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Abstract

This article focuses on the computing efficiency of the instantaneous cutter position error curve in computer numerical control cutter positioning, which reflects the positional relationship between the cutter and the desired surface and leads to the strip width of current positioning. The directed projection is proposed to measure the distance of a discrete point to the cutter surface. Two models using fitting techniques are established to compute the instantaneous cutter position error curve. The fitting technique used in this article is based on the quartic polynomial model. In addition, to enhance the accuracy in the nonsymmetric case, the nonsymmetric quartic polynomial model is established, and it induces a more adaptable method. Illustrated experiments show good performance of the proposed methods.

Keywords

CNC tool geometric computation cutter positioning optimization directed projection point-vector instantaneous cutter position error curve strip width

Introduction

The optimization of cutter positioning has been a great interest for scientists in computer numerical control (CNC) machining, especially in five-axis machining.^1,2 In order to estimate the adjustment, it needs to compute the envelope at current cutter positioning. However, the envelope of the cutter is unreachable in mathematics until all the positionings in a path are determined. Scientists take heroic efforts to approach the envelope. A classical idea is to measure the curvature of both the desired surface and the cutter at the cutter contact (CC) point, which focuses only on the geometric properties of one point and cost dinky in time.^3–6 But the curvature-based methods commonly become unstable while the desired surface rapidly rises and falls.

Another feasible strategy is to investigate the distances between the characteristic curve on the envelope and the orthogonal curve on the desired surface. The distribution of the mentioned distances forms a curve in two-dimensional (2D) plane, and the curve is defined to be the instantaneous cutter position error curve (ICPEC).⁷ The curvature-based methods are mostly carried out under the assumption that there exists only one CC point, while Warkentin et al. proposed the multi-point method (MPM) in that there exists more than one CC point to obtain wider cutting strip and enhance surface quality.^8–11 As the curvature model becomes complicated in mathematics under MPM, the model of the ICPEC stably measures the positioning error under arbitrary CC points or particularly with no CC point. The ICPEC model is used to measure the strip width of cutting and to check the gouging in a given cutter positioning as described in Wang et al.,¹² Xu et al.,^13–16 and Yan et al.¹⁷

A distinct drawback of ICPEC model is the low computation efficiency. The ICPEC is described by a set of discrete sample points in Li and Chen⁷ and Xu et al.^13–16 To acquire the sample points, it needs to measure the distances of discrete points on the cutter to the desired surface, which involves complicated computation of a point orthogonally projected onto a tanglesome surface.

Under the consideration of the simple expression of the cutter, this article measures the distances of discrete points on the desired surface to the cutter, instead of computing the distances of discrete points on the cutter to the desired surface. And the distance defined in the article is the projective distance, which is deduced from a classical method called “point-vector” commonly used in error detecting and simulation. The “point-vector” method was proposed by Chappel¹⁸ in 1983, and the process is similar to mowing a field of grass. The key idea of the point-vector method is to intersect a vector with the cutter, while the vector starts from a point on the desired surface. In Chappel,¹⁸ the cutter is represented by an oriented cylinder, which would cause previously unimagined error in cutter positioning optimization. The literature^19,20 describes the intersection method for flat-end-, ball-end-, and fillet-end-shaped cutters in detail, but limited to the case that all the chosen vectors are parallel to the z-axis of the world coordinate system. This article decomposes a cutter into patches with simple expressions and concerns about the directed intersection for different patches.

Based on the quartic polynomial expression of ICPEC in Engeli et al.,²¹ this article proposed a fast fitting method to reduce the time cost of computation. The MPM pursues the existence of two CC points and supposes that the two CC points are symmetrical on ICPEC. However, the assumption is sometimes difficult to hold. Actually, it is quite common that there exists only one CC point during the process of optimization. This article overthrows the assumption of the symmetry of ICPEC and proposed a nonsymmetric model. And the number of CC points would not affect the calculation of ICPEC. In order to reduce the scale of the data set that is used for fitting, an algorithm for searching the optimal step length is established.

This article is organized as follows. Section “Directed distance of a point to a cutter” describes the directed distances of a discrete point to a CNC cutter. Section “The ICPEC” introduces the ICPEC. In section “Fast computing of ICPEC,” two methods for computing ICPEC are proposed. In section “Illustrated examples and discussions,” we give two illustrated examples. The whole article is concisely concluded in section “Conclusion.”

