Optimized Tool Motion Symmetry for Strip-Width-Max Mfg of Sculptured Surfaces with Non-Ball Tools Based on Envelope Approximation

Abstract

The problem of machining complex surfaces with non-ball-end cutters by strip-width-maximization machining is formulated as a kind of surface fitting problem in which the tool surface envelope feature line approximates the design surface under the movement transform. The theory of surface envelope−approximation is proposed as a general method for optimizing tool movement in single-contact strip-width-maximization machining of sculptured surfaces with non-ball-end cutters. Based on the surface moving frame, the velocity equations and transformation matrices for the tool motion relative to the workpiece, described by the motion-invariant parameters of the tool surface and design surface, are derived. A functional extremum model for optimizing the tool position ensures continuous and symmetrical motion relative to the workpiece to achieve the highest machining efficiency and accuracy. Finally, a Matlab-based simulation example verifies the machining efficiency and accuracy of the envelope approximation theory.

1. Introduction

Complex surfaces, such as aviation structural components, gas turbine blades, and ship propellers, are widely employed in the aerospace, energy, and national defense industries. These products are directly related to the national economy and national defense security, reflect the major strategic needs of the country, and their manufacturing level represents the core competitiveness of the national manufacturing industry. Complex surfaces include complex parts and large-scale complex surfaces. Such surfaces exhibit the following characteristics: intricate shapes, thin-walled structures, low rigidity, high material removal rates, numerous hard-to-cut materials, stringent requirements for contour accuracy and surface quality, and a wide variety of small batch production [1]. Complex surfaces, characterized by intricate geometric shapes encompassing curves, curved surfaces, and irregular forms, often pose challenges in terms of machining complexity [2]. The machining process for these surfaces necessitates a comprehensive approach, encompassing tool path planning, tool selection, and optimization of machining parameters [3,4]. In 2018, the German Research Foundation launched the SCMP project to study the machined surface integrity of complex parts [5]. Later, the EU Horizon 2020 research and innovation program launched the M-AID/DIH4CPS project [6], which is dedicated to process optimization, predictive maintenance, and production planning for multi-axis machining. Strip-width-maximization machining technology utilizes the multi-degree-of-freedom motion capabilities of multi-axis CNC machines. By making appropriate adjustments to the tool positions and orientations of non-ball-end cutters, optimal machining efficiency and precision can be achieved. This technology has significant applications in industries such as aircraft, automotive, naval, and mold manufacturing. However, due to the absence of a unified standardized description of the complex tool motion in multi-axis CNC machining, coupled with the fact that the position planning, accuracy control, and interference checking for non-ball-end cutters are significantly more intricate than for spherical surface tools, past research in this field has been limited to specific tool types. The use of a single tool position, under the approximation of the tool surface and the local structure of the workpiece surface, has resulted in the simplification of the optimization process for tool position planning. There is no generalized, realistic optimization theory of tool motion control relative to the workpiece that can truly reflect the actual machining process [7,8,9,10,11,12,13].

In recent years, research on strip-width-maximization machining using non-ball-end cutters has primarily focused on rotary cutting tools, higher-order tangent approximations, and accurate computation of overall surface errors without simplifying the design surface. Zha, J, et al. [14] propose a practical accuracy-enhancement method for multi-axis machining. This method seeks to improve the machining accuracy without requiring complex models. The methodology includes an iterative process that treats the problem as a one-degree-of-freedom system. This method is called the accuracy-based evolutive method (ABEM). Sliusarenko O, et al. [15] aim is to perform a precision measurement of a machined 3D free-form surface using a (3 + 2)-axis contact-based CMM instead of the five continuous axes, commonly required for these types of complex surfaces, ensuring the machining time and final measuring deviations. Gdula [16] proposed an adaptive five-axis machining strategy for turbine rotor blades, considering the variable curvature radius and change in the mill axis orientation. A selection method for milling tool paths using the predicted value of the cutting forces as a decision criterion was proposed to minimize the dimensional errors of a complex surface [17]. Gong et al. [18] utilize the isometric inclusion principle to transform the overall error calculation of the tool envelope and design surface into a least-squares approximation problem of the isometric surface between the axial trajectory surface and the design surface. They further investigate the overall optimization of the tool position for machining non-expandable straight faces using a general rotary tool, taking into account the optimization of discrete tool positions as well as constraints at the end and those that prevent overcutting. Chen Zhidong et al. [19,20] proposed a unified method for generating tool paths for machining monotonically convex or concave sculptured surfaces by wide-row end milling. Zou Qi xiao et al. [21] and Dong Lei et al. [22] attribute the theoretical model of cylindrical and conical cutter side milling non-expandable rectilinear surfaces to the problem of best square approximation between the cutter surface and the design surface. In 2021, Wu et al. [23] reviewed the prediction and compensation of geometric errors during the machining of thin-walled parts.

The current approach to machining sculptured surfaces with non-ball-end cutters in a strip-width-maximization process faces three key issues: 1. Lack of Standardized Motion Description: The absence of a standardized method to describe tool motion relative to the workpiece impedes the coordinated optimization of tool position planning across local and global levels. 2. Unstandardized Envelope Surface Approximation: Using unstandardized methods to approximate tool envelope surfaces limits the practicability and stability of approximating design surfaces with tool envelopes. 3. Inadequate Tool Position Planning: The tool position planning is formulated as a single-position approximation optimization, artificially imposing smoothness conditions between positions. This fails to fully align with the machining objectives, specifically achieving a continuous and smooth equation of motion relative to the workpiece, hindering accuracy and efficiency enhancements. These issues complicate multi-axis CNC machining with non-ball-end cutters and impede the achievement of the desired machining accuracy and efficiency levels.

To address the issues in strip-width-maximization machining of complex surfaces with non-ball-end cutters, this paper proposes a generalized method based on the surface envelope approximation using differential geometry’s natural moving frames theory. This approach addresses the problems through: 1. Continuous Smooth Motion: Deriving the velocity equation and transformation matrix for the tool relative to the workpiece ensures continuous smoothness by maintaining geometric structure continuity. This avoids artificial continuity conditions that can lead to suboptimal solutions. 2. Standardized Tool Position Planning: Reducing the tool position planning problem to tool curvature, using oriented surface patches on simple tool surfaces as spline curves. The spline curve approximation theory enables the accurate and standardized approximation of design surfaces with tool envelope surfaces, based on interpolation conditions and accuracy requirements. 3. Generalized Polar Model for Optimization: Establishing a generalized polar model optimizes the machining efficiency and accuracy by ensuring continuous and smooth tool position optimization relative to workpiece movement. This facilitates coordinated optimization from local to overall levels, yielding equations that accurately reflect the actual machining process and ensure continuous, smooth tool motion.

The adoption of spiral or circular shapes for design surfaces, tool contact point trajectories, and machining boundaries is grounded in several pivotal reasons: 1. Continuity: The spiral trajectory’s seamless nature eliminates tool engagement and disengagement intervals, reducing pauses and directional shifts, thereby boosting processing efficiency. 2. Smoothness: The exceptional smoothness of the spiral trajectory minimizes vibrations and impacts from trajectory changes, safeguarding the tool and workpiece while enhancing the machined surface quality. 3. High-Speed Machining Adaptability: The spiral trajectory’s ability to maintain stable cutting speeds and forces during high-speed machining minimizes fluctuations, enhancing both speed and precision. 4. Uniform Stress Distribution: The circular trajectory ensures an even stress distribution, preventing stress concentration and deformation, thereby benefiting both the tool and workpiece. 5. Simplified Calculations: The circular trajectory’s geometric simplicity streamlines mathematical computations and programming, reducing complexity and errors during machining.

This paper initially introduces the theory and technology pertaining to the machining of complex surfaces, with a particular emphasis on strip-width-maximization machining. It presents a solution grounded in the principle of surface envelope approximation. The subsequent sections are structured as follows: Section 2 and Section 3 elaborate on the principle of envelope approximation for surfaces and formulate mathematical models for two distinct problems encountered in strip-width-maximization machining utilizing non-ball-end cutters. These sections meticulously define and analyze the relevant parameters and constraints. Section 4 details the simulation experiments conducted on the blade profiles using MATLAB. A comparative analysis of the bandwidths achieved by the three other processing methods underscores the advantages and efficiency of the method proposed in this paper. Section 5 concludes the paper by summarizing the key findings and outlining future research directions. This section provides a concise overview of the contributions and potential implications of this work, fostering further discussion and exploration in the field.

2. Envelope Approximation Principle for Surfaces

The theory of polynomial spline approximation for surfaces is a mathematical method that utilizes simple (polynomial) functions as basis functions to approximate sculptured surfaces. It represents a spatial surface as a linear space composed of polynomial spline curves. In multi-axis CNC machining, the tool surface envelopes the spatial envelope motion along the trajectory line of the tool contact point. At each moment, a feature line is enveloped. These feature lines are distributed along the trajectory line of the tool contact point to form the tool surface envelope in the coordinate system of the design surface (workpiece) based on the motion relationship. The tool surface envelopes the feature lines. The surface approximation model employs envelope feature lines as spline curves, the design surface as the target approximation surface, and the tool contact point trajectory as the boundary condition for surface approximation. By controlling the envelope motion of the tool relative to the workpiece, the envelope feature lines and their spatial distribution can be manipulated, thus controlling the error of the tool envelope surface approximation to the design surface. This is the principle of surface envelope approximation. The surface envelope approximation theory is described below in terms of spline (basis function) curves, interpolation (boundary) conditions, and approximation errors.

