# Tool Path Generation Algorithm Based on Rough Machining of Jade Carving Surface

Authors: Xiong Huang; QingLan Li; JieLie Chen
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

To improve the quality of tool paths in surface machining, different machining requirements are considered in the tool path generation algorithm. In the traditional tool path generation algorithm, due to the lack of a unified mathematical calculation framework, these requirements often need to be considered separately, but they often compete with each other, so that different requirements cannot be satisfied at the same time and the tool path cannot achieve the optimal. To resolve this issue, an extensive tool path generation algorithm framework for arbitrary surfaces was proposed. The proposed algorithm transforms the tool path calculation framework into a constrained functional optimization problem by establishing an optimization function based on the potential energy field and an inequality function based on the gradient field in the machining area. By establishing finite element approximation, the functional optimization problem is transformed into a numerical optimization problem, which reduces the difficulty of solving. According to the machining requirements of die surface such as tool path smoothness, line width constraint and cutting direction constraints were transformed into the constraints of potential energy field isopotential line, and the method of generating reciprocating cutter path and cutter contact path isosection cutter path was proposed by optimizing the method. It is evident from the simulation and machining experiments results that continuous and smooth tool paths can be generated by using this algorithm, and machining quality and machining efficiency can be improved. By adding different machining requirements into the framework, a tool path with different optimization objectives can be generated.


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## Body

## 1. Introduction

Numerical control (NC) machining is widely used in the machining of free-form surfaces, which is an important part of CAD/CAM, and the generation of the tool path is the key link of NC machining. The tool path generation algorithm directly affects the machining quality and efficiency. The generation of tool path in NC machining mainly includes parameter line method, sectional plane method, projection method, and equal residual height method [1]. According to the characteristics of the workpiece, a combination of one or several different methods can be used to complete the rough machining and fine machining. Roughing is the main purpose of fast cutting blank, usually rough processing in the removal of material volume is 5 to 10 times of finishing and roughing took up most of the machining hours, so improve the efficiency of roughing for improving the efficiency of the whole computer numerical control (CNC) machining process. With the development of machine tool manufacturing technology and the emergence of high-quality machining tools, CNC machining appeared as a kind of efficient rough machining method, that is, drilling type rough machining (Plunge), generally called knife processing (hereinafter called knife processing). It is the milling cutter like drill along with the curved face drilling type processing, this method of machine bed and tool have higher requirements, usually can provide this kind of processing side CNC machine tools have very good stability and overall rigidity, the tool used has high hardness, very good toughness, and wear resistance. This machining method has been successfully applied to practical NC machining, and CNC machine tools are supporting this machining method. The practice has proved that the biggest advantage of the slotting tool machining method is that its machining efficiency is several times higher than that of ordinary surface rough machining.Multiaxis NC machining is an important means of machining complex curved surface parts. However, due to the multiaxis NC machining involved multiaxis linkage, interpolation, and other related algorithms involved in high-dimensional space, and nonlinear factors, the algorithm complexity is high, and the calculation becomes large. China's current research in this field cannot meet the requirements of advanced manufacturing to high-speed and high-precision development. At the same time, because they are multiobjective optimization problems, including machining efficiency, surface quality, machine tool, and tool life, safety and energy-saving, and other requirements, it is necessary to comprehensively apply the theory of machining technology, complex numerical (optimization) algorithms, advanced computational geometry, and multibody dynamics. The traditional method can only be optimized independently for one or several requirements, and the tool path cannot take into account different machining requirements. With the development of major national projects such as “High-grade CNC Machine Tools and Basic Manufacturing Equipment”, relevant research on multiaxis CNC machining theory and technology is increasingly imperative [1].At present, there are few research works in the literature about the calculation of tool paths in tool slotting. In some CAM software, such as MasterCAM, there is a tool path calculation method, but this tool path generation uses the uniform distribution method in the projection plane. The analysis and experiment results show that this uniform distribution point in the projection plane is more suitable for the surface processing of parts with a relatively flat surface. For the parts whose surface curvature distribution changes greatly, the distribution of residual height may be seriously uneven, and the subsequent semi-finishing will be adversely affected. Given the characteristics of cutter machining methods, and based on the analysis of the conditions suitable for cutter machining surface, a cutter roughing method based on residual height constraint was proposed, which not only guaranteed the machining efficiency of cutter machining but also provided the blank with uniform residual height distribution for subsequent machining.Stereolithography (STL) has been applied largely in numerical control (NC) processing of sculpting surfaces. Compared with parametric surfaces, the sculpting surfaces based on the triangular polyhedron model are applied in computer-aided design. CAD/computer-aided manufacturing (CAM) data exchange is easier to achieve. Lasemi et al. [1] were the first in this field to study the tool path generation algorithm based on the polyhedron model. In a three-axis, NC machining system, carved on ball head tool surface cutting tool path diameter, the authors [2, 3] conducted more research on flat-bottomed cutter rarely used for three-axis sculpture surface processing, but the flat cutter has the advantages of high machining efficiency. In some of the concave compound curved surface machining spherical cutter can produce residues, choosing a suitable flat cutter can effectively avoid cutting. Zou and Zhao [4] studied the trajectory generation algorithm of flat bottom Cutter, but this algorithm is obtained on the assumption that the triangular data in each Cutter Contact (CC) area is known, and the triangular data search algorithm in the CC area is not given. Feng and Teng [5] put forward buckets' data search acceleration algorithm, which only pushes the problem to the higher level of data output and fails to achieve acceleration in the whole process from CAD to CAM tool path generation algorithm. Ding et al. [6] proposed changing the density and size of triangular slices of the 3D model and reducing the total number of triangular slices to improve the efficiency of machining simulation. Park and Choi [7] worked on the processing method based on the point model, avoiding the polyhedron model. Literature [8] presents a localization method based on the R tree to improve search efficiency. In this paper, a triangular slice data search algorithm in the CC region is proposed for the NC system of three-axis swing grinding, and the algorithm in Zou and Zhao [4] is improved and optimized, and a more perfect tool path generation algorithm suitable for the flat bottom tool is given.Due to the mold, blade, propeller, and other free-form contour parts of the profile being complex need to use CNC milling machine, machining center, and other equipment, in the two-axis semi-coordinate milling machine for two-coordinate linkage machining or three-coordinate CNC milling machine for three-coordinate linkage machining. No matter which processing method is adopted, it is necessary to first determine the cutter head in the curved surface profile of the path, that is, theX direction is divided into several sections, round head milling cutter along the curve of the YZ surface for milling, each section of the processing is completed into X, and then processing another adjacent curve, such as this, in turn, can be machined out of the entire surface. However, in many cases, the free-form surface cannot be accurately described by analytical relations and can only be determined by fitting some discrete type value points on a given surface.This paper presents a method to calculate tool path curves directly from free-form value points. The basic idea of this method is that the arbitrary distribution of type value points affects and determines the surface shape of each position in the control area, and the size of the influence is related to the distance between the type value points and the position.


