Computer-aided interactive design method capable of side-milling B-spline curved surface

Abstract

The invention discloses a computer-aided interactive design method capable of performing side milling on a B-spline curved surface, and the interactive design process is based on the B-spline curved surface S and follows the following steps: firstly, modifying the curved surface S by a user through operation; then, the curved surface S is optimized, and the curved surface is adjusted to ensure that the curved surface has the side milling machining capacity. Thereafter, the user can continue to modify the curved surface through further operation and optimization, so that the curved surface design becomes an increment process, and the increment process is continued until a satisfactory result of the user is achieved. In order to solve the constraint of a design space, an optimization framework is introduced, and a cutter shape, a motion path and a design curved surface are regarded as variables. A designed curved surface, a cutting tool and a motion path are regarded as variables in an optimization frame to be optimized, so that the curved surface can be precisely machined through side milling. Experiments show that a user can design various types of curved surfaces based on the frame, so that the method becomes a design and manufacturing integrated tool for effectively integrating curved surface design and machining.

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 product manufacturing technology, which is usually divided into spot milling and side milling. Side milling can provide higher processing efficiency than spot milling due to the larger contact area between the cutting tool and the surface, obtain better processing quality, and significantly reduce the tooth ridge problem in milling.

[0003] Path planning determines the motion path of the cutting tool, which directly affects the machining efficiency and accuracy. Reasonable path planning strategies usually require a detailed analysis of the geometric characteristics of the designed surface. Compared with point milling, path planning for side milling of free-form surfaces is much more difficult because a continuous tangential contact curve between the tool and the target surface needs to 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 try to find a large range of machinable areas on the designed surface. In order 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. This method is different from the existing path planning method for fixed design surfaces. Instead, it chooses to build a new framework to design surfaces that can be processed with high precision by side milling, and at the same time determine 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 to ensure that the surface can be processed with high precision by five-axis CNC side milling using these tools and paths.

[0005] The objective 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, and at the same time, S is c Apply node insertion algorithm to update;

[0011] Step 3: Initialize the motion path

[0012] Calculate N initial motion paths, denoted 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 rotation surface 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. Define the rotation surface Φ 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 will form 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-curve noodle constraints and adjacent curve noodle G 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 is a i (u) and b i (u) The corresponding ruled surface R i (u),v), by maintaining a constant length of the tool axis ||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 The 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. The motion smoothness is constrained on 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 isoparametric curves 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 by 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 isoparametric lines 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 by Calculate sampling point b i (u j ) on the curve μ i+1 The foot point of the perpendicular on (u) is denoted by 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, at parameter t ij The offset curve constraints are defined as follows:

[0034]

[0035] in It is in G 1 Continuity constraint μ 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-curved noodle constraint and adjacent curved noodle 1 Variables of continuity constraints and regularization constraints. In the process of solving optimization problems, 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 The G1 continuity constraint in the continuity constraint, Rigid corresponds to the rigid motion constraint in the single-curved noodle constraint, PD corresponds to the point distance constraint in the single-curved noodle constraint, TD corresponds to the tangent distance constraint in the single-curved noodle constraint, and Offset corresponds to the G1 continuity constraint of the 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 machining path planning for designed surfaces, 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 machining strips with smooth transitions between adjacent strips to ensure high-quality surface machining results. In addition, in order to improve computational efficiency and support interactive design processes, the constraints are restated as quadratic equations and optimized using guided projection methods. Experimental results show that the method of the present invention can effectively design complex free-form surfaces and provides an innovative core technology for computer-aided design software.