2.1. Sample Curve for Envelope Approximation: Envelope Feature Line Description

The envelope of a surface in space at a given time (position of motion) is related to the manner in which the surface moves in space and is determined by the engagement equations. The set of envelope feature lines formed by a surface at all positions of motion constitutes the envelope feature line space (envelope surface). Obviously, the crux of determining the envelope feature lines is determining how the surface moves in space. To render the theory of surface envelope approximation general and universal, the description of the envelope motion is conducted in the intrinsic coordinate system of the surface, specifically the surface moving frame, and is represented by the motion invariants of the surface, independent of the other coordinate systems and parameters.

2.1.1. Describe the Curvilinear Motion of Moving Frame along and on Surfaces

By applying the moving frame theory of differential geometry to describe the motion of the tool surface relative to the design surface, it is possible to establish a direct relationship between the envelope motion of the tool surface and invariant parameters of the tool surface, as well as the local motion of the design surface. This approach forms a universal, standardized method for describing the motion of the tool relative to the workpiece, which provides a convenient way to coordinate and optimize the planning of the tool position from local to overall [24].

As shown in Figure 1, there are (tool) surfaces ${\mathit{\Sigma }}_{\mathit{t}}:{\mathit{r}}_{\mathit{t}}={\mathit{r}}_{\mathit{t}}\left(u_t,{\upsilon }_t\right)\in C^2$, whose parameters ${ u}_t ,{ \upsilon }_t$ form an orthogonal parameter network, then at any point $M\left(u_{tM},{\upsilon }_{tM}\right)$ on the surface ${\Sigma }_t$ a right-handed unit orthogonal active scalar field is taken.

$$\begin{array}{l}{S}_{ft}\left[M; {\mathit{e}}_1={\displaystyle \frac{{\mathit{r}}_1}{\sqrt{E}}} {\mathit{e}}_2={\displaystyle \frac{{\mathit{r}}_2}{\sqrt{G}}} {\mathit{e}}_3={\mathit{n}}_{tM}={\mathit{e}}_1\times {\mathit{e}}_2\right],\\ {\mathit{r}}_1={\displaystyle \frac{d{\mathit{r}}_t}{d{u}_{tM}}},{\mathit{r}}_2={\displaystyle \frac{d{\mathit{r}}_t}{d{\upsilon }_{tM}}},E={\mathit{r}}_1\cdot {\mathit{r}}_1,G={\mathit{r}}_2\cdot {\mathit{r}}_2.\end{array}$$

When point M moves along the orthogonal parametric meshes $u_{tM}$ and ${\upsilon }_{tM}$ with the velocity${\mathit{V}}_{Mt}={\displaystyle \frac{d{u}_{tM}}{dt}}{\mathit{e}}_1+{\displaystyle \frac{d{\upsilon }_{tM}}{dt}}{\mathit{e}}_2$ relative to the (tool) surface solidification $\left.S_t\left[\begin{array}{cccc}{O}^{\left(t\right)};& x^{\left(t\right)}& y^{\left(t\right)}& z^{\left(t\right)}\end{array}\right.\right]$ (tool coordinate system), the moving frame $S_{ft}$ also moves and rotates around M. According to the fundamental theorem of the surface theory of differential geometry, the equation of motion of the moving frame $S_{ft}$ with respect to the tool coordinate system $S_t$ (Unit Vector Representation of Velocity along a Coordinate Axis) is given by

$$\left[\begin{array}{c}{\dot{e}}_1\\ {\dot{e}}_2\\ {\dot{n}}_{tM}\end{array}\right]=\left[\begin{array}{ccc}0& {\omega }_{12}& {\omega }_{13}\\ -{\omega }_{12}& 0& {\omega }_{23}\\ -{\omega }_{13}& -{\omega }_{23}& 0\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\\ n_{tM}\end{array}\right]$$

(1)

where the square matrix is the square form of the angular velocity vector ${\mathit{\omega }}_t$ of the moving frame $S_{ft}$ rotating about the point M, written in vector form as

$${\mathit{\omega }}_t={\omega }_{23}{\mathit{e}}_1-{\omega }_{13}{\mathit{e}}_2+{\omega }_{12}{\mathit{n}}_{tM}$$

(2)

Among them,

$$\left\{\begin{array}{l}{\omega }_{12}={\displaystyle \frac{-E_{2}d{u}_{tM}/dt+G_{1}d{\upsilon }_{tM}/dt}{2\sqrt{EG}}};E_2={\displaystyle \frac{dE}{d{\upsilon }_{tM}}};G_1={\displaystyle \frac{dG}{d{u}_{tM}}}\\ {\omega }_{23}={\displaystyle \frac{Md{u}_{tM}/dt+Nd{\upsilon }_{tM}/dt}{\sqrt{G}}};M={\mathit{n}}_{tM}·{\displaystyle \frac{{\partial }^2{\mathit{r}}_t}{\partial u_{tM}\partial {\upsilon }_{tM}}};N={\mathit{n}}_{tM}·{\displaystyle \frac{{\partial }^2{\mathit{r}}_t}{\partial {\upsilon }_{tM}^2}}\\ {\omega }_{13}={\displaystyle \frac{Ld{u}_{tM}/dt+Md{\upsilon }_{tM}/dt}{\sqrt{E}}};L={\mathit{n}}_{tM}·{\displaystyle \frac{{\partial }^2{\mathit{r}}_t}{\partial u_{tM}^2}}\end{array}\right.$$

They are all motion-invariant parameters of the surface.

As shown in Figure 2, $L_p$ is a curve (tool contact point trajectory) on the (design) surface ${\Sigma }_p:{\mathit{r}}_p={\mathit{r}}_p\left(u_p,{\upsilon }_p\right)\in C^2$, whose equation is ${\mathit{r}}_{pM}={\mathit{r}}_{pM}\left(u_{pM}\left(s_p\right),u_{pM}\left(s_p\right)\right)$, $S_p$ is the arc length parameter of the curve $L_p$; If the unit tangent vector at any point M on the curve $L_p$ is $\mathit{\alpha }$, and the unit normal vector of the surface at that point is ${\mathit{n}}_{pM}$, then a right-handed unit orthogonal active scalar field $L_p$ at point M on the curve $S_{fp}\left[\begin{array}{cccc}M;& \mathit{\alpha }& \mathit{\nu }={\mathit{n}}_{pM}\times \mathit{\alpha }& {\mathit{n}}_{pM}\end{array}\right]$. As M moves along $L_p$ with velocity ${\mathit{V}}_{Mp}={\displaystyle \frac{d{s}_p}{dt}}\mathit{\alpha }$ relative to the coordinate system $S_p\left[\begin{array}{cccc}{O}^{\left(p\right)};& x^{\left(p\right)}& y^{\left(p\right)}& z^{\left(p\right)}\end{array}\right]$ (workpiece coordinate system) which is fixed to the design surface ${\Sigma }_p$, the moving frame $S_{fp}$ moves and rotates around M. The equation for the motion of the moving frame $S_{fp}$ relative to the coordinate system $S_p$ (unit vector final velocity of the coordinate axis) is given by from the fundamental theorem of the surface theory of differential geometry, the equation for the motion of the moving frame $S_{fp}$ relative to the coordinate system $S_p$ (unit vector final velocity of the axes) is given by

$$\left[\begin{array}{c}\dot{\mathit{\alpha }}\\ \dot{\mathit{\nu }}\\ {\dot{\mathit{n}}}_{pM}\end{array}\right]={\displaystyle \frac{d{s}_p}{dt}}\left[\begin{array}{ccc}0& {\kappa }_{g\alpha }^{\left(p\right)}& {\kappa }_{n\alpha }^{\left(p\right)}\\ -{\kappa }_{g\alpha }^{\left(p\right)}& 0& {\tau }_{g\alpha }^{\left(p\right)}\\ -{\kappa }_{n\alpha }^{\left(p\right)}& -{\tau }_{g\alpha }^{\left(p\right)}& 0\end{array}\right]\left[\begin{array}{c}\mathit{\alpha }\\ \mathit{\nu }\\ {\mathit{n}}_{pM}\end{array}\right]$$

(3)

where the square is the square form of the angular velocity vector ${\omega }_p$ of the moving frame $S_{fp}$ rotating about the point M, written in vector form as

$${\mathit{\omega }}_p={\displaystyle \frac{d{s}_p}{dt}}\left({\tau }_{g\alpha }^{\left(p\right)}\mathit{\alpha }-{\kappa }_{n\alpha }^{\left(p\right)}\mathit{v}+{\kappa }_{g\alpha }^{\left(p\right)}{\mathit{n}}_{pM}\right)$$

(4)

${\tau }_{g\alpha }^{\left(p\right)}$, ${\kappa }_{n\alpha }^{\left(p\right)}$ are the short-range deflection and normal curvature of the surface ${\Sigma }_p$ at point M along the $\mathit{\alpha }$ direction, respectively, and ${\kappa }_{g\alpha }^{\left(p\right)}$ is the short-range curvature of the surface ${\Sigma }_p$ at point M along the $\mathit{\alpha }$ direction; both are motion-invariant parameters of the curve $L_p$ on surface ${\Sigma }_p$.

Theorems 1 and 2 are both derived from the theory of moving frames of surfaces in differential geometry, which describes the motion of rigid bodies (moving frames) along surfaces and curves on surfaces in terms of the invariant parameters of the motions of the surfaces and curves on surfaces in the natural coordinate system of the surfaces and curves on surfaces (moving frames), respectively. At the same time, the differential structure of surfaces and curves on surfaces is described from the perspective of rigid body kinematics, which establishes a direct connection between the motions of a rigid body (moving frames) along surfaces and curves on surfaces and the differential structure of surfaces and curves on surfaces, and provides a versatile and convenient method for describing the motion of tool surfaces that are developed into envelopes of the sculptured surface.

