Optimizing tool size and tool path of five-axis flank milling with bounded constraints via normal mapping

Abstract

This article presents an approach to generate optimal couples of tool size and tool path for flank milling ruled surface with a conical cutter based on normal mapping, while at the same time keeping this couples within the given bounded constraints. For given initial pairs of taper tool and tool path, tool envelope surface is calculated. Then, the tool axis trajectory surface is related to envelope surface through the normal mapping. On this basis, the correspondence between the surface pair “design ruled surface–envelope surface–tool axis trajectory surface” is established, and it is used to obtain the signed flank milling error. By using this signed error, tool size and tool path optimization for flank milling is formulated as an optimization model subjected to bounds, and the trust-region-reflective least-squares method is used to solve this problem. Numerical example is given to confirm the validity of the proposed method.

Keywords

Flank milling tool path envelope surface optimization model

Introduction

Flank milling is an operation to machine the workpiece with the side part of a cutter. This milling method is highly recommended for machining blades with ruled surface, because it allows access to areas that would be inaccessible to end milling. In addition, flank milling removes a considerable amount of material in one pass, and this leads to shorter machining times and greater productivity. However, when the non-developable ruled surface is flank-milled with cylindrical or conical tool, interference between the tool and the machined surface inevitably exists.^1,2 This is the starting point of research on flank milling, and more methods are proposed to reduce this error.³

First, the flank milling error is reduced from the perspective of tool path optimization, including local optimization and global optimization. Rubio et al.⁴ developed an error distribution–based method for tool positioning, in which they make the tool axis parallel to the ruling line considered and position the cutter by letting errors equal on the two directrices of the surface. Bedi et al.⁵ located the cutter by making it tangent to the two directrices at both ends of the rule surface. Then, the improved positioning method is presented in some papers,^6–8 in which the tool is tangent with two directrices and in contact with the ruling involved at one point. Chiou⁹ suggested adjusting the initial cutter location to reduce flank milling error by comparing the swept profile with the design surface. The above methods are local and discrete in nature since they focus on reducing error at one cutter location each time.

Compared to local optimization method, global tool path optimization no longer insists on optimal tool positioning at each single cutter location. Lartigue et al.¹⁰ gave a new method to deform the tool axis trajectory surface, a ruled surface defined by two guiding curves, so that the tool envelope surface fitted the design surface as much as possible. Based on the conclusion that the envelope surface of cylindrical cutter is the offset surface of tool axis trajectory surface, Gong et al.¹¹ transformed the problem of reducing error between envelope surface of cylindrical tool and the designed surface into that of reducing deviation between the tool axis trajectory surface and the offset surface of the design surface, and a least-squares model is presented to obtain optimal tool axis trajectory surface. In a subsequent work, Gong and Wang¹² extended the idea to handle generic revolving tools, and formulated the optimization problem as that of least-squares fitting of tool axis trajectory surface to map surface. Zhu et al.^13,14 proposed the signed point-to-surface distance function, whose first-order differential increment with respect to the differential deformation of the tool axis trajectory surface is analytical, and then Ding and Zhu¹⁵ used the differential properties of the signed point-to-surface distance function to optimize tool path for five-axis flank milling with a cylindrical cutter. Subsequently, Zhu et al.¹⁶ extended the work to deal with conical tool by considering the cutter as the envelope surface of a one-parameter family of spheres.¹⁷ In addition, Li and Zhu¹⁸ and Guo et al.¹⁹ integrated the cutter run-out effect into tool path optimization for five-axis flank milling. However, all the above methods are used to optimize tool path for five-axis flank milling when the tool size is fixed.

Second, some studies have made efforts to reduce flank milling error from the perspective of optimizing tool geometry. Zheng et al.²⁰ proposed an approach to optimize the bottom radius of a taper tool with the half taper angle being fixed. Monies et al.²¹ introduced an algorithm to obtain optimal couples of radius and half taper angle in order to respect the tolerance. But these methods focused on optimizing tool geometry with the tool path being fixed.

