Article

# An adaptive trajectory planning algorithm for robotic belt grinding of blade leading and trailing edges based on material removal profile model

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## Abstract

Robotic belt grinding of the leading and trailing edges of complex blades is considered to be a challenging task, since the microscopic material removal mechanism is complicated due to the flexible contact state accompanied with greatly varying curvature that finally affects the machined profile accuracy. The resulting poor accuracy of blade edges, to a great extent, is attributed to the trajectory planning method which less considers the dynamics. In this paper, an iso-scallop height algorithm based on the material removal profile (MRP) model is developed to plan the tool paths by taking into consideration the elastic deformation at contact wheel-workpiece interface. An improved constant chord-height error method considering the influence of elastic deformation is then proposed to adaptively plan the grinding points according to the curvature change characteristics of the free-form surface. Based on these two steps, a MRP model based adaptive trajectory planning algorithm is constructed to enhance the profile accuracy facing the robotic belt grinding operation. Simulation and experimental results demonstrate the effectiveness of the proposed trajectory planning algorithm for the robotic belt grinding of blades from the perspectives of surface roughness, profile accuracy and processing efficiency. Particularly this technology serves to solve the problem of over-cutting at the blade leading and trailing edges.

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... A numerical iteration method was developed by Han et al. [10] to determine the path spacing that achieves physically uniform coverage of a NURBS surface. A path spacing planning algorithm for belt polishing of a blade surface is established by Lv et al. [25]. The geometric derivation of the path spacing based on the constant chord-height error constraint is quite complicated. ...
... In this research, the abrasive disc is chosen as the polishing tool, which has merits such as light weight and high polishing efficiency [13]. It should be noted that this model can be generalized to other types of polishing tools such as the spherical tool [12] or belt tool [25]. When the polishing disc presses on the workpiece with a tilt angle, it deforms and generates a contact area wherein the contact pressure is positive. ...
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Off-line programming of the polishing tool trajectory for complex workpieces is challenging due to the non-trivial material removal model and the polishing accuracy requirement. Current tool trajectory planning methods are mainly developed for some simple surfaces, but cannot handle the increasingly complicated industrial parts, such as the wheel hubs. This paper first develops a numerical contact mechanics model for the point-sampled complex workpieces. The contact pressure distribution and the material removal depths on the workpiece point cloud can be predicted efficiently. A novel high-priority subregion searching algorithm is developed to track the most-worth-polishing workpiece points. By selecting the path pattern as direction-parallel, the path direction, tool dwell times, and the path spacings inside each extracted subregion are optimized to minimize the deviation from the desired material removal depths. The effectiveness of the proposed method is verified by performing disc polishing simulations on workpieces with different shapes. A robotic polishing experiment is also conducted on a wheel hub. Both simulation and experimental results show that reasonable tool trajectories can be generated on the workpiece, and desired material removal depths can be achieved.
... Nowadays, due to significant improvements of both advanced abrasive belt and processing equipment, belt grinding (BG) or belt finishing (BF) -a kind of abrasive machining methods-has been widely implemented. The application fields of that are expanding to aerospace manufacturing [4][5][6], mobile industry [7][8][9], railway maintenance [10][11][12], and robot-assisted technology [13][14][15], etc. Abrasive belt finishing has demonstrated excellent polishing performance of inducing fine surface roughness, suitable flat BAC shape, high grinding ratio, and great consistency, benefiting from its features of compliantly soft contact [16] and one-time use abrasive belt [3]. Thus, some researchers have successfully introduced BF to improve surface integrity after HT. ...
Article
Considering the wide application of dry belt fining and the importance of residual stress for bearing parts, this paper has characterized the residual stresses distribution along the depth generated by dry belt finishing after hard turning on hard materials AISI52100. The influences of applied force and finishing duration on residual stresses were experimentally investigated. The results were then theoretically discussed and further analyzed through 2D multi-scratching finite element model. Experimental measurements showed dry belt finishing produced considerable compressive residual stresses in the external layer. The insignificant effect of applied force on residual stress was caused by an essential reduction of sharing individual force on every useful local grit. Although increasing scratching numbers could intend the amplitude of residual stresses slightly as well as the influenced field depth range, the material removal process on the superficial layer would partly reset them, resulting in an ignorable effect of finishing duration. Besides, an improved measuring method with sufficient penetration depth resolution of both X-ray and electrochemical removal is expected to characterize residual stresses distribution of belt finishing more appropriately.
... Abrasive belt grinding, which is an efficient ultra-precision processing method, is frequently used in the last step of parts processing [1][2][3]. Different from rigid contact machining, such as turning and milling, because of the flexible contact characteristics between the abrasive belt and the workpiece, belt grinding offers remarkable advantages in processing complex surfaces and difficult-to-process materials. Some examples of the latter are titanium alloys, nickel alloys, and other new composite materials [4,5]. ...
Article
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In the abrasive belt grinding process, actual material removal is an important parameter that affects its accuracy. At present, for obtaining the actual material removal, offline measurements are required to establish the mathematical prediction model. To improve the accuracy and efficiency of abrasive belt machine grinding, this paper proposes a novel method for monitoring material removal using multiple sensors and a two-dimensional (2D) convolutional neural network (2D-CNN) learning algorithm. In this method, features of multiple types (color, texture, and shape) are extracted from vision signals, and that of multiple domains (time, frequency, and time–frequency domain) are extracted from sound and tactile signals. These features are constructed into a 2D feature matrix as the input model, and the 2D-CNN prediction model is established between the multisensor features and the material removal rate of the abrasive belt grinding process. An experimental dataset is used to train and verify the established model. The results show that the proposed method can identify that sensor signals are sensitive to the material removal rate. After optimizing and tuning the model parameters, the coefficient of determination of the prediction results is as high as 94.5% and the root mean square error is 0.017. Therefore, the proposed method can be employed for the prediction of material removal rate for different belt specifications and different grinding parameters. Compared to traditional machine learning methods, this method can yield better training results without feature selection and optimization.
... Wang et al. [3] proposed an abrasive belt polishing path planning method for precise material removal based on Hertz contact theory. Yuan et al. [4] carried out adaptive trajectory planning for the leading and trailing edges of the blades by using robot abrasive belt polishing. ...
