# Ultra precision analytical toolpath calculation for aspherical mirror surface machining

Eloïse Jeanroy, Julien Chaves-Jacob, Jean-Marc Linares, Santiago Arroyave-Tobón, Stephan Imperiali

► To cite this version:

Eloïse Jeanroy, Julien Chaves-Jacob, Jean-Marc Linares, Santiago Arroyave-Tobón, Stephan Imperiali. Ultra precision analytical toolpath calculation for aspherical mirror surface machining. CIRP Annals - Manufacturing Technology, 2025, 74 (1), pp.731-735. 10.1016/j.cirp.2025.04.047. hal-05486176

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# Ultra precision analytical toolpath calculation for aspherical mirror surface machining

Eloïse Jeanroy a, b, Julien Chaves-Jacob a, Jean-Marc Linares (1) a, Santiago Arroyave-Tobon a, Stephan Imperiali b

a Aix Marseille Univ, CNRS, ISM, Marseille, France

b Thales SESO, Aix-en-Provence, France

Submitted by Linares Jean Marc (1)

Ultra-precise machining of aspherical mirrors requires high surface quality with form defects below 10 µm. This study assessed the performance of toolpaths generated by commercial CAD/CAM software, focusing on the distance between consecutive toolpath points, variations in these distances, and geometric accuracy. The evaluation showed that commercial tools hinder high-precision toolpath generation due to surface faceting, even with minimized chord error. To mitigate this, a new analytical toolpath calculation method was proposed. Two Zerodur® mirrors were machined using this method and commercial CAM. These tests highlight a reduction of 52.8% in form defect by analytical method.

Computer aided manufacturing (CAM), Tool path, Ultra precision

## 1. Introduction

Manufacturing ultra-precision glass-ceramic components, especially from Zerodur®, is technically challenging. As explained by Kitajima et al. [1], these components require exceptionally high precision, with form tolerances of less than 10 µm, which is harder to achieve in the case of large components (several meters in diameter). Achieving high precision machining requires careful control: the machining and grinding processes, cutting parameters, toolpath, measurement devices and machine errors. The grinding process was studied by Henkel et al. [2] in the case of aspherical mirrors. They show the importance of implementing a semi-finishing step to improve the surface roughness of mirrors. Some tools were developed by Chaves-Jacob et al. [3] to simulate the influence of machining process on shape, waviness, and surface roughness defects. Cutting parameters were studied by Wang et al. [4] to limit the damage in glass-ceramic components. They showed that the most influential parameters were depth of cut, tool abrasive geometry and tool cutting speed. Measurement devices for surfaces were studied by Gao et al. [5]. This work focuses on on-machine and in-process metrology techniques to ensure precision manufacturing, using advanced measurement devices. The machine errors were studied by Gao et al. [6] using a classification-based approach to analyze geometric, thermal, and load-induced errors, distinguishing intra-axis, inter-axis, and volumetric errors. This paper explores measurement, modeling, and compensation techniques for machine errors. The toolpath is the result of the entire CAD/CAM process chain. This process starts with the Computer-Aided Design (CAD) modeling of the desired surface. This 3D model then guides Computer-Aided Manufacturing (CAM) to generate the necessary toolpaths for machining. The main limitation of commercial CAD/CAM tools is the restricted generation of high-precision toolpaths due to surface faceting, even with small chord error values. To reduce the faceting problems, Beudaert et al. [7] proposed smooth interpolation strategies.

The aim of this paper is to study the influence of the CAD/CAM process chain on the accuracy of machined aspherical mirrors. An analytical toolpath generator (ATG) is proposed to avoid faceting problems of the CAD/CAM process and improve surface geometrical quality. In the first section, the mathematical model of aspherical mirrors is defined, along with three toolpath performance criteria. These criteria are defined to evaluate the impact of the toolpath on the accuracy and stability of the machining. Three commercial CAM software were analyzed according to the defined criteria. In the second section, an analytical toolpath generator was developed to meet the requirements of high-precision machining. In the final section, a comparison is made between the toolpaths generated by the commercial CAMs and the analytical model.