Flank milling error is the deviation between the design surface and tool envelope surface. The above methods are used to reduce error by optimizing tool path or tool geometry such that the associated envelope surface approaches to the design surface as much as possible. In other words, for a given “tool geometry–design surface” pair, tool path offers a degree of freedom to reduce flank milling error; for a given “tool path–design surface” pair, tool geometry becomes the optimized variable to improve machining accuracy. However, the tool path or tool geometry is optimized separately in these methods. Obviously, for a given designed surface, when the tool path and tool geometry are considered as shape control variables of the tool envelope surface at the same time, the optimal couples of tool geometry and tool path further add a new degree of freedom to reduce the flank milling error. In addition, in actual machining, the tool size, which should be neither too small nor too big, must be within a reasonable range, and the tool axis trajectory surface should also be adjusted in a limited space.

However, to the best of our knowledge, the research on simultaneous optimization of tool path and shape for five-axis flank milling was addressed little. Zhu et al.²² extended their previous works^15,16 and developed the model and algorithm to simultaneously optimize the tool path and shape for five-axis flank milling. Then, in a subsequent work,²³ they considered both stiffness and geometric constraints, aiming to improve the rigidity of the conical tool while satisfying the machining accuracy. Bo et al.²⁴ proposed an optimization-based framework to seek both the optimal tool shape and its optimal tool path. By virtue of normal mapping and associated signed flank milling error, we presented an alternative way to deal with this problem.

The objective of this article is to develop a method to synchronously optimize tool geometry and tool axis trajectory surface while keeping both of them located in a given range. In this article, normal mapping is used to obtain analytical expression of signed flank milling error, which directly relates tool axis trajectory surface and tool geometry to error and provides the basis of the proposed optimization model. The remainder of this article is organized as follows. In section “Normal mapping between tool axis trajectory surface and envelope surface,” the normal mapping between tool axis trajectory surface and tool envelope surface is presented. The analytical expression of flank milling error is developed in section “Analytical expression of flank milling error under normal mapping.” In section “Model for synchronous optimization of tool size and tool path with bounded constraints,” synchronous optimization of tool size and tool axis trajectory surface is formulated as a nonlinear least-squares optimization problem with constraints. Numerical examples and conclusion are presented in sections “Numerical examples” and “Conclusion,” respectively.

Normal mapping between tool axis trajectory surface and envelope surface

Flank milling error is the deviation between the envelope surface and the designed surface. Because the deviation between the two surfaces is difficult to directly compare, one surface is usually discretized and the error at the discrete point is defined along the normal vector of the surface. For this reason, this section studies the normal mapping relationship between the envelope surface and the tool axis trajectory surface, which is the basis for calculating the flank milling error in the next section.

Normal mapping surface

The normal mapping surface contains the correspondence between two surfaces. Using this correspondence, the calculation of flank milling error will be very convenient. First, the definition of normal mapping surface is given as follows.

Definition

For a given smooth surface $S : r = r (u, v)$ , small straight line segments with length of $λ (u, v)$ along its normal vector are considered, and the trajectory of the endpoints of all these straight lines constitutes a new surface $\tilde{S} : \tilde{r} = \tilde{r} (u, v)$ , which is called the normal mapping surface of the original surface. $\tilde{S}$ is written as

\tilde{r} (u, v) = r (u, v) + λ (u, v) \cdot n (u, v)

(1)

where $λ (u, v)$ is the mapping length and $n (u, v)$ , the normal vector of surface $S$ , is the mapping direction.

The above normal mapping establishes the correspondence between points on original surface $r (u, v)$ and the mapping surface $\tilde{r} (u, v)$ . Furthermore, this correspondence is used to connect the tool axis trajectory surface, tool envelope surface and the designed surface. The three surfaces play an important role in optimization of tool size and tool path, and we will show this in section “Analytical expression of flank milling error under normal mapping.”

Tool envelope surface

Consider a taper tool as shown in Figure 1, which has a parametric form

\begin{matrix} r (v, θ) = [\begin{matrix} (r_{0} + v \cdot H \cdot tg α) \cdot \cos θ \\ (r_{0} + v \cdot H \cdot tg α) \cdot \sin θ \\ v \cdot H \end{matrix}] \overset{Δ}{=} [\begin{matrix} f (v) \cos θ \\ f (v) \sin θ \\ g (v) \end{matrix}], \\ v \times θ \in [0, 1] \times [0, 2 π) \end{matrix}

(2)

where $r_{0}$ is the radius of the bottom circle, $α$ is the half taper angle, and $H$ is the tool length.

