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A novel method for improving optical component smoothing quality in robotic smoothing systems by compensating path errors

Yonghong Deng, Xi Hou, Bincheng Li, Jia Wang, and Yun Zhang

Open Access Open Access

Abstract

Path deviations caused by geometrical errors in machining equipment significantly affect the machining quality of optical components. To enhance the quality and efficiency of optical component processing, this paper presents a Chebyshev interpolated Levenberg-Marquardt algorithm (CILM) aimed at compensating for path deviations in a robotic smoothing system utilized for optical component processing. First, the positioning accuracy of the robotic smoothing system is measured using a laser tracker. Subsequently, an objective function is constructed based on robot kinematics and error models to optimize the geometric errors in the system. Then, the proposed method is adopted to identify the geometric parameters of the robotic smoothing system to compensate for the smoothing path deviations. The compensation results confirm the effectiveness of the proposed method in enhancing the absolute positioning accuracy of the robotic smoothing system. Additionally, experimental verification is conducted to validate the effectiveness of the proposed method in improving the surface quality of optical components through smoothing path compensation. The results of the three experiments indicate that the proposed CILM achieves optical components with peak-to-valley values 15.70%, 28.7%, and 4.01% lower than those obtained before compensation, along with root mean square of 33.67%, 21.57%, and 10.23% lower than before compensation values, respectively. Moreover, the power spectral density curves of CILM exhibit smoother characteristics in comparison to the curves before compensation.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Optical components’ mid-frequency errors significantly affect imaging systems, resulting in small-angle scatter. Smoothing plays a crucial role in reducing mid-frequency errors in optical components [1]. Industrial robots, with their high flexibility, large working space, and cost-effectiveness, are increasingly being utilized in optical component manufacturing. Specifically, they have emerged as the primary processing equipment for optical components grinding, polishing, and smoothing stages, surpassing traditional CNC machines. With the increasing quality standards of optical components, the precision requirements for industrial robots are becoming increasingly stringent. Consequently, the precision requirements for industrial robots are rising accordingly [2–4]. Robots encounter two main origins of errors: geometric errors resulting from machining, manufacturing, assembly, and wear; and non-geometric errors arising from factors such as the accuracy of the robot servo system, the gearbox, the stiffness of the link, and environmental conditions [5–7]. In the identification process, our primary focus lies on geometric errors since they account for 80% of all error sources [8]. Despite exhibiting high repeatability, most robots suffer from low absolute positioning accuracy, thereby constraining their utility in high-precision manufacturing applications. The geometric errors of the robot and, consequently, its absolute positioning accuracy are directly influenced by the robot's link parameters. The absolute positioning accuracy of the robot can be significantly enhanced by identifying and compensating for deviations in its link parameters, consequently improving the accuracy of optical component processing.

Extensive studies have been conducted by numerous scholars in the field of robot error identification and compensation, to address this significant issue. Various intelligent algorithms have been proposed up until now to identify actual geometric errors, such as an improved Levenberg-Marquardt (ILM) algorithm [9], improved particle swarm optimization (IPSO) [10], genetic algorithm cascade simulated annealing algorithm (GA-MSAA) [11,12], or a denoising cascade of multiple algorithms (an extended Kalman filter integrated quadratic interpolated-beetle antennae search algorithm (EKF-QIBAS)) [13]. Chen et al. [14] introduced an improved beetle swarm optimization algorithm-based kinematic calibration method for an industrial robot aimed at performing drilling and riveting tasks. Experimental results indicate a significant enhancement in the accuracy of robot calibration achieved by the proposed method. Zhong et al. [15] put forth a novel kinematic calibration method, referred to as the improved whale swarm algorithm (IWSA), for enhancing the accuracy of robot calibration. Furthermore, a subsequent case study validated the efficacy of the method. Li et al. [16] employed a particle swarm optimization-Gaussian process method to enhance calibration efficiency. Experimental results demonstrate the superiority of the proposed method over traditional calibration methods. Cao et al. [17] presented a robot calibration method that integrates an extended Kalman filter and an artificial neural network within a butterfly and flower pollination algorithm to enhance the accuracy of the robot's absolute pose. The experimental results substantiated a substantial enhancement in the robot's pose accuracy achieved by the proposed method, outperforming previous techniques. The mentioned algorithms collectively showcase a noteworthy improvement in the identification accuracy of robots. Such as, the improved LM algorithm moderately refines identification accuracy through damping factor adjustments, albeit some truncation errors persist. The IPSO algorithm enhances identification accuracy by adapting particle weights, but it still exhibits slow convergence speed and susceptibility to getting trapped in local optima. While the cascading approach of GA-MSSA amalgamates algorithmic advantages to bolster identification accuracy, it does so at the cost of intricate architecture and slower convergence. The EKF-QIBAS denoising cascade method primarily removes Gaussian noise from measurements and subsequently heightens identification accuracy with an improved BAS. However, EKF's efficacy is limited by initial conditions and relatively inferior accuracy in resolving nonlinear high-dimensional challenges. Additionally, the cascade algorithm entails intricate structure and substantial time investment. Regrettably, these algorithms often encounter delayed convergence rates and subpar identification accuracy. Furthermore, their experimental code exhibits heightened complexity, inadequately addressing the issue of entrapment in local optima during the identification process.

