Open AccessArticle

A Five-Axis Toolpath Corner-Smoothing Method Based on the Space of Master–Slave Movement

Song Gao

^1,2,

Haiming Zhang

¹,

Jianzhong Yang

^1,2,

Jiejun Xie

^1,* and

Wanqiang Zhu

National NC System Engineering Research Center, Huazhong University of Science and Technology, Wuhan 430074, China

National Center of Technology Innovation for Intelligent Design and Numerical Control, Wuhan 430074, China

Author to whom correspondence should be addressed.

Machines 2024, 12(12), 834; https://doi.org/10.3390/machines12120834

Submission received: 15 October 2024 / Revised: 19 November 2024 / Accepted: 20 November 2024 / Published: 21 November 2024

(This article belongs to the Section Advanced Manufacturing)

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Abstract

The smoothing of linear toolpaths plays is critical in improving machining quality and efficiency in five-axis CNC machining. Existing corner-smoothing methods often overlook the impact of spline curvature fluctuations, which may lead to acceleration variations, hindering surface quality improvements. The paper presents a five-axis toolpath corner-smoothing method based on the space of master–slave movement (SMM), aiming to minimize curvature fluctuations in five-axis machining and improve surface quality. The concept of movement space in master–slave cooperative motion is introduced, where the tool tip position and tool orientation are decoupled into a main motion trajectory and two master–slave movement space trajectories. By deriving the curvature monotony conditions of a dual Bézier spline, a G2-continuous tool tip corner-smoothing curve with minimal curvature fluctuations is constructed in real-time. Subsequently, using the SMM and the asymmetric dual Bézier spline, a high-order continuous synchronization relationship between the tool tip position and tool orientation is established. Simulation tests and machining experiments show that with our smoothing algorithm, maximum acceleration values for each axis were reduced by 21.05%, while jerk was lowered by 22.31%. These results indicate that trajectory smoothing significantly reduces mechanical vibrations and improves surface quality.

Keywords:

master–slave movement; five-axis machining; corner smoothing; minimal curvature fluctuation; interpolation synchronization

1. Introduction

Recent advancements in multi-axis machining have greatly enhanced productivity in the production of complex components such as impellers, blades, and aircraft wings [1]. However, challenges remain in achieving smooth machining trajectories, which are crucial for optimizing machining quality and efficiency. Despite improvements in CAD models and the ability to define curved surfaces more accurately, real-time interpolation of these models still presents a significant computational challenge [2,3]. The need for smoother, more precise trajectories is vital to prevent issues like vibrations and mechanical wear that can negatively impact machining performance [4]. Existing methods for refining these motion paths fall into two categories, global smoothing and local smoothing, each offering distinct advantages in improving trajectory continuity [5].

The global smoothing method approximates multiple linear toolpath segments by substituting them with continuous parametric spline curves, as depicted in Figure 1. This method effectively reduces the number of data points needed to represent complex trajectory information. However, controlling the approximation error necessitates intricate iterative computations [6]. Additionally, the use of high-order parametric curves demands careful consideration to avoid over-fitting, which can lead to fluctuations and oscillations [7]. Yang et al. [8] tackled an optimization problem for quadratic B-spline curves, ensuring that the Hausdorff distance from a given polyline toolpath remains within a specified tolerance. Parque et al. [9] proposed a path-smoothing method using differential evolution with optimization fitting and smoothing criteria, incorporating different initial values and parameters such as pressure. Zhao et al. [10] developed a NURBS interpolation scheme that utilizes mixed-integer programming (MIP) for optimal polygonal approximation to effectively fit the NURBS curve. Global smoothing excels in optimizing the overall smoothness of the path, thanks to the high continuity typically found in parametric curves, which results in a desirable overall smoothing effect. Nonetheless, global smoothing methods heavily depend on iterative processes or optimization algorithms to control errors, which can make it challenging to manage errors in specific local regions. This may lead to unpredictable error accumulation and extended computation times, presenting difficulties for real-time implementation in CNC systems.

The local smoothing method involves replacing sharp transitions between linear segments with smooth curves to ensure continuous transitions, thereby eliminating abrupt corners, as illustrated in Figure 2. Local smoothing methods are advantageous because they can analytically determine the smoothing error at the tool tip position, which has led to their widespread use in three-axis machining. Techniques such as Bézier curves, B-spline curves, and PH curves have been employed to replace sharp corners in three-axis toolpaths [11]. In response to the challenge of sharp corners in finishing processes, researchers have developed corner-smoothing methods that take into account manufacturing tolerances and kinematic constraints [12]. Fan et al. [13] utilized a pair of quartic Bézier curves to link adjacent linear toolpaths, achieving minimal curvature fluctuation by maintaining curvature monotony for each Bézier segment. Zhang et al. [14] introduced a novel curvature-optimal smoothing transition algorithm, which creates smooth toolpaths using quintic B-spline curves in conjunction with an optimized feedrate strategy. Sencer et al. [15] proposed a G2 continuous corner-smoothing method based on quintic B-splines, focusing on optimizing the maximum curvature. The local smoothing method offers precise control over corner errors with a relatively low computational burden, making it well suited for CNC systems that demand high real-time performance. It finds extensive application in both three-axis and multi-axis systems. However, the majority of these methods primarily concentrate on enhancing trajectory continuity and constraining the maximum curvature of the splines, often neglecting the potential for acceleration fluctuations caused by variations in spline curvature.

In five-axis machining, toolpath corner smoothing presents two primary challenges: (1) smoothing the tool orientation and (2) synchronizing the interpolation of the tool tip position with the tool orientation [16]. Beudaert et al. [17] employed two cubic B-splines to smooth both the tool tip position and the path of the tool axis points. They introduced a third cubic B-spline to synchronize the curve parameters of these two smoothed paths, with the control points optimized using the Newton–Raphson method. Bi et al. [18] extended a three-axis local smoothing method to five-axis toolpath smoothing within the machine coordinate system (MCS), using Bézier curves to smooth both translational and rotational paths. The parametric synchronization of the two curves was achieved analytically, and the resulting error was mapped to the tool tip position error in the workpiece coordinate system (WCS) via forward kinematic transformation. Gao et al. [19] fitted the tool tip position and tool orientation as B-splines in Euclidean and spherical coordinate systems, respectively, achieving C2-continuous trajectories. Li et al. [20] approximated the tool tip position trajectory and the spherical coordinate trajectory of the tool orientation using NURBS curves in the WCS, establishing a parameter mapping relationship between the tool tip position and tool orientation. Huang et al. [21] proposed a real-time, G2-continuous local smoothing method, replacing the corners with cubic B-splines. Yang et al. [22] introduced a C3-continuous corner-smoothing method using quintic splines. Shi et al. [23] employed a pair of quintic PH curves to smooth the corners of five-axis toolpaths. In the current synchronous interpolation of tool position trajectory and tool axis direction trajectory, multiple splines are often used to smooth the tool tip position trajectory and tool direction trajectory separately. A synchronous interpolation relationship is then established between the two smoothed trajectories. This method results in a complex calculation process, impacting the real-time performance of the interpolation algorithm.

Building on an analysis of current research, this paper proposes a five-axis toolpath corner-smoothing method based on the master–slave motion space, aimed at minimizing curvature fluctuations in five-axis machining toolpaths and improving surface quality. First, a real-time method for constructing G2-continuous corner-smoothing curves with minimal curvature fluctuations is introduced. By deriving the conditions for curvature monotonicity in cubic Bézier curves, the control polygon constraints for Bézier transition curves with minimal fluctuations are determined, and a method for constructing a symmetric dual Bézier inscribed corner-smoothing curve is established. Next, the concept of master–slave coordinated motion is extended to trajectory smoothing, and a trajectory smoothing method based on the master–slave motion space is proposed. Using an asymmetric dual Bézier spline model, a higher-order continuous synchronization relationship between the tool tip position and tool orientation within the master–slave motion space is established.

The contributions of this paper are as follows: A spatial structure for master–slave cooperative motion, termed “master–slave cooperative space”, is introduced to model the relationship between tool tip position and tool orientation. Building on this structure, a toolpath smoothing and interpolation method for five-axis machining is proposed, grounded in the master–slave cooperative space. This method effectively resolves the issue of cross-dimensional synchronous interpolation continuity between linear and rotary axes. Theoretically, the proposed toolpath-smoothing approach offers broader applicability, not only fulfilling the requirements of five-axis systems but also extending to multi-axis CNC machines such as four-axis and six-axis, thereby overcoming the limitations of existing methods in multi-axis machining.

2. Space of Master–Slave Movement

A five-axis toolpath comprises both the tool tip position trajectory and the tool orientation trajectory. The dual challenges of smoothing these trajectories and synchronizing the interpolation between them impact feedrate and acceleration continuity. Most existing studies use multiple splines to smooth the tool tip position and tool orientation trajectories separately and then establish a synchronized interpolation relationship between the two smoothed paths. However, this approach often requires a complex derivation of continuity conditions, consuming significant computational resources and hindering the real-time performance of the interpolation algorithm. This paper introduces the concept of the master–slave motion space, which defines the relationship between the tool tip position and tool orientation within this space. Building on this, a toolpath smoothing and interpolation method for five-axis machining is proposed, addressing the issue of cross-dimensional synchronization between the machine’s linear and rotary axes.