Directed distance of a point to a cutter

To determine the error distribution of the desired surface and the cutter, one needs to measure the distance relationship between a discrete point and a rotary surface (namely, the cutter surface). In this section, we consider the distance of a point to the cutter via a specific vector and called this distance a directed distance.

Description of CNC cutters

Only the geometry of CNC cutters is taken into consideration in the following. A CNC cutter can be treated as a rotary surface generated by its generatrix. In general, the generatrices are in the form of lines and arcs. So the cutter is the combination of cones, cylinders, and torus patches. Figure 1 illustrates the generatrices of common CNC cutters. Figure 1(a) is the generatrix of a flat-end tool, Figure 1(b) is the generatrix of a ball-end tool, Figure 1(c) is the generatrix of a torus tool, and Figure 1(d) is the generatrix of a barrel tool.

Figure 1.

Generatrix of classical CNC cutters: (a) flat-end cutter, (b) ball-end cutter, (c) torus cutter, and (d) barrel cutter.

Directed distance of a point to a cutter

Usually, the distance of a point to a surface refers to orthogonal distance, which means the distance line between the point and the foot point is vertical to the surface (as shown in Figure 2). However, this article considers the distance that the distance line is not necessarily vertical to the surface. It is because in CNC machining, the allowance is defined to be the offset from the desired surface, not the cutter surface. So the error measure should be vertical to the desired surface, not the cutter surface. In detail, given a point $p$ and an unitized vector $\vec{n}$ , and the surface $τ (u, v)$ , the directed distance of p to $τ$ via $\vec{n}$ is defined to be the minimum $λ$ that satisfied

p + λ \vec{n} = τ (u, v), λ \in R

(1)

Figure 2.

The directed distance.

Denote the above $λ$ by $dis (p, \vec{n}, τ)$ .

When $τ (u, v)$ represents CNC cutters combined with cones, cylinders, and torus patches, equation (1) is a polynomial equations with at most degree 4, which can be solved by the known root-finding formulas.

The ICPEC

The process of CNC machining typically involves three surfaces, which are the desired surface, the cutter surface, and the envelope of the cutter surface. Let $S (u, v)$ be the desired surface, and $τ_{(u_{0}, v_{0})} (θ, ϕ)$ is the cutter surface when the cutter locates at a driving point (or a CC point) $S (u_{0}, v_{0})$ in a given pose. And the parametric $θ$ denotes the rotary angle of cutter generatrix, and $\vec{n} (u, v)$ denotes the unit normal vector of $S (u, v)$ . Then

S_{d} (u, v) = S (u, v) + d \cdot \vec{n} (u, v)

(2)

is called the offset surface of $S (u, v)$ , and d is called the offset distance. d is usually taken as the processing tolerance. The space between the desired surface and the offset surface forms the tolerance band. The intersection of the tolerance band and the cutter is the actual effective cutting area.

In the optimization of the cutter positioning, one needs to determine the actual effective cutting area by measuring the relationship between the desired surface and the envelope of the cutter. But inconsequently, the envelope of the cutter is unreachable before the optimization. While moving toward the cutting direction, the cutter surface instantaneously contacts the envelope at a curve, which is called the characteristic curve at current cutter positioning. The curve that the characteristic curve orthogonally projects on the desired surface is called the projective characteristic curve. To avoid the complex computation of envelope, we are trying to approach the distance between the two curves. A straightforward idea is to disperse the cutter surface and then compute the distances between the disperse points and the desired surface. Another strategy is to disperse the desired surface and then concern about the distances of the discrete points to the cutter. Since the former method involves the computation on the desired surface, which could be in high degree and complex, we follow the latter strategy. Then, the focus is transferred to the relationship between discrete points and the cutter surface.

In the rest of the section, we suppose that the driving point is $S (u_{0}, v_{0})$ . Let $ErrD (θ, ϕ)$ be the distance of the point $τ_{(u_{0}, v_{0})} (θ, ϕ)$ to the desired surface.