The LM algorithm offers a notable advantage in efficiently solving highly nonlinear robot kinematic models by leveraging gradients. However, it disregards second-order and higher terms in the Taylor expansion, leading to truncation errors and consequently impacting the accuracy of the solution [9,18]. Interpolation is a mathematical technique used to estimate values between known data points. It involves constructing a function or curve that passes through the given data points and can be used to approximate the value of the function at additional points within the data range. Chebyshev interpolation stands out as an optimal approach for determining point spacing. It is particularly effective in mitigating the occurrence of the Langer phenomenon, unlike Lagrange interpolation and Newton's difference quotient formula. Additionally, Chebyshev interpolation provides a better fit to the original function, resulting in more accurate predictions [19–21]. Inspired by this discovery, this paper presents a novel identification method called the Chebyshev interpolated Levenberg-Marquardt algorithm (CILM). This method integrates the LM algorithm with Chebyshev interpolation to effectively handle a robot's geometric errors and attain highly accurate identification results. Moreover, it efficiently compensates for the smoothing paths, leading to an enhancement in the quality of the optical components. The proposed method relies on the following key principles: The LM algorithm's efficient computational properties enable quick approximation of the geometric error, and constructing an interpolation interval using the approximate geometric error obtained from the LM algorithm, employing Chebyshev interpolation. Generating Chebyshev interpolation nodes within this interval, where a point exists that provides a closer approximation to the actual geometric error. Consequently, a more accurate estimation of the geometric error is obtained.

This paper presents the following unique contributions by incorporating the aforementioned concepts:

  • a) The amalgamation of Chebyshev interpolation and standard LM update rules yields a CILM algorithm that exhibits faster convergence and higher recognition accuracy. Consequently, it facilitates the identification of more accurate geometric errors within the robot smoothing system and enables more effective compensation of the smoothing paths.
  • b) It incorporates a highly efficient algorithm design, accompanied by a comprehensive application analysis. These aspects aim to offer researchers and engineers valuable guidance in utilizing CILM for identifying geometric errors in robotic smoothing systems and achieving smoothing paths. Moreover, the utilization of simple codes streamlines and clarifies the implementation process, making it more convenient.
  • c) An experimental study is conducted to investigate smoothing path compensation for optical components. The primary objective is to guide utilizing robotic smoothing systems to enhance the smoothing quality of optical components.

Section 2 develops an identification model for geometric error identification. Section 3 introduces the CILM algorithm to identify geometric errors. Section 4 provides detailed experiments on smoothing compensation. Section 5 presents the conclusions.

2. Constructing a model for identification

Figure 1 illustrates the robot smoothing system and the measuring device. The smoothing system comprises a Staubli TX200 six-joint robot with a smoothing device attached at its end, which accommodates a smoothing tool. Additionally, there is a workbench for supporting the components. The nominal geometric parameters of the robot are given in Table 1. In the realm of robot calibration, it is worth noting that geometric parameters and DH parameters are often considered to be consistent [22,23]. The measurement device includes an API laser tracker and a spherically mounted retroreflector (SMR). During the measurement process, the SMR replaces the smoothing tool and is mounted on the smoothing device.

 figure: Fig. 1.

Fig. 1. Robotic smoothing system and measuring device. {B} denotes robot base, {F} represents flange frame, {T} is tool frame and {L} stands for tracker frame.

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Table 1. Nominal geometric parameters of Staubli TX200 robot

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To achieve the highest possible absolute positioning accuracy in the robot smoothing system, it is crucial to minimize errors in the end position points. Consequently, the objective function f is formulated:

$$f = \min {||{\Delta {\mathbf E}} ||_2} = \min \left[ {\frac{1}{n}\sum\limits_{i = 1}^n {{{||{{{\mathbf P}_i} - {\mathbf P}_i^{\prime}} ||}_2}} } \right]$$
where ΔE denotes the pose deviation vector. Pi = [xi yi zi]T and Pi′ = [x′i y′i z′i]T are respectively the theoretical and measured position. n indicates the number of position points.
$$\begin{array}{l} {{\mathbf P}_i} = {{\mathbf T}_{\textrm{FT}}}{{\mathbf P}_N}\\ {{\mathbf P}_i}^{\prime} = {{\mathbf T}_{\textrm{BL}}}{{\mathbf P}_{\textrm{LT}}} \end{array}$$
where TBF, TFT and TBL are the transformation from {B} to {F}, {F} to {T} and {B} to {L} respectively. PLT is the position measured by the laser tracker. PN is the nominal position vector of the robot, which is calculated by the forward kinematic model:
$${{\mathbf T}_{\textrm{BF}}} = {{\mathbf A}_1}{{\mathbf A}_2}{{\mathbf A}_3}{{\mathbf A}_4}{{\mathbf A}_5}{{\mathbf A}_6} = \left[ {\begin{array}{{cc}} {{{\mathbf R}_N}}&{{{\mathbf P}_N}}\\ 1&1 \end{array}} \right]$$
$${{\mathbf A}_i} = \left[ {\begin{array}{{cccc}} {\cos {\theta_i}}&{ - \sin {\theta_i}\cos {\alpha_i}}&{\sin {\theta_i}\sin {\alpha_i}}&{{a_i}\cos {\theta_i}}\\ {\sin {\theta_i}}&{\cos {\theta_i}\cos {\alpha_i}}&{ - \cos {\theta_i}\sin {\alpha_i}}&{{a_i}\sin {\theta_i}}\\ 0&{\sin {\alpha_i}}&{\cos {\alpha_i}}&{{d_i}}\\ 0&0&0&1 \end{array}} \right]$$
where A denotes the transformation matrix of the robot. RN is the nominal rotation matrix. ai, di, θi, and αi represent link length, link offset, joint angle, and link twist angle of the i-th link in a robot, respectively, which are the robot’s D-H parameters as well as geometric parameters.

The system’s geometric parameters errors vector is given as:

$${\mathbf x} = {[\Delta {\alpha _1} \cdots \Delta {\alpha _6}\begin{array}{{c}} {} \end{array}\Delta {a_1} \cdots \Delta {a_6}\begin{array}{{c}} {} \end{array}\Delta {d_1} \cdots \Delta {d_6}\begin{array}{{c}} {} \end{array}\Delta {\theta _1} \cdots \Delta {\theta _6}]^T}$$

The Jacobian is calculated by:

$${\mathbf J} = [\frac{{\partial {{\mathbf T}_{\textrm{BF}}}}}{{\partial \alpha }},\frac{{\partial {{\mathbf T}_{\textrm{BF}}}}}{{\partial a}},\frac{{\partial {{\mathbf T}_{\textrm{BF}}}}}{{\partial d}},\frac{{\partial {{\mathbf T}_{\textrm{BF}}}}}{{\partial \theta }}]$$

By combining (3) and (6), the pose error model is achieved:

$$\Delta {\mathbf E} = {\mathbf{Jx}}$$

The robot's kinematic error model plays a crucial role in establishing the mapping relationship between position errors and parameter deviations. This model serves as the foundation for identifying kinematic parameters by comparing the theoretical position and the actual position of the robot's end effector. Through this comparison, the differences between the two positions can be analyzed, leading to a better understanding of the robot's kinematic characteristics and aiding in error identification.

3. Geometric errors identification and compensation

3.1 Geometric errors identification and compensation

Based on the kinematic and error models of the robotic smoothing system, the LM iterative equation can be formulated as:

$${{\mathbf x}_{k + 1}} = {{\mathbf x}_k} - {({\mathbf J}_k^T{{\mathbf J}_k} + \mu {\mathbf I})^{ - 1}}{\mathbf J}_k^Tf({{\mathbf x}_k})$$
where xk is the deviation of the geometric parameters vector at the k-th iteration. µ denotes the damping factor. f (•) indicates the objective function in Eq. (1).

The LM algorithm employs a first-order Taylor expansion to approximate the original function, leading to an inherent truncation error. Introducing Chebyshev interpolation nodes addresses this issue and enhances identification accuracy.

Using Eq. (8) iteratively for two steps to obtain xk1 and xk2:

$${{\mathbf x}_{k1}} = {{\mathbf x}_k} - {({\mathbf J}_k^T{{\mathbf J}_k} + \mu {\mathbf I})^{ - 1}}{\mathbf J}_k^Tf({{\mathbf x}_k})$$
$${{\mathbf x}_{k2}} = {{\mathbf x}_{k1}} - {({\mathbf J}_{k1}^T{{\mathbf J}_{k1}} + \mu {\mathbf I})^{ - 1}}{\mathbf J}_{k1}^Tf({{\mathbf x}_{k1}})$$

Building interpolation interval [a b]:

$$\begin{array}{l} {\mathbf a} = ({{\mathbf x}_k} + {{\mathbf x}_{k1}})/2\\ {\mathbf b} = {{\mathbf x}_{k2}} \end{array}$$

Since x, xk1, and xk2 are three consecutive adjacent approximate solutions, the constructed interval [a, b] contains at least one solution that is the closest to the actual geometric error.

Interpolation can effectively mitigate the Runge phenomenon and provide a closer approximation to the original function. Constructing the Chebyshev interpolation nodes within the interval [a, b] using the following equation:

$${{\mathbf x}_{kC}}(m) = \frac{{{\mathbf a} + {\mathbf b}}}{2} + \frac{{{\mathbf b} - {\mathbf a}}}{2}\cos \frac{{(2m - 1)\pi }}{{2M}}\begin{array}{{c}} {} \end{array}m = 1,2, \cdots ,M$$
where m denotes the m-th interpolated node. M is the total number of interpolated nodes.