2.1.2. Speed of Relative Movement of the Tool Surface and Its Envelope

Surface ${\Sigma }_t$ is the tool surface, ${\Sigma }_p$ is the design surface to be machined and $L_p$ is a tool contact point trajectory on ${\Sigma }_p$. The two surfaces shown in Figure 1 and Figure 2 are “assembled” together, along with their associated moving frames, curves, and coordinate systems, so that the two surfaces are tangent to a point M, forming a schematic diagram of the machining of a sculptured surface ${\Sigma }_p$ by a tool surface ${\Sigma }_t$, as shown in Figure 3, the tool surface ${ \Sigma }_t$ wrapping around the surface ${\Sigma }_{\mathrm{g}}$ by its movement $\Psi$ with respect to the design surface ${\Sigma }_p$. In Figure 3, $L_{\mathrm{g}}$ is the envelope feature line that forms the envelope surface ${\Sigma }_{\mathrm{g}}$. Therefore, at any time when the tool surface ${\Sigma }_t$ is moving along the tool contact point trajectory $L_p$ with respect to the design surface ${\Sigma }_p$, the surface ${\Sigma }_t$ is tangent to $L_{\mathrm{g}}$ at ${\Sigma }_{\mathrm{g}}$, and $L_{\mathrm{g}}$ and $L_p$ intersect at point M. If the motion $\Psi$ of the tool surface ${\Sigma }_t$ relative to the design surface ${\Sigma }_p$ is appropriately controlled so that the tool surface always makes a single contact cut with the design surface ${\Sigma }_p$ at a point M on ${\Sigma }_p$, then the envelope surface ${\Sigma }_{\mathrm{g}}$ formed by the tool surface ${\Sigma }_t$ under the envelope, motion $\Psi$ is tangent to the design surface ${ \Sigma }_p$ along the tool contact point trajectory $L_p$. From Theorem 1, we know the angular velocity ${\omega }_p$ of the coordinate system $S_t$ with respect to the moving frame $S_{ft}$, from Theorem 2, we know the angular velocity ${\omega }_t$ of the moving frame $S_{fp}$ with respect to the coordinate system $S_p$, and from Figure 3, we know that the angular velocity of the moving frame $S_{fp}$ with respect to the moving frame $S_{ft}$ is ${\mathit{\omega }}_{tp}={\displaystyle \frac{d\Delta }{dt}}{\mathit{n}}_{tM}$. Therefore, the angular velocity of the envelope motion $\Psi$ of the surface ${\Sigma }_t$ with respect to the surface ${\Sigma }_{\mathrm{g}}$ (or ${ \Sigma }_p$) is

$$\begin{array}{l}{\mathit{\omega }}_{tp}={\mathit{\omega }}_p-{\mathit{\omega }}_t-{\displaystyle \frac{d\Delta }{dt}}{\mathit{n}}_{tM}=-\left({\omega }_{23}{\mathit{e}}_1-{\omega }_{13}{\mathit{e}}_2+{\omega }_{12}{\mathit{n}}_{tM}\right)\\ +\left({\tau }_{g\alpha }^{\left(p\right)}\mathit{\alpha }-{\kappa }_{n\alpha }^{\left(p\right)}\mathit{\nu }+{\kappa }_{g\alpha }^{\left(p\right)}{\mathit{n}}_{pM}\right){\displaystyle \frac{d{s}_p}{dt}}-{\displaystyle \frac{d\Delta }{dt}}{\mathit{n}}_{tM}\end{array}$$

(5)

From the same analysis, the velocity of the surface ${\Sigma }_t$ with respect to the surface ${\Sigma }_{\mathrm{g}}$ at point M is

$$\begin{array}{c}{\mathit{V}}_{Mtp}={\mathit{V}}_{Mp}-{\mathit{V}}_{Mt}=-\left({\displaystyle \frac{d{u}_{tM}}{dt}}{\mathit{e}}_1+{\displaystyle \frac{d{\upsilon }_{tM}}{dt}}{\mathit{e}}_2\right)+\\ {\displaystyle \frac{d{s}_p}{dt}}\mathit{\alpha }=\left[-\left({\displaystyle \frac{d{u}_{tM}}{d{s}_p}}{\mathit{e}}_1+{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}}{\mathit{e}}_2\right)+\mathit{\alpha }\right]{\displaystyle \frac{d{s}_p}{dt}}\end{array}$$

(6)

In the manner of Equation (6), the factor ${\displaystyle \frac{d{s}_p}{dt}}$ in Equation (5) is represented and referred to as the reference rate, without loss of generality, so that ${\displaystyle \frac{d{s}_p}{dt}}=1$, eliminating the time parameter t, the velocity function of the surface ${\Sigma }_t$ with respect to the surface ${\Sigma }_{\mathrm{g}}$ at any point $P\left(u_t,{\upsilon }_t\right)$ on the surface ${\Sigma }_t$ is given by

$$\begin{array}{c}{\mathit{V}}_{tp}\left(u_{tM},{\upsilon }_{tM},s_p,{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\nu }_{tM}}{d{s}_p}},u_t,{\upsilon }_t,\Delta ,{\displaystyle \frac{d\Delta }{d{s}_p}}\right)={\mathit{\omega }}_{tp}\times {\mathit{r}}_t^{\left(ft\right)}+{\mathit{V}}_{Mtp}\end{array}$$

(7)

where $r_t^{\left(ft\right)}$ is the position vector of point $P\left(u_t,{\upsilon }_t\right)$under the moving scale $S_{ft}$; $\mathit{\Delta }$ is the angle between $S_{ft}$ and $S_{fp}$, as shown in Figure 3.

Obviously, when the movement of the tool relative to the workpiece $\Psi$ is controlled according to Equations (5)–(7), the envelope surface ${\Sigma }_{\mathrm{g}}$ of the tool surface ${\Sigma }_t$ is tangent to the design surface ${\Sigma }_p$ along the trajectory line $L_p$ of the tool contact (because the normal vectors of the above three surfaces ${\Sigma }_t$, ${\Sigma }_{\mathrm{g}}$ and ${\Sigma }_p$ always coincide at any one of the tool contacts M), and therefore, each point M of the surface ${\Sigma }_{\mathrm{g}}$ on the curve ${ L}_p$ has the same short-range deflection ${\tau }_{g\alpha }^{\left(p\right)}$, normal curvature ${\kappa }_{n\alpha }^{\left(p\right)}$ and short-range curvature ${\kappa }_{g\alpha }^{\left(p\right)}$ along the direction of the unit tangent vector $\mathit{\alpha }$ of the curve $L_p$ as that of the design surface ${\Sigma }_p$. Since the tool surface ${ \Sigma }_t$ and the design surface ${\Sigma }_p$ are both continuous second-order microsurfaces, not only is the smooth continuity of the motion $\Psi$ described in Equations (5)–(7) guaranteed, but also the feasible region of the motion $\Psi$ are completely determined according to the structural requirements of the tool surface ${\Sigma }_t$ and the design surface ${\Sigma }_p$. Therefore, it is possible to get rid of the limitations of external conditions, such as the structure of machine tools and to establish a global optimization model for the motion of the tool relative to the workpiece according to the structural characteristics of the tool surface ${\Sigma }_t$ and the structural requirements of the design surface ${\Sigma }_p$.

In fact, the motion $\Psi$ also represents a transformation of the motion from the surface ${\Sigma }_t$ to surface ${\Sigma }_p$. If the tool surface ${\Sigma }_t$ is a simple surface (e.g., planes, cylindrical surfaces, conical surfaces and helicoids), then by controlling this transformation, it is possible to envelop (tensor into) a complex family of surfaces (spaces) ${ \Sigma }_{\mathrm{g}}$; by optimizing the kinematic transformation parameters, it is possible to select a surface ${ \Sigma }_{\mathrm{g}}$ in this family of surfaces such that it is tangent to the design surface ${\Sigma }_p$ along a specified tool contact point trajectory $L_p$, and the short-range deflection and normal curvature are equal at each point on the curve $L_p$ in any direction (i.e., the interpolation condition), and it approximates the design surface with a specified (minimum) fitting error. This is the principle of the envelopment-approximation.

2.1.3. Motion Transformations from a Surface ${\Sigma }_t$ to Its Envelope ${\Sigma }_{\mathrm{g}}$

From Figure 3, the coordinate transformation path from the surface ${\Sigma }_t$ to surface ${\Sigma }_{\mathrm{g}}$ is shown as the path above the dotted line in Figure 4, and the significance of the kinematic transformation chi-squared matrices $M_{ftt}$, $M_{fpft}$ and $M_{pfp}$ in Figure 4, which are completely described by the invariant parameters of the surface motion of the tool surface ${\Sigma }_t$ and the design surface ${\Sigma }_P$ along the tool contact point trajectory $L_p$, and the motion of the tool with respect to the workpiece during the development of the surface ${\Sigma }_{\mathrm{g}}$ has been designed using the method of embedded differential geometry based on the surface moving frame $\Psi$. In this way of describing the motion, $\Psi$ is completely determined by the differential geometry of the surface ${\Sigma }_t$ and the design surface ${\Sigma }_P$. The transformation matrix from the surface ${\Sigma }_t$ to the design surface ${\Sigma }_P$ is shown in Figure 3 and Figure 4. As can be seen in Figure 3 and Figure 4, the transformation matrix from surface ${\Sigma }_t$ to design surface ${\Sigma }_P$ is

$${\mathit{M}}_{pt}={\mathit{M}}_{pfp}{\mathit{M}}_{fpft}{\mathit{M}}_{ftt}$$

(8)

