements and is widely used.

[0079] The 7-segment polynomial feed rate curve cannot guarantee the global continuity of the jerk, which will lead to machine vibration and poor machining results. To address this problem, a method to limit the size of the jump (the derivative of the jerk with respect to time) is proposed.

[0080] In addition to polynomial functions, trigonometric functions can also be used to construct feed rate curves. The latter can construct feed rate curves with continuous acceleration, more stable force and higher processing quality.

[0081] However, all the above methods have the defect of not being able to strictly control the bow height error. Bow height error is the distance between the line segment connecting two adjacent interpolation points and the curved path during the machining process. Because the tool cuts along the broken line segment connecting the interpolation points, the bow height error describes the geometric appearance error of the workpiece, which directly affects the accuracy and quality of the machining. There are two reasons why the previous acceleration and deceleration methods will cause the bow height error to exceed the limit: 1. The bow height error between two points on the path is estimated based on the curvature circle, not the actual value; 2. In speed planning, only the feed speed of the segment point takes into account the bow height error limit.

[0082] In the technical solution of this application, it is first proved that the bow height error between the line segment connecting two points on the non-uniform rational B-spline (NURBS) path and the curved path can be converted into the real root solution of a univariate algebraic equation, thereby designing a numerical calculation method. Then, the concept of feasibility analysis is proposed, which includes reachability analysis based on kinematic constraints and bow height error detection. Thus, the actual bow height error can be calculated and controlled. In addition, an optimization strategy that is conducive to improving processing quality is proposed.

[0083] 2. Feasibility analysis:

[0084] Because a cubic NURBS curve can represent a satisfactory tool path, it is assumed that the input path is a cubic NURBS curve.

[0085] 2.1 Some definitions and explanations

[0086] In order to make the description of the technical solution of the present application clearer and more accurate and avoid ambiguity, the meanings of some terms mentioned in the technical solution of the present application are first defined and explained below, so as to provide a unified basis for subsequent calculations and analyses.

[0087] In feed rate planning, the curve will be divided into several small segments by the so-called pre-interpolation. This process also records the maximum possible feed rate v at the point with curve parameter u. A sequence pair (u, v) is called an information vector.

[0088] The two information vectors at the boundary of a small curve segment provide the necessary information for subsequent feed rate planning, so the following second-order square matrix is ​​defined as the information matrix of this curve segment:

[0089]

[0090] Among them, the symbol u s Indicates the starting parameter of the curve segment, symbol u e Indicates the termination parameter of the curve segment, u s e . A column of the information matrix is ​​an information vector.

[0091] A curve segment with an information matrix is ​​called a block. There are three types of blocks: s <v e , the block is called an A block if v s =v e , the block is called a U block if v s >v e , the block is called a D block.

[0092] After the pre-interpolation process, two situations may occur:

[0093] (1) The speed is unattainable, that is, the speed given by the information vector cannot be reached under kinematic constraints;

[0094] (2) Even if the maximum feed rate can be achieved, the geometric properties of the tool path may cause the bow height error to exceed the limit at some interpolation points.

[0095] In order to solve the above problems, feasibility analysis is needed. Reachability analysis will detect whether the speed is reachable and make some adjustments to make it reachable. Feasibility analysis includes reachability analysis and has the function of detecting whether the bow height error in a block exceeds the limit, so that further adjustments can be made.

[0096] The essence of feedrate planning can be viewed as using block operations to make an otherwise infeasible block feasible. There are three basic types of block operations:

[0097] (1) Modify the elements of the information matrix, which will not change the number of columns of the matrix;

[0098] (2) Delete the columns of the information matrix;

[0099] (3) Insert a new column into the information matrix.

[0100] Combining block operations provides ample space for implementing optimization strategies, but before discussing this, it is necessary to study how to perform feasibility analysis on a given block, which is the basis for feed rate planning.

[0101] 2.2 Reachability Analysis of a Block

[0102] ​2.2.1 Taking a specific S-shaped curve as an example, the technical solution of this application can achieve the control of kinematics and bow height errors

[0103] Use the 7-segment trigonometric function curve to generate the feed rate curve when accelerating or decelerating. The expression for the acceleration situation is given below. The deceleration situation can be regarded as the reverse acceleration situation.

[0104] The expression of the acceleration curve is:

[0105]

[0106] where τ i =tt i ,i=1,2,…,7 and J p is the maximum tangential jerk of the curve, sequence pair (T 1 ,T 2 ,T 3 ,J p ) determines the jerk curve of a block.

[0107] The expressions of acceleration curve a(t), velocity curve v(t), and distance curve d(t) can be obtained by continuous integration, and the constant term of the integration can be determined according to the boundary value. According to the planning method, the initial acceleration must be 0, Figure 1 (b)(c)(d) show the images of the curves.

[0108] Although the formula is complicated, only the following three formulas are needed in reachability analysis. The first one is

[0109]

[0110] Among them, A p is the maximum tangential acceleration of the curve. The second is

[0111] V p =v(t 8 )=v 0 +A p (2T 1 +T 2 +T 3 ),

[0112] Among them, V p is the maximum feed rate of the curve. The third one is

[0113]

[0114] Among them, D p is the maximum movement distance of the curve.

[0115] To reduce the number of unknown parameters, let T 1=αT 2 , where α is an artificial parameter. In addition, J p Set as the upper limit of the cutting speed to approach Bang-Bang control. Therefore, the sequence (T 2 ,T 3 ) defines the jerk curve of a block.

[0116] 2.2.2 Reachability Analysis

[0117] Consider a block whose information matrix is

[0118]

[0119] Using the composite Newton-Cotes formula, the arc length l of the curve segment can be calculated numerically.

[0120] Without loss of generality, assume that the block is an A block, that is, v s <v e ≤V m , where V m is the upper limit of feed speed. Remember that the upper limit of tangential acceleration is A m , the upper limit of tangential acceleration is J m The approach to reachability analysis is given below.

[0121] Ignore v for now e and V m , calculate when T 3 = 0, the maximum moving distance of the tool under the constraints of tangential acceleration and jerk is obtained.

[0122]

[0123] Compare l and There are two cases for the size of .

[0124] Case 1: It is necessary to consider the maximum increment of the feed rate when the distance is l. To do this, first calculate the maximum tangential acceleration of this block. It can be calculated

[0125]

[0126] It is about The cubic equation of .

[0127]

[0128] Thus we get

[0129]

[0130] So the equation has a unique real root, because f1 (0) = -l < ​​0, so this real root is a positive root, and its calculation formula is as follows

[0131]

[0132] Among them, a f ,c f ,d f They are the coefficient of the cubic term, the coefficient of the 1st term and the constant term of the equation respectively.