The objective function values at each interpolation node and the right endpoint are evaluated, and the optimal position is selected as the initial value for the subsequent iteration. Upon reaching the acceptable accuracy or the maximum number of iterations, the iteration process is achieved.

The compensated geometric parameter G* is obtained by subtracting the identified geometric error x from the nominal kinematic parameter G of the robot.

$${\mathbf G}\ast{=} {\mathbf G} - {\mathbf x}$$

Based on the aforementioned inferences, we present the detailed workflow of the Chebyshev interpolated Levenberg-Marquardt algorithm-based geometric parameter identification Algorithm I. CILM-GPI, which is in Table 2. Note that x0 denotes pre-set parameter errors. Qn = [q1n, q2n…, q6n]T, is the robot’s rotation angle vector can be obtained from the teach pendant. {PLT (n)} indicates the coordinates of the robot's end position measured by a laser tracker. n is the n-th sample point. K is the maximum number of iterations. M represents the number of samples.

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Table 2. Pseudocode for the CILM algorithm

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Additionally, Fig. 2 illustrates the workflow chart for compensating the robotic smoothing system. Through the above implementation, the improvement of the positioning accuracy of the smoothing robot is achieved.

 figure: Fig. 2.

Fig. 2. Flowchart for robotic smoothing system compensation.

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3.2 Analysis of results

3.2.1 General settings

Evaluation indicators. Various evaluation metrics, including root mean squared error (RMSE), average error (Mean), and maximum error (Max) [7–13], are employed:

$$\begin{array}{l} RMSE = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^n {||{{{\mathbf P}_i} - {\mathbf P}_i^{\prime}} ||_2^2} } \\ M\textrm{ean} = \frac{1}{n}\sum\limits_{i = 1}^n {\sqrt {||{{{\mathbf P}_i} - {\mathbf P}_i^{\prime}} ||_2^2} } \\ Max = \max \left\{ {\sqrt {||{{{\mathbf P}_i} - {\mathbf P}_i^{\prime}} ||_2^2} } \right\},i = 1,2, \cdots ,n \end{array}$$
Dataset: 1500 data samples are collected by our research group using the system in Fig. 1. Subsequently, Table 3 displays the five groups of sampling points. From these, 150 samples are randomly selected to create a test case with a uniform distribution. This process is repeated ten times to generate ten distinct data cases, reducing data bias. Each data case is evaluated using an 80%-20% training-test setup. The model's performance is assessed by calculating the mean and standard deviation across all ten data cases. Figure 3 illustrates the positions of the 1500 sample points within the 1400 mm × 2000mm × 900 mm-sized workspace of the smoothing robot. The blue points represent the training samples, and the red points denote the validation samples.

 figure: Fig. 3.

Fig. 3. Compensation and validation position in the workspace.

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Table 3. Detailed five samples

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Compared Models. To validate the effectiveness of the proposed identification method, a comparison is made with several state-of-the-art robot identification methods:

M1: The improved Levenberg-Marquardt (ILM) algorithm. The LM methodology is extensively employed in the domain of identifying errors in robots [9]. The ILM algorithm represents an enhanced version of the LM approach, featuring an automatic adjustment of the damping factor. Similarly, the CILM is an improved iteration of LM. To reflect performance disparities between the proposed CILM and the same type of ILM, the ILM algorithm is utilized as a benchmark for comparison.

M2: IPSO denotes the improved particle swarm optimization algorithm, featuring adaptive inertia weights, while PSO stands as a widely utilized evolutionary computation method in various engineering domains [10]. IPSO serves as a comparative benchmark, illustrating performance differences between the proposed CILM technique and evolutionary computation methods.

M3: A genetic algorithm with a modified simulated annealing algorithm (GA-MSAA) [11,12] for identification parameters. GA-MSAA refers to a cascade of the GA and the MSAA. Both GA and SAA are prevalent computational techniques frequently employed in engineering domains. GA-MSAA is utilized as a comparative reference, shedding light on performance distinctions between the proposed CILM method and the cascade approach.

M4: An extended Kalman filter integrated quadratic interpolated-beetle antennae search algorithm (EKF-QIBAS) -based identification model [12]. EKF-QIBAS involves the cascading of EKF with the QIBAS. EKF is employed to mitigate the Gaussian noise present in measurements, while QIBAS additionally identifies geometric errors. Positioned as one of the latest advancements, EKF-QIBAS serves as a representative instance of an alternative computational approach. It is utilized as a benchmark for demonstrating the performance disparity between the proposed CILM method and the latest denoising cascade identification technique.

M5: The CILM model proposed in this study.

In conclusion, the chosen M1∼M4 methods embody distinct computational methodologies. These methods serve as comparative benchmarks for the CILM technique, facilitating the assessment of performance disparities between the proposed CILM and various multi-type algorithms.