Definition 1

Denote

ErrD (θ) = \min_{ϕ} ErrD (θ, ϕ)

(3)

$ErrD (θ)$ is called the instantaneous cutter position error curve.⁷

Usually, the cutter and the desired surface contact at one or several points. Let $p_{0} = τ_{(u_{0}, v_{0})} (θ_{0}, ϕ_{0})$ and $p_{2} = τ_{(u_{0}, v_{0})} (θ_{2}, ϕ_{2})$ be the two points on the cutter surface. If $p_{0}$ and $p_{2}$ locate on the desired surface at the same time, then $p_{0}$ and $p_{2}$ are both CC points in current cutter positioning. According to the literature,²¹ we have

ErrD (θ) = a {({(θ - θ_{0} - \frac{θ_{2} - θ_{0}}{2})}^{2} - {(\frac{θ_{2} - θ_{0}}{2})}^{2})}^{2} + O (θ^{5})

(4)

The above expression of the ICPEC indicates the instantaneous positioning deviation of the cutter from the desired surface, but it is not so evident for computing the cutting strip width. Without loss of generality, suppose that the cutter cuts toward the u iso-parametric lines and the presumption holds in the rest of the article without a particular declaration. The CC points p₀ and p₂ satisfy $p_{0} = S (u_{0}, v_{0})$ and $p_{2} = S (u_{2}, v_{2})$ . Note that locally there exists one-to-one correspondence between the parameter $θ$ and the parameter u. Assume $θ = L (u)$ , where $L (u)$ is a linear mapping. Then, we have

\begin{matrix} ErrD (u) & = \min_{v} ErrD (u, v) \\ = a {({(u - u_{0} - \frac{u_{2} - u_{0}}{2})}^{2} - {(\frac{u_{2} - u_{0}}{2})}^{2})}^{2} + O (u^{5}) \end{matrix}

(5)

where $ErrD (u, v)$ denotes the distance measurement between the point $S (u, v)$ and the cutter surface.

Equation (5) shows that the ICPEC could be approximately treated as a quartic polynomial function.

Let $Stol \geq 0$ be the cutting allowance. Define the solution of

ErrD (u) \leq Stol

(6)

to be the effective cutting interval (shown in Figure 3). The length of the effective cutting interval is called the strip width. Let

\begin{matrix} V_{l} = \min {u \in R | ErrD (u) = Stol} \\ V_{r} = \max {u \in R | ErrD (u) = Stol} \end{matrix}

$V_{l}$ and $V_{r}$ denote the verges of a strip and are used in computing the overlaps of different strips.

Figure 3.

The illustration of the ICPEC.

Remark 1

If the cutter cuts toward the $v$ iso-parametric lines, then similarly the ICPEC becomes a quartic polynomial function of $v$ . Since $ErrD (u, v)$ is the measurement of distance of a point to a surface, it can be appropriately defined in several ways. For instance, $ErrD (u, v)$ can be defined as the orthogonal projective distance of a point $S (u, v)$ to the cutter surface. Under the consideration of the definition of the offset surface, in this article, we define

ErrD (u, v) = dis (S (u, v), \vec{n} (u, v), τ_{(u_{0}, v_{0})})

(7)

Fast computing of ICPEC

In order to approach the ICPEC, one usually needs to obtain a series of sampling points located in $[V_{l}, V_{r}]$ , which leads to evident inefficiency.

Symmetrical ICPEC

Let

\begin{matrix} ErrD (x) & = a {({(x - x_{0} - \frac{x_{2} - x_{0}}{2})}^{2} - {(\frac{x_{2} - x_{0}}{2})}^{2})}^{2} + O (x^{5}) \\ = a {({(x - x_{0} - \frac{x_{2} - x_{0}}{2})}^{2} - {(\frac{x_{2} - x_{0}}{2})}^{2})}^{2} + b \end{matrix}

(8)

where $a, b \in R$ and $a \neq 0$ , $x_{0} \leq x_{2}$ . Denote $x_{1} = (x_{0} + x_{2}) / 2$ , then $ErrD (x)$ is symmetrical about $x_{1}$ . Particularly, when $x_{0} = x_{2}$ , we have

ErrD (x) = a {(x - x_{0})}^{4} + b

(9)

Denote

\begin{matrix} x_{l} = \min {x \in R | ErrD (x) = Stol} \\ x_{r} = \max {x \in R | ErrD (x) = Stol} \end{matrix}

By substituting $x_{2} = 2 x_{1} - x_{0}$ into equation (8), one can acquire

ErrD (x) = a {({(x - x_{1})}^{2} - {(x_{1} - x_{0})}^{2})}^{2} + b

(10)