Figure 1.

Taper tool.

Based on tangency condition in envelope theory and the body velocity representation in spatial kinematics, the analytical solution of the envelope surface of the taper tool can be derived. We refer the reader to Zhu et al.’s²⁵ work or our previous work²⁶ for more details. For completeness, the jth discrete characteristic point on envelope surface at the instant $t_{i}$ is given as follows

\begin{matrix} O_{i, j} (t_{i}, v_{j}, θ_{i, j}) = R (t_{i}) \cdot r (v_{j}, θ (t_{i}, v_{j})) + P (t_{i}) \\ = (r_{0} + v_{j} \cdot H \cdot tg α) \cdot [\cos θ_{i, j} \cdot X (t_{i}) + \sin θ_{i, j} \cdot Y (t_{i})] \\ + v_{j} \cdot H \cdot Z (t_{i}) + P (t_{i}) \end{matrix}

(3)

where $R (t) = [\begin{matrix} X (t) & Y (t) & Z (t) \end{matrix}] \in SO (3)$ is a unit orthogonal matrix describing the orientation of tool with respect to workpiece coordination system, and $P (t) \in ℝ^{3}$ is the cutter location in workpiece frame. $θ_{i, j}$ is determined by the equation

a (t_{i}, v_{j}) \cdot \cos θ_{i, j} + b (t_{i}, v_{j}) \cdot \sin θ_{i, j} = c (t_{i}, v_{j})

(4)

where

{\begin{matrix} a (t_{i}, v_{j}) = (P' (t_{i}) • X (t_{i})) \cdot H + (ω (t_{i}) • Y (t_{i})) \cdot (r_{0} \cdot H \cdot tg α + v_{j} \cdot H^{2} \cdot \sec^{2} α) \\ b (t_{i}, v_{j}) = (P' (t_{i}) • Y (t_{i})) \cdot H - (ω (t_{i}) • X (t_{i})) \cdot (r_{0} \cdot H \cdot tg α + v_{j} \cdot H^{2} \cdot \sec^{2} α) \\ c (t_{i}, v_{j}) = (P' (t_{i}) • Z (t_{i})) \cdot H \cdot tg α \end{matrix}

(5)

If $| c (t_{i}, v_{j}) / c (t_{i}, v_{j}) \sqrt{a^{2} (t_{i}, v_{j}) + b^{2} (t_{i}, v_{j})} \sqrt{a^{2} (t_{i}, v_{j}) + b^{2} (t_{i}, v_{j})} | \leq 1$ , we get $θ_{i, j}$ from equation (4) as follows

θ_{i, j} = {\begin{matrix} \sin^{- 1} (\frac{c (t_{i}, v_{j})}{\sqrt{a^{2} (t_{i}, v_{j}) + b^{2} (t_{i}, v_{j})}}) - ϕ \\ π - \sin^{- 1} (\frac{c (t_{i}, v_{j})}{\sqrt{a^{2} (t_{i}, v_{j}) + b^{2} (t_{i}, v_{j})}}) - ϕ \end{matrix}

(6)

where

ϕ = a \tan 2 (\frac{a (t_{i}, v_{j})}{\sqrt{a^{2} (t_{i}, v_{j}) + b^{2} (t_{i}, v_{j})}}, \frac{b (t_{i}, v_{j})}{\sqrt{a^{2} (t_{i}, v_{j}) + b^{2} (t_{i}, v_{j})}})

(7)

Normal mapping surface of the envelope surface

A one-parameter family of tool surfaces is generated when the tool undergoes a one-parameter spatial motion. According to the definition of envelope,²⁷ for any given moment, there exists a tool surface in the surface family which is tangent to the envelope surface along a characteristic curve. In addition, equation (3) shows that characteristic curve is composed of those points on the tool surface that satisfy the envelope condition. Therefore, envelope surface and tool surface have a common normal vector at characteristic points.