## 2. Impact of commercial CAD/CAM process chain on toolpath accuracy

The CAD/CAM process chain plays a critical role in the accuracy of the computed toolpath, which directly affects the geometrical quality of the machined surfaces. Approximations made at each stage can lead to deviations that affect the accuracy of the machined surface. These deviations can result from the strategies used in the toolpath generation algorithms, projection methods, or numerical tolerances within the CAM software. This section examines the extent to which the process chain influences these deviations, with a focus on understanding how each CAM system's approach affects the accuracy of the toolpath. The aim is to identify where quality loss occurs in the process chain and to propose improvements to minimize these deviations, thereby increasing the fidelity of the machined surface to its nominal model.

### 2.1. Materials and methods

The whole study would be carried out on aspheric optical mirrors as defined in section 2.1.1.

#### 2.1.1. Aspherical mirror geometry

Optical aspheric surfaces are primarily defined by Eq. (1), as explained by Sun et al. [8] or by Li et al. [9] (Fig. 1.).

$$Z = \frac{\frac{r^2}{Rc}}{1 + \sqrt{1 - (1 + k) \frac{r^2}{Rc^2}}} + Ar^4 + Br^6 + Cr^8 \quad (1)$$

where Z is the sag of the surface, $R_c$ is the vertex radius, k is the conic constant, and r is the radial distance from the Z-axis, $r = \sqrt{x^2 + y^2}$, A, B, C are aspheric coefficients used to refine the surface profile.

In some cases, the defined surface may not be centered on the axis of revolution. Such a surface is called "off-axis". This surface, also known as the "daughter" surface, is actually a section cut from the original surface of revolution, known as the "mother" surface.
### 2.1.2. Tested CAM systems

Three commercial CAM systems were tested to analyze their toolpaths using the same nominal surface of an industrial mirror with a diameter of around 1,200 mm. The nominal surface has an $R_c = 3,500$ mm and $k = -1$. The coefficients A, B and C are zero. Toolpaths were generated as 2 mm pitch spirals, with a toolpath tolerance of 0.005 mm. The tool radius must be less than $R_c$, therefore, the tool used in these tests is part of a hemispherical cutter with a radius of 500 mm.

* CAM 1 provides an option to generate a spiral toolpath directly on the desired surface.

* CAM 2 generates the spiral on the sphere with the best fit relative to the surface and then projects it onto the surface.

* CAM 3 generates the spiral in the XY plane and then subsequently projects it in the Z direction onto the surface.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Aspherical mirror geometry</figcaption></figure>

### 2.1.3. Definition of performance criteria of a toolpath

In order to analyze the different CAM systems, it is essential to define performance criteria to evaluate the impact of the toolpath on the accuracy and stability of the machining:

**C1:** This criterion evaluates the distance between points on the toolpath. First, all the distances between the points are calculated. If the distance is too small, the machine will have to make micro-stops because it does not have time to read the following points during this small movement. The criterion is then a minimum distance not to be exceeded as defined by Lavernhe et al. [10] or by Beudaert et al. [7]. The minimum allowable distance is calculated using Eq. (2):

$$D_{\text{mini}} = Vf * \text{cycle} \qquad (2)$$

where $Vf$ is the feed rate, and cycle is the interpolation cycle time, which ranges from 0.5 to 4 ms depending on the machine CNC. In

<figure><img src="image_3.png" /></figure>

our case, the cycle time is 2 ms and the feed rate is $Vf=1000$ mm/min, giving $D_{\text{mini}} = 0.03$ mm. Therefore, if the calculated distances are greater than 0.03 mm, C1 is satisfied.

**C2:** The distances between points are retrieved, and the percentage variation of the distance between consecutive points is calculated. As shown by Kopac et al. [11], this parameter is crucial for ensuring a good surface finish. In this paper, several experimental tests were carried out with different trajectories, each with a different point distribution. For each test, the quality of the resulting surface was specified. C2 was built around this information by measuring the distance between each point on the different trajectories and calculating the variation in these distances. It was determined that if the variation exceeded approximately 30%, the toolpath would create defects on the surface of the component. Therefore, C2 is satisfied if the variation does not exceed 30%.

**C3:** This criterion measures the deviation between the CAM generated points and the nominal surface. The coordinates of the toolpath points are collected. Using Python code, these coordinates are reworked to shift from the tool tip points ($C_t$) to the point of contact ($C_c$) between the tool and the surface. Then, the Z-coordinate of the contact point ($Zcc$) is compared to the theoretical Z-coordinate ($Zth$). The greater the deviation, the less accurate the surface geometry.

