Computer-aided interactive design method for B-spline surfaces capable of side milling

Abstract

The present invention discloses a computer-aided interactive design method for a B-spline surface that can be side-milled. The interactive design process is based on a B-spline surface S and follows the following steps: First, the user modifies the surface S through operations. Then, the surface S is optimized and adjusted to ensure that it has the ability to be side-milled. Thereafter, the user can continue to modify the surface through further operations and optimizations, making the surface design an incremental process that continues until a result that satisfies the user is achieved. In order to address the constraints of the design space, the present invention introduces an optimization framework that regards the tool shape, motion path, and designed surface as variables. By optimizing the designed surface, cutting tool, and motion path as variables in the optimization framework, the surface can be accurately processed by side milling. Experiments show that users can design various types of surfaces based on the framework, making this method an integrated design and manufacturing tool that effectively integrates surface design and processing.

Description

Technical Field

[0001] The invention relates to an interactive design method for a B-spline surface, in particular to a computer-aided interactive design method for a B-spline surface capable of being processed by side milling. Background Art

[0002] Five-axis CNC milling is a widely used manufacturing technology, typically divided into spot milling and side milling. Side milling offers higher machining efficiency and better machining quality than spot milling due to the larger contact area between the cutting tool and the surface, significantly reducing the tooth ridge problem during milling.

[0003] Path planning determines the motion path of the cutting tool and directly affects machining efficiency and accuracy. A reasonable path planning strategy usually requires a detailed analysis of the geometric characteristics of the designed surface. Compared with point milling, side milling path planning for free-form surfaces is much more difficult because a continuous tangential contact curve between the tool and the target surface must be maintained during the machining process. Therefore, the range of surfaces that can be accurately manufactured by side milling is inherently limited. Existing methods are either restricted to ruled surfaces or strive to find a large range of machinable areas on the designed surface. To enhance the capabilities of side milling technology, some works consider the shape of the cutting tool as a variable and calculate the optimal tool shape to adapt to the designed surface. Summary of the Invention

[0004] The present invention provides a computer-aided interactive design method for side-milling B-spline surfaces. Unlike existing path planning methods for fixed design surfaces, this method chooses to build a new framework to design surfaces that can be machined with high precision by side milling, while also determining the optimal tool shape and motion path. Specifically, the framework integrates the requirements of high-precision side milling into the surface design process and provides an interactive tool for designing B-spline surfaces. The framework not only outputs the designed surface, but also generates a set of motion paths and corresponding cutting tools, ensuring that the surface can be machined with high precision by five-axis CNC side milling using these tools and paths.

[0005] The purpose of the present invention is achieved through the following technical solutions:

[0006] A computer-aided interactive design method for side milling of B-spline surfaces comprises the following steps:

[0007] Step 1: Initialize S,S c and Φ

[0008] Initialize the design surface S and its control surface S c , both of which have the same simple shape and, at the same time, provide an initial cutting tool Φ;

[0009] Step 2: Modify Sc Surface

[0010] The user adjusts S by modifying the control points c shape, while S is c Apply node insertion algorithm to update;

[0011] Step 3: Initialize the motion path

[0012] Calculate N initial motion paths, recorded as

[0013] Step 4: Optimize the surface for side milling

[0014] Optimize the side milling processability of S while optimizing the cutting tool shape Φ and motion path This process requires ensuring that S and S c Approximately, the specific optimization method is as follows:

[0015] Step 41: Design a surface S consisting of M×N control points P ij , B-spline surface defined by i=1,...,M,j=1,...,N;

[0016] Step 42: The cutting tool is represented as a surface of revolution Φ, which is defined as the envelope of a family of single-parameter spheres centered at a point on the axis l. Let l(s) be the arc length parameter of l. The surface of revolution Φ is defined by the radius function r(s) of the surface. The r(s) function is represented in discrete form as a vector of radius values ​​[r k =r(v k )], where v k Defined as v k =(k-1) / (K-1), k=1,…,K, K is the number of sampling points;

[0017] Step 43: A tool motion path R i Defined by the trajectory of the tool axis l, the trajectory forms a surface R(s,t). Let t represent the time parameter, a(t) and b(t) are the motion trajectory curves of the two endpoints of l. By expressing a(t) and b(t) as B-spline curves of order d, the motion trajectory is defined as a B-spline surface of order d×1:

[0018] R(s,t)=a(t)(1-s)+b(t)s,s,t∈[0,1]

[0019] Step 44: Define single noodle constraints and G of adjacent noodle 1 Continuity constraints and regularization constraints are used to ensure that the design surface can approximate the envelope of the moving tool, where:

[0020] The single noodle constraints include the following constraints:

[0021] Rigid motion constraint: Assume that the length L of the tool axis is fixed and for each boundary curve a i (u) and b i (u) The corresponding ruled surface R i (u),v), by keeping the tool axis constant in length ||a throughout the motion i (u)-b i (u)||=L to ensure the rigidity of the tool motion. The rigid motion constraint is obtained by a set of sampling parameters u j Evaluate this condition to define:

[0022] (a i (u j )-b i (u j )) T (a i (u j )-b i (u j ))=L 2

[0023] Distance constraint: Distance constraint is applied to ruled surface R i Selected sampling points on (u,v) where u j and v k is a sampling point in the parameter domain, and v k Matching the sampling of the tool radius function, for each sample point Calculate its foot point on S and in Surface normal at Then about the point The point distance constraint and tangent distance constraint are:

[0024]

[0025] During the optimization process, each iteration Treated as a constant value;

[0026] Motion smoothness constraint: To ensure smooth motion of the tool, the ruled surface R i (u,v) boundary curve a i (u) and b i The smoothness of the curve is evaluated by the size of its first and second order derivatives, and the motion smoothness is constrained by the sampling parameter u j The definition is as follows:

[0027] a″i (u j )=0,a′ i (u j )=0

[0028] b″ i (u j )=0,b′ i (u j )=0

[0029] The G of the adjacent curved surface 1 Continuity constraints consider the following constraints:

[0030] G 1 Continuity constraint: requires the isoparametric curve μ of S i (u) is the envelope strip E i and E i+1 The common boundary curve between the two strips is to ensure that the G 1 Continuity, E i and E i+1 Must be in μ i (u) have the same normal vector. In addition, since S needs to approximate the envelope strip, E i and E i+1 In μ i The normal vector on (u) should be as consistent as possible with the normal vector on S, and the normal function is denoted as n i (u); based on μ i (u) Introducing the offset curve τ i (u), the curve is a B-spline curve, which is obtained by interpolating the data points τ i (u j )=μ i (u j )+∈ n n i (u j ) construction, where n i (u j ) is in μ i (u j ) at the surface normal,∈ n is the offset distance;

[0031] Envelope strip E i G of adjacent envelope strips 1 Continuity is defined as follows: Let a i (u) and b i (u) is the envelope surface E i The corresponding boundary curve of the ruled surface of the tool motion path, and the corresponding isoparameters of the surface S are μ i (u) and μ i+1 (u), the related offset curve is τi (u) and τ i+1 (u), considering the sampling parameter u j , calculate the sampling point a i (u j ) on the curve μ i The foot point of the perpendicular on (u) is denoted as Calculate sampling point b i (u j ) on the curve μ i+1 The foot point of the perpendicular on (u) is denoted as Then the sampling parameter u j Department, G 1 The continuity constraints are defined as follows:

[0032]

[0033] Offset curve constraint: To ensure τ i (u) is the offset curve along the normal direction, and the parameter t ij The offset curve constraints are defined as follows:

[0034]

[0035] in It is in G 1 μ in continuity constraints i (u) is the set of footpoint parameters on the graph;

[0036] The regularization constraints include the following constraints:

[0037] Surface smoothness constraint: For the sampling parameter (u j ,v i ), the surface smoothness constraint is defined as follows:

[0038] S uu (u j ,v i )=0,S u (u j ,v i )=0

[0039] S vv (u j ,v i )=0,S v (u j ,v i )=0

[0040] Shape deviation constraint: Assume Represents the position of the S control point on the surface before optimization. The shape deviation constraint is defined as follows:

[0041]