The unit normal vector of tool surface at the characteristic point is

\begin{matrix} n (v_{j}, θ_{i, j}) = \frac{r_{v} (v_{j}, θ_{i, j}) \times r_{θ} (v_{j}, θ_{i, j})}{| r_{v} (v_{j}, θ_{i, j}) \times r_{θ} (v_{j}, θ_{i, j}) |} \\ = \frac{1}{\sqrt{1 + t g^{2} α}} \cdot [\begin{matrix} - \cos θ_{i, j} \\ - \sin θ_{i, j} \\ t g^{2} α \end{matrix}] \end{matrix}

(8)

Then, it is transformed into workpiece frame

\begin{matrix} n_{w} (t_{i}, v_{j}, θ_{i, j}) = R (t_{i}) \cdot n (v_{j}, θ_{i, j}) \\ = \frac{1}{\sqrt{1 + t g^{2} α}} \cdot R (t_{i}) \cdot [\begin{matrix} - \cos θ_{i, j} \\ - \sin θ_{i, j} \\ t g^{2} α \end{matrix}] \end{matrix}

(9)

Equation (9) is also the mapping direction at characteristic point $O_{i, j} (t_{i}, v_{j}, θ_{i, j})$ .

However, as shown in Figure 1, the normal line of the tool surface at any point $B$ intersects the tool axis at point $E$ , and the distance between the two points is the mapping length

λ (v_{j}) = \sqrt{1 + t g^{2} α} \cdot (r_{0} + v_{j} \cdot H \cdot tg α)

(10)

Thus, if we pick any characteristic point $O_{i, j} (t_{i}, v_{j}, θ_{i, j})$ on the envelope surface, the mapping direction $n_{w} (t_{i}, v_{j}, θ_{i, j})$ and the mapping length $λ (v_{j})$ together determine the corresponding normal mapping point on tool axis trajectory surface

S_{axis} (ω_{i}, η_{j}) = O_{i, j} (t_{i}, v_{j}, θ_{i, j}) + λ (v_{j}) \cdot n_{w} (t_{i}, v_{j}, θ_{i, j})

(11)

where $S_{axis} (ω, η)$ is the tool axis trajectory surface, which is a ruled surface formed by a moving tool axis and is usually defined by two guiding curves

\begin{matrix} S_{axis} (ω, η) = (1 - η) \cdot B (ω) \\ + η \cdot T (ω), (ω, η) \in [0, 1] \times [0, 1] \end{matrix}

(12)

where both $B (ω)$ and $T (ω)$ are B-spline curves of order 3

B (ω) = \sum_{k = 0}^{n} N_{k, 3} (ω) \cdot B_{k}, T (ω) = \sum_{k = 0}^{n} N_{k, 3} (ω) \cdot T_{k}

(13)

where ${B_{k}}$ and ${T_{k}}$ are the control points of two guiding curves, respectively.

Equations (9)–(11) together determine the normal mapping between envelope surface and tool axis trajectory surface. In other words, the tool axis trajectory surface is the normal mapping surface of the tool envelope surface.

Note that although the right-hand side of equation (11) determines a point on tool axis trajectory surface $S_{axis} (ω, η)$ , parameters $(ω_{i}, η_{j})$ are still unknown. Given a point $p = (x, y, z)$ on a surface $S (u, v)$ , and find the corresponding parameter $(u^{*}, v^{*})$ such that $S (u^{*}, v^{*}) = p$ , which is called point inverse. This problem is solved using Newton’s iteration to minimize the distance between $p$ and $S (u, v)$ . We refer the readers to Piegl and Tiller’s²⁸ work for more details.

Analytical expression of flank milling error under normal mapping

The characteristic point $O_{i, j} (t_{i}, v_{j}, θ_{i, j})$ relates to a map point $S_{axis} (ω_{i}, η_{j})$ on tool axis trajectory surface through normal mapping, as shown in Figure 2. Given any characteristic point and corresponding map point, and then obtain a line by joining the two points. This line intersects the design surface at point $I_{i, j}$ , as shown in Figure 3. In addition, as explained in section “Normal mapping surface of the envelope surface,” the line is perpendicular to the envelope surface at characteristic point $O_{i, j} (t_{i}, v_{j}, θ_{i, j})$ . Thus, the distance between $I_{i, j}$ and $O_{i, j} (t_{i}, v_{j}, θ_{i, j})$ is the geometric error of flank milling

ε_{i, j} = ‖ O_{i, j} - I_{i, j} ‖

(14)

One may note that the error in this article is defined as the distance of considered point on design surface to envelope surface, which is measured along the normal direction of the envelope surface.