In the process of computing ICPEC, if $x_{0}$ and $x_{1}$ are given, one needs only to determine a and b. ${(X_{i}, Y_{i})}_{0 \leq i \leq n}$ is a set of given sampling points on ICPEC, and $X_{0} < X_{1} < \dots < X_{n}$ . Denote

f_{1} (x) = {({(x - x_{1})}^{2} - {(x_{1} - x_{0})}^{2})}^{2}

(11)

We determine a and b by following the principle of least squares. Let

g (a, b) = \sum_{0 \leq i \leq n} {({af}_{1} (X_{i}) + b - Y_{i})}^{2}

(12)

Solve the optimization model

Minimize g (a, b)

(13)

Then

\begin{matrix} a & = \frac{\sum_{i} f_{1} (X_{i}) \sum_{i} Y_{i} - (n + 1) \sum_{i} f_{1} (X_{i}) Y_{i}}{{(\sum_{i} f_{1} (X_{i}))}^{2} - (n + 1) \sum_{i} f_{1} {(X_{i})}^{2}} \\ b & = \frac{\sum_{i} f_{1} (X_{i}) Y_{i} \sum_{i} f_{1} (X_{i}) - \sum_{i} Y_{i} \sum_{i} f_{1} {(X_{i})}^{2}}{{(\sum_{i} f_{1} (X_{i}))}^{2} - (n + 1) \sum_{i} f_{1} {(X_{i})}^{2}} \end{matrix}

It shows that one needs only to find a set of reliable sampling points in priority to determine ICPEC. And simultaneously, the scale of the data set directly affects the efficiency of the computation. We prescribe that

{\begin{matrix} x_{0} = X_{0} < \dots < X_{n} = x_{1} or x_{1} = X_{0} < \dots < X_{n} = x_{2}, & x_{0} \neq x_{2} \\ x_{l} = X_{0} < \dots < X_{n} = x_{1} or x_{1} = X_{0} < \dots < X_{n} = x_{r}, & x_{0} = x_{2} \end{matrix}

(14)

And there exists a constant $step$ , such that

\begin{matrix} X_{i} = X_{0} + i \cdot step, 1 \leq i < n \\ Y_{i} = \min_{u} ErrD (u, X_{i}), 1 \leq i \leq n \end{matrix}

Remark 2

In the initial positioning of a cutter in CNC machining, $x_{1}$ can be easily determined since it is connected to the given driving point on the desired surface, then one could obtain ${(X_{i}, Y_{i})}_{0 \leq i \leq n}$ by searching from $X_{0} = x_{1}$ or $X_{n} = x_{1}$ .

Nonsymmetric ICPEC

Since ICPEC is not always symmetric in practice, the method based on the symmetric model may lead to prodigious errors in some ordinary cases. In this section, we overthrow the symmetric assumption and establish a nonsymmetric model from two approximate CC points.

Assume that $ErrD (x)$ is still a quartic polynomial and achieves extremum at x₀, x₁, and x₂. Then, let

ErrD' (x) = 12 a (x - x_{0}) (x - x_{1}) (x - x_{2})

(15)

By integration of both sides in equation (15), we have

\begin{matrix} ErrD (x) & = \int ErrD' (x) dx \\ = {af}_{2} (x) + b \end{matrix}

where

\begin{matrix} f_{2} (x) = & 3 x^{4} - 4 (x_{0} + x_{1} + x_{2}) x^{3} \\ + 6 (x_{0} x_{1} + x_{1} x_{2} + x_{2} x_{0}) x^{2} - 12 x_{0} x_{1} x_{2} x \end{matrix}

(16)

x₀, x₁, and x₂ are given, and ${(X_{i}, Y_{i})}_{0 \leq i \leq n}$ are the sampling points from ICPEC. According to the principle of least squares, we have

\begin{matrix} a = \frac{\sum_{i} f_{2} (X_{i}) \sum_{i} Y_{i} - (n + 1) \sum_{i} f_{2} (X_{i}) Y_{i}}{{(\sum_{i} f_{2} (X_{i}))}^{2} - (n + 1) \sum_{i} f_{2} {(X_{i})}^{2}} \\ b = \frac{\sum_{i} f_{2} (X_{i}) Y_{i} \sum_{i} f_{2} (X_{i}) - \sum_{i} Y_{i} \sum_{i} f_{2} {(X_{i})}^{2}}{{(\sum_{i} f_{2} (X_{i}))}^{2} - (n + 1) \sum_{i} f_{2} {(X_{i})}^{2}} \end{matrix}