Article
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Industrial robot-assisted abrasive cloth wheel (ACW) accurately polish blades is considered to be a challenging task, and it is necessary to realize the digitalization of the process. Due to the flexible contact properties of the abrasive cloth wheel and the curvature change of the blade surface, the microscopic material removal is not uniform and the blade surface roughness value is large. In this paper, a finite element simulation model of the contact between the abrasive cloth wheel and the blade is established, and analyze the contact profile and pressure distribution pattern in the contact area. Then use NURBS curve to extract the blade polishing area curve, and considering the flexible contact deformation between the abrasive cloth wheel and the blade surface when planning the step length and row spacing. The flexible adaptive trajectory planning method is simulated by offline programming software. Finally, experiments were carried out on a four-station wheel changing polishing platform. Simulation and experiments results show that the proposed flexible adaptive trajectory planning method can make the surface roughness of the convex and the concave Ra≤0.3μm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{a} \le 0.3\;{\mu m}$$\end{document}, the surface roughness of the leading and trailing edges Ra≤0.2μm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{a} \le 0.2\;{\mu m}$$\end{document}, and the total polishing efficiency increased by about 9.4%.
... As a type of precision machining technology, belt grinding is commonly used for the grinding of free-form surfaces in aerospace and naval ships [1][2][3]. With the uninterrupted development of these industrial fields and the continuous improvement of the machining precision and surface quality requirements for component fabrication, the requirements of high abrasive belt grinding accuracy are also increasing [4,5]. The precision of a grinding process depends on the precision of the material removal model, which is typically affected by many factors such as the grinding parameters, workpiece parameters, and abrasive belt wear status [6]. ...
Article
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The wear state of an abrasive belt is one of the important factors affecting the grinding precision of belt grinding processes. At present, there are two problems associated with the monitoring method of the wear condition of abrasive belts: (1) there are no uniform wear criteria to classify the wear condition of abrasive belts, and the segmentation threshold of the wear condition is affected by the change in the grinding parameters; and (2) an abrasive belt wear model based on indirect sensor monitoring of signals is affected by the change in the grinding parameters of abrasive belts; therefore, it is only suitable for abrasive belt wear monitoring under specific grinding parameters. This paper introduces a method of belt wear state monitoring based on machine vision and image processing. Surface images of an abrasive belt during its entire life are captured using a noncontact electron microscope. Three image features related to the wear state are selected: first-order distance of color component R, entropy of the horizontal subgraph, and vertical subgraph of the texture feature. Moreover, the wear state is classified into three categories based on the selected features. Using the selected features and the random forest classification algorithm, an abrasive belt wear state classifier is established. The performance of the classifier is verified and evaluated using a data subset of different images. The results show that the proposed method has high recognition accuracy for belt wear state, which can reach 99% in the accelerated wear stage. The proposed method solves the problem of the dependence and sensitivity of the monitoring model on the variation in the grinding parameters in the process of abrasive belt wear monitoring, and it improves the adaptability and versatility of the monitoring method.
... For example, Borges et al. [17] applied the Gauss-Newton method to fitting calculation to solve the accuracy problem, and the result indicated that faster convergence could be obtained with this method. Zhu and Lv et al. [18,19] used an improved constant chord-height error method to enhance the profile accuracy in the robotic belt grinding operation, and the result demonstrated the effectiveness of the proposed path planning algorithm. Park et al. [20] selected value points according to the complexity of the curve by improving the least-squares minimization method, resulting in the reduction of the calculation amount. ...
Article
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The processing efficiency of dentures is generally quite low due to the complex profile form and high brittleness of the dentures. However, there is little research for efficient path optimization and look-ahead speed control algorithm in denture machining field, which hinders the improvement of the machining efficiency and denture quality. In this work, an optimization method for both straight transition and curve paths was proposed, and look-ahead speed control algorithms for different machining paths were designed. The anomalous points were deleted according to the preprocessing algorithm before the path optimization. The straight transition path and curve path were fitted by the quasi-parabola model and cubic B-spline curve, respectively. The error between the optimized path and original path was analyzed, and the results demonstrated that the path optimization methods were reliable. A look-ahead speed control model based on S-type speed control was proposed, and the boundary conditions for different paths were calculated. The simulated and experimental results indicated that optimized algorithm significantly improved the machining efficiency and reduced the vibration of the machine tool. The accuracy and surface roughness of the denture processed using the optimized algorithm met the medical standard. This work can provide a theoretical guidance for high-efficiency and precision machining of dentures of glass ceramics.
... As a kind of precision machining technology, belt grinding is widely used for the grinding of free-form surfaces in aerospace and naval ships area [1][2][3]. With the successive development of industrial field and the continuous improvement of machining precision and surface quality requirements for parts, the requirements of abrasive belt grinding accuracy are also higher and higher [4][5]. The precision of grinding process depends on the precise material removal model which are usually affected by many factors, such as grinding parameters, workpiece parameters and abrasive belt wear status [6]. ...
Preprint
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The wear state of abrasive belt is one of the important factors affecting the grinding precision of belt grinding processes. Accurate monitoring of abrasive belt wear can not only provide the basis for accurate material removal model to improve grinding accuracy, but also can replace the belt to avoid surface burn in time. However, most of the existing abrasive belt wear monitoring methods are only suitable for monitoring the belt wear state under specific grinding parameters, are not universal. This paper introduces a method of belt wear state monitoring based on machine vision and image-processing. All the surface images of the belt were obtained from the new belt to the worn-out of the belt by a non-contact electron microscope. The features of abrasive belt surface images are extracted from RGB color space and wavelet texture. By analyzing the trendency of the extracted features in the whole grinding process, the wear state is divided into three categories. Three image features related to the wear state are selected: the first order distance of color component R, the entropy of horizontal subgraph, and vertical subgraph of texture feature. Based on the selected features and the random forest classification algorithm, the wear state classifier of abrasive belt is established. The performance of the classifier is verified and evaluated by using the data subset of different images. The results show that the proposed method has high recognition accuracy for the belt wear state, and the accuracy can reach 99% in the accelerated wear stage. The proposed method is suitable for the monitoring of the belt wear state by the surface images of the abrasive belt measured under different grinding parameters and different measurement parameters.
... The influence mechanism of surface generation and the generation mechanism of residual stress in the grinding process have attracted the attention of many researchers Ding et al., 2020;Sun et al., 2021;Xiao et al., 2021). For the surface quality requirements of precision machining of complex parts, robotic grinding is also used as a method to control the surface integrity (Lv et al., 2020;Xie et al., 2020;Zhu et al., 2020). The grain orientation is a key parameter in the analysis of the microstructure effect. ...