Figure 2.

Characteristic points and corresponding map points.

Figure 3.

Error of flank milling.

However, the point pair “intersection point $I_{i, j}$ –characteristic point $O_{i, j}$ –map point $S_{axis} (ω_{i}, η_{j})$ ” also determines the flank milling error

ε_{i, j} = ‖ I_{i, j} - S_{axis} (ω_{i}, η_{j}) ‖ - λ (v_{j})

(15)

From the perspective of calculation, there is no difference between the two definition of flank milling error, that is, equations (14) and (15). However, the meaning behind is different. First, the error defined in equation (14) is unsigned, but equation (15) determines a signed flank milling error. As shown in Figure 3, the error is negative if the characteristic point is located inside the envelope surface and positive if the characteristic point lies on the exterior of the envelope surface. Second, shape control parameters of tool axis trajectory surface and tool geometry are built into signed error defined by equation (15), which provides a great convenience for the subsequent optimization of tool size and tool path.

In addition, it is necessary to calculate the intersection of a designed surface and a line to obtain the point $I_{i, j}$ . To solve this problem, the line is usually considered to be the intersection of two planes, and then the designed surface equation is substituted into the two plane equations. This gives us two equations defining the intersection curve formed by the intersection of the designed surface with the two planes. The intersection of these two curves gives the point at which the line intersects the designed surface. We refer the readers to Schneider and Eberly’s²⁹ work for more details.

Model for synchronous optimization of tool size and tool path with bounded constraints

Basic idea of optimizing tool size and tool path

Based on normal mapping, the correspondence between the surface pair “design ruled surface–envelope surface–tool axis trajectory surface” is established in section “Normal mapping between tool axis trajectory surface and envelope surface,” and then, this correspondence is utilized to obtain the signed flank milling error. This signed error is related to both tool size and tool path, which facilitates the optimization of tool geometry and its trajectory surface.

The basic idea is to modify control points of tool axis trajectory surface and tool size (bottom radius and half taper angle) simultaneously such that the envelope surface fitted the designed surface as much as possible. For more clarity, equations (10) and (12) are substituted into equation (15)

\begin{matrix} ε_{i, j} = \\ ‖ I_{i, j} - [(1 - η_{j}) \cdot \sum_{k = 0}^{n} N_{k, 3} (ω_{i}) \cdot B_{k} + η_{j} \cdot \sum_{k = 0}^{n} N_{k, 3} (ω_{i}) \cdot T_{k}] ‖ \\ - \sqrt{1 + {(a_{0})}^{2}} \cdot (r_{0} + v_{j} \cdot H \cdot a_{0}) \end{matrix}

(16)

where $a_{0} = tg α$ .

For any given sampling point $O_{i, j}$ on envelope surface, the corresponding map point $S_{axis} (ω_{i}, η_{j})$ on tool axis trajectory surface and intersection point $I_{i, j}$ on designed surface are determined using the normal mapping. Moreover, the parameters $(ω_{i}, η_{j})$ are obtained by solving the problem of point inversion in section “Normal mapping surface of the envelope surface.” By treating parameters $(ω_{i}, η_{j})$ and intersection point $I_{i, j}$ as known values, the control points ${B_{k}}$ and ${T_{k}}$ , bottom radius $r_{0}$ , and the tangent value of half taper angle $a_{0}$ are optimized so that the sum of squares of errors at sampling points is minimized. This formulates the objective function of optimization problem. In addition, due to the constraints in the actual machining, the optimization variables must be limited to a certain range. The optimization model with constraints and its solution will be described in detail in the subsequent section.