[0133] Then the corresponding maximum feed speed is obtained The calculation is as follows:

[0134]

[0135] Compare and v e There are three cases for the size of:

[0136] Situation 1.1: At this time v e Unreachable, but as long as v e Modified to It can be transformed into situation 1.2.

[0137] Case 1.2: At this time v e reachable, and

[0138]

[0139] T 3 =0.

[0140] Case 1.3: At this time v e is reachable, but the maximum tangential acceleration is no longer The corresponding parameters need to be recalculated. Let δ v =v e -v s Next, we can establish the following about A p and T 3 System of linear equations in two variables:

[0141]

[0142] From the second equation, we get

[0143]

[0144] Substituting it into the first equation, we get p The quadratic equation of

[0145]

[0146] Since the discriminant can be calculated to be The equation has two real roots.

[0147] make

[0148]

[0149] Then its image is a parabola opening upward, and its axis of symmetry is Because f 2 (0)=δv>0, the equation has two positive roots. And because T 3 ≥0, so According to the expression of the axis of symmetry, A p is the smaller real root of the equation, so

[0150]

[0151] So we can calculate T 1 , T 2 and T 3 .

[0152] Case 2: Ignore v for now e and V m , consider again the maximum increment of feed speed when the distance is l. p =A m , and get T 3 The quadratic equation

[0153]

[0154] make

[0155]

[0156] Then its image is a parabola that opens upward, and its axis of symmetry is on the left side of the y-axis. Since the parabola intersects the y-axis at the negative semi-axis, the equation has a unique positive root, and T 3 To take the larger root. 3 After that, we get

[0157]

[0158] Compare and v e , there are 3 cases:

[0159] Case 2.1: At this time v e Unreachable, but only need to v e Change to The speed can be achieved.

[0160] Case 2.2: At this time v e reachable.

[0161] Case 2.3: Calculate the corresponding parameters according to the method of case 1.3.

[0162] 2.3 Feasibility Analysis of a Block

[0163] 2.3.1 Calculation method of bow height error

[0164] Many works use the curvature circle to approximate the bow height error, but this algorithm is inaccurate and may cause the bow height error to exceed the bound significantly. Before giving a feasibility analysis, we first study how to accurately calculate the bow height error.

[0165] Assume that Represents a cubic NURBS toolpath with a node vector of U.

[0166] The bow height error between two points on a curve is the Hausdorff distance between the arc and the chord, and the chord and Arc The Hausdorff distance between can be calculated by the following formula:

[0167] d H =max{d H,1 ,d H,2 ,d H,3},

[0168]

[0169] where g(u) is the foot point of C(u).

[0170] It is easy to find that for any k=1,2,3, d H,k There is only one equality constraint, which is a one-variable equation about u. Therefore, we can solve this equation first, then discard the inappropriate roots according to the inequality constraint, and finally get d H .

[0171] Since NURBS curves are piecewise curves, they can be analyzed node by node interval. Then we can get several intervals, namely So we can rewrite d H,k ,k=1,2,3, for processing.

[0172]

[0173] The problem is thus transformed into how to calculate d H,m,k,m=1,2,3,k=n i ,…,n j Next, we prove that H,n,k The equality constraint is a polynomial equation about u, which means that the essence of bow height error calculation is to solve a univariate polynomial equation, which has been widely studied.

[0174] Consider d H,1,k When u∈S k When C(u) is For convenience, remember

[0175]

[0176] in

[0177]

[0178] Q=C(u i ).

[0179] Then there is

[0180]

[0181] in

[0182]

[0183] Rewrite the above formula into the following form:

[0184]

[0185] in, and M i,j is an element of M. Regardless of the degree of the NURBS toolpath, it is always a univariate polynomial equation.

[0186] Next calculate W k According to the iterative definition of B-spline curves, their specific expressions can be calculated. Note that they can be regarded as 1-dimensional B-spline curves. Let the control point of W (the weight factor of the path) be w i , the control point of X is w i x i , and so on.

[0187] by For example, in the process of calculating this quantity, we can get W k and From now on, we assume that the number of toolpaths is 3. According to the iteration definition of NURBS, we have

[0188]

[0189] in

[0190]

[0191] Then simplify to get the power basis expression

[0192]

[0193] in

[0194] b X1 =c X1 +c X2 +c X3 +c X4 ,

[0195] b X2 =-3(c X1 U k +c X2 U k-1 +c X3 U k+2 +c X4 U k+1 ),

[0196]

[0197] b X4 =w i-2 x i-2 -c X5 (U k-1 +U k +U k+2 ),

[0198] b W1 =c W1 +c W2 +c W3 +c W4 ,

[0199] b W2 =-3(c W1 U k +c W2 U k-1 +c W3 U k+2 +c W4 U k+1 ),

[0200]

[0201] b W4 =w i-2 -c W5 (U k-1 +Uk +U k+2 ).

[0202] Then get The expression

[0203]

[0204] in

[0205] a X1 =b X1 b W2 -b X2 b W1 ,

[0206] a X2 =2(b X1 b W3 -b X3 b W1 ),

[0207] a X3 =3(b X1 b W4 -b X4 b W1 )+b X2 b W3 -b X3 b W2 ,

[0208] a X4 =2(b X2 b W4 -b X4 b W2 ),

[0209] a X5 =b X3 b W4 -b X4 b W3 .

[0210] According to the above derivation, we can get and Then calculate d H,1,k The equation in is a seventh-order equation about u

[0211] (a M1 a X1 +a M2 a Y1 +a M3 a Z1 ) 7 +(a M1 a X2 +b M1 a X1 +aM2 a Y2 +b M2 a Y1

[0212] +a M3 a Z2 +b M3 a Z1 )u 6 +(a M1 a X3 +b M1 a X2 +c M1 a X1 +a M2 a Y3 +b M2 a Y2

[0213] +c M2 a Y1 +a M3 a Z3 +b M3 a Z2 +c M3 a Z1 )u 5 +(a M1 a X4 +b M1 a X3 +c M1 a X2

[0214] +d M1 a X1 +a M2 a Y4 +b M2 a Y3 +c M2 a Y2 +d M2 a Y1 +a M3 a Z4 +b M3 a Z3

[0215] +c M3 a Z2 +d M3 a Z1 )u 4 +(a M1 a X5 +b M1 a X4 +c M1 a X3 +d M1 a X2 +aM2 a Y5

[0216] +b M2 a Y4 +c M2 a Y3 +d M2 a Y2 +a M3 a Z5 +b M3 a Z4 +c M3 a Z3 +d M3 a Z2 )u 3

[0217] +(b M1 a X5 +c M1 a X4 +d M1 a X3 +b M2 a Y5 +c M2 a Y4 +d M2 a Y3 +b M3 a Z5

[0218] +c M3 a Z4 +d M3 a Z3 )u 2 +(c M1 a X5 +d M1 a X4 +c M2 a Y5 +d M2 a Y4 +c M3 a Z5

[0219] +d M3 a Z4 )u + d M1 a X5 +d M2 a Y5 +d M3 a Z5 =0,

[0220] wherein

[0221] a Mi =M i,1 b X1 +M i,2b Y1 +M i,3 b Z1 +v i b W1 ,

[0222] b Mi =M i,1 b X2 +M i,2 b Y2 +M i,3 b Z2 +v i b W2 ,

[0223] c Mi =M i,1 b X3 +M i,2 b Y3 +M i,3 b Z3 +v i b W3 ,

[0224] d Mi =M i,1 b X4 +M i,2 b Y4 +M i,3 b Z4 +v i b W4 ,

[0225] i=1,2,3.

