1. Introduction

The demand for ultra-precision optical components has experienced significant growth across diverse areas in the past two decades, such as space telescopes for astronomical exploration [1], high-NA EUV lithograph optics [2], and reflective mirrors for synchrotron radiation [3] and free-electron laser [4] facilities. These precision requirements vary, with maximum surface shape deviations ranging from microns for visible light applications to sub-nanometer levels for X-ray imaging at the diffraction limit.

In the manufacture of these optical components, deterministic Computer Numerical Controlled (CNC) finishing processes are required in the final stages to achieve such high level of precision. Typically, to correct the final residual errors without damaging the optical component, sub-aperture compliant machine tools [5], such as polishing pads [6], bonnets [7], Magneto-Rheological Fluids (MRF) [8] and energy beams (e.g., fluid jet [9], Ion Beam Figuring (IBF) [3], and air-pressure plasma [10] ), have been widely employed.

As shown in Fig. 1, deterministic CNC finishing comprises several key stages, including metrology, Tool Influence Function (TIF) determination, tool path planning, dwell time optimization, and velocity scheduling. The effectiveness of each stage directly influences the level of convergence.

 

Fig. 1. Schematic of a CNC finishing process: After taking measurements and planning a tool path, dwell time optimization is performed to calculate the precise duration the machine tool should dwell at each point along the tool path to correct surface errors. Afterwards, dwell time is transformed into continuously varying velocities, reducing the dynamic stress imposed on the motion system and resulting in a smoother and more accurate surface generation. This procedure is repeated until the residual surface error is reduced to the specified level.

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Initially, precision measurements are taken, from which the residual error, $z_r(\bf {x})$, is calculated as the difference between the measured shape, $z_a(\bf {x})$, and the target shape, $z_t(\bf {x})$, respectively. Meanwhile, the TIF, which characterizes both the shape of the machine tool and the rate of material removal, can be determined through two methods. Theoretical modeling of the TIF can be achieved by employing the classical Preston equation [11]. Alternatively, it can be extracted from the measurements obtained during either single-spot polishing [3] or scanning trenches at a constant feed rate [12].

Subsequently, tool path planning is another essential element of CNC finishing. Raster paths are the most used tool paths, in which the machine tool scans the optical surface along a single direction. However, these regular paths may leave repetitive tool marks on the surface if the tool spacing is sparse or the processing time is long [13]. This issue can be addressed by introducing a sparse bi-step raster path to reduce these tool marks [14]. Alternatively, tool path randomization has been explored, including pseudorandom tool paths [15], unicursal maze paths [16] , and random adaptive paths [17]. To further mitigate the dynamics stress on the machine during cornering, the circular-random path [18] and tree-shaped random path [19] can be used.

Using the measurement data, TIF, and the planned tool path, dwell time optimization is carried out to determine the precise duration for which the machine tool should dwell at each point along the tool path to correct surface errors. Knowing the TIF and $z_r(\bf {x})$, the dwell time, $t(\mathbf {u}_i)$ (for the $\mathbf {u}_i$ along the tool path) is estimated through deconvolution. However, deconvolution is an ill-posed inverse problem, leading to non-uniqueness in its solutions. While deconvolution has been extensively studied in the field of image super-resolution, various concepts and algorithms have been applied to dwell time optimization. Yet, as depicted in Fig. 1, dwell time optimization has specific objectives on its accuracy, feasibility, flexibility, and efficiency. Researchers have proposed a range of dwell time optimization methods that partially addressed these objectives. These methods can be classified into iterative [20], matrix-form [21], Fourier transform [22] and Bayesian [23] methods. Although these methods and their variants [24–27] were developed independently and prioritized different objectives, they share similar principles and strategies. The performances of existing dwell time optimization methods have been comprehensively studied [28]. Additionally, advanced methods [29,30], which combined the merits of existing methods to achieved all the specified objectives, have been proposed. These developments indicate a significant leap forward in dwell time optimization for CNC finishing. With the optimized dwell time along a well-defined tool path, the next challenge is to accurately implement the dwell time in the machine.

