Symmetrical curved surface modeling method based on lattice type origami and structure and application thereof

Abstract

The invention discloses a symmetric curved surface modeling method based on lattice type origami and a structure and application thereof. The method comprises the following steps: S1, generating a strip contour of a target curved surface by designing an NURBS curve, and adding seven-fold Shuelai pattern information to form a basic left half strip; s2, defining an expandable constraint, a coplanar constraint and a side length constraint, and realizing a left half strip meeting geometric constraints; s3, for the generalized cylindrical curved surface, designing a lattice pattern, and generating a lattice strip and a generalized cylindrical model; s4, designing a lattice pattern for the axisymmetric curved surface, and generating a lattice strip and an axisymmetric model; s5, model folding information is calculated, and a rigid folding simulation sequence is obtained. According to the method, a grid-type unit approximation generalized cylinder and an axisymmetric curved surface are used, and a symmetric target curved surface can be fitted through an NURBS curve and model parameter adjustment. The lattice pattern designed by the method can improve the stability of the paper folding structure, and can be used for paper folding pattern research, three-dimensional structure modeling, folding robots and the like.

Description

Technical Field

[0001] The present invention relates to the fields of computer graphics and origami three-dimensional structure modeling, and in particular to a symmetrical surface modeling method based on lattice origami and its structure and application. Background Art

[0002] Origami is an art form that constructs a target shape by folding flat materials. Origami technology provides scientists and engineers with a design solution and has received widespread attention in fields such as engineering, computer graphics, and materials science. Origami structures can be designed for a variety of scenarios due to their flexibility and variability. Paper cutting is a derivative of origami technology that allows for more structural design forms by allowing cuts in flat materials. Patterned structures can be applied to foldable aerospace structures, lightweight folding robots, mechanical material structures, architecture, and art.

[0003] The six-fold mine pattern has high degrees of freedom and adaptability to different Gaussian curvatures. It is composed of a triangular structure and can be flexibly reverse-engineered to approximate various target shapes. However, there is a risk of self-intersection at its center point, which limits its application in certain structures. Paper-cutting technology allows cutting and gluing operations, realizes complex surface fitting and increases the freedom of the structure. Box-type paper-cutting can be used to approximate models of chessboard structures, realize surfaces that approximate lattice structures, and have certain load strength in terms of mechanical properties. However, the paper-cutting method introduces cuts and does not have the material integrity and model watertightness of the origami method. Summary of the invention

[0004] The present invention proposes a symmetrical surface modeling method based on lattice origami, its structure and application. First, NURBS curves are interactively generated to obtain the target surface contour, and a seven-fold torpedo pattern is added to obtain the basic left half strip. Secondly, expandable constraints, coplanar constraints and edge length constraints are defined and applied to the basic left half strip to satisfy the geometric constraints. On the basis of the seven-fold torpedo pattern, the central edge is expanded into a surface, and a lattice origami pattern is proposed. A generalized cylindrical target surface is constructed by copying and moving, and an axisymmetric target surface is constructed by copying, moving and rotating, so as to realize model fitting of two types of symmetrical surfaces. Based on the symmetry of model construction, a lattice pattern model folding calculation method is further provided to obtain a rigid folding simulation sequence.

[0005] In order to achieve the above object, the technical solution provided by the present invention is as follows:

[0006] A symmetrical surface modeling method based on lattice origami comprises the following steps:

[0007] S1. Generate the strip outline of the target surface by designing NURBS curves, and add the seven-fold mine pattern information to form the basic left half strip B h ;

[0008] S2. Define expandable constraints, coplanar constraints, and edge length constraints to achieve the left half strip S that meets the geometric constraints. h ;

[0009] S3. For generalized cylindrical surfaces, design grid patterns and generate grid strips S c , generalized cylindrical model M c ;

[0010] S4. For axisymmetric surfaces, design grid patterns and generate grid strips S a , axisymmetric model M a ;

[0011] S5. Calculate the folding information of two types of models to obtain a rigid folding simulation sequence.