[0226] Following the above steps, we can write d H,2,k and d H,3,k The remaining problem is how to solve the univariate polynomial equation, which can be solved by using existing numerical methods such as the Rpoly algorithm.

[0227] 2.3.2 Feasibility Analysis

[0228] Without loss of generality, consider the information matrix as A block. Even if v e If it is unreachable, it can also be modified to ensure reachability. Therefore, it is always possible to conduct further feasibility analysis after implementing reachability analysis and possibly modifying the information matrix elements. The process is as follows:

[0229] (1) Perform reachability analysis and modify elements of the information matrix if necessary;

[0230] (2)u=u s ,flag=1;

[0231] (3) Calculate the curve parameter of the next interpolation point, denoted as u'. For the specific process, see the description of the subsequent pre-interpolation part.

[0232] (4) Calculate the bow height error between C(u) and C(u'). If the bow height error does not exceed the bow height error limit δ, and u' e , then u=u+Δu, go to step 3. If the bow height error exceeds the limit, flag=0 and terminate the process.

[0233] After the feasibility analysis, 4 situations may occur:

[0234] (1) Feasible: The speed is achievable and the bow height error is within the limit;

[0235] (2) Achievable but out-of-bounds error: The speed is achievable, but the bow height error is out-of-bounds;

[0236] (3) Error within bounds but unreachable: The speed is originally unreachable. After the reachability analysis modifies the elements of the information matrix to make the speed reachable, the bow height error does not exceed bounds.

[0237] (4) Unreachable and out-of-bounds error: The speed is inherently unreachable. Even after reaching the speed through reachability analysis, the bow height error still exceeds the bounds.

[0238] Based on the above content, the content of a CNC interpolation method for controlling bow height error applied to a CNC machining system for a cylinder block of an automobile engine provided by the present invention is described below. The method may include the following steps:

[0239] S1. Path reception and analysis, including:

[0240] ​During the CNC machining of the cylinder block of an automobile engine, the CNC system receives a cylinder block machining path instruction represented by a cubic non-uniform rational B-spline NURBS curve; the control point coordinates, node vectors and weight factors in the cylinder block machining path instruction are parsed and converted into a data format that can be recognized and processed by the CNC machining system, thereby constructing a mathematical model of the cylinder block machining path. Among them, the control point coordinates represent the position of a specific point in three-dimensional space used to determine the shape of the curve. The CNC system outlines the approximate path of the tool movement based on the coordinates of these control points. For example, when machining a complex curved surface of an engine cylinder block, the reasonable distribution of multiple control points can allow the curve to accurately simulate the shape of the cylinder block surface, thereby guiding the tool to perform precise machining. The node vector specifies the segmentation method of the curve and the parameter range of each segment. It determines the characteristics of the curve in different intervals and affects the local shape and overall continuity of the curve. In cylinder block machining, a reasonable node vector setting can enable the curve to achieve a smooth transition in different machining areas, avoiding the problem of discontinuous machining traces or uneven surfaces. Each control point corresponds to a weight factor, and the size of the weight factor affects the "closeness" of the curve to the control point. The larger the weight factor, the closer the curve is to the corresponding control point; the smaller the weight factor, the less the curve is affected by the control point. By adjusting the weight factor, the shape of the curve can be fine-tuned to meet the requirements of accuracy and shape of different parts in cylinder processing.

[0241] The cylinder block machining path instruction in this step is the basis for the CNC system to control the tool movement. The CNC system calculates the specific position and movement direction of the tool at each moment based on the curve information in the instruction, thereby driving the tool to perform machining along the predetermined path. This ensures that the tool can accurately cut the cylinder block material and process an engine block that meets the design requirements.

[0242] 3. Algorithm Process

[0243] The algorithm flow is as follows Figure 2 As shown, in the preprocessing part, each block will be generated and some columns of the path information matrix will be deleted through pre-scanning. In the feed speed planning part, the speed curve will be generated using scanning technology.

[0244] 3.1 Preprocessing

[0245] Preprocessing consists of 3 main steps:

[0246] (1) Pre-interpolation;

[0247] (2) Curve segmentation;

[0248] (3) Pre-scan.

[0249] Preprocessing uses the interpolation point method to calculate the maximum allowable feed rate at each sampling point while ignoring the tangential acceleration and jerk. The result helps to segment the curve and generate the initial information matrix of the entire path. The specific process of pre-interpolation is as follows:

[0250] (1)i=1,u i =U 1 , where U 1 is the left boundary of the path definition domain;

[0251] (2) Calculate u i The radius of curvature at i , and then according to the estimation formula

[0252]

[0253] as well as

[0254]

[0255] Calculate u i The maximum allowable feed rate at A n and J n are the normal acceleration and the upper jerk limits respectively.

[0256] (3) Use the second-order Taylor expansion to calculate the parameters of the next interpolation point, that is,

[0257]

[0258] (4) Calculate C(u i ) and C(u i+1 ), denoted as δ i ;

[0259] (5) If δ i ≤δandu i+1 In the path definition domain, i = i + 1 and go to step 2. If δ i ≤δandu i+1 is not within the path definition domain, terminate pre-interpolation. Otherwise, δ i >δ, go to step 6;

[0260] (6) Due to u i The feed rate at has an upper bound v i and the lower bound is 0, the feasible feed rate can be calculated using the bisection method. i Then, go to step 5.

[0261] In fact, the result of pre-interpolation is a series of information vectors, and the second elements of the first and last information vectors need to be changed to 0.

[0262] The process of curve segmentation is as follows:

[0263] (1) i = 2;

[0264] (2) Check u i In other words, check whether

[0265] av i-1 ≤(<)v i ,v i >(≥)v i+1

[0266] bv i-1 ≥(>)v i ,v i <(≤)v i+1

[0267] If one of these occurs, record it;

[0268] (3) If i <n pre -1, go to step 2, where n pre is the number of pre-interpolation points. Otherwise, the segmentation is terminated.

