Approaching the characteristic curve of the cutter’s envelope based on a velocity field

Abstract

The characteristic curve of the envelope generated by the tool motions is an important medium for measuring the distance between the tool and the desired surface. Via the single parametric surface envelope theory in differential geometry, this article proved the correctness of the characteristic curve obtained by minimum distance pairs. Simultaneously, combined with the existing envelope theory and the longitude method, an algorithm based on a velocity field is proposed to approach the characteristic curve. The torus cutter is first dispersed into longitudes and then the characteristic point on each longitude is determined with the estimated velocity. In the experiments, the actual velocity of the estimated characteristic point is computed by the true trajectory simulation of the tool. Therefore, the deviation of the proposed method could be measured comparing with the actual velocity. In the implementations, the proposed method is validated in machining a flat, a spiral, and a blade surfaces. In machining the spiral, our algorithm improves approximately 7 times accuracy with 1/7 time cost compared with the existing methods. The results in the blade surface example prove the stableness of the proposed algorithm.

Introduction

Tool orientation optimization is a critical step in computer numerical control (CNC) machining. When the cutter positions on the desired surface, to adjust the cutter orientation, the measurement of the error between the cutter and the desired surface is very necessary. In Cai and Xi, ¹ a interferential-free cutter location is generated by computing the projection of points on the surface to the axis of cutter. In He and Chen, ² the cutter orientation is adjusted based on the error distribution between the cutter and the desired surface. Ying and Zhitong ³ generate quasi constant machining strip width according to the error computation. Fan and Xi ⁴ compute the scallop height tool path based on the estimated tool path interval. In Zheng et al., ⁵ the geometric error is measured by the signed distance. In Fan and Ball, ⁶ in order to compute the error between the cutter and the design surface, the design surface is approximated locally by a quadric.

The cutter motions in CNC machining generate the machined surface, just as surface motions generate the swept profile in mathematical envelope theory. So, the problem of the machined surface can be transformed into the swept profile problem. In ordinary envelope theory, when the cutter motions are determined, the swept profile is reachable by solving the sweep–envelope equation. The cutter swept profile is commonly used in machining verification and CNC simulation.^7,8 Chiou and Lee ⁹ proposed an approach of three-dimensional (3D) shape-generating profiles of different types of cutters for constructing the G-buffer models for five-axis machining. In Du et al., ¹⁰ using envelope theory and swept volume generation, an algorithm is presented for analytically formulating the swept profiles and generating topological valid swept volume of five-axis tool motions. In Zhiqiang and Wuyi,^11,12 they studied the cutting edge of torus with translational motions. A closed-form solution of five-axis swept profile for a generalized cutter is deduced in Chiou and Lee. ¹³ In Bi et al., ¹⁴ the analytical representation of a conical cutter used for flank milling with rational motion is presented. In Jung et al., ¹⁵ based on the triangle strip representation, an algorithm is presented for generating the swept volume with self-intersecting.

Analytical ways for cutter envelope surface are high accurate and efficient because it directly yields the exact result by a deduced formula. But the analytical method requires determined cutter motions. In five-axis machining, the orientations in a path are optimized independently according to the error distribution between the swept profile and the desired surface. Therefore, the swept profile is hard to be reached in optimizing. Lartigue et al. ¹⁶ proposed a method to deform the tool path in five-axis flank milling, in which the assessment is calculated by a kinematics approach envelope surface.

In the previous studies,^17–19 the swept profile is simplified as an ellipse while verifying the errors of an orientation in narrow strip machining. This method works well when the machining strip is within the neighborhood of the cutter contact (CC) point. However, Warkentin et al.^20,21 proposed that there can be more than one CC points and deduced a positioning method called multiple-point method to enhance the machining strip width. The local contacting problem is turned into global contacting problem in one sense. The ellipse method loses its effectiveness in wide strip machining.

Characteristic curve of a cutter is an instantaneous curve on the cutter that actually grazes the machined surface. For a given cutter orientation, the error distribution would be denoted by the distances between characteristic points and the design surface. Thus, the orientation can be optimized according to the error distribution. So, it becomes important to approach to characteristic points first. In Tournier and Duc, ²² the scallop height is performed by intersecting the constant height surface with a circle, which represents the characteristic curve of a ball-end mill. With fixed tool axis orientations, the scallop point is determined by the envelope equation in Tournier and Duc. ²³ The literature^24–26 proposed the minimum distance pairs method to measure the error distributions based on the principle of global curvature matching. The method is developed and proved to be an applicable tool in cutter orientation optimization.^27–29 Particularly, Jiayong et al. ³⁰ improved the computing efficiency of the minimum distance pair method via envelope theory. But because the desired surface is treated as points, the solution of envelope equation is numerical and does not cover all the characteristic points. In Zhengqiang et al., ³¹ the strategy of dispersing the cutter to compute the minimum distance pairs is established for drum-like cutter. Fanjun et al. ³² proposed a latitude method to compute the error distribution and showed a static comparison with Jiayong et al.’s method and Zhengqiang et al.’s method. Rufeng et al. ³³ used a similar method, separating the cutter by longitudes for torus tool. Although the strategy of dispersing a cutter is fast, it would involve unnecessary errors, which will be discussed later.