[0042] Step 45: Use the guided projection method to solve G that satisfies the single-curve noodle constraint and adjacent curved noodle 1 Variables of continuity constraints and regularization constraints. In the process of solving the optimization problem, each type of constraint sets a weight value. The specific parameter values ​​are as follows: G1 =1.0,λ Rigid =λ PD =λ TD =0.1,λ Smooth =λ Offset =λ RegSurface =λ RegControl =10 -5 , each weight corresponds to a type of constraint, where the subscript G1 corresponds to the G of the adjacent curve surface 1 G1 continuity constraint in continuity constraint, Rigid corresponds to rigid motion constraint in single-curved noodle constraint, PD corresponds to point distance constraint in single-curved noodle constraint, TD corresponds to tangential distance constraint in single-curved noodle constraint, Offset corresponds to G1 of adjacent curved noodle 1 The offset curve constraint in the continuity constraint, Smooth corresponds to the motion smoothness constraint in the single-curved noodle constraint, RegSurface corresponds to the surface smoothness constraint in the regularization constraint, and RegControl corresponds to the shape deviation constraint in the regularization constraint;

[0043] Step 5: Update the control surface

[0044] In order to maintain S and S c The approximate relationship between them is taken as the target, and S is updated by solving a surface approximation problem. c , if the surface design has not been completed, proceed to step 2, if the surface design is completed, proceed to step 6;

[0045] Step 6: Output

[0046] Generate the final design surface S, optimized cutting tool Φ and tool motion path

[0047] Compared with the prior art, the present invention has the following advantages:

[0048] 1. In order to avoid errors caused by planning the processing path of the designed surface, the present invention provides a technology for integrating computer-aided surface design and manufacturing. In order to enhance the flexibility of surface modeling, an optimization framework is provided, which regards the designed surface, cutting tool shape and tool motion path as optimization variables. The designed surface consists of a series of processing strips with smooth transitions between adjacent strips to ensure high-quality surface processing results. In addition, in order to improve computational efficiency and support interactive design processes, the constraints are restated as quadratic equations, and the guided projection method is used for optimization. Experimental results show that the method of the present invention can effectively design complex free-form surfaces, providing an innovative core technology for computer-aided design software.

[0049] 2. This paper presents the first interactive design method for highly complex side-millable B-spline surfaces. A series of experiments demonstrate the method's ability to design complex shapes suitable for practical applications. Furthermore, this paper provides an important reference for modeling the shape of side-millable surfaces, including but not limited to sequentially arranged machining strips. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 It is a B-spline surface S (semi-transparent light blue surface with grid lines) and a set of envelope surfaces E i (white and dark blue) approximation, these envelope surfaces are generated by the cutting tool moving along a set of motion paths;

[0051] Figure 2 It is an interactive design framework for side-milling B-spline surfaces;

[0052] Figure 3 is the motion path R and the envelope surface Ω generated by the cutting tool Φ - ;

[0053] Figure 4 The sampling points on S are translated along the normal direction to obtain interpolation points, and the offset curve is constructed based on the interpolation point fitting;

[0054] Figure 5 Is the processing strip E i G of the boundary curve 1 Continuity;

[0055] Figure 6 This is a surface design. In each single-step design, only one control point is modified. (ag) shows the design process starting from the initial plane and step by step. N = 10 strips are used. Each design step shows the shape of the milling cutter, as well as the distance error and G. 1 Color coding of continuity errors; (h) is the final design result after multiple interactive design steps;

[0056] Figure 7 It is a complex surface design (free-form machining surface) using N = 20 envelope strips. (ac) shows the three key steps from the initial plane to the final design result, as well as the distance error and G 1 Color coding of continuity errors; (d) shows the final design surface;

[0057] Figure 8 The complex surface design (mask) uses N = 30 envelope strips. (ac) shows the three key steps from the initial plane to the final design result, as well as the distance error and G 1 Color coding of continuity errors; (d) shows the final design surface. DETAILED DESCRIPTION

[0058] The technical solution of the present invention is further described below with reference to the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention that does not depart from the spirit and scope of the technical solution of the present invention should be included in the scope of protection of the present invention.