[0049] 2. The present invention proposes for the first time an interactive design method for highly complex side-millable B-spline surfaces. Through a series of experiments, it is proved that the method can design complex shapes that meet practical applications. In addition, including but not limited to sequentially arranged processing strips, the present invention provides an important reference for shape modeling of side-millable surfaces. 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) are approximated, and these envelope surfaces are generated by the movement of the cutting tool 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 It is a surface design. Only one control point is modified in each single-step design. (ag) shows the design process starting from the initial plane and step by step. N = 10 strips are used. The shape of the milling cutter, as well as the distance error and G are shown in each design step. 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 is a complex surface design (mask), using 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 in conjunction with the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be included in the protection scope 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 accuracy. 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 processed strips, which requires that the corresponding envelope surfaces have G 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-Aided Design 163, (2023), 103555.), but the study regards the designed surface as a fixed target shape, and the prior art 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] The present invention proposes for the first time an interactive design method for highly complex side-millable B-spline surfaces, which interactively modifies a B-spline surface S and follows the following steps: First, the user modifies the surface S. Then, the surface S is optimized to ensure that it has the ability to be side-milled. Thereafter, the user can 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, 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 shape 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 of which have the same shape and take simple shapes (such as planes or cylinders). c The number of control points of S is small, while S c At the same time, an initial cutting tool Φ is provided, usually a cylindrical tool, whose size and length are set by user-defined parameters.

[0064] Step 2: Modify S c The user adjusts S by modifying the control points c At the same time, the design surface S is c Apply the node insertion algorithm to 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 uses incremental modifications, the motion path results of the previous iteration can be used as a good initial value for the next round of design iterations, 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 Approximately, 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 , a B-spline surface defined by i=1,...,M,j=1,...,N.

[0069] (2) The cutting tool is represented as a rotation surface 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, then the rotation surface 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, the trajectory forms a surface R(s,t). Let t represent the time parameter, and 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:

[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 upper and lower envelope surfaces 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 obtained by 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 The 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. 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 magnitude 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, which can be optimized, while in the existing surface approximation methods, the foot point in the distance error function remains fixed in each iteration. This improvement allows the algorithm to explore the optimal shape of the millable surface in a restricted 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 motion path R i and R i+1 The resulting envelope E i and E i+1 G 1 Continuity definition. 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 by n i (u).

[0086] However, directly applying 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 constraints, the present invention is based on the boundary curve μ i (u) Introduce an auxiliary curve (called offset curve) to simplify μ i (u) to simplify 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 ), 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 seen 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 isoparametric lines 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 by Similarly, calculate the sampling point b i (u j ) on the curve μ i The foot point of the perpendicular on (u) is denoted by Without loss of generality, the present invention assumes that the cutting tool is oriented such 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, at parameter t ij The offset curve constraints are defined as follows:

[0092]

[0093] in It's G 1 Continuity constraint μ 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. For the sampling parameter (u 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 interactive 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 satisfy the above constraints. In the process of solving the optimization problem, each type of constraint is set with a weight value, and 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 The G1 continuity constraint in the continuity constraint, Rigid corresponds to the rigid motion constraint in the single-curved noodle constraint, PD corresponds to the point distance constraint in the single-curved noodle constraint, TD corresponds to the tangent distance constraint in the single-curved noodle constraint, and Offset corresponds to the G1 continuity constraint of the 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] When the optimization is completed and the side-millable surface is obtained, the user may need to further adjust the shape of the designed 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 generate 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-millable B-spline surfaces is implemented based on C++ and uses the libigl graphics library. All experiments are performed on a desktop computer equipped with 16GB memory and a 3.8GHz eight-core Intel Core i7 CPU.

[0109] 1. Evaluation Metrics

[0110] In order to evaluate the quality of the results, this embodiment needs to evaluate the approximation error between the designed surface S and the sequence of envelope strips 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 a signed distance error, and the model is normalized so that the bounding box has a unit diagonal. This measurement is used on the ruled surface R i The 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 is the sampling number.

[0111] In addition, the present invention also requires measuring the G between adjacent processed 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 specific sampling points and is defined as follows:

[0112]

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

[0114] 2. Experimental results and evaluation

[0115] This section presents experimental results of multiple rounds of the single-step surface design process. The experiments start with a simple initial surface, given a B-spline surface S, the milling direction and the number of strips as input. The experimental 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 shows the optimized tool shape for each critical step. Figure 6 to Figure 8 Different types of surface design cases are shown in the experiment. Each example includes 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 in 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 method proposed in the present invention realizes the interactive design of B-spline surfaces that can be processed by side milling. In order to solve the constraints of the design space, the present invention introduces an optimization framework that regards the tool shape, motion path and design surface as variables. By optimizing the design 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.