## 2.2. Results

The analyses were performed on a mirror section to reduce the computational load, focusing on the section where $400<r<600$ mm. All results are summarized in Fig. 2. C1 is met by all 3 CAMs, ensuring that the machine does not experience stuttering movements during operation. C2 is exceeded in all three CAM systems for CAM 1 on the whole section examined, for CAM 2 on the outer part of the section

<caption>Fig. 2. Analysis of the 3 commercial CAM tools and the analytical toolpath generator according to defined performance criteria</caption>
(r>530mm) and for CAM 3 when r>480mm. For C3, the results are within the tolerance specified during toolpath generation. The maximum deviation is 1.8 µm for CAM systems 1 and 3, and 3.5 µm for CAM system 2.

## 2.3. Discussion

The results show that criteria C2 and C3 are not met by any of the CAMs analyzed, leading us to believe that the surface finish of the parts will be poor when they leave the machining area.

To explain the previous problems, our hypothesis is that the commercial CAM systems rely on a triangular mesh to use a surface in toolpath calculation. To support this hypothesis, a numerical test was carried out to compare the location of points on a triangular mesh with the location of toolpath points. Both geometric elements are calculated from the same nominal surface (Fig.3). The CAD1 software (associated with CAM1) is used to design a nominal surface (portion of a sphere with Rc=-100 mm, k=0 and A, B, C = 0 mm). Firstly, with the nominal surface, the CAD1 was used to generate a triangular mesh with a tolerance of ±0.2 mm (Fig. 3a). Figure 3b zooms in a portion of the triangularly meshed surface. Secondly, with CAM1, a toolpath was generated using the nominal surface. This toolpath was a raster with a tolerance of 0.4 mm in Fig. 3c (which corresponds to a tolerance of ±0.2 mm to match the tolerance of the triangular mesh). Figure 3d zooms in a portion of the toolpath. In red color arrows represent the tool direction vectors at contact points. Both geometric elements were superimposed in figure 3e. It can be seen that the toolpath contact points lie on the edges of the triangularly meshed surface. This experiment was extended to the two others CAM systems. The same phenomenon was observed. In conclusion, commercial CAM systems calculate their toolpaths via a meshed surface.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Faceting. a) Triangular mesh of the surface, b) Zoom on 2 triangles of mesh, c) Nominal surface with toolpath, d) Zoom on the toolpath, e) superposition of the triangularly meshed surface and the toolpath</figcaption></figure>

This faceting step performed by commercial CAMs generates errors. The faceting step of the source into triangles is explained by Rypl *et al.* [12]. The size of the triangles depends on the tolerance required for the toolpath, as the edges of the triangles must remain within the tolerance range. Calculated toolpath points are placed along the edges of triangles that are on the nominal surface but within the specified tolerance range. This explains the difference calculated in criterion 3. Bouchenitfa *et al.* [13] further demonstrate how the distribution of points on an STL explains the variation in the distance between consecutive points. A possible solution to improve surface quality would be to reduce the toolpath tolerance, thereby bringing it closer to the theoretical surface geometry. However, this approach also reduces the size of the triangles and the distance between points which can lead micro-stops of the machine. In addition, it could overload the machine's RAM due to the excessive number of points in the toolpath. Finding a balance between tolerance reduction and system performance is therefore essential to avoid these problems. With the different results obtained, commercial CAMs do not meet accurate requirements of the ultra-high precision mirror application and stability of the machining. The potential solution of reducing the toolpath tolerance to reduce the deviation of the geometric shape will not work due to the reduced distance between points. Another solution is to develop an ATG to meet the various criteria of the domain.

# 3. Analytic toolpath generator

To meet precision requirements and eliminate dependency on a purely geometric model, an ATG was developed and implemented in Python language.

