Implicit voxel Boolean subtraction optimization method in five-axis machining simulation

Abstract

The invention relates to an implicit voxel Boolean subtraction optimization method in five-axis machining simulation. According to the method, an implicit tool geometric expression model based on a symbolic distance function is constructed, and a dynamic voxel modeling technology with a layered structure is combined, so that efficient interactive modeling of a material removal process between a tool and a workpiece is realized. The whole process of the method comprises four key stages of spatial positioning, vertex interference detection, projection correction and grid reconstruction, and can cope with geometric modeling challenges caused by continuous motion of a multi-degree-of-freedom and multi-attitude cutter. A dynamic block loading mechanism and a local state synchronization strategy are introduced, so that the real-time response capability of Boolean subtraction operation under complex working conditions is effectively improved, and meanwhile, the continuity and integrity of machining geometric boundaries are kept. The method has good expandability and robustness, obviously reduces space-time complexity of calculation on the basis of keeping geometric surface continuity and topological correctness, realizes millisecond-level response time in an experiment, and has a good application prospect. The method is widely applied to virtual machining simulation and intelligent process verification systems of complex free-form surface parts such as aero-engine blades and automobile structural parts.

Description

Technical Field

[0001] The present invention relates to the field of CNC machining simulation and geometric modeling, and specifically to an implicit voxel Boolean subtraction operation optimization method in five-axis machining simulation, which is suitable for multi-degree-of-freedom machining path geometry simulation and real-time visualization system of complex free-form surface parts in aerospace, automobile manufacturing, etc. Background Art

[0002] With the surge in demand for complex parts processing in fields such as aerospace and automotive manufacturing, five-axis machining technology has become a core process due to its multi-degree-of-freedom cutting capabilities. In five-axis machining, path verification is an important step to ensure machining accuracy and avoid potential defects. However, traditional path verification methods, such as trial cutting, are often faced with high costs and high risks. Computer simulation methods, through virtual simulation technology, can verify machining paths in a digital environment in advance, avoiding the high costs and risks associated with physical trial cutting.

[0003] In the machining simulation technology system, workpiece geometry modeling and tool scanning volume construction constitute the two core links. Existing research mainly develops along the following paths:

[0004] (1) In terms of workpiece modeling, early discrete mesh generation methods relied on uniform sampling or fixed subdivision rules, resulting in redundant calculations in flat areas and loss of details in high curvature areas. Although precise modeling methods (such as B-rep and CSG) can describe complex topological changes, Boolean subtraction operations face the challenges of high computational complexity and long time consumption in complex geometric models, making it difficult to meet the real-time simulation requirements of five-axis machining.

[0005] (2) In terms of tool sweep volume construction, explicit parameterization methods directly express the tool sweep volume boundary based on parameterized surfaces or discrete grids, achieving phased breakthroughs in piecewise linear interpolation of free-form surface tool trajectories and geometric description of complex tools. However, the dynamic changes in tool posture in five-axis machining cause the complexity of solving parametric equations to increase exponentially with the increase in the degrees of freedom of motion. The implicit method significantly reduces the time complexity of Boolean subtraction operations and naturally avoids the problem of geometric self-intersection. However, it lacks an analytical expression for the nonlinear deformation caused by rotational motion, resulting in a high contour error rate.

[0006] In addition, the geometric representation of the material removal process mostly uses a one-way discretization model, such as the Dexel model, which cannot capture the three-dimensional dynamic boundary evolution caused by the tool's multi-posture cutting, and is difficult to adapt to the three-dimensional dynamic boundary evolution problem caused by the tool's multi-degree-of-freedom posture changes in five-axis linkage machining.

[0007] Therefore, there is an urgent need for a material removal simulation method that can adapt to five-axis dynamic tool motion, support high-speed Boolean operations, and have good space management capabilities to solve the problems of existing technologies. Summary of the Invention

[0008] In view of this, the present invention proposes an efficient Boolean subtraction operation method based on a layered dynamic voxel model and implicit tool scanning surface theory. The purpose of the present invention is to achieve high-fidelity, real-time modeling of the material removal process between the tool and the workpiece in five-axis CNC machining simulation through the deep integration of the axial layered discretization strategy and the implicit signed distance field.