## 2. Tool Path Generation Strategy

In this section, we present some basic concepts related to tool path and tool path generation strategy that are as follows:(i)
Cutting Contact Point (abbreviated as CC-Point) refers to the Cutting Contact Point between the tool and the part surface in the processing process.(ii)
Cutting Contact Path (CC-Path) refers to the curve formed by the tool contacts during machining. The tool contact trajectory can be explicitly defined, such as the isoparametric line of a surface, the intersection line of two surfaces, etc., or implicitly defined, such as the constrained tool movement within the range of the guiding surface, and the beam tool movement along the guiding line.(iii)
Cutter Location Data: refers to the data required to accurately determine each position of the tool in the processing process. In general, the position of the tool can be determined by the coordinate of the tool point and the direction of the tool axis. Usually, in CAM systems, knife point refers to the point of the knife.(iv)
Cutter Location Path (simply called CL—Path) refers to the curve formed by cutter points in the process of machining. The tool position trajectory is usually calculated from the tool contact trajectory based on bias.(v)
Cutter Location Surface (Cutter Location Surface) refers to the set of all cutter locations in the processing process.Tool path generation is the most important and most widely studied content in NC machining of free-form surface parts. In multi-coordinate NC machining of free-form surface, the quality of tool path directly affects its machining accuracy and efficiency. The length, continuity, and direction consistency of the tool path are the indexes of evaluating the tool path.The generation algorithm of the tool path is the core part of the NC programming system. There are many types of research on this aspect, mainly for three-axis NC machining and five-axis NC machining. In conclusion, for the machining of free-form surface, there are mainly the following strategies to generate interference-free tool path from the surface model:(a)
Surface model—noninterference tool contact trajectory—tool location trajectory. This strategy generates the tool contact path directly from the surface model and is only applicable to the three-axis NC machining with a fixed tool axis vector. Due to the need for numerical iteration, it is difficult to ensure the stability of the plan.(b)
Surface model—cutter contact track-cutter position track—no interference. The basic idea is to first generate the tool contact trajectory without considering the interference problem, and then generate the tool location trajectory without interference through the interference check and processing. This strategy is very suitable for four and five-axis NC machining, because in four and five-axis NC machining, due to the flexibility of the tool axis control, it is difficult to determine the optimal deflection Angle of the tool contact and the tool axis by the tool point, so four and five-axis NC machining, especially nonball head tool machining into the tool path of the generation algorithm adopts this strategy.(c)
The surface model—bias surface model—does not interfere with the tool locus. According to the machining parameters and the tool selected, the cutter center point surface corresponding to the original surface is calculated, which is called the cutter location surface or the offset surface. If a ball cutter is used for machining, whether it is three-axis NC machining or four or five-axis NC machining, the offset surface is the isometric surface of the original surface. However, if the annular cutter or end milling cutter is used, only the fixed three-axis CNC machining of the cutter axis vector can be constructed according to the original surface of the cutter center surface. For four and five-axis CNC machining, because the orientation of the cutter axis vector is difficult to determine, causing the generation and processing of the offset surface of the nonspherical cutter machining is difficult.(d)
Surface model one lattice model does not interfere with the path of cutter position. The basic idea is to replace the machined surface with a lattice within the precision control range and establish a lattice model. The z-buffer data of the surface can be obtained by the canonical Zmap, and then the tool path can be generated from the Z-buffer data of the surface through the “reverse bias” method. This strategy is simple, stable, and suitable for three-axis NC machining.