Of these,

$$\begin{array}{c}{\mathit{M}}_{ftt}=\left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1z}& -\left(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z}\right)\\ e_{2x}& e_{2y}& e_{2z}& -\left(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z}\right)\\ n_{tMx}& n_{tMy}& n_{tMz}& -\left(x_{tM}{n}_{tx}+y_{tM}{n}_{ty}+z_{tM}{n}_{tz}\right)\\ 0& 0& 0& 1\end{array}\right],\end{array}$$

$$\left.{\mathit{M}}_{fpft}=\left[\begin{array}{cccc}cos\Delta & -sin\Delta & 0& 0\\ sin\Delta & cos\Delta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right.\right]$$

$${\mathit{M}}_{pfp}=\left[\begin{array}{cccc}{\alpha }_x& {\nu }_x& n_{pMx}& x_{pM}\\ {\alpha }_y& {\nu }_y& n_{pMy}& y_{pM}\\ {\alpha }_z& {\nu }_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right]$$

It should be noted that,

$${\mathit{r}}_{tM}^{\left(t\right)}\left(u_{tM},{\upsilon }_{tM}\right)={\left[\begin{array}{cccc}{x}_{tM}\left(u_{tM},{\upsilon }_{tM}\right)& y_{tM}\left(u_{tM},{\upsilon }_{tM}\right)& z_{tM}\left(u_{tM},{\upsilon }_{tM}\right)& 1\end{array}\right]}^{\mathit{T}}$$

denotes the position vector function of any point M on the surface ${\Sigma }_t$ in the tool coordinate system $S_t$, which is conjugated to the point M on the curve $L_p$ on the surface ${\Sigma }_P$, with respect to the motion $\Psi$. ${\mathit{r}}_{pM}^{\left(p\right)}\left(s_p\right)={\left[\begin{array}{cccc}{x}_{pM}\left(s_p\right)& y_{pM}\left(s_p\right)& z_{pM}\left(s_p\right)& 1\end{array}\right]}^{\mathit{T}}$ is the vector parametric equation of the curve $L_p$ in the working coordinate system $S_p\left[O^{\left(p\right)}; x^{\left(p\right)} y^{\left(p\right)} z^{\left(p\right)}\right]$, which is also a vector function of the position of the point M on the curve $L_p$ on the surface ${\Sigma }_P$.

It is also exceedingly important to note that the coordinate transformation paths below the dotted line in Figure 4 are the coordinate transformation paths used to implement the tool envelope motion on a multi-axis CNC machine. Where $S_m\left[\begin{array}{cccc}{O}^{\left(m\right)};& x^{\left(m\right)}& y^{\left(m\right)}& z^{\left(m\right)}\end{array}\right]$ is the fixed coordinate system of the machine, fixed to the machine frame, and the kinematic transformation matrices $M_{mt}$ and $M_{pm}$ in this coordinate transformation path are determined only by the structural parameters of the machine for the specific surface to be machined and by the machining parameters set in the machining software. In Figure 4, the two coordinate transformation paths above and below the dashed line are completely equivalent, and one is directly described by the surface motion-invariant parameters of the tool surface ${\Sigma }_t$ and the design surface ${\Sigma }_P$ to be machined along the tool contact point trajectory $L_p$, which is universal and simple in tool envelope motion design; a description of the machine structure parameters of a specific surface and the machining parameters set in the machining software is practical and relevant to the realization of the tool envelope motion. This method of designing and implementing the tool motion relative to the workpiece during multi-axis CNC machining of sculptured surface, as shown in Figure 4, is the strength and characteristic of the methodology of this paper.

2.1.4. Parametric Description of Surface Motion Invariants for Tool Surface Envelopes

Equations (5)–(8) are the tool surface envelope motions $\Psi$ described by the invariant parameters of the tool surface and the design surface motions, which are applied to the tool surface envelope description to obtain the tool surface envelope feature lines and envelope surfaces described by the invariant parameters of the tool surface and the design surface motions.

From the engagement equations and the relative motion transformations, the equation for the envelope ${\Sigma }_{\mathrm{g}}$ of the surface ${ \Sigma }_t$ in the coordinate system $S_p\left[O^{\left(p\right)}; x^{\left(p\right)} y^{\left(p\right)} z^{\left(p\right)}\right]$, which is cemented to the design surface ${\Sigma }_P$ (workpiece coordinate system), is

$$\left\{\begin{array}{l}{\mathit{r}}_g^{\left(p\right)}={\mathit{M}}_{pt}{\mathit{r}}_t^{\left(t\right)}\\ {\mathit{N}}_t·{\mathit{V}}_{tp}=0\end{array}\right.$$

(9)

where, ${\mathit{r}}_t^{\left(t\right)}\left(u_t,{\upsilon }_t\right)={\left[\begin{array}{cccc}{x}_t\left(u_t,{\upsilon }_t\right)& y_t\left(u_t,{\upsilon }_t\right)& z_t\left(u_t,{\upsilon }_t\right)& 1\end{array}\right]}^T$ is the position vector function of any point P on the tool surface ${\Sigma }_t$ in the tool coordinate system $S_t$; $N_t\left(u_t,{\upsilon }_t\right)$ is the vector function of the unit normal vector at any point P on the surface ${\Sigma }_t$, which is any point on the surface ${\Sigma }_t$ other than point M, as shown in Figure 3.

The analyses of Equations (4)–(9) show that the tool envelope ${\Sigma }_{\mathrm{g}}$ determined by Equation (9) is described by the surface ${\Sigma }_t$, the curve $L_p$ and the corresponding kinematic invariant parameters $u_{tM}\left(s_p\right),{\upsilon }_{tM}\left(s_p\right)$ and $\Delta$ of ${\Sigma }_P$, which is tangent to the design surface ${\Sigma }_P$ along the tool contact trajectory curve${ L}_p$. This makes it possible to model the optimization of the tool motion in relation to the workpiece without taking into account external factors, such as the specific structure of the machine tool, so that the tool envelope meets the predefined local structural requirements (interpolation conditions) with respect to the design surface along the tool contact point trajectory curve $L_p$; at the same time, the error of the tool envelope surface relative to the design surface is globally minimized in a wide range of regions beyond the tool contact point trajectory curve $L_p$, thus achieving coordinated optimization of tool position planning from local to global. The following is discussed in the case that the envelope motion $\Psi$ satisfies Equations (5)–(8): (1) If the parameter $\Delta$ is set appropriately, the normal curvature ${\kappa }_{n\nu }^{\left(g\right)}$ of the tool surface envelope ${\Sigma }_{\mathrm{g}}$ in the perpendicular ($\mathit{\nu }$) direction of the tangent to the tool contact point trajectory ${ L}_p$ in the plane of common tangency between the tool surface ${ \Sigma }_t$ and the design surface ${\Sigma }_P$ can also be made equal to the normal curvature $K_{n\nu }^{\left(p\right)}$ of the design surface ${\Sigma }_P$ in the same direction, Combined with the condition that the tool envelope ${ \Sigma }_{\mathrm{g}}$ is tangent to the design surface ${\Sigma }_P$ along the tool contact point trajectory curve ${ L}_p$, we obtain the interpolation condition that the short-range deflection and normal curvature of the tool envelope ${\Sigma }_{\mathrm{g}}$ and the design surface ${\Sigma }_P$ are equal at any point on the curve ${ L}_p$ in any direction. (2) By adjusting the kinematic invariants $u_{tM}\left(s_p\right)$ and ${\upsilon }_{tM}\left(s_p\right)$, the tool envelope error with respect to the design surface is minimized over a wide range beyond the tool contact point trajectory curve $L_p$.

2.2. ${\Sigma }_g$ Approximates the Interpolated Boundary Conditions for ${\Sigma }_P$

In order to compare the error of a tool envelope with respect to a design surface, it is necessary to specify the relative spatial position of the surfaces. For different machining methods and different surface requirements, the tool surface envelope ${\Sigma }_{\mathrm{g}}$ (the actual machined surface) may have different positions with respect to the design surface ${\Sigma }_P$, which is actually a reflection of the local structure of the two surfaces. From a surface approximation point of view, this is the interpolated boundary condition of the tool envelope approximation with respect to the design surface, i.e., the constraints used in previous studies of this type [14,15].

As mentioned earlier, the problem of designing non-ball-end cutters into sculptured surfaces is essentially a class of surface interpolation approximation problems. Theoretically, the interpolation conditions can be specified for any important geometric structure and its dimensions that need to be accurately guaranteed. For example, when machining surfaces with single-point tangential contact, the geometrically localized structure along the tool contact point trajectory is usually specified as the interpolation condition that must be guaranteed precisely. Of course, some surface machining does not require a precise local structure, such as when machining a surface with the Multi-Point Method that allows overcutting [11], and the tool envelope intersects the design surface. In this case, the interpolating boundary condition for the tool envelope approximation with respect to the design surface is the intersection of the two surfaces. Therefore, the same approach can be used for strip-width-maximization machining for the surface envelope approximation described in this article. The following discussion focuses only on the case of machining sculptured surfaces with single-point tangential contact.

The local geometric structure along the tool contact path is interpolated so that the tool surface envelope is tangent to the design surface along the tool contact path, and the short-range deflection and normal curvature are equal at each point on the curve in any direction. This not only allows the actual machined surface to approximate the design surface within a given line width around the tool contact point trajectory with minimal error but also avoids local gouging and design surface damage caused by rear gouging.

As shown in Figure 3, $\Delta$ is the angle between the tool surface and the moving frame of the design surface (tool envelope surface), whose value at any moment can be theoretically independent of the geometric structure of the tool surface and the design surface (since the differential structure of the two surfaces can be unrelated) and can be defined arbitrarily to satisfy the important geometric structure and its dimensions (interpolation conditions) required for machining. It is shown that if the tool envelope ${\Sigma }_{\mathrm{g}}$ is tangent to the design surface ${ \Sigma }_P$ along the curve ${ L}_p$, and the short-range deflection and normal curvature are equal at each point on the curve ${ L}_p$ in any direction, as an interpolation condition for ${\Sigma }_{\mathrm{g}}$ to approximate the design surface ${\Sigma }_P$, then $\Delta$ can also be described by the surface ${ \Sigma }_t$, the curve ${ L}_p$ and the corresponding kinematic invariants of ${\Sigma }_P$, $u_{tM}\left(s_p\right)$ and ${\upsilon }_{tM}\left(s_p\right)$.