[0059] In surface design, multiple factors jointly determine the range of surfaces that can be side milled in the design space. In addition to the shape and motion path of the cutting tool, the arrangement of the machining strips is also crucial. The machining strips are formed by the cutting tool along a single path, creating an envelope surface that approximates the design surface. In order to achieve high-precision machining, the envelope surface must approximate the design surface with high precision. The machining strips along all motion paths together cover the entire design surface. Theoretically, the arrangement of these strips can be very complex and may involve topological singularities. However, the present invention focuses on a simple case where the machining strips are arranged in sequence (see Figure 1 ). In order to further improve the surface quality, the present invention requires a smooth transition between adjacent processing strips, which requires G between the corresponding envelope surfaces. 1 Continuity.

[0060] The key algorithm of the present invention involves combining the design surface S with a set of G 1The problem of fitting a continuous envelope surface has been partially explored in existing research (Kanika Rajain, Michal Bizzarri, Miroslav Lavicka, Jiri Kosinka, and Michael Barton. 2023. Towards G1-continuous multi-strip path-planning for 5-axis flank CNC machining of free-form surfaces using conicalcutting tools. Computer-AidedDesign 163, (2023), 103555.), but the study regards the designed surface as a fixed target shape, and the existing technology does not consider the machinability of the surface in surface design. In the interactive surface design method of the present invention, the designed surface S can be operated and optimized by the user so that it has the property of being able to be side milled. In addition, in order to provide the greatest possible flexibility, the present invention also optimizes the tool shape.

[0061] This paper proposes, for the first time, an interactive design method for highly complex side-millable B-spline surfaces. This method interactively modifies a B-spline surface S, following these steps: First, the user modifies surface S. Then, surface S is optimized to ensure it is side-millable. The user can then continue to update the surface through further modifications and optimizations. This method makes surface design an incremental process that continues until a satisfactory result is achieved.

[0062] The present invention regards the design surface S as a series of processing strips ε={E i}, each processed strip E i is the cutting tool Φ along the motion path R i In order to optimize the side milling capability of the surface S, the present invention not only considers the surface S itself during the optimization process, but also considers the cutting tool Φ and a set of motion paths. As an optimization variable. In order to achieve high-quality machined surfaces, the present invention requires tangent continuity between adjacent machined zones. The final output of the algorithm includes the designed B-spline surface, the cutting tool shape, and the corresponding motion path set. The present invention also uses a control surface to modify the surface S. The control surface is a B-spline surface with fewer control points, which approximates the surface S during the design process. Other shape deformation techniques, such as free-form deformation (FFD), can also be used. Figure 2 As shown, the interactive design method of the present invention mainly includes the following steps:

[0063] Step 1: Initialize S,S c Initialize the design surface S and its control surface S c , both have the same shape and are simple shapes (such as planes or cylinders). S c The number of control points of S is small, while S c At the same time, an initial cutting tool Φ is provided, which is usually a cylindrical tool, and its size and length are set by user-defined parameters.

[0064] Step 2: Modify S c Surface. The user adjusts S by modifying the control points c At the same time, the design surface S is obtained by c Apply the node insertion algorithm to perform the update.

[0065] Step 3: Initialize the motion path. Calculate N initial motion paths, denoted as These paths will need to be further optimized later. This step is necessary when the user first edits, but in subsequent iterations, if the user makes incremental changes, the motion path results of the previous iteration can serve as a good starting value for the next round of design iteration, and there is no need to perform this initialization operation again.

[0066] Step 4: Optimize the side milling surface. Optimize the machinability of S, while optimizing the cutting tool shape Φ and motion path. This process requires ensuring that S and S c Approximate, and S is capable of high-precision side milling.

[0067] The present invention designs a surface in a constrained space and solves an optimization problem. The variables in the optimization include the designed surface S, the cutting tool Φ, and a set of motion paths of the tool. The details are as follows:

[0068] (1) The design surface S is a surface consisting of M×N control points P ij , B-spline surface defined by i=1,...,M,j=1,...,N.

[0069] (2) The cutting tool is represented as a surface of revolution Φ, which is defined as the envelope of a family of single-parameter spheres with a point on the axis l as the sphere center. Let l(s) be the arc length parameter of l, then the surface of revolution Φ can be defined by the radius function r(s) of the surface. In the implementation of the algorithm, the r(s) function is represented in discrete form as a vector of radius values ​​[r k =r(v k )], where v k Defined as v k=(k-1) / (K-1), k=1,…,K, K is the number of sampling points.