[0009] The technical solution adopted by the present invention to achieve the above-mentioned purpose is: an implicit voxel Boolean subtraction operation optimization method in five-axis machining simulation, comprising the following steps:

[0010] S1. Spatial positioning: Construct the tool AABB bounding box, build a layered dynamic voxel model through axial layered discretization and dynamic block collaborative optimization, and map the tool AABB bounding box to the index space of the layered dynamic voxel model to determine the voxel range to be processed;

[0011] S2. Vertex interference detection: Through geometric transformation and space decomposition strategy, combined with implicit signed distance function (SDF), voxel vertex interference detection is performed to determine the interference state between the workpiece vertex and the tool;

[0012] S3, vertex projection correction: the interference vertices are projected and corrected by regional classification to simulate the material removal effect during the cutting process;

[0013] S4, state synchronization and mesh update: Update the voxel state according to the vertex state matrix, generate incremental mesh reconstruction and release the completely cut voxel resources.

[0014] The hierarchical dynamic voxel model is constructed by using hierarchical voxel units (LVUs) as basic geometric expression units to dynamically segment the range of the workpiece to be processed; in each hierarchical voxel unit (LVU), vertices of each layer are stored by layer and vertices of adjacent layers are connected by side quadrilaterals to ensure geometric continuity between layers; and the cutting status of the voxels is recorded in real time, so that they are dynamically loaded and instantiated when the tool may interfere, generating an index space; the possible interference is the possible occurrence of processing operations when the tool is within the AABB bounding box.

[0015] The hierarchical voxel unit LVU formula is as follows:

[0016]

[0017] where N x ,N y ,N z are discretization parameters, corresponding to the X / Y grid resolution and the Z-level number of layers; the geometric feature set V ijk ∈R3 represents the coordinates of the 8 vertices of the kth layer; the state mark vector S ij=(isBoundary, isCut, isEmit), isBoundary marks whether it is a boundary voxel, isCut marks whether the current voxel interferes with the tool, and isEmit indicates that the voxel has been completely cut; the vertex state vector T ijk ∈{0,1} 8 , used to store the cutting status of each vertex.

[0018] The method of constructing the tool AABB bounding box includes: calculating the maximum and minimum coordinate values ​​of the workpiece and the tool in the directions of each coordinate axis; constructing the AABB bounding box, and screening potential interference voxels based on the tool bounding box.

[0019] The implicit signed distance function SDF is used to express the geometric shape of the ball-cylinder composite tool, which is composed of the ball domain distance function and the cylinder domain distance function, and the Boolean union is realized by taking the minimum value; wherein SDF ball (P) is calculated based on the Euclidean distance between point P and the center c, SDF cyl (P) Based on the joint determination of the projection of point P in the direction of the tool axis and the radial offset, this combined SDF expression is used to achieve real-time expression and calculation of complex tool surfaces and postures without explicit intersection.

[0020] The vertex interference detection comprises the following steps:

[0021] The workpiece vertex v world Mapping from the world coordinate system to the tool local coordinate system, the vertex position v is obtained local , used for calculation based on the tool's own axial direction;

[0022] In the tool local coordinate system, the vertex position v local Decomposed into the longitudinal projection along the tool axis and the offset in the radial plane (l, r);

[0023] Based on the tool's geometric characteristics and kinematic constraints, the three-dimensional space is divided into several characteristic regions. The axial projection component of the vertex in the tool's local coordinate system is used to determine whether the vertex is interfering. The formula is as follows: when SDF(p) < 0, it indicates that the vertex is interfering, and SDF(p) > = 0, it indicates that there is no interference.

[0024]

[0025] SDF(P)=min(SDF ball (p), SDF cyl (p))

[0026] Where c∈R 3 is the coordinate of the tool center, R is the radius of the ball head, d∈S 2is the unit vector in the axial direction of the tool, L is the length of the cylindrical blade, and l is the projection length of point P along the axial direction of the tool.

[0027] The vertex projection correction comprises the following steps:

[0028] Based on the judgment results of the multi-region state machine, when the vertex is located inside the tool, it is projected onto the tool surface to simulate material removal;

[0029] Region-by-region projection: The ball head area is normalized and scaled along the direction from the center of the ball to the vertex, so that the vertex is accurately projected onto the ball head surface; the cylindrical area maintains the vertex position l in the tool axis and scales it to the cylindrical surface in the radial plane.

[0030] The vertex projection correction is calculated as follows:

[0031]

[0032] Where R tool 、R work are the rotation matrices of the tool and workpiece to the world coordinate system, c∈R 3 is the coordinate of the tool center, R is the radius of the ball head, d∈S 2 is the unit vector in the axial direction of the tool, L is the length of the cylindrical blade, and (Vx, Vy, Vz) represents the coordinates of the vertex.