Article
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Crystallographic texture is related to the anisotropy or isotropy of material physical properties, including mechanical performance. The crystallographic effect in micromachining is more significant than that in macro-processing owing to that the depth of the cut and the grain size are in the same order. It is of great significance to model the crystallographic texture evolution induced by mechanical and thermal load during micro-machining to investigate the surface integrity and performance of the finished product. This study performed hot deformation experiments of Al alloy 7075 (AA7075) under various input parameters, including the temperature, temperature rate, stain rate, and strain, which was designed using the Taguchi method. Following that, crystallographic orientation of the samples before and after the deformation was tested using electron back-scattered diffraction (EBSD). Then, the crystallographic texture evolution was modeled with the parameters obtained by fitting a part of the experimental data. The crystallographic texture evolution of AA7075 under different levels of input parameters is studied and analyzed. Finally, the sensitivity of crystallographic orientation evolution to the process parameter is analyzed. The results indicate that these four input parameters have a significant impact on some crystallographic texture of the specimens. The proposed model is instructive in the future investigation of micromachining and microstructure evolution.
... Wang et al. [3] proposed an abrasive belt polishing path planning method for precise material removal based on Hertz contact theory. Lv et al. [4] carried out adaptive trajectory planning for the leading and trailing edges of the blades by using robot abrasive belt polishing. ...
Preprint
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... where a p,total is the totally removal depth, which can be obtained by Preston equation, as Eq. (11) [20]. ...
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In the robotic belt grinding process, the elastic contact condition between the flexible tool and the workpiece is a critical issue which extremely influences the surface quality of the manufactured part. The existing analysis of elastic removal mechanism is based on the statistic contact condition but ignoring the dynamic removal phenomenon. In this paper, we discussed the dynamic contact pressure distribution caused by the non-unique removal depth in the grinding process. Based on the analysis of the removal depth of a single grit, we obtained the topology of a single groove. With the coupling analysis of the topology single groove and grit trajectories in elastic removing procedure, an elastic grinding surface topography model was established with the consideration of the dynamic contact condition in the removing process. Robotic belt grinding experiments were accomplished to validate the precision of this model, while the result showed that the surface roughness prediction error could be confined to 11.6%, which meant this model provided higher accuracy than the traditional predicting methods.
... However, the force controllers have common problems on the response speed, systems stability, and control accuracy due to complicated models. In order to overcome these problems, compliant end-effectors are presented to keep grinding force stable [15]. Chaoui et al. [16] designed a pneumatic compliant actuator to achieve a high-quality final grinding surface. ...
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Uneven surface quality usually occurs when grinding welds offline, which results non-uniform stress and then would damage the workpiece. In this paper, the robotic welding seam online grinding system based on laser vision sensor was proposed and built. A weld seam tracking software was developed and the data online interaction method of grinding system based on XML (Extensible Markup Language) file was applied. Firstly, hand-eye calibration model was built to convert data in the robot coordinate system. Then the weld profile information was extracted and stored in the data buffer area, and the coordinates of the robotic grinding point were transmitted through the self-developed weld grinding software. Finally, the vision system and the self-made grinding system were integrated at the end of the robot. The experiments were conducted to verify the reliability and practicality of this system and the proposed data interaction online method.
... Rail grinding has been applied worldwide as the most common means of routine maintenance [2]. Due to the merits of compliant grinding, efficient grinding, and cold grinding compared with the traditional abrasive wheel rail grinding, abrasive belt grinding technology [3,4] has recently become a novel rail grinding method [5,6]. Fig. 1 compares abrasive wheel and abrasive belt grinding technologies, illustrating the virtues of using an abrasive belt. ...
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Accurate material removal modeling is the basis for optimizing the surface quality and improving the performance of equipment components. In this study, a multi-sensor fusion method of vision and sound is used to monitor in-process grinding material removal rate (MRR). First, belt grinding experiments are conducted using different grinding parameters, and vision and sound signals are captured by industrial CCD cameras and an omnidirectional condenser microphone, respectively. Second, the features of the captured grinding spark images are extracted based on two aspects: color and texture, and those of the grinding sound are investigated in the time, frequency, and time–frequency domains. Moreover, the complementarity between the vision and sound signals and their sensitivity to different grinding parameters are discussed. Finally, based on feature-level fusion strategies, the Pearson correlation coefficient and the sequential backward selection algorithms are jointly used to select the optimal feature subsets. MRR prediction models are established using the selected feature subsets and an improved light gradient boosting machine (LightGBM) algorithm. The test results show that the error in the MRR prediction model of same-specification abrasive belts is less than 3 %, and the coefficient, R², is as high as 99.2 %. The proposed method can be used to predict the MRR resulting from a single grinding parameter and multiple ones, using the same-specification abrasive belts. Compared to other prediction models, the improved LightGBM model is superior in terms of the time factor without reduction in the accuracy of the model.
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Chapter
Large forging parts widely adopted in oil rigs, wind mills, large vessels and other complex equipment often carry random forging defects such as parting lines, burrs and high islands, which are traditionally removed through manual grinding by skilled operators. These random defects pose a big challenge to researchers interested in large forging parts grinding path generation by a CAD/CAM system. This paper proposes a new path generation method based on intelligent defect recognition of robotic grinding for large forging parts. A point cloud is first constructed of random defects on the surfaces of the to-be-ground parts from the unmatched points that emerge from the matching of the point clouds captured by a laser camera from the parts against those from the standard parts, a process employing both a random sample consensus algorithm and a modified iterative closest point algorithm. Then, the grinding path generation strategy is established by sorting the random defects according to a law of area on the fitting surface and robotic grinding motion programs are generated by transferring the coordinates of the random defects from the laser camera frame into the robot base frame. Finally, robotic grinding tests are conducted to verify the identification accuracy of the proposed new method. Results of the tests indicate that the method has accurately identified all random defects on a 10-m long forging part and intelligently generated subsequent robotic grinding paths according to the identified random feature categories. This study therefore provides an intelligent tool for finishing large forging parts.
