Optimization of a Redundant Freedom Machining Toolpath for Scroll Profile Machining

Abstract

The scroll disc is a critical functional component of the scroll compression mechanism, and its machining precision and quality directly impact the performance and longevity of the compressor. Current machining methods for scroll profiles face challenges in simultaneously achieving wide applicability, high precision, and high efficiency. This paper addresses issues related to unsmooth toolpaths of machine tool axes and high acceleration in the rotary axis during redundant degrees of freedom scroll profile machining. This paper proposes a toolpath optimization method for redundant axes, with optimization objectives focused on reducing the counts of directional changes in the linear axes and smoothing the trajectories of all axes. Experimental results demonstrate that the proposed method offers higher machining efficiency compared to traditional polar coordinate machining.

1. Introduction

Scroll compressors are widely used due to advantages such as fewer parts, compact size, light weight, low noise, and high efficiency [1,2]. Traditionally, the spiral profile of the scroll plate is milled using a three-axis CNC machine. Frequent changes in the movement direction of the X and Y linear axes can lead to surface defects, such as overcut marks, which are typically mitigated by adjusting the feedrate [3,4]. Advances in multi-axis control technology have facilitated the use of four-axis machining centers with three linear axes (X, Y, Z) and one rotary axis, effectively eliminating the need for directional changes in linear axis movement. For machining the sidewalls of the scroll plate, a polar coordinate mode is often selected, utilizing simultaneous control of one linear and one rotary axis. However, this approach applies only to single involute curves and is unsuitable for complex, combined profiles. This paper aims to investigate a scroll profile milling method on machines with a rotary axis capable of handling diverse combined profiles while ensuring processing accuracy and surface quality.

Four-axis machining introduces the issue of redundant freedom. Redundant degrees of freedom allow the motion system to perform secondary tasks or optimize its working performance beyond the primary task [5,6]. Maamar et al. [7] eliminated redundancy with the optimization goals of high stiffness and good dynamic behavior when using a serial robot for precision machining tasks. Guo et al. [8] aimed to increase positioning accuracy and machining quality by eliminating redundancy with high stiffness as the optimization goal. Lin et al. [9] optimized the machining toolpath by eliminating redundancy with high motion performance, static performance, and dynamic performance as the optimization goals. Tao et al. [10,11] proposed detecting adverse vibrations during machining and eliminating redundancy with the goal of suppressing vibrations, thereby avoiding machining issues caused by chatter. When operating in complex environments, redundant motion systems can avoid collisions with obstacles by adjusting the redundant motions [12,13,14]. Although redundant degree-of-freedom machining methods have been increasingly applied in various machining scenarios, no research has yet explored their application in the study of scroll discs.

When optimizing machining toolpaths through redundant degrees of freedom, there are generally two approaches: One involves establishing optimization models at each cutter location along the machining toolpath. Liao et al. [15] formulated optimization models at each cutter contact point on sub-regions of free-form surfaces and solved them using a simulated annealing algorithm. Li et al. [16] established a virtual dynamic model through a virtual repulsive potential field method and solved the model using an optimized Runge–Kutta method. LUKASZ et al. [17] constructed a nonlinear optimization model with equality and inequality constraints, using one-dimensional redundant acceleration as the control variable, and solved it via a quadratic programming approach. Liu et al. [18] built a relationship between cutter locations and redundant variables through a Legendre surrogate model and optimized the surrogate model using a tailorable sequential linear programming (SLP) method. Zhang et al. [19] obtained the current optimal redundant solution through genetic algorithms and feedforward neural networks. The method of establishing optimization models at cutter locations typically involves models that are not complex to solve but have limited optimization space.

The other approach involves establishing an optimization model over an entire segment of the machining toolpath to directly optimize that segment. Liu et al. [20] compressed the cutter location points of the machining toolpath using an improved Douglas–Peucker (DP) algorithm and solved the optimization model using the sine cosine algorithm (SCA). Lu et al. [21] divided the complete machining toolpath into multiple sub-paths and solved them using a differential evolution algorithm. Mario et al. [22] proposed a general framework for layered redundancy resolution under arbitrary constraints. Chen et al. [23] formulated a non-convex optimization model that considered joint constraints and avoided gouging. Minsung et al. [24] introduced a method for generating initial toolpaths based on reinforcement learning. McAvoy et al. [25] solved point-to-point motion planning using a genetic algorithm, with toolpaths expressed through B-splines. Peng et al. [26] proposed an adaptive simulated annealing genetic algorithm (ASAGA) for toolpath planning of redundant manipulators. In the context of scroll profile machining, when the redundant variables over the entire toolpath segment are used as control variables in the optimization model, the model becomes complex. Therefore, it is necessary to find a method that can quickly converge when solving for the optimization of redundant degrees of freedom in scroll profile machining.

Addressing the aforementioned issues, this paper proposes a redundant degree-of-freedom machining method and toolpath optimization approach tailored for scroll profile machining. Focusing on the problem of redundant toolpath optimization in scroll profile milling, research is conducted on the establishment, simplification, and solution methods of the optimization model. This has led to the optimization of redundant axis toolpaths, a reduction in the counts of direction changes for the machine tool’s linear axes, and an improvement in the smoothness of the machining toolpath.

The remainder of this paper is organized as follows: Section 2 introduces the four-axis redundant degree-of-freedom machining method for scroll profiles. Section 3 presents the toolpath optimization for scroll profile machining with redundant degrees of freedom. Then, Section 4 provides simulation and experimental verification. Finally, the conclusions are presented in Section 5.

2. Processing Method for Scroll Profile

2.1. Description of the Proposed Method

To address the optimization of redundant axis toolpaths in scroll profile milling, this paper proposes three evaluation criteria: counts of directional changes for the linear axes, motion magnitude of machine axes, and smoothness of curve in space of master–slave movement (SMM). Based on these criteria, the redundant axis toolpath is generated and optimized, as illustrated in Figure 1.