[0033] The state synchronization and grid update include the following steps:

[0034] Use the error threshold ∈ to judge the difference between the front and back displacements of each vertex. If it exceeds the threshold, update its corresponding state matrix to the "cut" state;

[0035] For each voxel, the state vector T of all vertices ijk Traverse, if all are "cut", mark the voxel as "unloadable" and release it from the computing resources;

[0036] A spatial hash table is used to incrementally reconstruct the topology of all voxels marked as "cut" but not unloaded, remove duplicate vertices, and generate patches;

[0037] Perform local grid updates based on voxel block indexing mechanism to optimize index space.

[0038] The present invention achieves high-precision and high-efficiency modeling of material removal processes under complex tool trajectories by fusing a hierarchical dynamic voxel model with a signed distance field, and has the following advantages:

[0039] 1. The introduction of signed distance function and regional state machine can adapt to any tool posture changes;

[0040] 2. The layered dynamic voxel model supports on-demand loading and real-time updates, significantly reducing computational and memory overhead;

[0041] 3. Using a local vertex correction algorithm to ensure geometric continuity while avoiding topological problems such as cracks and broken surfaces that occur in traditional Boolean algorithms;

[0042] 4. The entire process supports dynamic simulation of multi-degree-of-freedom tool trajectories, meeting the dual requirements of real-time performance and accuracy in industrial scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 The figure is a flow chart of the overall method of Boolean subtraction operation.

[0044] Figure 2 Corrected flowchart for vertex projection.

[0045] Figure 3 This is a simulation diagram of the experimental environment.

[0046] Figure 4 This is the vertical feed simulation result diagram.

[0047] Figure 5 This is the simulation result of inclined feeding.

[0048] Figure 6 This is the simulation result diagram of circumferential rotation.

[0049] Figure 7 This is a graph showing the time consumption of Boolean subtraction operations. DETAILED DESCRIPTION

[0050] To make the above-mentioned objects, features, and advantages of the present invention more readily apparent, the specific implementation methods of the present invention are described in detail below with reference to the accompanying drawings. The following description sets forth many specific details to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways than those described herein, and those skilled in the art may make similar modifications without departing from the scope of the invention. Therefore, the present invention is not limited to the specific implementation methods disclosed below.

[0051] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art of the present invention. The terms used in the specification of the invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention.

[0052] A method for optimizing implicit voxel Boolean subtraction operations in five-axis machining simulation includes the following steps:

[0053] Step 1) Spatial positioning: Construct an AABB bounding box and map the tool bounding box to the index space of the hierarchical dynamic voxel model to determine the voxel range to be processed.

[0054] Step 2) Vertex interference detection: Through geometric transformation and space decomposition strategy, voxel vertex interference detection is performed based on the implicit signed distance function (SDF) to efficiently determine the interference state between the workpiece vertex and the tool scanning volume.

[0055] Step 3) Vertex projection correction: The interfering vertices are projected and corrected by regional classification to simulate the material removal effect during the cutting process;

[0056] Step 4) State synchronization and mesh update: Update the voxel state according to the vertex state matrix, generate incremental mesh reconstruction and release the completely cut voxel resources.

[0057] In the further step 1), the layered dynamic voxel model breaks through the accuracy-efficiency contradiction of the traditional method through the collaborative optimization of axial layered discretization and dynamic block management. Its core innovation is to decouple the cubic complexity problem of the traditional uniform three-dimensional voxel model into the collaborative optimization of two-dimensional plane discretization and axial layering, thereby achieving a dynamic balance between geometric representation accuracy and computational efficiency. The layered voxel unit (LVU) is the basic geometric expression unit of the model, and its mathematical formalization definition formula is as follows. Each voxel block is dynamically loaded and instantiated only when the tool may interfere. The cutting state of the voxel is recorded in real time through the vertex state matrix, and the cutting feature of the Z-axis direction is expressed in fine-grained manner in combination with the layered structure, effectively supporting the modeling and simulation of five-axis cutting trajectories in arbitrary directions and multiple degrees of freedom in three-dimensional space.

[0058]

[0059] where N x ,N y ,N z are discretization parameters corresponding to the X / Y grid resolution and the number of layers in the Z direction. Each LVU unit contains the following items:

[0060] Geometric feature set V ijk ∈R3: represents the coordinates of the eight vertices of the kth layer. The vertices of each LVU are stored by layer, and the vertices of adjacent layers are connected by side quadrilaterals to ensure geometric continuity between layers, achieving intra-layer topological invariance and inter-layer continuity constraints.