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In order to solve the problem of fluctuation in the six-axis NC polishing machining, in this paper, a post-processing optimization algorithm for six-axis NC polishing machining is proposed, which can solve the problem of fluctuation in the machining process of six-axis NC machining. First, the initial A-axis angle value corresponding to the tool position is solved according to the tool position information contained in the toolpath file; Second, use B-spline to smooth the data of A-axis, and solve the corresponding B-axis and C-axis angle value according to the angle value of A-axis; Third, calculate the moving value of X-axis Y-axis and Z-axis to realize the post-processing of the toolpath file; Finally, the algorithm is verified that it can solve the machine trajectory fluctuation in the current six-axis post-processing using VERICUT and the machining experiments, and realize the high quality six-axis NC polishing machining.
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The effect of abrasive grain size on the material removal performances of the grinding surface was studied by theoretical modeling and grinding experiments. The results indicted that a smaller abrasive grain size of the abrasive belts led to smaller microscopic contour height and surface roughness of the ground surfaces, fewer curl chips, and more spherical chips. The smaller grain sizealso led to smaller macroscopic grinding depth and material removal rate. There was an approximately linear relationship between the macroscopic grinding depth and material removal rate during plane grinding. The slope was related to the abrasive grain size. Grinding force, vibration, vision, and sound signals could distinguish the macroscopic material removal changes caused by abrasive belt particle sizes.
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Double-sided tools system is a promising design to counteract the forces caused by tools, and has become a practical way to machine thin-walled parts with the twisted structure. However, it is still difficult to obtain uniform machining effects on the part surfaces. This paper proposes a coordinated motion planning method for coordinating two tools in the double-sided tools system. The tools are devoted to guaranteeing the uniform surface quality while meeting the constraints about tools' velocity, tools' inclination and the system structure. At first, a velocity-dependent model is presented to describe the two tools' motions along the predefined paths. Then, the motion-planning problem of two tools is formulated as a constrained velocity-optimization problem, and the genetic algorithm is proposed to achieve the optimization target of the surface uniformity. Finally, experiments are performed on a real blade in a self-developed double-sided ultrasonic surface rolling process machine tool. The results demonstrate that the proposed coordinated motion planning method can effectively improve the surface uniformity in the double-sided tools system.
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The problem of the path planning method is regarded as one of the key bottlenecks in the robotic belt grinding system. It has a significant impact on the surface quality of processed workpieces with complex surfaces. To improve the surface quality, a novel path planning method is proposed in this article, which generates the grinding path containing both the grinding position and orientation for an industrial robot. First, the theoretical grinding trajectory is obtained by selecting the key contact points on the 3-D model and calculating the inverse solutions of the B-spline curves. Next, based on the curvature variation rate and curve length criterion, the key contact points are further refined to produce the target points in the sensitive area of the workpiece where surface curvature changes abruptly or surfaces intersect. Finally, the grinding orientation at each target point location is obtained according to the solutions of the surface equation constructed using bicubic B-spline interpolation. To validate the method, a faucet is used for contrast simulation and experiment. The results demonstrate that the proposed method performs better than the section method in terms of both planning efficiency and surface quality. With this method, the surface roughness value $R_{a}$ of the processed workpiece can be controlled within 0.086 $\mu$ m without the risk of overcut, which is ideal for robotic belt grinding.
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The elastic state at contact wheel–workpiece interface is a critical issue during robotic belt grinding process that significantly influences the finishing profile accuracy. Establishing a reasonable undeformed chip-thickness (UCT) model that suits to this operation is considered a feasible approach to clarify the cutting mechanisms. In the present paper, an elastic state–driven robotic belt grinding chip-thickness model is established to predict the workpiece surface roughness. In this new model, the combined modulus of elasticity of the contact wheel is calculated according to the formula of Young’s modulus, and the exponent with respect to the effects of linear and nonlinear deflection is further determined based on the energy balance hypothesis. Experiments are conducted to verify the reasonability of the improved chip-thickness model from the perspective of surface roughness, and the findings are likely to clarify the differences in material removal mechanism between wheel grinding and robotic belt grinding essentially.
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A tentative study from the perspective of abrasive grain geometry in this paper is conducted to investigate the specific energy and energy efficiency for clarifying the robotic belt grinding mechanisms. The energy efficiency model is established based on the friction coefficient model of the single spherical grain, then the experiments and simulation are implemented to energetically evaluate the microscale material removal mechanisms from the specific energy contributions. It has been demonstrated that the specific plowing energy is more predominant than both the specific cutting and sliding energy in robotic belt grinding, resulting in the energy efficiency ranges between 17 and 41 %. Both the large grain size and normal contact force can be taken as optimization strategies to maximize the energy efficiency for material removal.
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Vibratory finishing is widely used in mass-finishing of components. However, it is difficult to guarantee shape accuracy and predict surface roughness evolution. Here, a material removal model is proposed that is based on Preston's law and statistical overlapping of particle trajectories computed by Discrete Element Method (DEM). After validating DEM motion by particle image velocimetry and rope experiment, the occupancy ratio is studied as function of media type and mixing. Predictions of removal volume and surface roughness agree with experimental results within 12% on average, making this simulation method a very practical tool that reduces the need for extensive trial runs and barrel reloading.
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Robotic belt grinding has emerged as a finishing process in recent years for machining components with high surface finish and flexibility. The surface machining consistency, however, is difficult to be guaranteed in such a process. To overcome this problem, a method of hybrid force-position control combined with PI/PD control is proposed to be applied in robotic abrasive belt grinding of complex geometries. Voltage signals are firstly obtained and transformed to force information with signal conditioning methods. Secondly, zero drift and gravity compensation algorithms are presented to calibrate the F/T transducer which is installed on the robot end-effector. Next, a force control strategy combining hybrid force-position control with PI/PD control is introduced to be employed in robotic abrasive belt grinding operations where the force control law is applied to the Z direction of the tool frame and the positon control law is used in the X direction of the tool frame. Then, the accuracy of the F/T transducer and the robotic force control system is analyzed to ensure the stability and reliability of force control in the robotic grinding process. Finally, two typical cases on robotic belt grinding of a test workpiece and an aero-engine blade are conducted to validate the practicality and effectiveness of the force control technology proposed.