[0012] To further illustrate, the above step S1 is specifically as follows:

[0013] S11. First, determine the shape of the target surface, interactively input the two-dimensional contour points of the target surface in the program, and construct the NURBS curve Γ. The target surface is composed of a grid of unit patterns, and the number of rows is N. r , the number of columns is N c , a column in the surface mesh is considered as a strip.

[0014] S12. The target surface is a two-type symmetrical structure. For the generalized cylindrical model, the curve Γ is translated to the left along the y-axis by half the width of the model strip, denoted as w h , and get the contour curve From the curve Γ and The left half strip profile B that constitutes the basic generalized cylindrical model Γ When constructing an axisymmetric model, the curve Γ is rotated counterclockwise around the z axis by Θ = π / N c Angle, forming a contour curve From the curve Γ and The left half strip profile B that constitutes the basic axisymmetric model Γ .

[0015] S13. According to the number of model rows, Γ Horizontally divided into N r For the 70% fold mine pattern, Figure 1 As shown in (a)(b)(c), based on the six-fold mine pattern, its center point can be stretched into an edge to generate a seven-fold mine pattern structure with five valley folds and two mountain folds. Fill the seven-fold mine pattern and apply an offset of half the strip width between adjacent rows to obtain the basic left half strip B h . Among them B h The plane where the left point of is located is Φ L , Bh The plane where the right point of is located is Φ C ,like Figure 2 (a) to (g) and Figure 3 (a) to (g). Due to the symmetry of the surface generation, B h The upper and lower edges of are boundary points, and the remaining points are internal points.

[0016] To further illustrate, the above step S2 specifically includes the following contents:

[0017] S21. The present invention takes symmetry into consideration and the optimization only needs to be applied to the left half strip B. h Considering the expandability of internal points, the model must be able to be fully unfolded into a plane without tearing or crossing. Figure 4 As shown in (a), the expandable residual d of the i-th internal vertex of the model i It can be defined as the sum of the angles around the point minus π, as shown in formula (1):

[0018]

[0019] Among them, N V Indicates the number of internal vertices in the left half strip, n i represents the number of adjacent vertices of the i-th internal vertex, α i,k Represents the arc between the i-th internal vertex and the adjacent k-th face. The residual E of the expandable constraint d Defined by formula (2):

[0020]

[0021] For B h The quadrilateral structure in Figure 4 (b) The coplanarity condition must be met to ensure that the component surface structures of the model are all located in the same plane. The coplanarity residual p of the i-th quadrilateral structure i Defined by formula (3):

[0022]

[0023] Among them, vertex subtraction represents vector, and Represents two vectors in the plane, and their cross product is the same as the result of the point The normal vectors of the planes are in the same direction. represents another vector, and the angle symbol ∠ indicates the calculation of the direction of the normal vector and the vector The angle between them, the number of units in the left half of the basic strip is N r , the coordinates of the quadrilateral vertices of the i-th face are expressed as Residual E of plane constraintp Defined by formula (4):

[0024]

[0025] For the pattern units that make up the model, during the optimization process, the center face and the quadrilateral faces on both sides of the pattern have the risk of degenerating to the edges, such as Figure 4 The upper side in (c) corresponds to side h i,a , the center plane corresponds to the edge h i,c The side h corresponding to the lower side i,b , for the left strip B of the foundation h The length constraints of the three types of edges of the i-th unit are imposed, and the length residual l of the edge of type t i,t Defined by formula (5):

[0026]

[0027] where h i,t Represents three different edges in the unit pattern, and the subscript t takes values ​​of a, b, and c, that is, h i,t Corresponding h i,a 、h i,c 、h i,b These three sides, for N r The edge length constraints are imposed on each element. represents the average value of the same type of edge, j represents the traversal sequence number of all edges, and the goal is to make the length of the edge close to the average value of the same type. The residual error E of the edge length is obtained l is the accumulation of the residual values ​​of the three types of edges, defined by formula (6):

[0028]

[0029] The iterative optimization process of geometric constraints can be summarized as minimizing the residual value through gradient descent, and cyclically verifying whether the geometric error of the current mesh model meets the given range. If the condition is not met, continue to iterate until the condition is met. When the condition is met or the upper limit of the iteration number is reached, the optimization is completed.