3.2.2 Compensation performance

We compare the compensation performance of the developed algorithm with advanced algorithms. Figure 4 depicts the compensation performance. Figure 5 depicts the achieved compensation accuracy. Figure 6 illustrates the convergence curves. Table 4 shows the geometric parameters after compensation through M5.

 figure: Fig. 4.

Fig. 4. Comparison of performance among M1∼M5. (a) RMSE error; (b) mean error; (c) max error; (d) time cost. Note that BC denotes before compensation.

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 figure: Fig. 5.

Fig. 5. Measuring point's position accuracy achieved by M1-5. (a) BC, M1, M2, and M5; (b) M3, M4, and M5. Note that BC denotes before compensation.

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 figure: Fig. 6.

Fig. 6. MRSE convergence curves of M1-5.

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Table 4. The geometric parameters after compensation by M5

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Based on these results, we have the following crucial discoveries:

The CILM achieves the highest compensation accuracy among M1-M4. As shown in Figs. 4(a)∼(c), M5’s RMSE is 0.3332 mm, which is respectively 66.1%, 17.2%, 41.9%, 12.9%, and 11.1% lower than BC’s 0.9832 mm (before compensation), M1’s 0.4027 mm, M2’s 0.5734 mm, M3’s 0.3823 mm, and M4’s 0.3751 mm. Similar outcomes are also observed when Mean and Max are utilized as the assessment metrics. Therefore, the proposed approach proves effective in enhancing the compensation accuracy of robotic smoothing systems.

The CILM exhibits the fastest convergence rate. Figure 6 illustrates that M5 reached RMSE convergence after only 32 iterations. In contrast, M1 requires 36 iterations to achieve RMSE convergence. Furthermore, both M2∼4 require more iterations to attain convergence. Therefore, integrating Chebyshev interpolation into the updating rule of the LM algorithm significantly enhances its convergence rate.

The CILM model demonstrates a lower time cost compared to the most effective compensation models. Figure 4 (d) illustrates that while the proposed CILM model (M5) has a higher time cost than M1, it generally remains lower than M2-4. This disparity is primarily attributed to the considerable time required by CILM for generating interpolation nodes and evaluating fitness values. Despite the additional time required, CILM exhibits a 17.2% improvement in accuracy compared to M1, thereby justifying the investment of additional seconds to achieve superior compensation accuracy. Notably, the implementation of efficient model parallelization with GPUs or other computing frameworks can significantly reduce these time costs. Overall, CILM provides superior compensation accuracy at an acceptable cost in terms of time, making it a worthwhile approach.

4. Smoothing compensation experiment

4.1 Compensation for smoothing paths

Owing to the inherent limitations of the industrial robot control system, direct modification of the geometric parameters within the control system is not feasible. Therefore, to minimize the deviation between the actual smoothing paths and the theoretical smoothing paths, we compensate for the recognized geometric errors by adjusting the paths based on the error distribution. The process of smoothing optical components using a smoothing robot involves several steps. First, a machining path is generated based on the equation of the component surface. Next, the machining path is converted into NC code through post-processing. The robot then executes the NC code to machine the workpiece. However, the absolute positioning accuracy of the robot is reduced due to its geometric errors. This deviation leads to differences between the theoretical smoothing path and the actual smoothing path in the NC code.

Figure 7 illustrates the smooth paths of an aspheric optical component for the uncompensated path, CILM compensated path and theoretical path. In Fig. 8, a comparison is presented for the cutting contact points on the machined face of the component in the theoretical case, before compensation, and after CILM compensation. Additionally, Fig. 9 shows the distribution of tool path point errors on the surface before compensation and after CILM compensation. Enhancing the absolute positioning accuracy of the robot helps to keep the deviation of the smoothing path within the acceptable tolerance range. However, the actual tool walking path may deviate considerably from the theoretical path. To address this issue, an identification and compensation method for robot geometric errors can be employed. This method further enhances the accuracy of the path, bringing the contour of the polishing tool walking path closer to the theoretical contour. By achieving this, stable contact between the polishing tool and the workpiece is maintained, ensuring high-quality machining.

 figure: Fig. 7.

Fig. 7. Comparison of smoothing paths before and after compensation.

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 figure: Fig. 8.

Fig. 8. Comparison of dwell points before and after compensation.

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 figure: Fig. 9.

Fig. 9. Comparison of dwell points error distribution on the curved surface before and after compensation. (a) Before compensation; (b) CILM compensation.

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4.2 Smoothing compensation experiment

4.2.1 Compensation experiment I

In Experiment I, we select before compensation, M1 (a widely adopted and representative method for identifying robot geometric errors, belonging to the same category as M5), and M3 (as representatives of the cascade method) for comparison with M5 in smoothing compensation experiments. The goal is to assess the smoothing quality of optical components obtained under various absolute positioning accuracy of the robot. Figure 10 presents the experimental equipment for the optical smoothing compensation investigation. The aspherical equations and parameters adopted in experiment I are as follows:

$$z(x) = \frac{{c{x^2}}}{{1 + \sqrt {1 - (k + 1){c^2}{x^2}} }} + {A_4}{x^4} + \cdots + {A_{2n}}{x^{2n}},n = 2,3,4, \cdots$$
where c stands for the vertex curvature, k denotes the quadric coefficient, and A2nis the high-order aspherical coefficient. Some parameters are as follows: c = 306.203, k = 0.

 figure: Fig. 10.