Dwell time is not directly implemented, as it incurs frequent acceleration and deceleration when the machine tool is moved to the next dwell point, introducing heavy dynamics stress to the CNC controller and resulting in extra dwell time and inaccurate surface generation. In practice, dwell time $t(\mathbf {u}_i)$ is always converted to velocities $v(\mathbf {\mathbf {u}}_i)$ and the CNC machine is driven by these continuously varying velocities along the tool path [13]. As shown in Fig. 1, we summarize the four requirements of a reliable velocity scheduling method:

One simple velocity model, as shown in Fig. 2(a), is the constant-acceleration model [31], where velocities in each machining interval consist of a constant-acceleration segment followed by a constant velocity segment. To adhere to machine dynamics constraints, Particle Swarm Optimization (PSO) has been employed to adjust the velocities before feeding them to the machine controller [32].

 

Fig. 2. The constant-acceleration model (a) cannot provide smooth velocity profiles. The PVT model (b) generates smooth velocity profile in each segment, but fails to ensure the continuity of acceleration. The E-PVT model (c) not only generates smooth velocity profiles but also guarantees velocity and acceleration continuities.

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Alternatively, an analytical model has been proposed to directly derive velocities within dynamics constraints by adaptive machining intervals for raster paths [33]. While the constant-acceleration model is easy to implement, it falls short in providing smooth and continuous velocity profiles. This limitation restricts its widespread use in the finishing processes, where velocity smoothness and continuity play a crucial role in fine-correcting the small residual surface errors [13].

Position-Velocity-Time (PVT) is a more advanced velocity model widely supported by numerous CNC controllers. It describes the positions and velocities between consecutive tool path points using piecewise cubic and quadratic polynomials, respectively. As a result, the velocity profiles within the PVT model are notably smoother. In our previous research, we introduced an innovative PVT-based velocity scheduling method [13], effectively solving velocities within the PVT framework. By employing a constrained linear least-squares (CLLS) solver, we enforced dynamics constraints and velocity continuity. Integrated with our in-house developed Ion Beam Figuring (IBF) system [3], this approach achieved sub-nanometer level convergence. Nonetheless, three challenges still hindered its broader application.

Firstly, solving the PVT problem becomes excessively time-consuming when dealing with dense tool paths. The time complexity of resolving a CLLS system is $O(n^3)$, with $n$ representing the number of tool path points. Secondly, the full potential of PVT-based controllers remains unexplored, indicating an opportunity for further optimization. On one hand, the consideration of dynamics constraints is incomplete. The velocity extrema may not only occur at the endpoints of each machining interval, but also arise when acceleration equals zero in between. On the other hand, as shown in Fig. 2(b), there is a lack of continuous velocity and acceleration across machining intervals. Lastly, no mechanism existed to provide compensation when the dynamics constraints are exceeded. The feed drive controller can only run correctly when the system’s dynamics can respond to the entered PVT data in a reasonable manner.

To answer these challenges, this study proposes an Enhanced PVT (E-PVT) velocity scheduling method. Firstly, E-PVT reduces the time complexity of the existing PVT method from $O(n^3)$ to $O(n)$ by incorporating cubic splines [34]. It has been demonstrated that the cubic spline model is equivalent to the PVT model [13]. Secondly, as depicted in Fig. 2(c), E-PVT comprehensively takes into account dynamics constraints and continuities. Lastly, E-PVT compensates the dynamics constraints by adding a constant extra dwell time, $\Delta t$, to each tool path point [31]. A novel compensation method driven by PSO is employed to minimize $\Delta t$, ensuring the minimal impact to the overall processing efficiency.

The rest of this paper is organized as follows. Section 2 reviews the existing constant-acceleration and PVT velocity scheduling methods. Section 3 demonstrates the proposed E-PVT method. Simulation studies of the performance of E-PVT are then presented in Section 4, followed by an IBF finishing experiment using E-PVT in Section 5. Section 6 concludes the paper.

2. Scheduling velocities from dwell time

A feed drive system comprises the control of positions, velocities and accelerations. Figure 2 illustrates the motions along an arbitrary $\chi$-axis, where $p_{i}$, $v_{i}$, $a_{i}$ and $t_{i}$ represent the positions, velocities, accelerations, and time at the $i$th velocity control points for $i=0,1,2,{\ldots},n$, where $n$ is the number of dwell points. The relationship between velocity control points and dwell points is further elucidated in Fig. 3, where dwell points are typically positioned in the middle of consecutive velocity control points [31], denoted as $\tau _{i}=t_{i+1}-t_i$.

 

Fig. 3. Dwell points are positioned the middle of consecutive velocity control points. A constant $\Delta t$ is added as a compensation when dynamics constraints are exceeded.