We prescribe that

{\begin{matrix} x_{l} = X_{0} < \dots < X_{n} = x_{r}, & x_{0} = x_{1} = x_{2} \\ x_{0} = X_{0} < \dots < X_{n} = x_{2}, & otherwise \end{matrix}

(17)

And there exists a constant $step$ , such that

\begin{matrix} X_{i} = X_{0} + i \cdot step, 1 \leq i < n \\ Y_{i} = \min_{u} ErrD (u, X_{i}), 1 \leq i \leq n \end{matrix}

Calculation of strip width

Once $ErrD (x)$ is achieved, then by solving the quartic equation

ErrD (x) = stol

(18)

via the root-finding formula, one can obtain four roots. Denote

JWidth (x, y) = {\begin{matrix} y - x, & ErrD (\frac{x + y}{2}) \leq stol \\ 0, & ErrD (\frac{x + y}{2}) > stol \end{matrix}

(19)

If equation (18) has only one or less than one real root, then $width = 0$ . When equation (18) has k real roots $v_{1}, \dots, v_{k}$ with $1 < k \leq 4$ , $v_{1} \leq \dots \leq v_{k}$ . Then, the strip width is

Width = \sum_{2 \leq i \leq k} JWidth (v_{i - 1}, v_{i})

(20)

Choice of the step length

The choice of the constant $step$ mentioned in section “Symmetrical ICPEC” and section “Nonsymmetric ICPEC” directly determines the computational efficiency. On the premise of given accuracy, the largest $step$ should be chosen to reduce the scale of the data set. Before the calculation of cutter spacing for the whole desired surface, it is necessary to analyze the curvature distribution of the desired surface first and isolate the abnormal areas. Characteristic points picked up from the abnormal areas of the desired surface can be used as driving points to determine the step length step. As we can see, when the driving point P is given and the cutter positioning CP is fixed, a determined step leads to a determined estimated strip width according to section “Calculation of strip width.” Hence, strip width can be denoted by $width (step, P, CP)$ . Algorithm $OptimalStep$ outputs the optimal step length by analyzing a set of characteristic points.

Algorithm OptimalStep
Input An acceptable initial step $s_{0}$ , a set of characteristic points ${P_{i}}_{1 \leq i \leq n}$ , a set of typical cutter positioning ${{CP}_{j}}_{1 \leq j \leq m}$ , a given accuracy $ε$ of width.
Output The optimal step $s$ .
for i from 1 to n
for j from 1 to m
$\begin{matrix} w = width (s_{0}, P_{i}, {CP}_{j}); \\ s_{i, j} = s_{0}; \\ k = 2; \end{matrix}$
while $\frac{\| w - width ({ks}_{0}, P_{i}, {CP}_{j}) \|}{w} < ε$ do
$\begin{matrix} s_{i, j} = {ks}_{0}; \\ k = k + 1 \end{matrix}$
return $\min_{i, j} s_{i, j}$ ;

Illustrated examples and discussions

Illustrated examples

Example 1

The desired surface S is chosen to be the concave side of a cylindrical surface with radius $CR = 30 mm$ and height $h = 100 mm$ (see Figure 4). As shown in Figure 5, the cutter $τ$ is chosen to be a flat-end torus tool, where $L_{1} = 6.5 mm$ and the radius of C₁ is $r = 1.5 mm$ . The tolerance of machining is set to be $Stol = 0.02 mm$ , and the allowance for overcut is $0.01 mm$ . The cutter moves toward the minimal principal direction f of the surface with an inclination angle $θ = 10 . 55^{\circ}$ and a tilt angle $ϕ = 0^{\circ}$ . As shown in Figure 6, the cutter initially positions on the surface with cutter axis parallel to the normal vector n of a driving point p with parameters $u = 0.5$ and $v = 0.46$ on the desired surface, and the cutting direction that starts from the driving point lies on the same flat with the end of the cutter. The cutter is uplifted by Stol from the cutting path toward the normal vector at the driving point p. Let the x-axis parallel with $f$ and the z-axis parallel with n, $O = p + (r + Stol) \cdot n$ , establish the coordinate system Oxyz that follows the right-hand rule. The cutter is first rotated around the z-axis by $ϕ$ and then rotated around the y-axis by $θ$ . The cutter positioning strategy used here is called the middle point error control method in the literature.²²

Figure 4.