## 3. Related Work

Computer numerical control (CNC) machining technology is one of the most important technologies in modern machining technology. The quality of tool tracks in NC machining has an important influence on machining quality and efficiency. For example, to ensure the processing quality, the maximum stepover should be less than the maximum allowable value. In order to improve machining efficiency, the tool path should be as smooth as possible and the path length should be as small as possible. Due to the lack of a unified mathematical framework, the traditional tool path generation algorithm is often optimized independently for different requirements. However, different machining requirements may conflict with each other, so the tool path generated by the traditional method can only meet certain requirements and cannot achieve the optimal tool path [5–7].When generating the tool path of multiaxis NC machine tool to process complex curved parts, the geometric constraints such as tool point, tool axis vector, tool shape and size, and tool velocity between tool points should be considered comprehensively. Because these geometric constraints are interrelated, it is difficult to consider them comprehensively. Therefore, many researchers try to generate knife paths by considering the constraints of each [8].The traditional tool path generation method of complex parts surface can be divided into equal parameter method, equal section method, equal residual height method, offset surface method, main curvature matching method, and so on. These traditional methods can only meet some processing requirements and have some defects. In order to solve the problems existing in traditional machining methods, many new tool path generation methods are proposed. Lee [9] proposed a tool path generation method that can reduce the amount of tool position file data transmitted to the NC system, and this method can also keep the required feed speed and position accuracy of NC machine tools.To improve the quality of the cutter path, many innovative methods have been put forward. Zhu et al. [10] published a patent that can generate a spiral cutter path for machining blades, which generates a virtual potential field in the machining area and uses it to generate an auxiliary curve that can transition into a spiral cutter path. This patent can only deal with the case that the machining area is flat. Kim [11] constructed a path scalar field to represent the tool path, to optimize the smoothness of the tool path. The optimization problem is transformed into the elliptic partial differential equation with boundary conditions, and these methods are only used when the machining area is flat. Kim and Yang [12] transformed the tool path generation problem into solving the elastic deformation minimization problem of the curvilinear coordinate systems. Senatore et al. [13] proposed a tool path generation method that considered multiple targets and constraints and was suitable for ball-end knives. Both of these methods use the method of solving two potential energy fields, and the two potential energy fields may conflict with each other, that is, the conflict between objective functions. The research work conducted by Wright et al. [14] described part of a “Pipeline of Design and Manufacturing Tools” for product designers who are driven by short delivery times. The overall project has been called CyberCut because the modules can interoperate in a distributed, Internet-based environment. This particular paper focuses on tool path generation of sculptured surfaces using 3-axis CNC machines. Mladenović et al. [15] proposed a tool path generation algorithm based on the criterion of maintaining the cutting forces at a constant predefined level for 3-axis ball-end milling processes of free form surface with feed rate scheduling. It has been experimentally confirmed that the proposed algorithm offers a number of advantages over the machining strategies used in commercial CAD/CAM software.Because there is no unified mathematical model in the existing tool path generation methods, they are not extensible and cannot produce the optimal tool path satisfying multiple constraints simultaneously. The tool path generated by computer-aided manufacturing (CAM) system only contains geometric information but does not contain physical information (such as speed, acceleration, and acceleration) except the upper limit of feed speed. The speed planning of the machine tool is accomplished by the CNC system. To minimize the processing time, the machine should theoretically move as fast as possible. However, the speed of each axis of the machine tool is also limited by physical constraints: the upper limit of the speed of each axis, the upper limit of acceleration, the upper limit of acceleration, the following error, and the maximum cutting force and so on. Researchers have proposed several tool path generation methods for complex surface multiaxis machining with physical constraints. The research topics of these methods can be divided into three categories: (i) prediction and optimization of cutting forces, (ii) speed planning of CNC machine tools, and (iii) curve error compensation.


