Study on optimization of the dynamic performance of the robot bonnet polishing system

To solve the problem of processing quality degradation due to vibration in the robot bonnet polishing system (RBPS), the vibration generation mechanism of the polishing system is revealed based on the modal analysis and dynamic modeling of the RBPS. The modal analysis results showed that the operating frequency of the system is near the natural frequency, which makes the system susceptible to resonance. On the other hand, the forced vibration caused by the polishing force leads to high vibration at the end of the robot during the polishing process. In order to suppress the vibration generated by the polishing system and optimize the dynamic characteristics of the polishing system, a vibration suppression method to increase the damping ratio of the system was proposed. The experiments of SiC fixed point polishing were performed using both vibration suppression bonnet and original bonnet. The surfaces RMS and PV of the parts polished by the vibration suppression bonnet were generally better than those of the original bonnet. The vibration suppression bonnet has improved the convergence ratio of RMS and PV by 42.59% and 19.56% respectively over the original bonnet in the whole surface polishing experiments. The PSD analysis illustrated that the vibration suppression bonnet can better suppress the intermediate frequency errors. The effectiveness of the method in suppressing vibration and improving machining quality is demonstrated.


Introduction
Bonnet polishing technology with CNC machine tools has been widely used in the manufacturing of optical components for large optical engineering projects [1][2][3], but there are outstanding problems in the use of bonnet polishing machine tools, such as high development costs, large floor space, and insufficient adaptability to the size of the processing workpiece. The bonnet polishing system with industrial robot can better solve the above problems [4,5]; however, the robot system is easy to produce vibration compared to the machine tools, which is a common problem. In order to determine the causes of vibration, the vibration generation mechanism has been studied. The vibration types can be classified as free vibration, self-excited vibration, and forced vibration. Many vibration suppression methods were put forward to improve the stability and processing accuracy of the robot processing system [6,7].
Guo et al. [8,9] studied the vibration mechanism of robot boring processing considering the characteristics and stiffness characteristics of robot boring. The mechanism showed that self-excited vibration and forced vibration of the boring bar radial direction are the causes of system vibration. In order to suppress vibration, a presser foot mechanism was designed to offset the effect of cutting force and suppress system vibration.
Li et al. [10,11] studied the relationship between the force on the end of the robot and vibration generation during robotic milling machining through modal analysis experiments. The machining effect was improved by optimizing the tool path and workpiece clamping position.
Li et al. [12] divided the robotic arm into a robotic arm subsystem and a hydraulic subsystem. Based on the Lagrange principle and the hypothetical modal method, the dynamics model of each subsystem was established. The overall dynamics model of the robotic arm system was obtained according to the Jacobian matrix in the subsystem model. The global sliding mode control method and the adaptive inverse sliding mode control method were used to suppress the vibration at the end of the system.
Chen et al. [13] designed an integrated force controller to ensure a constant contact force between the workpiece and the end of the robot through force sensor, gravity compensator, and PI controller. Their experimental results showed that the range of variation of the contact force was reduced from 8 N to 1 N and the machining quality of the workpiece was improved by adding this controller.
Sun et al. [14] designed a robot ultrasonic milling system. At the end of the robot, an ultrasonic device was added to reduce the force and vibration generated by the self-excited vibration of the system. The feasibility of the method was verified by comparing the machining results without the adding of the ultrasonic device.
Wang et al. [15] designed an active force control device for the grinding of thin-walled parts to control the constant contact force between the workpiece and the end of the robot and to suppress system vibration. The effect of the device on the suppression of system vibration was verified by modal experiment and finite element analysis method. The relationship between the grinding force and the system vibration response was also established. The effect of the device on the improvement of machining quality was verified by experiments.
The above papers reveal that the self-excited vibration and the forced vibration are the main causes of vibration in robot systems. The effects of self-excited vibration and forced vibration on the system are usually investigated separately by means of modal analysis and dynamic modeling. Bonnet polishing has a higher offset during the polishing process, and the characteristics of the precession motion result in a different force situation at the end of the robot than in the papers mentioned above. Moreover, the SiC material is a hard and brittle material, which leads to a higher polishing force that is required to remove the material. All of these will cause that the dynamic characteristic of the entire polishing system is different significantly from other systems and a specific analysis of the causes of vibration generation is necessary. Furthermore, the works in the papers mentioned above, vibration suppression is achieved by adding specific mechanisms, force control devices, and ultrasonic devices to the processing system, which will result in an increase in the cost and size of the processing system development and may cause a change in the inherent characteristics of the processing system. Therefore, this paper combines the characteristics of the SiC polishing process to investigate the vibration generation mechanism of the RBPS, proposes a vibration suppression method to reduce the vibration at the end of the system, and improve the stability of the polishing force and material removal during the polishing process so that the system processing accuracy can be improved. The framework of the study is shown in Fig. 1.
In Section 2, the vibration mechanism of the RBPS is studied, and the reasons for the vibration of the system and the impact of vibration on the machining are analyzed from the two aspects, the self-excited vibration and forced vibration of the system.
In Section 4, a vibration suppression method is proposed to optimize the dynamic characteristic of the system. Finally, the optimization effect is verified by experiments. Fig. 1 The framework of the study 2 Study of the vibration mechanism of the RBPS