The desired surface for experiments.

Figure 5.

The generatrix of the cutter for experiments.

Figure 6.

Cutter positioning.

$P = (55.596602, 0.106186, - 30.019854)$ is a point on the desired surface with parameters $u = 0.500808$ and $v = 0.444$ . The normal vector of S at P is

\vec{n} = (55.596602, 0.106133, - 29.999709)

(21)

We have $dis (P, \vec{n}, τ) = 0.020146 mm$ . So

Err (0.500808, 0.444) = 0.020146 mm

(22)

The time cost for computing $Err (0.500808, 0.444)$ is $0.01854 ms$ . The directed projective point of P on $τ$ via $\vec{n}$ is $q = (55.596602, 0.106133, - 29.999709)$ . As a comparison, the time cost for computing the closest point of q on S is $0.343 ms$ using the iterative algorithm offered by computer-aided manufacturing (CAM) system UGS NX 4.0. And the outcome closest point exactly matches P. It demonstrates that our strategy leads to little time cost with high accuracy.

Remark 3

The cutter positioning method has no effects on the computation of ICPEC. The middle point error control method is only used to describe the posture of the cutter.

Example 2

Example goes on under the same condition with example 1. The initial step length is set to be $step = 0.002122$ . By Method 1 (the method mentioned in section “symmetrical ICPEC”), the ICPEC is

f (x) = {Ax}^{4} + {Bx}^{3} + {Cx}^{2} + Dx + E

(23)

where $A \approx 7667.220645, B \approx - 15, 359.214413, C \approx 11, 507.56805, D \approx - 3821.718382, E \approx 474.688382$ . Then, by solving $f (x) = Stol,$ $Width = 0.125670, V_{l} = 0.438, and V_{r} = 0.564$

Using the algorithm OptimalStep with accuracy $ε = 0.001$ and the cutter positioning in example 1, the optimal step length $step = 0.016976$ is achieved. Method 1 returns $A \approx 7610.143519, B \approx - 15, 244.875743, C \approx 11, 421.902209, D \approx - 3793.268346, E \approx 471.154595$ . $Width = 0.125783, V_{l} = 0.437916, V_{r} = 0.563699$ .

In Figure 7, the solid circles denote the fitted points that used to approach the ICPEC, the curve is the approximate ICPEC acquired from Method 1, and the solid squares are sampling points on the ICPEC. As we can see, the curve in Figure 7(b) is fitted by only four points, while the curve in Figure 7(a) is fitted by 22 points.

Figure 7.

Experimental results of Method 1 for Example 2: (a) using the initial step length and (b) using the optimal step length.

Similarly, the performances of Method 2 (the method mentioned in section “Nonsymmetric ICPEC”) are shown in Figure 8.

Figure 8.

Experimental results of Method 2 for Example 2: (a) using the initial step length and (b) using the optimal step length.

For a point $p : (X, Y)$ and a curve $γ : (x (t), y (t))$ in 2D space, define the error of the point p to the curve $γ$

E (p, γ) = \min_{t} \sqrt{{(X - x (t))}^{2} {(CR \cdot π)}^{2} + {(Y - y (t))}^{2}}

(24)

The errors of the sampling points to the approximate ICPEC are shown in Figure 9. The errors of the two methods are almost equal. The average errors in Table 1 are obtained by the average errors of the points shown in Figures 7 and 8 to the approximate ICPEC computed by the two methods. And t in time cost refers to the total time cost of the method fitting the ICPEC by all the points located in $[V_{l}, V_{r}]$ . As the optimal step length is much larger than the initial step length, Table 1 comprehensively shows that both Method 1 and Method 2 vastly reduce the time cost with small numerical errors using the optimal step length.

Figure 9.

Errors of different methods for Example 2.

Table 1.

Performance of two methods in Example 2.