As shown in Figure 1, first, a sequence of tool location points $\left\{CL\right\}$ in the workpiece coordinate system (WCS) is generated from the scroll part model, including the X-axis and Y-axis positions. An optimization model is established along each tool location point on the machining toolpath, with the objectives of minimizing directional changes in the machine’s linear axes and reducing the movement magnitude of each axis. The position sequence of the redundant C-axis $\left\{C\right\}$ is then solved using a greedy algorithm.

Next, regions of unsmooth machine axis toolpaths are identified and compressed using piecewise cubic splines. In these regions, a nonlinear optimization model is established, aiming to minimize the directional changes in linear axes and enhance the smoothness of the curves of SMM. The compressed redundant axis toolpath is then optimized using particle swarm optimization (PSO).

Finally, velocity planning and real-time interpolation are applied to the optimized tool location point sequence $\left\{{CL}^{\prime }\right\}$ in the WCS and the optimized rotary axis position sequence $\left\{C^{\prime }\right\}$ in the machine coordinate system (MCS), generating interpolated motion commands. The incremental values $\{\Delta X_i,\Delta Y_i,\Delta C_i\}$ for the X, Y, and C axes at each interpolation cycle are output to the servo system.

2.2. Kinematic Model of the Machine Tool

The XYZC machine tool consists of linear guideways along the X, Y, and Z axes, combined with a high-precision rotary C-axis, as shown in Figure 2.

When using the rotational tool center point control method for machining, the numerical control system directly tracks the tool location point toolpath and the tool axis vector toolpath in the WCS. It then allocates the motion of each axis of the machine tool through the kinematic transformation model of the machine tool. Typically, the machining toolpath data in the numerical control program include the tool location point coordinates ${[x,y,z]}^T$ in the WCS and the tool axis vector direction ${[i,j,k]}^T$. The current tool position $P$ in the WCS $\left\{O_w\right\}$ can thus be expressed as ${}^{W}P$:

$${}^{W}P=\left[\begin{array}{cc}i& x\\ j& y\\ k& z\\ 0& 1\end{array}\right]$$

(1)

During machining, the current tool position $P$ in the tool coordinate system $\left\{O_t\right\}$ can thus be expressed as ${}^{T}P$:

$${}^{T}P=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]$$

(2)

According to the motion chain of the machine tool, transforming the tool position point from the WCS $\left\{O_w\right\}$ to the MCS $\left\{O_m\right\}$, followed by a transformation from the tool coordinate system $\left\{O_t\right\}$ to the MCS $\left\{O_m\right\}$, Equation (3) can be derived:

$${}_Y{}^{M}T·{}_X{}^{Y}T·{}_C{}^{X}T·{}_W{}^{C}T·{}^{W}P={}_Z{}^{M}T·{}_T{}^{Z}T·{}^{T}P$$

(3)

where the transformation matrix ${}_{\beta }{}^{\alpha }T$ represents the transformation from coordinate system $\beta$ to coordinate system $\alpha$. If only a translational transformation exists between the two coordinate systems, the corresponding transformation matrix can be expressed as Equation (4):

$${}_{\beta }{}^{\alpha }T=Trans(d_{\alpha \beta x},d_{\alpha \beta y},d_{\alpha \beta z})=\left[\begin{array}{cccc}1& 0& 0& d_{\alpha \beta x}\\ 0& 1& 0& d_{\alpha \beta y}\\ 0& 0& 1& d_{\alpha \beta z}\\ 0& 0& 0& 1\end{array}\right]$$

(4)

where $\left[d_{\alpha \beta x},d_{\alpha \beta y},d_{\alpha \beta z}\right]$ represents the vector from the origin of coordinate system $\alpha$ to the origin of coordinate system $\beta$. In addition to the translational transformation, the transformation between the coordinate systems of the X-axis and the C-axis also includes a rotation about the $\overrightarrow{\mathbf{z}}$ direction. ${}_C{}^{X}T$ can be expressed as

$${}_C{}^{X}T=Trans(d_{XCx},d_{XCy},d_{XCz})·Rot(\overrightarrow{\mathbf{z}},{\Phi }_C)$$

(5)

where

$$Rot(\overrightarrow{\mathbf{z}},{\Phi }_C)=\left[\begin{array}{cccc}\cos ({\Phi }_C)& -\sin ({\Phi }_C)& 0& 0\\ \sin ({\Phi }_C)& \cos ({\Phi }_C)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(6)

By combining Equations (1) through (6), the forward kinematics transformation matrix of the machine tool can be derived as

$$\begin{array}{cc}\hfill {}^{W}P& =\left[\begin{array}{cc}i& x\\ j& y\\ k& z\\ 0& 1\end{array}\right]={}_W{}^{C}T^{-1}·{}_C{}^{X}T^{-1}·{}_X{}^{Y}T^{-1}·{}_Y{}^{M}T^{-1}·{}_Z{}^{M}T·{}_T{}^{Z}T·{}^{T}P\hfill \\ & =\left[\begin{array}{cc}0& (Y+A_0)·\sin (C)-(X+A_1)·\cos (C)+A_2\\ 0& -(Y+A_0)·\cos (C)-(X+A_1)·\sin (C)+A_3\\ 1& Z+d_{ZTz}+d_{MZz}-d_{CWz}-d_{XCz}-d_{YXz}-d_{MYz}\\ 0& 1\end{array}\right]\hfill \end{array}$$

(7)

where the coefficients $A_0$, $A_1$, $A_2$, and $A_3$ are only related to the structural parameters of the machine tool, as shown below:

$$\begin{array}{l}{A}_0=d_{MYy}+d_{XCy}+d_{YXy}-d_{MZy}-d_{ZTy}\\ A_1=d_{MYx}+d_{XCx}+d_{YXx}-d_{MZx}-d_{ZTx}\\ A_2=-d_{CWx}\\ A_3=-d_{CWy}\end{array}$$

(8)

In the milling process of scroll profiles, the tool axis vector is always parallel to the Z-axis, meaning the tool axis vector ${[i,j,k]}^T$ is consistently ${[0,0,1]}^T$. According to Equation (7), it is not possible to obtain the coordinates of the machine’s rotary C-axis in the machine coordinate system through inverse kinematics transformation.