Modal analysis of the RBPS
The RBPS studied in this paper is shown in Fig. 2. The robot model is ABB IRB6700 with a polishing tool at the end of the robot.
The FEA model of the RBPS established during modal analysis is shown in Fig. 3, which is the same as in the literature [16], with the difference being the foundation material and the boundary conditions imposed.
(1) Material. The virtual plane of the bolt connection is represented by virtual material 1. The foundation layered soil is represented by virtual material 2. The material parameters are shown in Table 1.
(2) Constraint. The connection between each joint of the robot and the drive motor of the polishing tool is set to revolute joint, and the rest of the connections are bonded contact. The joint stiffness values are shown in Table 2 [17].
(3) Mesh. The robot body is meshed to a size of 20 mm, the elastic foundation is meshed to a size of 40 mm, and the element type is solid185. (4) Load. Fix support is used on the sides and bottom of the robot's elastic base. Two sinusoidal excitations of 60 N in the X and Y directions with a phase difference of 90° are loaded to the end of the robot in the harmonic response to simulate centrifugal forces.
The deformation results for each modal order are shown in Fig. 4. The results of the modal analysis are shown in Table 3.
Combining the results shown in Fig. 4 and Table 3, it can be seen that there are six orders of modes within the operating frequency range of polishing (250 rpm to 1500 rpm is the common rotational speed for bonnet polishing), each of

Gray Cast Iron
Concrete+Dinas which may cause resonance in the system during the polishing process. In order to determine the natural frequency of the polishing system in practice, it was experimentally demonstrated that the system operating frequency is in the range of the system's natural frequency.

Vibration measurement experiment
A laser vibrometer and an oscilloscope were used to measure the displacement response amplitude of the key joints of the bonnet polishing system at idle, as shown in Fig. 5. The robot was idled at a variable speed of 50 rpm across the span, and the displacement amplitude of the key joints of the robot was measured from three directions using the laser vibrometer. Meanwhile, an oscilloscope was used to capture the voltage signal from the laser vibrometer to display the displacement waveforms of the measured joints. The results are shown in Fig. 6.
The experimental results are compared with the simulation results as shown in Table 4. In the actual measurements, the first-and third-order modes of the polishing system could not be collected.
From the results in Table 4, it can be seen that the error between the simulation results and the experimental results is less than 18%. The error between experimental and numerical method is very small in determining the other modes given in Table 4, but the error between experimental and numerical in the second mode is 17.9%. There are three presumptions for the error. First, the calculated value is the relative error of a particular order of mode. Comparisons of errors between different modes are not meaningful. Second, the absolute error in the second mode is 1.58 Hz. The difference in absolute error between the different modes is not large. The relative error is larger due to the lower frequency of the second mode. Third, errors due to the accuracy of the measurement instruments, the accuracy of the simulation model and the measurement conditions also have an impact on the results. The materials, load conditions, and structure of the robot in the simulation differ from those that exist in reality. The error is within a reasonable range, which proves the correctness of the obtained natural frequency of the system. Therefore, the system operating frequency near the natural frequency is one of the reasons for the vibration during polishing. In addition, the low stiffness of the system is also responsible for the vibration due to the forced vibration. Therefore, it is significant to study the dynamics of the system.