Optimization model and its solution

For a set of points ${O_{i, j} \in ℝ^{3}, 1 \leq i \leq M, 1 \leq j \leq N}$ sampled from envelope surface, the corresponding parameters ${(ω_{i}, η_{j})}$ of map points ${S_{axis} (ω_{i}, η_{j})}$ and intersection points ${I_{i, j}}$ can be determined using normal mapping. Then, let the sum of squares of errors at sampling points be minimal while limiting the range of optimization variables. Thus, this leads to the following nonlinear least-squares optimization problem subjected to bounds

\begin{array}{l} \min E (x) = \sum_{i, j} ε_{i, j}^{2} \\ \begin{matrix} = {\sum_{i, j} (\begin{array}{l} ‖ I_{i, j} - [(1 - η_{j}) \cdot \sum_{k = 0}^{n} N_{k, 3} (ω_{i}) \cdot B_{k} + η_{j} \cdot \sum_{k = 0}^{n} N_{k, 3} (ω_{i}) \cdot T_{k}] ‖ \\ - \sqrt{1 + {(a_{0})}^{2}} \cdot (r_{0} + v_{j} \cdot H \cdot a_{0}) \end{array})}^{2} & \begin{array}{l} s . t . \\ r_{1} \leq r_{0} \leq r_{2} \\ a_{1} < a_{0} < a_{2} \\ B_{k}^{0} - Δ < B_{k} < B_{k}^{0} + Δ, k = 0, …, n \\ T_{k}^{0} - Δ < T_{k} < T_{k}^{0} + Δ, k = 0, …, n \end{array} \end{matrix} \end{array}

The bottom radius $r_{0}$ and the tangent value of half taper angle $a_{0}$ are limited to $[r_{1}, r_{2}]$ and $[a_{1}, a_{2}]$ , respectively. According to the strong convex hull property of B-spline curve,²⁸ that is, the curve is contained in the convex hull of its control polygon, the adjustment range of tool axis trajectory surface is limited by restricting the variation of the control points ${B_{k}}$ and ${T_{k}}$ . $B_{k}^{0}$ and $T_{k}^{0}$ are the initial control points of tool axis trajectory surface, and $Δ$ is the variation range of these control points. In addition, the initial taper tool usually chooses a relatively large size, and the variation range of the control points is selected as the initial tool radius. In this optimization problem, the optimization variables are $x = {r_{0}, a_{0}, B_{k}, T_{k}, k = 0, \dots, n}$ , and these variables are all bounded by upper bound and lower bound constrains.

The trust-region reflective optimization method,³⁰ which uses the interior reflective algorithm to solve the trust-region sub-problem, is a powerful approach to solve bound-constrained nonlinear minimization problems.³¹ The basic idea of trust-region method is to approximate objective function $E (x)$ with a quadratic function $q (x)$ , which reflects the behavior of function $E (x)$ in a neighborhood $N$ around the point $x$ . A trial step $s$ is computed by minimizing the trust-region sub-problem over $N$ . The current point is updated to be $x + s$ if $E (x + s) < E (x)$ ; otherwise, the current point remains unchanged and $N$ is shrunk for the next step. The interior reflective method is used to solve its sub-problem to generate $x^{k}$ which is located in the interior of constraint box defined by the upper bound and lower bound. In this study, the MATLAB routine lsqnonlin, which provides the trust-region-reflective least-squares algorithm for solving least-squares problem subjected to bounds, is used to seek for optimal tool size and tool path.

Now, we present the following algorithm to optimize tool size and tool path simultaneously for five-axis flank milling with a taper tool.

Numerical examples

In order to demonstrate the validity of the proposed method, an example of its application on flank milling of ruled surface with taper tool is supplied. The test ruled surface is taken from Zhu et al.,²⁵ and we refer the readers to Zhu et al.’s²⁵ work for the control points of the ruled surface. The parameters of taper tool chosen for simulation are as follows: bottom radius $r_{0} = 5 mm$ , half taper angle $φ = 5 \circ$ , the length of the cutting edge along the tool axis $H = 40 mm$ , and the total length of tool $L = 60 mm$ . Cutter locations are determined using Chiou’s⁹ method. By interpolating points on two endpoints of tool axis with B-spline curve of order 3, we get the two guiding curves defining the initial tool axis trajectory surface. The two directrices are defined in knot vector U = [0, 0, 0, 0.205293, 0.209603, 0.307501, 0.335242, 0.409771, 0.465196, 0.517712, 0.59396, 0.624008, 0.707803, 0.728316, 0.80713, 0.82531, 0.887203, 1, 1, 1, 1], and its control points are listed in Table 1. The bounded constraints are $r_{0} \in [1, 5]$ , $a_{0} \in [0.02, 0.5]$ , and $Δ = (5, 5, 5)$ .