2.1. Definition

In multi-axis motion involving more than two axes, a two-dimensional space

\{s_{p}, s_{q}\}

is established, where

s_{p}

represents the travel of the master motion and

s_{q}

represents the travel of the slave motion. This space is referred to as the SMM. The master motion refers to the motion that is directly controlled by feedrate scheduling. In five-axis machining, due to the requirement to control the relative motion speed between the tool and the workpiece, the tool tip position

(x, y, z)

in the WCS is considered the master motion unit, while the rotational axes

(A, B, C)

are considered slave motion units.

In the SMM

\{s_{p}, s_{q}\}

, the five-axis machining linear trajectory is represented as shown in Figure 3. Here,

s_{p}

represents the cumulative travel of the tool tip position in the WCS, while

s_{q}, (q = A, B, C)

represents the cumulative travel of the rotational axes. The

i

-th tool tip position in the SMM is denoted as

{C L}_{i}

, for example,

{C L}_{1} = (s_{p 1}, s_{q 1})

. The travels

s_{p}

and

s_{q}

are defined by the following equations:

\{\begin{array}{l} s_{p i} = \sum_{1}^{i} \sqrt{{∆ x}_{i}^{2} + {∆ y}_{i}^{2} + {∆ z}_{i}^{2}} \\ s_{q i} = \sum_{1}^{i} ∆ q_{i} \end{array}

(1)

It is evident that

s_{p i}

is an increasing function of

i

. When the initial position of

q

is set to zero,

s_{q i}

represents the actual position of the rotational axis

q (q = A, B, C)

. The trajectory formed by any three tool positions can exist in the SMM as follows, depending on the quadrant in which

(s_{p 2}, s_{q 2})

is located, as presented in Figure 4. In Figure 4, (c) can be viewed as the mirror image of (a), and (d) can be seen as the mirror image of (b). In (e) and (f), there is a segment of the trajectory where the master motion travel is zero while the slave motion travel is not. In this case, the speed of the master motion must be reduced to zero to ensure the continuity of velocity and acceleration.

2.2. Trajectory Error Constraint

In three-dimensional Cartesian space, trajectory error refers to the degree of deviation between the toolpath and the programmed path. This is typically quantified by the Hausdorff distance between the smoothed and original paths, with the maximum design tolerance of the workpiece serving as the error constraint for trajectory smoothing. In the SMM, however, trajectory error primarily refers to synchronization error in the slave motion displacement when the master motion displacement is fixed. Therefore, the error

ε

in the SMM can be defined as follows:

ε (S_{p}) = |S_{q}^{c m d} (S_{p}) - S_{q}^{s m t} (S_{p})|

(2)

In this context,

S_{q}^{c m d} (S_{p})

represents the value of the vertical coordinate

S_{q}

for the programmed trajectory when the horizontal coordinate in the SMM is

S_{p}

. Similarly,

S_{q}^{s m t} (S_{p})

represents the value of the vertical coordinate

S_{q}

for the smoothed trajectory when the horizontal coordinate in the SMM is

S_{p}

. In five-axis machining, the error

ε (0)

indicates the deviation in the tool axis direction at the programmed tool position, as shown in Figure 5.

In five-axis machining, the tool direction tolerance region is typically represented as a conical area centered on the tool direction at the cutter location. It can be described as the angle

θ

between the actual tool axis vector

v

and the ideal tool axis vector

n

being smaller than the tolerance

θ^{*}

, meaning the tool direction tolerance region forms a cone with an apex angle of

2 θ^{*}

. As shown in Figure 6, the tool direction at the cutter location is

n

, and

c_{n}

is the generatrix where the conical surface, with

n

as the rotation axis and the cutter location as the apex, intersects the unit Gaussian sphere. The area enclosed by the generatrix

c_{n}

represents the tool axis direction tolerance region.

This tolerance region in the WCS can be mapped to the MCS to obtain the error constraints on the rotational axis angles.

‖(∆ α_{1}, ∆ α_{2})‖ \leq \frac{\sin (θ^{*})}{‖T‖} = ϵ_{M}

(3)

where

‖T‖ = ‖{[\begin{matrix} t & b \end{matrix}]}^{T} \cdot J_{22}‖

(4)

In Equation (4),

{t, b, n}

is the expression of the tool coordinate system posture in the WCS, and

J_{22}

is the Jacobian matrix, which relates to the kinematic chain of the rotational axes in a five-axis machine tool.

2.3. Trajectory Smoothness

When the master motion satisfies the acceleration continuity condition, the slave motion also adheres to the acceleration continuity constraint. Since feedrate scheduling only considers the continuity of the master motion, it is necessary to enforce the continuity of the slave motion by constraining the trajectory continuity in SMM.

When the velocity of the slave motion is continuous, the left and right limits of the velocity of the slave motion are equal.

{v_{q} |}_{l e f t} = {v_{q} |}_{r i g h t}

(5)

The master motion

p

and the slave motion

q

have a continuous correspondence, thus:

{q_{p} v_{p} |}_{l e f t} = {q_{p} v_{p} |}_{r i g h t}

(6)

where

q_{p} = \frac{d s_{q}}{d s_{p}}

represents the first-order differential of the slave motion relative to the master motion. Additionally, the continuity of velocity and acceleration in the master motion is ensured by the feedrate scheduling:

\begin{matrix} {v_{p}|}_{l e f t} = {v_{p}|}_{r i g h t} \\ {a_{p}|}_{l e f t} = {a_{p}|}_{r i g h t} \end{matrix}

(7)

Substituting into Equation (6) gives:

{q_{p} |}_{l e f t} = {q_{p} |}_{r i g h t}

(8)

When the acceleration of the slave motion is continuous, the left and right limits of the acceleration of the slave motion are equal, as shown in Equation (9).

{a_{q} |}_{l e f t} = {a_{q} |}_{r i g h t}

(9)

Considering the relationship between

p

and

q

, Equation (9) can be expressed as

{q_{p p} {v_{p}}^{2} + q_{p} a_{p}|}_{l e f t} = {q_{p p} {v_{p}}^{2} + q_{p} a_{p}|}_{r i g h t}

(10)

where

q_{p p} = \frac{d^{2} s_{q}}{d s_{p}^{2}}

represents the second-order differential of the slave motion relative to the master motion. Together with Equations (7) and (8), there exists a necessary and sufficient condition for the continuity of velocity and acceleration in the slave motion:

\{\begin{matrix} {q_{p} |}_{l e f t} = {q_{p} |}_{r i g h t} \\ {q_{p p} |}_{l e f t} = {q_{p p} |}_{r i g h t} \end{matrix}

(11)

According to Equation (11), using

κ_{r}

to denote the relative curvature of the curve in the SMM, the following equation can be easily derived:

\{\begin{matrix} κ_{r} = {\frac{q_{p p}}{{(1 + q_{p}^{2})}^{1.5}}|}_{l e f t} = {\frac{q_{p p}}{{(1 + q_{p}^{2})}^{1.5}}|}_{r i g h t} \\ {q_{p} |}_{l e f t} = {q_{p} |}_{r i g h t} \end{matrix}

(12)

Equation (12) represents the expression for G2 continuity (curvature continuity). Therefore, when the velocity and acceleration of the master motion are continuous, the necessary and sufficient condition for the continuity of the velocity and acceleration of the slave motion is that the trajectory in the SMM is G2 continuous.

3. Tool Tip Position Smoothing with Minimal Curvature Fluctuation

3.1. Symmetric Dual Bézier Spline

To ensure continuous velocity and acceleration in the master motion and reduce acceleration fluctuations during spline interpolation, this paper employs the G2-continuous cubic Bézier spline to construct the transition curve for the master motion trajectory. The expression is as follows:

C_{1} = \sum_{j = 0}^{3} B_{j, 3} (u) a_{j}, 0 \leq u \leq 1

(13)

C_{2} = \sum_{j = 0}^{3} B_{j, 3} (u) b_{j}, 0 \leq u \leq 1

(14)

where

B_{j, 3} (u)

is the cubic Bernstein basis function, and

a_{j}

and

b_{j}

are the control points of curves

C_{1}

and

C_{2}

, respectively. As shown in Figure 7,

{C L}_{0}

{C L}_{1}

, and

{C L}_{2}

are three adjacent tool tip points.

q = a_{0} = b_{0}

represents the connection point of the two spline curves. Points

a_{1}

a_{2}

, and

a_{3}

lie on

{C L}_{0} {C L}_{1}

, and points

b_{1}

b_{2}

, and

b_{3}

lie on

{C L}_{1} {C L}_{2}

. The distances between points

{C L}_{1}

a_{1}

a_{2}

, and

a_{3}

are denoted as

l_{1}

l_{2}

, and

l_{3}

, respectively. Similarly, the distances between points

{C L}_{2}

b_{1}

b_{2}

, and

b_{3}

are denoted as

{l_{1}}^{'}

{l_{2}}^{'}

, and

{l_{3}}^{'}

. Additionally,

l_{2} = k_{1} \cdot l_{1}

and

l_{3} = k_{2} \cdot l_{1}

Given the connection point

q

, the spline curves

C_{1}

and

C_{2}

can be fully determined by specifying the lengths

l_{1}

l_{2}

, and

l_{3}

, as well as

{l_{1}}^{'}

{l_{2}}^{'}

, and

{l_{3}}^{'}

. Furthermore, the G2 continuity constraint at the connection point

q

can be expressed as follows:

(1): $C_{1}$ and $C_{2}$ share the same tangent direction at point $q$ . According to the properties of Bézier splines, the control points $a_{1}$ , $q$ , and $b_{1}$ are collinear.
(2): $C_{1}$ and $C_{2}$ have the same curvature at point $q$ .