To ensure the machining accuracy of the workpiece toolpath, this paper expresses the scroll profile machining toolpath through the tool location point toolpath in the WCS and the rotary axis toolpath in the MCS. In this paper, the tool location point toolpath in the WCS and the rotary axis toolpath in the MCS are represented by sequences of tool location points $\left\{CL\right\}$ and rotary axis positions $\left\{C\right\}$ along the machining toolpath.

2.3. Inverse Kinematics Solving Strategy for Machine Tools

2.3.1. Toolpath in the Space of Master–Slave Movement

In the process of multi-axis simultaneous movement, the motion of the tool tip point $\left[x,y,z\right]$ in the WCS can be regarded as the primary motion, while the motions of the various machine tool axes can be considered as the secondary motions. In the machining of scroll profiles, the motion quantity of the tool tip point $\left[x,y,z\right]$ in the WCS, i.e., the cumulative arc length along the machining toolpath in the WCS, is taken as the horizontal coordinate $s$, and the coordinate values of the rotary axes in the MCS are taken as the vertical coordinate $q$, forming a space of master–slave movement $\{s,q\}$, abbreviated as SMM. Since the horizontal coordinate represents the travel along the machining toolpath in the WCS, it is referred to as the travel space in this paper.

In the numerical control program, the tool location point toolpath in the WCS and the rotary axis toolpath in the MCS are described in the form of straight line segments (G01). Within each segment, the relationship between the travel $s$ and the rotary axis toolpath in the MCS can be established, as shown in Figure 3.

The toolpath $q(s)$ of the curve in SMM is

$$q(s)=q_i+{\displaystyle \frac{s-s_i}{s_{i+1}-s_i}}·(q_{i+1}-q_i)·s_i\le s\le s_{i+1}$$

(9)

where $i=1,2,\dots ,n-1$, n is number of tool location points; $q_i$ is the coordinate of the rotary axis in the MCS at the $i$-th tool location point; and $s_i$ is the arc length from the $i$-th tool location point to the starting point of the tool location point toolpath in the WCS.

2.3.2. Inverse Kinematics Solving for Machine Tool Linear Axes

During the machining process, the tool location point toolpath in the WCS is first interpolated. Based on the cumulative travel s along the toolpath in the WCS, the coordinates of the rotary axis in the MCS can be directly calculated using Equation (9). Subsequently, using the forward kinematics Equation (7) of the machine tool, the coordinates of the linear axes X and Y of the machine tool can be obtained:

$$\begin{array}{l}{X}_i=-(x_i-A_2)·\cos (C)-(y_i-A_3)·\sin (C)-A_1\\ Y_i=-(y_i-A_3)·\cos (C)+(x_i-A_2)·\sin (C)-A_0\end{array}$$

(10)

In summary, based on Equations (9) and (10), the coordinates of the rotary axis C and the coordinates of the linear axes of the machine tool can be obtained, eliminating the redundancy issue associated with using the inverse kinematics model of the machine tool.

3. Generation and Optimization of the Redundant Axis Toolpath

In scroll profile machining, the tool axis vector is always parallel to the Z-axis, making it impossible to directly generate the machine coordinate system’s rotary axis coordinates at each tool position point through inverse kinematics transformation. This section introduces an optimization index for the redundant axis and applies this index to generate and optimize the redundant axis trajectory.

3.1. Redundant Axis Optimization Indices

3.1.1. Counts of Direction Changes for Linear Axes

The linear axes of the machine tool convert the rotational motion of the servo drive motor into linear motion through ball screws. Due to the existence of backlash, changes in the direction of motion of the machine tool’s linear axes can lead to the formation of reversal marks on the machined surface. Therefore, the counts of direction changes are used as an index for optimizing the redundant axis.

For scroll profile machining toolpaths that rotate around the Z-axis in a spiral manner, it is possible to reduce the counts of direction changes in the motion of the machine tool’s linear axes by distributing some of the feed motion to the rotary axis C. Based on Equation (10), we can obtain

$${(X_i+A_1)}^2+{(Y_i+A_0)}^2=\sqrt{{(x_i-A_2)}^2+{(y_i-A_3)}^2}$$

(11)

From the above derivation, it can be seen that at the $i$-th tool location point, the feasible values of the linear axes X and Y can be represented as a circle with center $(-A_1,-A_0)$ and radius $\sqrt{{(x_i-A_2)}^2+{(y_i-A_3)}^2}$. Since $A_0$, $A_1$, $A_2$, and $A_3$ are only related to the machine tool structure, the range of values for the linear axes X and Y is a set of concentric circles in the XY plane projection for all tool location points along the machining toolpath. Figure 4 shows the feasible region for the linear axes X and Y along the machining toolpath and illustrates two possible toolpaths:

In Toolpath 1, the total counts of direction changes for the linear axes X and Y of the machine tool are 20, while in Toolpath 2, the counts of direction changes are only 2. Therefore, by optimizing the redundant axis toolpath, the counts of direction changes in the motion of the machine tool’s linear axes can be reduced. An index $E_R$ related to the counts of direction changes in the motion of the machine tool’s linear axes is proposed:

$$E_R=k·P_R$$

(12)

where $k$ represents the counts of directional changes in the movement of the machine’s linear axis. $P_R$ denotes the predefined penalty for linear axis directional changes, set to −100. The evaluation index $E_R$ increases with the counts of direction changes in the linear axes.