Dynamic modeling of the RBPS under force
As shown in Fig. 7, which is the schematic figure of bonnet polishing, O is the center of polishing tool, h is offset, ρ is precession angle, and n is rotational speed.
From the theoretical analysis of static stiffness in the literature [16] and Fig. 7, it can be seen that the dynamic polishing force can be decomposed into a normal force and a tangential frictional force in the process of polishing, which acts at the end of the robot as the reaction force. Therefore, the vibration of the RBPS can be simplified to the plane where the polishing reaction force F n and friction force F t are located and analyzed as a two-degree-of-freedom problem.
As shown in Fig. 8, the robot's stiffness model is an ellipsoid in three dimensions when it is in a certain position, and then it will behave as an ellipse in two dimensions. By simplifying the robot stiffness ellipsoid shown in Fig. 8a to the plane in which F n and F t are shown in Fig. 8b, the maximum and minimum values of the robot stiffness are obtained in the direction of the semi-axes O x1 and O x2 of the stiffness ellipse respectively, corresponding to the two semi-axis lengths k 1 and k 2 of the ellipse.
According to the literature [8], it is known that the stiffness matrix corresponding to the semi-axes O x1 and O x2 directions of the ellipse at this point is a diagonal array, so the RBPS is set up with the semi-axes O x1 and O x2 directions of the ellipse as axes of coordinates for the main stiffness coordinate system, thus eliminating its coupling, as shown in Fig. 9.
In Fig. 9, δ, which is the angle between O x2 and F n , can be solved based on the robot's stiffness ellipse. From the static stiffness theory of the robot in [16], the stiffness matrix K of the robot can be decomposed into four threedimensional sub-matrices in a certain position, of which K fd is the force-linear displacement stiffness matrix. Since where e O2 is the directional vector of O x2 and e F n is the directional vector of F n .
Thus, in the main stiffness coordinate system in the x 1 Ox 2 plane shown in Fig. 9, when the system is subjected to a polishing force, the robot will generate forced vibrations in two mutually perpendicular directions, O x1 and O x2 , which can be expressed in terms of the dynamic equations as  where m is the equivalent mass of the system; c 1 and c 2 are the damping coefficients of the system in both directions O x1 and O x2 along the main stiffness coordinate system, respectively; k 1 and k 2 are the stiffness of the system in both directions O x1 and O x2 along the main stiffness coordinate system, respectively; F 1 and F 2 is the polishing force in both directions O x1 and O x2 along the main stiffness coordinate system, respectively.  The dynamic model is shown in Fig. 10.
Considering the influence of the spindle rotation on the direction of offset, a sinusoidal fluctuation of a fix amplitude is introduced to the polishing pressure, by substituting the polishing pressure from the dynamic model into the sinusoidal force F n sin ωt for analysis, where ω is the operating frequency and is related to the rotational speed.
The frictional force F t at this point is calculated according to the following equation according to the literature [18]: where μ is the interfacial friction coefficient. According to the dynamics model shown in Fig. 10, if the angle between the polishing pressure of the robot in the current position and the semi-axis O x2 of the stiffness ellipse is δ, the geometric relationship in the figure gives the two polishing (4) F t = F n sin t forces F 1 and F 2 in the direction of the two axes O x1 and O x2 of the main stiffness coordinate system as F n sin δ sin ωt + F t cos δ and F n cos δ sin ωt − F t sin δ, respectively, i.e., x 3 Fig. 9 The main stiffness coordinate system ξ 1 , ξ 2 , ω 1 , and ω 2 respectively can be calculated according to the following equation: From the forced vibration theory of vibration mechanics, the steady-state response of Equation (6) can be obtained as where X 1 and X 2 are the amplitudes of steady-state response and φ 1 and φ 2 are the phase differences of the steady-state response. X 1 , X 2 , φ 1 , and φ 2 respectively can be calculated according to the following equation: where λ 1 and λ 2 are the frequency ratios, which can be calculated as The dynamic model of the polishing system is established. According to the simulation analysis of the dynamic model, the steady-state response curves of the system under different process parameters can be obtained.