[0061] State mark vector S ij =(isBoundary, isCut, isEmit), isBoundary marks whether it is a boundary voxel, isCut marks whether the current voxel interferes with the tool, and isEmit indicates that the voxel has been completely cut.

[0062] Vertex state vector T ijk∈{0,1} 8 , used to store the cutting status of each vertex (0 for uncut, 1 for cut).

[0063] In the further step 1), the spatial positioning comprises the following steps:

[0064] (1) Tool bounding box calculation: Calculate the maximum and minimum coordinate values ​​of the workpiece and tool scanning volume in each coordinate axis direction, use them as parameters to construct an AABB bounding box, and quickly screen potential interference voxels based on the tool bounding box.

[0065] (2) Voxel index range calculation: The tool AABB bounding box is mapped to the index space of the hierarchical dynamic voxel model to determine the voxel range to be processed.

[0066] In the further step 2) vertex interference detection, the signed distance function SDF uniformly expresses the geometric form of the ball-cylinder composite tool, which is composed of the ball domain distance function and the cylinder domain distance function, and the Boolean union is realized by taking the minimum value; wherein SDF ball (P) is calculated based on the Euclidean distance between point P and the center c, SDF cyl (P) Based on the projection of point P on the tool axis and the joint determination of the radial offset, this combined SDF expression can achieve real-time expression and calculation support for complex tool surfaces and postures without explicit intersection. It includes the following steps:

[0067] (1) Coordinate transformation: transform the workpiece vertex v world Mapping from the world coordinate system to the tool local coordinate system allows subsequent calculations to be performed based on the tool's own axial direction, eliminating the complexity caused by posture changes and simplifying kinematic coupling problems.

[0068] (2) Axial projection decomposition: In the local coordinate system of the tool, the vertex position v local Decomposed into the longitudinal projection along the tool axis and the offset in the radial plane (l, r). This decomposition simplifies 3D collision detection into geometric relationship judgment within a 2D plane, improving computational efficiency.

[0069] l=v local ·d

[0070]

[0071] Where d, l, and r are the tool axial unit vector, projection length, and radial distance, respectively.

[0072] (3) Multi-region state machine: Based on the tool's geometric features and kinematic constraints, the three-dimensional space is divided into several characteristic regions. The axial projection component of the vertex in the tool's local coordinate system is used to determine whether the vertex is interfering. The formula is as follows: when SDF(p) < 0, the vertex is interfering, and SDF(p) > = 0, the vertex is not interfering.

[0073]

[0074] SDF(P)=min(SDF ball (p), SDF cyl (p))

[0075] Where c∈R 3 is the coordinate of the tool center, and R is the radius of the ball head. 2 is the unit vector in the axial direction of the tool, L is the length of the cylindrical blade, and l is the projection length of point P along the axial direction of the tool.

[0076] In the further step 3), the vertex projection correction comprises the following steps:

[0077] Based on the results of the multi-region state machine, when a vertex is located inside the tool, it is projected onto the tool surface to simulate material removal. Projection strategies for different regions: The spherical region is scaled along the direction from the sphere center to the vertex, ensuring that the vertex is accurately projected onto the spherical region surface. The cylindrical region first maintains the vertex's position l in the tool's axial direction and then scales it to the cylindrical surface in the radial plane.

[0078]

[0079] Where R tool 、R work are the rotation matrices of the tool and workpiece to the world coordinate system. c∈R 3 is the coordinate of the tool center, and R is the radius of the ball head. 2 is the unit vector in the axial direction of the tool, L is the length of the cylindrical blade, (V x , V y , V z ) represents the coordinates of the vertex, and clamp is a mathematical function, which is mainly used to limit the value within a fixed range.

[0080] In the further step 4), the state synchronization and grid update include the following steps:

[0081] (1) After the vertex projection correction is completed, the error threshold ∈ is used to judge the difference between the front and back displacement of each layered voxel unit (LVU). If it exceeds the threshold, the corresponding state matrix is ​​updated to the "cut" state;

[0082] (2) For all vertex state vectors T of each voxel ijk Traverse, if all are "cut", mark the voxel as "unloadable" and release it from the computing resources;

[0083] (3) Use the spatial hash table to perform incremental topology reconstruction on all voxels marked as “cut” but not unloaded, remove duplicate vertices, generate facets, and ensure mesh continuity and visualization quality;

[0084] (4) Local grid updates are performed based on the voxel block indexing mechanism, and the block strategy is combined to reduce overhead and ensure that Boolean operations have real-time performance and geometric accuracy.