3.1.2. Motion Magnitude of Machine Axes

In this paper, the toolpath of the rotary axis is directly planned along the machining toolpath. Taking a four-axis machining center with linear axes XYZ and a rotary axis C as an example, according to the kinematic transformation equation of the machine tool, and considering two adjacent tool location points $C{L}_i$ and $C{L}_{i+1}$ in the sequence of tool location points, the motions of the linear axes X, Y, and the rotary axis C are as follows:

$$\left\{\begin{array}{cc}\hfill \Delta X_i& =(x_{i+1}-A_2)·\cos (C_{i+1})+(y_{i+1}-A_3)·\sin (C_{i+1})\hfill \\ & -(x_i-A_2)·\cos (C_i)-(y_i-A_3)·\sin (C_i)\hfill \\ \hfill \Delta Y_i& =-(y_{i+1}-A_3)·\cos (C_{i+1})+(x_{i+1}-A_2)·\sin (C_{i+1})\hfill \\ & +(y_i-A_3)·\cos (C_i)-(x_i-A_2)·\sin (C_i)\hfill \\ \hfill \Delta C_i& =C_{i+1}-C_i\hfill \end{array}\right.$$

(13)

The parameters $A_2$ and $A_3$ are referred to in Equation (8). As shown in Equation (13), if the coordinates of the rotary axis at adjacent tool location points along the machining toolpath are determined, then the motions $\Delta X_i$ and $\Delta Y_i$ of the machine tool’s linear axes are also determined. This paper is based on the assumption that the motions of each machine axis should be matched with the performance of the corresponding drive motor. Therefore, the motion magnitude of each machine axis is weighted using Equation (14), with the highest weighted value taken as the evaluation index:

$$E_b^i=\max \left({\varpi }_0·\Delta X_i,{\varpi }_1·\Delta Y_i,{\varpi }_2·\Delta C_i\right)$$

(14)

where $E_b^i$ represents the evaluation index for the motion of the machine axis along the segment $\overline{C{L}_{i}C{L}_{i+1}}$ and $[{\varpi }_0,{\varpi }_1,{\varpi }_2]$ are coefficients related to the drive performance of the X-, Y-, and C-axis motors, respectively. When selecting drive motors for a four-axis machining center, the same model is typically chosen for the linear axes. Therefore, to simplify the expression of Equation (14), let ${\varpi }_0={\varpi }_1=1$. ${\varpi }_2$ can be simplified as

$${\varpi }_2={\displaystyle \frac{V_X}{V_C}}$$

(15)

where $V_X$ and $V_C$ represent the maximum machining velocity imposed by the drive motors of the linear axis X and the rotary axis C. To alleviate the load on each machine axis’ drive motor, it is necessary to minimize the motion of each axis. Consequently, for any pair of adjacent tool location points, a smaller value of the evaluation index $E_b^i$ for the machine axis motion is preferred.

3.1.3. Smoothness of Curve in SMM

In this paper, we establish a space of master–slave movement $\{s,q\}$, utilizing the arc length s of the toolpath on the workpiece as the horizontal axis and the position $q$ of each machine axis as the vertical axis. For any given space of the master–slave movement $\{s,q\}$, the displacement curve of the machine axes within this space can be described by $q=f(s)$. By taking the first and second derivatives with respect to time $t$ on both sides of the equation, we obtain

$$\begin{array}{l}{q}_t=s_t·f^{\prime }(s)\\ q_{tt}=s_{tt}·f^{\prime }(s)+{s_t}^2·f^{″}(s)\end{array}$$

(16)

where $q_t$ and $q_{tt}$ represent the velocity and acceleration of the machine tool axis q, $s_t$ and $s_{tt}$ represent the feed rate and acceleration of the workpiece system, and $f^{\prime }(s)$ and $f^{″}(s)$ represent the first and second derivatives of the travel space curve.

For any specific machine tool, the velocity and acceleration of each machine axis are constrained by the performance limits of the drive motors. Typically, these limits are pre-calibrated by the machine operator and remain unchanged during the machining process. Therefore, during machining, the motion of each machine axis is

$$\left\{\begin{array}{l}\left|q_t\right|\le v{q}_{\lim it}\\ \left|q_{tt}\right|\le a{q}_{\lim it}\end{array}\right.$$

(17)

where $v{q}_{limit}$ and $a{q}_{limit}$ represent the maximum machining velocity and maximum machining acceleration of each machine axis, respectively, as constrained by the performance of the drive motors and calibrated by the machine operator. Combining Equations (16) and (17), we obtain

$$\left\{\begin{array}{l}\left|s_t·f^{\prime }(s)\right|\le v{q}_{\lim it}\\ \left|s_{tt}·f^{\prime }(s)+{s_t}^2·f^{″}(s)\right|\le a{q}_{\lim it}\end{array}\right.$$

(18)

From the aforementioned equation, it can be inferred that as the derivatives of various orders in the travel space decrease, the velocity and acceleration of the workpiece system increase. Therefore, curve smoothness is adopted as the optimization index for the redundant axis. In the travel space, the sequence of machine axes is not sampled at equal distances along the horizontal axis (i.e., the machining toolpath). Hence, in this paper, the first derivative $fdq$ and the second derivative $sdq$ of the travel space curve $f(s)$ for each machine axis are estimated along the toolpath point sequence using numerical differentiation, as follows:

$$\left\{\begin{array}{l}fd{q}_j^i={\displaystyle \frac{\Delta s_{i-1}^2{q}_j^{i+1}+(\Delta s_i^2-\Delta s_{i-1}^2)q_j^i-\Delta s_i^2{q}_j^{i-1}}{(\Delta s_i+\Delta s_{i-1})\Delta s_{i-1}\Delta s_i}}\\ sd{q}_j^i={\displaystyle \frac{2q_j^{i-1}}{\Delta s_{i-1}(\Delta s_i+\Delta s_{i-1})}}-{\displaystyle \frac{2q_j^i}{\Delta s_{i-1}\Delta s_i}}+{\displaystyle \frac{2q_j^{i+1}}{\Delta s_i(\Delta s_i+\Delta s_{i-1})}}\end{array}\right.2\le i\le n-1;j=\left\{X,\mathrm{Y},\mathrm{C}\right\}$$