This article first introduces the minimum distance pair method and gives a static proof. The longitude method is described as a dual method of dispersing the desired surface, and the error is discussed concisely. A key element in the envelope equation is the velocity of the characteristic point. Nevertheless, the velocity of a point on the cutter is difficult to compute while only knowing the current orientation. This article projects the center of the longitude onto a vector field and chooses the vector of the foot point to be the velocity of characteristic points on current longitude. The envelope equation is accurately solved using the estimated velocity. Then the orientation error is valued by the distances of the characteristic points to the design surface. Experimental results show improvements of our method respect to the longitude method and Jiayong et al.’s method. The error of the estimated velocity will be discussed by comparing with the actual velocity in a determined path.

Compared with the existing methods, the main contributions of this article are listed as follows:

1. A theoretical proof for the minimum distance pair method is presented.

2. A method based on the envelope theory and longitude method is proposed. As shown in the implements, the proposed method significantly enhances the computing efficiency and reduces the time cost, compared with the existing methods.

3. To measure the errors of the characteristic curve, the dot products of normal vectors and velocities are used in the validation.

In what follows, section “Preliminaries on the cutter’s envelope” describes the preliminaries on envelope theory. Section “Generalized envelope theory” defines the minimum distance pair and demonstrates a concise proof, and the longitude method is also introduced in this section. A new method to compute the characteristic curve is proposed in section “Approaching the characteristic curve based on a vector field.” Two examples are illustrated in section “Numerical experiments and discussion.” Section “Conclusion” concludes the whole article concisely.

Preliminaries on the cutter’s envelope

Definition 1

$\{\mathit{t}(\alpha )\}$ is a family of surfaces in a 3D space with at least G1 continuous and S is a specific surface. Then, S is called the envelope of $\{\mathit{t}(\alpha )\}$ , if and only the following conditions hold:

• $\forall p\in S$ , $\exists \alpha$ , such that $p\in \mathit{t}(\alpha )$ , and ${\mathit{n}}_S(p)\times {\mathit{n}}_{t(\alpha )}(p)=0$ ;

• $\forall \alpha$ , $\exists p\in \mathit{t}(\alpha )$ , such that $p\in S$ , and ${\mathit{n}}_S(p)\times {\mathit{n}}_{t(\alpha )}(p)=0$ .

where ${\mathit{n}}_S(p)$ and ${\mathit{n}}_{t(\alpha )}(p)$ denote the unitized normal vector of S and $\mathit{t}(\alpha )$ at p, respectively.

In general, any surface in the family contacts the envelope at a curve. For a given $\alpha$ , the contacted tangent curve of $\mathit{t}(\alpha )$ and S is called the characteristic curve of $t(\alpha )$ or simply referred to as the characteristic curve without ambiguity. A point on the characteristic curve can be called a characteristic point. In CNC machining, the characteristic curve represents the actual cutting part of the cutter surface.

In what follows, an important necessary condition of the characteristic curve will be deduced. Let $\mathit{S}(u,v)$ be the envelope surface of the single parameter surface family $\{\mathit{t}(\theta ,\phi ,\alpha )\}$ . $p=\mathit{t}({\theta }_p,{\phi }_p,{\alpha }_p)=\mathit{S}(u_p,v_p)$ is a point that lies on the characteristic curve of $\mathit{t}(\theta ,\phi ,{\alpha }_p)$ . It is indicated from definition 1b that

$${\mathit{n}}_S\left(p\right)\times {\mathit{n}}_{t(\alpha )}\left(p\right)=0$$

Implies

$${\mathit{S}}_u(u_p,v_p)\times {\mathit{S}}_v(u_p,v_p)\times {\mathit{n}}_{t(\alpha )}\left(p\right)=0$$

It is from the property of cross product that

$$\begin{array}{c}({\mathit{S}}_u(u_p,v_p)·{\mathit{n}}_{t(\alpha )}\left(p\right)){\mathit{S}}_v(u_p,v_p)\hfill \\ -({\mathit{S}}_v(u_p,v_p)·{\mathit{n}}_{t(\alpha )}\left(p\right)){\mathit{S}}_u(u_p,v_p)=0\hfill \end{array}$$

Because ${\mathit{S}}_u(u_p,v_p)$ and ${\mathit{S}}_v(u_p,v_p)$ are linear independent, we have

$${\mathit{S}}_u(u_p,v_p)·{\mathit{n}}_{t(\alpha )}\left(p\right)={\mathit{S}}_v(u_p,v_p)·{\mathit{n}}_{t(\alpha )}\left(p\right)=0$$

The above equation is equivalent to

$$(a{\mathit{S}}_u(u_p,v_p)+b{\mathit{S}}_v(u_p,v_p))·{\mathit{n}}_{t(\alpha )}\left(p\right)=0,\forall a,b\in \mathbb{R}$$

(1)

The above equation describes a necessary condition of p that becomes a characteristic point and shows the way to compute the characteristic curve as well. It should be emphasized that since it is not a sufficient condition, the point satisfied the condition is not necessary to be the characteristic point. To take advantages of ${\mathit{S}}_u(u_p,v_p)$ and ${\mathit{S}}_v(u_p,v_p)$ , it needs to obtain the envelope surface S at first, but S cannot be determined before all the cutter orientations in a path are adjusted. As the machining process can be treated as the motion of the cutter surface, let the velocity direction at p of the cutter surface be $\mathit{v}(p)$ . Here, it is claimed that $\mathit{v}(p)$ is a unitized vector. As p lies on the envelope of the cutter surface, $\mathit{v}(p)$ landed on the tangent plane of S. Then, $\exists a,b\in \mathbb{R}$ , such that $\mathit{v}(p)=a{\mathit{S}}_u(u_p,v_p)+b{\mathit{S}}_v(u_p,v_p)$ , substituting in equation (1), we have

$$\mathit{v}\left(p\right)·{\mathit{n}}_{t(\alpha )}\left(p\right)=0$$

(2)

Although equation (2) is still a necessary condition, all the characteristic points have to be contained in the solutions. In this article, a point p that satisfied equation (2) is indiscriminately called the characteristic point. If given a point p on the cutter surface, it could be verified whether p is a characteristic point or not by equation (2). In the process of three-axis machining, as the velocities of each point on the cutter surface are exactly the same, it is easy to judge the characteristic of a point. However, it becomes difficult to calculate the velocities of points on the cutter surface in multiple-axis motions, especially in five-axis motions, on account of the following problems:

1. The cutter surface could rotate around an axis in the space. It leads to different directions of the point velocities.