## 4. Blade Roughing Tool Path Generation Algorithm

In this section, according to the machining technology and machining characteristics of blade surface roughing, and combined with the algorithm framework of tool path generation for surface machining proposed in Section2, the spiral path of blade surface roughing using the “cross section method” is proposed. Because the rough machining adopts a plane cutter path, the objective function of the framework can be simplified, which makes the calculation simpler and more efficient. To improve the machining efficiency, a zonal and stepwise mixed tool path was proposed. Simulation and machining verification are provided to prove the effectiveness of the algorithm.


### 4.1. Processing Methods of Rough Machining of Blades

The blade surface can be roughed by the “cross section method”. The blank is square steel, the cutter moves along the axis of square steel, and the blade is processed cross section by section. During the whole machining process, the cutter shaft is within the machining section. The method is usually four-axis linkage machining, that is, the cutter shaft has only rake angle without side angle. When machining A certain section, theY, Z translational axis, and A rotary axis of the machine tool are linked with three axes.


#### 4.1.1. Simplification of Target Functional Equation in Two-Dimensional Case

When the processing region (potential field region) is flat, the degree matrix tensor is the identity matrix, which can be substituted into the gradient vector of the two-dimensional potential field:(1)∇ϕ=∂ϕ∂x,∂ϕ∂yT.Substitute the gradient of the two-dimensional vector field into the row-width optimization objective equation, and get(2)Fϕ=∫∂ϕ∂x2+∂ϕ∂y2dxdy.Substitute the formula into the line width optimization objective function of the two-dimensional vector field:(3)δFϕ=δ∫∂ϕ∂x2+∂ϕ∂y2dxdy=0.The rectangular L1 cut from the square steel blank and the plane where the cutter shaft is located is taken as the outer boundary, and aAAA is fixed. The curve L2 cut from the blade surface and the plane where the cutter shaft is located is taken as the inner boundary and BBBB (n is a positive integer) is fixed.The solution of the functional can be simplified as solving the Laplace equation with Dirichlet constraints as follows:(4)∇2ϕ=0,ϕ|L1=0,ϕ|L2=n.We generate triangular meshes in the machining area and solve the equations using MATLAB software. When the optimized potential energy field is obtained, we can find the isopotential line satisfying the requirement of maximum row width by using the method. The optimized potential energy field isopotential lines are shown in Figure1. Each isopotential line is smooth, and the distance between two adjacent isopotential lines is less than the maximum allowable line width Hmax.Figure 1
Isopotential line of potential energy field satisfying Laplace equation.