(19)

where $n$ represents the number of points on the machining toolpath, $\Delta s_i=‖C{L}_{i+1}-C{L}_i‖$ (where $C{L}_i$ denotes the $i$-th point in the machining toolpath sequence), $q_j^i$ signifies the coordinate of the $j$-th axis at the $i$-th point, and $fd{q}_j^i$ and $sd{q}_j^i$ represent the first and second derivatives, respectively, at the $i$-th point in the travel space $\{s,q_j\}$.

For the initial and final points of the sequence, the first derivative $fdq$ and the second derivative $sdq$ can be estimated as follows:

$$\begin{array}{l}fd{q}_j^1={\displaystyle \frac{q_j^2-q_j^1}{\Delta s_1}}\\ fd{q}_j^n={\displaystyle \frac{q_j^n-q_j^{n-1}}{\Delta s_{n-1}}}\\ sd{q}_j^1={\displaystyle \frac{q_j^3-q_j^2}{\Delta s_1\Delta s_2}}+{\displaystyle \frac{q_j^1-q_j^2}{\Delta {s_1}^2}}\\ sd{q}_j^n={\displaystyle \frac{q_j^{n-2}-q_j^{n-1}}{\Delta s_{n-1}\Delta s_{n-2}}}+{\displaystyle \frac{q_j^n-q_j^{n-1}}{\Delta {s_{n-1}}^2}}\end{array}$$

(20)

Based on the aforementioned analysis, under the premise of fulfilling the workpiece machining tasks, if the extremum values of $f^{\prime }\left(s\right)$ and $f^{″}\left(s\right)$ in each travel space within the numerical control program are low and there are no sudden changes, then the velocity and acceleration limit values outputted by the velocity verification will be high. This allows for the planning of a velocity curve with high machining efficiency. The smoothness index for the collaborative motion space curve of the master and slave axes is as follows:

$$E_M={\alpha }_0·{\displaystyle \sum _{j=0}^2{\displaystyle \sum _{i=1}^n{\varpi }_j·{(fd{q}_j^i)}^2}}+{\alpha }_1·{\displaystyle \sum _{j=0}^2{\displaystyle \sum _{i=1}^n{\varpi }_j·{(sd{q}_j^i)}^2}}$$

(21)

where ${\alpha }_0$ and ${\alpha }_1$ represent the importance of the first and second derivatives and ${\varpi }_0$, ${\varpi }_1$, and ${\varpi }_2$ denote the importance of the SMM for the machine axes X, Y, and C. As mentioned above, for the evaluation index $E_M$ of curve smoothness in the SMM, the smaller the value, the better.

3.2. Redundant Axis Toolpath Generation Based on the Greedy Algorithm

Taking the rotation axis sequence $\left\{r\right\}$ along the machining toolpath as the control variable and aiming to minimize the performance index of the curve in SMM as the optimization objective, the optimization model is established as follows:

$$\underset{\left\{r\right\}}{\min }({\alpha }_0·{\displaystyle \sum _{j=0}^2{\displaystyle \sum _{i=1}^n{\varpi }_j·{(fd{q}_j^i)}^2}}+{\alpha }_1·{\displaystyle \sum _{j=0}^2{\displaystyle \sum _{i=1}^n{\varpi }_j·{(sd{q}_j^i)}^2}})$$

(22)

For a four-axis machining center, in the absence of additional constraints, the value range of any element in the control variable rotation axis sequence $\left\{r\right\}$ is $[-\pi ,\pi ]$. The dimension of the rotation axis sequence $\left\{r\right\}$ is equal to the number of cutter location points n along the machining toolpath. Control variables located in high-dimensional space are typically difficult to search for an optimal solution that satisfies the constraints, and they are prone to converging to the initial solution or a local optimum, leading to the curse of dimensionality.

To address the issue of the curse of dimensionality, this paper proposes a method based on the greedy algorithm to sequentially assign redundant axis coordinates along the machining toolpath. Specifically, along the machining toolpath, the redundant variable with the optimal performance indicator at each cutter location point is selected, thereby reducing the originally n-dimensional control variable to just one dimension. This approach can effectively avoid the problem of the curse of dimensionality.

The weighted values of the index $E_R$, which is related to the counts of direction changes in the linear axis movement of the machine tool, and the index $E_b^i$, which represents the amount of machine tool movement, are taken as the objectives of the optimization model. The goal is to search for the redundant variables at the current cutter location that minimize the objective function. Therefore, the current optimization model can be described as follows:

$$\underset{r}{\min }(E_R^i+E_b^i)$$

(23)

In actual machining, according to Equation (18), when the slope of the SMM is high, the main movement is significantly velocity-limited. Therefore, this paper proposes to constrain the machine tool axis movement between adjacent cutter locations by limiting the increment of machine tool axis movement per unit arc length, thereby reducing the range of redundant axes. This is expressed as follows:

$$f{d}_{\min }·\Delta s_i\le \Delta r_i\le f{d}_{\max }·\Delta s_i$$

(24)

where $\Delta s_i=‖C{L}_{i+1}-C{L}_i‖$, and $C{L}_i$ represents the $i$-th cutter location on the machining toolpath; $f{d}_{\min }$ and $f{d}_{\max }$ are the minimum and maximum amounts of machine tool rotary axis movement per unit arc length set by the process planner. Since the redundant variable is one-dimensional, the optimization model at the current cutter location can be quickly solved through a traversal method. The solution steps are as follows:

Step 1: Calculate the range of the rotation axis at the current tool position based on the rotation axis coordinate $r_{i-1}$ of the previous tool position and its allowed increment $\Delta r_{i-1}$, and form a candidate set ${\left\{r\right\}}_C^i$ by discretizing it at equal intervals.