[0269] Curve segmentation can find the segmentation points of the path, thereby generating an information matrix of the path according to the corresponding feed rate. The starting point and end point of the curve are also considered as segmentation points. In the "curve segmentation" step, the CNC system determines u i When the speed at the extreme point is determined, the change in speed will be combined to make a comprehensive judgment; when the change in speed exceeds the preset acceleration threshold, it is determined that u i The speed at the extreme point is used to identify the key points of speed change in the cylinder machining path and improve the accuracy of curve segmentation.

[0270] Before velocity planning, a pre-scan is performed. Under the existing information vector, some AD or AUD structures cannot satisfy the tangential kinematics or bow height error constraints while retaining the A block and D block. At this time, some columns of the information matrix will be deleted. Using forward scanning, AD and AUD structures can be found and checked whether columns need to be deleted.

[0271] 3.2 Scanning Strategy

[0272] Before showing how to plan the feed rate for the block structure, the scanning strategy is briefly introduced first.

[0273] The algorithm only needs to process four basic block structures. After planning the feed rate for them, the feed rate planning for the entire path is completed.

[0274] (1) Structure A: Two consecutive A blocks. When planning the feed rate for this structure, only the first A block is considered.

[0275] (2) AU structure: 1 A block and 1 U block arranged in consecutive order.

[0276] (3) AD structure: 1 A block and 1 D block arranged in consecutive order.

[0277] (4) AUD structure: 1 A block, 1 U block, and 1 D block arranged in consecutive order.

[0278] In this algorithm flow, bidirectional scanning will be performed after the preprocessing part is executed. The forward scan will process the A structure and AU structure, while the reverse scan will process the D structure and UD structure. Then another forward scan will be performed to process the remaining AD and AUD structures, which is called the final scan. The final scan will check and optimize the continuity of the entire machining path. It will ensure smooth transitions between each block structure to avoid sudden changes in speed or discontinuous paths. This is very important for improving machining quality and reducing tool wear. For example, if there is no good transition between adjacent acceleration and deceleration sections, it may cause the tool to vibrate during movement, affecting the finish of the machined surface. The final scan can make these transitions more natural and smooth by adjusting the feed rate and path parameters. The final scan can generate a complete feed rate planning for the motion trajectory of the cylinder machining tool; the parallel computing resources of the CNC system can be used during the final scan, because feasible speed curves can be generated for these structures while ensuring that the boundary information vector is fixed, thereby improving the computing efficiency.

[0279] 3.3 Planning of basic block structure to ensure the processing quality of automobile engine cylinder block

[0280] 3.3.1A Structure

[0281] Assume that the information matrix of the block structure is After conducting a feasibility analysis, there are 3 possible scenarios.

[0282] If one is feasible, then the next block structure can be considered.

[0283] The second is that the error does not exceed the bound but is unreachable, so modify v e The value of .

[0284] The third is that the error exceeds the bound. In this case, if necessary, first modify v according to the accessibility e Next, the proposed ladder construction method is used.

[0285] The goal of the ladder construction method is: when overcoming the extra processing time caused by the bow height error exceeding the limit, you can try to divide a block into several small blocks to improve the processing efficiency.

[0286] Figure 3 The idea of ​​the ladder construction method is shown. At the dot in Figure (a), the bow height error exceeds the limit, so the red solid line is adjusted to a red dotted line, resulting in an increase in processing time. However, if the segmentation is performed as in Figure (b), although the speed is even lower at the beginning, the increase in speed in the latter small segment will improve the processing efficiency of the entire curve segment.

[0287] Although the staircase construction method can be used iteratively, in order to reduce the computation time, it can be assumed that a block is divided into at most two small blocks.

[0288] The process of the ladder construction method is as follows. Assume that the artificial parameter β<1 and the positive integer γ 1 . Generate an arithmetic progression

[0289] s=[u 1 ,u 1 +Δu,…,u 1 +(u 2 -u 1 )β]

[0290] in Use s j Represents the j-th element in the array.

[0291] When j = 1, the block will not be split. Use binary search to find the appropriate So that v 2 Replace with If the binary search does not terminate after several iterations,

[0292] When j>1, an extra column will be inserted into the information matrix, becoming where v j is C(s j ) at the feed rate. First, use the binary method to find the appropriate v j ∈[v 1 ,v 2 ] makes the information matrix (v 1 ,v' 1 ) is feasible. Then consider the information matrix as (v' 1 ,v 2 ) is processed according to the above method.

[0293] One j corresponds to one processing time. Find the j corresponding to the shortest processing time and update the information matrix M.

[0294] 3.3.2A-U structure

[0295] Assume the information matrix is

[0296] First, consider the information matrix (v 1 ,v 2 ) is feasible, so there are two situations.

[0297] If this works, no block operations are needed.

[0298] Otherwise, consider the information matrix (v 1 ,v 3 ) is feasible. If feasible, the binary search method can be used to find the appropriate And modify u 2 Make the block structure feasible. If the error is within the bound but not reachable, M becomes in Calculated based on accessibility. If the bow height error is still out of bounds even after modifying the elements based on accessibility, use the step construction method and generate an arithmetic progression. The separation position should be in the interval [u 1 ,u 2 ]Inside.

[0299] When j = 1, the AU structure degenerates into an A block. When j>1, the information matrix of the block structure is M = (v 1 ,v' 1 ,v 2 ,v 3 ). First generate a suitable v' 1 The information matrix is ​​(v 1 ,v' 1 ) is feasible, and then the information matrix is ​​(v' 1 ,v 2 ,v 3 )'s AU structure is planned based on the above discussion.

[0300] 3.3.3A-D Structure

[0301] Assume the information matrix is

[0302] First, for the matrix (v 1 ,v 2 ) to perform feasibility analysis on block A and find a feasible solution. Similarly, for the matrix (v 2 ,v 3 ) to perform feasibility analysis on the D block and find a feasible solution.

[0303] if The AD structure cannot be maintained, so the block structure will degenerate into a matrix (v1 ,v 3 ) block.

[0304] Otherwise the AD structure can be maintained, which again has 4 possible cases:

[0305] (1) Both blocks can ensure that the bow height error does not exceed the limit after reachability analysis;

[0306] (2) Only the first block has an out-of-bound bow height error after accessibility analysis;

[0307] (3) Only the second block has an out-of-bound bow height error after accessibility analysis;

[0308] (4) After the accessibility analysis, the bow height errors of both blocks exceeded the limit;

[0309] For case (1), if Just need to modify v 2 If v 2 The value of is changed to This results in a decrease in the processing efficiency of block D. In order to improve the processing efficiency, the uniform speed segment generation method is used. 2 Then, try to add a column to the matrix to generate a uniform speed segment. Modified to Figure 4 Demonstrates the idea of ​​the method.

[0310] The process of the uniform speed segment generation method is as follows: First, take a very small trial step Where Δu is small enough. If the block structure corresponding to this matrix is ​​feasible, the binary search method can be used to find the appropriate u' 2 Otherwise, it is considered that a high-speed uniform speed segment that significantly improves the processing efficiency cannot be generated, so no additional U block is generated.