Method	Step	Average error	Time cost	Width	$V_{l}$	$V_{r}$
Method 1	0.002122	0.008546	0.385965t	0.125670	0.438	0.564
Method 1	0.016976	0.00857	0.070175t	0.125783	0.437916	0.563699
Method 2	0.002122	0.007607	0.736842t	0.122893	0.438	0.561
Method 2	0.036075	0.007657	0.070175t	0.122974	0.437905	0.560879

Example 3

Continue from Example 2, the inclination angle changes to $θ = 11 . 17^{\circ}$ and the tilt angle of the cutter is set to be $ϕ = 5^{\circ}$ . Then, the ICPEC becomes nonsymmetric.

The comprehensive contrast of the performance is listed in Table 2. Figures 10 and 11 show the acquired approximate ICPEC. The numerical errors of the two methods are illustrated in Figure 12. From Figure 10, the symmetric model involves unpredictable errors, and Figure 11 shows that the proposed nonsymmetric model stably solves the problem.

Table 2.

Performance of two methods in Example 3.

Method	Step	Average error	Time cost	Width	$V_{l}$	$V_{r}$
Method 1	0.002122	0.021177	0.294118t	0.083636	0.459	0.543
Method 1	0.019098	0.02133	0.058824t	0.083708	0.458954	0.542662
Method 2	0.002122	0.004618	0.72949t	0.107416	0.458	0.566
Method 2	0.029709	0.004649	0.078431t	0.107476	0.458441	0.565917

Figure 10.

Experimental results of Method 1 for Example 3: (a) using the initial step length and (b) using the optimal step length.

Figure 11.

Experimental results of Method 2 for Example 3: (a) using the initial step length and (b) using the optimal step length.

Figure 12.

Errors of different methods for Example 3.

Example 4

To verify the accuracy of the strip width estimated from Method 1 and Method 2, we check the tolerance of the path overlaps in simulating software VERICUT 6.0.3. Figure 13 shows the simulating machined surfaces with two different methods in Example 2. Figure 14 shows the simulating machined surfaces with the same cutter positioning in Example 3. The green part of the figures denotes that the tolerance is within $0.03 mm$ . And the deep pink part means that the tolerance is within $0.02 mm$ , which is the designed allowance in configuration. Both Figures 13 and 14(b) barely have green part, and this illustrates the strip widths estimated from Method 1 and Method 2 are of high precision. Figure 14(a) illustrates that Method 1 leads to unexpected undercut while the ICPEC is nonsymmetric.

Figure 13.

Simulating results for Example 2: (a) the machined surface of Method 1 and (b) the machined surface of Method 2.

Figure 14.

Simulating results for Example 3: (a) the machined surface of Method 1 and (b) the machined surface of Method 2.

Discussions

This article first proposed the concept of directed distance, which originated from a classical method called the “point-vector method.” The orthogonal restriction is removed, and the distance calculation is turned into a line-surface intersection problem. Taking advantages of the simple expression of CNC rotary tools, the proposed strategy greatly reduces the computing time, without losing the accuracy, which is illustrated in example. The error distribution between the CNC cutter and the desired surface needs to compute the distances of the point cloud to the cutter. The less the number of points in the cloud, the less time that the computation costs. Thus, the computing time could be valued by the actual points that are involved in the distance computation. We proposed to reduce the points by two fitting models. The symmetric model works well and fast when the ICPEC is symmetric, while it involves computing error if the ICPEC is nonsymmetric. The nonsymmetric model is always effective, both in symmetric and nonsymmetric cases.

Conclusion

This article proposes two fast calculating algorithms of the ICPEC in CNC machining. Both the algorithms are based on the fitting techniques and to reduce the points involved in the distance computation. In order to accurately solve the positional relationship between the desired surface and the cutter surface, the strategy of computing the distances between discrete points on the desired surface and the cutter is used. And the directed projection is established to measure the exact distance of a point and the cutter. The experimental results show that the proposed algorithms greatly reduce the computation time under acceptable accuracy.

Footnotes

Academic Editor: Michal Kuciej

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was supported by Chinese National Science and Technology Major Project 2013ZX04011031.

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Fast computing of instantaneous cutter position error curve

Abstract

Keywords

Introduction

Directed distance of a point to a cutter

Description of CNC cutters

Directed distance of a point to a cutter

The ICPEC

Definition 1

Remark 1

Fast computing of ICPEC

Symmetrical ICPEC

Remark 2

Nonsymmetric ICPEC

Calculation of strip width

Choice of the step length

Illustrated examples and discussions

Illustrated examples

Example 1

Remark 3

Example 2

Example 3

Example 4

Discussions

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References