Step 2: Traverse the candidate set ${\left\{r\right\}}_C^i$ and calculate the movement amounts $(\Delta X,\Delta Y,\Delta r)$ of each machine tool axis between the current and the $(i-1)$-th tool positions based on the kinematic equations of the machine tool.

Step 3: Calculate the comprehensive evaluation index $E=E_R^i+E_b^i$ corresponding to the movement amounts of each machine tool axis between adjacent tool positions using Equations (12) and (14).

Step 4: For the redundant axis at the current tool position, select the element in ${\left\{r\right\}}_C^i$ that minimizes the comprehensive evaluation index E. If the current tool position is the end point of the machining toolpath, then the entire process ends; otherwise, set $i=i+1$ and return to Step 1.

Based on the optimization model established using a greedy algorithm, a model with the objective function of machine tool movement indicators is created at each tool position, and the current optimal value is solved sequentially along the machining toolpath. This approach only considers local optimality and does not take into account global optimality, so there is still room for optimization in the current redundant axis toolpath.

3.3. Redundant Axis Toolpath Optimization Based on PSO

3.3.1. Toolpath Optimization Model

Based on the greedy algorithm, the method for generating the redundant axis sequence can obtain the initial rotation axis toolpath along the machining toolpath. Currently, this rotation axis toolpath is still described by a discrete sequence with a length equal to the number of tool positions $n$ in the machining toolpath. By employing spline interpolation for toolpath compression, it is possible to retain the crucial data points that describe the rotation axis toolpath. When the spline is only position continuous and the motion along the machining toolpath is uniform, the toolpath of the inflection points in the rotation axis toolpath of the SMM is illustrated as shown in Figure 5:

In Figure 5, $k_1$ and $k_2$ represent the slopes of the rotation axis toolpaths in regions ① and ②, respectively, and F denotes the feedrate along the machining toolpath. Based on the curve of the rotation axis along the machining toolpath, the acceleration $a$ at the inflection point can be determined as follows:

$$a=\left(k_2-k_1\right)·{\displaystyle \frac{F}{\Delta t}}$$

(25)

As indicated in Equation (25), even when moving at a constant velocity along the machining toolpath, a high acceleration will occur when passing through the inflection points of the rotation axis toolpath. Therefore, considering the smoothness of processing, this paper proposes to compress the toolpath through segmented cubic spline. The cubic spline curve between adjacent data points is as follows:

$$q=a_0·s^3+a_1·s^2+a_2·s+a_3$$

(26)

where, $a_0$, $a_1$, $a_2$, and $a_3$ are the coefficients of the cubic spline. If the segmented cubic spline consists of a total of $n_{ip}$ data points, the expression for the cubic spline starting from the $j$-th data point is given by

$$\begin{array}{l}{q}^j(s)=a_0^j·s^3+a_1^j·s^2+a_2^j·s+a_3^j\\ q^j{(s)}^{\prime }=3·a_0^j·s^2+2·a_1^j·s+a_2^j\\ q^j{(s)}^{″}=6·a_0^j·s+2·a_1^j\end{array}\hspace{1em}j=1,2,\dots ,n_{ip}-1$$

(27)

In Equation (27), $q^j(s)$, $q^j{(s)}^{\prime }$, and $q^j{(s)}^{″}$ represent the rotation axis toolpath equation, its first derivative, and its second derivative, respectively, starting from the $j$-th data point. Combining the position constraints, first-derivative constraints, and second-derivative constraints mentioned above, we can establish the following system of constraint equations:

$$\begin{array}{l}{q}^j(s_j)=q_j\\ q^j(s_{j+1})=q_{j+1}\\ q^j{(s_{j+1})}^{\prime }=q^{j+1}{(s_{j+1})}^{\prime }\\ q^j{(s_{j+1})}^{″}=q^{j+1}{(s_{j+1})}^{″}\end{array}\hspace{1em}j=1,2,\dots ,n_{ip}-1$$

(28)

where $(s_j,q_j)$ represents the coordinates of the $j$-th data point. By solving the aforementioned system of constraint equations, we can determine the polynomial coefficients for each segment of the piecewise spline.

By constructing a sequence of discrete points for the rotation axis toolpath in SMM using segmented cubic splines that satisfy the aforementioned boundary conditions, the original rotation axis sequence along the machining toolpath is transformed into segmented cubic splines. To minimize the number of data points required for the segmented cubic splines while meeting the toolpath error requirements, we initially use only the start and end points of the toolpath as the data point sequence for the segmented cubic splines. If the generated segmented cubic spline does not meet the toolpath error requirements, the toolpath discrete point with the largest error is inserted into the data point sequence of the segmented cubic spline.

Originally, the number of elements in the rotation axis sequence was the same as the number of cutter location points in the machining toolpath, both being n. However, after using piecewise cubic splines for description, only a portion of the cutter location points, denoted as $n_{ip}$ (which is less than or equal to $n$), need to be retained. If the data points of the piecewise cubic splines are taken as control variables, then the dimension of the control variables is $n_{ip}$, thereby reducing the dimensionality of the search space.