### 4.2. The Method of Obtaining Spiral Trajectory by Equipotential Line Insertion

Isopotential lines generated by the method in Section3 can be used to generate spiral paths. In order to meet the processing requirements, the screw knife path should meet the following conditions:(I)
The distance between two adjacent turns of the screw is less than the maximum allowable row width.(II)
The knife path should be as smooth as possible.(III)
The knife path is not self-intersecting. In order to satisfy the above three requirements, we use the B-spline curve to fit the isopotential line and use the linear interpolation method to generate the spiral path from the isopotential line. The spiral cutter path produced by this method can meet the above three requirements.We generate a spiral path from two adjacent equipotential lines in turn. This section of spiral knife path is in a circle composed of two adjacent equipotential lines, starting from a point of the equipotential line of the outer ring and ending at a point of the equipotential line of the inner ring, and gradually smooth transition from the curve of the outer ring to the curve of the inner ring. All sections of spiral knife paths generated by two adjacent isopotential lines are connected sequentially to obtain the spiral knife paths that meet the requirements of maximum row width. The spiral knife paths start from the outer boundary of the blank to the inner boundary of the blade body and transition smoothly in the middle.When the circular ring composed of two adjacent equipotential lines (hereafter referred to as circular ring) has a bend, the spiral knife Louie is generated from the intersection. In order to avoid self-intersection of spiral knife paths, we divide the circle where it bends into subregions and generates spiral knife paths in each subregion, respectively. As shown in Figures2 and 3, we define boundary points with included angles less than 1200 as partition points. From each partition point, we can generate a gradient line starting from the outer boundary and ending at the inner boundary and use these gradient lines as the dividing line of the ring region. In the example shown in the figure, four partition points divide the annular region formed by two adjacent equipotential lines into four subregions, and we generate helices in each subregion in turn.Figure 2
Search segmentation point algorithm.Figure 3
Blade machining tool paths on 9 sections.Let the parametric equationDi(u) represent the ith isopotential line, adjacent to Di+1(u). Di,k(u) is the k segment on the ith equipotential line. For each subregion, we make curves Mi,k(u) and Ni,k(u), which generate the inner and outer boundaries of the spiral trajectory for the subregion, respectively.We defineα as the coefficient determining the positions of Mi, K (u), and Ni, k(u) in the subregion, which can be calculated by the proportion of LK to L:(5)αk=∑ikliL,k∈1,n,α0=0.The two boundariesMi, k(u), and Ni, k(u) of the subregion can be obtained by linear interpolation of Di(u) and Di + 1(u):(6)Mi,ku=1−αk−1Di,ku+αk−1Di+1,ku,Ni,ku=1−αkDi,ku+αkDi+1,ku.In order to connect the spiral trajectory of different subsections end to end, we require that the upper boundary of the previous subregionNi, the end point of k − 1(u) is connected with the starting point of the lower boundary of the next subregion Mi, k(u):(7)Ni,k1=Mi,k0.The spiral trajectory is obtained by linear interpolation ofMi, K, and Ni, k:(8)Ci,ku=1−uMi,ku+uNi,ku.In each subregion, the boundary condition of this linear interpolation is that the starting point of the spiral trajectory coincides with the starting pointMi and k(0) of the lower boundary, and the ending point coincides with the ending point Ni and k − 1(1) of the upper boundary, as shown in Figure 4:(9)Ci0=Mi,k0,Ci1=Ni,k1.Figure 4
Partition of equivalence lines and spiral trajectories generated by equivalence lines.A complete spiral trajectory can be obtained by connecting the spiral trajectories of different subregions sequentially, and a complete spiral trajectory can be obtained by connecting the spiral trajectories of the circle regions surrounded by all equipotential lines.As can be seen in Figure2, the spiral trajectory smoothly transitions from the outer boundary of the blank and the cutter axis plane to the inner boundary of the blade surface and the cutter axis plane. Since each spiral trajectory is obtained by linear interpolation of two equipotential lines, and the distance between adjacent equipotential lines is less than the maximum row width, the spiral trajectory can meet the requirement of maximum row width. Since the annular region is divided into subregions by equipotential lines, the distortion of the annular region is avoided, so the spiral trajectory will not intersect with itself.


### 4.3. Mixed Knife Path Method

As shown in Figure2, the outer boundary of the processing area is a rectangle, and the inner boundary is a blade curve. In different areas, the distance between the inner and outer boundaries varies greatly, resulting in uneven row width in the processing course. Because the number of isopotential lines is determined by the maximum distance between the inner and outer boundaries, it will lead to the area where the inner and outer boundaries are smaller isopotential lines are too dense, that is, the line width is too small, which reduces the machining efficiency. In order to improve the machining efficiency, we propose a method combining Zig and spiral tool paths. In this method, the machining area is partitioned, and then the reciprocating cutter path and spiral cutter path are, respectively, used for machining. This method combines the advantages of the two tool paths to make the line width uniform and improve the machining efficiency.The advantages of reciprocating tool path include (i) the tool path is straight line, the tool speed can be larger; (ii) uniform line width, uniform cutter force; (iii) line width can be maintained as a large value, high processing efficiency. Spiral knife advantages include (i) less empty knife (only a back and forth knife), continuous cutting tool, high processing efficiency; (ii) the tool path is smooth, and there is no obvious acceleration and deceleration during the tool movement; and (iii) the shape of the blade path is variable, gradually transiting from the shape close to the outer contour to the blade surface shape. The hybrid tool path combines the advantages of the two tools and can improve the machining efficiency.For rough machining of blades, the biggest distance between inner and outer boundaries is usually the blank corner. The mixed knife path method divides the processing into two steps. The first step is to cutoff the corner with the reciprocating knife path to form a new blank for the second step of processing. The distance between the new blank and blade surface is evenly distributed, and the spiral cutter path is generated between the new inner and outer blank by using the method based on the potential energy field mentioned above, and the row width will become even. The area cut by the reciprocating cutter path in the first step is called the “reciprocating processing zone”, and the area cut by the spiral cutter path in the second step is called the “spiral processing zone”. First, the reciprocating cutter path is used to cut the blank corner with a larger row width and then the spiral cutter path with higher processing efficiency is used to transition from the boundary shape of the new blank to the shape of the blade surface.As the machining area is a curved surface, it is easy to generate a reciprocating cutter path, so the maximum allowable line width can be used to make parallel lines from the corner point as the starting point. The generation method of the spiral knife path can follow the above-described method based on the potential field equipotential line. The difficulty of the method of the mixed tool path is to determine the dividing line of the machining area of the two tool paths.