Figure 5 shows the situation when the tool surface ${ \Sigma }_t$ is developed to a point M on the tool contact point trajectory $L_p$ and the tool surface ${\Sigma }_t$ and the design surface ${\Sigma }_P$ are in the plane of common tangency. When the envelope motion $\Psi$ of the developed tool envelope surface ${\Sigma }_{\mathrm{g}}$ satisfies Equations (5)–(8), if the tool envelope ${\Sigma }_{\mathrm{g}}$ is tangent to the design surface ${\Sigma }_P$ along the curve ${ L}_p$, and the short-range deflection and normal curvature of both are equal at any point on the curve $L_p$ in any direction, then the normal curvature ${\kappa }_{nm}^{\left(i\right)}\left(i=t,g,p\right)$ and the short-range deflection ${\tau }_{gm}^{\left(i\right)}\left(i=t,g,p\right)$ of the surfaces ${\Sigma }_{\mathrm{g}}$ and ${\Sigma }_P$ are related to the surface along the tangent direction ${\alpha }_m$ of the eigenline $L_{\mathrm{g}}$ as follows.

$${\kappa }_{nm}^{\left(t\right)}\left(={\kappa }_{nm}^{\left(g\right)}\right)={\kappa }_{nm}^{\left(p\right)}$$

(10)

$${\tau }_{gm}^{\left(t\right)}\left(={\tau }_{gm}^{\left(g\right)}\right)={\tau }_{gm}^{\left(p\right)}$$

(11)

In addition, from the Euler-Bertrand formula for the tool surface ${ \Sigma }_t$, the

$${\kappa }_{nm}^{\left(t\right)}={\kappa }_{n1}^{\left(t\right)}{cos}^2{\phi }_1+2{\tau }_{g1}^{\left(t\right)}sin{\phi }_{1}cos{\phi }_1+{\kappa }_{n2}^{\left(t\right)}{sin}^2{\phi }_1$$

(12)

$${\tau }_{gm}^{\left(t\right)}=-\left({\kappa }_{n1}^{\left(t\right)}-{\kappa }_{n2}^{\left(t\right)}\right)sin{\phi }_{1}cos{\phi }_1+{\tau }_{g1}^{\left(t\right)}\left({cos}^2{\phi }_1-{sin}^2{\phi }_1\right)$$

(13)

where ${\tau }_{gi}^{\left(t\right)}$, ${\kappa }_{ni}^{\left(t\right)}$ are the short-range deflection and normal curvature of the surface${ \Sigma }_t$ at point M along the direction ${\mathit{e}}_i\left(i=\mathrm{1,2}\right)$, respectively.

For the design surface ${ \Sigma }_P$, which is the tool surface envelope ${ \Sigma }_{\mathrm{g}}$ (interpolating boundary condition), the

$${\kappa }_{nm}^{\left(p\right)}={\kappa }_{n\alpha }^{\left(p\right)}{cos}^2{\phi }_2+2{\tau }_{g\alpha }^{\left(p\right)}sin{\phi }_{2}cos{\phi }_2+{\kappa }_{n\nu }^{\left(p\right)}{sin}^2{\phi }_2$$

(14)

$${\tau }_{gm}^{\left(p\right)}=-\left({\kappa }_{n\alpha }^{\left(p\right)}-{\kappa }_{n\nu }^{\left(p\right)}\right)sin{\phi }_{2}cos{\phi }_2+{\tau }_{g\alpha }^{\left(p\right)}\left({cos}^2{\phi }_2-{sin}^2{\phi }_2\right)$$

(15)

where ${\kappa }_{n\nu }^{\left(p\right)}$ is the normal curvature of the design surface ${\Sigma }_P$ and also the tool surface envelope ${\Sigma }_{\mathrm{g}}$ in the perpendicular direction ($\mathit{\upsilon }$ direction) of the tangent to the tool contact point trajectory $L_p$ in the plane of common tangency between the tool surface ${\Sigma }_t$ and the design surface${ \Sigma }_P$. As shown in Figure 5, there are

$$\Delta \left(u_{tM},{\upsilon }_{tM},s_p\right)={\phi }_2\left(u_{tM},{\upsilon }_{tM}\right)-{\phi }_1\left(s_p\right)$$

(16)

From Equations (10)–(15), we can find ${\phi }_1\left(u_{tM},{\upsilon }_{tM}\right)$, ${\phi }_2\left(s_p\right)$. Substituting them into Equation (16), $\Delta$ can be expressed as a function of the embedded parameters $u_{tM},$${\upsilon }_{tM},$$s_p, \mathrm{i}.\mathrm{e}.,$ when the parameters $\Delta$ to be determined in the tool surface envelope motion equation from Equations (5)–(8) satisfy Equations (10)–(16), the tool envelope ${\Sigma }_{\mathrm{g}}$ is tangent to the design surface ${\Sigma }_P$ along the curve $L_p$, and the short-range deflection and normal curvature of both are equal to each point of the curve $L_p$ in any direction.

It is necessary to further explain that when the tool surface according to Equations (5)–(8) for the envelope motion, the tool envelope ${\Sigma }_{\mathrm{g}}$ is tangent to the design surface ${\Sigma }_P$ along the curve $L_p$, that is, the tool envelope ${\Sigma }_{\mathrm{g}}$ on the curve $L_p$ at each point M along the curve $L_p$ unit tangent vector $\mathit{\alpha }$ direction and the design surface${ \Sigma }_P$ have equal short-range deflection ${\tau }_{g\alpha }^{\left(p\right)}$, normal curvature ${\kappa }_{n\alpha }^{\left(p\right)}$ and short-range curvature ${\kappa }_{g\alpha }^{\left(p\right)}$; when the pending parameters $\Delta$ in Equations (5)–(8) satisfy Equations (10)–(16), the tool envelope ${\Sigma }_{\mathrm{g}}$ also satisfies the interpolation boundary condition of the envelope approximation, i.e., the tool envelope ${\Sigma }_{\mathrm{g}}$ is tangent to the design surface ${\Sigma }_P$ along the curve $L_p$, and the short-range deflection and normal curvature of the two are equal at any point on the curve $L_p$ in any direction.

Substituting $\Delta$ obtained by the above method into Equations (7) and (8), we obtain the velocity function of the surface ${\Sigma }_t$ with respect to the surface ${\Sigma }_{\mathrm{g}}$ at any point $P\left(u_t,{\upsilon }_t\right)$ and the transformation matrix function from the surface ${\Sigma }_t$ to surface ${\Sigma }_{\mathrm{g}}$. Substituting Equations (7) and (8) into Equation (9) and eliminating the parameter $u_t$, we obtain the equation of the vector function of the enveloping surface ${\Sigma }_{\mathrm{g}}$ of the tool surface ${\Sigma }_t$ under the motion $\Psi$, in the coordinate system $S_p\left[O^{\left(p\right)};x^{\left(p\right)} y^{\left(p\right)} z^{\left(p\right)}\right]$ fixed with the design surface ${\Sigma }_P$ (the workpiece coordinate system).

$${\mathit{r}}_{\mathrm{g}}^{\left(p\right)}={\mathit{r}}_{\mathrm{g}}\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)$$

(17)

From Equation (17), when the tool surface${ \Sigma }_t$ moves relative to the design surface ${\Sigma }_P$ according to the motion $\Psi$ described in Equations (7) and (8), the enveloping surface ${\Sigma }_{\mathrm{g}}$ is a family of surfaces with ($s_p,{\upsilon }_t$) as the curvilinear coordinate parameter, and all surfaces in this family of surfaces are tangent to the design surface ${\Sigma }_P$ along the tool contact point trajectory curve $L_p$, and the short-range deflection and normal curvature of the two surfaces are equal at any point on the curve $L_p$ in any direction; In the surface area beyond the tool contact point trajectory${ L}_p$, the surface ${\Sigma }_{\mathrm{g}}$ in the surface, family has a different error from the design surface ${\Sigma }_P$. In Equation (17), the parameters ${\upsilon }_t$ and $s_p$ are the curvilinear coordinates of the surface ${\Sigma }_{\mathrm{g}}$, and $u_{tM}\left(s_p\right)$ and ${\upsilon }_{tM}\left(s_p\right)$ are the variable parameters that form the surface family. Therefore, it is possible to determine the motion-invariant parameters $u_{tM}\left(s_p\right)$ and ${\upsilon }_{tM}\left(s_p\right)$ by further optimizing the tool motion (7) and (8) with respect to the workpiece, i.e., by determining the conjugate curve of the tool contact point trajectory ${ L}_p$ on the tool surface ${ \Sigma }_t$ under the envelope motion $\Psi$, in order to achieve a global minimum of the error of the tool envelope surface with respect to the design surface in a wide range of regions other than the tool contact point trajectory curve ${ L}_p$; From this family of surfaces, a surface ${\Sigma }_{\mathrm{g}}$ that minimizes the error with the design surface ${\Sigma }_P$ is selected as the envelope of the tool surface ${\Sigma }_t$ (the actual machining surface). In the following, we describe the method of estimating the error between the envelope surface and the design surface in the theory of envelope−approximation, which serves as the theoretical basis for the practical application of machining.