Simulation of the steady-state response of the RBPS
The specific vibration analysis is carried out in combination with the process parameters in the actual polishing process. The offset h is taken as 0.6 mm and 1 mm, the rotational speed n is taken as 500 rpm and 1000 rpm, and the corresponding F n and interfacial friction coefficient μ are taken with reference to the experimental values in the literature [19]. K fd in a certain position can be calculated according to the robotic posture. According to Equation (9), the maximum and minimum values in the XZ plane, k 1 and k 2 , are obtained as 1.476 × 10 3 N/mm and 3.289 × 10 2 N/mm, respectively. δ between the direction corresponding to k 2 and the Z axis is 13.2°. The damping ratio and the equivalent mass m are determined by reference to the calculation method in the literature [20], where the equivalent mass is calculated by the following equation: where |K fd | is the Euclidean parameterization of K fd and ω n is the natural frequency of the system.
The results of the steady-state response curve for the RBPS in the O x1 direction are shown in Fig. 11.
The results in Fig. 11 showed that the amplitude of the forced vibration generated by the force on the robot system is on the order of 10 -2 mm. From Equation (9), it can be seen that the amplitude of the steady-state response of the forced vibration is more significant in the operating band around the system's natural frequency, which seriously affects the polishing effect and requires suppressing the vibration.
According to Equation (8), the main factors affecting the amplitude of the steady-state response of the system include the stiffness, equivalent damping ratio, and equivalent mass of the system. Considering that the equivalent damping of the robot system is relatively small, the vibration suppression idea of this paper for the vibration generated by the RBPS near the natural frequency will start by improving the equivalent damping ratio of the system.

Vibration suppression study based on system equivalent damping ratio
For the vibration generated in the operating band near the natural frequency of the robot system, a vibration suppression method based on viscoelastic damping material properties and damping theory of structural dynamics commonly used in the study of damping structures for process systems is proposed, as shown in Fig. 12.
According to the analysis in Section 2.2.1, when the wave spring-rubber elastic element is incorporated into the system, the dynamic equations can be expressed as where C ex 1 is the equivalent damping coefficient of the wave spring-rubber elastic element along the O x1 direction in the main stiffness coordinate system and C ex 2 is the equivalent damping coefficient of the wave spring-rubber elastic element along the O x2 direction in the main stiffness coordinate system. Then, the amplitudes X 1 and X 2 and phase differences φ 1 and φ 2 of the steady-state response of Equation (12) can be expressed respectively as (12) mẍ 1 + c 1 + c ex 1 ̇x 1 + k 1 x 1 = F n sin sin t + F n cos sin t mẍ 2 + c 2 + c ex 2 ̇x 2 + k 2 x 2 = F n cos sin t − F n sin sin t (13) where ζ e1 is the damping ratio of the system along the O x1 direction after the addition of the wave spring-rubber elastic element and ζ e2 is the damping ratio of the system along the O x2 direction after the addition of the wave spring-rubber elastic element. ζ e1 and ζ e2 are calculated according to the following equation: Comparing Equations (9) and (13), it can be seen that when the wave spring-rubber elastic element is added to the RBPS, the amplitude of the vibration generated by the system in the operating band around the natural frequency will be reduced due to the increase in the overall damping ratio. The vibrations of the RBPS will also be suppressed.
In order to analyze the damping effect of this elastic element from a theoretical point of view, it is first necessary to obtain the equivalent damping ratio of the wave springrubber elastic element. A simplified simulation model of this structure is shown in Fig. 13.
In this case, the Neo-Hookean model [21] was chosen as the hyperelastic instanton model for the rubber ring. After the simulation run, the displacement of the upper compression plate surface was viewed in ANSYS post-processing, and the hysteresis loop was plotted based on the load and displacement data. The dynamic hysteresis loop of the wave spring-rubber elastic element for excitation of 20 Hz frequency and 4 mm amplitude can be obtained, as shown in Fig. 14.
Combining Equation (15) with the method shown in Fig. 15 [20], the loss factor of the wave spring-rubber elastic element η = 0.115 is calculated as where B is the transfer force corresponding to the maximum displacement deformation of the hysteresis loop and C is the absolute length of the load on the hysteresis return line when the simulation model displacement is zero.
According to the relationship between the material damping ratio and the loss factor in structural dynamics analysis as in Equation (16), the equivalent damping ratio ζ e of the wave spring-rubber elastic element at different excitation frequencies can be obtained as Finally, by converting ζ e into two directional components in the main stiffness coordinate system and substituting them into Equations (13) and (14), the steady-state response curve  of the RBPS can be obtained after the addition of the wave spring-rubber elastic element. The steady-state response of the RBPS is shown in Fig. 16, where the offset is taken as h = 0.6 mm and h = 1 mm respectively, and the rotational speed is taken as n = 250 rpm, 500 rpm, 1000 rpm, 1250 rpm, and 1250 rpm, respectively. As shown in Fig. 16, in the operating frequency range corresponding to the rotational speed n = 250~1500 rpm, the elastic element can suppress the vibration to a certain extent when the operating frequency is close to the natural frequency of the system. However, due to the difference in operating frequency caused by the difference in rotational speed, there is also a certain difference in the vibration damping capacity of the vibration suppression structure to the system.
From an overall perspective, as the equivalent damping of the original RBPS is relatively low, the damping performance of the vibration suppression structure is better than that of the original structure when the operating frequency of the system is around the natural frequency, which is more beneficial for improving the dynamic performance of the polishing system. In order to verify the optimized performance of the vibration suppression bonnet, the original bonnet and the vibration suppression bonnet are used respectively to perform polishing experiments on SiC to verify the optimized effect of the vibration suppression bonnet on the dynamic performance of the RBPS.  Figure 17a shows the original bonnet, and Fig. 17b is the vibration suppression bonnet. According to the experimental scheme shown in Table 5, the SiC fixed spot polishing experiments were carried out using the original bonnet and the vibration suppression bonnet, respectively. The surface shapes of SiC were measured with ZYGO NV7300 after polishing. The part experimental results are shown in Fig. 18.

