Triangulation intersection approach from Poisson’s equation applied to automatic tool selection in computer numerical control mill-lathe

Abstract

This article presents a new method for automatic tool selection using discrete techniques based on Poisson’s equations for computer numerical control lathe-mill machine operations and rapid tooling. This method proposes to generate linear triangulation in the boundary of the piece and tool, after creating moves of tool around the piece to detect intersection; thus, this creates directional field through mesh allowing selecting the type of tool in agreement with the size, magnitude and direction of tool. The proposed methodology has been implemented in computer-aided manufacturing software, in different types for two-dimensional, two and a half–dimensional and three-dimensional pieces computer-aided design, showing the correct selection of the cutting tools. The novelty of this article is the use of discrete technique applying Poisson’s equation; the piece thus checks the intersection between the piece and tool when the tool is moving.

Keywords

Cutting manufacturing management optimization forming rapid prototyping optimization production machines

Introduction

Nowadays, the demand for better productivity with high quality is increasing. The computer numerical control (CNC) machines at present and in the next generation should take into account numerous variables that affect the machining process, for example, tool materials, total error compensation, concepts of autonomous manufacturing, process condition monitoring, to develop commercial computer-aided manufacturing (CAM), the development of a system capable of recognizing complex features, cutting fluids and tool selection, among others.¹ Certain contributions on these topics are the investigations conducted by Abou-El-Hossein² and Eladawi et al.³ Automatic tool selection in turning and milling operation is one of the most important steps in process planning; moreover, at present, CAM software transfers this task to the worker who does it based on his or her own experience, ability and knowledge. In machining processes, commonly tool selection is made by a human operator; if this is incorrect, it can produce errors in workpiece dimensional, such as possible crashes, and consequently rejecting the piece. In this sense, researchers have developed for tool selection in milling processes such as in the work presented by Bouaziz and Zghal.¹ Ahmad et al.⁴ conducted a study based on analytical models using optimization algorithm only for prismatic pockets machining. In the same way, Ramaswami et al.⁵ applied dynamic programming to determine the optimal tools for machining only polygonal pockets. Another approach for tool selection is proposed by Lim et al.⁶ who developed an experimental algorithm based on mathematical Boolean.

On the other hand, in the case of tool selection problems in CNC machines, Oral and Cakir⁷ and Edalew et al.⁸ developed an approach of tool selection based on mathematical methods and data composed of years of heuristic knowledge. The use of dynamic SQL and criteria of tool select from database to recognize characteristics of the piece shape in .DXF format is presented by You et al.⁹ Finally, the case of tool sequence problems by Im and Walczyk,¹⁰ the cutting condition effects as roughness presented by Sarma et al.,¹¹ the prediction of cutting conditions by Gadelmawla et al.¹² and path generation are included for milling CNC machines by Eladawi et al.³

The novelty of this research is a new method based on triangulation intersection approach from Poisson’s equation applied to automatic selection of the tool CNC mill-lathe machine, showing an easy way to simulate pieces, eliminating the designer subjective decisions, to reduce tool selection mistakes, to make the selection process and to design faster, and hence, it does not require a priori knowledge in operations of milling and turning.

Methodology

The application of triangulation and rapid tooling (RT)¹⁰ to determine the intersection using Poisson’s equations to solve manufacturing problems is a new approach in the literature. Automatic tool selection is an important cutting parameter to reduce machining time, errors, human interaction, costs and better product quality.¹³ Sutheebanjard and Premchaiswadi¹⁴ developed new methodologies by the widely use of CNC machines in the industry. The development of diamond shape of cutting tools for CNC lathe and ball nose for CNC mill or machining center to perform operations is varied; using this research data handbook and catalogs of tool, manufacturers used to select the proper tool for turning and milling operations.