2.3. Error When the Envelope ${\Sigma }_g$ of the Tool Surface ${ \Sigma }_t$ Approaches ${ \Sigma }_P$

For graphical convenience, it is assumed that the tool surface envelope ${\Sigma }_{\mathrm{g}}$, shown in Figure 6, has been selected from the family of envelope surfaces shown in Equation (17), which must be tangent to the design surface ${ \Sigma }_P$ along the tool contact point trajectory curve $L_P$, and that the short-range deflection and normal curvature of the two surfaces are equal at each point on the curve ${ L}_P$ in each direction. $L_{\mathrm{g}}$ is the envelope eigenline of ${\Sigma }_{\mathrm{g}}$ over any point M on $L_P$, and ${\mathcal{Q}}_{\mathrm{g}}$ is an arbitrary point on it through which a plumb line of surface ${\Sigma }_P$ intersects surface ${\Sigma }_P$ at ${\mathcal{Q}}_p$. Then the deviation ${\delta }_k$ of surface ${\Sigma }_{\mathrm{g}}$ from surface ${\Sigma }_P$ at ${\mathcal{Q}}_{\mathrm{g}}$ is

$${\delta }_k{\mathit{N}}_p^{\left(p\right)}={\delta }_k{\mathit{N}}_p\left(u_p,{\upsilon }_p\right)={\mathit{Q}}_p{\mathit{Q}}_{\mathrm{g}}={\mathit{r}}_{\mathrm{g}}^{\left(p\right)}-{\mathit{r}}_p^{\left(p\right)}={\mathit{r}}_{\mathrm{g}}\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)-{\mathit{r}}_p\left(u_p,{\upsilon }_p\right)$$

(18)

Where ${\mathit{N}}_p\left(u_p,{\upsilon }_p\right)$ is the vector parametric equation of the unit normal vector at any point ${\mathcal{Q}}_p$ on the surface${ \Sigma }_P$. Obviously, if the point ${\mathcal{Q}}_{\mathrm{g}}$ is chosen, i.e., the vector ${\mathit{r}}_g^{\left(p\right)}$ is determined according to Equation (17), then the three scalar equations expressed in Equation (18) can be solved for the parameters $u_p$, ${\upsilon }_p$ and ${\delta }_k$ and expressed as

$$u_p=u_p\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)$$

(19)

$${\upsilon }_p={\upsilon }_p\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)$$

(20)

$${\delta }_k={\delta }_k\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)$$

(21)

Equations (19) and (20) determine the mapping relationship between the curvilinear coordinate $\left(s_p,{\upsilon }_t\right)$ of any point ${\mathcal{Q}}_{\mathrm{g}}$ on the tool surface envelope ${\Sigma }_{\mathrm{g}}$ and the curvilinear coordinate $\left(u_p,{\upsilon }_p\right)$ of point ${\mathcal{Q}}_p$ on the design surface ${ \Sigma }_P$, and Equation (21) determines the deviation ${\delta }_k$ between points ${\mathcal{Q}}_{\mathrm{g}}$ and ${\mathcal{Q}}_p$. Equations (19)–(21) actually determine the mapping relationship between the actual machining surface, i.e., the envelope surface ${\Sigma }_{\mathrm{g}}$ of the tool surface and the machining target surface, i.e., the design surface ${\Sigma }_P$. This provides a theoretical basis for the family of envelope surfaces (space ${\Sigma }_{\mathrm{g}}$ of the tool surface to approximate the machining target surface ${ \Sigma }_P$. The principle of envelope approximation of the tool surface ${ \Sigma }_t$ to the design surface ${ \Sigma }_P$ by the motion transformation, $\Psi$ has thus been fully explained. In the following, the principle of envelope approximation is applied to obtain the optimized tool motion with the optimization objectives of maximizing machining efficiency and minimizing machining error.

3. Mathematical Model of 2 Types of Problems in Strip-Width-Maximization Machining with Non-Ball-End Cutter

In the engineering practice of strip-width-maximization machining, the control problems of machining accuracy and machining efficiency have always been two kinds of core problems in the field of CNC machining technology. Among them, CNC machining accuracy (error) calculation and control problems can be solved by Equation (21), and CNC machining efficiency calculation and control problems are related to the calculation of CNC machining line width, but the line width is not a standard formal mathematical concept. In the following, the line width in CNC machining is defined as a kind of surface implicit motion-invariant, and its calculation method is given; then, the optimal control model of tool motion is established with the aim of optimal accuracy and efficiency of CNC machining.

3.1. Definition and Calculation of Line Width in CNC Machining

Figure 7 shows the case of single-contact strip-width-maximization machining of a sculptured surface. At each tool contact M, the tool surface expands into an envelope (feature line) ${ L}_g$, For the tool surface envelope ${\Sigma }_{\mathrm{g}}$, each tool contact M (with curvilinear coordinates${(s}_p,{\upsilon }_{tM})$on the tool contact point trajectory ${ L}_P$ has a unique value of the ${ S}_P$ coordinate, which corresponds to the unique envelope feature line $L_{\mathrm{g}}$, i.e., the ${\upsilon }_t$ line of surface ${\Sigma }_{\mathrm{g}}$. Let ${\mathcal{Q}}_{\mathrm{g}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{g}}^{\left(2\right)}$ be two points on the envelope feature line $L_{\mathrm{g}}$ located on either side of the tool contact M, and their corresponding curve coordinates can be defined as $\left(s_p,{\upsilon }_{tM}+{\upsilon }_{t1}\right)$ (located to the right of the point M ${\upsilon }_{t1}\ge 0$), $\left(s_p,{\upsilon }_{tM}-{\upsilon }_{t2}\right)$ (located to the left of the point M ${\upsilon }_{t2}\ge 0$), the curve $L_{\mathrm{g}\mathrm{p}}$, points ${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$ are the curves $L_{\mathrm{g}}$, the projections of points ${\mathcal{Q}}_{\mathrm{g}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{g}}^{\left(2\right)}$ on the surface ${ \Sigma }_P$, respectively. If the distance ${\delta }_k$ between points ${\mathcal{Q}}_{\mathrm{g}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{g}}^{\left(2\right)}$ on $L_{\mathrm{g}}$ to the surface ${\Sigma }_P$ is not greater than the specified error $\epsilon$, then the arc length $s_p^{\left(1\right)\left(2\right)}$ of the curve $L_{\mathrm{g}\mathrm{p}}$ on the surface ${\Sigma }_P$ between points ${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$ is the so-called line width of strip-width-maximization machining when the tool develops the surface at tool contact M under the error $\epsilon$. Evidently, the line width can only be measured with the specified machining accuracy. When the machining error ${\delta }_k\le \epsilon$ is specified, the line width is not the same for each of the tool contacts at M to form a surface.

As shown in Figure 7, the curve $L_{gp}$, points ${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$ are the projections of the curve ${ L}_g$, points ${\mathcal{Q}}_{\mathrm{g}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{g}}^{\left(2\right)}$ on the surface ${ \Sigma }_P$. This projection relationship is actually determined by Equations (19) and (20). The curvilinear coordinates of the projection point ${\mathcal{Q}}_{\mathrm{p}}$ (${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$) of any point ${\mathcal{Q}}_{\mathrm{g}}$ (including points ${\mathcal{Q}}_{\mathrm{g}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{g}}^{\left(2\right)}$) of surface ${\Sigma }_{\mathrm{g}}$ onto surface ${ \Sigma }_P$ can be calculated by Equations (19) and (20), and by substituting Equations (19) and (20) into the parametric equations ${\mathit{r}}_p={\mathit{r}}_p\left(u_p,{\upsilon }_p\right)$ of the design surface ${ \Sigma }_P$ we obtain the equations of the projections of the surface ${\Sigma }_{\mathrm{g}}$ onto surface ${ \Sigma }_P$.

$$\begin{array}{c}{\mathit{r}}_{\mathrm{g}p}=\left(u_{tM},{\upsilon }_M,{\displaystyle \frac{d{u}_M}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)=\end{array}\left(u_p\left(u_M,{\upsilon }_M,{\displaystyle \frac{d{u}_M}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_M}{d{s}_p}},s_p,{\upsilon }_t\right),{\upsilon }_p\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_t\right)\right)$$

(22)

where the fixed parameter${ S}_P$ is not changed, the parametric equation of the curve $L_{\mathrm{g}p}$ is obtained. The arc length $s_p^{\left(1\right)\left(2\right)}$ of the curve${ L}_{\mathrm{g}p}$ between points ${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$, i.e., the line width is

$$s_p^{\left(1\right)\left(2\right)}={\int }_{{\upsilon }_{tM}-{\upsilon }_{t2}}^{{\upsilon }_{tM}+{\upsilon }_{tl}}\left|{\displaystyle \frac{\partial {\mathit{r}}_{\mathrm{g}p}}{\partial u_t}}\right|\mathrm{d}{\nu }_t$$

(23)

In practice, the curvilinear coordinates $\left(u_p,{\upsilon }_p\right)$ of the design surface ${ \Sigma }_P$ are usually orthogonal. In this case, Equation (23) can be simplified as

$$s_p^{\left(1\right)\left(2\right)}={\int }_{{\upsilon }_{tM}-{\upsilon }_{t2}}^{{\upsilon }_{tM}+{\upsilon }_{tl}}\sqrt{E_p{\left({\displaystyle \frac{\partial u_p}{\partial {\upsilon }_t}}\right)}^2+G_p{\left({\displaystyle \frac{\partial {\upsilon }_p}{\partial {\upsilon }_t}}\right)}^2}d{\upsilon }_t$$

(24)

where $E_p={\displaystyle \frac{\partial {\mathit{r}}_p}{\partial u_p}}$, $G_p={\displaystyle \frac{\partial {\mathit{r}}_p}{\partial {\upsilon }_p}}$, they are the first class of fundamental quantities of the surface ${ \Sigma }_P$. When the error of strip-width-maximization machining is specified not to exceed $\epsilon$, from Equation (21), ${\nu }_{t1}$ and ${\nu }_{t2}$ in the upper and lower bounds of the integrals of Equations (23) and (24) are the solutions of Equations (25) and (26), respectively (as shown in Figure 7).

$${\delta }_k\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_{tM}+{\upsilon }_{t1}\right)=\epsilon$$

(25)

$${\delta }_k\left(u_{tM},{\upsilon }_{tM},{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},s_p,{\upsilon }_{tM}-{\upsilon }_{t2}\right)=\epsilon$$

(26)

where $\epsilon$ is the machining error required for machining. Thus, the line width is defined as the arc length of the curve on the design surface ${\Sigma }_P$, and is calculated according to Equation (23). However, it cannot be used as an objective function for the overall optimization of tool motion because the line widths of the tool are not equal when it is extended over a surface at each tool contact M.