[0030] For this purpose, an objective function needs to be constructed to represent the total residual value E defined by formula (7):

[0031] E=ω d E d +ω p E p +ω l E l (7)

[0032] Among them, ω d ,ω p ,ω lThey represent the weights of expandability optimization, plane optimization, and length optimization, respectively. E represents the total iterative residual. Since expandability and coplanarity are the conditions for the model to be unfolded into a plane, they are set to 1. The side length residual only needs to meet the lower requirement. Through experiments, it is found that it can be set to 1 and halved successively during the iterative optimization process to obtain the left half strip S that meets the geometric constraints. h .

[0033] To further illustrate, the above step S3 specifically includes the following contents:

[0034] S31. For generalized cylindrical surface modeling, record w c is the center structure width of the grid pattern, w h is the left half strip width. First, S h Translate w along the y axis c / 2 length, Φ L Translate w along the y axis c The length, Φ c Keep still, then S h With plane Φ L and Φ C Formed between c / 2 width, fill this part with a quadrilateral structure to form a half strip of the generalized cylindrical model of the grid mine pattern, and then symmetric along the xz plane to obtain the following Figure 2 (d) The generalized cylindrical strip S of the complete grid-type mine pattern c .

[0035] S32, strip S c Copy N c -1 times, and then translate along the y axis (2w h +2w c )×N i The length of N i For the i-th strip, we get Figure 2 (g) The generalized cylindrical model of the lattice mine pattern shown.

[0036] To further illustrate, the above step S4 specifically includes the following contents:

[0037] S41. For axisymmetric surface modeling, such as Figure 3 The left half strip of c, plane Φ C With Φ L The angle θ is π / N c , translate the left half strip along the bisector of its corresponding fan, and fill the translated strip and plane Φ with a quadrilateral structure C With Φ L The gap is then symmetrically obtained along the xz plane as follows Figure 3 (d) The axisymmetric strip S of the complete grid-type mine patterna .

[0038] S42, strip S a Copy N c -1 times, and rotate around the z axis successively (2π / N c )×N i angle, forming Figure 3 (g) The axisymmetric model of the grid-type mine pattern shown.

[0039] To further illustrate, the above step S5 specifically includes the following contents:

[0040] S51. For the generalized cylindrical model, each folded state remains perpendicular to the xz plane during the entire movement, and the distance and relative position between vertices remain unchanged, which can maintain the rigidity of the folding. The folding method of the symmetric model defines the folding rate as Ψ C , as shown in formula (8):

[0041]

[0042] Where D is the left half strip width w of the generalized cylindrical model h , D′ is the current compression length, and the value range of D′ is from 0 to D, that is, from the deployed state to the fully compressed state. Since the quadrilateral structure in the center of the lattice pattern cannot be compressed, without considering the material thickness, the fully compressed state will have a width of 2w h ×N c lattice structure.

[0043] For axisymmetric models, the central axis needs to be dynamically calculated during the folding process to ensure that the final state remains rigid and parallel to the y-axis. A , as shown below:

[0044]

[0045] Where θ is the left half strip angle π / N c , Θ′ is the fan-shaped angle formed by the edges of the strips when folded, and the value range is 0 to Θ, that is, from the deployed state to the fully compressed state. The axisymmetric model changes the position of the central axis while compressing, so that the fully compressed state obtains a vertical structure. Since the quadrilateral structure in the center of the lattice pattern cannot be compressed, without considering the material thickness, the fully compressed state will obtain a width of 2w h ×N c lattice structure.

[0046] Based on the above method, the present invention also proposes an origami pattern structure, which is obtained by the symmetrical surface modeling method based on lattice origami.

[0047] The origami pattern structure proposed by the present invention can be applied to foldable aerospace structures, lightweight folding robots, mechanical material structures, architecture and art fields.