Symmetric Bézier splines refer to the control points of curves

C_{1}

and

C_{2}

being symmetrically distributed along two adjacent trajectory segments. The control points adhere to the following geometric symmetry constraints:

\{\begin{matrix} l_{1} = {l_{1}}^{'} \\ l_{2} = {l_{2}}^{'} \\ l_{3} = {l_{3}}^{'} \\ ‖a_{1} q‖ = ‖b_{1} q‖ \end{matrix}

(15)

Under the premise of satisfying the geometric symmetry constraints, based on the convex hull property of Bézier spline control points, it is known that the distance from the connection point

q

{C L}_{1}

represents the maximum error

ε

between the smoothed trajectory and the programmed trajectory. Once

ε

is determined, the connection point

q

and the segment length

l_{1}

can be defined along the angle bisector of the trajectory. Next, by determining

l_{2}

and

l_{3}

, the spline curves

C_{1}

and

C_{2}

can be uniquely defined. Since

a_{1}

q

, and

b_{1}

are collinear, the two curves share the same normal vector

\vec{N}

at connection point

q

. Due to the symmetry of the curves, they have equal curvature at

q

. According to the differential equation of the curve’s arc length, as shown in Equation (16),

C_{1}

and

C_{2}

satisfy the G2 continuity condition at connection point

q

\frac{d^{2} C}{{d s}^{2}} = κ \vec{N}

(16)

From the properties of Bézier splines, it is also known that the control points

a_{1}

a_{2}

, and

a_{3}

are collinear, and the curvature at endpoint

a_{3}

is zero. Similarly, the curvature at endpoint

b_{3}

is also zero. According to the equation for endpoint curvature, the curvature at connection point

q

is given by the following equation:

κ_{q} = \frac{2}{3} \frac{\sin (θ) \cos (θ) k_{1}}{l_{1}}

(17)

Based on the characteristics of the symmetric dual Bézier transition curve, the following geometric constraint exists:

ε = \sin (θ) l_{1}

(18)

Under the condition of monotonic curvature variation for the Bézier spline, the trajectory error will also vary monotonically. Therefore,

ε

represents the maximum trajectory error. By combining this with Equation (17), the relationship between curvature and the maximum trajectory error can be established as follows:

ε = \frac{2}{3} \frac{\sin^{2} (θ) \cos (θ) k_{1}}{κ_{q}}

(19)

3.2. Minimal Curvature Fluctuation Conditions

Since curvature represents the rate of change of the curve’s tangential direction, restricting the curvature of

C_{1}

and

C_{2}

to vary monotonically along the parameter will help improve trajectory smoothness, reduce fluctuation characteristics, and enhance the stability of machine motion. Under the condition of monotonic curvature variation for

C_{1}

and

C_{2}

, Equation (17) provides the method for calculating the maximum curvature of the dual Bézier spline, which is independent of the control polygon parameters

l_{2}

and

l_{3}

. However, it is evident that

l_{2}

and

l_{3}

affect the monotonicity of the Bézier spline.

For the dual Bézier spline, a hypothesis can be proven to hold (with the proof provided in Appendix A): When the following conditions are met, the curvatures of curves

C_{1}

and

C_{2}

are monotonic, and the curvature

κ_{q}

at the connecting point

q

is maximal.

(\{k_{2} \geq k_{1}^{2} - \frac{1}{10} {(\frac{8}{5} - \frac{b}{10} + \frac{d}{10})}^{2}\} \cup (\{k_{1} \leq 1\} \cap \{k_{2} \geq - 1.5 k_{1}^{2} + 5 k_{1} - 2.5\})) \cap \{k_{2} \leq 6 k_{1}^{2} - 4 k_{1}\}

(20)

where

\{\begin{matrix} a = \sqrt{10000 k_{1}^{4} - 20000 k_{1}^{3} + 15240 k_{1}^{2} - 5240 k_{1} + 625} \\ b = \sqrt{500 k_{1}^{2} - 5 a - 500 k_{1} + 131} \\ d = \sqrt{500 k_{1}^{2} + 5 a - 500 k_{1} + 137 - \frac{192}{b}} \end{matrix}

From Equation (17), to minimize

κ_{q}

k_{1}

should take its smallest value. Analyzing Equation (20) yields the smallest value of

k_{1}

within the range of

\frac{1}{2} + \frac{\sqrt{- 30 + 80 \sqrt{6}}}{50}

. For ease of calculation, rounding

k_{1}

to two decimal places yields an approximate minimum value of 0.76.

Under the condition

k_{1} = 0.76

, when the error

ε

remains constant, the smaller the value of

k_{2}

, the closer the control points

a_{3}

and

b_{3}

are to the tool position point

{C L}_{1}

. This results in a shorter segment length required for trajectory smoothing, enhancing algorithm compatibility. Therefore, the approximate minimum value for

k_{2}

is set to 0.37. Thus, the default initial parameters are

(k_{1}, k_{2}) = (0.76, 0.37)

. Once

k_{1}

and

k_{2}

are determined, the control points are dependent solely on the length of

l_{1}

3.3. Tool Position Smoothing Process

When the programmed segment is too short, maintaining the maximum contour error can lead to the intersection of consecutive corner transition curves on the straight section. As shown in Figure 8, the transition curves at tool tip points

{C L}_{2}

and

{C L}_{3}

are

C_{1}

and

C_{2}

, respectively. The intersection points of

C_{1}

and

C_{2}

with the trajectory segment

{C L}_{2} {C L}_{3}

are

P_{1}

and

P_{2}

, where a crossing occurs, making the smoothed trajectory discontinuous along the trajectory segment

{C L}_{2} {C L}_{3}

. Let the lengths of

{C L}_{2} P_{1}

and

{C L}_{3} P_{2}

L_{2}

and

L_{3}

, respectively, and the length of trajectory segment

{C L}_{2} {C L}_{3}

be L. To avoid the intersection of

C_{1}

and

C_{2}

, the following constraint must be satisfied:

L \geq L_{2} + L_{3}

(21)

The values of

L_{1}

and

L_{2}

can be adjusted by modifying the contour error of the transition spline. Thus, the constraint condition is established as follows, and the smoothing distance should not exceed half of the segment length:

L_{i} \leq \frac{L}{2}

(22)

where

L

represents the shortest length of the trajectory line segment adjacent to the tool position

{C L}_{i}

, and

L_{i}

denotes the spline smoothing distance corresponding to the tool position

{C L}_{i}

In summary, the primary process for smoothing the corner of the tool tip position with minimal curvature fluctuation is illustrated in Algorithm 1.

Algorithm 1. Smoothing of tool tip position.
Input:	$Tool position points p_{1}$ $, p_{2}$ $, p_{3}$ $Maximum contour tolerance ε^{*}$ $Length of the linear trajectory L = m i n (‖p_{1} - p_{2}‖, ‖p_{3} - p_{2}‖)$ $Parameters k_{1} = 0.76$ $, k_{2} = 0.37$
Output:	$B é zier splines C_{1}$ $and C_{2}$
Step 1:	Calculate vectors: $v_{1} = \frac{p_{1} - p_{2}}{‖p_{1} - p_{2}‖}$ $v_{2} = \frac{p_{3} - p_{2}}{‖p_{3} - p_{2}‖}$ Calculate the angle: $ρ = \arccos (v_{1} \cdot v_{2})$ $θ = \frac{π - ρ}{2}$
Step 2:	Calculate contour error: $ε = \min (\frac{L \sin (θ)}{2 (1 + k_{1} + k_{2})}, ε^{}) = \min (\frac{L \sin (θ)}{4.26}, ε^{})$ Calculate control polygon dimensions: $l_{1} = \frac{ε}{\sin (θ)}$ $l_{2} = k_{1} * l_{1}$ $l_{3} = k_{2} * l_{1}$
Step 3:	Calculate control points: $\{\begin{matrix} a_{i} = P + l_{i} τ_{1} \\ b_{i} = P + l_{i} τ_{2} \end{matrix} (i = 1, 2, 3)$ $a_{0} = b_{0} = q = \frac{(b_{1} + a_{1})}{2}$
Step 4:	Construct Bézier spline using control points: $C_{1} = \sum_{j = 0}^{3} B_{j, 3} (u) a_{j}, 0 \leq u \leq 1$ $C_{2} = \sum_{j = 0}^{3} B_{j, 3} (u) b_{j}, 0 \leq u \leq 1$

4. Trajectory Smoothing and Synchronous Interpolation of SMM

4.1. Asymmetric Dual Bézier Corner Smoothing

4.1.1. Asymmetric Dual Bézier Spline

Similar to symmetric dual Bézier splines, let the spline control points for two Bézier curves on the segments

{C L}_{1} P

and

{C L}_{2} P

be denoted as

[a_{1} a_{2} a_{3}]

and

[b_{1} b_{2} b_{3}]

, respectively. The two splines are G2 continuous at the control point

q

, as illustrated in Figure 9. The G2 continuity constraints can be represented using Beta constraint. Thus, continuous Bézier spline shape parameters