3.2. An Optimization Model for Tool Motion Efficiency

In engineering practice, there are two main types of single-contact CNC machining problems for sculptured surfaces, which are related to the surface approximation problems described in the previous section: the first type aims at the highest machining efficiency. As shown in Figure 7, given the error requirements ${\delta }_k\le \epsilon$ of the actual machining surface (tool surface envelope ${\Sigma }_{\mathrm{g}}$) and the design surface ${\Sigma }_P$, under the condition of satisfying the deviation requirements, the tool surface envelope ${\Sigma }_{\mathrm{g}}$ is approximated to the design surface ${\Sigma }_P$ within the maximum range by determining the appropriate motion transformation $\Psi$, i.e., the highest machining efficiency problem. The mathematical model for this type of problem is described as follows:

Since the line widths of the tool to form a surface at each tool contact M cannot be equal if the accuracy condition ${\delta }_k\le \epsilon$ is satisfied, the optimization objective of maximizing the line widths at each tool contact may result in discontinuous tool movements, which cannot be achieved in practical machining. However, optimizing for the maximum area of the surface formed by the set of curves $L_{\mathrm{g}p}$ within the row widths of all tool contacts (as shown in the shaded area of Figure 7) ensures that a continuous and smooth tool motion is obtained [25]. Therefore, given the error requirements for two surfaces in case ${\mathsf{\delta }}_{\mathrm{k}}\le \mathsf{\epsilon }$, the objective function of the mathematical optimization model, aiming to approximate the design surface ${\Sigma }_P$ with the maximum projected area of the tool surface envelope ${\Sigma }_{\mathrm{g}}$, is formulated as follows.

$$S=max\left({\int }_{s_{p1}}^{s_{P2}}d{s}_p{\int }_{{\upsilon }_{tM}-{\upsilon }_{t2}}^{{\upsilon }_{tM}+{\upsilon }_{tI}}\sqrt{\left[{\left({\mathit{r}}_{gp}\right)}_{{\upsilon }_t}·{\left({\mathit{r}}_{gp}\right)}_{{\upsilon }_t}\right]\left[{\left({\mathit{r}}_{gp}\right)}_{s_p}·{\left({\mathit{r}}_{gp}\right)}_{s_p}\right]-{\left[{\left({\mathit{r}}_{gp}\right)}_{{\upsilon }_t}·{\left({\mathit{r}}_{gp}\right)}_{s_p}\right]}^2}d{\upsilon }_t\right)$$

(27)

where ${\left(r_{gp}\right)}_{{\nu }_t}={\displaystyle \frac{\partial r_{gp}}{\partial {\upsilon }_t}},{\left(r_{gp}\right)}_{s_p}={\displaystyle \frac{\partial r_{gp}}{\partial s_p}}$; $s_{p1}$ and $s_{p2}$ are the arc length coordinates of the start point M₁ and end point M₂ of the tool contact point trajectory, respectively.

To ensure that the surface ${\Sigma }_{\mathrm{g}}$ approaches the target surface ${ \Sigma }_P$ from outside the target surface ${ \Sigma }_P$, and to avoid rear gouging, constraints must be added.

$${\delta }_k={\delta }_k\left(u_{tM},\hspace{0.25em}{\upsilon }_{tM},\hspace{0.25em}{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},\hspace{0.25em}{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},\hspace{0.25em}{s}_p,\hspace{0.25em}{\upsilon }_t\right)\ge 0$$

(28)

Equations (25), (26) and (28) are the constraints of such optimization models.

3.3. An Optimization Model for Tool Motion Precision

The second type of problem in single-contact CNC machining of sculptured surfaces is minimizing the machining error. As shown in Figure 7, given the surface boundaries $L_{p1}$ and $L_{p2}$ on the design surface ${\Sigma }_P$, it is necessary to make the tool surface envelope ${\Sigma }_{\mathrm{g}}$ approximate the design surface ${\Sigma }_P$, with the minimum error by determining the appropriate kinematic variations $\Psi$ within the boundaries, i.e., the maximum machining accuracy problem. The mathematical model for this type of problem is described as follows:

Let points ${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$ be any two points of $L_{p1}$ and $L_{p2}$ on the boundary of the design surface ${\Sigma }_P$ on either side of $L_p$. The normals of the design surface ${\Sigma }_P$ at points ${\mathcal{Q}}_{\mathrm{P}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{P}}^{\left(2\right)}$ intersect the envelope ${\Sigma }_{\mathrm{g}}$ at points ${\mathcal{Q}}_{\mathrm{g}}^{\left(1\right)}$ and ${\mathcal{Q}}_{\mathrm{g}}^{\left(2\right)}$, respectively, and the deviation of the surface ${\Sigma }_{\mathrm{g}}$ from the target surface ${\Sigma }_P$ at any one of its points can be found in Equation (18). In single-point tangent machining, the deviation of the surface ${\Sigma }_{\mathrm{g}}$ from the design surface ${\Sigma }_P$ at any point along the envelope feature line $L_{\mathrm{g}}$ increases monotonically with the distance from the point to the tool contact, and the deviations ${\delta }_{kmax}^{\left(1\right)}$ and ${\delta }_{kmax}^{\left(2\right)}$ are greatest at the two points ${\mathcal{Q}}_P^{\left(1\right)}$ and ${\mathcal{Q}}_P^{\left(2\right)}$ on the boundary of the design surface ${\Sigma }_P$ (as shown in Figure 7). By simultaneously minimizing ${\delta }_{kmax}^{\left(1\right)}$ and ${\delta }_{kmax}^{\left(2\right)}$ can we ensure that the surface ${\Sigma }_{\mathrm{g}}$ approaches the target surface ${\Sigma }_P$ with the highest accuracy. However, the deviations ${\delta }_{kmax}^{\left(1\right)}$ and ${\delta }_{kmax}^{\left(2\right)}$ of each feature line are not equal, so if the deviation ${\delta }_{kmax}^{\left(1\right)}+{\delta }_{kmax}^{\left(2\right)}$ of each feature, line is controlled one by one to minimize the deviation ${\delta }_{kmax}^{\left(1\right)}+{\delta }_{kmax}^{\left(2\right)}$, the tool motion that satisfies the control conditions may be discontinuous and smooth. Therefore, in order to obtain a continuous and smooth tool motion, the optimization objective should be to minimize the sum of the mean values of the deviations ${\delta }_{kmax}^{\left(1\right)}$ and ${\delta }_{kmax}^{\left(1\right)}$ along the surface boundaries $L_{p1}$ and ${ L}_{p2}$, respectively. Therefore, the optimal model for the approximation of the target surface ${\Sigma }_P$ by the envelope ${\Sigma }_{\mathrm{g}}$ is as follows:

Objective function

$$F=min\left({\displaystyle \frac{{\int }_{L_{p1}}{\delta }_{kmax}^{\left(1\right)}d{s}_{p1}}{{\int }_{L_{p1}}d{s}_{p1}}}+{\displaystyle \frac{{\int }_{L_{p2}}{\delta }_{kmax}^{\left(2\right)}d{s}_{p2}}{{\int }_{L_{p2}}d{s}_{p2}}}\right)$$

(29)

Constraint condition

$${\delta }_k={\delta }_k\left(u_{tM},\hspace{0.25em}{\upsilon }_{tM},\hspace{0.25em}{\displaystyle \frac{d{u}_{tM}}{d{s}_p}},\hspace{0.25em}{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}},\hspace{0.25em}{s}_p,\hspace{0.25em}{\upsilon }_t\right)\ge 0$$

(30)

The constraint in Equation (30) ensures that the surface ${\Sigma }_{\mathrm{g}}$ approaches the design surface ${\Sigma }_P$ from outside the design surface${ \Sigma }_P$ to avoid overcutting. The $s_{p1}$ and $s_{p2}$ in Equation (29) are the arc length coordinates of the boundary curves $L_{p1}$ and $L_{p2}$, respectively.

The functions ${ u}_{tM}\left(s_p\right)$ and ${ \upsilon }_{tM}\left(s_p\right)$ can be determined by solving the optimization model consisting of Equations (25)–(28) or Equations (29) and (30), and the equations of motion and transformation matrix equations of a continuous smooth tool relative to the workpiece satisfying the optimization objective can be obtained from Equations (7) and (8).

It is important to note that the optimization model described by Equation (27) or Equation (29) is a class of generalized extremal models with functions $u_{tM}\left(s_p\right)$ and ${\upsilon }_{tM}\left(s_p\right)$ and their first-order derivatives${\displaystyle \frac{d{u}_{tM}}{d{s}_p}},{\displaystyle \frac{d{\upsilon }_{tM}}{d{s}_p}}$as the functions to be determined, rather than a functional extremal model. After satisfying the interpolation condition of the approximation between the tool envelope and the design surface, it can be used to optimize the tool movement relative to the workpiece. The variables controlling the total error between the tool envelope and the design surface are used to determine the two functions $u_{tM}\left(s_p\right)$ and ${\upsilon }_{tM}\left(s_p\right)$ of the tool position, i.e., the conjugate curves of the tool contact point trajectory $L_p$ on the tool surface ${\Sigma }_t$ under the enveloping motion $\Psi$, which actually determines the relative positions of the tool and the workpiece (the design surface) at each instant and is the essence of the motion optimization model mentioned above. From the theory of bounded linear operators of generalized functional analysis, it can be proved that the functions $u_{tM}\left(s_p\right)$ and $u_{tM}\left(s_p\right)$ satisfying Equation (27) or Equation (29) are continuously smooth [22]. This guarantees that the equations of motion of the tool relative to the workpiece and the equations of the transformation matrix are also continuously smooth.