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For a specific feed axis, its motion is constrained by its maximum velocity and acceleration as

2.1 Constant-acceleration models

The most naïve velocity model is the linear polynomial model, described as

To address these issues, a constant-acceleration model has been introduced [31]. As depicted in Fig. 2(a), in this model, the trapezoidal velocity profile for each machining interval consists of an acceleration segment with a fixed acceleration followed by a segment of constant velocity, described as

This model has been successfully applied in correction polishing with IBF [31] and MRF [35]. However, two remaining challenges in applying this model to finishing processes with higher precision requirements are evident. First, as shown in Fig. 2(a), the constant-acceleration model, while achieving velocity continuity, fails to ensure continuity in acceleration, which leads to unsmooth velocity profiles along the machining intervals. Additionally, the actual $\tau _i$ may not be achievable when $a$ is small. From Eq. (3), velocities can be solved as

When $\left (v_{i}^2+a\tau _{i}\right )^{2}<\left (v_{i}^2+2as_{i}\right )$, which means that $a$ is so small that $s_{i}$ cannot be achieved within $\tau _{i}$, $v_i$ has to be calculated as the maximum achievable velocity in $s_i$ as

2.2 Position-velocity-time model

PVT is a model that has been supported by many modern CNC controllers. As shown in Eq. (6), it describes the motion for the segment $\left [p_{i},p_{i+1}\right ]$ using the positions, time, and velocities at the endpoints of the segment as,

To solve $v_{i}$ and $\bf {b}_i$ from this model, our previous research work [13] introduced an additional pair of acceleration equations to each segment and added four boundary conditions as

Also, we imposed the dynamics (see Eq. (7)) and acceleration continuity (i.e., $6t_i\left (b_{i,4}-b_{i+1,4}\right )-2\left (b_{i+1,3}-b_{i,3}\right )=0$) constraints, hoping that the obtained velocities and accelerations are valid within the machine’s constraints and continuous across the segments. The resulting system is expressed as

As outlined in Section 1, the current PVT method exhibits three significant weaknesses: it is time-consuming, inadequately considers machine constraints, and lacks compensation for dynamics constraints when these limits are exceeded. Specifically, regarding dynamics constraints, Eq. (8) addresses only the situations where velocity extrema occur at velocity control points, as illustrated in Fig. 4(a). However, as shown in Fig. 4(b), velocity extrema may also occur when acceleration equals zero between points. In Section 3, we will present a comprehensive explanation of E-PVT, which effectively resolves all these issues.

 

Fig. 4. Maximum velocities may be attained at either the endpoints of each segment (a) or where the acceleration equals zero between the endpoints (b). Our conventional PVT only considers (a), while E-PVT adds (b).

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3. Enhanced Position-Velocity-Time (E-PVT) velocity scheduler

Differing from PVT, which is defined at the velocity control points, E-PVT utilizes a cubic spline that characterizes the motion between them [34]. In this cubic spline representation, positions within each segment between consecutive velocity control points are defined as a function of $t$ by

Therefore, the velocities solved from E-PVT can be directly fed into a PVT controller.

3.1 Solving E-PVT in $O(n)$ while maintaining the continuity of velocity and acceleration

In E-PVT, the choice of a cubic spline formulation offers the inherent maintenance of continuity in velocity and acceleration. We describe the positions, velocities and accelerations in $\left [p_i,p_{i+1}\right ]$ as

Among the possible choices, we adopted the boundary conditions presented in Eq. (7) from our previous PVT-based method. These conditions are considered appropriate because it is expected that the initial and final velocities and accelerations will be zero in practical CNC tool path implementations. With these conditions, we have a total of $4n+2$ conditions, while there are $4n$ unknowns. To ensure that the system is determinate, two more equations are required.

To address this, as depicted in Fig. 2(c), two additional points, $p_{a1}$ and $p_{a2}$, are introduced to the segments $\left [p_0,p_1\right ]$ and $\left [p_{n-1},p_{n}\right ]$, respectively. In E-PVT, the positions of these two points are controlled through the corresponding time, $t_{a1}$ and $t_{a2}$, expressed as

With all these conditions described above, E-PVT can be solved by substituting $c_{i,3}=a_{i}/2$ to Eq. (9), resulting in the position and velocity expression in terms of accelerations:

By incorporating the conditions introduced in Eqs. (7) and (12), we obtain:

Let $\mathbf {a}$ represent the vector of unknown accelerations as $\mathbf {a} = {\left [ a_{a1},{a_1},{\ldots},{a_{n - 1}}, a_{a2} \right ]^{\rm {T}}}$, where $a_{a1}$ and $a_{a2}$ are the accelerations of the two added points, respectively, the following linear system can be derived:

It is evident that the matrix $\mathbf {C}$ is sparse and tridiagonal, allowing for the utilization of the efficient Thomas algorithm [36] to solve the system in Eq. (17). This results in a computational complexity of $O(n)$, which is significantly more efficient compared to the conventional PVT method with a computational complexity of $O(n^3)$. Subsequently, by substituting the derived accelerations $a_{i}$ back into Eq. (11), the cubic spline coefficients are obtained as

The velocities $v_{i}$ are obtained by substituting Eq. (20) to Eq. (11).

To summarize, the E-PVT model described so far offers two distinct advantages over the conventional PVT method. Firstly, this model inherently maintains the continuity of velocity and acceleration, thanks to the cubic spline formulation. As visualized in Fig. 2(c), these continuity conditions guarantee smooth connections of position, velocity, and acceleration between segments, providing a seamless and well-behaved trajectory. Secondly, the remarkable computational efficiency achieved through the utilization of the Thomas algorithm makes E-PVT suitable for tackling large-scale problems when dealing with dense tool paths or large optical components.

The final challenge for E-PVT lies in the complete identification of the dynamics constraints, as depicted in Fig. 4, and the subsequent compensation for these constrains. In the following subsection, we will explore the process of pinpointing points where dynamic constraints are exceeded and present a method for effectively resolving these issues.

3.2 Identification and compensation of dynamics constraints with particle swarm optimization

In actual CNC finishing processes, when the machine tool exceeds its maximum allowable velocity or acceleration, machining errors can occur.

To ensure that machining dynamics align with the machine tool’s maximum capabilities, several methods have been attempted in the literature. First, a small time interval, $\Delta t$, representing the time required for the machine to traverse the maximum machining interval with a certain speed, can be added to $t_i$ [31], as illustrated in Fig. 3. In this way, the requirements of the maximum velocity and acceleration are reduced. In the context of finishing, having each dwell point remain for an additional duration of $\Delta t$ is essentially equivalent to conducting extra constant material removal on a surface. However, in this approach, $\Delta t$ was added by trial-and-error, without careful consideration of the machine’s dynamics constraints. In practice, it is expected to minimize $\Delta t$ to reduce its impacts on machining accuracy and efficiency.

An alternative method is the interior point method, which takes into account dynamics constraints in the dwell time optimization process [37]. However, the non-linearity nature of this algorithm limits computational efficiency. Similarly, a CLLS method was employed to solve dwell time while considering dynamics constraints [38]. Nevertheless, as mentioned in Section 2.2, a CLLS solver still necessitates a substantial amount of computation time.

It is also worth noting that these studies primarily focus on the considerations of dynamics constraints under the constant-acceleration model described in Section 2.1, with relatively limited research on the PVT model.

In E-PVT, as depicted in Fig. 3, we adopt the same method of extending the dwell time at each dwell point by $\Delta t$ [31]. However, unlike the trial-and-error method used previously, E-PVT aims to search for the shortest $\Delta t$ while identifying and considering the machine’s dynamics constraints.

To fully identify the dynamics constraints under the E-PVT framework, we extract the velocity and acceleration formula from Eq. (11) as

Given the monotonic nature of acceleration in Eq. (21), it is evident that acceleration extrema only occur at the endpoints. Consequently, the acceleration constraints can be straightforwardly defined as the constraints at the endpoints, where

However, velocity is a quadratic model thus its extrema are obtained in two different situations. As illustrated in Fig. 4(a), when $a_{i}a_{i+1}\geq 0$, the velocity profile takes on a monotonic form, in which case it is adequate to solely consider the velocity at the endpoints as

When $a_{i}a_{i+1}<0$, as shown in Fig. 4(b), the velocity extrema not only occur at the endpoints but may also arise at time $\bar t$, where $a(\bar t)=0$. In this case, the velocity constraints should be expressed as

By combining Eqs. (22), (23) and (24), the complete dynamics constraints in the E-PVT model are identified as

Here, when $\gamma =1$, it indicates that all the dynamics constraints are satisfied, whereas $\gamma =0$ indicates non-satisfaction of any of these constraints. With Eq. (25), we define an optimization problem to search for the minimum $\Delta t$ as

Equations (26) and (27) indicate that, in the optimization, $\Delta t$ will only be updated when all the dynamics constraints, as identified in Eq. (25), are met.