[0070] (3) A tool motion path R i Defined by the trajectory of the tool axis l, this trajectory forms a surface R(s,t). Let t be the time parameter, and a(t) and b(t) be the motion trajectory curves of the two endpoints of l. By expressing a(t) and b(t) as B-spline curves of degree d, the motion trajectory is defined as a B-spline surface of degree d×1:

[0071] R(s,t)=a(t)(1-s)+b(t)s,s,t∈[0,1] (1)

[0072] As the tool moves along its motion path, two envelope surfaces, an upper envelope surface and a lower envelope surface, are formed. The present invention only focuses on the envelope surface that is closest to the designed curved surface. Figure 3 The motion of the cutting tool and an envelope surface generated by it are shown.

[0073] The present invention defines a series of constraints to ensure that the designed surface can approximate the envelope surface of the moving tool.

[0074] 1. Single noodle constraint:

[0075] (1) Rigid motion constraint. Assume that the length L of the tool axis is fixed, and for each boundary curve a i (u) and b i (u) The corresponding ruled surface R i (u,v), by keeping the tool axis constant in length ||a throughout the motion i (u)-||b i (u)||=L to ensure the rigidity of the tool motion. The rigid motion constraint is achieved by a set of sampling parameters u j Evaluate this condition to define:

[0076] (a i (u j )-b i (u j )) T (a i (u j )-b i (u j ))=L 2 (2)

[0077] (2) Distance constraint. In order to ensure that the envelope surface generated by the moving tool is as close as possible to the design surface S, a soft constraint corresponding to the distance error is defined. The distance constraint is applied to the ruled surface R i Selected sampling points on (u,v) where uj and v k is a sampling point in the parameter domain, and v k Matches the sampling of the tool radius function. For each sample point Calculate its foot point on S and in Surface normal at About Dot The point distance constraint and tangent distance constraint are:

[0078]

[0079] During the optimization process, each iteration It should be noted that the sampling points at both ends of each tool axis (corresponding to v1 = 0 and v K = 1), as they will be handled in the G1 continuity constraints.

[0080] (3) Motion smoothness constraint. In order to ensure the smooth motion of the tool, the algorithm evaluates the ruled surface R i (u,v) boundary curve a i (u) and b i The smoothness of the curve is evaluated by the size of its first and second order derivatives. The motion smoothness constraint is on the sampling parameter u j The definition is as follows:

[0081]

[0082] The above constraints have also been discussed in the existing surface approximation work. The difference between the present invention and the existing method is that the present invention uses the surface S as a variable in the optimization process. For example, in the method of the present invention, the foot point in equations (3) and (4) is is a function of the control points of the designed surface S and can be optimized. In existing surface approximation methods, the foot point of the distance error function remains fixed in each iteration. This improvement allows the algorithm to explore the optimal shape of the millable surface within a confined space.

[0083] 2. G of adjacent curved surfaces 1 Continuity constraints:

[0084] It is expected that there will be a smooth transition between two adjacent processing strips. This smooth transition is specifically determined by the tool moving along the path R i and R i+1 The resulting envelope E i and E i+1 G between 1 Continuity definition. G 1 Continuity Requirements i and Ei+1 Share a common boundary curve C and have the same normal vector on C. Curve C needs to lie on the surface S, and E i and E i+1 The normal vector at C should match the surface normal of S. In surface design, the curve C is aligned with the isoparm μ of S. i (u) alignment and maintain modeling flexibility. Consider the following constraints:

[0085] (1)G 1 Continuity constraint. Consider μ i (u) is the envelope strip E i and E i+1 To ensure that the G 1 Continuity, E i and E i+1 Must be in μ i (u) have the same normal vector. In addition, since S needs to approximate the envelope strip, E i and E i+1 In μ i The normal vector on (u) should be as consistent as possible with the normal vector on S, and the normal function is denoted as n i (u).

[0086] However, directly converting the normal vector n i (u) is used as a function of the surface S, which will lead to a more complex constraint expression, making the optimization more difficult and the optimization may converge to a local minimum. In order to reduce the complexity of the constraint, the present invention is based on the boundary curve μ i (u) Introduce an auxiliary curve (called offset curve) to simplify μ i (u) is expressed as the normal vector, thereby simplifying the constraints.