In this article, the original file is two-dimensional (2D) or three-dimensional (3D) model taken from computer-aided design (CAD) software, and it validates in CAM–computer-aided engineering (CAE) systems whether graphics, designs and simulates, respectively, facilitate the user to do the tasks.^14–16 This piece is transformed to a .DXF file format by the easy and low cost in terms of computational work.^17,18 After this, a method of discretization (triangulation) is applied on the piece and tool using Poisson’s equations to obtain the mesh, based on the Galerkin scheme by Sulaiman et al.¹⁹ Many algorithms for computing Delaunay triangulations in 2D rely on fast operations for detecting when a point is within a triangle; however, it is necessary to modify the method.^20–22 After obtaining features of the mesh such as the gradient to find the slope, edge detect for the position of the gradient vector is applied. Finally, a structural element (tool discretized) with the shape of the cutting is displaced through workspace with the length values of the boundary piece and the lengths of the cutting tool from right to left and from bottom to top creating roughing paths for 2D models and zig-zag, and zig-zag constant with contour for 3D model, changing the dimensions of the tool, if there is an intersection²³ between the model and the tool when it is moving (see Figure 1). The advantages of this algorithm are eliminates the designer subject decisions, reduces tool select mistakes, is relatively simple and fast to be implemented, does not require a priori knowledge and the designed tool selection can be integrated within a CAD/CAM system to use in other applications.

Figure 1.

General diagram.

Step 1: design original of piece CAD

In this step, the original file (2D, 2.5D and 3D models) is designed or taken from CAD; in order to analyze the method, two pieces are used: the first piece is a mold and the second piece is a pawn of chess.

Step 2: acquisition of .DXF file

The original design (Figure 2(a)) is obtained from CAD; after this, upper profile model is converted to .DXF file.²⁴ This file contains information about the drawing, definitions of items and the entities that make up each block as shown in Figure 2(b).

Figure 2.

Original files: (a) design from CAD and (b) .DXF file.

Step 3: three-node triangular elements for 2D piece and four-node triangular elements for 3D piece

The following step is the discretization²⁵ using three-node triangular element in 2D elements as shown in Figure 3. This is also known as linear triangular element and is simpler than others;²⁶ it has three nodes at the vertices of the triangle with $XY$ as linear elements for 2D piece and $XYZ$ linear elements for 3D piece as shown in equation (1)

f_{V_{1} V_{2} V_{3}} = ℓ_{1} + ℓ_{2} x + ℓ_{3} y = \sum_{i = 1}^{3} {[ℓ_{i}]}^{T} \times [1 x y] = [1 x y] {\begin{matrix} ℓ_{1} \\ ℓ_{2} \\ ℓ_{3} \end{matrix}}

(1)

where $V_{1}$ , $V_{2}$ and $V_{3}$ are the vertices in the x and y coordinates, $f_{V_{1} V_{2} V_{3}}$ is the triangular surface bidimentional and $ℓ_{i}$ is the constant to be determined; the first iteration the vertices are shown in the part see Figure 3(b). Equation (1) represents the nodal variables at the three nodal points as interpolation function, and substituting the $x$ and $y$ values at each nodal point gives equation (2)

\sum_{i = 1}^{3} {{V_{i}} = [1 x_{i} y_{i}] \times \sum_{j = 1}^{3} ℓ_{j}}

(2)

where $V_{i}$ is the nodal variable, and $x_{i}$ and $y_{i}$ are the coordinate values at the $i th$ node (Figure 3). Using inverse of matrix in equation (2) and rewriting

{\begin{matrix} ℓ_{1} \\ ℓ_{2} \\ ℓ_{3} \end{matrix}} = \frac{1}{2 (Area)} [\begin{matrix} x_{2} y_{3} - x_{3} y_{2} x_{3} y_{1} - x_{1} y_{3} x_{1} y_{2} - x_{2} y_{1} \\ y_{2} - y_{3} y_{3} - y_{1} y_{1} - y_{2} \\ x_{3} - x_{2} x_{1} - x_{3} x_{2} - x_{1} \end{matrix}] {\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix}}

(3)

Figure 3.

Implementation of three-node triangular element: (a) discretization in two-dimensional elements and (b) discretization in three-dimensional elements.