SiC fixed spot polishing experiments
The numerical results are shown in Fig. 19.
According to the results shown in Fig. 19, the surface quality of SiC after the vibration suppression bonnet process is significantly better than that of the original bonnet. In     From the figure and the results of the analysis above, it can be seen that the values fluctuate over a wide range when the inner pressure is a variable. The inference is that inflation causes deformation of the bonnet, resulting in a change in bonnet stiffness, which ultimately affects material removal.
The next experiment was carried out to verify the optimization of the vibration suppression bonnet in the whole surface polishing of SiC.

SiC whole surface polishing experiments
The tool path was grating path and the other experimental process parameters were set as shown in In addition to the main influencing factors of the tool path and tool influence function, the vibration during the polishing process is also a key factor in generating the intermediate frequency error on the surface of SiC. Therefore, to verify that the vibration suppression bonnet can suppress the intermediate frequency error, the power spectral density (PSD) of the SiC surface shapes after polishing was extracted and calculated for the two bonnets, respectively. The result is shown in Fig. 22.
According to the result shown in Fig. 22, the X-direction is the scanning direction of the machining path, and the Y-direction is the feed direction. As the raster path is used for the whole surface polishing, a significant ripple error is caused in the Y-direction. As shown in the partial enlargement of the figure, the magnitude of the ripple error caused by polishing the entire surface of the ordinary bonnet is greater than that caused by the vibration suppression bonnet. The vibration suppression bonnet has a certain improvement effect on the intermediate frequency error generated by the polishing process.
In summary, the optimization effect of vibration suppression bonnet in SiC whole surface polishing is not as noticeable as that of spot polishing. The reason for this analysis is the low motion accuracy of the robot. Due to more joints and low stiffness of the robot, the robot generated large motion errors during the motion, which caused the instability of factors such as the size of the polishing spot and the polishing force, and finally led to low processing accuracy. The vibration suppression method can be proved to be feasible and effective from the above experiment results.

Conclusion
In this paper, the dynamic performance of the RBPS was analyzed. A method for optimizing the dynamic performance of the polishing system based on wave spring-rubber elastic element was proposed. The vibration of the robot system near the natural frequency is suppressed. The main contents are as follows.
For the self-excited vibration of the RBPS, based on the FEA method and modal measurement experiments, the system modal parameters were obtained, thus clarifying that the operating frequency of the system near the natural frequency is one of the reasons for the vibration of the system.
The dynamics model was established for the forced vibration of the RBPS. The steady-state response curves of the system subjected to dynamic polishing forces are obtained. The analysis results show that the vibration has a large impact on the polishing effect and needs to be suppressed.
The suppression method based on a wave spring-rubber elastic element was proposed for the vibration generated near the natural frequency of the robot system. By increasing the damping coefficient of the system, the vibration generated at the end of the robot is suppressed. The experimental results show that the vibration suppression bonnet has a better ability to optimize the dynamic performance of the RBPS compared to the original bonnet.

Data availability
We declare that some of the data of this study are available within the article.