2. The velocity of current orientation cannot be determined until the adjacent orientations in the same path are ascertained.

Generalized envelope theory

Minimum distance pairs

In engineering practice, it is usually unable to compute the exact characteristic points due to complex cutter motions. Within the acceptable machining accuracy, if a point p on the cutter surface satisfies

$$\mathit{v}\left(p\right)·{\mathit{n}}_{t(\alpha )}\left(p\right)\approx 0$$

(3)

then p is called the approximate characteristic point of the envelope.

• Lemma 1. Let $\mathit{r}(t)$ be a parametric curve in a 3D space, $\mathbf{\tau }(u,v)$ be a parametric surface. Denote

$$\mathrm{dis}(\mathit{r},\mathbf{\tau })=\mathrm{minimiz}{\mathrm{e}}_{(t,u,v)}\Vert \mathit{r}\left(t\right)-\mathbf{\tau }(u,v)\Vert$$

(4)

to be the distance of $\mathit{r}(t)$ to $\mathbf{\tau }(u,v)$ . If the solution of the optimizing problem (4) is $(t_0,u_0,v_0)$ , and $p=\mathit{r}(t_0)$ and $q=\mathbf{\tau }(u_0,v_0)$ , then

$$(p-q)\times {\mathit{n}}_{\mathbf{\tau }}\left(q\right)=0$$

$$(p-q)·\mathit{r}\prime \left(p\right)=0$$

In which ${\mathit{n}}_{\tau }(q)$ denotes the normal vector of $\mathbf{\tau }(u,v)$ at q, $\mathit{r}\prime (p)$ is the tangent vector of $\mathit{r}(t)$ at p. This implies that the line that connects p and q is perpendicular to the surface $\mathbf{\tau }(u,v)$ and the tangent vector of $\mathit{r}(t)$ at the same time. $(p,q)$ is called the minimum distance pair of $\mathit{r}(t)$ to $\mathbf{\tau }(u,v)$ .

Proof

The objective function of equation (4) can be written as

$$g(t,u,v)=(\mathit{r}-\mathbf{\tau })·(\mathit{r}-\mathbf{\tau })=\mathit{r}·\mathit{r}-2\mathit{r}·\mathbf{\tau }+\mathbf{\tau }·\mathbf{\tau }$$

When the $g(t,u,v)$ achieves the optimal solution, it is deduced from the condition of the first-order optimality that

$$\{\begin{array}{c}{g}_t=2r·\mathit{r}\prime -2\tau ·\mathit{r}\prime =2(\mathit{r}-\mathbf{\tau })·\mathit{r}\prime =0\hfill \\ \multicolumn{1}{c}{g_u=2\tau ·{\mathbf{\tau }}_u-2r·{\mathbf{\tau }}_u=2(\mathbf{\tau }-\mathit{r})·{\mathbf{\tau }}_u=0}\\ \multicolumn{1}{c}{g_v=2\tau ·{\mathbf{\tau }}_v-2r·{\mathbf{\tau }}_v=2(\mathbf{\tau }-\mathit{r})·{\mathbf{\tau }}_v=0}\end{array}$$

Implies

$$\begin{array}{c}(p-q)·\mathit{r}\prime \left(t_0\right)=(p-q)·{\mathbf{\tau }}_u(u_0,v_0)\hfill \\ =(p-q)·{\mathbf{\tau }}_v(u_0,v_0)=0\hfill \end{array}$$

Then, the line that connects p and q is perpendicular to the surface $\mathbf{\tau }(u,v)$ and the tangent vector of $\mathit{r}(t)$ at the same time.

Theorem 2.(the principle of the minimum distance pairs)

• Given a desired surface $\mathit{s}(u,v)$ for machining, we establish a series of surfaces along the feeding direction (or the tool path direction). A series of sectional lines $\{{\mathit{r}}_{\alpha }(t)|\alpha \in [a,b]\}$ can be obtained by intersecting the established surfaces and $\mathit{s}(u,v)$ (see Figure 1). If the minimum distance pair of ${\mathit{r}}_{\alpha }(t)$ to the cutter surface $\mathbf{\tau }(u,v)$ is $(p_{\alpha },q_{\alpha })$ , then $q_{\alpha }$ is an approximate characteristic point of the envelope, and $\{q_{\alpha }\}$ composes the approximate characteristic curve on the cutter surface.

Figure 1.

Illustration for sectional lines.