The Area corresponding is obtained by equation (4)

Area = \frac{1}{2} {x_{1} \times (y_{2} - y_{3}) + x_{2} \times (y_{3} - y_{1}) + x_{3} \times (y_{1} - y_{2})}

(4)

The processes of computation of the element nodal sequence must be in the same direction for every element. Substituting equation (3) into equation (1) produces equation (5)

f_{V_{1} V_{2} V_{3}} = \sum_{i = 1}^{3} {G_{i} (x, y) \times V_{i}}

(5)

In the above, $G_{i} (x, y)$ are functions that generate the shape for a linear triangular element, and they are given by equations (6)–(8)

G_{1} = \frac{1}{2 (Area)} [(x_{2} y_{3} - x_{3} y_{2}) + (y_{2} - y_{3}) x + (x_{3} - x_{2}) y]

(6)

G_{2} = \frac{1}{2 (Area)} [(x_{3} y_{1} - x_{1} y_{3}) + (y_{3} - y_{1}) x + (x_{1} - x_{3}) y]

(7)

G_{3} = \frac{1}{2 (Area)} [(x_{1} y_{2} - x_{2} y_{1}) + (y_{1} - y_{2}) x + (x_{2} - x_{1}) y]

(8)

Figure 4 shows an example of mesh discretization, where $V_{1}$ , $V_{2}$ and $V_{3}$ are the vertices for a specific triangle.

Figure 4.

Example of mesh discretization.

As seen in the discretization of Figure 4, the conditional point of vertices and the actual curved boundary are approximated by a piecewise linear boundary. In 3D analysis the Poisson’s equation,²⁵ can be extended directly into equation (9)

\nabla^{2} f = 3 D (x, y, z) = M = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}} = - \int [\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}] dxyz

(9)

where $f$ is the surface in the plane $(x, y, z)$ and $M$ is the linear matrix of mesh in the piece; moreover, applying a discretization of the piece in 3D on the surface and computing (this equation has an analytical solution) give element matrices and vectors. the tetrahedral element is used in 3D pieces as shown in Figure 5; increasing a variable and a constant, this variable interpolation for an element is assumed as

\sum_{i = 1}^{4} {V_{i}} = {[1 x y z]}^{T} \times \sum_{k = 1}^{4} {C_{k}}^{T}

(10)

Figure 5.

Implementation of four-node triangular element (discretization) in three-dimensional elements.

Otherwise, the interpolation function is linear in each axis evaluated at the nodal points, to obtain equation (11)

\sum_{i = 1}^{4} {V_{i}} = \sum_{j = 1}^{4} [1 x_{j} y_{j} z_{j}] \times \sum_{k = 1}^{4} {C_{k}}

(11)

Since $V_{1}, V_{2}, V_{3}, and V_{4}$ vertices are measured from the origin coordinate, equation (11) can be rewritten in the following way

\sum_{i = 1}^{4} {f_{V_{i}}} = \sum_{j = 1}^{4} {V_{j}}^{T} = {\bar{X}} {C}

(12)

Step 4: determination of M matrix of triangulation for 2D and 3D

The standard triangulation of the piece may be refined using linear triangular elements. Other alternatives are using higher-order elements of polynomial expressions¹⁹ or Delaunay.^20,21 For a linear element of 2D triangular piece shown in Figure 4, the matrix element is calculated as

[M] = \int (\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}) dxy = \int {\sum_{i = 1}^{3} [\frac{\partial G_{i}}{\partial x}] \times \sum_{i = 1}^{3} {[\frac{\partial G_{j}}{\partial x}]}^{T} + \sum_{j = 1}^{3} [\frac{\partial G_{i}}{\partial y}] \times \sum_{j = 1}^{3} {[\frac{\partial G_{j}}{\partial y}]}^{T}} dxy

(13)

where $dxy$ is the field in the plane bidimentional and $f$ the surface in the plane $(x, y)$ . The stage after this is integration, and substituting the shape functions in equations (6) through (8) into equation (13) gives