[0311] The process of generating the constant speed segment on the left is similar. Therefore, if v 2 The value of Then use the uniform velocity segment generation method on the right side. If v 2 The value of Then use the uniform speed segment generation method on the left.

[0312] For case (2), use the ladder construction method for the first block and 1 and v 2 Insert a new column v' 1 For a given v' 1 , consider (v' 1 ,v 3) is feasible. If not feasible, it degenerates to the result of j=1. Otherwise, update v as needed 2 The value of and use the uniform velocity segment construction method on the required side. Case (3) is similar to case (2).

[0313] For case (4), we need to use the ladder construction method for both blocks to obtain two arithmetic progressions:

[0314] (a)s l =[u 1 ,u 1 +Δ 1 u,…,u 1 +(u 2 -u 1 )β],

[0315] (b)s r =[u 3 ,u 3 -Δ 2 u,…,u 3 -(u 3 -u 2 )β],

[0316] Use j i Represents the serial number of the item in the i-th sequence, thus obtaining the sequence pair (j 1 ,j 2 ). Using the bidirectional symmetrical ladder construction method, it specifies j 1 +j 2 =γ 1 +2, the number of sequence pairs from (γ 1 +1) 2 Reduced to γ 1 +1, thereby improving the computational efficiency. In addition, parallel computation can be used for different order pairs. Figure 5 The idea of ​​bidirectional symmetrical ladder construction method is demonstrated.

[0317] After using the bidirectional symmetric ladder construction method, the information matrix may become (v 1 ,v' 1 ,v 2 ,v' 3 ,v 3 ). In determining the newly inserted column v' 1 and v' 3 After that, consider the matrix (v' 1 ,v 2 ,v' 3 ) block structure. Follow the above discussion to plan. If (v' 1 ,v' 3 ) is not feasible, it degenerates into (v1 ,v 3 )The result of planning.

[0318] 3.3.4A-UD structure

[0319] Assume the information matrix is

[0320] Check the matrices respectively (v 1 ,v 2 ) and (v 3 ,v 4 )’s feasibility. If all are feasible, no block operation is required.

[0321] If at least one block is infeasible, perform the following process on the infeasible blocks. Let's assume that block A is infeasible. The detection matrix is ​​(v 1 ,v 3 ) is feasible. If it is feasible, a feasible A block can be generated by bisection, and its matrix is make If this is not possible, a r =u 3 -u 2 In addition, if (v 1 ,v 2 ) is feasible, stipulate a r = 0. Similarly, the feasibility of block D can be calculated a r +d r with u 3 -u 2 The relationship between the size of determines the feed speed v 2 Whether the U block can exist, there are two possible situations.

[0322] The first case is a r +d r ≤u 3 -u 2 This means that the planning of the acceleration side and the deceleration side is independent to some extent. Specifically, if a r >0, we only need to pay attention to the matrix (v 1 ,v 2 ,v' 2 ) of the AU structure, where v' 2 =(u 2 +a r ,v 2 ) T Here, it is required that not only the first column of the information matrix of the AU structure should be fixed, but also the last column. When the ladder construction method is used, if the matrix is ​​(v 1 ,v' 1 ,v2 ,v' 2 ) structure cannot find a feasible solution, and can always be degenerated into a matrix (v 1 ,v 2 ,v' 2 ) AU structure. For the matrix (v 3 ,v' 3 ,v 4 ) is made in the same way, where v' 3 =(u 3 -d r ,v 2 ) T . Finally, M was replace.

[0323] The second case is a r +d r >u 3 -u 2 First, the block structure must be degenerated into an AD structure, and then the feed rate planning is performed using the process of planning the AD structure. The key issue is to determine the segmentation point of the AD structure. Only by determining it can the information matrix (v 1 ,v',v 4 ). Let the artificial parameter γ 2 , generating the first term as u 1 +δ 1 (u 2 -u 1 ), the last term is u 4 -δ 2 (u 4 -u 3 ) t , where 0<δ 1 ,δ 2 <1. When solving, we can let δ 1 =δ 2 .

[0324] 4. Simulation Results

[0325] This section uses three examples of comparative tests with existing high-level ACC / DEC to verify that the algorithm of the technical solution of this application can ensure processing efficiency while satisfying the bow height error constraint and tangential kinematics constraint. Existing high-level ACC / DEC methods all use trigonometric functions to construct a continuous feed rate curve for tangential jerk. Among them, the algorithm of Hu et al. was recently published, and the algorithm of Wang et al. is the only known algorithm that attempts to control the bow height error. The relevant parameters of the experiment are shown in the following table.

[0326] Table 1 Necessary parameters

[0327] Parameter name Value Feed speed limit 200mm / s Tangential acceleration limit 1000mm / s^2 Centripetal acceleration limit 1000mm / s^2 Tangential jerk limit 80000mm / s^3 Centripetal jerk limit 80000mm / s^3 Bow height error limit 0.001mm Interpolation cycle 0.001s

[0328] 4.1 Trident Path

[0329] This path is a simple closed NURBS curve with only 7 control points, so only a small amount of acceleration and deceleration is required within a relatively large bow height error limit.

[0330] Figure 6 The simulation results are shown. First, two phenomena can be found by observing the feed speed curve:

[0331] 1. There are 4 maximum feed speed points, and the result of the technical solution of this application is closest to the upper limit (black dotted line);

[0332] 2. All three methods can obtain two uniform feed speed segments, but the segment length of Wang et al.'s result is significantly shorter, which affects the processing efficiency.

[0333] Next, let's look at the tangential acceleration. The algorithm of the technical solution of the present application and the algorithm of Hu et al. can maintain the maximum tangential acceleration for a period of time, while the algorithm of Wang et al. cannot do this. As for the tangential jerk, only the algorithm of the technical solution of the present application can maintain the maximum value for a period of time, because the function used in the technical solution of the present application has stronger expressive power and is closer to the so-called Bang-Bang control, that is, at any time during the processing, a certain amount of motion will reach the upper limit. At the same time, the results of Wang et al. cannot even reach the maximum tangential jerk, because the function they use does not have a strong shape expression ability. In summary, it is not surprising that the technical solution of the present application has the highest processing efficiency, see Table 2.

[0334] Table 2 Simulation results of trident path

[0335] method Maximum bow height error (mm) Processing time(s) The technical solution of this application 1.676*10^(-4) 2.730 Hu et al. 1.595*10^(-4) 2.762 Wang et al. 1.254*10^(-4) 3.424

[0336] 4.2 Butterfly Path

[0337] This path is relatively complex, and the presence of sharp corners may make it difficult to control the bow height error. Although the processing time of Hu et al.'s result is the shortest (see Table 3), the bow height error exceeds the limit. Although the processing time of the other two methods is slightly longer, they can control the bow height error. In addition, the processing efficiency of the technical solution of this application is significantly higher than that of Wang et al.