It should be noted that the aforementioned functional extremum model for maximizing strip width in the machining of sculptured surfaces is primarily suited for large, monotonically convex (or concave) surfaces with small processing errors and line widths. However, surfaces that exhibit both concave and convex features can be segmented into monotonically convex and concave parts, which can then be processed individually according to the aforementioned method. In more complex surface machining scenarios, such as global collisions, machining errors, or line width distortions, it is necessary to incorporate constraints into the functional extremum model based on specific circumstances. Nonetheless, the method described in this paper, which utilizes surface motion invariants to describe and control the tool motion relative to the workpiece within a naturally moving frame, remains applicable. Due to space constraints, the details will not be reiterated here.

4. Simulation Examples

Example 1: Taking a propeller blade surface as an example, simulate the machining in Matlab and compare the calculation results with the machining method (second-order Taylor approximation) in the literature [13] to verify the validity and accuracy of the envelope approximation principle.

The design surface ${\Sigma }_P$ (propeller blade) is a free-form surface with discrete points, as shown in Figure 8; the * type line represents the tool contact point trajectory $L_p$, which is the interpolation condition for the envelope approximation: the tool envelope ${\Sigma }_{\mathrm{g}}$ obtained by the method of this paper is tangent to ${\Sigma }_P$ along the tool contact point trajectory curve $L_p$, and the short-range deflection and normal curvature of both are equal to each point along the curve $L_p$ in any direction (the interpolation condition); The • type lines on either side of $L_p$ represent the boundary lines $L_{p1}$ and $L_{p2}$, respectively, which are two parametric curves equidistant from the tool contact point trajectory on the design surface ${\Sigma }_P$. Their distance from the arc length (i.e., line width) is 12.23 mm. The Tapered surfaces Tool half taper angle $\varphi$ = 22.5 mm, the bottom radius R = 150 mm and the height H = 32 mm. In order to minimize machining errors, these tool surface parameters were obtained as follows: (1) avoid local interference between the tool surface and the design surface; (2) minimize the deviation of the tool surface from the design surface when the center point of the tool surface is tangent to the design surface ${\Sigma }_P$ at the center (point of interest) M of the tool contact point trajectory $L_p$. Figure 9 shows the simulated machining results, and Figure 10 shows the situation near the tool contact point M during machining: The envelope feature line $L_{\mathrm{g}}$ intersects the tool contact point trajectory $L_p$ and its conjugate line $L_t$ on the tool surface at the tool contact M, indicating that the tool and the workpiece are in the correct position relative to each other during the strip-width-maximization machining. Figure 11 shows the variation in the tool envelope to the design surface along the selected machining area boundary lines $L_{p1}$ and ${ L}_{p2}$, showing that the machining error is different at each point along the selected boundary lines. The machining error of the same tool position on the two boundary lines is generally not guaranteed to be minimized simultaneously, but the sum of the two varies slightly. Figure 12 shows the error distribution between the tool envelope and the design surface obtained by the second-order Taylor approximation [13] and the surface envelope approximation method for a given strip-width-maximization machining range in the form of a topological map. Figure 13 compares the variation in the machining bandwidth of the two methods for the same machining error ($\epsilon$ = 0.01 mm) when the design surface is formed along the same tool contact point trajectory. Figure 12 and Figure 13 show that compared to the second-order Taylor method, the envelope approximation method improves significantly in terms of line width and accuracy for the same machining error and machining area requirements. This is mainly due to the fact that the second-order Taylor method uses a second-order approximation to the design surface and expresses the surface structure near the tool contact as an elliptical paraboloid, which does not take into account the actual approximation of the surface in the line width region beyond the tool contact point; On the other hand, the envelope approximation method follows the real geometry of the design surface and calculates the error between the tool envelope and the design surface in the area outside the tool contact path, based on the interpolation conditions on the tool contact path. The tool motion relative to the workpiece is determined with the aim of minimizing the error or maximizing the line width, and the structural parameters of the tool surface are designed and determined with the aim of minimizing the deviation from the surface being machined. Therefore, the envelope approximation method has clear advantages over the second-order Taylor method in terms of machining accuracy and efficiency.

Example 2: A simulation example of machining a helical surface with a circular tool from the literature [15] is used to verify the correctness and effectiveness of the surface envelope approximation method. The parametric equation for the helical surface is

$$S\left(u,v\right)=\left[\begin{array}{c}ucosv-\left(a+u\sqrt{{\displaystyle \frac{a+u}{a-u}}}\right)sinv\\ usinv+\left(a+u\sqrt{{\displaystyle \frac{a+u}{a-u}}}\right)cosv\\ -pv\end{array}\right]$$

(31)

where, $-6 \mathrm{m}\mathrm{m}\le u\le 8 \mathrm{m}\mathrm{m},0\le v\le 1$.

The helical surface parameters are a = 10 mm and p = 100 mm, and the parametric curve with u = −2 is chosen as the cutter contact curve. The design surface and its tool contact path are shown in Figure 14, where the radius of the torus of the circular cutter is R = 10 mm, and the radius of the corner is 2.5 mm. Figure 15 shows the simulation results of machining the design surface with the tool position and posture determined by the envelope approximation method.

In order to verify the machining accuracy and efficiency of the method in this paper, firstly, according to the type 1 optimization model (i.e., with the maximum line width as the optimization objective), the maximum machining line widths are calculated when the given machining error is $\epsilon$ = 0.005 mm and $\epsilon$ = 0.01 mm, respectively, and compared with the existing machining methods. The calculation results show that the maximum width of the line width is 6.63 mm when the tool contact point is located at the curvilinear coordinates (2, 0.376) of the helical surface to be machined and the direction of the tool axis in the coordinate system of the surface to be machined is (0.9374, 0.2811, −0.2057) with a machining error of $\epsilon$ = 0.01 mm. In the case of a machining error of $\epsilon$ = 0.005 mm, the maximum width of the line width is 5.87 mm when the tool contact point is located at the curvilinear coordinates of the helical surface to be machined (2, 0.563), and the direction of the tool axis in the coordinate system of the surface to be machined is (0.8968, 0.3301, −0.245). Then, according to the type 2 optimization model (i.e., the optimization objective is to minimize the machining error), a machining area with a line width of 6.07 mm is selected on both sides of the tool contact point trajectory, the machining area boundaries are set (the curves of $u$ = −0.5 and$u$ = −4.5 shown in Figure 14), and the deviations of the tool envelope from the design surface are calculated along the machining area boundaries for the tool contact points. Figure 16 shows the variation in the deviation of the tool envelope from the design surface at each point along the two boundaries.

Table 1. Ball-end cutter and toroidal cutter processing bandwidth under different tool position planning mm.

	Method	Ball-End Cutter R = 5.5 mm	Second-Order Taylor Approximation $\mathit{\lambda }=25$°	Third-Order Taylor Approximation	Envelope Approximation
$\mathit{\epsilon }/\mathit{m}\mathit{m}$
0.01		0.98	0.312	0.614	0.663
0.005		0.69	0.248	0.528	0.587

shows the machining bandwidths of the above surfaces when machined with ball-end cutters and circular knives with different tool planning. It can be seen that, with the same machining accuracy and proper tool position planning, the machining range of circular-faced tools with non-ball-end is much greater than that of ball-end tools, demonstrating the high efficiency of the non-ball-end cutter for strip-width-maximization machining. As shown in Table 1, Figure 15, and Figure 16, the envelope approximation method can achieve the best approximation between the tool envelope and the design surface in a given machining area and has higher machining accuracy and efficiency than the existing machining methods (e.g., second-or third-order Taylor approximation) in the case of CNC machining with circular surface tools.

However, the efficiency of this method is lower than that of the multi-point tangential approach due to the fact that the multi-point tangential approach [13] allows for a certain amount of overcutting.

5. Conclusions

This paper presents a generalized canonical framework for describing tool motion in multi-axis CNC machining of sculptured surfaces, drawing upon the theory of surface natural moving frames from differential geometry. The envelope motion of the tool relative to the workpiece is characterized by invariant parameters that encapsulate both the tool surface and the designed surface motions. Simulation outcomes validate that the proposed methodology, when utilized to govern the relative motion between the tool and workpiece, ensures consistent variations in the machining line width and error along the tool’s contact trajectory for each tool position, thereby satisfying practical machining standards.

In comparison to second-order and third-order Taylor approximations, as well as traditional ball-end cutter approaches, the study reveals that when $\epsilon$ is 0.01, the bandwidth of the non-spherical cutter is 0.98, the second-order Taylor approximation’s bandwidth is 0.312, the third-order Taylor approximation’s bandwidth is 0.614, while the envelope approximation introduced in this paper achieves a bandwidth of 0.663. Similarly, at $\epsilon$ = 0.005, the bandwidths are 0.69, 0.248, 0.528, and 0.5827, respectively. Notably, the envelope approximation consistently demonstrates the widest bandwidth, indicating its potential to significantly enhance both the processing accuracy and efficiency, particularly in the machining of symmetrical workpieces.

However, the research is based on idealized conditions, and its conclusions may deviate in non-ideal machining scenarios. This limitation and its implications should be investigated in future studies.