[M] = [M_{3 x 3}]

(14)

whose entries are displayed as follows

M_{11} = \frac{1}{4 (Area)} [{(x_{3} - x_{2})}^{2} + {(y_{2} - y_{3})}^{3}]

(15)

M_{12} = \frac{1}{4 (Area)} [(x_{3} - x_{2}) (x_{1} - x_{3}) + (y_{2} - y_{3}) (y_{3} - y_{1})]

(16)

M_{13} = \frac{1}{4 (Area)} [(x_{3} - x_{2}) (x_{2} - x_{1}) + (y_{2} - y_{3}) (y_{1} - y_{2})]

(17)

M_{21} = M_{12}; M_{31} = M_{13}; M_{32} = M_{23}

(18)

M_{22} = \frac{1}{4 (Area)} [{(x_{1} - x_{3})}^{2} + {(y_{3} - y_{1})}^{2}]

(19)

M_{23} = \frac{1}{4 (Area)} [(x_{1} - x_{3}) (x_{2} - x_{1}) + (y_{3} - y_{1}) (y_{1} - y_{2})]

(20)

M_{33} = \frac{1}{4 (Area)} [{(x_{2} - x_{1})}^{2} + {(y_{1} - y_{2})}^{2}]

(21)

where $\partial G_{i} / \partial x$ , $\partial G_{i} / \partial y$ and $\partial G_{i} / \partial z$ are the shape functions for a linear triangular element with tree nodes ( $V_{1} V_{2} V_{3}$ ) and tetrahedral element with four nodes ( $V_{1} V_{2} V_{3} V_{4}$ ). The integrand in equation (13) is constant and becomes the integrand multiplied by the area of the element domain defined by equations (15) through (21). For determination of the surface $f_{V_{1}, V_{2}, V_{3}, V_{4}}$ of a tetrahedral triangulation in 3D, the inverse of matrix $[\bar{X}]$ is multiplied by equation (12) and defined by equations (15) through (21). For determination of the surface $f_{V_{1}, V_{2}, V_{3}, V_{4}}$ of a tetrahedral triangulation in 3D, the inverse of matrix $[\bar{X}]$ is multiplied by equation (12)

{f_{V_{1}, V_{2}, V_{3}, V_{4}}} = {X}^{T} {[\bar{X}]}^{- 1} {f_{V_{1}, V_{2}, V_{3}, V_{4}}} = {G_{1} G_{2} G_{3} G_{4}}^{T} {f_{V_{1}, V_{2}, V_{3}, V_{4}}}

(22)

Here $[\bar{X}]$ is the variable of the equation lineal $(A \bar{x} = C \leftarrow Ax + By + C = 0)$ defined above and $f_{V_{1} V_{2} V_{3} V_{4}}$ is the triangular surface 3D

{[\bar{X}]}^{- 1} = \sum_{i = 1}^{4} \sum_{j = 1}^{4} [ℓ_{ij}]

(23)

Converting equation (9) and substituting the shape functions $G_{i}$ for the discretization, the result is an element matrix $[M]$ defined by

[M] = \int {\sum_{i = 1}^{4} [\frac{\partial G_{i}}{\partial x}] \times \sum_{i = 1}^{4} {[\frac{\partial G_{j}}{\partial x}]}^{T} + \sum_{j = 1}^{4} [\frac{\partial G_{i}}{\partial y}] \times \sum_{j = 1}^{4} {[\frac{\partial G_{j}}{\partial y}]}^{T} + \sum_{k = 1}^{4} [\frac{\partial G_{i}}{\partial z}] \times \sum_{k = 1}^{4} {[\frac{\partial G_{j}}{\partial z}]}^{T}} dxyz

(24)

where $G_{i} (x, y, z)$ is the shape functions written by equation (25)

\sum_{i = 1}^{4} G_{i} (x, y, z) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} {(ℓ_{ji}) \times {[1 x y z]}^{T}}

(25)

Finally, Figure 6 shows the discretization (meshing) results of the piece, with element width of 3.3, minimum angle between triangles of 175°, approximation factors (between 0 and 1) of 0.001 and number of smoothing iterations of 500. Figure 7 shows the discretization of the piece with element width of 1.1, minimum angle between triangles of 175°, approximation factors (between 0 and 1) of 0.001 and number of smoothing iterations of 500.