Subsequently, to effectively solve the optimization problem presented in Eq. (26), a derivative-free optimizer is necessary, as the fitness function given in Eq. (27) is not a continuous function. E-PVT employs PSO [39] to tackle Eq. (26), which is a heuristic algorithm inspired by the foraging behavior of birds. The core idea of PSO is to seek the optimal value through collaboration and information sharing among individuals within a swarm. It is extremely effective in derivative-free optimization tasks where the number of optimization variables is small. During each iteration of PSO, dynamics constraints are identified by Eq. (25) after solving the cubic spline coefficients as per Eq. (17). Thereafter, the values of $(\Delta t,r_1,r_2)$ are updated within each particle of the swarm using the fitness function. Once all particles complete their search, the results are combined to determine the final $\Delta {\hat t}$.

The incorporation of PSO in E-PVT results in a time complexity of $O(p\cdot m \cdot n)$, where $p$ and $m$ denote the numbers of particles and PSO iterations, respectively. In practical applications, we have observed that by setting $p=20$, we consistently achieved $\Delta {\hat t}$ with fewer than 10 iterations ($m<10$). Considering that $p\cdot m=200\ll n$ in actual CNC finishing processes and the calculations within $p$ particles are always parallelized, the computational complexity of E-PVT thus still remains $O(n)$.

Finally, it is worth mentioning that the values of the ratios, $r_1$ and $r_2$, introduced in Eq. (13) for determining the positions of the two additional points, are incorporated as parameters into PSO but remain unoptimized, as illustrated in Eq. (27). Their specific values are arbitrarily chosen during each PSO iteration, which provides additional degrees of freedom in tuning $\Delta t$. In other words, the inclusion of $r_1$ and $r_2$ serves solely to ensure that their values may help the minimization of $\Delta t$.

3.3 E-PVT enabled dwell time optimization and velocity scheduling strategy

Once we have determined $\Delta {\hat t}$ through E-PVT, the next step is to apply this adjustment to the velocity control points to effectively compensate for any exceeded dynamics constraints, as demonstrated in Fig. 3. However, it is worth mentioning that direct addition of $\Delta {\hat t}$ may affect the accuracy of surface generation and processing efficiency if $\Delta {\hat t}$ is large [30]. This issue becomes especially critical when the machining interval is big and the machine tool’s TIF is not axial symmetric due to the discretized convolution process [26].

In fact, a more appropriate strategy is to feed $\Delta {\hat t}$ into dwell time optimization to ensure that smallest dwell time entry, $\tau _{min}$, is adjusted to $\tau _{min}=\tau _{min}+\Delta {\hat t}$ during the optimization. This strategy is straightforward to implement, as several advanced dwell time optimization algorithms [26,29,30] have the capability to optimize dwell time while considering $\tau _{min}$. In this study, our recently proposed Robust Iterative Surface Extension (RISE) algorithm [30] is employed, and the E-PVT enabled RISE and velocity scheduling strategy is schematically illustrated in Fig. 5.

 

Fig. 5. Flow of the E-PVT enabled RISE and velocity scheduling strategy.

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Initially, $\Delta {\hat t}=0$ and RISE is performed with $\tau _{min}=\mathrm {max}\left |p_{i+1}-p_{i}\right |/\left |v_{max}\right |$, which is the time need for the machine tool to traverse the largest machining interval with the maximum allowed speed. Subsequently, dwell time $\tau _{i}$ is calculated, followed by the E-PVT to identify the exceeded dynamics and search for $\Delta {\hat t}$. If $\Delta {\hat t}=0$ is obtained, indicating no violation of dynamics constraints, the velocities $v_i$ are computed. Otherwise, $\Delta {\hat t}$ is added to $\tau _{min}$ and the above steps are repeated.

It is worth highlighting that the process shown in Fig. 5 only requires a single repetition when dynamics constraints are exceeded, indicating the effectiveness of this E-PVT enabled RISE and velocity scheduling strategy. This strategy has been employed in the following simulation and experimental studies of E-PVT.

4. Simulation study on the performance of E-PVT

The performance of E-PVT in terms of accuracy, compliance with dynamic constraints, continuity, and efficiency is studied through simulation. Figure 6 provides an overview of the simulation setup.