## 3.1. Input data

A human-machine interface allows the user to input the necessary information to run the program (part geometry, tool geometry, machining strategy). The nominal surface is defined by the same equation as before (Eq. 1). The user must enter the coefficients A, B, and C, the spherical radius Rc, and the conic constant k. They also need to specify the part shape (circular or rectangular) and its dimensions. The tool geometry follows De San Román *et al.* [14], requiring the major radius Ro and minor radius ro of the torus. The toolpath strategy is chosen based on its importance for achieving the desired surface quality and accuracy. Two toolpath strategies are considered:

* The spiral toolpath follows a continuous, expanding or contracting path across the surface, maintaining consistent tool contact and reducing wear. It's ideal for circular or rotationally symmetric surfaces needing uniform material removal.

* The rastering toolpath uses parallel, linear movements over the surface, offering precise texture control and uniform coverage, especially suited for rectangular or planar surfaces.

In addition, tool inclination—both the degree and direction of tilt—is critical for effective material removal and surface accuracy. There are two main adjustments:

* Longitudinal inclination aligns the tilt with the cutting direction, improving surface finish and potentially reducing tool pressure.

* Transverse inclination tilts the tool perpendicular to the cutting direction, aiding chip evacuation and maintaining stability on certain geometries.

Finally, setting an appropriate chordal error (or toolpath tolerance) ensures the toolpath closely follows the intended surface profile, defining the maximum deviation between the toolpath and surface geometry. Fig. 4 illustrates the chordal error for the spiral (Fig. 4a and 4b) and rastering cases (Fig. 4c and 4d). The error occurs because CNC machines use line segments between points instead of arcs. A smaller chordal error yields a more accurate toolpath but requires longer processing times due to a higher density of toolpath points.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Machining strategies. a) 2D spiral, b) 3D spiral, c) 2D raster, d) 3D raster</figcaption></figure>

## 3.2. Toolpath calculation

Once all necessary data has been entered, the computation can be initiated. The 5 steps of the toolpath point calculation process are summarized in Fig. 5.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Flowchart of the ATG</figcaption></figure>

* The toolpath generation in XY plane: the toolpath is defined within the XY plane as shown in Fig. 4. For the spiral toolpath, a specific formula is applied to control the spacing and starting point of the spiral to optimize the tool's entry and movement across the component. The spiral is defined by the Eq. (3):

$$r = -\frac{\theta}{2\pi} + \left(\frac{De}{2} + 0.8\frac{Do}{2}\right) \qquad (3)$$

where, r is the radial distance from the center at a given angle θ, p represents the pitch of the spiral, De is the outer diameter of the component, and Do is the tool's diameter. This approach begins the spiral from the outside of the component. The second term creates an initial offset allowing the tool to approach the component gradually. For the raster (scanning) toolpath, the tool moves back and forth in a linear and parallel path across the surface. Each toolpath is followed by an external retraction to clear the edge of the piece before beginning the next pass.

* Toolpath discretization: once generated, the toolpath must be discretized into a set of points. This step depends on the desired chordal error/tolerance, as explained by Li et al. [15], and the surface curvature in rastering or spiral toolpaths. For a spiral toolpath, the first point is placed outside the component (Section 3.2.1). The subsequent points are determined by calculating the angle β between two points around the Z-axis using Eq. (4).

$$\beta = 2 \cos \left( 1 - \frac{tol}{r_t} \right) \qquad (4)$$

where tol represents the chordal error, and r_t is the local radius. For rastering, the radius considered is no longer the distance of the point from the center but the local curvature radius of the component surface. Both cases are illustrated in Fig. 4b) and 4d).

* Toolpath projection on the surface: in the case of a revolution surface, the toolpath projection is made directly with Eq. (1). In the case of the daughter's surface, Eq. (1) represents the shape of the mother. A change of reference frame is therefore required to obtain the "daughter" reference frame. The resulting system consists of two-equation system shown in Eq.(5):

$$\begin{cases} r_m = r_d \cos \gamma + Z_d \sin \gamma + r_{dec} \\ Z_m = -r_d \sin \gamma + Z_d \cos \gamma + Z_{dec} \end{cases} \qquad (5)$$

where γ the rotation angle between the mother and daughter frames, r_m and Z_m are the coordinates in the mother frame, r_d and Z_d are the coordinates in the daughter frame and r_dec and Z_dec represent the translation between the mother and daughter frames.

This two-equation system has three unknowns, r_m, Z_m and Z_d which is solved with an iteration step implemented to guarantee an error of 1e-4 mm.