[0338] The reason why Hu et al.'s algorithm cannot control the bow height error may be that a large amount of information obtained by pre-interpolation is not used in speed planning, and only the feed speed information at the segmentation point is retained. The bow height errors at these positions may not be effectively controlled because Hu et al.'s algorithm uses the curvature circle estimation formula to estimate the bow height error and does not calculate the accurate value.

[0339] This example shows that feasibility analysis is indispensable to control bow height error.

[0340] Table 3 Simulation results of butterfly path

[0341] method Maximum bow height error (mm) Processing time(s) The technical solution of this application 9.985*10^(-4) 3.164 Hu et al. 0.0038 3.008 Wang et al. 1.695*10^(-4) 3.605

[0342] 4.3 Omega curve

[0343] This curve has two extremely sharp corners.

[0344] As shown in Table 4, Hu et al. have the highest processing efficiency, but the cost is that the bow height error is significantly out of bounds. Although the three methods can find sharp corners and correctly segment the curve during the pre-interpolation process, only the algorithm of the technical solution of the present application can reduce the feed speed sufficiently at sharp corners to control the bow height error.

[0345] This example shows that the bow height error estimation based on the curvature circle may produce completely wrong results. Compared with the other two unreliable algorithms, the algorithm of the technical solution of the present application can control the bow height error to ensure the processing quality and is practical.

[0346] Table 4 Simulation results of Omega type path

[0347] method Maximum bow height error (mm) Processing time(s) The technical solution of this application 9.978*10^(-4) 1.165 Hu et al. 0.0436 0.814 Wang et al. 0.0420 0.953

[0348] The technical solution of this application ensures that the bow height error in the processing process is always strictly controlled within the set error limit through pre-interpolation calculation, accurate calculation of curvature radius and feed speed, real-time monitoring and adjustment of bow height error, etc., avoiding the problem of bow height error exceeding the limit in traditional methods, significantly improving the processing accuracy of the workpiece, and is particularly suitable for aerospace, precision instruments and other fields with extremely high precision requirements. At the same time, by segmenting the curve and generating the information matrix, combined with bidirectional scanning and a variety of block structure planning methods, the motion path and feed speed of the tool can be accurately planned according to the actual situation of the curve, so that the tool can always meet the kinematic limits and bow height error limits set by the user, and the degree of fit can always be guaranteed to be within the range given by the user. Moreover, for different block structures (such as A, AU, AD, AUD structures), appropriate planning methods are adopted, such as uniform speed segment generation method, ladder construction method, etc., under the premise of ensuring accuracy, the feed speed is reasonably adjusted to improve the processing efficiency, the speed and acceleration are continuous, so that these two kinematic quantities will not mutate, and the acceleration is bounded, and the two together make the tool motion stable, thereby avoiding damage to the machine tool or reduction in processing quality due to excessive motion. During the planning process, the tangential kinematic constraints of the CNC system (such as tangential acceleration and tangential jerk upper limit) are fully considered to ensure that the tool movement meets the performance requirements of the machine tool, avoid damage to the machine tool or degradation of processing quality due to excessive movement, and enhance the compatibility and reliability of the system.

[0349] The technical solution of this application can also be used in the processing of complex curved surface parts, the manufacturing of aircraft engine blades, the processing of automobile parts, mold manufacturing, etc.

[0350] When machining parts with complex curved surfaces, such as in the aerospace field, it is often necessary to machine parts with complex curved surfaces, such as the leading edge of an aircraft wing, the air intake of an engine, etc. The curved shapes of these parts can be described by NURBS surfaces, and the paths on the surfaces can be converted into cubic UNRBS curves within a given tolerance.

[0351] When the technical solution of this application is used for the processing of complex curved surface parts, the path input and analysis are performed first: the engineer inputs the designed third-order NURBS curve instruction into the CNC system, and the system analyzes the control point coordinates, node vectors and weight factors in the instruction. For example, for the leading edge of the wing of an aircraft, the surface shape of the leading edge of the wing can be described by a NURBS surface, and the computer-aided manufacturing software can generate a processing path on the surface. These paths can be converted into a number of third-order NURBS curve paths, and then the CNC interpolation is started. During the pre-interpolation process, the system calculates the radius of curvature of each point on the surface. The change in the curvature of the surface causes the curvature of the path located on the surface to change. The system determines the maximum allowable feed speed of each point based on parameters such as the radius of curvature, the bow height error limit, and the upper limit of the feed speed. In areas with large curvature, in order to ensure the processing accuracy, the feed speed will be reduced accordingly; while in areas with small curvature, the feed speed can be appropriately increased. The parameters of the next interpolation point are calculated by the second-order Taylor expansion, and the interpolation process is continuously promoted to generate an information vector containing the curve parameters and the corresponding maximum allowable feed speed. Finally, curve segmentation and planning are performed: the pre-interpolated curve is segmented to find the speed extreme points. For example, in certain transition areas of the wing leading edge surface, the speed may be extreme. An information matrix is ​​generated based on these extreme points and the starting and ending points of the path, and then pre-scanning, bidirectional scanning and planning of different block structures are performed. Speed ​​optimization and bow height error control are performed for A structure, AU structure, etc. to ensure that the processed parts meet the design requirements.

[0352] In the manufacturing scenario of aircraft engine blades, aircraft engine blades are one of the key components of aircraft engines. They have complex shapes and extremely high precision requirements. The surface of the blade is usually designed using a third-order NURBS surface to achieve efficient airflow control and energy conversion. At this time, the path input and analysis are first performed: the design data of the blade is input into the CNC machining system in the form of a third-order NURBS curve instruction. The system parses the instruction, accurately converts and verifies the control point coordinates to ensure the shape accuracy of the blade. For example, the twisted shape of the blade requires accurate control point coordinates to define, and the CNC system will carefully process these coordinate information to ensure the accuracy of subsequent processing. Then pre-interpolation processing is performed: in the pre-interpolation stage, the system calculates the curvature radius of each point on the blade surface. Due to the complex shape of the blade and the drastic change of curvature in different parts, the calculation of the curvature radius is particularly important. According to the curvature radius and the system preset parameters, the maximum allowable feed speed of each point is determined. For example, at the tip and root of the blade, the curvature is large, and the feed speed will be strictly limited to avoid excessive bow height errors and affect the performance of the blade. The parameters of the next interpolation point are calculated through the second-order Taylor expansion, and the information vector is gradually generated to provide a basis for subsequent processing planning. Finally, planning and control are carried out: the blade profile curve is segmented, the velocity extreme points are found, and the information matrix is ​​generated. During the pre-scan and bidirectional scanning process, different block structures are processed. For example, for the AD structure on the blade profile, the system will perform feasibility analysis on blocks A and D respectively, and adjust the speed planning according to the analysis results to ensure the processing accuracy and surface quality of the blade. At the same time, the bow height error is strictly controlled during the processing to ensure the aerodynamic performance of the blade.