Figure 6.

Discretization of the piece, width of 3.3 and angle of 175°.

Figure 7.

Discretization of the piece, width of 1.1 and angle of 175°.

Step 5: detection of gradient of the piece in the triangles

Determination of gradient of the piece in the triangles is obtained by applying the normal vector at a point of the triangulation obtained from the discretization; the gradient gives the position and direction of the piece in the workspace.^27,28 The normal vector at a point $(x_{0}, y_{0}, z_{0})$ on a surface $S = f (x, y, z)$ is given by equation (26)

S = \nabla f = [\begin{matrix} f_{x} (x_{0}, y_{0}, z_{0}) \\ f_{y} (x_{0}, y_{0}, z_{0}) \\ f_{z} (x_{0}, y_{0}, z_{0}) \end{matrix}] = [\begin{matrix} \partial f / \partial x \\ \partial f / \partial y \\ \partial f / \partial z \end{matrix}]

(26)

Figure 8 shows the directional vectors of gradient in the edge of the piece for 2D and directional vectors of gradient in the edge of the triangles for 3D, for CNC mill-lathe, and the algorithm is shown in Table 1.

Figure 8.

Directional vectors (gradient) of the triangles: (a) piece in 3D and (b) piece in 2D.

Table 1.

Algorithm for determining the vectors of gradient in the edge of the triangles.

Begin

//Select a triangle whit vertices V1, V2, V3

V_{1, 2, 3} \Leftarrow f_{T} [V_{1} (x, y, z), V_{2} (x, y, z), V_{3} (x, y, z)]

//Calculate the edge of triangle

V_{A} (x, y, z) \Leftarrow V_{2} (x, y, z) - V_{1} (x, y, z)

V_{B} (x, y, z) \Leftarrow V_{3} (x, y, z) - V_{1} (x, y, z)

//Calculate the component of vector gradient

X \leftarrow V_{A} (y) * V_{B} (z) - V_{A} (z) * V_{B} (y)

Y \leftarrow V_{A} (z) * V_{B} (x) - V_{A} (x) * V_{B} (z)

Z \leftarrow V_{A} (x) * V_{B} (y) - V_{A} (y) * V_{B} (x)

//Position of gradient vector

Gradiente = S \leftarrow \nabla f (X, Y, Z)

End

Step 6: tool selection

The cutting tool is in a turret and during cutting operation is necessary consider other crash with machine tool body, the workspace with the structural element of the turret was implemented, however, lengths of all machines have different dimensions, in this article, own software let the user input of data for reconstruction of tools for a special example. The direction of the gradient in the edge allows selecting the type, size and the angle of the tool, and the common types of cutting tools used in turning operations are left (L), right (R), neuter (N), knife (K), drilling (D) and holder (H) for external and internal works, and the common cutting tool (structural element) used is shown in Table 2.

Table 2.

Common cutting tool (structural element) used.