 

Fig. 6. Specifications for the simulation using E-PVT: an off-axis elliptical cylindrical mirror with the clear aperture size of 540 mm $\times$ 10 mm is fabricated from its best-fit spherical mirror. The target ellipse (a) is fit to its best-fit circle (b), which has a RoC of 149 m and the minimum material removal from the target ellipse. The initial height error from the target elliptical cylindrical mirror to the best-fit spherical mirror is 2.07 $\mu$m RMS (c). A random rough surface of a 0.55 nm RMS is added to mimic the influence of middle-to-high-frequency errors. The Gaussian TIF of the IBF system used to correct this height error has a FWHM of 5 mm and a PRR of 6.36 nm/s (e).

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The simulation presents the IBF process of a Vertical Kirkpatrick-Baez (VKB) mirror intended for one of the beamlines at the National Synchrotron Light Source II (NSLS-II). The VKB mirror is an off-axis elliptical cylindrical mirror, as depicted in Fig. 6(a), with an object distance of $20583.6$ mm, an image distance of $2170.8$ mm, and a grazing angle of $1.5^{\circ }$. The Clear Aperture (CA) (i.e effective length) measures 540 mm $\times$ 10 mm.

To enhance processing efficiency, we fit this elliptical shape to its best-fit circle by minimizing the material removal from the circle to the ellipse. The resulting circle possesses a Radius of Curvature (RoC) of 149 m, shown in Fig. 6(b). The initial residual height error from the target VKB mirror surface to the best-fit spherical mirror within the 540 mm $\times$ 10 mm CA measures 2.07 $\mu$m RMS, as illustrated in Fig. 6(c). A random rough surface of a 0.55 nm RMS, as shown in Fig. 6(d), is generated using the surface synthesis method [40] and added to the height error map to mimic the influence of middle-to-high-frequency errors.

Figure 6(e) illustrates the TIF of the IBF system, extracted from an ion beam footprint on a Silicon surface. It has a Full Width Half Maximum (FWHM) of 5 mm and a Peak Removal Rate (PRR) of 6.36 nm/s. A raster tool path, larger than the CA’s outermost perimeter by the TIF radius, is employed to fabricate this mirror. To demonstrate the computational efficiency of E-PVT, the machining interval in the raster path is set at 0.1 mm, resulting in a total of $n$=1,100,000 dwell points in a 550 mm $\times$ 20 mm dwell grid. The dynamics constraints of the IBF machine are $\left |v_{max}\right |=0.25$ m/s and $\left |a_{max}\right |=2~\mathrm {m/s^2}$, as outlined in [13].

4.1 Study on the accuracy, compliance with dynamics constraints, and process efficiency of E-PVT

Figure 7 demonstrates the application of the E-PVT enabled dwell time optimization and velocity scheduling process illustrated in Fig. 5 to simulate the IBF of the VKB mirror shown in Fig. 6(c).

 

Fig. 7. The initial dwell time (a) leads both the velocities and accelerations exceeding the maximum limits (b). The PSO algorithm proposed in E-PVT finds $\Delta {\hat t}=3.2$ ms that can be added as a compensation, resulting in an updated dwell time (c). The E-PVT calculation based on this dwell time provides velocities and accelerations within the maximum allowed values (d), confirming the compliance of dynamics constraints of the proposed E-PVT method. The estimated residual height error after the compensation is 0.62 nm RMS, which is close to the added middle-to-high-frequency errors (e).

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Initially, as shown in Fig. 7(a), we employed the RISE dwell time optimization algorithm [29] to correct the height error within the CA. The total dwell time optimized using this algorithm was 9156.57 min. Subsequently, the computed dwell time was utilized in the E-PVT algorithm to schedule the velocities, and the resulting velocity and acceleration profiles are visualized in Fig. 7(b). It is evident that, some velocity and acceleration profiles significantly exceeded the permissible limits $\left |v_{max}\right |$ and $\left |a_{max}\right |$, respectively, indicating that the dwell time solution in Fig. 7(a) could not be correctly implemented within the IBF machine’s dynamics constraints.

Consequently, we adopted the PSO method, as proposed in Section 3.2, to search for an appropriate $\Delta {\hat t}$ value for compensation. After applying PSO, we determined that $\Delta {\hat t} = 3.2$ ms, which was then incorporated into the RISE algorithm for another round of dwell time optimization. The updated total dwell time, shown in Fig. 7(c), stood at 9311.47 min. This optimized dwell time solution was then integrated into the E-PVT algorithm for velocity scheduling. This time, depicted in Fig. 7(d), the maximum speed and acceleration were 0.03 m/s and 2.00 $\mathrm {m/s^2}$, respectively, both of which remained within the dynamics constraints.