With the performance index of the SMM in the machining toolpath as the optimization objective and the sequence of data points for the segmented cubic spline as the control variables, we construct the optimization model for the rotation axis toolpath as follows:

$$\underset{\left\{r_{ip}\right\}}{\min }({\alpha }_0·{\displaystyle \sum _{j=0}^2{\displaystyle \sum _{i=1}^n{\varpi }_j·{(fd{q}_j^i)}^2}}+{\alpha }_1·{\displaystyle \sum _{j=0}^2{\displaystyle \sum _{i=1}^n{\varpi }_j·{(sd{q}_j^i)}^2+E_R}})$$

(29)

where $\left\{r_{ip}\right\}$ denotes the sequence of data points for the segmented cubic spline. The optimization objective includes both the performance index of the curve in SMM and the counts of direction changes in the motion of the machine tool’s linear axes. This implies that while minimizing the counts of direction changes in the motion of the machine tool’s linear axes, the processing performance of the curve in SMM is improved, with the aim of achieving smoother and more efficient machining. Here, ${\omega }_j$ is the weighted coefficients for each axis of the machine tool, which in this paper are taken as the normalized values of the ratios of the highest processing velocity of each axis; ${\alpha }_0$ and ${\alpha }_1$ are the weighted coefficients for the first and second derivatives, both of which are set to 1 in this paper.

3.3.2. Division of Optimization Regions

For scroll profiles, the derivatives of various orders of the travel space for each machine tool axis are typically not excessively large or exhibit sudden changes along the entire machining toolpath. The first-order and second-order derivatives of the travel space for a certain scroll profile processing toolpath are as follows:

As shown in Figure 6, when the machining toolpath travel $s$ is between 200 and 220 mm, the first and second derivatives of the SMM for the linear axes X, Y, and the rotation axis C are all relatively large and exhibit fluctuations; there are no obvious maximum derivatives in other regions. Therefore, when establishing the optimization model, it is only necessary to separately optimize the regions where the travel space along the machine tool axes is not smooth.

By dividing the regions to be optimized, the dimension of the control variables can be further reduced to avoid the curse of dimensionality as much as possible during the optimization process. It should be noted that when optimizing, it is necessary to ensure the continuity of the toolpath position and the continuity of the differential (first derivative) at the segmentation points before and after the region to be optimized in order to prevent sudden changes at the breakpoints.

3.3.3. Model Solution Method

PSO is an algorithm that solves optimization problems by simulating the social behavior of bird flocks. In the search process of PSO, particles update their velocities according to the following Equation (30):

$$v_i=w·v_i+c_1·rand·(pbes{t}_i-x_i)+c_2·rand·(gbest-x_i)$$

(30)

where $x_i$ and $v_i$ are the position and velocity of the $i$-th particle, $pbes{t}_i$ is the personal best position of the $i$-th particle, $gbest$ is the global best position, $\varpi$ is the inertia weight, $c_1$ and $c_2$ are the learning factors, and rand is a random variable. If the population size is set to M and the maximum number of iterations to N, the basic algorithm of the PSO for the redundant axis toolpath optimization model is as shown below:

Step 1: Randomly generate M sets of segmented cubic spline data point sequences $\left\{r_{ip}\right\}$ for the redundant axis, where multiple sets are assigned as the initial segmented cubic data point sequences generated by the toolpath compression algorithm. Also, randomly generate M sets of velocities.

Step 2: For the M sets of particles, calculate the fitness value for each and update the current particle’s historical best position and the population’s best position.

Step 3: For the M sets of particles, update the velocity according to the population’s best position and the particle’s own best historical position using Equation (30), and calculate the next position of the current particle based on the velocity.

Step 4: Determine whether the algorithm has converged or reached the maximum number of iterations. If it has converged or reached the maximum number of iterations, the process ends; otherwise, return to Step 2.

By following the aforementioned process, the redundant axis toolpath optimization model can be solved. By splicing the optimized regions with the original regions, the redundant rotation axis toolpath can be generated. Along the machining toolpath, at each tool location point, by substituting the travel $s$ of the machining toolpath into the segmented cubic spline curve, the corresponding redundant rotation axis coordinates for the current tool location point can be calculated.

4. Simulations and Experiments

4.1. Simulation of Scroll Disc Machining

Taking the scroll disk part of an automotive air conditioning compressor as a case study, we generate both a three-axis toolpath (TAT) and the scroll profile machining toolpath proposed in this paper, which is expressed through the tool location point toolpath in the WCS and the redundant axis toolpath in the MCS (redundant toolpath, RT). For convenience, in this chapter, the machining path generated by the polar coordinate YC-linked redundant axis sequence allocation method is denoted as RT1; the machining toolpath generated by the redundant axis based on the greedy algorithm is denoted as RT2; and the machining toolpath optimized by the redundant axis based on the PSO is denoted as RT3. The relevant parameters of PSO are shown in

Table 1. PSO parameters.

Parameter	Value
Particle dimension	$n_{ip}$
Population size	100
Maximum number of iterations	1500
Individual learning factor	2
Social learning factor	4
Inertia weight	0.4~0.9
Velocity range	−0.1~0.1

.

The simulation machining experiments in this section were conducted on a Huazhong CNC system equipped with a redundant degree-of-freedom scroll profile toolpath interpolation module. The feedrate is set to F = 10000 mm/min. The maximum machining velocity and maximum machining acceleration, constrained by the performance of each axis’ motor drive in the machine tool, are shown in

Table 2. Constraints on machine tool axis motor drive performance.

	X	Y	Z	C
Max velocity (mm/min)	15,000	15,000	15,000	30,000
Max acceleration (mm/s2)	5550	5550	5550	3000

.

Below are the simulation test results for different toolpaths. The counts of direction changes for the linear axis movement of the machine tool are presented in

Table 3. Counts of direction changes for machine tool linear axis movement.