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For the past three decades, robotic machining has attracted a large amount of research interest owning to the benefit of cost efficiency, high flexibility and multi-functionality of industrial robot. Covering articles published on the subjects of robotic machining in the past 30 years or so; this paper aims to provide an up-to-date review of robotic machining research works, a critical analysis of publications that publish the research works, and an understanding of the future directions in the field. The research works are organised into two operation categories, low material removal rate (MRR) and high MRR, according their machining properties, and the research topics are reviewed and highlighted separately. Then, a set of statistical analysis is carried out in terms of published years and countries. Towards an applicable robotic machining, the future trends and key research points are identified at the end of this paper.
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In order to improve the efficiency and precision of abrasive belt grinding of the free-form surface, a novel trajectory planning approach based on machining accuracy control is proposed in this paper. At the same time, a method to optimize the size of the contact wheel based on the diploid genetic algorithm is also presented. Then, the effectiveness of the method is necessary to demonstrate through an abrasive belt grinding experiment of the aero-engine blade. The results show that the blade profile accuracy after grinding by using the proposed method can better meet the corresponding tolerance requirements, and the surface quality and accuracy of blade profile are improved effectively.
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Robotic abrasive belt grinding has emerged as a finishing process in recent years for machining components with high surface finish owing to its advantages of excellent flexibility and high efficiency. The profile accuracy of components, however, is difficult to be guaranteed due to the contact wheel deformation in conformity with the surface of workpiece. To overcome this problem, an improved cutting force model is developed to analyze and assess the force-controlled robotic abrasive belt grinding mechanisms based on the experimental observation of the over- and under-cutting phenomenon on the cut-in and cut-off paths. Specifically, a material removal rate model considering the effects of cut-in and cut-off paths is firstly built to demonstrate the real robotic belt grinding process. The microscopic scale of cutting force model consisting of sliding, ploughing and cutting components is then introduced and its specific grinding energy is calculated to evaluate the cutting efficiency with orthogonal experiments. Finally the reasonable grinding parameters are obtained to improve the machining stability and energy efficiency, and a typical case on the robotic belt grinding of test workpiece is conducted to validate the practicality and effectiveness of the force model.
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This paper presents a novel two degrees of freedom (DOFs) contact force control method for robotic blisk grinding. The grinding tool is controlled to automatically adapt to the curvity change of the blisk blade and maintain a constant contact force as expected. A smart end effector is used as the actuating device for contact force control. The proposed force controller includes a gravity compensation module, a force prediction module, and a force-position controller. The direction and amplitude of the contact force are predicted with the force prediction module and are controlled with the force-position controller. The tool path of the robotic blisk grinding process is generated and optimized so that the contact points between the tool tip and the workpiece are evenly distributed along the grinding path. Both simulations and experiments are carried out to validate the effectiveness of the proposed method. The results show that the proposed method provides a good contact force control performance, with less than 1 N force fluctuation. The surface finish and roughness are significantly improved compared to the case without force control. The grinding efficiency is raised by about sixfold compared to the case with one DOF force control.
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As robots are increasingly involved in industry, robotic manufacturing techniques arise in order to replace their conventional counterpart. Advances in robotics led to improvements in the robot internal mechanics and in their control to be able to address more challenging applications such as friction stir welding, jet cutting, casting, etc... Robotic machining is also one of them as there is still much research to be done in the area. Combining the agility and the flexibility of the industrial robot, this economical solution may represent an alternative for finishing operations, especially for large workpieces coming from aeronautics or foundry industry. Nevertheless, robotic machining encounters some limitations coming from the lack of joint stiffness which influence the positioning accuracy of the cutting tool. This paper focusses on the modelling aspects of a flexible industrial robot compelled to machining operations. The latter was modelled as a multibody system comprising six degrees of freedom whose end-effector carried a payload representing the spindle holding the cutting tool. Moreover, each axis was supplemented with torsional springs and dampers to capture the joint and drive flexibil-ities. Milling forces were then computed and applied at the tool tip. Robot dimensions were recovered from CAD models and were used to analytically compute its inverse kinematics. After a description of the robotic machining environment, the paper compares cutting force results coming from model variants taking into account different effects such as gravity and backlash. Simulated milling operations are conducted in aluminium and steel along a straight line toolpath and results are compared with experimental data.
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Surface quality and profile accuracy of the leading and trailing edges (LTE) have a direct influence on the performance and lifetime of an aero-engine. The artificial buffing polishing for LTE has the drawbacks of heavy workload, poor consistency, experience-dependence, and so on. To solve these problems, the five-axis abrasive belt flap wheel (ABFW) polishing method for LTE of an aero-engine blade is proposed in this manuscript. Firstly, the flexible deformation law of ABFW is studied based on the Hertz elastic contact theory. Then, the control system of the contact force is established to keep the stability of the material removal rate in ABFW polishing. Furthermore, the key technologies of LTE polishing path planning with ABFW such as path direction determining, row space determining, and tool posture calculating are studied. After that, the polishing path of LTE with ABFW is generated. Finally, the verification experiments are carried out. The results suggest that the ABFW flexible deformation law is reliable, and the proposed polishing method for LTE is feasible and effective.
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Robotic belt grinding systems can be used not only to replace low efficiency, high pollution manual finishing operations but also to improve production rate and manufacturing flexibility, especially for grinding small batches of workpieces with complicated features. The contact wheel is made from soft material with significant elasticity and is tensioned by a grinding belt. Soft contact between the workpiece and contact wheel provides the benefits of high surface quality but reduces the dimensional accuracy of the finished workpiece. This paper analyzes the contact wheel’s deformation caused by belt tension in order to accurately predict the depth of cut. The elastic mechanics based on the power series method is employed to establish and solve the tension model, and the deformation of the contact wheel is obtained. The validity of the analytical model is verified by a finite element software. Then, two modified models of grinding stress distribution are developed, and the distribution of depth of grinding is predicted. Tests are running and showing that the prediction error is less than 3.1% on a given grinding path. An accurate, fast method is thus developed to predict the depth of cut for belt grinding.
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Calibration in the robot-assisted belt grinding of complex blades is regarded as one of the key bottlenecks of measurement accuracy. To enhance the accuracy, a TCP-based (tool center point) calibration method is proposed in this paper to calibrate the relationship between the precalibrated 3D laser scanner and the robot end-effector by using criterion spheres as the calibration object. Based on the description of the robot-scanning system from the perspectives of coordinates and scanner measurement modes, the calibration strategies on translational and rotational motions of the robot are provided to determine the translation vector and the orientation matrix. Calibration experiments on the criterion sphere are performed, both the calibration errors (positioning error and orientation error) and sphere fitting error are calculated. A typical case on the robotic belt grinding of 84K-2R1 aviation blade is conducted to validate the calibration results. Finally, the key factors influencing the calibration accuracy are analyzed. It has been demonstrated that the TCP-based calibration method proposed is effective, concise, and time-saving, and can be widely applied in the robot-assisted belt grinding operation.