[0087] With μ i The corresponding offset curve of (u) is denoted as τ i (u), using a B-spline curve, whose initial value is constructed by interpolating the following data points: τ i (u j )=μ i (u j )+∈ n n i (u j ), where n i (u j ) is in μ i (u j ) at the surface normal, and the offset distance ∈ n In the implementation of the algorithm, it is set to 0.1. A schematic diagram of the offset curve can be found in Figure 4 In addition, it is important to note that μi (u) and τ i (u) will share the same parameter u.

[0088] Each strip E i G between adjacent stripes 1 Continuity is defined as follows. Figure 5 . Let a i (u) and b i (u) is the envelope surface E i The corresponding boundary curve of the ruled surface of the tool motion path, and the corresponding isoparameters of the surface S are μ i (u) and μ i+1 (u), the corresponding offset curves are τ i (u) and τ i+1 (u). Calculate the sampling point a i (u j ) on the curve μ i The foot point of the perpendicular on (u) is denoted as Similarly, calculate the sampling point b i (u j ) on the curve μ i The foot point of the perpendicular on (u) is denoted as Without loss of generality, the present invention assumes that the cutting tool is oriented so that a i (u) is aligned with the starting point of the radius function (e.g. v=v1). j Department, G 1 The continuity constraints are defined as follows:

[0089]

[0090] (2) Offset curve constraint

[0091] In order to ensure τ i (u) is the offset curve along the normal direction, and the parameter t ij The offset curve constraints are defined as follows:

[0092]

[0093] in It's G 1 μ in continuity constraints i The set of footpoint parameters on (u).

[0094] 3. Regularization constraints

[0095] (1) Surface smoothness constraint

[0096] In order to ensure the smoothness of the designed surface S, the present invention adopts approximate bending energy. This method is often used in surface approximation work. j ,v i ), the surface smoothness constraint is defined as follows:

[0097]

[0098] (2) Shape deviation constraint

[0099] In order to provide an incremental surface interaction design framework, it is necessary to control the difference between the optimized surface and its current state. To this end, the present invention introduces a soft constraint based on the change of control points. Represents the position of the S control point on the surface before optimization. The shape deviation constraint is defined as follows:

[0100]

[0101] 4. Optimization solution

[0102] The present invention adopts the text-guided projection method to solve the variables that meet the above constraints. In the process of solving the optimization problem, each type of constraint is set with a weight value. The specific parameter values ​​are as follows: G1 =1.0,λ Rigid =λ PD =λ TD =0.1,λ Smooth =λ Offset =λ RegSurface =λ RegControl =10 -5 Each weight corresponds to a type of constraint, where the subscript G1 corresponds to the G of the adjacent curve surface. 1 G1 continuity constraint in continuity constraint, Rigid corresponds to rigid motion constraint in single-curved noodle constraint, PD corresponds to point distance constraint in single-curved noodle constraint, TD corresponds to tangential distance constraint in single-curved noodle constraint, Offset corresponds to G1 of adjacent curved noodle 1 The offset curve constraint in the continuity constraint, Smooth corresponds to the motion smoothness constraint in the single-curved noodle constraint, RegSurface corresponds to the surface smoothness constraint in the regularization constraint, and RegControl corresponds to the shape deviation constraint in the regularization constraint.

[0103] Step 5: Update the control surface. In order to maintain S and S c The algorithm takes S as the target and updates S by solving a surface approximation problem. c If the surface design is not yet complete, proceed to step 2.

[0104] After the optimization is completed and the side milling surface is obtained, the user may need to further adjust the shape of the design surface, which requires another round of optimization. c As a tool for users to manipulate the design surface. In the method of the present invention, S c It is necessary to approximate S throughout the design process. This approximation is achieved by solving a surface approximation problem for the target shape S.

[0105] Step 6: Output. A series of design iterations generates the final design surface S, the optimized cutting tool Φ, and the tool motion path The design surface S may need to be trimmed depending on the specific application.

[0106] A single design step includes shape modification, optimization, and control surface update. After each design step, the surface S becomes a surface that can be processed by side milling, and the cutting tool Φ and the corresponding motion path are generated. The complete design process typically involves a sequence of these steps to produce the final design surfaces, tool shapes, and motion paths.