Tool for milling			Type	Size (mm)
Name/letter		Shape	Cut	$D$	$d$	$L$	$l$	$FI$	$Sd$	$ϕ (^{\circ})$
Ball nose	B		Milling	0.015	0.016	40.149	15.181	20.417	0.015	0.009
Ball nose	B		Milling	0.200	0.199	20.149	15.181	17.417	0.210	0.099
Endmill	E		Drilling	0.150	0.015	20.149	14.181	16.417	0.015	0.749
Conic	C		Milling	0.099	0.099	25.149	14.181	20.417	0.100	0.049
Conic				0.049	0.049	15.149	7.181	12.417	0.050	0.024
Corner				0.049	0.049	15.149	7.181	12.417	0.050	0.024
Dovetail	D		Facing	0.009	0.005	10.14	5.181	7.417	0.010	0.004
Dovetail	D			0.309	0.309	30.00	19.99	22.99	0.400	0.200
Chamfering	CH			0.349	0.349	40.149	15.181	20.417	0.350	0.210
				0.399	0.399	45.149	10.181	15.417	0.380	0.230
				0.599	0.599	25.149	10.181	15.417	0.390	0.250
Counter bore	CB
Holder	H
Right	R		Turning							35
Right	R			–	–		–	–	0.406	125
Left	L					19.049				197
Left	L			–	–		–	–	0.406	19
Neuter	N		Turning	–	–		–	–	0.507	90
Neuter	N			–	–	12.699	–	–	–	93
Knife	K									90
Knife	K									95

The library of the cutting tool is generated from manuals and tools stored in the warehouse; these data (diameter, length, angle and nose radius of the tool) are introduced into the software; to identify the most suitable tool, the decision is given in Table 2.

The common types of cutting tools used in milling operations are shown in Table 1; the tool is moved through the edge in a way such that the cornering of piece, which represents the cutting tool, coincides with each triangle of the edge. If at any triangle of the edge there is an intersection between the piece and tool (Figure 6(a)), the tool is not correct and another tool is tried (Figure 6(b)); to identify the most suitable tool, the decision is given in Table 3. The procedure of intersection of the tool with piece is checked using triangle–triangle intersection techniques with the own software as shown in Table 4.

Table 3.

Algorithm for automatic tool select in CNC lathe.

Begin
//Compare the directional vectors of gradient in the edge of the piece $(∠ \nabla f)$ with the direction vector of tool
if $180^{\circ} < ∠ \nabla f < 270^{\circ}$ then
use Left insert
if $270^{\circ} < ∠ \nabla f < 360^{\circ}$ then
use Right insert
if $∠ \nabla f = = 270^{\circ}$ then
Use Neuter insert
else //The gradient is $∠ \nabla f = = 180^{\circ} or ∠ \nabla f = = 360^{\circ}$
use Knife insert
end
end
end
End

Table 4.

Algorithm for automatic tool select tool in CNC mill.

Begin
$f_{n} \leftarrow f (x, y, z)$	//Original piece
$D \leftarrow d_{k} (mm) \|_{k = 1}^{k = n}$	//Select diameter of tool cutting
$f_{V_{i + 1} V_{i + 2} V_{i + 3}} \leftarrow ℓ_{i + 1} + ℓ_{i + 2} x + ℓ_{i + 3} y \|_{i = 0, 3, 7, \dots}^{n = triangles}$	//Discretization using Poisson’s equation for the piece and tool
$n_{A} \leftarrow f_{V_{1} V_{2} V_{3} \dots}$	//linear triangular element for piece
$n_{B} \leftarrow f_{D_{1} \dots}$	//linear triangular element for tool
for i=1 to n_A do	// $n_{A}$ are triangles in the piece
for j=1 to n_B do	// $n_{B}$ are the triangles in the tool
intersection triangle-triangles ( $n_{A}$ , $n_{B}$ )	//program triangle-triangles
if (Intersection) then
$D (mm) \leftarrow d_{k} (mm)$	//Diameter change
$k = k + 1$
else
$Trajectory (k) \leftarrow {Pos}_{tool} (k)$	//Generate Trajectory
end
end
end
End

Step 7: triangle–triangle intersection

In this step, the workpiece and the cutting tool were already discretized (triangulation) according to desired dimensions (angle and size of the triangles) as shown in Figure 9. As a result of the previous steps, the data of input are the number of triangles of the cutting tool and the piece to be machined (n_A , n_B ); with the fixed piece and moving tool, a triangle of tool is selected and compared with each triangle of the piece; the comparison allows to determine the normal for these two triangles and check that they are not degenerative, and finally, the distances of each triangle are compared with respect to a reference framework; if these distances are the same or coincide, then there is intersection and the dimensions of the tool are changed; otherwise, other triangles are selected, if the comparison ends and there is no intersection, then the tool selection is correct, and this process is shown in Table 5.