[0048] Beneficial effects of the present invention:

[0049] The present invention proposes a grid-type derived pattern based on the seven-fold mine pattern, and proposes a fitting symmetric surface and optimization method. The seven-fold mine contains seven folds, and the central structure is an edge. The present invention expands the central edge of the seven-fold mine into a surface, and proposes a grid-type origami pattern, which can be used to approximate the target surface and prevent the center point from self-intersecting. The present invention also provides two symmetric surface modeling and optimization methods, namely generalized cylindrical surfaces and axisymmetric surfaces. For models that meet geometric constraints, the present invention simulates a rigid folding process, which can show the folding deployment form of the model and provide a basis for the production of physical objects. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is the basic pattern unit and its corresponding 3 rows and 2 columns crease pattern;

[0051] Figure 1 (a) The basic mine pattern and 3 rows and 2 columns of creases;

[0052] Figure 1 (b) is the expansion of the six-fold mine pattern center point to a length of h c , and obtain the seven-fold torpedo pattern and the corresponding 3-row and 2-column crease pattern;

[0053] Figure 1 (c) The center edge of the seven-fold torpedo pattern is expanded to a width of w c , obtain the grid-type mine pattern and the corresponding 3-row and 2-column crease pattern;

[0054] Figure 2 It is a generalized cylindrical model constructed based on the 60%-fold mine, 70%-fold mine and grid-type mine patterns;

[0055] Figure 2 (a) NURBS curve Γ constructed interactively by the user;

[0056] Figure 2 (b) Six-fold mine strip constructed based on Γ;

[0057] Figure 2 (c) A seven-fold mine strip constructed based on Γ;

[0058] Figure 2 (d) A grid-shaped mine strip constructed using Γ as the basis;

[0059] Figure 2 (e) is a generalized cylindrical model generated using a six-fold mine strip;

[0060] Figure 2 (f) is a generalized cylindrical model generated using a seven-fold mine strip;

[0061] Figure 2 (g) is a generalized cylindrical model generated using a grid-shaped mine strip;

[0062] Figure 3 It is an axisymmetric model based on the 60%-fold mine, 70%-fold mine and grid-type mine patterns;

[0063] Figure 3 (a) NURBS curve Γ constructed interactively by the user;

[0064] Figure 3 (b) is the basic six-fold mine strip constructed using Γ;

[0065] Figure 3 (c) A seven-fold mine strip constructed based on Γ;

[0066] Figure 3 (d) A grid-shaped mine strip constructed using Γ as the basis;

[0067] Figure 3 (e) is an axisymmetric model generated using a six-fold mine strip;

[0068] Figure 3 (f) is an axisymmetric model generated using a seven-fold mine strip;

[0069] Figure 3 (g) is an axisymmetric model generated using a grid-shaped mine strip;

[0070] Figure 4 Geometry optimization for modeling based on lattice mine patterns;

[0071] Figure 4 (a) The angle relationship and expandable optimization between the grid-shaped mine pattern and the seven-fold mine pattern;

[0072] Figure 4 (b) The common quadrilateral structure and coplanar optimization of the grid-type mine pattern and the seven-fold mine pattern;

[0073] Figure 4 (c) The side length information and side length optimization of the grid-type mine pattern;

[0074] Figure 5 Schematic diagram of the process for implementing the present invention. DETAILED DESCRIPTION

[0075] The present invention will be further described below in conjunction with the accompanying drawings.

[0076] The present invention provides a symmetrical surface modeling method based on lattice origami, and its structure and application. A computer program can be used to fit the target surface, complete three-dimensional modeling and folding simulation, and memory materials, 3D printing and other manufacturing methods can be used.

[0077] like Figure 5 As shown, the present invention is a symmetrical surface modeling method based on lattice origami, comprising the following steps:

[0078] S1. Generate the strip outline of the target surface by designing NURBS curves, and add the seven-fold mine pattern information to form the basic left half strip B h ;

[0079] S2. Define expandable constraints, coplanar constraints, and edge length constraints to achieve the left half strip S that meets the geometric constraints. h ;

[0080] S3. For generalized cylindrical surfaces, design grid patterns and generate grid strips S c , generalized cylindrical model M c ;

[0081] S4. For axisymmetric surfaces, design grid patterns and generate grid strips S a , axisymmetric model M a ;

[0082] S5. Calculate the model folding information and obtain a rigid folding simulation sequence.