[0107] Example:

[0108] In this embodiment, the computer-aided interactive design method for side-milling B-spline surfaces is implemented in C++ and uses the libigl graphics library. All experiments were performed on a desktop computer equipped with 16GB of memory and a 3.8GHz eight-core Intel Core i7 CPU.

[0109] 1. Evaluation Metrics

[0110] To evaluate the quality of the results, this embodiment needs to evaluate the approximation error between the designed surface S and the envelope strip sequence generated by the cutting tool. These indicators can measure the geometric error when side milling with the optimized cutting tool along the motion path. The approximation error is measured using the signed distance error, and the model is normalized so that the bounding box has a unit diagonal. This measurement is performed on the ruled surface R i Sampling points on Calculated at, defined as in yes The foot point on S, r k =r(t k ) is the corresponding radius of the tool, j, k are the sampling numbers.

[0111] In addition, the present invention also needs to measure the G between adjacent processing strips. 1Continuity error. Since both strips are optimized, the shared curve μ(u) is aligned with the common normal vector n(u) on S, so G is evaluated by measuring the angle between the normal vector on S and the normal vector of the corresponding boundary curve of the machined strip. 1 Continuity error. For the motion path R i , whose two boundary curves a i (u) and b i (u)G 1 The error is evaluated at a specific sampling point and is defined as follows:

[0112]

[0113] in is a i (u j ) in μ i (u) is the parameter of the foot point, j is the sampling number, n i (u) is the curve μ i (u) on the normal vector. For b i The corresponding symbols of (u) are defined in a similar way. θ(v1,v2) represents the angle (in degrees) between the two vectors v1 and v2.

[0114] 2. Experimental Results and Evaluation

[0115] This section presents experimental results for a multi-pass, single-step surface design process. Starting with a simple initial surface, the experiment is given a B-spline surface S, the milling direction, and the number of strips as input. The results show several intermediate key stages and the final optimized surface.

[0116] In each single-step design, the surface is represented by a series of envelope strips generated by moving the optimized cutting tool along the optimized motion path. The experiment uses color coding to visualize the distance error and G 1 Continuity error and displays the optimized tool shape for each critical step. Figures 6 to 8 Different types of surface design cases are shown in the experiment. Each example consists of an initial surface, several intermediate processes of single-step design stages, and the final design surface. In each single-step design process, the user modifies a control point and then performs an optimization step. After multiple rounds of single-step design processes, the desired design surface is finally obtained. In addition, the experiment shows the shape of the milling cutter during each round of optimization. The experiment uses color coding to visualize the distance error and G 1 Continuity error and showing the optimized cutting tool shape for each intermediate step.

[0117] The proposed method enables the interactive design of B-spline surfaces suitable for side milling. To address the constraints of the design space, the present invention introduces an optimization framework that treats the tool shape, motion path, and designed surface as variables. By optimizing the designed surface, cutting tool, and motion path as variables within the optimization framework, the surface can be precisely machined using side milling. Experiments have shown that users can design a variety of surface types based on the framework, making this method an effective design and manufacturing tool that integrates surface design and machining.

Claims

1. A computer-aided interactive design method for side-milling B-spline surfaces, characterized in that The method comprises the following steps: Step 1: Initialization , and Initialize the design surface and its control surface , both of which have the same simple shape and provide an initial cutting tool ; Step 2: Modify Surface The user adjusts the control points by modifying shape, and at the same time, Through Apply node insertion algorithm to update; Step 3: Initialize the motion path calculate The initial motion path is recorded as ; Step 4: Optimize the surface for side milling optimization Excellent side milling machinability while optimizing cutting tool shape and motion paths This process needs to ensure and Approximately, the specific optimization method is as follows: Step 41: Designing the Surface is a control points , B-spline surface defined by i=1,...,M, j=1,...,N; Step 42: Cutting Tool Represented as a Surface of Revolution , defined as the axis The envelope of a family of single-parameter spheres with the points on the sphere as the center, let for The arc length parameterization defines the rotation surface through the radius function r(s) , the r(s) function is represented in discrete form as a vector of radius values ,in Defined as ,K is the number of sampling points; Step 43: A tool motion path R i By tool axis The trajectory is defined, which forms a surface R(s,t). Let represents the time parameter, and yes The motion trajectory curves of the two endpoints are and Expressed as the order The B-spline curve of the motion trajectory is defined as B-spline surface: Step 44: Define single noodle constraints and adjacent noodle Continuity constraints and regularization constraints to ensure that the designed surface can approximate the envelope of the moving tool; Step 45: Use guided projection method to solve the problem of satisfying the single curved noodle constraint and adjacent curved noodle. Variables with continuity constraints and regularization constraints; Step 5: Update the control surface To maintain and The approximate relationship between As the goal, we update , if the surface design has not been completed, proceed to step 2, if the surface design is completed, proceed to step 6; Step 6: Output Generate the final design surface , optimized cutting tools and tool motion path .