## 4.1. Processing Methods of Rough Machining of Blades

The blade surface can be roughed by the “cross section method”. The blank is square steel, the cutter moves along the axis of square steel, and the blade is processed cross section by section. During the whole machining process, the cutter shaft is within the machining section. The method is usually four-axis linkage machining, that is, the cutter shaft has only rake angle without side angle. When machining A certain section, theY, Z translational axis, and A rotary axis of the machine tool are linked with three axes.


### 4.1.1. Simplification of Target Functional Equation in Two-Dimensional Case

When the processing region (potential field region) is flat, the degree matrix tensor is the identity matrix, which can be substituted into the gradient vector of the two-dimensional potential field:(1)∇ϕ=∂ϕ∂x,∂ϕ∂yT.Substitute the gradient of the two-dimensional vector field into the row-width optimization objective equation, and get(2)Fϕ=∫∂ϕ∂x2+∂ϕ∂y2dxdy.Substitute the formula into the line width optimization objective function of the two-dimensional vector field:(3)δFϕ=δ∫∂ϕ∂x2+∂ϕ∂y2dxdy=0.The rectangular L1 cut from the square steel blank and the plane where the cutter shaft is located is taken as the outer boundary, and aAAA is fixed. The curve L2 cut from the blade surface and the plane where the cutter shaft is located is taken as the inner boundary and BBBB (n is a positive integer) is fixed.The solution of the functional can be simplified as solving the Laplace equation with Dirichlet constraints as follows:(4)∇2ϕ=0,ϕ|L1=0,ϕ|L2=n.We generate triangular meshes in the machining area and solve the equations using MATLAB software. When the optimized potential energy field is obtained, we can find the isopotential line satisfying the requirement of maximum row width by using the method. The optimized potential energy field isopotential lines are shown in Figure1. Each isopotential line is smooth, and the distance between two adjacent isopotential lines is less than the maximum allowable line width Hmax.Figure 1
Isopotential line of potential energy field satisfying Laplace equation.


## 4.1.1. Simplification of Target Functional Equation in Two-Dimensional Case

When the processing region (potential field region) is flat, the degree matrix tensor is the identity matrix, which can be substituted into the gradient vector of the two-dimensional potential field:(1)∇ϕ=∂ϕ∂x,∂ϕ∂yT.Substitute the gradient of the two-dimensional vector field into the row-width optimization objective equation, and get(2)Fϕ=∫∂ϕ∂x2+∂ϕ∂y2dxdy.Substitute the formula into the line width optimization objective function of the two-dimensional vector field:(3)δFϕ=δ∫∂ϕ∂x2+∂ϕ∂y2dxdy=0.The rectangular L1 cut from the square steel blank and the plane where the cutter shaft is located is taken as the outer boundary, and aAAA is fixed. The curve L2 cut from the blade surface and the plane where the cutter shaft is located is taken as the inner boundary and BBBB (n is a positive integer) is fixed.The solution of the functional can be simplified as solving the Laplace equation with Dirichlet constraints as follows:(4)∇2ϕ=0,ϕ|L1=0,ϕ|L2=n.We generate triangular meshes in the machining area and solve the equations using MATLAB software. When the optimized potential energy field is obtained, we can find the isopotential line satisfying the requirement of maximum row width by using the method. The optimized potential energy field isopotential lines are shown in Figure1. Each isopotential line is smooth, and the distance between two adjacent isopotential lines is less than the maximum allowable line width Hmax.Figure 1
Isopotential line of potential energy field satisfying Laplace equation.