[0083] The specific contents of step S1 of the present invention include:

[0084] S11. First, determine the shape of the target surface, interactively input the two-dimensional contour points of the target surface in the program, and construct the NURBS curve Γ. The target surface is composed of a grid of unit patterns, and the number of rows is N. r , the number of columns is N c , a column in the surface mesh is considered as a strip.

[0085] S12. The target surface is a two-type symmetrical structure. For the generalized cylindrical model, the curve Γ is translated to the left along the y-axis by half the width of the model strip, denoted as w h , and get the contour curve Γ l When constructing an axisymmetric model, the curve Γ is rotated counterclockwise around the z axis by Θ = π / N c Angle, forming the contour curve Γ l . From the curves Γ and Γ l Constitute the basic left half strip outline B Γ .

[0086] S13. According to the number of model rows, Γ Horizontally divided into N rFor the 70% fold mine pattern, Figure 1 As shown, based on the six-fold mine pattern, its center point can be stretched into an edge to generate a seven-fold mine pattern structure with five valley folds and two mountain folds. Fill the seven-fold mine pattern and apply an offset of half the strip width between adjacent rows to obtain the basic left half strip B h . Among them B h The left point is at Φ L Plane, B h The right side point of C Flat Figure 2 and Figure 3 As shown. Due to the symmetry of the surface generation, B h The upper and lower edges of are boundary points, and the remaining points are internal points.

[0087] Step S2 of the present invention specifically includes the following contents:

[0088] S21. The present invention takes symmetry into consideration and the optimization only needs to be applied to the left half strip B. h Considering the expandability of internal points, the model must be able to be fully unfolded into a plane without tearing or crossing. Figure 4 As shown in (a), the expandable residual d of the i-th internal vertex of the model i It can be defined as the sum of the angles around the point minus π, as shown in formula (1):

[0089]

[0090] Among them, N V Indicates the number of internal vertices in the left half strip, n i represents the number of adjacent vertices of the i-th internal vertex, α i,k Represents the arc between the I-th internal vertex and the adjacent k-th face. The residual E of the expandable constraint d Defined by formula (2):

[0091]

[0092] For B h The quadrilateral structure in Figure 4 (b) The coplanarity condition must be met to ensure that the component surface structures of the model are all located in the same plane. The coplanarity residual p of the i-th quadrilateral structure i Defined by formula (3):

[0093]

[0094] Among them, vertex subtraction represents vector, and Represents two vectors in the plane, and the cross product is the same as the result of the point The normal vectors of the planes are in the same direction. represents another vector, and the angle symbol ∠ indicates the calculation of the direction of the normal vector and the vector The angle between them, the number of units in the left half of the basic strip is N r , the coordinates of the quadrilateral vertices of the Ith face are expressed as Residual E of plane constraint p Defined by formula (4):

[0095]

[0096] For the pattern units that make up the model, during the optimization process, the center face and the quadrilateral faces on both sides of the pattern have the risk of degenerating to the edges. Figure 4 The upper side in (c) corresponds to side h i,a , the center plane corresponds to the edge h i,c The side h corresponding to the lower side i,b , for the left strip B of the foundation h The length constraints of the three types of edges of the i-th unit are imposed, and the length residual l of the edge of type t i,t Defined by formula (5):

[0097]

[0098] where h i,t Represents three different edges in the unit pattern, for N r The edge length constraints are imposed on each element. Represents the average value of the same type of edge, and the goal is to make the length of the edge close to the mean of the same type. Get the edge length residual E l is the accumulation of the residual values ​​of the three types of edges, defined by formula (6):

[0099]

[0100] The iterative optimization process of geometric constraints can be summarized as minimizing the residual value through gradient descent, and cyclically verifying whether the geometric error of the current mesh model meets the given range. If the condition is not met, continue to iterate until the condition is met. When the condition is met or the upper limit of the iteration number is reached, the optimization is completed.