[0085] Example 1: Boolean subtraction process for five-axis machining simulation

[0086] like Figure 1 As shown, the Boolean subtraction operation consists of the following four stages:

[0087] 1) Spatial positioning: Calculate the AABB bounding box of the tool based on its current position and posture, map the AABB to the voxel model index space, locate the index range of the affected block, and only activate the Block instance corresponding to this area.

[0088] 2) Vertex interference detection: transform the voxel vertex v world Convert to v local , calculate its projection length l in the direction of the tool axis and radial offset r; if SDF(P)<0, it is determined that the point is inside the tool.

[0089] 3) Vertex projection correction: For points that are judged to be interference, different strategies are used according to the area where they are located ( Figure 2 ): The spherical head domain is normalized and scaled along the radial direction and projected onto the spherical surface; the cylindrical domain maintains the axial position l and is compressed to the cylindrical surface in the radial plane to ensure the physical consistency and surface continuity of the processing.

[0090] 4) State synchronization and mesh update: Update the vertex state vector T according to the displacement threshold ∈ ijk , updates the voxel cutting status; if all vertices have been cut, the voxel is marked as "unloadable"; otherwise, the mesh reconstruction process begins. A spatial hash table is used to record the vertices of the activated voxels, and local triangles are regenerated through indexing to achieve incremental updates of visualization.

[0091] Example 2: Simulation Verification and Performance Evaluation

[0092] Experimental environment construction: Build a real-time machining simulation system on the Unity platform. The experimental environment is as follows: Figure 3The experiments were run on conventional computing devices, focusing on verifying the robustness and practicality of the method without dedicated hardware acceleration, and providing a reference for generalized deployment in industrial scenarios.

[0093] Simulation verification:

[0094] When the tool is pressed down along the Z axis, the cutting area forms a regular circular groove on the upper surface of the workpiece. The transition between the side wall and the bottom surface of the groove is continuous and smooth, and there are no holes or local geometric damage. The boundary contour is clear, the surface smoothness is strictly consistent with the tool geometric characteristics, and there is no surface tearing problem ( Figure 4 ).

[0095] When the tool cuts obliquely from the upper side of the workpiece at a specific inclination angle, the cutting depth changes continuously along the tool axis direction, forming an asymmetric groove with a smooth and continuous groove surface, no local distortion or curvature mutation, and no step artifacts ( Figure 5 ).

[0096] When the tool rotates circumferentially to cut the side surface, the cutting groove boundary is clear and sharp, the transition between the groove bottom and the side wall is smooth, and there is no overcutting phenomenon caused by topological lag in traditional explicit modeling ( Figure 6 ).

[0097] Performance evaluation:

[0098] The Boolean subtraction operation time changes dynamically with the number of tool positions and is stable at 10-20ms throughout the entire process. The simple plane cutting time is stable within 10ms, and the complex trajectory can be controlled within 20ms. Figure 7 ).

[0099] Experimental results show that the present invention can still significantly reduce the computational load under complex motion trajectories, meeting the stringent requirements of the CNC system for real-time simulation. The algorithm can meet the real-time requirements of scanning surface construction in five-axis machining solid geometry simulation.

[0100] The above is a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should be regarded as within the scope of protection of the present invention.

Claims

1. An implicit voxel Boolean subtraction optimization method in five-axis machining simulation, characterized in that: The following steps are involved: S1. Spatial positioning: Construct the tool AABB bounding box, build a layered dynamic voxel model through axial layered discretization and dynamic block collaborative optimization, and map the tool AABB bounding box to the index space of the layered dynamic voxel model to determine the voxel range to be processed; S2. Vertex interference detection: Through geometric transformation and space decomposition strategy, combined with implicit signed distance function (SDF), voxel vertex interference detection is performed to determine the interference state between the workpiece vertex and the tool; S3, vertex projection correction: the interference vertices are projected and corrected by regional classification to simulate the material removal effect during the cutting process; S4, state synchronization and mesh update: Update the voxel state according to the vertex state matrix, generate incremental mesh reconstruction and release the completely cut voxel resources.

2. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 1 is characterized in that: The hierarchical dynamic voxel model is constructed by using hierarchical voxel units (LVUs) as basic geometric expression units to dynamically segment the range of the workpiece to be processed; in each hierarchical voxel unit (LVU), vertices of each layer are stored by layer and vertices of adjacent layers are connected by side quadrilaterals to ensure geometric continuity between layers; and the cutting status of the voxels is recorded in real time, so that they are dynamically loaded and instantiated when the tool may interfere, generating an index space; the possible interference is the possible occurrence of processing operations when the tool is within the AABB bounding box.

3. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 2, characterized in that: The hierarchical voxel unit LVU formula is as follows: where N x ,N y ,N z are discretization parameters, corresponding to the X / Y grid resolution and the Z-level number of layers; the geometric feature set V ijk ∈R3 represents the coordinates of the 8 vertices of the kth layer; the state mark vector S ij =(isBoundary, isCut, isEmit), isBoundary marks whether it is a boundary voxel, isCut marks whether the current voxel interferes with the tool, and isEmit indicates that the voxel has been completely cut; Vertex state vector T ijk ∈{0,1} 8 , used to store the cutting status of each vertex.

4. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 1, characterized in that: The method of constructing the tool AABB bounding box includes: calculating the maximum and minimum coordinate values ​​of the workpiece and the tool in the directions of each coordinate axis; constructing the AABB bounding box, and screening potential interference voxels based on the tool bounding box.

5. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 1, characterized in that: The implicit signed distance function SDF is used to express the geometric shape of the ball-cylinder composite tool, which is composed of the ball domain distance function and the cylinder domain distance function, and the Boolean union is realized by taking the minimum value; wherein SDF ball (P) is calculated based on the Euclidean distance between point P and the center c, SDF cyl (P) Based on the joint determination of the projection of point P in the direction of the tool axis and the radial offset, this combined SDF expression is used to achieve real-time expression and calculation of complex tool surfaces and postures without explicit intersection.

6. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 1 or 5, characterized in that: The vertex interference detection comprises the following steps: The workpiece vertex v world Mapping from the world coordinate system to the tool local coordinate system, the vertex position v is obtained local , used for calculation based on the tool's own axial direction; In the tool local coordinate system, the vertex position v local Decomposed into the longitudinal projection along the tool axis and the offset in the radial plane (l, r); Based on the tool's geometric characteristics and kinematic constraints, the three-dimensional space is divided into several characteristic regions. The axial projection component of the vertex in the tool's local coordinate system is used to determine whether the vertex is interfering. The formula is as follows: when SDF(p) < 0, it indicates that the vertex is interfering, and SDF(p) > = 0, it indicates that there is no interference. SDF(P)=min(SDF ball (p),SDF cyl (p)) Where c∈R 3 is the coordinate of the tool center, R is the radius of the ball head, d∈S 2 is the unit vector in the axial direction of the tool, L is the length of the cylindrical blade, and l is the projection length of point P along the axial direction of the tool.

7. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 1, characterized in that: The vertex projection correction comprises the following steps: Based on the judgment results of the multi-region state machine, when the vertex is located inside the tool, it is projected onto the tool surface to simulate material removal; Region-by-region projection: The ball head area is normalized and scaled along the direction from the center of the ball to the vertex, so that the vertex is accurately projected onto the ball head surface; the cylindrical area maintains the vertex position l in the tool axis and scales it to the cylindrical surface in the radial plane.

8. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 7, characterized in that: The vertex projection correction is calculated as follows: Where R tool 、R work are the rotation matrices of the tool and workpiece to the world coordinate system, c∈R 3 is the coordinate of the tool center, R is the radius of the ball head, d∈S 2 is the unit vector in the axial direction of the tool, L is the length of the cylindrical blade, and (Vx, Vy, Vz) represents the coordinates of the vertex.

9. The implicit voxel Boolean subtraction optimization method in five-axis machining simulation according to claim 1, characterized in that: The state synchronization and grid update include the following steps: Use the error threshold ∈ to judge the difference between the front and back displacements of each vertex. If it exceeds the threshold, update its corresponding state matrix to the "cut" state; For each voxel, the state vector T of all vertices ijk Traverse, if all voxels are "cut", mark the voxel as "unloadable" and release it from computing resources; A spatial hash table is used to incrementally reconstruct the topology of all voxels marked as "cut" but not unloaded, remove duplicate vertices, and generate patches; Perform local grid updates based on voxel block indexing mechanism to optimize index space.

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