Importantly, the maximum acceleration of 2.00 $\mathrm {m/s^2}$ equals $\left |a_{max}\right |$ of the IBF machine, confirming that the added $\Delta {\hat t} = 3.2$ ms is the minimum extra time needed to such a compensation, thus guaranteeing maximum possible process efficiency. Additionally, as illustrated in Fig. 7(e), the estimated residual height error from the compensated dwell time solution achieved 0.62 nm RMS, which closely matched the added middle-to-high-frequency errors, further validating that the dynamics compensation did not compromise the optimization accuracy.

4.2 Study on the continuity and computational efficiency of E-PVT

To assess the continuity of velocity and acceleration in the E-PVT solution, we performed a comparative analysis between the velocity and acceleration profiles generated by E-PVT and those obtained from the traditional PVT model. This evaluation relies on the compensated dwell time map depicted in Fig. 7(c). For clarity, we have replicated Fig. 7(c) in Fig. 8(a).

 

Fig. 8. The center line of the dwell time (a) is converted to velocities using E-PVT (b). E-PVT achieves both velocity (c) and acceleration (d) continuities. Discontinuities in acceleration calculated by PVT are represented in dashed lines.

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Figure 8(b) illustrates the velocity profiles of the raster line at $y=0$ computed using both the conventional PVT and E-PVT methods. While both velocity profiles exhibit similar trends, the one derived from the conventional PVT displays significantly greater dispersion compared to that derived from E-PVT. This discrepancy stems from the instability of the CLLS system described in Eq. (8) when the bounding or equality constraints become too stringent.

This issue becomes more apparent in Fig. 8(c), which zooms in on a small portion of the segments along the raster tool path, providing a clearer visualization of the velocity and acceleration profiles computed from the two models. It is evident that, E-PVT achieves both velocity and acceleration continuity across these segments. However, as acceleration continuity was not ensured in the conventional PVT model, it struggles to find valid accelerations within the range of negative and positive maximum allowed acceleration. Furthermore, the acceleration profiles indicate that E-PVT exhibits the least variation in acceleration, resulting in the least dynamic stress on the actual IBF machine.

The computational efficiency of E-PVT was examined through a comparison with the constant acceleration model and the conventional PVT model in scheduling the velocities for the $n$=1,100,000 (5500 $\times$ 200) points in the dwell time map shown in Fig. 7(c). All tests were conducted on a Dell workstation equipped with an Intel Xeon CPU E5-1603 v4 (2.80 GHz, 4 cores) and 32 GB RAM. The programs were implemented and tested using MATLAB R2019b. To ensure a fair comparison, each method was executed three times, and the computation time for each was averaged from these runs. The results, along with considerations regarding accuracy, compliance with dynamic constraints, and continuity, are summarized in Table 1.

Table 1. Comparison of the PVT and E-PVT velocity scheduling methods in terms of accuracy, compliant with dynamics constraints, continuity, and efficiency.^a

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The constant-acceleration method is the simplest and most expedient approach, taking only 0.14 s to schedule velocities for all $n$=1,100,000 points along the tool path. It proves suitable for CNC machines lacking PVT-based control support.

In contrast, the conventional PVT method exhibits extreme computational inefficiency. It took 6309.55 s to compute velocities for just one line out of the 200 lines. Given that lines in a raster path are planned independently, the approximate total computation time for velocities across 200 lines would exceed 14 days, which is even longer than the total dwell time (9311.45 min), rendering it impractical for any optical finishing planning tasks.

Conversely, the E-PVT method proposed in this study achieves a significantly reduced computation time of 571.44 s, approximately 2,230 times faster than the conventional PVT method. This enhanced performance, coupled with the advantages of compliance with dynamics and comprehensive continuity considerations outlined in this study, confirms the efficacy of E-PVT for CNC optical finishing. Additionally, for raster tool paths, it is worth noting that multi-core optimization can further accelerate computation by a factor equal to the number of physical cores of the CPU.

5. Experimental verification of E-PVT

To further verify the feasibility of the E-PVT method in actual computer-controlled optical finishing applications, we adopted the E-PVT-scheduled velocities in our post-production correction of a lamellar grating surface illustrated in Fig. 9(a).