Table 1.

Control points of the two directrices of the initial tool axis trajectory surface.

Bottom directrix			Top directrix
x₀ (mm)	y₀ (mm)	z₀ (mm)	x₁ (mm)	y₁ (mm)	z₁ (mm)
84.2159	52.1901	42.0454	127.9180	68.0928	79.9557
91.2586	46.6713	37.9686	133.9130	62.9318	77.1222
98.3338	41.5487	32.9949	139.3640	57.0819	74.0410
109.0040	34.8319	25.4109	147.3440	48.4969	69.5121
113.5760	32.1579	22.1510	150.7180	44.8720	67.5976
120.8520	28.3514	17.2123	155.9270	39.3005	64.6503
126.7620	25.6060	13.3980	160.0420	34.9201	62.3300
133.9740	23.0109	9.1901	164.8320	29.8469	59.7040
141.6200	20.7258	5.4194	169.7020	24.7110	57.1380
148.6420	18.9500	2.9555	174.0100	20.2656	55.1887
157.2510	16.7123	0.9782	179.3860	14.9250	53.3682
163.4260	14.9552	0.3741	183.3370	11.1583	52.5481
171.7210	12.1216	−0.0155	188.8640	6.1138	52.2869
176.9360	10.0025	−0.0838	192.4570	2.9144	52.6126
183.8230	6.7768	−0.0637	197.1830	−1.4483	53.5528
192.0120	2.5086	0.0437	202.7800	−6.7720	55.1763
199.0660	−2.0059	0.1167	207.5390	−11.5887	57.4739
203.4480	−5.1551	0.0105	210.7110	−14.6516	58.8073

50 × 30 points are sampled from the envelope surface formed by cutting tool. The optimal bottom radius is $r^{*} = 3.437 mm$ , and the optimal half taper angle is $α^{*} = 5.664 \circ$ . Figure 4 describes the straight generatrix of the taper tool before and after optimization, and it can be seen that the optimal taper tool has a radius that is less than the radius at the same height before optimization. The control points of bottom and top directrices that determine the optimal tool axis trajectory surface are listed in Table 2. Figure 5 shows the optimal tool axis trajectory surface moves toward the ruled surface except for the outward purple area.

Figure 4.

Straight generatrix of taper tool before and after optimization.

Table 2.

Control points of the two directrices of the tool axis trajectory surface after optimization.

Bottom directrix			Top directrix
x₀ (mm)	y₀ (mm)	z₀ (mm)	x₁ (mm)	y₁ (mm)	z₁ (mm)
85.4623	53.0910	41.0377	128.7756	68.5263	79.4082
93.0022	47.4862	36.9273	134.4636	63.6592	76.6528
100.2394	42.3749	31.9175	140.0817	57.6189	73.4390
110.9643	35.7414	24.4542	149.9936	47.1865	68.0359
115.1862	33.0222	21.0068	154.4403	42.5317	65.7636
121.2364	29.5465	16.1512	160.4752	35.4641	61.5785
126.8493	27.1350	12.8795	164.9683	29.9201	57.8182
132.4173	25.0515	9.6943	169.7709	24.8479	54.9544
138.8539	23.0345	6.2998	174.6837	19.7113	52.3616
145.4118	21.4543	4.3422	179.0099	15.4744	50.1895
155.0837	18.9368	1.9400	184.3854	10.1549	48.3721
160.5293	17.4483	1.1521	188.3370	6.7851	47.5481
169.5539	14.5796	−0.1461	193.8417	1.9477	47.3580
176.1526	12.0533	−0.8019	197.4126	−1.2708	47.6127
184.8457	8.0023	−0.9215	201.9908	−5.2994	48.5540
193.6407	3.3371	−1.0558	206.8576	−9.6433	51.8176
200.4723	−1.1237	−1.2644	210.3634	−13.0895	55.8193
204.6758	−4.1506	−0.8296	211.3670	−14.1804	58.5526