β_{1}

and

β_{2}

are introduced. When the connection point

q

is specified, the shape parameters

β_{1}

and

β_{2}

determine the positions of the control vertices

q

a_{1}

b_{1}

a_{2}

, and

b_{2}

Let the unit direction vector of

\vec{P {C L}_{1}}

be denoted as

v_{1}

, the unit direction vector of

\vec{P {C L}_{2}}

v_{2}

, and the unit direction vector of

\vec{P q}

v_{ε}

. It is evident that

\{\begin{matrix} a_{i} = P + \sum_{j = 1}^{i} l_{j} v_{1} \\ b_{i} = P + \sum_{j = 1}^{i} {l_{i}}^{'} v_{2} \end{matrix} (i = 1, 2, 3)

(23)

(1): Condition for G2 continuity at point $q$

According to the Beta constraint of Bézier curves, the following geometric relationships exist:

b_{1} - q = β_{1} (q - a_{1})

(24)

l_{2} β_{1} γ = l_{1}

(25)

{l_{2}}^{'} γ = {l_{1}}^{'}

(26)

γ = \frac{2 (1 + β_{1})}{β_{2} + 2 (1 + β_{1})} (β_{1} > 0)

(27)

Combining Equations (23) and (24), a linear equation regarding

l_{1}

and

{l_{1}}^{'}

can be obtained:

(β_{1} + 1) q = (β_{1} + 1) P + {l_{1}}^{'} v_{2} + β_{1} l_{1} v_{1}

(28)

We present the following equations:

l_{2} = k_{1} \cdot l_{1}

(29)

l_{3} = k_{2} \cdot l_{1}

(30)

l_{2}^{'} = k_{1}^{'} \cdot l_{1}^{'}

(31)

l_{3}^{'} = k_{2}^{'} \cdot l_{1}^{'}

(32)

It is evident that when

k_{1}

k_{2}

k_{1}^{'}

, and

k_{2}^{'}

satisfy the conditions in Equation (20), the curvature variation in the curve exhibits monotonicity. Based on Equations (25) and (26), we have the following:

\{\begin{matrix} k_{1} = \frac{1}{β_{1}^{2} γ} \\ k_{1}^{'} = \frac{1}{γ} \end{matrix}

(33)

From Equation (33), the necessary condition for the G2 continuity can be derived:

k_{1}^{'} = β_{1}^{2} k_{1}

(34)

(2): Condition for minimal curvature fluctuation

Let

η

denote the distance between points

P

and

q

, and let

θ_{1}

and

θ_{2}

represent the angles between vectors

v_{1}

and

\vec{b_{1} a_{1}}

and between

v_{2}

and

\vec{a_{1} b_{1}}

, respectively. As illustrated in Figure 10, the tool position

{C L}_{1}

is reflected across the vector

\vec{P q}

to yield point

S {C L}_{1}

, while the tool position

{C L}_{2}

is reflected across

\vec{P q}

to yield point

S {C L}_{2}

. When the vector

\vec{a_{1} b_{1}}

is orthogonal to

\vec{P q}

, the Bézier spline is mirrored along

\vec{P q}

, forming a symmetric dual Bézier spline in conjunction with the original spline, thereby allowing for the application of the minimal curvature fluctuation condition given in Equation (20). To ensure that both Bézier curves in the asymmetric spline satisfy the conditions specified in Equation (20), the following constraint must be established:

\min (k_{1}, k_{1}^{'}) = 0.76

(35)

By combining Equation (34) and considering the value of

β_{1}

, we have the following:

0.76 = \{\begin{matrix} \frac{1}{β_{1}^{2} γ}, & β_{1} > 1 \\ \frac{1}{γ}, & β_{1} \leq 1 \end{matrix}

(36)

4.1.2. Corner Smoothing in SMM

As illustrated in Figure 11, consider the machining trajectory formed by the tool position points

{C L}_{0}

{C L}_{1}

, and

{C L}_{2}

. A smooth trajectory is established within the SMM

{S_{p}, S_{q}}

. Points

a_{1}

a_{2}

, and

a_{3}

represent the control points of the spline on the line segment

{C L}_{0} {C L}_{1}

, while points

b_{1}

b_{2}

, and

b_{3}

serve as the control points of the spline on the line segment

{C L}_{2} {C L}_{1}

. The distance from

q

{C L}_{1}

represents the error in the rotational axis

α

at the tool position

{C L}_{1}

. Clearly, for the trajectory near

{C L}_{1}

, the transitional spline exhibits the maximum rotational axis error at this point.

(1): Condition for minimal curvature fluctuation

A line

l_{c}

is established at the connection point

q

that is perpendicular to the line segment

a_{1} b_{1}

, intersecting lines

{C L}_{0} {C L}_{1}

and

{C L}_{1} {C L}_{2}

at points

c_{1}

and

c_{2}

, respectively. Clearly, there exists a line that is symmetric to line

{C L}_{0} {C L}_{1}

with respect to line

l_{c}

, which intersects line

{C L}_{0} {C L}_{1}

at point

c_{1}

. When the length of segment

c_{1} a_{1}

is denoted as

{l c}_{1}

, then the monotonicity of the curvature of the spline formed by control points

a_{1}

a_{2}

a_{3}

, and

q

satisfies the constraint given by Equation (20). Similarly, when the length of segment

c_{2} b_{1}

is denoted as

{l c}_{1}'

, then the monotonicity of the curvature of the spline formed by control points

b_{1}

b_{2}

b_{3}

, and

q

also satisfies the constraint in Equation (20). The lengths

{l c}_{1}

and

{l c}_{1}'

satisfy the following equations:

\{\begin{matrix} {l c}_{1} = ‖a_{1} - c_{1}‖ = \frac{‖a_{1} - q‖}{\cos (θ_{1})} \\ {l c}_{1}' = ‖b_{1} - c_{1}‖ = \frac{‖b_{1} - q‖}{\cos (θ_{2})} \end{matrix}

(37)

(2): Endpoint constraint

To ensure that the transition curves at two adjacent tool tip points do not interfere with each other, the following constraint conditions in the SMM are established:

\{\begin{matrix} ‖a_{3}‖ \leq \frac{L_{1}}{2} \\ ‖b_{3}‖ \leq \frac{L_{2}}{2} \end{matrix}

(38)

where

L_{1} = ‖{C L}_{0} - {C L}_{1}‖ L_{2} = ‖{C L}_{2} - {C L}_{1}‖

(39)

Based on the fact that

a_{1}

b_{1}

, and

{C L}_{1}

form a triangle, the following geometric relationship is established:

\frac{‖a_{1}‖}{\sin (θ_{2})} = \frac{‖b_{1}‖}{\sin (θ_{1})}

(40)

(3): q and tangent at q

As illustrated in Figure 12, let the midpoint of

{C L}_{0} {C L}_{1}

be denoted as

p_{a}

and the midpoint of

{C L}_{2} {C L}_{1}

be denoted as

p_{b}

. The intersection of the line

p_{a} p_{b}

with the vertical axis is defined as

q^{'}

, and the distance from

q^{'}

{C L}_{1}

is denoted as

ε^{'}

. When

ε^{'} \leq ε^{*}

, the smoothed trajectory will meet the error constraints. The line at point

q

in the direction of

τ

serves as the tangent to the transition curve, intersecting

p_{a} {C L}_{1}

and

p_{b} {C L}_{1}

at points

a_{1}

and

b_{1}

, respectively. To ensure G2 continuity, the necessary condition must be satisfied: the line along the tangent direction that passes through the connection point

q

must intersect the segments

p_{a} {C L}_{1}

and

p_{b} {C L}_{1}

at points that are not endpoints. Therefore, let

τ

be in the same direction as

\vec{p_{a} p_{b}}

and

q

be positioned between

{C L}_{1}

and

q^{'}

(4): Control polygon proportional coefficients

The following variables are defined:

p_{a} = (x_{a}, y_{a})

(41)

p_{b} = (x_{b}, y_{b})

(42)

τ = (x_{τ}, y_{τ}) = p_{b} - p_{a}

(43)

q = (0, s i g n * ε)

(44)

The tangent equation at point

q

can be expressed as follows:

y = \frac{y_{τ}}{x_{τ}} x + s i g n * ε

(45)

where

s i g n

is related to the positive or negative sign of

q

on the vertical axis and can be determined by the slopes of the lines

p_{a} {C L}_{1}

and

p_{b} {C L}_{1}

s i g n = \{\begin{matrix} - 1, \frac{y_{a}}{x_{a}} > \frac{y_{b}}{x_{b}} \\ 1, \frac{y_{a}}{x_{a}} \leq \frac{y_{b}}{x_{b}} \end{matrix}

(46)

The linear equation of

p_{a} {C L}_{1}

is as follows:

y = \frac{y_{a}}{x_{a}} x

(47)

The linear equation of

p_{b} {C L}_{1}

is as follows:

y = \frac{y_{b}}{x_{b}} x

(48)

Based on Equations (45) and (47), the expression for

a_{1}

can be derived as

a_{1} = s i g n * ε * (\frac{x_{τ} x_{a}}{x_{τ} y_{a} - x_{a} y_{τ}}, \frac{y_{a} x_{τ}}{x_{τ} y_{a} - x_{a} y_{τ}})

(49)

Based on Equations (45) and (48), the expression for

b_{1}

can be derived as

b_{1} = s i g n * ε * (\frac{x_{τ} x_{b}}{x_{τ} y_{b} - x_{b} y_{τ}}, \frac{y_{b} x_{τ}}{x_{τ} y_{b} - x_{b} y_{τ}})

(50)

Based on the G2-continuous Beta constraint model, the parameter

β_{1}

is calculated as follows:

β_{1} = \frac{‖b_{1} - q‖}{‖a_{1} - q‖}

(51)