Proof

It is according to lemma 1 that

$$(p_{\alpha }-q_{\alpha })·\mathit{r}\prime \left(p_{\alpha }\right)=0$$

Since the sectional lines $\{{\mathit{r}}_{\alpha }(t)|\alpha \in [a,b]\}$ are chosen to be parallel with the feeding direction, it is obvious that $\mathit{v}\left(q_{\alpha }\right)\approx \mathit{r}\prime \left(p_{\alpha }\right)$ when $p_{\alpha }$ and $q_{\alpha }$ stay closely enough. It becomes

$$\mathit{v}\left(q_{\alpha }\right)·{\mathit{n}}_{\tau }\left(q_{\alpha }\right)\approx \mathit{r}\prime \left(p_{\alpha }\right)·{\mathit{n}}_{\tau }\left(q_{\alpha }\right)=\mathit{r}\prime \left(p_{\alpha }\right)·\frac{p_{\alpha }-q_{\alpha }}{\Vert p_{\alpha }-q_{\alpha }\Vert }=0$$

(5)

The above equation shows that the point $q_{\alpha }$ in the minimum distance pair $(p_{\alpha },q_{\alpha })$ satisfies condition (3) and so is an approximate characteristic point, and it results in $\{q_{\alpha }\}$ composing the approximate characteristic curve.

Remark

In machining, when $\Vert p_{\alpha }-q_{\alpha }\Vert <<\epsilon$ with a fixed $\epsilon$ , and maintain that the cutter would not gouge the desired surface, the desired surface locally tangents the cutter. For this purpose

$$\mathit{v}\left(q_{\alpha }\right)\approx \mathit{r}\prime \left(p_{\alpha }\right)$$

When the denominator $\Vert p_{\alpha }-q_{\alpha }\Vert$ in equation (5) tends to 0, we have

$${\mathbf{r}}^{\prime }\left(p_{\alpha }\right)·\frac{p_{\alpha }-q_{\alpha }}{\Vert p_{\alpha }-q_{\alpha }\Vert }=\frac{r^{\prime }\left(p_{\alpha }\right)·\left(p_{\alpha }-q_{\alpha }\right)}{\Vert p_{\alpha }-q_{\alpha }\Vert }=\mathrm{Lemma}1\frac{0}{\Vert p_{\alpha }-q_{\alpha }\Vert }=0$$

The last equality of the above equation holds because 0 is the higher-order infinitesimal quantity of any infinitesimal quantity. So, equation (5) is numerical stable.

Define

$$f\left(\alpha \right)=\Vert p_{\alpha }-q_{\alpha }\Vert$$

to be the error distribution function. For a given allowance $\mathrm{stol}$ , let $\mathrm{Intv}(\mathrm{stol})=\{\alpha |f(\alpha )\le \mathrm{stol}\}$ , then the Lebesgue measure ³⁴ of the set $\mathrm{Intv}(\mathrm{stol})$ is called the machining strip width.

Longitude method

In manufacturing process, both the design surface and the cutter surface are known. The process can be regarded as the envelope movement of the tool surface over the design surface, that is, the tool surface machines the design surface. Dually, the process can also be treated as the envelope movement of the design surface over the tool surface, that is, processing the tool surface by the design surface.

Another equivalent description of minimum distance pair is that given a cutter surface $\mathbf{\tau }(u,v)$ for machining, we establish a series of surfaces along the feeding direction (or the tool path direction). A series of sectional lines $\{{\mathit{r}}_{\alpha }(t)|\alpha \in [a,b]\}$ can be obtained by intersecting the established surfaces and $\tau (u,v)$ . If the minimum distance pair of ${\mathit{r}}_{\alpha }(t)$ to the design surface $\mathit{s}(u,v)$ is $(p_{\alpha },q_{\alpha })$ , then similarly, $q_{\alpha }$ is an approximate characteristic point on the design surface in this sense. We call $p_{\alpha }$ the approximate dual characteristic point, and $\{p_{\alpha }\}$ comprises the approximate dual characteristic curve on the cutter surface.

Within the neighborhood of the CC point, the connective line of the minimum distance pair is almost vertical to the cutter surface; thus, the approximate dual characteristic point on the cutter surface mostly coincides with the approximate characteristic point. Based on this fact, the above definition offers another way to reach the minimum distance pair. That is to disperse the cutter surface into sectional lines and then compute the distances of the sectional lines to the desired surface, respectively. As the expressions of common CNC tool surfaces are simple with respect to the design surface, the discrete tool strategy has distinct advantages in computing efficiency.

For instance, when the cutter surface has the following torus expression

$$\begin{array}{c}\mathbf{\tau }(\theta ,\phi )=((R+r\cos \left(\theta \right))\cos \left(\phi \right)\hfill \\ \mathit{t}(R+r\cos \left(\theta \right))\sin \left(\phi \right),r\sin \left(\theta \right))\hfill \end{array}$$

It should be noted that it requests that the sectional lines are calculated based on the feeding direction in the definition, which involves tangle intersection computation of surfaces with the cutter. In order to improve the computational efficiency, the sectional lines are fixed to be the longitudes when the cutter is a rotary surface, instead of lines along the feeding direction. For the torus tool, first equidistantly dividing $[0,2\pi ]$ by $\{{\phi }_{\alpha }|0<\alpha \le n\}$ , one can obtain the set of sectional lines $\{{\mathit{r}}_{\alpha }(t)=\mathbf{\tau }(t,{\phi }_{\alpha })\}$ and then the minimum distance pair $(p_{\alpha },q_{\alpha })$ can be computed.

The longitude method avoids the calculation of sectional lines. The expressions of longitudes are simple and easily processed. But since the longitude has nothing to do with the feeding direction (see Figure 2), the longitude method would not guarantee

$$\mathit{v}\left(p_{\alpha }\right)\approx \mathit{r}\prime \left(q_{\alpha }\right)$$

Figure 2.

Comparison of two division methods.