[0101] For this purpose, an objective function needs to be constructed to represent the total residual value E defined by formula (7):

[0102] E=ω d E d +ω p E p +ω l E l (7)

[0103] Among them, ωd ,ω p ,ω l They represent the weights of expandability optimization, plane optimization, and length optimization, respectively. E represents the total iterative residual. Since expandability and coplanarity are the conditions for the model to be unfolded into a plane, they are set to 1. The side length residual only needs to meet the lower requirement. Through experiments, it is found that it can be set to 1 and halved successively during the iterative optimization process to obtain the left half strip S that meets the geometric constraints. h .

[0104] Step S3 of the present invention specifically includes the following contents:

[0105] S31. For generalized cylindrical surface modeling, firstly transform S h Translate w along the y axis c / 2 length, Φ L Translate w along the y axis c The length, Φ C Keep still, then S h With plane Φ L and Φ C Formed between c / 2 width, fill this part with a quadrilateral structure to form a half strip of the generalized cylindrical model of the grid mine pattern, and then symmetric along the xz plane to obtain the following Figure 2 (d) The generalized cylindrical strip S of the complete grid-type mine pattern c .

[0106] S32, strip S c Copy N c -1 times, and then translate along the y axis (2w h +2w c )×N i The length of Figure 2 (g) The generalized cylindrical model of the lattice mine pattern shown.

[0107] To further illustrate, the above step S4 specifically includes the following contents:

[0108] S41. For axisymmetric surface modeling, such as Figure 3 The left half strip of c Φ C With Φ L The angle θ is π / N c , translate the left half strip along the bisector of its corresponding fan, and fill the translated strip and plane Φ with a quadrilateral structure C With Φ L The gap is then symmetrically obtained along the xz plane as follows Figure 3 (d) The axisymmetric strip S of the complete grid-type mine pattern a .

[0109] S42, strip Sa Copy N c -1 times, and rotate around the z axis successively (2π / N c )×N i angle, forming Figure 3 (g) The axisymmetric model of the grid-type mine pattern shown.

[0110] Step S5 of the present invention specifically includes the following contents:

[0111] S51. For the generalized cylindrical model, each folded state remains perpendicular to the xz plane during the entire movement, and the distance and relative position between vertices remain unchanged, which can maintain the rigidity of the folding. The folding method of the symmetric model defines the folding rate as Ψ c , as shown in formula (8):

[0112]

[0113] Where D is the left half strip width w of the generalized cylindrical model h , D′ is the current compression length, and the value range of D′ is from 0 to D, that is, from the deployed state to the fully compressed state. Since the quadrilateral structure in the center of the lattice pattern cannot be compressed, without considering the material thickness, the fully compressed state will have a width of 2w h ×N c lattice structure.

[0114] For axisymmetric models, the central axis needs to be dynamically calculated during the folding process to ensure that the final state remains rigid and parallel to the y-axis. The axisymmetric model motion folding rate is as follows:

[0115]

[0116] Where θ is the left half strip angle π / N c , Θ′ is the fan-shaped angle formed by the edges of the strips when folded, and the value range is 0 to Θ, that is, from the deployed state to the fully compressed state. The axisymmetric model changes the position of the central axis while compressing, so that the fully compressed state obtains a vertical structure. Since the quadrilateral structure in the center of the lattice pattern cannot be compressed, without considering the material thickness, the fully compressed state will obtain a width of 2w h ×N c lattice structure.

[0117] Based on the above method, the present invention also proposes an origami pattern structure, which is obtained by the symmetrical surface modeling method based on lattice origami.

[0118] The series of detailed descriptions listed above are only specific descriptions of feasible implementation methods of the present invention. They are not intended to limit the scope of protection of the present invention. All equivalent methods or changes that do not deviate from the technical creation of the present invention should be included in the scope of protection of the present invention.