Figure 9.

Tool select, (a) incorrect tool and (b) correct tool.

Table 5.

Algorithm of triangle–triangle intersection.

Begin
Calculate number of triangles, n_A , n_B	//Poisson’s equation (discretization)
if intersection then
end
else
for i=1 to n_A do
for j=1 to n_B do
Select triangles T_Ai , T_Bj
Calculate the normal of the triangles N ₁<= T_Ai and N ₂<= T_Bj
Check that the triangles are not degenerative “N₁≠0, N₂≠0”
Calculate a distance (Adist(x, y, z)<=) of the points of B against the plane of A
and (Bdist(x, y, z)<=) of the points of A against the plane of B
if (Adist ≈ 0) or (Bdist ≈ 0) then
Intersection
end
end
end
end
End

Step 8: tool path generation strategy

The methods of trajectories generation are zig (one way), zig-zag (two ways), zig with contour, zig-zag with contour, follow-periphery, trochoidal and profile.²⁹ In this article, the zig and zig with contour are used because of easy generation of trajectories. The tool path is obtained with an algorithm (zig), and a tool (structural element) with the shape of the cutting tool is displaced through all workspaces from right to left and finally from bottom to top in the $X$ direction; if this is approaching to the boundary of piece, the tool is back to the start, and the first machining is presented in Figure 10, and the strategy is linear zig.

Figure 10.

Tool path generation strategy, zig with contour.

Results and discussion

With the aim to validate the proposed methodology, two pieces were processed to automatically select the correct tool. Eight tools are available for machining, one piece for 3D (milling) and four for 2D (turning), including operations such as external rough cutting, internal rough and thread for turning and pocket, thread, helicoidal, extrusion, profile and section, fillet and chamfer for rough mill and finish mill.

The first piece is shown in Figure 11. The selection of the cutting tools is presented in Table 6, and the cutting path for rough cutting is shown in Figure 11(a) and (b), and Table 6 shows the results of the selected tools by applying the proposed method.

Figure 11.

Tool selection (final piece): (a) rough turn (zig with contour) and (b) rough turn (zig-zag).

Table 6.

Selected inserts for piece of Figure 11.

Rough turn (zig with contour and zig-zag)
Area	Type
A	L
B	K
C	R
D	N

L: left; R: right; N: neuter; K: knife.

The second piece is shown in Figure 12; the result of the selected tool by the proposed method is presented in Table 7, and the cutting path for spiral zig with contour is shown in Figure 12(a) and for constant Z in Figure 12(b). The developed software is shown in Figure 13. It was developed in Microsoft Visual C++ 2010 and tested with 2D, 2.5D and 3D pieces generated in any CAD software.

Figure 12.

Tool selection using rough mill (final piece): (a) spiral zig with contour and (b) constant Z.

Figure 13.

Final software for automatic tool selection.

Table 7.

Selected inserts for piece of Figure 12.

Rough turn (zig with contour and zig-zag)
Area	Spiral	Constant Z
1	B	B
2	C	C
3	E	E
4	D	D

B: ball nose; C: conic; E: endmill; D: dovetail.

Conclusion

In this work, a new method for automatic tool selection using triangulation intersection approach from Poisson’s equation applied to automatic selection for CNC mill-lathe machines has been presented. This method is relatively simple and fast to be implemented. The proposed methodology has been implemented in pieces generated by CAD software and exported to .DXF file. Also, a higher resolution transformation of CAD file to .DXF file can be utilized to improve both the resolution and accuracy of this system. The methodology was performed and applied to two mechanical pieces in order to find the cutting tools and the respective paths. Results show that the methodology can select the correct tool.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

The authors wish to acknowledge the financial support of UAQ and the financial support by CONACYT (grant number 209333).

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