Hence, the result does not satisfy equation (3). Furthermore, the error between the dual characteristic point and the characteristic point on the cutter increases while the dual characteristic point is getting away from the CC point. In a word, despite high efficiency, the longitude method sacrifices accuracy that cannot be estimated.

Approaching the characteristic curve based on a vector field

In CNC machining, the cutter is usually orientated at a drive point on the desired surface, and the tool path is planned on the desired surface as well. So, there exists a vector field that denotes the tool path direction on the desired surface after the tool path is planned, corresponding to the feed direction of each driving point. As it concerns only about the direction of the velocity in envelope theory, velocity refers to unitized vector in this article.

Definition 2

$\mathit{S}(u,v)$ denotes a surface, $\mathbf{\Lambda }(u,v)\in {\mathbb{R}}^3$ is a unitized vector, and $\mathbf{\Lambda }(u,v)$ satisfies

$$\mathbf{\Lambda }(u,v)=a(u,v){\mathit{S}}_u(u,v)+b(u,v){\mathit{S}}_v(u,v)$$

where $a(u,v),b(u,v)\in \mathbb{R}$ . Then, $\mathbf{\Lambda }(u,v)$ is called a velocity field defined on $\mathit{S}(u,v)$ . If $p=\mathit{S}(u_0,v_0)$ , denote $\mathbf{\Lambda }(p)=\mathbf{\Lambda }(u_0,v_0)$ .

Suppose the cutter surface is a torus. In the world coordinate system, the center of the torus is $O_{\tau }$ , the unitized tool axis is ${\mathit{e}}_2$ , the major radius of the torus is R, and the corner radius is r. If the center of the longitude circle that is associated with the CC point is $O_c$ , then let

$${\mathit{e}\prime }_1=\frac{O_c-O_{\tau }}{\Vert O_c-O_{\tau }\Vert }$$

$${\mathit{e}\prime }_3={\mathit{e}\prime }_1\times {\mathit{e}}_2$$

Then

$$O_{\alpha }=O_{\tau }+R(\cos \left(\alpha \right){\mathit{e}\prime }_3+\sin \left(\alpha \right){\mathit{e}\prime }_1)$$

expresses the center of ${\mathit{r}}_{\alpha }$ , which is one of the longitude circles. Let

$${\mathit{e}}_1\left(\alpha \right)=\frac{O_{\alpha }-O_{\tau }}{\Vert O_{\alpha }-O_{\tau }\Vert }$$

$${\mathit{e}}_3\left(\alpha \right)={\mathit{e}}_1\left(\alpha \right)\times {\mathit{e}}_2$$

Establish a coordinate system $\mathrm{LC}{\mathrm{S}}_{\alpha }$ (as shown in Figure 3) with the origin $O_{\alpha }$ and the axes ${\mathit{e}}_1(\alpha )$ , ${\mathit{e}}_2$ , and ${\mathit{e}}_3(\alpha )$ . The following discussion is taken in $\mathrm{LC}{\mathrm{S}}_{\alpha }$ .

Figure 3.

Coordinate system LCS _α .

If the orthogonal projective point of $O_{\alpha }$ on the desired surface is ${\omega }_{\alpha }$ , then the corresponding velocity in the velocity field of ${\omega }_{\alpha }$ is $\mathbf{\Lambda }({\omega }_{\alpha })$ . The coordinate of $\mathbf{\Lambda }({\omega }_{\alpha })$ in $\mathrm{LC}{\mathrm{S}}_{\alpha }$ is

$${\mathbf{\Lambda }}_{Lcs}\left({\omega }_{\alpha }\right)=\left(x_{\alpha },y_{\alpha },z_{\alpha }\right)=\left(\mathbf{\Lambda }\left({\omega }_{\alpha }\right)·{\mathbf{e}}_1\left(\alpha \right),\mathbf{\Lambda }\left({\omega }_{\alpha }\right)·{\mathbf{e}}_2,\mathbf{\Lambda }\left({\omega }_{\alpha }\right)·{\mathbf{e}}_3\left(\alpha \right)\right)$$

Assume that $\mathbf{\Lambda }({\omega }_{\alpha })$ is also the instantaneous velocity of the characteristic point on the current longitude circle, and

$$P_{\alpha }\left(\theta \right)=r(\cos \left(\theta \right),\sin \left(\theta \right),0)$$

is a characteristic point on current longitude circle ${\mathit{r}}_{\alpha }$ . Hence, the normal vector of the torus at $P_{\alpha }(\theta )$ is

$${\mathit{n}}_{\mathrm{LCS}}\left(P_{\alpha }\left(\theta \right)\right)=(\cos \left(\theta \right),\sin \left(\theta \right),0)$$

Solve the following equation

$${\mathbf{\Lambda }}_{\mathrm{Lcs}}\left({\omega }_{\alpha }\right)·{\mathit{n}}_{\mathrm{LCS}}\left(P_{\alpha }\left(\theta \right)\right)=0$$

(6)

and acquire the solution

$${\theta }_1\left(\alpha \right)=\arctan (-\frac{x_{\alpha }}{y_{\alpha }})$$

Note that ${\theta }_2(\alpha )={\theta }_1(\alpha )+\pi$ is the other solution of equation (6). Let $p_{\alpha }(\theta )=O_{\alpha }+r\cos (\theta ){\mathit{e}}_1(\alpha )+r\sin (\theta ){\mathit{e}}_2$ , thereby the two characteristic points on ${\mathit{r}}_{\alpha }$ are $p_{\alpha }({\theta }_1(\alpha ))$ and $p_{\alpha }({\theta }_2(\alpha ))$ in the world coordinate system, respectively. Hence, $p_{\alpha }({\theta }_1(\alpha ))$ is called the outer characteristic point of the torus, and $p_{\alpha }({\theta }_2(\alpha ))$ is the inner characteristic point. Accordingly, the approximate outer characteristic curve of the torus is $C_1(\alpha )=p_{\alpha }({\theta }_1(\alpha ))$ , and the approximate inner characteristic curve is $C_2(\alpha )=p_{\alpha }({\theta }_2(\alpha ))$ . The outer characteristic curve refers to the intersection of the cutter and the machined surface. So, it is used to compute minimum distance pairs.