The proportional parameters of the control polygon are established as

\{\begin{matrix} \begin{matrix} k_{1} = \frac{‖a_{2} - a_{1}‖}{‖a_{1}‖} \\ k_{2} = \frac{‖a_{3} - a_{2}‖}{‖a_{1}‖} \end{matrix} \\ \begin{matrix} k_{1}^{'} = \frac{‖b_{2} - b_{1}‖}{‖b_{1}‖} \\ k_{2}^{'} = \frac{‖b_{3} - b_{2}‖}{‖b_{1}‖} \end{matrix} \end{matrix}

(52)

By introducing

c_{1}

and

c_{2}

, the proportional parameters of the control polygon are transformed into the proportional parameters of the symmetric spline:

\{\begin{matrix} \begin{matrix} {k k}_{1} = \frac{‖a_{2} - a_{1}‖}{‖a_{1} - c_{1}‖} \\ {k k}_{1}^{'} = \frac{‖b_{2} - b_{1}‖}{‖b_{1} - c_{2}‖} \end{matrix} \\ \begin{matrix} {k k}_{2} = \frac{‖a_{3} - a_{2}‖}{‖a_{1} - c_{1}‖} \\ {k k}_{2}^{'} = \frac{‖b_{3} - b_{2}‖}{‖b_{1} - c_{2}‖} \end{matrix} \end{matrix}

(53)

According to Equations (39), (52), and (53), we obtain

\{\begin{matrix} \begin{matrix} k k_{1} = f_{a} k_{1} \\ k k_{2} = f_{a} k_{2} \end{matrix} \\ \begin{matrix} k k_{1}^{'} = f_{b} k_{1}^{'} \\ k k_{2}^{'} = f_{b} k_{2}^{'} \end{matrix} \end{matrix}

(54)

where

f_{a} = \frac{‖a_{1}‖ \cos (θ_{1})}{‖a_{1} - q‖} f_{b} = \frac{‖b_{1}‖ \cos (θ_{2})}{‖b_{1} - q‖}

According to Equations (34) and (55), we obtain

{k k}_{1} = g * {k k}_{1}^{'}

(55)

where

g = \frac{‖a_{1}‖ \cos (θ_{1})}{‖b_{1}‖ \cos (θ_{2}) β_{1}} = \frac{\sin (θ_{2}) \cos (θ_{1})}{\sin (θ_{1}) \cos (θ_{2}) β_{1}}

To ensure the monotonicity of the curvature of Bézier splines, the constraint equation between

k k_{1}

and

k k_{2}

is established as written in Equation (56) and shown in Figure 13:

\begin{matrix} \frac{k k_{2}}{k k_{1}} = \frac{k k_{2}^{'}}{k k_{1}^{'}} = \frac{0.37}{0.76} \\ {k k}_{1} \geq 0.76 \\ {k k}_{1}^{'} \geq 0.76 \end{matrix}

(56)

Based on Equation (35), let

\min ({k k}_{1}, {k k}_{1}^{'}) = 0.76

(57)

(5): Error adjustment

The proportional coefficients from the edge lengths of the control points to the maximum edge length can be calculated based on the proportional coefficients of the control polygon:

h_{1} = \frac{k_{1} + k_{2} + 1}{σ_{1}} h_{2} = \frac{k_{1}^{'} + k_{2}^{'} + 1}{σ_{2}}

(58)

where

σ_{1} = \frac{‖p_{a}‖}{‖a_{1}‖} σ_{2} = \frac{‖p_{b}‖}{‖b_{1}‖}

We define

h = m a x (h_{1}, h_{2})

. When the value of h is less than or equal to 1, it indicates that the edge length of the control polygon meets the allowable maximum value. When the value

h

is greater than 1, it indicates that the edge length of the control polygon exceeds the allowable maximum value. In this case, the error

ε

is reduced by a factor of

h

, as shown in Equation (59). The proportional coefficients of the control polygon are then recalculated, resulting in a transition curve with monotonic curvature for both segments.

ε^{'} = \frac{ε}{h}

(59)

In summary, the calculation method for the control points of the transition curve is presented in Algorithm 2.

Algorithm 2. Smoothing of SMM
Input:	$C L = (x, y)$ $, p_{a} = C L + (x_{a}, y_{a})$ $, p_{b} = C L + (x_{b}, y_{b})$ $, ε = θ^{*}$ $s i g n = \{\begin{matrix} - 1, \frac{y_{a}}{x_{a}} > \frac{y_{b}}{x_{b}} \\ 1, \frac{y_{a}}{x_{a}} \leq \frac{y_{b}}{x_{b}} \end{matrix}$ $τ = (x_{τ}, y_{τ}) = p_{b} - p_{a}$ $θ_{1} = < - τ, (x_{a}, y_{a}) >$ $θ_{2} = < τ, (x_{b}, y_{b}) >$
Output:	$B é zier splines C_{1}$ $and C_{2}$
01:	while true do
02:	$q = C L + (0, s i g n * ε)$
03:	$a_{1} = s i g n * ε * (\frac{x_{τ} x_{a}}{x_{τ} y_{a} - x_{a} y_{τ}}, \frac{y_{a} x_{τ}}{x_{τ} y_{a} - x_{a} y_{τ}})$
04:	$b_{1} = s i g n * ε * (\frac{x_{τ} x_{b}}{x_{τ} y_{b} - x_{b} y_{τ}}, \frac{y_{b} x_{τ}}{x_{τ} y_{b} - x_{b} y_{τ}})$
05:	$β_{1} = \frac{‖b_{1} - q‖}{‖a_{1} - q‖}$
06:	$f_{a} = \frac{‖a_{1}‖ \cos (θ_{1})}{‖a_{1} - q‖}$ $, f_{b} = \frac{‖b_{1}‖ \cos (θ_{2})}{‖b_{1} - q‖}$
07:	$g = \frac{f_{a}}{f_{b} β_{1}}$
08:	$if g \geq 1$
09:	${k k}_{1}^{'} = 0.76$ $, {k k}_{2}^{'} = 0.37$
10:	$k_{1}^{'} = f_{b} k k_{1}^{'}$ $, k_{2}^{'} = f_{b} k k_{2}^{'}$
11:	$k_{1} = \frac{k_{1}^{'}}{β_{1}^{2}}$
12:	$k k_{1} = f_{a} k_{1}$
13:	$k k_{2} = \frac{0.37}{0.76} k k_{1}$
14:	$k_{2} = \frac{k k_{2}}{f_{a}}$
15:	else
16:	$k k_{1} = 0.76$ $, k k_{2} = 0.37$
17:	$k_{1} = \frac{k k_{1}}{f_{a}}$ $, k_{2} = \frac{{k k}_{2}}{f_{a}}$
18:	$k_{1}^{'} = β_{1}^{2} k_{1}$
19:	$k k_{1}^{'} = f_{b} k_{1}^{'}$
20:	$k k_{2}^{'} = \frac{0.37}{0.76} k k_{1}^{'}$
21:	$k_{2}^{'} = f_{b} k k_{2}^{'}$
22:	end if
23:	$h_{1} = \frac{k_{1} + k_{2} + 1}{σ_{1}}$ $, h_{2} = \frac{k_{1}^{'} + k_{2}^{'} + 1}{σ_{2}}$
24:	$h = m a x (h_{1}, h_{2})$
25:	$if h > 1$
26:	$ε^{'} = \frac{ε}{h}$
27:	else
28:	exit
29:	end if
30:	end
31:	$l_{1} = ‖a_{1}‖$ $, l_{1}^{'} = ‖b_{1}‖$
32:	$l_{2} = k_{1} \cdot l_{1}$ $, l_{3} = k_{2} \cdot l_{1}$ $, l_{2}^{'} = k_{1}^{'} \cdot l_{1}^{'}$ $, l_{3}^{'} = k_{2}^{'} \cdot l_{1}^{'}$
33:	$a_{1} = a_{1} + C L$ $, b_{1} = b_{1} + C L$
34:	$a_{2} = \frac{a_{1}}{‖a_{1}‖} (l_{1} + l_{2}) + C L$ $, b_{2} = \frac{b_{1}}{‖b_{1}‖} (l_{1}^{'} + l_{2}^{'}) + C L$
35:	$a_{3} = \frac{a_{1}}{‖a_{1}‖} (l_{1} + l_{2} + l_{3}) + C L$ $, b_{3} = \frac{b_{1}}{‖b_{1}‖} (l_{1}^{'} + l_{2}^{'} + l_{3}^{'}) + C L$
36:	$C_{1} = \sum_{j = 0}^{3} B_{j, 3} (u) a_{j}, 0 \leq u \leq 1$ $, C_{2} = \sum_{j = 0}^{3} B_{j, 3} (u) b_{j}, 0 \leq u \leq 1$

4.2. Synchronization and Interpolation of Five-Axis Machining Toolpaths

The toolpaths for five-axis machining involve the motion of two rotational axes, generating two SMMs. Let the travel distances of the two rotational axes

α_{1}

and

α_{2}

on the five-axis machine tool be denoted as

s_{q}

and

s_{r}

, respectively. As shown in Figure 14, the two-dimensional SMMs

{s_{p}, s_{q}}

and

{s_{p}, s_{r}}

can be combined to form a three-dimensional space

{s_{p}, s_{q}, s_{r}}

. Assuming there is a trajectory

p_{1} p_{2} p_{3}

, a local coordinate system is established at point

p_{2}

. Consequently, points

p_{1}

and

p_{2}

have projection points on the planar spaces

{s_{p}, s_{q}}

and

{s_{p}, s_{r}}

, respectively. In the decomposed two-dimensional SMMs, the transition curves between the tool tip trajectory and the rotational axes can be initially constructed using the method outlined in the previous section.