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Belt grinding is commonly used in the process of machining complex surface. However, due to the elasticity of the grinding belt, it needs repeated or longer dwell-time grinding in order to meet the required machining precision, which is inefficient, time-consuming, and always ended up with poor surface quality. So, this paper focuses on a machining method so as to improve machining efficiency and accuracy. First, considering the elastic deformation of the contact wheel and characteristics of the workpiece, the global and local material removal processes of belt grinding are modeled to calculate the acting force. Then, based on the analysis of rigid-flexible coupling, a controlling strategy is proposed to control the acting force and grinding dwell time. The variable feed grinding experiments were carried out on the developed five-axis CNC belt grinding machine integrated with measuring and machining. The ladder type workpiece surface and free-form workpiece surface were employed to validate the proposed controllable material removal strategy. The results verify that the proposed strategy is feasible and efficient.
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In this paper, a prediction model of the material removal depth for the polishing process is developed from the microscopic point of view. Based on the statistics analysis, and by the use of the elastic contact theory and the plastic contact theory, the relationship between the pressure and the depth of indentation is obtained. Moreover, the calculation equation for the linear removal intensity, which is the material removal depth per unit contact length along the polishing path, is presented. Finally, by integrating the linear removal intensity, the micro-model of the material removal depth for the polishing process is developed. Analyzed from the perspective of the abrasive grains, the model takes the grit designation and the structural number as the two basic variables for abrasive grains characteristics, and it is assumed that the shape of an abrasive grain is conic with spherical tip and the distribution of its protrusion heights is taken to be Gaussian distribution, which fully takes into account the impact of the abrasive grains characteristics on the depth of removal. In the model, different stages of the polishing process are decomposed in detail, and the reality that the plastic deformation is accompanied by the presence of elastic deformation is taken into consideration, which makes the model more realistic. Experimental results are compared with the prediction results to verify the theoretical model. The model can be used as the theoretical foundation for the selection of abrasive grains and the process parameters.
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Because of the special characteristics of aero-engine precision-forged blade edges such as heat-resistant and high-strength materials, the unequal distribution and small removal allowance, and easy deformation in processing, edge profile shape errors are easily caused by using conventional processing methods. These errors include chamfered, sharp, flat, and obtuse edges and cervical part shrinkage, so the real-R shape of the edge is difficult to guarantee, and the aero-engine air dynamic performance is seriously affected. The purpose of this research is to improve the dimensional precision, profile shape errors, and surface quality of the real-R edge of the precision-forged blade. With these aims, the method of equivalent self-adaptive belt grinding (ESBG) using a cubic boron nitride (CBN) belt is developed in this study. First, the grinding characteristics and process planning to the blade real-R edge are analyzed. The methodologies of equivalent belt grinding (EBG) and ESBG using a CBN belt are then illustrated, the control equation for EBG is established, and the method of force adaptive control is introduced. Finally, the ESBG method using the CBN belt is verified by experimental investigation of the aero-engine precision-forged blade and comparison with manual and EBG methods. The experimental results showed that the ESBG method using the CBN belt achieved the requirements of the aero-engine precision-forged blade real-R edge. It was verified that the method of ESBG using the CBN belt has the advantages of real-R shape dimensional precision and surface quality for the blade edge compared with the conventional method.
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Early studies on robot machining were reported in the 1990s. Even though there are continuous worldwide researches on robot machining ever since, the potential of robot applications in machining has yet to be realized. In this paper, the authors will first look into recent development of robot machining. Such development can be roughly categorized into researches on robot machining system development, robot machining path planning, vibration/chatter analysis including path tracking and compensation, dynamic, or stiffness modeling. These researches will obviously improve the accuracy and efficiency of robot machining and provide useful references for developing robot machining systems for tasks once thought to only be capable by CNC machines. In order to advance the technology of robot machining to the next level so that more practical and competitive systems could be developed, the authors suggest that future researches on robot machining should also focus on robot machining efficiency analysis, stiffness map-based path planning, robotic arm link optimization, planning, and scheduling for a line of machining robots.
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The flexible contact and machining with wide strip are two prominent advantages for the robotic belt grinding system, which can be widely used to improve the surface quality and machining efficiency while finishing the workpieces with sculptured surfaces. There lacks research on grinding path planning with the constraint of curvature. With complicated contact between the contact wheel and the workpiece, the grinding paths for robot can be obtained by the theory of contact kinematics. The grinding process must satisfy the universal demands of the belt grinding technologies, and the most important thing is to make the contact wheel conform to the local geometrical features on the contact area. For the local surfaces with small curvature, the curve length between the neighboring cutting locations becomes longer to ensure processing efficiency. Otherwise, for the local areas with large curvature, the curve length becomes shorter to ensure machining accuracy. A series of planes are created to intersect with the target surface to be ground, and the corresponding sectional profile curves are obtained. For each curve, the curve length between the neighboring cutting points is optimized by inserting a cutter location at the local area with large curvatures. A method of generating the grinding paths including curve length spacing optimization is set up. The validity is completely approved by the off-line simulation, and during the grinding experiments with the method, the quality of surface is improved. The path planning method provides a theoretical support for the smooth and accuracy path of robotic surface grinding.
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The paper compares three approximate methods of finding the area of contact, the contact pressure and the deformation in elliptical Hertzian contacts.