The above procedure to computing minimum distance pairs is shown as the flowchart in Figure 4. In computing the minimum distance pairs, the advantage of the longitude method is that it is easy to acquire the sectional lines, while the disadvantage is that it causes relatively large error. So, the proposed method disperses the tool as the longitude method does but determines the characteristic point by the envelope equation. The accuracy of the proposed method is controlled by the chosen reliable velocity. Here, the velocity field is established exactly along the feed directions, believed to be reliable. This strategy results in both high accuracy and less time cost.

Figure 4.

Flowchart of error distribution computation.

This article focuses only on torus cutters for convenience. It is implicitly required that the characteristic curve intersects each longitude. The results can be easily generated to other rotary cutters by revising the expression of the longitude.

Numerical experiments and discussion

Machining a flat surface

Given the design surface s under the world coordinate system with expression

$$\mathit{s}(u,v)=(100-100v,60u,0),u,v\in [0,1]$$

The tool surface is chosen to be a torus $\tau$ , whose major circle radius is $R=6.5\mathrm{mm}$ , and minor circle radius is $r=1.5\mathrm{mm}$ . If the cutter moves along the u isoparametric lines, then the velocity field is defined to be

$$\mathbf{\Lambda }(u,v)=\frac{{\mathit{s}}_v(u,v)}{\Vert {\mathit{s}}_v(u,v)\Vert }=(-1,0,0)$$

Locate the torus such that it is tangent to the design flat at the drive point $p=\mathit{s}(0.5,0.5)$ , the contact point on the torus is denoted by q. Maintain the flat, in which is the major circle of the torus located, parallel with the tangent flat of the desired surface at p. Denote the center of the longitude circle corresponding to q by $O_c$ , the center of the torus is $O_{\tau }$ . Position the torus such that $O_{\tau }{O}_c\times \mathbf{\Lambda }(p)=0$ and $O_{\tau }{O}_c·\mathbf{\Lambda }(p)>0$ . Therefore, the torus is initially positioned on the desired surface at the drive point p.

Let $O_c$ be the origin, $\mathbf{\Lambda }(p)$ be x-axis, ${\mathit{n}}_s(p)$ be y-axis, we establish the space right-handed Cartesian coordinate $O_c\mathrm{xyz}$ . The cutter first rotates around the y-axis and then around z-axis. Tilt angle denotes that the rotary angle of the cutter rotates around the z-axis, and incline angle is the rotary angle of the cutter that rotates around the y-axis.

Set the incline angle to be $\beta =5\circ$ , and the tilt angle $\gamma =0\circ$ . Let $\alpha =5\circ$ , in the coordinate system $\mathrm{LC}{\mathrm{S}}_{\alpha }$ (defined in previous section)

$$\mathbf{\Lambda }\left({\omega }_{\alpha }\right)=(0.99240388,0.08715574,0.08682409)$$

Solve the equation

$${\mathbf{\Lambda }}_{\mathrm{LCS}}\left({\omega }_{\alpha }\right)·{\mathit{n}}_{\mathrm{LCS}}\left(P_{\alpha }\left(\theta \right)\right)=0$$

and obtain the solution of $\theta$

$${\theta }_1\left(\alpha \right)\approx -1.483198,{\theta }_2\left(\alpha \right)\approx 1.658395$$

Thus, the characteristic points on current longitude ${\mathit{r}}_{\alpha }$ are

$$\begin{array}{c}{p}_{\alpha }\left({\theta }_1\left(\alpha \right)\right)=(50.02464,29.42205,0.002199)\hfill \\ p_{\alpha }\left({\theta }_2\left(\alpha \right)\right)=(50.02464,29.44493,3.002112)\hfill \end{array}$$

Since the cutter motion in the example is a translational motion, the acquired characteristic points exactly match the theoretical points. As a comparison, the solution of $\theta$ calculated from the longitude method is

$$\theta \left(\alpha \right)\approx -1.487893$$

So, the characteristic point computed by the longitude method is

$$p_{\alpha }\left(\theta \left(\alpha \right)\right)=(50.03166,29.42266,0.002211)$$

The error of $p_{\alpha }\left(\theta \left(\alpha \right)\right)$ is estimated by

$$\mathrm{Err}\left(\alpha \right)=\Vert p_{\alpha }\left({\theta }_1\left(\alpha \right)\right)-p_{\alpha }\left(\theta \left(\alpha \right)\right)\Vert =0.007042652$$

The distance of $p_{\alpha }({\theta }_1(\alpha ))$ to the desired surface is

$$\mathrm{dis}\left(p_{\alpha }\left({\theta }_1\left(\alpha \right)\right)\right)=0.002199$$

And similarly

$$\mathrm{dis}\left(p_{\alpha }\left(\theta \left(\alpha \right)\right)\right)=0.002211$$