Since the points of maximum error along the transition curves on

{s_{p}, s_{q}}

and

{s_{p}, s_{r}}

are the connection points of the asymmetric dual Bézier splines, they lie within the same normal plane

{p_{2} - s_{q} s_{r}}

of the

s_{p}

axis. It is sufficient to ensure that the maximum error

∆ α_{1}

{s_{p}, s_{q}}

and the maximum error

∆ α_{2}

{s_{p}, s_{r}}

satisfy Equation (3) to meet the tolerance requirements for the toolpath.

Therefore, after initially constructing the transition trajectories in

{s_{p}, s_{q}}

and

{s_{p}, s_{r}}

, the overall profile error must be verified and the spline reconstructed:

(1) First, the error adjustment coefficients are calculated. Let the coordinates of the connecting point

q

{s_{p}, s_{q}}

and

{s_{p}, s_{r}}

(0, ε_{α_{1}})

and

(0, ε_{α_{2}})

, respectively.

δ = \frac{ϵ_{M}}{‖(ε_{α_{1}}, ε_{α_{2}})‖}

(60)

(2) If

δ < 1

, Equation (3) is not satisfied, and the position of the connection point

q

{s_{p}, s_{q}}

and

{s_{p}, s_{r}}

needs to be adjusted simultaneously:

\{\begin{matrix} ε_{α_{1}} = δ * ε_{α_{1}} \\ ε_{α_{2}} = δ * ε_{α_{2}} \end{matrix}

(61)

(3) The transition curve should be reconstructed using the method outlined in Section 4.1.2.

Unlike interpolation in Cartesian space, the SMM does not require the trajectory to be discretized according to arc length. As shown in Figure 15, we assume the motion from point

p_{1}

to point

p_{2}

. The displacement increment corresponding to the i-th interpolation cycle is denoted as

∆ s_{i}

, and the total displacement is given by

s_{i} = \sum_{j = 1}^{i} ∆ s_{j}

(62)

In the local coordinate system of the SMM

{s_{p}, s_{q}}

at point

p_{1}

, the parameter model of spline trajectory 1 is as follows:

[M (u), N (u)] = \sum_{j = 0}^{3} B_{j, 3} (u) b_{j}, 0 \leq u \leq 1

(63)

The parameter model of spline trajectory 2 is as follows:

[M (u), N (u)] = \sum_{j = 0}^{3} B_{j, 3} (u) a_{j}, 0 \leq u \leq 1

(64)

We define

b_{3} = (x_{b}, y_{b})

and

a_{3} = (x_{a}, y_{a})

. The target point for this interpolation cycle, denoted as

(m_{i}, n_{i})

, may lie on spline trajectory 1, spline trajectory 2, or the line segment, depending on the value of

m_{i}

When

m_{i} \leq x_{b}

, the interpolation point lies on the spline trajectory at

p_{1}

. By setting

m_{i} = s_{i}

and substituting it into the Bézier spline model, we can solve the univariate cubic equation for the parameter u, yielding the corresponding parameter

u_{i}

. Then,

n_{i}

can be obtained by substituting

u_{i}

back into Equation (63).

When

x_{b} < m_{i} < x_{a}

, the interpolation point lies on the line segment. The expression for the line segment is as follows:

\frac{m_{i} - x_{b}}{n_{i} - y_{b}} = \frac{x_{a} - x_{b}}{y_{a} - y_{b}}

(65)

Setting

m_{i} = s_{i}

and substituting this into the previous equation allows us to solve for

n_{i}

When

m_{i} \geq x_{a}

, the interpolation point lies on the spline trajectory at

p_{2}

. Again, we let

m_{i} = s_{i}

and substitute it into the Bézier spline model to solve the univariate cubic equation for the parameter u, yielding the corresponding parameter

u_{i}

. Then,

n_{i}

can be obtained by substituting

u_{i}

back into Equation (64).

Once

n_{i}

is determined, the motion increment of the rotational axis

α_{1}

for the i-th interpolation cycle can be calculated as follows:

∆ α_{1} = n_{i} - n_{i - 1}

(66)

Similarly, the motion increment

∆ α_{2}

of the rotational axis

α_{2}

for the i-th interpolation cycle can be calculated in the SMM

{s_{p}, s_{r}}

5. Simulation and Experiment

To validate the effectiveness of the proposed method, we conducted numerical simulations in the MATLAB (R2022a) software environment and machining experiments on a five-axis CNC machine in the laboratory.

5.1. Numerical Simulation

This section employs a five-point toolpath for numerical simulation, with the tool tip position and tool orientation coordinates as follows:

P_{1} = [100 200 200]

O_{1} = [- 0.5 - 0.51]

P_{2} = [0 200 100]

O_{2} = [- 2 - 1 1]

P_{3} = [0 100 0]

O_{3} = [- 2 - 2 1]

P_{4} = [100 0 0]

O_{4} = [- 2 - 2.5 1]

P_{5} = [200 100 100]

O_{5} = [- 0.5 - 1.5 1]

The error tolerances for the tool tip position and tool orientation are set to 10 mm and

10 °

, respectively. The smoothed trajectory is shown in Figure 16, where it is evident that both the smoothness of the tool tip position and the tool orientation have improved. The corresponding trajectory in the SMM is illustrated in Figure 17, which also demonstrates a noticeable smoothing effect on the SMM trajectory.

We employ the B-spline transition method proposed by Zhao et al. [7] as the control sample in this paper, which is referred to as the B-spline method, and the method in this paper is referred to as dual Bézier method. When the tool tip position smoothing error is set to 10 mm, the local trajectory of the two methods is shown in Figure 18. The curvature distribution for the dual Bézier method is shown in Figure 19a, with a maximum curvature of 0.017. The curvature distribution for the B-spline method is shown in Figure 19b, with a maximum curvature of 0.022. The dual Bézier method presented in this paper reduces the maximum curvature by about 23%.

5.2. Machining Experiment

To further demonstrate the effectiveness of the proposed method, this section presents CNC machining experiments using the side milling toolpath for an impeller, as shown in Figure 20a. In the experiment, the forming surface of the impeller blade was designed as a straight line, constituting a ruled surface. The experiments were carried out on a VMC-C30H five-axis CNC machine manufactured by Shanghai TOPNC (Shanghai, China) and equipped with an HNC-8 CNC system, as illustrated in Figure 20b. The actual axial position and velocity during the machining process can be determined in a closed loop by the servo control system. The finish machining allowance was set at a uniform distribution of 0.4 mm across the entire workpiece. Detailed information about the specific machining tool and cutting parameters can be found in Table 1.

The maximum allowable tolerance of the tool tip position is set to 0.005 mm, while the maximum allowable tolerance of the tool orientation is set to 1°. The trajectory smoothing results are shown in Figure 21.

The feedrate was set to

F = 3000

mm/min and the spindle speed to

S = 15000 r p m

. Cutting processes were conducted under both the proposed smoothing function and the original smoothing function of the HNC-8 CNC system, with the results summarized in Table 2. The comparative curves for speed, acceleration, and jerk of each machine axis are illustrated in Figure 22, Figure 23 and Figure 24. As indicated by the figures and table, compared to the original smoothing method of the HNC-8 CNC system, the proposed method achieves higher speeds and shorter machining times, resulting in a 17.36% increase in efficiency. Additionally, the maximum acceleration values for each axis using our smoothing algorithm were reduced by 21.05%, and jerk decreased by 22.31%, demonstrating that trajectory smoothing effectively mitigates mechanical vibrations and enhances surface quality.

The machining results of the original HNC-8 CNC system and proposed smoothing method are shown in Figure 25.

A comparison of surface quality on the pressure side near the trailing edge (Region A), where defects commonly occur during blade machining, is presented in Figure 26. It is evident that when the original smoothing function of the system is used, the surface exhibits noticeable vertical striations, particularly pronounced near the trailing edge of the blade. And the machine experiences significant vibration, vibration patterns with intervals of 0.16 mm are observed on the surface of the workpiece. When the smoothing function proposed in this paper is applied, the vertical striations on the processed surface are significantly reduced, indicating that the smoothed toolpath decreases tool vibrations during the machining process and improves surface quality. The obtained result demonstrates that the method proposed in the paper can effectively enhance the machining quality of five-axis machining.

The impeller processed with the proposed smoothing method was subjected to error measurement using a coordinate measuring machine (model LH108, WENZEL Präzision GmbH, Ltd, Wiesthal, Bavaria, Germany) equipped with a Renishaw probe (model REVO-2, Renishaw plc, Ltd., Gloucestershire, UK), as shown in Figure 27. Measurement results from four selected sections are presented in Table 3.

As shown in Table 3, the maximum deviation of the impeller is 0.0493 mm, and the average deviation does not exceed 0.035 mm, meeting the requirements for impeller machining.