Algorithm (tool size and tool path optimization)

Input. Initial control points of tool axis trajectory surface

{B_{k}^{0}}

and

{T_{k}^{0}}

, initial bottom radius

r^{0}

of taper tool, initial tangent value of half taper angle

a^{0}

, design surface, and

δ

the desired accuracy of the algorithm.
Output. Optimal control points of tool axis trajectory surface

{B_{k}^{*}}

and

{T_{k}^{*}}

, optimal bottom radius

r^{*}

, and optimal tangent value of half taper angle

a^{*}

.
Step 1
(1) Set k = 0;
(2) Compute

E (x^{0})

, where

x^{0} = {r^{0}, a^{0}, B_{l}^{0}, T_{l}^{0}, l = 0, \dots, n}

(3) Set

ξ^{0} = E (x^{0})

;
Step 2
Solve least-squares problem with bounded constraints to determine the optimum solution

x^{k + 1} = {r^{(k + 1)}, a^{(k + 1)}, B_{l}^{(k + 1)}, T_{l}^{(k + 1)}, l = 0, \dots, n}

;
Step 3
(1) Update characteristic points

{O_{i, j}^{(k + 1)}}

;
(2) Update map points

{S_{axis}^{(k + 1)}}

and corresponding parameters

{({ω_{i}}^{(k + 1)}, {η_{j}}^{(k + 1)})}

;
(3) Update intersection points

{I_{i, j}^{(k)}}

;
(4) Update

E (x^{k + 1})

;
(5) Update

ξ^{k + 1} = E (x^{k + 1})

;
(6) If

| 1 - \frac{ξ^{k}}{ξ^{k + 1}} | > δ

, set

k = k + 1

, and go to step 2. Otherwise, exit and output the solution

x^{*} = x^{k + 1}

Figure 5.

Tool axis trajectory surface before and after optimization.

Figures 6(a) and (b) describe the distribution of the geometric error before and after optimization. The maximum overcut error and the maximum undercut error at each cutter location before and after optimization are shown in Figures 7 and 8, respectively. It is observed that the error before optimization varies in range [–0.117113, 0.001958], whose interval length is 0.119071 mm, and the error after optimization varies in range [–0.019159, 0.007724], whose interval length is 0.026883 mm. It should be noted that the negative sign is used only to indicate overcut. We can see that the global maximum overcut error reduces from 0.117113 to 0.019159 mm, as shown in Figure 7. In addition, Figure 6(a) shows that the overcut error before optimization is dominant, and this is the reason for the smaller undercut error before optimization, as shown in Figure 8. Although the global maximum undercut error increases slightly from 0.001958 to 0.007724 mm, the length of error interval greatly decreases from 0.119071 to 0.026883 mm.

Figure 6.

Distribution of geometric error (a) before and (b) after optimization.

Figure 7.

Maximum overcut error at each cutter location before and after optimization.

Figure 8.

Maximum undercut error at each cutter location before and after optimization.

Conclusion

In this article, an approach is presented to simultaneously optimize the tool size and tool path for five-axis flank milling of a ruled surface with a taper tool. The normal mapping is introduced to associate the tool axis trajectory surface with tool envelope surface, and we draw the conclusion that tool axis trajectory surface is the normal mapping surface of tool envelope surface. On this basis, the correspondence between the surface pair “design ruled surface–envelope surface–tool axis trajectory surface” is established, and then, it is used to obtain the signed flank milling error, which incorporates the shape control parameters of both tool path and tool geometry. In addition, the basic idea of reducing flank milling error is to deform tool path or tool size so that the associated tool envelope surface approximates to the designed surface as closely as possible. Based on this idea, the signed flank milling error is utilized to formulate the mathematical model for synchronous optimization of tool size and tool path with bounded constraints, and the trust-region-reflective least-squares method is used to solve this constrained optimization problem.

In addition, if the B-spline curve is used to represent the generatrix of tool and the associated cutter rotation surface, the proposed method can be extended to general rotary tools.

Footnotes

Academic Editor: Kuei Hu Chang

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

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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