Fig. 10. Smoothing compensation experimental equipment.

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Four distinct groups are implemented. Robotic polishing is employed to achieve a sine or cosine initial surface form for each set of experimental components. The surface has a spatial period of 4 mm and an amplitude PV of approximately 300 nm. The first group generates a scanning path without compensation, while the remaining three groups apply path compensation using M1, M3, and M5, respectively. The specific experimental conditions are provided in Table 5. Each group utilized a robotic system for smoothing. Additionally, the surfaces are measured with a LUPHOScan 600 HD, and the corresponding Power Spectral Density (PSD) curves are calculated. Figure 11 presents the measurement results, while Fig. 12 displays the variation of peak-to-valley (PV) and root mean square (rms) after smoothing. Additionally, Fig. 13 illustrates the PSD curve.

 figure: Fig. 11.

Fig. 11. Surface smoothing result with different methods (80% of full aperture:150 mm, FFT fixed 1∼10 mm band pass filter).

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 figure: Fig. 12.

Fig. 12. Comparison of changes in PV and rms. (a) PV; (b) rms.

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 figure: Fig. 13.

Fig. 13. Comparison of PSD curve. (a) Initial surface; (b) after smoothing.

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Table 5. The experimental conditions for experiment I

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Based on these experimental results, we have made the following intriguing observations:

The proposed CILM method achieves a substantial enhancement in the smoothing quality of optical surfaces. Figures 11 and 12 demonstrate M5's 15.70% and 33.67% reductions in PV (232.472 nm) and rms (13.490 nm), respectively, compared to the before compensation’s 275.778 nm and 20.339 nm after smoothing. Figure 13 shows a smoother PSD curve for M5 compared to the before compensation curve.

The CILM method produced superior surface quality compared to M1 and M3. As depicted in Figs. 11 and 12, the M5'PV is 232.472 nm after smoothing, which is 12.06% and 5.91% lower than M1’s 264.364 nm and M3’s 247.072 nm, respectively. Similar outcomes are observed when utilizing rms as the evaluation metrics. Furthermore, the M5'PSD curve is smoother compared to the PSD curves of M1 and M3, as illustrated in Fig. 13.

The CILM method is more efficient in smoothing. The initial surface PV and rms of M5 are higher than BC, M1, and M3, while its PSD curves are inferior to them as well, as shown in Figs. 11–13. However, after smoothing, M5 achieves a better surface quality than the other methods.

4.2.2 Compensation experiment II

Experiment II utilized aspherical mirror blanks made of ultra-low expansion (ULE) glass material, with equations and parameters consistent with those employed in Experiment I. It consisted of two groups: one with before compensation smoothing paths, and the other with CILM-compensated smoothing paths. The experimental components’ initial surface, achieved through robotic machining, displays sine-cosine ripples on a sinusoidal pattern with a spatial period of approximately 4 mm and an amplitude of approximately 150 nm, as shown in Fig. 14. The remaining experimental conditions can be found in Table 6.

 figure: Fig. 14.

Fig. 14. Initial surface in experiment II (60% of full aperture:140 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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Table 6. Experiment II smoothing parameter table

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The measurement results of the smoothed optical components are presented in Fig. 15, while Fig. 16 illustrates the PSD curves of the initial and smoothed surfaces. Table 7 provides a summary of the measurement results for both the initial and smoothed surfaces. From these results, it can be obtained that:

 figure: Fig. 15.

Fig. 15. Surface after smoothing in experiment II (60% of full aperture:140 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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 figure: Fig. 16.

Fig. 16. Comparison of PSD curves in experiment II. (a) Initial surface; (b) after smoothing.

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Table 7. Results of experiment II

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Obtaining an optical component's surface form through smoothing using the CILM compensation method demonstrates superiority over the surface form of an optical component obtained before compensation. The comparison presented in Fig. 15 and Table 7 illustrates significant improvements. Specifically, the CILM’s PV is 360.469, which is 28.7% smaller than the 506.001 before compensation. Additionally, the CILM’s rms is 25.329, showing a 21.57% reduction compared to the 32.297 before compensation. Moreover, the PV of the cross-section obtained through CILM is 47.748, representing a 64.86% decrease from the 135.895 before compensation. Furthermore, the cross-section rms of 12.354 is 29.44% smaller than the before compensation of 17.509. Additionally, one of the peak-to-valley in the cross-section obtained by CILM has a distance of 20.374, which is 34.02% lower than the before compensation of 30.879. The PSD curves, depicted in Fig. 16, also demonstrate that the CILM-obtained optics have smoother profiles compared to those before compensation. These findings provide strong evidence that the CILM compensation method yields optical components of higher quality compared to before compensation.