 

Fig. 9. A 200 lines/mm lamellar grating (a) was measured by our in-house developed FSI system (b). The initial height error was 11.06 nm RMS (c). Our in-house developed IBF system was utilized to correct this height error (d).

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In a prior research work, we presented the successful application of IBF in correcting the residual height error of a blazed X-ray grating surface without compromising the grating structure [41]. Recently, we received another lamellar-type X-ray grating with a density of 200 lines/mm from the Advanced Light Source (ALS) for correcting its residual height error.

The center 95 mm $\times$ 30 mm region of the grating surface was measured by our in-house developed Fizeau Stitching Interferometry (FSI) system [42], as depicted in Fig. 9(b). The interferometer used in this FSI system is a 4-inch Zygo Verifire MST with a 776 $\times$ 576 pixels CCD camera. An 80% overlap of the subsets in the stitching data acquisition was applied and the measurement was repeated 5 times. The total data acquisition time was about 20 mins. The averaged residual height error of the grating surface, shown in Fig. 9(c), measured 11.06 nm RMS. The repeatability of these 5 measurements achieved 0.12 nm RMS. The same IBF TIF shown in Fig. 6(e) was utilized in the experiment and the machining interval in the raster tool path was 0.8 mm.

Our in-house developed IBF system given in Fig. 9(d) was utilized to correct this residual height error with the E-PVT scheduled velocities. They IBF system [3] involved in the experiment is equipped with a KDC10 ion source from Kaufman & Robinson. It operated with a beam voltage of 600 V and a beam current of 10 mA during the experiment. The CNC unit in the IBF system is composed of three translation stages (one NLS8-500 and two NLE-50) from Newmark and an NSC-G3-E41$\times$ feed drive controller equipped with a CMC-41$\times$3 motion card from Galil, which supports PVT-based control.

The process of E-PVT with PSO compensation illustrated in Fig. 5 was performed in the experiment. As depicted in Fig. 10(a), before dynamics compensation, the total dwell time was 15.22 min and the maximum acceleration was exceeded. Therefore, we adopted the PSO method and found that a $\Delta {\hat t} = 6.06$ ms should be added to the dwell time optimization. Consequently, the updated total dwell time, shown in Fig. 10(b), became 15.76 min, and the maximum speed and acceleration were 139.58 mm/s and 2.00 m/s$^2$, respectively, indicating that $\Delta {\hat t} = 6.06$ ms is the minimum extra time needed for dynamics compensation. Therefore, these scheduled velocities were fed into the IBF machine to finish the grating surface.

 

Fig. 10. A 200 lines/mm lamellar grating (a) was measured by our in-house developed FSI system (b). The initial height error was 11.06 nm RMS (c). Our in-house developed IBF system was utilized to correct this height error (d).

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The experimental result is illustrated in Fig. 10(c). Based on the E-PVT scheduled velocities, the estimated residual height error was 0.36 nm RMS. After one IBF process, the actual measured residual height error was reduced from 11.06 nm RMS (see Fig. 6) to 0.57 nm RMS. The discrepancy between the measured residual height error after IBF and the estimation is only 0.21 nm RMS, confirming the feasibility of applying the proposed E-PVT based velocity scheduling method to sub-nanometer-level finishing processes.

6. Concluding remarks

In this study, we introduced the Enhanced Position-Velocity-Time (E-PVT) modulated velocity scheduling method. Compared to our previous PVT method, E-PVT achieves the following enhancements.

Firstly, E-PVT efficiently computes accurate and smooth velocity profiles with a time complexity of only $O(n)$, which is notably faster than the PVT method with a time complexity of $O(n^3)$, ensuring remarkable computational efficiency.

Secondly, E-PVT comprehensively considers CNC dynamics constraints and ensures both velocity and acceleration continuity. The simulation results, as illustrated in Figs. 8(b) and 8(c), demonstrates that the E-PVT solution is stable even for scheduling velocities for an extremely large number of dwell points.

Thirdly, E-PVT incorporates an innovative compensation algorithm utilizing particle swarm optimization (PSO) for addressing situations where dynamics constraints are exceeded. This algorithm guarantees that the added extra dwell time for compensation is minimized, optimizing process efficiency. While the introduction of PSO parameters, such as the number of particles and PSO iterations, contributes to increased computational complexity, the overall efficiency is substantially improved compared to the PVT method, as demonstrated in Table 1.

Lastly, the experimental results, coupled with PVT-enabled machine controllers, demonstrate that the E-PVT scheduler reliably achieves sub-nanometer-level finishing capability.