Claims

1. A symmetrical surface modeling method based on lattice origami, characterized in that: The steps include: S1. Generate the strip outline of the target surface by designing NURBS curves, and add the seven-fold mine pattern information to form the basic left half strip B h ; Among them, the target surface includes two types of symmetric structures: generalized cylindrical surface and axisymmetric surface; S2. Define expandable constraints, coplanar constraints, and edge length constraints to achieve the left half strip S that meets the geometric constraints. h ; S3. For generalized cylindrical surfaces, design grid patterns and generate grid strips S c , we get the generalized cylindrical model M c ; S4. For axisymmetric surfaces, design grid patterns and generate grid strips S a , we get the axisymmetric model M a ; S5. Calculate the folding information of the two types of models to obtain a rigid folding simulation sequence.

2. A symmetrical surface modeling method based on lattice origami according to claim 1, characterized in that: The step S1 is specifically as follows: S11. First, determine the shape of the target surface, interactively input the two-dimensional contour points of the target surface in the program, and construct the NURBS curve Γ. The target surface is composed of a grid of unit patterns, and the number of rows is N. r , the number of columns is N c , a column in the surface mesh is considered as a strip; S12. The target surface includes two types of symmetrical structures. For the generalized cylindrical surface, the curve Γ is translated to the left along the y-axis by half the width of the model strip, denoted as w h , and get the contour curve From the curve Γ and The left half strip profile B that constitutes the basic generalized cylindrical model Γ ; When constructing an axisymmetric model, rotate the curve Γ counterclockwise around the z axis by Θ=π / N c Angle, forming a contour curve From the curve Γ and The left half strip profile B that constitutes the basic axisymmetric model Γ ,; S13. According to the number of model rows, Γ Divide horizontally into N r For the seven-fold mine pattern, based on the six-fold mine pattern, stretch its center point into an edge to generate a seven-fold mine pattern structure with five valley folds and two mountain folds; fill the seven-fold mine pattern and apply an offset of half the strip width between adjacent rows to obtain the basic left half strip B h , where B h The plane where the left point of is located is Φ L , B h The plane where the right point of is located is Φ C ; Due to the symmetry of the surface generation, B h The upper and lower edges of are boundary points, and the remaining points are internal points.

3. A symmetrical surface modeling method based on lattice origami according to claim 1, characterized in that: The step S2 comprises the following: S21. Considering the symmetry, the optimization only needs to be applied to the left half strip B h Considering the expandability of internal points, the model is required to be able to be fully unfolded into a plane without tearing or crossing, as follows: The expandable residual d of the i-th internal vertex of the model i It can be defined as the sum of the angles around the point minus π, as shown in formula (1): Among them, N V Indicates the number of internal vertices in the left half strip, n i represents the number of adjacent vertices of the i-th internal vertex, α i,k Represents the arc between the i-th internal vertex and the adjacent k-th face, which can be expanded to constrain the residual E d Defined by formula (2): For B h The quadrilateral structures in must satisfy the coplanarity condition to ensure that the component surface structures of the model are located in the same plane. The coplanarity residual p of the i-th quadrilateral structure i Defined by formula (3): Among them, vertex subtraction represents vector, and Represents two vectors in the plane, and the cross product is the same as the result of the point The normal vectors of the planes are in the same direction. represents another vector, and the angle symbol ∠ indicates the calculation of the direction of the normal vector and the vector The angle between them, the number of units in the left half of the basic strip is N r , the coordinates of the quadrilateral vertices of the i-th face are expressed as Residual E of plane constraint p Defined by formula (4): For the pattern units that make up the model, during the optimization process, the center face and the quadrilateral faces on both sides of the pattern have the risk of degenerating to the edges, so the left strip B of the foundation h The length constraints of the three types of edges of the i-th unit are imposed, and the length residual l of the edge of type t i,t Defined by formula (5): where h i,t Represents three different edges in the unit pattern, for N r The edge length constraints are imposed on each element. Represents the average value of the same type of edge. The goal is to make the length of the edge close to the mean of the same type, and obtain the edge length residual E l is the accumulation of the residual values ​​of the three types of edges, defined by formula (6):