Note that the initial surface PV, with 435.378, is used before compensation. However, after the smoothing process, the PV increased to 506.001. This increment is highly likely attributed to the geometric error of the robot, causing the smoothing tool to deviate and inadvertently resulting in local surface form damage to the optical components.

4.2.3 Compensation experiment III

Experiment III employs K9 aspherical components, the initial surface of the aspherical components utilized in experiment III featured a sine-cosine surface form with a spatial period of approximately 4.4 mm and an amplitude of around 150 nm, achieved through robotic polishing, as shown in Fig. 17. The detailed experimental details are shown in Table 8. Some aspherical parameters for experiment III are as follows: c = 152.26682266, k = 0.

 figure: Fig. 17.

Fig. 17. Initial surface in experiment III (75% of full aperture:80 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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Table 8. Experiment III smoothing conditions

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The measurement results of the smoothed surface are presented in Fig. 18, and the comparison of the PSD curves is shown in Fig. 19. Additionally, the experimental findings are summarized in Table 9. Based on these results, noteworthy observations can be made:

 figure: Fig. 18.

Fig. 18. Surface after smoothing in experiment III (75% of full aperture:80 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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 figure: Fig. 19.

Fig. 19. Comparison of PSD curves in experiment III. (a) Initial surface; (b) after smoothing.

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Table 9. Results of experiment III

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The surface quality of an optical component achieved through smoothing using the CILM compensation method surpasses that of an optical component smoothed without compensation. Figure 18 and Table 9 present the results, revealing that after smoothing, the PV and rms values of the CILM are 791.454 and 56.826, respectively. These values are 4.01% and 10.23% lower, respectively, compared to 824.544 and 63.304 before compensation. Similar improvements are observed when comparing cross-section PV and rms. Moreover, as depicted in Fig. 19, optics obtained using CILM exhibit smoother PSD curves compared to before compensation.

The CILM compensation method for smoothing exhibits superior efficiency. Figures 17 and 18, along with Table 9, illustrate the PV and rms values after using the CILM compensation method, which are 791.454 and 56.826, respectively. These values demonstrate reductions of 25.70% and 31.30% compared to the initial surface PV (1065.166) and rms (82.716), respectively. Additionally, the smoothed PV and rms values before compensation, 824.544 and 63.304, are 11.67% and 28.85% lower, respectively, compared to the initial surface PV (933.509) and rms (88.971). These results directly highlight the higher smoothing efficiency of the CILM compensation method in comparison to that before compensation.

4.2.4 Summary

Based on the above extensive experimental results, we can draw the following conclusions:

  • a) The proposed CILM algorithm achieves higher compensation accuracy and utilizes it to compensate for processing paths, resulting in optical components with excellent surface quality compared to alternative methods.
  • b) The proposed CILM compensation method has higher smoothing efficiency compared to the before compensation.
  • c) The reduction in absolute positioning accuracy caused by geometric errors in the robot can lead to a significant offset between the actual and theoretical dwell points during smoothing. This misalignment, in turn, has the potential to damage the local Surface form of the optics. However, this issue can be partially addressed by identifying and compensating for robot geometry errors along the smoothing path.

5. Conclusion

In this paper, we propose a CILM compensation method to enhance the absolute positioning accuracy of the robotic system, thereby improving the surface processing quality of optical components. To validate the algorithm, we performed a series of analyses on compensation accuracy and conducted experiments on optical processing compensation. The summarized results are as follows:

  • 1) The RMSE of the compensation accuracy using the CILM method has decreased from 0.9832 mm to 0.3332 mm. Compared to before compensation, the compensation accuracy has improved by 66.1%. In comparison, the methods of ILM, IPSO, GA-MSAA, and EKF-QIBAS show respective increases of 17.2%, 41.9%, 12.9%, and 11.1%. These results serve as a direct confirmation of the correctness and effectiveness of the proposed method.
  • 2) The proposed CILM method significantly enhances the smoothness of optical surfaces compared to the before compensation. In three experiments, the PV values of the optical components obtained by CILM are 15.70%, 28.7%, and 4.01% lower than those before compensation, respectively, while the rms values are 33.67%, 21.57%, and 10.23% lower than the values before compensation. Moreover, the PSD curves of CILM exhibit smoother characteristics compared to the before compensation curves. These results provide substantial evidence of the significant progress achieved by the CILM method in smoothing optical surfaces.

Moreover, our future research encompasses online identification and real-time compensation of geometric and thermal errors in smoothing robots, as well as high-precision vision-guided localization and machining, among other areas of interest. The primary objective is to enhance the precision and efficiency of optical component processing by robots.

Funding

National Natural Science Foundation of China (62275249).