6. Conclusions

In summary, this paper presents a non-symmetric corner-smoothing method for five-axis toolpaths based on the SMM. The main conclusions are as follows:

(1): Unlike traditional corner-smoothing methods, this approach achieves trajectory smoothing and synchronous interpolation of the tool tip position and tool orientation through the master–slave cooperative space. Symmetric dual Bézier spline smoothing is used for the tool tip position trajectory as the primary motion, and sufficient conditions for monotonic curvature in the transition curve are derived, minimizing curvature fluctuation.
(2): Asymmetric dual Bézier spline smoothing is applied to the master–slave cooperative space trajectory, establishing a G2-continuous synchronization relationship between the tool tip position and tool orientation. This effectively resolves the continuity issue in cross-dimensional synchronous interpolation between linear and rotary axes, simplifying the computation process and improving real-time performance.
(3): Simulation tests and machining experiments show that, with the proposed method, the tool tip acceleration curve is smoother during part machining; maximum acceleration values for each axis were reduced by 21.05%, while jerk was lowered by 22.31%, leading to a significant reduction in machining marks. This indicates that the method effectively reduces machine vibration and enhances surface quality.
(4): Theoretically, the proposed toolpath-smoothing method has broad applicability, meeting the requirements of five-axis systems and extending to multi-axis CNC machines such as four-axis and six-axis systems. This overcomes the limitations of current methods in multi-axis machining. In the future, this method will be further extended to other multi-axis CNC systems, including four- and six-axis machines. Additionally, further research will focus on feedrate planning methods that consider machine drive performance to enhance the efficiency of multi-axis CNC machining.

Author Contributions

Conceptualization, S.G. and J.X.; methodology, S.G. and J.X.; software, S.G. and H.Z.; validation, S.G. and H.Z.; formal analysis, S.G.; investigation, S.G. and H.Z.; resources, J.Y. and W.Z.; data curation, W.Z.; writing—original draft preparation, S.G.; writing—review and editing, S.G. and J.X.; supervision, J.X.; project administration, J.X.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Key Research and Development Program of China (Project No. 2022YFF0605201).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

MIP	Mixed-integer programming
MCS	Machine coordinate system
WCS	Workpiece coordinate system
SMM	Space of master–slave movement

Appendix A

Hypothesis A1.

When the following conditions are met, the curvatures of curves

C_{1}

and

C_{2}

are monotonic, and the curvature

κ_{q}

at the connecting point

q

is maximal.

(\{k_{2} \geq k_{1}^{2} - \frac{1}{10} {(\frac{8}{5} - \frac{b}{10} + \frac{d}{10})}^{2}\} \cup (\{k_{1} \leq 1\} \cap \{k_{2} \geq - 1.5 k_{1}^{2} + 5 k_{1} - 2.5\})) \cap \{k_{2} \leq 6 k_{1}^{2} - 4 k_{1}\}

where

\{\begin{matrix} a = \sqrt{10000 k_{1}^{4} - 20000 k_{1}^{3} + 15240 k_{1}^{2} - 5240 k_{1} + 625} \\ b = \sqrt{500 k_{1}^{2} - 5 a - 500 k_{1} + 131} \\ d = \sqrt{500 k_{1}^{2} + 5 a - 500 k_{1} + 137 - \frac{192}{b}} \end{matrix}

Proof.

Since Bézier splines possess geometric invariance, affine invariance, and symmetry, without a loss of generality, we assume that

\vec{{C L}_{1} {C L}_{0}}

and

\vec{{C L}_{2} {C L}_{1}}

are symmetric with respect to the vertical axis in a Cartesian coordinate system, with

\vec{{C L}_{1} {C L}_{0}}

located in the third quadrant. Therefore, we have

\{\begin{matrix} a_{i} = (- x_{i}, - \tan (θ) \cdot x_{i}) i = 1, 2, 3 \\ q = (0, - \tan (θ) \cdot x_{1}) \end{matrix}

(A1)

where

x_{i} = c o s (θ) \cdot \sum_{n = 1}^{i} l_{n}

We define the parameter values of the endpoints of the Bézier spline

C_{1} (x, y)

as follows:

u_{q} = 0

(A2)

u_{a_{3}} = 1

(A3)

The curvature can then be calculated using the following equation:

κ = - \frac{l_{1}}{3} \cdot K

(A4)

where

\begin{matrix} K = \frac{m}{{(n)}^{1.5}} \\ m = u S_{2 θ} (k_{1} u - k_{2} u + k_{2}) \\ n = C_{θ}^{2} {(u^{2} k_{2} + 2 k_{1} u + 2 k_{2} u + u^{2} + k_{2} - 2 k_{1} u^{2})}^{2} + S_{θ}^{2} {(u - 1)}^{2} {(2 k_{1} u - k_{2} u + k_{2})}^{2} \end{matrix}

For ease of notation, let the sine and cosine of the angle

θ

be represented as

S_{θ}

and

C_{θ}

, respectively. To further study the characteristics of curvature variation, the differential of the curvature is derived as follows:

\frac{d κ}{d u} = - \frac{l_{1}}{3} \cdot \frac{m^{'} n - 1.5 m n'}{n^{2.5}}

(A5)

We define U as follows:

U = m^{'} n - 1.5 m n^{'}

(A6)

Since

n \geq 0

always holds, the curvature of the curve varies monotonically when

U \geq 0

. Substituting the expressions for

m

and

n

yields

U = S_{θ} C_{θ} (C_{θ}^{2} p + q)

(A7)

where

\begin{array}{l} p = (16 k_{2} + (8 - 48 k_{1}) k_{2} + 32 k_{1}^{2} - 8 k_{1}) u^{5} \\ + (- 40 k_{2}^{2} + (- 10 + 80 k_{1}) k_{2} - 20 k_{1}^{2}) u^{4} + 32 (k_{2}^{2} - k_{2} k_{1}) u^{3} \\ - 8 k_{2}^{2} u^{2} \\ q = (- 32 k_{1}^{3} + 8 k_{2}^{3} + 64 k_{1}^{2} k_{2} - 40 k_{1} k_{2}^{2}) u^{5} \\ + (40 k_{1}^{3} - 30 k_{2}^{3} - 140 k_{1}^{2} k_{2} + 120 k_{1} k_{2}^{2}) u^{4} \\ - (- 8 k_{1}^{3} + 40 k_{2}^{3} + 88 k_{1}^{2} k_{2} - 120 k_{1} k_{2}^{2}) u^{3} \\ + (- 20 k_{2}^{3} - 12 k_{1}^{2} k_{2} + 40 k_{1} k_{2}^{2}) u^{2} + 2 k_{2}^{3} \end{array}

When

q \geq 0

, the following inequality is clearly satisfied:

C_{θ}^{2} p + q \geq C_{θ}^{2} (p + q)

. Thus, there exists a sufficient condition for

U \geq 0

, which is also a sufficient condition for

\frac{d κ}{d u} \leq 0

\{\begin{matrix} q \geq 0 \\ p + q \geq 0 \end{matrix}

(A8)

(1) It is easy to derive that the sufficient condition for

q \geq 0

k_{1} \leq 6.99 k_{2}

(A9)

(2) We derive the sufficient condition for

p + q \geq 0

Consider the following condition:

\frac{d (p + q)}{d u} = 0

(A10)

Solve for the four stationary points where

p (u) + q (u) = 0

\begin{matrix} u_{1} = 0 \\ u_{2} = - \frac{k_{2}}{k_{1} - k_{2}} \\ u_{3} = \frac{k_{1} - k_{2} + 0.632 \sqrt{k_{1}^{2} - k_{2}}}{2 k_{1} - k_{2} - 1} \\ u_{4} = \frac{k_{1} - k_{2} - 0.632 \sqrt{k_{1}^{2} - k_{2}}}{2 k_{1} - k_{2} - 1} \end{matrix}

Additionally, since

p (0) + q (0) = 2 k_{2}^{3} \geq 0

, the necessary and sufficient condition for

p (1) + q (1) = 12 k_{1}^{2} - 8 k_{1} - 2 k_{2} \geq 0

k_{2} \leq 6 k_{1}^{2} - 4 k_{1}

(A11)

It is evident that when condition (A11) is satisfied, the following sufficient condition exists for

p (u) + q (u) \geq 0, u \in [0, 1]

u_{j} \notin (0, 1) \cup p (u_{j}) + q (u_{j}) \geq 0 j = 1, 2, 3, 4

(A12)

Since

u_{1}, u_{2} \notin (0, 1)

always holds, it is sufficient to analyze the influence of the stationary points

u_{3}

and

u_{4}

on the sign of

q

. By analyzing in the

{k_{1}, k_{2}}

space, we can easily obtain the following.