2. The computer-aided interactive design method for side-milling B-spline surfaces according to claim 1, characterized in that The single noodle constraints include the following constraints: Rigid motion constraint: Assume that the tool axis L is fixed, and for each boundary curve and Corresponding ruled surface , by maintaining a constant length of the tool axis throughout the entire motion To ensure the rigidity of tool motion, the rigid motion constraint is implemented by a set of sampling parameters Evaluate this condition to define: Distance Constraint: Distance constraint applied to ruled surfaces Selected sampling points on ,in and are sampling points in the parameter domain, and Matching the sampling of the tool radius function, for each sample point , calculate its The foot point on , and in Surface normal at , then about the point The point distance constraint and tangent distance constraint are: During the optimization process, each iteration Treated as a constant value; Motion smoothness constraint: To ensure smooth motion of the tool, the ruled surface is evaluated Boundary Curve and The smoothness of the curve is evaluated by the size of its first and second order derivatives, and the motion smoothness is constrained on the sampling parameters The definition is as follows: 。 3. The computer-aided interactive design method for side-milling B-spline surfaces according to claim 2, characterized in that The adjacent curved strips Continuity constraints consider the following constraints: Continuity Constraints: Requirements It is an envelope strip and The common boundary curve between the two strips is to ensure Continuity, and Must be in have the same normal vector, and since The envelope strip needs to be approximated, so and exist The normal vector on The normal vectors on the surface are as consistent as possible, and the normal function is recorded as ;based on Introducing an offset curve , the curve is a B-spline curve, which is obtained by interpolating the data points structure, where is The surface normal at , is the offset distance; Envelope Strip With adjacent envelope strips Continuity is defined as follows: and Envelope The corresponding boundary curves of the ruled surface of the tool motion path and the corresponding isoparametric lines of the surface S are and , the related offset curve is and , considering the sampling parameter u j , calculate the sampling points On the curve The foot point of the vertical , calculate the sampling points On the curve The foot point of the vertical , then in the sampling parameters Department, The continuity constraints are defined as follows: Offset Curve Constraint: To ensure is the offset curve along the normal direction, with parameter The offset curve constraints are defined as follows: in , yes Continuity constraints The set of perpendicular point parameters on .

4. The computer-aided interactive design method for side-milling B-spline surfaces according to claim 3, characterized in that The regularization constraints include the following constraints: Surface smoothness constraint: For sampling parameters , the surface smoothness constraint is defined as follows: Shape deviation constraint: Assume Represents the optimized front surface The position and shape deviation constraints of the control points are defined as follows: 。 5. The computer-aided interactive design method for side-milling B-spline surfaces according to claim 4, characterized in that In step 45, in the process of solving the optimization problem, a weight value is set for each type of constraint. The specific parameter values ​​are as follows: , each weight corresponds to a type of constraint, where the subscript G1 corresponds to the adjacent curve surface G1 continuity constraint in continuity constraint, Rigid corresponds to rigid motion constraint in single noodle constraint, PD corresponds to point distance constraint in single noodle constraint, TD corresponds to tangential distance constraint in single noodle constraint, Offset corresponds to adjacent noodle The offset curve constraint in the continuity constraint, Smooth corresponds to the motion smoothness constraint in the single-curved noodle constraint, RegSurface corresponds to the surface smoothness constraint in the regularization constraint, and RegControl corresponds to the shape deviation constraint in the regularization constraint.

Images