## 4.2. The Method of Obtaining Spiral Trajectory by Equipotential Line Insertion

Isopotential lines generated by the method in Section3 can be used to generate spiral paths. In order to meet the processing requirements, the screw knife path should meet the following conditions:(I)
The distance between two adjacent turns of the screw is less than the maximum allowable row width.(II)
The knife path should be as smooth as possible.(III)
The knife path is not self-intersecting. In order to satisfy the above three requirements, we use the B-spline curve to fit the isopotential line and use the linear interpolation method to generate the spiral path from the isopotential line. The spiral cutter path produced by this method can meet the above three requirements.We generate a spiral path from two adjacent equipotential lines in turn. This section of spiral knife path is in a circle composed of two adjacent equipotential lines, starting from a point of the equipotential line of the outer ring and ending at a point of the equipotential line of the inner ring, and gradually smooth transition from the curve of the outer ring to the curve of the inner ring. All sections of spiral knife paths generated by two adjacent isopotential lines are connected sequentially to obtain the spiral knife paths that meet the requirements of maximum row width. The spiral knife paths start from the outer boundary of the blank to the inner boundary of the blade body and transition smoothly in the middle.When the circular ring composed of two adjacent equipotential lines (hereafter referred to as circular ring) has a bend, the spiral knife Louie is generated from the intersection. In order to avoid self-intersection of spiral knife paths, we divide the circle where it bends into subregions and generates spiral knife paths in each subregion, respectively. As shown in Figures2 and 3, we define boundary points with included angles less than 1200 as partition points. From each partition point, we can generate a gradient line starting from the outer boundary and ending at the inner boundary and use these gradient lines as the dividing line of the ring region. In the example shown in the figure, four partition points divide the annular region formed by two adjacent equipotential lines into four subregions, and we generate helices in each subregion in turn.Figure 2
Search segmentation point algorithm.Figure 3
Blade machining tool paths on 9 sections.Let the parametric equationDi(u) represent the ith isopotential line, adjacent to Di+1(u). Di,k(u) is the k segment on the ith equipotential line. For each subregion, we make curves Mi,k(u) and Ni,k(u), which generate the inner and outer boundaries of the spiral trajectory for the subregion, respectively.We defineα as the coefficient determining the positions of Mi, K (u), and Ni, k(u) in the subregion, which can be calculated by the proportion of LK to L:(5)αk=∑ikliL,k∈1,n,α0=0.The two boundariesMi, k(u), and Ni, k(u) of the subregion can be obtained by linear interpolation of Di(u) and Di + 1(u):(6)Mi,ku=1−αk−1Di,ku+αk−1Di+1,ku,Ni,ku=1−αkDi,ku+αkDi+1,ku.In order to connect the spiral trajectory of different subsections end to end, we require that the upper boundary of the previous subregionNi, the end point of k − 1(u) is connected with the starting point of the lower boundary of the next subregion Mi, k(u):(7)Ni,k1=Mi,k0.The spiral trajectory is obtained by linear interpolation ofMi, K, and Ni, k:(8)Ci,ku=1−uMi,ku+uNi,ku.In each subregion, the boundary condition of this linear interpolation is that the starting point of the spiral trajectory coincides with the starting pointMi and k(0) of the lower boundary, and the ending point coincides with the ending point Ni and k − 1(1) of the upper boundary, as shown in Figure 4:(9)Ci0=Mi,k0,Ci1=Ni,k1.Figure 4
Partition of equivalence lines and spiral trajectories generated by equivalence lines.A complete spiral trajectory can be obtained by connecting the spiral trajectories of different subregions sequentially, and a complete spiral trajectory can be obtained by connecting the spiral trajectories of the circle regions surrounded by all equipotential lines.As can be seen in Figure2, the spiral trajectory smoothly transitions from the outer boundary of the blank and the cutter axis plane to the inner boundary of the blade surface and the cutter axis plane. Since each spiral trajectory is obtained by linear interpolation of two equipotential lines, and the distance between adjacent equipotential lines is less than the maximum row width, the spiral trajectory can meet the requirement of maximum row width. Since the annular region is divided into subregions by equipotential lines, the distortion of the annular region is avoided, so the spiral trajectory will not intersect with itself.


## 4.3. Mixed Knife Path Method

As shown in Figure2, the outer boundary of the processing area is a rectangle, and the inner boundary is a blade curve. In different areas, the distance between the inner and outer boundaries varies greatly, resulting in uneven row width in the processing course. Because the number of isopotential lines is determined by the maximum distance between the inner and outer boundaries, it will lead to the area where the inner and outer boundaries are smaller isopotential lines are too dense, that is, the line width is too small, which reduces the machining efficiency. In order to improve the machining efficiency, we propose a method combining Zig and spiral tool paths. In this method, the machining area is partitioned, and then the reciprocating cutter path and spiral cutter path are, respectively, used for machining. This method combines the advantages of the two tool paths to make the line width uniform and improve the machining efficiency.The advantages of reciprocating tool path include (i) the tool path is straight line, the tool speed can be larger; (ii) uniform line width, uniform cutter force; (iii) line width can be maintained as a large value, high processing efficiency. Spiral knife advantages include (i) less empty knife (only a back and forth knife), continuous cutting tool, high processing efficiency; (ii) the tool path is smooth, and there is no obvious acceleration and deceleration during the tool movement; and (iii) the shape of the blade path is variable, gradually transiting from the shape close to the outer contour to the blade surface shape. The hybrid tool path combines the advantages of the two tools and can improve the machining efficiency.For rough machining of blades, the biggest distance between inner and outer boundaries is usually the blank corner. The mixed knife path method divides the processing into two steps. The first step is to cutoff the corner with the reciprocating knife path to form a new blank for the second step of processing. The distance between the new blank and blade surface is evenly distributed, and the spiral cutter path is generated between the new inner and outer blank by using the method based on the potential energy field mentioned above, and the row width will become even. The area cut by the reciprocating cutter path in the first step is called the “reciprocating processing zone”, and the area cut by the spiral cutter path in the second step is called the “spiral processing zone”. First, the reciprocating cutter path is used to cut the blank corner with a larger row width and then the spiral cutter path with higher processing efficiency is used to transition from the boundary shape of the new blank to the shape of the blade surface.As the machining area is a curved surface, it is easy to generate a reciprocating cutter path, so the maximum allowable line width can be used to make parallel lines from the corner point as the starting point. The generation method of the spiral knife path can follow the above-described method based on the potential field equipotential line. The difficulty of the method of the mixed tool path is to determine the dividing line of the machining area of the two tool paths.