[0353] In the automotive parts processing scenario, in automobile manufacturing, the processing accuracy of some key parts such as engine cylinders and crankshafts directly affects the performance and reliability of the car. Some contour curves of these parts can also be described by 3rd order NURBS curves to achieve more accurate design and processing. At this time, path input and analysis are first performed: the design data of automotive parts is input into the CNC system in the form of 3rd order NURBS curve instructions. The system parses the instructions and processes the control point coordinates, node vectors and weight factors. For example, the inner wall contour curve of the engine cylinder may be defined by a 3rd order NURBS curve. The CNC system will accurately convert the design data into information that it can process, preparing for subsequent processing. Then pre-interpolation processing is performed: During the pre-interpolation process, the system calculates the curvature radius of each point on the inner wall curve of the cylinder body. According to the curvature radius and the system preset parameters, the maximum allowable feed speed of each point is determined. In some special parts of the cylinder body, such as the transition area of ​​the cylinder bore, the curvature changes greatly, and the feed speed will be adjusted accordingly. The parameters of the next interpolation point are calculated by the second-order Taylor expansion to generate an information vector. For example, when machining cylinder holes, in order to ensure the roundness and cylindricity of the cylinder holes, the system will strictly control the bow height error to ensure machining accuracy. Finally, machining planning and optimization are carried out: the inner wall curve of the cylinder body is segmented to generate an information matrix. During the pre-scan and bidirectional scanning process, different block structures are planned and processed. For example, for the AU structure, the system will adjust the speed and optimize the structure according to its feasibility analysis results. Through reasonable speed planning and bow height error control, the machining quality and production efficiency of automotive parts can be improved.

[0354] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features thereof may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A numerical control interpolation method for controlling bow height error, applied to the numerical control machining system of automobile engine cylinder, characterized in that: The following steps are involved: S1. Path reception and analysis, including: During the CNC machining process of the automobile engine cylinder block, the CNC system receives a cylinder block machining path instruction represented by a cubic non-uniform rational B-spline NURBS curve; parses the control point coordinates, node vectors and weight factors in the cylinder block machining path instruction, and converts them into a data format that can be recognized and processed by the CNC machining system, thereby constructing a mathematical model of the cylinder block machining path; S2, pre-interpolation processing, specifically including: S21, initialization operation, including: setting the initial index i=1, setting the curve parameter u i Set to the left boundary U1 of the path definition domain; S22, calculation of curvature radius, including: based on the mathematical characteristics of the cubic NURBS curve, the CNC system calculates u based on the curve parameter equation i The radius of curvature at i , the radius of curvature ρ i Used to reflect the bending degree of the cylinder body processing path at the corresponding point; S23, determining the maximum allowable feed speed, including: according to the calculated curvature radius ρ i , and the bow height error limit δ and feed speed upper limit V pre-set by the CNC system m , Normal acceleration upper limit A n And the upper limit of jerk J n , first use the formula Preliminary estimate of speed value, and then through Determine i The maximum permissible feed speed v at i ; S24, calculation of the next interpolation point parameters, including: using the second-order Taylor expansion Calculate the curve parameter u of the next interpolation point i+1 ; where the symbol T s Indicates the interpolation cycle set by the numerical control machining system; S25. Calculation and judgment of bow height error, including: calculation of C(u i ) and C(u i+1 ) i ; If δ i ≤δ and u i+1 In the path definition domain, the index i is incremented by 1 and the above calculation steps are repeated; if δ i ≤δ but u i+1 Not within the path definition domain, terminate pre-interpolation; if δ i >δ, using the dichotomy method in the speed range [0,v i ] Adjust v i After the pre-interpolation is completed, multiple information vectors containing curve parameters u and the corresponding maximum allowable feed speed v are generated, and the second element of the first and last information vectors is set to 0; S26, curve segmentation, including: analyzing the cylinder machining curve after pre-interpolation, starting from i=2, checking u i Is the velocity at the extreme point, that is, whether it satisfies v i-1 ≤(<)v i ,v i >(≥)v i+1 , or v i-1 ≥(>)v i ,v i <(≤)v i+1 If the condition is met, record the point; when i <n pre -1, continue to check the next point; if i ≥ n pre -1, the check is terminated; the curve is segmented according to the recorded extreme points and the starting and ending points of the path, and the segmentation points are determined, where the symbol n pre Indicates the number of pre-interpolation points; S26, generating an information matrix, including: generating an information matrix of the cylinder body processing path according to the segmentation points obtained by segmenting the curve and the corresponding maximum allowable feed speed, wherein each column of the matrix represents a segmentation point, the first row element is the curve parameter u corresponding to the segmentation point, the second row element is the maximum allowable feed speed v corresponding to the segmentation point, and the matrix is ​​arranged in ascending order of the curve parameter u; S26, pre-scanning processing, including: forward pre-scanning the generated information matrix to find the AD and AUD structures therein; feasibility testing the found structures according to the tangential kinematic constraints and bow height error limits set by the numerical control system; if the structure does not meet the constraints, deleting the corresponding columns in the information matrix to ensure that the information matrix meets the requirements of subsequent cylinder body processing; S27, bidirectional scanning and feeding speed planning, specifically including the following steps: S271, forward scanning, including: processing the A structure and the AU structure in the information matrix, determining the reasonable feed speed of each point, so as to improve the efficiency and accuracy of cylinder body processing; S272, reverse scanning, including: processing D structure and UD structure, further optimizing feed speed planning, making the processing process more stable; S273, final scanning, including: performing forward scanning again, processing the residual AD and AUD structures, and generating a complete feed speed planning of the cylinder body machining tool motion trajectory; in the final scanning process, the parallel computing resources of the numerical control system are utilized to improve the computing efficiency; S274. Planning and processing of different block structures, including: for A structure, AU structure, AD structure and AUD structure, different planning strategies are respectively adopted to further optimize the feed speed and control the bow height error to ensure the processing quality of the automobile engine cylinder block.

2. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: In step S1, when the CNC system performs coordinate transformation on the control point coordinates, it transforms them from the coordinate system of the design software to the working coordinate system of the CNC machining system, and at the same time performs data integrity and validity checks on the node vectors and weight factors to ensure that they comply with the mathematical definition of the cubic NURBS curve.

3. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: In the "pre-interpolation process" of step S2, the binary method is used to adjust v i When the speed interval [0,v i ] the middle value v mid , calculate v mid The bow height error δ is the feed speed mid ; If δ mid >δ, then update the new speed range to 0,v mid ]; if δ mid ≤δ, then update the new speed range to [v mid , v i ], repeat this process until v that meets the bow height error requirements is found i value, so as to accurately control the bow height error during the cylinder body processing.

4. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: In step S26 of "curve segmentation", the numerical control system determines u i When the speed at the extreme point is determined, a comprehensive judgment is made based on the change in speed. When the change in speed exceeds the preset acceleration threshold, u i The speed at the extreme point is used to identify the key points of speed change in the cylinder machining path and improve the accuracy of curve segmentation.

5. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: After the feasibility check of the AD and AUD structures in the "pre-scan process", if it is found to be infeasible, the information matrix is ​​modified by deleting columns and modifying elements.

6. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: When processing structure A in the forward scan of "Bidirectional Scanning and Feed Speed ​​Planning", if the information matrix is After conducting a feasibility analysis, (1) If feasible, consider the next block structure; (2) If the error is within the limit but the speed is unreachable, the CNC system uses an optimization algorithm to adjust v according to the dynamic model of the machine tool and the process requirements of the cylinder body processing. e The value of (3) If the error exceeds the limit, first calculate v according to the reachability. e Modify it and then use the step construction method; specifically, set the artificial parameter β < 1 and the positive integer γ1 to generate an arithmetic sequence s = [u1, u1 + Δu, …, u1 + (u2-u1) β], where, Use s j Represents the jth element in the array; When j = 1, the block will not be split, and the binary search method is used to find a suitable Make the structure feasible after replacing v2 with this value. If the binary search does not terminate after several iterations, let When j>1, an extra column is inserted into the information matrix to form In the form of; first use the binary search method to determine the appropriate v in the interval [v1,v2] j , so that the information matrix It is feasible, and then the information matrix is The A block is processed according to the above method; by comparing the processing times corresponding to different j values, the j corresponding to the shortest processing time is selected, and the information matrix M is updated accordingly to improve the processing efficiency and accuracy of the cylinder body A structure part.

7. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: When processing the AU structure in the forward scan of "bidirectional scanning and feed speed planning", the information matrix is ​​assumed to be First, determine whether the block with information matrix (v1, v2) is feasible; If feasible, no block operation is required; If it is not feasible, then further determine whether the block with information matrix (v1, v3) is feasible; if the block is feasible, use the binary search method to find a suitable And modify u2 to make the block structure feasible; if the error does not exceed the bound but is not reachable, transform M into in Calculated based on accessibility; if the bow height error still exceeds the limit even after modifying the elements based on accessibility, the step construction method is used to generate an arithmetic sequence, and the separation position of the arithmetic sequence should be limited to the interval [u1u2]; When j=1, the AU structure degenerates into an A block; when j>1, the information matrix of the block structure becomes M=v1,v'1,v2,v3); at this time, first generate a suitable v'1 to make the block with the information matrix (v1,v'1) feasible, and then plan and process the AU structure with the information matrix (v'1,v2,v3) according to the above method to optimize the processing path and speed of the AU structure part of the cylinder body.

8. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: When processing the AUD structure in the reverse scan of "Bidirectional Scanning and Feed Speed ​​Planning", if the information matrix is Check the feasibility of the blocks with matrices (v1, v2) and (v3, v4) respectively. If both are feasible, no block operation is required. If at least one block is infeasible, perform the following process on the infeasible blocks: suppose block A is infeasible, and check whether the block with matrix (v1, v3) is feasible; If feasible, generate a feasible A block by bisection, whose matrix is make If this is not possible, a r =u3-u2; In addition, if (v1, v2) is feasible, specify a r =0; at the same time, according to the feasibility calculation of block D a r +d r The relationship between u3-u2 determines whether the U block with a feed rate of v2 can exist. There are two situations: The first case is a r +d r ≤u3-u2, in which case the planning for the acceleration side and the deceleration side is independent; specifically, if a r >0, we only need to focus on the AU structure with the matrix (v1,v2,v'2), where v'2=(u2+a r ,v2) T ; In this case, it is required that not only the first column of the information matrix of the AU structure be fixed, but also the last column. When the ladder construction method is used, if a feasible solution cannot be found for the structure with a matrix of (v1, v'1, v2, v'2), it can be degenerated into an AU structure with a matrix of (v1, v2, v'2). The same method is used for the UD structure with a matrix of (v3, v'3, v4), where v'3 = (u3-d r ,v2) T ; Finally, M was replace; The second case is a r +d r >u3-u2, at this time, the block structure must first be degenerated into an AD structure, and then the feed speed planning is carried out using the process for planning the AD structure.

9. The numerical control interpolation method for controlling bow height error according to claim 1, characterized in that: When processing the residual AD structure in the final scan of "bidirectional scanning and feed rate planning", the information matrix is For the matrix A block of the feasibility analysis to find a feasible For the matrix The D block is feasibility analyzed to find a feasible if The AD structure cannot be maintained, so the block structure will degenerate into a block with matrix (v1, v3); If min(v 2,1 ,v 2,2 )≥max(v1,v3), the AD structure can be maintained. At this time, the following processing is performed according to the bow height error of the two blocks: (1) If both blocks can ensure that the bow height error does not exceed the limit after the accessibility analysis; like Only the value of v2 needs to be modified; like The value of v2 is changed to This results in a decrease in the processing efficiency of block D. In order to improve the processing efficiency, the uniform speed segment generation method is used. After modifying v2, a column is added to the matrix to generate a uniform speed segment. Modified to The process of the uniform speed segment generation method is as follows: First, take a trial step Where Δu is less than a predetermined value, if the block structure corresponding to this matrix is ​​feasible, a suitable u'2 is found by bisection; otherwise, it is considered that a high-speed uniform speed segment that improves processing efficiency cannot be generated, so no additional U blocks are generated; When generating a constant speed segment on the left, if Change the value of v2 to Then use the uniform speed segment generation method on the right side. If Change the value of v2 to Then use the uniform speed segment generation method on the left side; (2) Only the first block has an out-of-bound bow height error after accessibility analysis; For the first block, use the ladder construction method and insert a new column v'1 between v1 and v2. For a given v'1, consider the feasibility of 9v'1,v3); if it is not feasible, degenerate to the result of j=1, otherwise, update the value of v2 as needed and use the uniform speed segment construction method on the required side; (3) Only the second block has an out-of-bound bow height error after accessibility analysis; (4) If the height error of both blocks exceeds the limit after the accessibility analysis, the ladder construction method is used for both blocks to obtain two arithmetic progressions: (a) (b) Use j i Represents the serial number of the item in the i-th sequence, thus obtaining the sequence pair (j1, j2). Using the bidirectional symmetric ladder construction method, by stipulating j1+j2=γ1+2, the number of sequence pairs is increased from (γ1+1) 2 It is reduced to γ1+1, thereby improving the computational efficiency.

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