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This paper presents the enhanced stiffness modeling and analysis of robot manipulators, and a methodology for their stiffness identification and characterization. Assuming that the manipulator links are infinitely stiff, the enhanced stiffness model contains: 1) the passive and active stiffness of the joints and 2) the active stiffness created by the change in the manipulator configuration, and by external force vector acting upon the manipulator end point. The stiffness formulation not accounting for the latter is known as conventional stiffness formulation, which is obviously not complete and is valid only when: 1) the manipulator is in an unloaded quasistatic configuration and 2) the manipulator Jacobian matrix is constant throughout the workspace. The experimental system considered in this study is a Motoman SK 120 robot manipulator with a closed-chain mechanism. While the deflection of the manipulator end point under a range of external forces is provided by a high precision laser measurement system, a wrist force/torque sensor measures the external forces. Based on the experimental data and the enhanced stiffness model, the joint stiffness values are first identified. These stiffness values are then used to prove that conventional stiffness modeling is incomplete. Finally, they are employed to characterize stiffness properties of the robot manipulator. It has been found that although the component of the stiffness matrix differentiating the enhanced stiffness model from the conventional one is not always positive definite, the resulting stiffness matrix can still be positive definite. This follows that stability of the stiffness matrix is not influenced by this stiffness component. This study contributes to the previously reported work from the point of view of using the enhanced stiffness model for stiffness identification, verification and characterization, and of new experimental results proving that the conventional stiffness matrix is not complete and is valid under certain assumptions.
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Robotic grinding is considered as an alternative towards the efficient and intelligent machining of complex components by virtue of its flexibility, intelligence and cost efficiency, particularly in comparison with the current mainstream manufacturing modes. The advances in robotic grinding during the past one to two decades present two extremes: one aims to solve the problem of precision machining of small-scale complex surfaces, the other emphasizes on the efficient machining of large-scale complex structures. To achieve efficient and intelligent grinding of these two different types of complex components, researchers have attempted to conquer key technologies and develop relevant machining system. The aim of this paper is to present a systematic, critical, and comprehensively review of all aspects of robotic grinding of complex components, especially focusing on three research objectives. For the first research objective, the problems and challenges arising out of robotic grinding of complex components are identified from three aspects of accuracy control, compliance control and cooperative control, and their impact on the machined workpiece geometrical accuracy, surface integrity and machining efficiency are also identified. For the second aim of this review, the relevant research work in the field of robotic grinding till the date are organized, and the various strategies and alternative solutions to overcome the challenges are provided. The research perspectives are concentrated primarily on the high-precision online measurement, grinding allowance control, constant contact force control, and surface integrity from robotic grinding, thereby potentially constructing the integration of “measurement – manipulation – machining” for the robotic grinding system. For the third objective, typical applications of this research work to implement successful robotic grinding of turbine blades and large-scale complex structures are discussed. Some research interests for future work to promote robotic grinding of complex components towards more intelligent and efficient in practical applications are also suggested.
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Robotic machining centers offer diverse advantages: large operation reach with large reorientation capability, and a low cost, to name a few. Many challenges have slowed down the adoption or sometimes inhibited the use of robots for machining tasks. This paper deals with the current usage and status of robots in machining, as well as the necessary modelling and identification for enabling optimization, process planning and process control. Recent research addressing deburring, milling, incremental forming, polishing or thin wall machining is presented. We discuss various processes in which robots need to deal with significant process forces while fulfilling their machining task.
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Polishing of thin-walled blisks can easily lead to vibrations and affect the surface quality. To solve this problem, we present a novel smart end effector for active contact force control and vibration suppression in robotic polishing of thin-walled blisks. A gravity-compensated force controller is developed to maintain the contact force between the polishing tool and the workpiece to an expected value. Two novel eddy current dampers are designed and integrated into the smart end effector to improve the system dynamics and suppress the vibrations. The principles of the contact force control and vibration suppression with the smart end effector are explained. Experimental results show that the presented smart end effector reduces the contact force variation in the robotic blisk polishing from 8 N to less than 1 N, and significantly suppresses the spindle vibrations, therefore, leading to a better surface quality.
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The demand of improving the accuracy of leading/trailing edges of aero engine blades has increased continually. This paper proposes a method of electrochemical machining with tangential feeding in which the leading/trailing edges are electrochemically processed by the cathode tools feeding along the tangential direction of the mean camber line of blades. The modelling and simulation on the ECM process have been carried out. A specific experiment system has been developed. Theoretical and experimental studies have proved that the proposed technology of tangential feeding offered unique advantages such as short electrolyte path, stable machining current and so achieved high machining accuracy.
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A tentative work from the perspective of cutting forces is carried out in this paper to analyse and assess the robotic belt grinding mechanisms. Firstly, a microscopic scale of cutting force model consisting of sliding, ploughing and cutting components is introduced, then the effects of force components on the machined surface roughness are explored based on the force control experiments, and finally a typical case on the robotic belt grinding of aero-engine blade with constant contact force is conducted to validate the practicality and effectiveness of force control. The results reveal two significant findings with respect to the high sliding force percentage and low cutting efficiency in comparison with the robotic belt grinding without force control.
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Nowadays, with the large use of robot manipulators in the most different fields of industrial production, two main aims must be commonly reached: robot dynamic behavior improvement and end-effector position errors reduction. For a N DOF robot arm, in case of specific applications such as milling manufacturing, one of the main source of end-effector position errors can be identified with joint compliances. This aspect, well known in literature, has been confirmed by experimental tests and measurements carried out on a specific robot assigned to non-standard milling manufacturing of marble objects (sculptures realization). To approach and analyze this issue the authors chose the multibody simulation environment. Hence, the authors developed a parametric modelling procedure that, by determining the robot characteristics through CAD model and technical data sheet investigation, provides reliable multibody dynamic models of generic N DOF robot arms. In this modelling approach the robot geometry construction is based on a standard strategy typical of this research field (i.e. Denavit-Hartenberg, Veitschegger-Wu). The developed procedure enables to obtain robot representation at various complexity levels according to the number of modelled robot components and actuation typology (Motion laws defined both in displacement or applied torque). Eventually, for a specific test case, the authors were able to correctly simulate the robot dynamic behavior, as it was demonstrated by numerical/experimental comparison. In this way the influence of the joint compliance behavior and actuator rotational inertia effects on end-effector position accuracy was analyzed.
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Visually-guided robot grinding is a novel and promising automation technique for blade manufacturing. One common problem encountered in robot grinding is hand-eye calibration, which establishes the pose relationship between the end effector (hand) and the scanning sensor (eye). This paper proposes a new calibration approach for robot belt grinding. The main contribution of this paper is its consideration of both joint parameter errors and pose parameter errors in a hand-eye calibration equation. The objective function of the hand-eye calibration is built and solved, from which 30 compensated values (corresponding to 24 joint parameters and six pose parameters) are easily calculated in a closed solution. The proposed approach is economic and simple because only a criterion sphere is used to calculate the calibration parameters, avoiding the need for an expensive and complicated tracking process using a laser tracker. The effectiveness of this method is verified using a calibration experiment and a blade grinding experiment. The code used in this approach is attached in the Appendix.