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (19)
Fig. 1.
Fig. 1. Robotic smoothing system and measuring device. {B} denotes robot base, {F} represents flange frame, {T} is tool frame and {L} stands for tracker frame.

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Fig. 2.
Fig. 2. Flowchart for robotic smoothing system compensation.

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Fig. 3.
Fig. 3. Compensation and validation position in the workspace.

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Fig. 4.
Fig. 4. Comparison of performance among M1∼M5. (a) RMSE error; (b) mean error; (c) max error; (d) time cost. Note that BC denotes before compensation.

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Fig. 5.
Fig. 5. Measuring point's position accuracy achieved by M1-5. (a) BC, M1, M2, and M5; (b) M3, M4, and M5. Note that BC denotes before compensation.

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Fig. 6.
Fig. 6. MRSE convergence curves of M1-5.

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Fig. 7.
Fig. 7. Comparison of smoothing paths before and after compensation.

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Fig. 8.
Fig. 8. Comparison of dwell points before and after compensation.

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Fig. 9.
Fig. 9. Comparison of dwell points error distribution on the curved surface before and after compensation. (a) Before compensation; (b) CILM compensation.

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Fig. 10.
Fig. 10. Smoothing compensation experimental equipment.

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Fig. 11.
Fig. 11. Surface smoothing result with different methods (80% of full aperture:150 mm, FFT fixed 1∼10 mm band pass filter).

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Fig. 12.
Fig. 12. Comparison of changes in PV and rms. (a) PV; (b) rms.

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Fig. 13.
Fig. 13. Comparison of PSD curve. (a) Initial surface; (b) after smoothing.

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Fig. 14.
Fig. 14. Initial surface in experiment II (60% of full aperture:140 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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Fig. 15.
Fig. 15. Surface after smoothing in experiment II (60% of full aperture:140 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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Fig. 16.
Fig. 16. Comparison of PSD curves in experiment II. (a) Initial surface; (b) after smoothing.

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Fig. 17.
Fig. 17. Initial surface in experiment III (75% of full aperture:80 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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Fig. 18.
Fig. 18. Surface after smoothing in experiment III (75% of full aperture:80 mm, FFT fixed 1∼10 mm band pass filter). (a) Before compensation; (b) CILM compensation.

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Fig. 19.
Fig. 19. Comparison of PSD curves in experiment III. (a) Initial surface; (b) after smoothing.

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Tables (9)

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Table 1. Nominal geometric parameters of Staubli TX200 robot

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Table 2. Pseudocode for the CILM algorithm

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Table 3. Detailed five samples

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Table 4. The geometric parameters after compensation by M5

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Table 5. The experimental conditions for experiment I

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Table 6. Experiment II smoothing parameter table

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Table 7. Results of experiment II

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Table 8. Experiment III smoothing conditions

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Table 9. Results of experiment III

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Equations (15)

Equations on this page are rendered with MathJax. Learn more.

f = min | | Δ E | | 2 = min [ 1 n ∑ i = 1 n | | P i − P i ′ | | 2 ]
P i = T FT P N P i ′ = T BL P LT
T BF = A 1 A 2 A 3 A 4 A 5 A 6 = [ R N P N 1 1 ]
A i = [ cos ⁡ θ i − sin ⁡ θ i cos ⁡ α i sin ⁡ θ i sin ⁡ α i a i cos ⁡ θ i sin ⁡ θ i cos ⁡ θ i cos ⁡ α i − cos ⁡ θ i sin ⁡ α i a i sin ⁡ θ i 0 sin ⁡ α i cos ⁡ α i d i 0 0 0 1 ]
x = [ Δ α 1 ⋯ Δ α 6 Δ a 1 ⋯ Δ a 6 Δ d 1 ⋯ Δ d 6 Δ θ 1 ⋯ Δ θ 6 ] T
J = [ ∂ T BF ∂ α , ∂ T BF ∂ a , ∂ T BF ∂ d , ∂ T BF ∂ θ ]
Δ E = J x
x k + 1 = x k − ( J k T J k + μ I ) − 1 J k T f ( x k )
x k 1 = x k − ( J k T J k + μ I ) − 1 J k T f ( x k )
x k 2 = x k 1 − ( J k 1 T J k 1 + μ I ) − 1 J k 1 T f ( x k 1 )
a = ( x k + x k 1 ) / 2 b = x k 2
x k C ( m ) = a + b 2 + b − a 2 cos ⁡ ( 2 m − 1 ) π 2 M m = 1 , 2 , ⋯ , M
G ∗ = G − x
R M S E = 1 n ∑ i = 1 n | | P i − P i ′ | | 2 2 M ean = 1 n ∑ i = 1 n | | P i − P i ′ | | 2 2 M a x = max { | | P i − P i ′ | | 2 2 } , i = 1 , 2 , ⋯ , n
z ( x ) = c x 2 1 + 1 − ( k + 1 ) c 2 x 2 + A 4 x 4 + ⋯ + A 2 n x 2 n , n = 2 , 3 , 4 , ⋯
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