## 5. Processing Verification

In this section, we provide two groups of experiments to verify the algorithm. In Experiment1, we used this algorithm to generate the rough blade cutting path and processed the blade. The cutter path is a spiral cutter path. In Experiment 2, we generated λ=0,0.25,0.5,0.75,1 mixed knife paths, respectively. Since the feasibility of this algorithm has been verified in Experiment 1, this group of experiments did not carry out real machining but used VERICUT to do machining simulation.Experiment 1.
Processing Verification
In the processing of the first section, because the tool is full knife cutting (cutting width for tool diameter), we took a smaller cut depth than other sections (0.6 mm). When machining each section, we recorded the machining time and observed the machine load displayed by the CNC system. As the feed rate increases, the machining time of a section decreases correspondingly, while the machine load remains almost constant (about 7% of the rated machine load). This means that in the actual processing, the feed speed has a large space to improve.
If we adjust the spindle speed to 3200 r/min and the feed speed to 4000 mm/min, the total working time of 9 sections is about 60 minutes. The total processing time required by the traditional 3 + 2 machining method with the same processing parameters is about 90 minutes. Compared with the traditional machining method, the machining efficiency of the tool path generated by this algorithm is higher. As can be observed from the processing results, the leaf surface is smooth and the chip color is uniform.
The complete tool path is shown in Figures3 and 5, and the machining results are shown in Figures 6 and 7.Figure 5
Knife path on a section.Figure 6
Blade roughing results.Figure 7
Details of machining results and chips.Experiment 2.
Efficiency Verification of Mixed Knife Path
In order to verify the knife processing feasibility and validation of the road mix knife way when the uneven distribution of internal and external boundary improves the processing efficiency as produced by mixed knife withλ=0,0.25,0.5,0.75,1. The knife in the road cutting parameters (cutting depth, cutting width, etc.) and processing parameters of spiral cutter path in Experiment 1are completely consistent, the specific processing parameters are listed in Table 1. In order to verify the feasibility of the algorithm, practical machining was done with a 0.5 mixed tool path. The machining machine tools, cutting tools, and machining parameters are consistent with Experiment 1. Because the mixed blade path is still processed by spiral blade path, the mixed blade path does not affect the processing quality, but only the processing efficiency. The machining results show that the mixed tool path can be used in practical machining. In order to ensure safety, it is thought out that the feed speed was changed during the actual processing, and so the processing time was not recorded. The experimental results are listed in Table 2.Table 1
Processing parameters.
Cross section 1Cross section 2Cross section 3Cross section 4Cross section 5Cross section 6Cross section 7Cross section 8Cutting width (mm)3214.87514.87514.87514.87514.87514.87514.875Cutting depth (mm)0.61111111Spindle speed (r/min)28002800280030003200320032003200Feed speed (mm/min)20002000320032004000400040004000Processing time (min)138665555Table 2
Mixed tool paths atλ = 1.
λ00.250.50.751Total time (s)25892159213923002515Air time (s)91204318470645Tool path length (mm)133782134227137546153058172985


## 6. Conclusion

The paper presents a procedure for generating optimized tool paths for jade carving surfaces according to the criterion of keeping the cutting force constant by using the cutting force model created for simulating the process of machining.In order to improve the quality of tool paths in surface machining, different machining requirements should be considered in the tool path generation algorithm. In the traditional tool path generation algorithm, due to the lack of a unified mathematical calculation framework, these requirements often need to be considered separately, but different requirements often compete with each other, so that different requirements cannot be satisfied at the same time, the tool path cannot achieve the optimal. In order to solve the above problems, the preset article proposes a tool path generation algorithm for surface machining and applies the algorithm to the blade and die surface machining. This paper mainly introduces the algorithm to generate the spiral path for rough machining of the blade body. The algorithm is based on the tool path generation algorithm framework of surface machining proposed in Section2, and the framework is simplified according to the characteristics of the planar machining tool path. A hybrid machining method was proposed, which can automatically divide machining areas according to the shape of machining areas and combine reciprocating and spiral tool paths in different machining areas. It can be observed from the experimental and simulation results that the proposed algorithm for tool path generation is applicable in CAM software packages primarily for the reason of minimizing the total machining time for the case of rough cutting.



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*Source: 2518687-2022-04-15.xml*