4. A symmetrical surface modeling method based on lattice origami according to claim 3, characterized in that: The geometric constraints of step S2 are designed as follows: The iterative optimization process of geometric constraints is defined as: minimizing the residual value through gradient descent, and cyclically verifying whether the geometric error of the current mesh model meets the given range. If not, continue to iterate until the condition is met. When the condition is met or the upper limit of the number of iterations is reached, the optimization is completed; An objective function is constructed to represent the total residual value E, which is defined by formula (7): E=ω d E d +oh p E p +oh l E l (7) Among them, ω d ,ω p ,ω l They represent the weights of expandability optimization, plane optimization, and length optimization, respectively. E represents the total iterative residual. Since expandability and coplanarity are the conditions for the model to be unfolded into a plane, they are set to 1. The side length residual is set to 1 and is halved in the iterative optimization process to obtain the left half strip S that meets the geometric constraints. h .

5. The symmetrical surface modeling method based on lattice origami according to claim 1, characterized in that: The step S3 is specifically as follows: S31. For a generalized cylindrical surface, let w c is the center structure width of the grid pattern, w h is the left half strip width. First, S h Translate w along the y axis c / 2 length, Φ L The plane is translated along the y axis by w c The length, Φ C The plane remains stationary, then S h With plane Φ L and Φ C Formed between c / 2 width, fill this part with a quadrilateral structure to form a half strip of the generalized cylindrical model of the grid-type mine pattern, and then symmetrically along the xz plane to obtain the generalized cylindrical grid strip S of the complete grid-type mine pattern c ; S32, the strip S c Copy N c -1 times, and then translate along the y axis (2w h +2w c )×N i The length of N i For the i-th strip, we get the generalized cylindrical model M of the grid mine pattern c .

6. The symmetrical surface modeling method based on lattice origami according to claim 1, characterized in that: The step S4 is specifically as follows: S41. For an axisymmetric surface, the Φ of the left half strip C Plane and Φ L The angle θ between the planes is π / N c , translate the left half strip along the bisector of its corresponding fan, and fill the translated strip and plane Φ with a quadrilateral structure C With Φ L The gap is then symmetrically obtained along the xz plane to obtain the axisymmetric strip S of the complete grid mine pattern. a ; S42, strip S a Copy N c -1 times, and rotate around the z axis successively (2π / N c )×N i Angle, forming a grid-shaped mine pattern axisymmetric model M a .

7. The symmetrical surface modeling method based on lattice origami according to claim 1, characterized in that: The generalized cylindrical model M in step S5 c The folding information design is as follows: S51, for the generalized cylindrical model M c , each folding state remains perpendicular to the xz plane during the entire movement, the distance and relative position between vertices remain unchanged, and the rigidity of the folding can be maintained. The folding method of the symmetric model defines the folding rate as Ψ C , as shown in formula (8): Where D is the left half strip width w of the generalized cylindrical model h , D′ is the current compression length, and the value range of D′ is from 0 to D, that is, from the deployed state to the fully compressed state. Since the quadrilateral structure in the center of the grid pattern cannot be compressed, without considering the material thickness, the fully compressed state will have a width of 2w h ×N c lattice structure.

8. The symmetrical surface modeling method based on lattice origami according to claim 1, characterized in that: The axisymmetric model M in step S5 a The folding information design is as follows: For the axisymmetric model M a , the central axis needs to be dynamically calculated during the folding process to ensure that the final state remains rigid and parallel to the y-axis. The folding rate of the axisymmetric model is Ψ A , the calculation is as follows: Where θ is the left half strip angle π / N c , Θ′ is the fan-shaped angle formed by the edge of the strip when folded, and its value range is 0 to Θ, that is, from the deployed state to the fully compressed state, the axisymmetric model changes the position of the central axis while compressing, so that the fully compressed state obtains a vertical structure. Since the quadrilateral structure in the center of the lattice pattern cannot be compressed, without considering the material thickness, the fully compressed state will obtain a width of 2w h ×N c lattice structure.

9. An origami pattern structure of an industrial product, characterized in that: The origami pattern structure is obtained by the symmetrical surface modeling method based on lattice origami as described in any one of claims 1 to 8.

10. A foldable aerospace device, a lightweight folding robot, a mechanical material structure, a building structure, etc. using the origami pattern structure described in claim 9.

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