Condition 1: A sufficient condition for

u_{3} \notin (0, 1)

is as follows:

(k_{1}, k_{2}) \in Φ

(A13)

where

Φ

is a set in the

(k_{1}, k_{2})

space that satisfies

\{\begin{matrix} \frac{10 k_{1}^{2} - 10 k_{2}}{25} < {(k_{1} - 1)}^{2} \\ k_{1} > 1 \\ 2 k_{1} - k_{2} - 1 > 0 \end{matrix}

Condition 2: A sufficient condition for

u_{4} \notin (0, 1)

is as follows:

(k_{1}, k_{2}) \in Ω

(A14)

where

Ω

is a set in the {

k_{1}, k_{2}}

space, expressed as follows:

\begin{array}{l} Ω = Ω_{1} \cup Ω_{2} \cup Ω_{3} \cup Ω_{4} \\ Ω_{1} = \{1 < k_{1}, \frac{5}{3} k_{2} + \frac{1}{3} \sqrt{10 k_{2}^{2} - 6 k_{2}} < k_{1}\} \\ Ω_{2} = \{1 < k_{1}, k_{2} < 0.6\} \\ Ω_{3} = \{k_{1} < 1, k_{1} < \frac{5}{3} k_{2} - \frac{1}{3} \sqrt{10 k_{2}^{2} - 6 k_{2}}, k_{2} < - \frac{3}{2} k_{1}^{2} + 5 k_{1} - \frac{5}{2}\} \\ Ω_{4} = \{k_{1} < 1, k_{2} < 0.60, k_{2} < - \frac{3}{2} k_{1}^{2} + 5 k_{1} - \frac{5}{2}\} \end{array}

Condition 3: The sufficient condition for

p (u_{3}) + q (u_{3}) \geq 0

is as follows:

(k_{1}, k_{2}) \in Π = {2 k_{1} - 1 < k_{2}}

(A15)

Condition 4: The sufficient condition for

p (u_{4}) + q (u_{4}) \geq 0

is given by (A16):

\begin{matrix} (k_{1}, k_{2}) \in Ψ \\ Ψ = \{k_{1} < 1, k_{2} \leq k_{2_{2}}, k_{2} < 2 k_{1} - 1, k_{2_{3}} < k_{2}\} \cup \\ {1 \leq k_{1}, k_{2} < 2 k_{1} - 1, k_{2_{2}} < k_{2}} \end{matrix}

(A16)

According to (A13) and (A15),

Π \cap Φ = Φ

, which implies that

u_{3}

always satisfies (A13). Similarly, based on (A14) and (A16), we can conclude that

Ψ \cup \bar{Ω}

forms a set that satisfies (A12). In light of the limitations set forth in (A11), when considered in conjunction with

Ψ \cup \bar{Ω}

, it becomes evident that the sufficient condition for

p + q \geq 0

is as follows:

(\{k_{2} \geq k_{1}^{2} - \frac{1}{10} {(\frac{8}{5} - \frac{b}{10} + \frac{d}{10})}^{2}\} \cup (\{k_{1} \leq 1\} \cap \{k_{2} \geq - 1.5 k_{1}^{2} + 5 k_{1} - 2.5\})) \cap \{k_{2} \leq 6 k_{1}^{2} - 4 k_{1}\}

(A17)

Finally, since

Ψ \cup \bar{Ω}

is a subset of (A9), it follows that (A17) is a sufficient condition for

U \geq 0

. Therefore, when (A17) is satisfied, the curvature of the Bézier spline

C_{1} (x, y)

satisfies

κ \leq 0

and

\frac{d κ}{d u} \leq 0

, which means that at the connection point

q

, the curvature

κ_{q} = |κ|

is maximized. Since

C_{2} (x, y)

is symmetric to

C_{1} (x, y)

with respect to the y-axis, similarly, the curvature

κ_{q}

at the connection point

q

for

C_{2} (x, y)

is also maximized. Thus, Section 3.2 is proven. □

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Figure 1. Schematic of global smoothing for linear trajectories.

Figure 2. Schematic of local smoothing for linear trajectories.

Figure 3. A typical toolpath in the SMM.

Figure 4. Classification of trajectories in the SMM. (a)

s_{q}

monotonically increases; (b)

s_{q}

increases first and then decreases; (c) the mirror image of (a); (d) the mirror image of (b); (e) zero master movement between

{C L}_{1}

and

{C L}_{2}

; (f) zero master movement between

{C L}_{0}

and

{C L}_{1}

.Therefore, trajectory smoothing in the SMM primarily focuses on the cases of (a,b). By improving the continuity of trajectories in the SMM, the slave motion’s speed and acceleration can remain continuous.

Figure 4. Classification of trajectories in the SMM. (a)

s_{q}

monotonically increases; (b)

s_{q}

increases first and then decreases; (c) the mirror image of (a); (d) the mirror image of (b); (e) zero master movement between

{C L}_{1}

and

{C L}_{2}

; (f) zero master movement between

{C L}_{0}

and

{C L}_{1}

Figure 5. Smoothed trajectory in the SMM.

Figure 6. Tool direction tolerance for a five-axis toolpath.

Figure 7. G2-continuous symmetric Bézier spline.

Figure 8. Intersection of transition curves.

Figure 9. G2 continuous asymmetric dual Bézier spline.

Figure 10. Mirror image of the asymmetric dual Bézier spline.

Figure 11. Asymmetric dual Bézier spline trajectory in the SMM.

Figure 12. The spline connection point and tangent when

ε^{'} \leq ε^{*}

Figure 12. The spline connection point and tangent when

ε^{'} \leq ε^{*}

Figure 13. The strategy for selecting the values of

k k_{1}

and

k k_{2}

under the condition of curvature monotonicity.

Figure 13. The strategy for selecting the values of

k k_{1}

and

k k_{2}

under the condition of curvature monotonicity.

Figure 14. Trajectory smoothing in the three-dimensional SMM

{s_{p}, s_{q}, s_{r}}

Figure 14. Trajectory smoothing in the three-dimensional SMM

{s_{p}, s_{q}, s_{r}}

Figure 15. Schematic of the program segment interpolation.

Figure 16. Smoothing result of the toolpath for five-axis machining.

Figure 17. Trajectory in the SMM. (a)

{s_{p}, s_{q}}

; (b)

{s_{p}, s_{r}}

Figure 17. Trajectory in the SMM. (a)

{s_{p}, s_{q}}

; (b)

{s_{p}, s_{r}}

Figure 18. Comparison of trajectory shape between the B-spline method and dual Bézier method.

Figure 19. Comparison of curvature between the dual Bézier method and the B-spline method. (a) dual Bézier method; (b) B-spline method.

Figure 20. Machining experiment case and machine tool. (a) Side milling toolpath for the impeller; (b) AC dual rotary table five-axis CNC machine.

Figure 21. Smoothing result of the impeller machining trajectory.

Figure 22. Velocity of each machine axis. (a) Complete view; (b) local magnified view.

Figure 23. Acceleration of each machine axis. (a) Complete view; (b) local magnified view.

Figure 24. Jerk of each machine axis. (a) Complete view; (b) local magnified view.

Figure 25. Cutting results of the impeller blade. Region A: near the trailing edge. (a) Original HNC-8 smoothing function; (b) proposed smoothing method.

Figure 26. Comparison of surface quality in region A of the blade. (a) Original HNC-8 smoothing function; (b) proposed smoothing method.

Figure 27. Measurement equipment. (a) Coordinate measuring machine; (b) impeller measurement.

Table 1. Machining experiment parameters and tool configuration.

Item	Description
Cutter model	Tapered ball-end milling cutter Diameter: 2 mm, taper: 7°, fillet radius: 1 mm
Cutting parameters	Spindle speed: 20,000 rpm Feedrate: 3200 mm/min
Workpiece material	Al6061 alloy
Tool holder type	HSK-A63 shrink-fit tool holder

Table 2. Comparison results of smoothing methods.

	HNC-8 Method	Proposed Method
Time (s)	6.74	5.57
Vmax (mm/min)	18,632	21,745
Amax (mm/s²)	4546	3589
Jmax (mm/s³)	260,000	202,000

Table 3. Statistical analysis of deviations in selected sections.

Section	Min Dev (mm)	Max Dev (mm)	Mean Dev (mm)
1	0.0185	0.0476	0.0343
2	0.0195	0.0358	0.0240
3	0.0046	0.0493	0.0272
4	−0.0079	0.0293	0.0242

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Gao, S.; Zhang, H.; Yang, J.; Xie, J.; Zhu, W. A Five-Axis Toolpath Corner-Smoothing Method Based on the Space of Master–Slave Movement. Machines 2024, 12, 834. https://doi.org/10.3390/machines12120834

AMA Style

Gao S, Zhang H, Yang J, Xie J, Zhu W. A Five-Axis Toolpath Corner-Smoothing Method Based on the Space of Master–Slave Movement. Machines. 2024; 12(12):834. https://doi.org/10.3390/machines12120834

Chicago/Turabian Style

Gao, Song, Haiming Zhang, Jianzhong Yang, Jiejun Xie, and Wanqiang Zhu. 2024. "A Five-Axis Toolpath Corner-Smoothing Method Based on the Space of Master–Slave Movement" Machines 12, no. 12: 834. https://doi.org/10.3390/machines12120834

APA Style

Gao, S., Zhang, H., Yang, J., Xie, J., & Zhu, W. (2024). A Five-Axis Toolpath Corner-Smoothing Method Based on the Space of Master–Slave Movement. Machines, 12(12), 834. https://doi.org/10.3390/machines12120834

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A Five-Axis Toolpath Corner-Smoothing Method Based on the Space of Master–Slave Movement

Abstract

1. Introduction

2. Space of Master–Slave Movement

2.1. Definition

2.2. Trajectory Error Constraint

2.3. Trajectory Smoothness

3. Tool Tip Position Smoothing with Minimal Curvature Fluctuation

3.1. Symmetric Dual Bézier Spline

3.2. Minimal Curvature Fluctuation Conditions

3.3. Tool Position Smoothing Process

4. Trajectory Smoothing and Synchronous Interpolation of SMM

4.1. Asymmetric Dual Bézier Corner Smoothing

4.1.1. Asymmetric Dual Bézier Spline

4.1.2. Corner Smoothing in SMM

4.2. Synchronization and Interpolation of Five-Axis Machining Toolpaths

5. Simulation and Experiment

5.1. Numerical Simulation

5.2. Machining Experiment

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Abbreviations

Appendix A

References

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