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: Initialize S,S c and Φ 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; Step 2: Modify S c Surface The user adjusts S by modifying the control points c shape, and at the same time, S is c Apply node insertion algorithm to update; Step 3: Initialize the motion path Calculate N initial motion paths, denoted as Step 4: Optimize the surface for side milling 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 approximate; Step 5: Update the control surface 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; Step 6: Output Generate the final design surface S, optimized cutting tool Φ and tool motion path 2. The computer-aided interactive design method for side-millable B-spline surfaces according to claim 1, characterized in that In step 4, the specific optimization method is as follows: 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; Step 42: The cutting tool is represented as a rotation surface 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. Define the rotation surface Φ by the radius function r(s). 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; Step 43: A tool motion path R i Defined by the trajectory of the tool axis l, the trajectory will form 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: R(s,t)=a(t)(1-s)+b(t)s,s,t∈[0,1] Step 44: Define single-curve noodle constraints and adjacent curve noodle G 1 Continuity constraints and regularization constraints to ensure that the designed surface can approximate the envelope of the moving tool; Step 45: Use the guided projection method to solve G that satisfies the single-curved noodle constraint and adjacent curved noodle 1 Variables for continuity constraints and regularization constraints.

3. The computer-aided interactive design method for side-millable B-spline surfaces according to claim 2, characterized in that The single noodle constraints include the following constraints: Rigid motion constraint: Assume that the length L of the tool axis is fixed and for each boundary curve is 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: (a i (u j )-b i (u j )) T (a i (u j )-b i (u j ))=L 2 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 The surface normal at 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 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 is constrained on the sampling parameter u j The definition is as follows:

4. The computer-aided interactive design method for side-millable B-spline surfaces according to claim 3, characterized in that The G of the adjacent curved surface 1 Continuity constraints consider the following constraints: G 1 Continuity constraint: requires isoparametric curves 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 by 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; 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 isoparametric lines of the surface S are μ i (u) and μ i+1 (u), the related offset curve is τ i (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 by Calculate sampling point b i (u j (On the curve μ i+1 The foot point of the perpendicular on (u) is denoted by Then the sampling parameter u j Department, G 1 The continuity constraints are defined as follows: Offset curve constraint: To ensure τ i (u) is the offset curve along the normal direction, at parameter t ij The offset curve constraints are defined as follows: (τ i (t ij )-μ i (t ij )) T (S u (t ij ,v i ))=0 (τ i (t ij )-μ i (t ij )) T (S v (t ij ,v i ))=0 (t i (t ij )-m i (t ij )) T (t i (t ij )-m i (t ij ))=(∈ n ) 2 in It's G 1 Continuity constraint μ i The set of footpoint parameters on (u).

5. The computer-aided interactive design method for side-millable B-spline surfaces according to claim 4, characterized in that The regularization constraints include the following constraints: Surface smoothness constraint: For the sampling parameter (u j ,v i ), the surface smoothness constraint is defined as follows: S uu (u j ,v i )=0,S u (u j ,v i )=0 S vv (u j ,v i )=0,S v (u j ,v i )=0 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:

6. The computer-aided interactive design method for side-millable B-spline surfaces according to claim 5, characterized in that In step 45, in the process of solving the optimization problem, a weight value is set for each type of constraint, and 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 The G1 continuity constraint in the continuity constraint, Rigid corresponds to the rigid motion constraint in the single-curved noodle constraint, PD corresponds to the point distance constraint in the single-curved noodle constraint, TD corresponds to the tangent distance constraint in the single-curved noodle constraint, and Offset corresponds to the G1 continuity constraint of the 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.

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