Article
Belt grinding technology is used for machining the complex surface of a blade; however, it is difficult to ensure processing accuracy. To solve this problem, a surface removal contour (SRC) model for grinding the complex surface of a blade is proposed. First, this paper discusses why the normal contact pressure between the grinding wheel and workpiece surface accords with the Hertz contact theory, and further, the calculation method for the pressure distribution of the Hertz contact is given. Second, the SRC model is determined from the material removal rate (MRR) nonlinear model. To determine the parameters of the MRR nonlinear and linear models, an abrasive belt grinding experiment was performed, which showed the relative error for the MRR nonlinear model was −1.1∼1.4 % and for the linear model was −12∼8 %. Third, combined with the Hertz contact theory, a SRC model based on the MRR nonlinear model was built. The SRC experiment showed the model’s accumulative error was only ±1 %, but the accumulative error of the SRC model based on the MRR linear model was −11∼5 %. Finally, the application of abrasive belt grinding on the blade showed the SRC model based on the MRR nonlinear model was better in dimensional precision and consistency of surface quality than the MRR nonlinear model. This led to more than 17.5 % surface roughness over the processing requirement and, beyond a 30 % maximum error, exceeded the standard. The residual stress on the blade surface after grinding appeared as a tensile stress.
Article
In the finish machining of aero-engine blades, the error of some partial area may exceed the design tolerance forming error region. The paper presents a path planning method of belt grinding for error region. Firstly, the process of error region grinding and the configuration of abrasive belt wheel are indicated. Then, error distribution and error region are obtained based on design tolerance and data processing. There are leading angle and screw angle defined to determine abrasive belt wheel location and orientation due to contact situation between abrasive belt wheel and blade surface. In addition, the grinding path is corrected and the locations are interpolated to determine the final path which guarantees to cover the entire error region and grinding steadily. At last, a simulation on VERICUT software and a blade grinding experiment are implemented to verify the present method which improves machining efficiency and machining accuracy.
Article
This work is motivated by the overestimate of chip formation contribution due to the unspecified proportion of ploughing and chip formation in previous studies on mechanisms of belt grinding. In this paper, a microscopic scale of ploughing force model is introduced, and the characterization of cutting mechanisms generated by robot-assisted belt grinding of titanium alloys is provided from the energetic aspect based on experimental forces. It has been demonstrated that sliding is more predominant than ploughing and chip formation in robot-assisted belt grinding process, resulting in lower energy efficiency. Strategies for balancing energy consumption and energy efficiency are suggested.
Article
A stranded-wire helical spring is formed of a multilayer and coaxial strand of several wires twisted together with the same direction of spiral. Because of the multilayer structure, the wear on the local area of the steel wires’ surface is related to the elliptical contact between adjacent wires during working process. Based on Boussinesq potential functions and elastic half-space model, the contact area and surface pressure of elliptical Hertz contact were investigated. Moreover, these surface contact quantities are used to expand the calculation of subsurface stress relevant to elliptical contact. It is found that the greater contact angle of two contact wires, the smaller contact area and greater maximum contact pressure. Meanwhile, the magnitude of subsurface von Mises stress is much higher when the contact angle increases.
Article
Polishing operations are commonly carried out manually, thus inducing variability on the surface quality. The aim of this paper is to automate the polishing of free-form surfaces in order to obtain high quality surfaces. Tool wear and toolpath surface covering have a great impact on surface properties. The current work proposes therefore a toolpath which optimizes both tool wear and surface covering. This toolpath is composed of an optimized elementary pattern repeated along a 5-axis carrier trajectory. Usually, trochoid patterns are used. Non uniform wear of the tool and uneven probability density function of the surface covering are the main inconvenients of such pattern. So, this paper proposes two optimized patterns: Spade and Triangular. Both of them lead to uniform tool wear. Our paper also demonstrates that the second solution provides a uniform probability density function. All presented computations are validated experimentally
Article
As a kind of manufacturing system with a flexible grinder, the material removal of a robot belt grinding system is related to a variety of factors, such as workpiece shape, contact force, robot velocity, and belt wear. Some factors of the grinding process are time-variant. Therefore, it is a challenge to control grinding removal precisely for free-formed surfaces. To develop a high-quality robot grinding system, an off-line planning method for the control parameters of the grinding robot based on an adaptive modeling method is proposed in this paper. First, we built an adaptive model based on statistic machine learning. By transferring the old samples into the new samples space formed by the in-situ measurement data, the adaptive model can track the dynamic working conditions more rapidly. Based on the adaptive model the robot control parameters are calculated using the cooperative particle swarm optimization in this paper. The optimization method aims to smoothen the trajectories of the control parameters of the robot and shorten the response time in the transition process. The results of the blade grinding experiments demonstrate that this approach can control the material removal of the grinding system effectively.
Article
A linear regression by the method of least squares is made on the geometric variables that occur in the equation for elliptical-contact deformation. The ellipticity and the complete elliptic integrals of the first and second kind are expressed as a function of the x,y-plane principal radii. The ellipticity was varied from 1 (circular contact) to 10 (a configuration approaching line contact). The procedure for solving these variables without the use of charts or a high-speed computer would be quite tedious. These simplified equations enable one to calculate easily the elliptical-contact deformation to within 3 percent accuracy without resorting to charts or numerical methods.
Article
Recent papers [D.C.H. Yang & T. Kong (1994), Comput. Aided Design 26, 225--234; S--S. Yeh & P--L. Hsu (1999), Comput. Aided Design 31, 349--357] formulate real--time CNC interpolators for variable feedrates along parametric curves. These interpolators employ truncated Taylor series to compute successive reference--point parameter values, but in both papers an erroneous coefficient for the highest (quadratic) term is cited. The derivation of the proper coefficients is a straightforward, although somewhat convoluted, exercise in chain--rule differentiation. Compact recursive formulae are presented here to compute the correct coefficients, up to the cubic term, in cases where the feedrate depends on (i) elapsed time; (ii) curve arc length; or (iii) local path curvature. The local and cumulative effects of truncation errors on the accuracy of such interpolator schemes are also assessed, and compared with the essentially exact interpolators for Pythagorean--hodograph curves. Keywords: par...