Let

$$\mathrm{disErr}\left(\alpha \right)=|\mathrm{dis}\left(p_{\alpha }\left({\theta }_1\left(\alpha \right)\right)\right)-\mathrm{dis}\left(p_{\alpha }\left(\theta \left(\alpha \right)\right)\right)|=0.000012$$

By changing the value of $\alpha$ from $-180\circ$ to $180\circ$ , we can obtain the characteristic curves $C_1(\alpha )$ and $C_2(\alpha )$ , as the blue lines shown in Figure 5. The lavender dash in Figure 5 denotes the characteristic curve computed by the longitude method. Intuitively shown in Figure 5(a), the inner characteristic curve in this example is blocked, so it is actually not on the envelope surface. Figure 6 illustrates the error distribution of characteristic points about the parameter $\alpha$ . In Figure 6, the range of $\alpha$ is $[-180\circ ,0\circ ]$ in the left side and $[0\circ ,20\circ ]$ in the right side. It is obvious that the distribution is symmetric. So, the left side reflects the global distribution, while the right side illustrates the local distribution. It shows that when we measure the error distribution using two different methods, only very slight errors may be involved relative to the positional errors of the characteristic points. Inasmuch as the result of the longitude method is inaccurate but valuable in the example.

Figure 5.

Characteristic curves on the torus: (a) front view, (b) top view, and (c) left view.

Figure 6.

Error distribution results of example 1.

As the incline angle $\beta$ changes from $0\circ$ to $90\circ$ , the difference between the proposed method and the longitude method is intuitively illustrated in Figures 7 and 8. As we can see, the results of the two methods are close when the absolute of $\alpha$ is small. But the error of the longitude method becomes large while the absolute of $\alpha$ and $\beta$ turn large. Mostly, the absolute of α ranges from 0° to 30°. The results of the longitude method are still acceptable under this assumption.

Figure 7.

Outer characteristic curves computed by the proposed method as the incline angle changes: (a) front view, (b) top view, and (c) left view.

Figure 8.

Characteristic curves computed by the longitude method as the incline angle changes: (a) front view, (b) top view, and (c) left view.

Machining a spiral surface

Consider the following design surface

$$\mathit{s}(u,v)=(100u\cos \left(2\pi v\right),100u\sin \left(2\pi v\right),100v),u,v\in [0,1]$$

The cutter surface is still a torus $\tau$ with $R=8.5\mathrm{mm}$ and $r=2\mathrm{mm}$ . Define the velocity field to be

$$\mathbf{\Lambda }(u,v)=\frac{{\mathit{s}}_v(u,v)}{\Vert {\mathit{s}}_v(u,v)\Vert }$$

On the drive curve $\mathit{s}(0.5,v)$ , the incline angles of each cutter orientations are set to be $\beta =2.558594\circ$ , and the tilt angles are $\gamma =0\circ$ . At the drive point $\mathit{s}(0.5,0.5)$ , when $\alpha =5\circ$ , we have

$$\begin{array}{c}\mathbf{\Lambda }\left({\omega }_{\alpha }\right)=(-0.00044,-0.95155,0.307503)\hfill \\ \mathrm{ch}{P}_{\alpha }=p_{\alpha }\left({\theta }_1(5\circ )\right)=(0.749432,-8.79023,0.303003)\hfill \\ \mathit{n}\left(\mathrm{ch}{P}_{\alpha }\right)=(0.004285,-0.3075,-0.95154)\hfill \end{array}$$

Because all the cutter orientations in the path are determined, the cutter motion is clear in the example. The actual velocity of each point on the cutter can be computed. Assume that the drive curve for the tool path is $\mathit{sd}(t)$ , the local tool coordinate system is $\mathrm{LTS}(t)$ corresponding to the drive point $\mathit{sd}(t)$ . And $\mathit{Oc}(t)$ is the origin of $\mathrm{LTS}(t)$ , $\mathit{Rcs}(t)$ is an orthogonal matrix consist of the axes of $\mathrm{LTS}(t)$ . The motion equation of a point $p_{\mathrm{lts}}$ in $\mathrm{LTS}(t)$ is

$$p\left(t\right)=\mathit{Oc}\left(t\right)+\mathit{Rcs}\left(t\right)p_{\mathrm{lts}}$$

and then the instantaneous velocity of $p_{\mathrm{lts}}$ is

$$v_p\left(t\right)=\mathit{Oc}\prime \left(t\right)+\mathit{Rcs}\prime \left(t\right)p_{\mathrm{lts}}$$

(7)

According to equation (7), the velocity of $\mathrm{ch}{P}_{\alpha }$ is

$$\mathit{V}(\mathrm{ch}{P}_{\alpha })=(-0.14384,-309.153,99.98918)$$

Unitize $\mathit{V}(\mathrm{ch}{P}_{\alpha })$ and obtain

$$\mathit{v}(\mathrm{ch}{P}_{\alpha })=(-0.00044,-0.95147,0.307734)$$

We can see that there exists small difference between the estimated velocity $\mathbf{\Lambda }({\omega }_{\alpha })$ and the actual motion velocity $\mathit{v}(\mathrm{ch}{P}_{\alpha })$ . The error of $\mathrm{ch}{P}_{\alpha }$ can be verified by the dot product

$$\mathrm{dotP}\left(\alpha \right)=\mathit{v}\left(\mathrm{ch}{P}_{\alpha }\right)·\mathit{n}\left(\mathrm{ch}{P}_{\alpha }\right)=-0.000243$$