* Cutter reference point computation: The cutter contact point (C_c) is insufficient for directly controlling 5-axis CNC machines. To ensure accurate toolpath execution, the cutter reference point (C_l) point must be determined. This is done by reconstructing the C_l point using the tool vector IJK and the surface normal, as per Eq. (6):

$$C_l = C_c + R_o \mathbf{n} - R_o \mathbf{IJK} \qquad (6)$$

The surface normal is derived by defining a local grid around the C_l point using four surrounding points. These points form two surface vectors, and the normal is obtained from their cross product. This ensures the calculated normal accurately represents the surface
orientation at the C<sub>i</sub> point. The tool vector **IJK** is calculated by rotating the surface normal vector **n** into the appropriate tool orientation, as shown in Eq. (7). Rotation occurs in the plane through point C<sub>c</sub>, defined by the vectors **n** and **V**<sub>f</sub>. The guiding vector **V**<sub>f</sub>, representing the feed direction between two points, underpins the tool vector adjustment.

$$ \mathbf{IJK} = \cos(\alpha) \mathbf{n} + \sin(\alpha) \mathbf{V}_f \quad (7) $$

* In five-axis machining, angles A and C are calculated using the IJK vector of a CNC tool. Equation (8) determines the inclination angle (A-axis) relative to the vertical Z-axis, while Equation (9) calculates the azimuth angle (C-axis), representing the tool's rotation around the vertical Z-axis projected on the XY-plane. As shown by Cui et al. [16], these equations are derived from the trigonometric relationships used to determine the orientation of a tool in three-dimensional space relative to the component.

$$ A = \tan^{-1} \left( \frac{\sqrt{I^2 + J^2}}{K} \right) \quad (8) $$

$$ C = -\tan^{-1}\left(\frac{I}{J}\right) \quad (9) $$

where I, J and K are the components of tool vector **IJK**.

## 4. Experimental validation

Firstly, the ATG was tested numerically using the three previous performance criteria. Figure 2 summarizes the results obtained by the ATG. In comparison with CAM 1, 2 and 3, the distance between points was smoothed.

Secondly, experimental validation was carried out on two mirrors (Section 2.1.2, with a central hole of 300 mm diameter) machined from Zerodur®. The first was machined using a toolpath generated by CAM2 and the second by ATG. In order to focus on the toolpath effect, the two machining surfaces were carried out using the same machine-tool, cutting parameters and environmental conditions. After machining, the surfaces of the two mirrors were measured using ultra-precise non-contact 3D profilometer (±50 nm accuracy at 3σ). Using this profilometer, 64 radial profiles were measured on each machined surfaces. The profile derived of the nominal surface was removed from each measured profile to obtain the deviation-profile. Afterward, these 64 profiles were folded into a single plane. The mean deviation-profile was highlighted using darker color in Fig.6. The ranges of mean deviation-profiles are 15.19 µm for ATG (Fig. 6a) and 32.25 µm for CAM2 (Fig. 6). This represents a 52.8% reduction in form defect.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Deviation-profiles of measured 64 radial profiles a) Deviation-profile obtained with ATG, b) Deviation-profile obtained with CAM2</figcaption></figure>

## 5. CONCLUSION

In this paper, the influence of the CAD/CAM process chain on geometrical toolpath accuracy is evaluated in the context of high-precision mirror manufacturing. For this purpose, three performance criteria were defined to evaluate three commercial CAM software: C1 (distance between points), C2 (distance variation) and C3 (toolpath point/nominal surface error). This study concluded that these three commercial CAM software do not provide toolpaths meeting the three performance criteria limits (C1>0.03 mm, C2< 30%, C3≈0 µm). They are unable to machine mirrors with the expected form defects: less than 10µm and without undesirable traces of the machining process. Therefore, an analytical toolpath generator (ATG) was developed. This is based on the mathematical equation of the mirror surface. Thereafter, a machining of Zerodur® mirrors was carried out to evaluate the improvement on the form defect value. It showed that the ATG decreases the form defect by 52.8% compared to the commercial CAM software.

## Acknowledgement

The authors acknowledge the Association Nationale Recherche Technologie (France) and Thales SESO (France) for financial support (Cifre thesis n° 2023/0255).
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