	TAT	RT1	RT2	RT3
Counts of direction changes	18	1	0	0

.

As indicated in Table 3, the introduction of a redundant C-axis can reduce or eliminate the issue of direction changes in the linear axis movement of the machine tool compared to the three-axis machining toolpath.

Based on

Table 4. Statistical results of travel space for each machine tool axis.

	RT1	RT2	RT3
Max first derivative (X-axis) [mm0]	0	1.04	1.07
Max second derivative (X-axis) [mm−1]	0	1.56	1.51
Max first derivative (Y-axis) [mm0]	1.40	0.71	0.70
Max second derivative (Y-axis) [mm−1]	3.11	1.71	0.73
Max first derivative (C-axis) [°/mm]	37.08	36.43	36.45
Max second derivative (C-axis) [°/mm2]	86.60	40.30	21.77

, compared to the RT1 scroll profile machining toolpath, the RT2 scroll profile machining toolpath exhibits reduced maximum absolute values for both the first and second derivatives of the s-C travel space toolpath, as the linear axis X of the machine tool bears part of the motion. Furthermore, after optimizing the redundant C-axis toolpath with the goal of improving the smoothness of the curve in SMM of the machine tool axes, the RT3 toolpath is able to further reduce the peak value of the second derivative of the redundant C-axis. Therefore, the redundant axis toolpath optimization method proposed in this paper is effective.

In the aforementioned experimental environment, the simulated machining times for the three scroll profile machining toolpaths, RT1, RT2, and RT3, are summarized in

Table 5. Simulated machining times.

	RT1	RT2	RT3
Time (s)	6.605	5.758	5.508

.

According to Table 5, through the optimization of the redundant axis toolpath, the machining efficiency of the scroll profile can be improved, reducing the processing time from 6.6 s to 5.5 s, which is approximately a 16.7% reduction in machining time. The velocity and acceleration of each machine tool axis during the machining of the RT3 scroll profile toolpath are illustrated in Figure 7.

As shown in Figure 7, the velocities and accelerations of each axis in the machine tool during the machining of the RT1, RT2, and RT3 scroll profiles comply with the maximum velocity and acceleration limits imposed by the axis drive performance constraints. The RT3 method has the shortest processing time. In other words, the redundant axis toolpath optimization method proposed in this paper achieves higher machining efficiency while satisfying the constraints on the velocity and acceleration of each axis during the machining process.

4.2. Machining Experiment on the Scroll Disc

As shown in Figure 8, the experimental processing is conducted on the SV300 dual-spindle vertical scroll compressor plate machining special machine tool produced by Shanghai SmartState Technology. The interpolation period was set to 1 ms in the CNC platform, and the feedrate for the scroll profile machining toolpath in the CNC program was set to 10,000 mm/min. The maximum machining velocity and maximum machining acceleration, constrained by the performance of each axis’ motor drive in the machine tool, are presented in

Table 6. Constraints on machine tool axis motor drive performance.

	X	Y	Z	C
Max velocity (mm/min)	15,000	15,000	15,000	30,000
Max acceleration (mm/s2)	12,820	12,820	12,820	4160

. The remaining parameter settings were consistent with the simulation system.

The overall machining time for the scroll disk CNC program is 7.736 s. As shown in Figure 9, the measured line profile tolerance is less than 0.02 mm, which meets the requirements for the machining accuracy of automotive air conditioning scroll disks:

The surface finish of the machined scroll profile is smooth, with no visible tool marks caused by direction changes in the linear axis movement of the machine tool when passing through quadrants. The machined surface of the scroll disk part is shown in Figure 10.

Using an INSIZE Universal Auto-Focus Digital Microscope (model 5317-AF109, made by Suzhou INSIZE Measurement Technology Co., Ltd in Suzhou, China) to observe the machined surface of the part, when the magnification of the microscope is set to 4, the machined surface near the quadrant point is examined, and the results are shown in Figure 11.

In the red-framed area of Figure 11a, distortion of the cutting marks and the presence of noticeable pits can be observed. These distorted tool marks are clearly visible to the naked eye, as shown in Figure 12. In contrast, Figure 11b demonstrates a uniform distribution of cutting marks, with no apparent direction change marks typically seen in three-axis machining.

In summary, the scroll profile milling method based on a machine tool with redundant degrees of freedom proposed in this paper effectively ensures the accuracy of the machining toolpath for the workpiece in actual processing, controlling the actual toolpath error within 20 μm, which meets the requirements of automotive scroll compressors. Simultaneously, it avoids the issue of direction changes in the linear axis movement of the machine tool, eliminating tool marks at quadrant transitions. Furthermore, using the method proposed in this paper, the machining time for the scroll profile is only 7.736 s, satisfying the demand for high-velocity machining of scroll disks.

5. Conclusions

In conclusion, this paper presents a method for scroll profile machining and toolpath optimization based on redundant degrees of freedom. This method expresses the scroll profile machining toolpath through the tool position point toolpath in the WCS and the redundant axis toolpath in the MCS and directly interpolates the tool position point toolpath in the WCS in the CNC system to ensure its machining accuracy. Additionally, a method for generating and optimizing the redundant axis toolpath is proposed, which aims to minimize the counts of direction changes in the linear axis movement of the machine tool and ensure smooth toolpaths for each axis, thereby improving machining efficiency and surface quality. Finally, the results of simulation tests and machining experiments strongly demonstrate the good performance of the proposed method. Compared to the YC-linkage method for generating redundant axes, the redundant axis toolpath optimization method proposed in this paper effectively enhances the smoothness of the machine tool toolpath, reducing machining time by 1.1 s while each axis meets the constraints of axis drive performance. Within the same machining time, it can increase production by approximately 20%. In the future, research will continue on using higher-order continuous toolpaths to replace linear segment toolpaths, further improving machining accuracy and efficiency.