It is restated that $\mathrm{dotP}(\alpha )$ should be 0 when $\mathrm{ch}{P}_{\alpha }$ is a theoretical characteristic point referring to equation (2). The more the $\mathrm{dotP}(\alpha )$ away from 0, the larger the error $\mathrm{ch}{P}_{\alpha }$ involves. Furthermore, let

$$f\left(x\right)=\mathrm{dotP}(360\circ x-180\circ )$$

The mean value of the data can be computed by

$$E\left(f\right)=\underset{0}{\overset{1}{\int }}\mathit{s}f\left(x\right)\mathrm{dx}=-0.00117$$

And the variance is

$${\sigma }^2\left(f\right)=E\left({(f\left(x\right)-E\left(f\right))}^2\right)=1.83492\times {10}^{-5}$$

In Figure 9, the solid line is the distribution of $\mathrm{dotP}(\alpha )$ associated with $\alpha$ , and the dash denotes the dot product of the normal vector and the actual velocity of the characteristic point acquired from the longitude method. Figure 10 shows the dot product distribution of Jiayong et al.’s method with the same orientation. In Figure 9, the variance and the mean value of the dot product by the proposed method are much smaller than those of the longitude method, which implies that the proposed method is more accurate. Compared with Figure 9, the distribution in Figure 10 is messy; thus, the proposed method is stable in the result.

Figure 9.

Dot product distribution of the two methods.

Figure 10.

Dot product of Jiayong et al.’s method.

A direct comparison of the three methods can be made from Table 1, which lists in detail the accuracy and efficiency of the methods. As shown in Table 1, both the variance and the mean value of the error of the longitude method and Jiayong et al.’s method are several times larger than those of the proposed method, and the time cost of the proposed method is much smaller than the existing methods at the same time. In this example, the error of Jiayong et al.’s method is fluctuant and relatively large. It is because in Jiayong et al.’s method, each sectional line on the design surface is in discrete form, and the projective point have to be among the two alternative points. Since the longitude method involves iterative computations of points to the free-form surface, the time performance is the worst. It shows that our algorithm improves approximately 7 times accuracy with 1/7 time cost compared with the existing methods.

Table 1.

Experimental results of the three methods.

	Variance	Mean value	Time cost (ms)
Proposed method	$1.83492\times {10}^{-5}$	−0.00117	4.54
Longitude method	0.001051	−0.0237	74.06
Jiayong et al.’s method	0.001182	−0.0083	31

Figure 11 demonstrates the simulating results. The errors of the machined surface to the desired surface computed from the proposed method are in accordance with the simulating results. The error distributions computed by the three methods are drawn in Figure 12.

Figure 11.

Simulating result.

Figure 12.

Error distribution computed by the three methods.

Machining a blade

To verify the computing performance of the proposed method in machining free-form surfaces, in this section, experiments on a blade, which has both concave and convex areas, will be demonstrated.

The blade size is about 43 mm × 51 mm. The diameter of the torus cutter used in this experiment is 12 mm. The cutter corner radius is 1.5 mm. We generate the tool path by moving the cutter along the isoparametric lines. The machining allowance is 0.01 mm.

A total of 18 sample points are randomly chosen from the blade surface. Over the surface at each sample point, the cutter is positioned similarly with the way discussed in the previous sections. We recorded the computing time for the error distribution using the proposed method, the longitude method, and Jiayong et al.’s method. The variance and mean value absolute performances for the dot product distribution of each method are illustrated in Figures 13 and 14, respectively. Figure 15 shows the recorded time cost of each method. The results show that our method yields lower variance and mean value of dot product with lesser time cost. From Figure 15, one can see that the time performance of Jiayong et al.’s method is also excellent. In the computation of sample point 8, Jiayong et al.’s method shows the least time cost compared with the other two methods. Jiayong et al.’s method needs to calculate the distances of discrete points to the cutter surface, which has a simple expression. So, the time performance of Jiayong et al.’s method is good and stable. However, our method and the longitude method need to calculate the distances of discrete points to the desired surface, which has a complex expression. The convergence of the iterating method for distance calculation in the longitude method and our method is low at some sample points. But the precision of cutter dispersed strategy is better than Jiayong et al.’s method from the distribution shown in Figures 13 and 14.

Figure 13.

Variance comparison of the three methods.

Figure 14.

Mean value comparison of the three methods.

Figure 15.

Time performance of the three methods.

At each drive point of the path, the tilt angle $\gamma$ of the cutter is set to be $0\circ$ . The incline angle $\beta$ is optimized to be the minimum that would not involve gouging according to the error distribution computed by our method. The paths overlap each other once using the estimated strip width defined in section “Generalized envelope theory.” The verification in Figure 16 yields that there exist only slight excess, which is circled in the illustration.

Figure 16.

Verification of the proposed method on a blade.

Conclusion

This article presents a concise proof on the minimum distance pair method in the first place and introduces two ways to achieve the minimum distance pairs. To take advantages of the envelope theory and the longitude method, a method based on the velocity field, which describes the path, is proposed. The proposed method disperses the cutter as the longitude method does and determines the characteristic point by solving the envelope equation instead of computing the minimum distance. While machining a flat with translational motions, the proposed method generates the exact characteristic points. In multiple-axis machining, although the velocity of the point on the cutter is hardly judged in only one orientation, the estimated velocity obtained from the velocity field matches the actual velocity with small errors. The experimental results show that the proposed method maintains high precision and stability with less time cost.

The whole article is concluded as follows:

1. The minimum distance pair method is correct in theory.

2. The proposed method generates the exact characteristic curve with translational motions.

3. The proposed method is faster and more accurate than Jiayong et al.’s method and the longitude method.

4. The proposed method is stable in machining both concave and convex free-form surfaces.