Trajectory planning of robot-assisted abrasive cloth wheel polishing blade based on flexible contact

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%.


Introduction
The blade is the core component of the engine, and its polishing quality directly affects the service life and work efficiency of the engine. Due to the complex contour shape of the blade and high processing requirements, the traditional manual polishing is faced with high pollution, high working hours, high technical experience dependence, and low processing consistency. Although the multi-axis CNC machining technology improves the defects caused by manual polishing, its high cost, low flexibility, and limited working space restrict the diversified development of blade polishing. In recent years, industrial robots have been widely used in the processing field. For the polishing of complex free-form surface parts, the robot processing system has high efficiency, good flexibility, large operating space, and low cost, and combined force/torque sensor can ensure the processing quality and consistency of the blade. Many scholars have studied the polishing blades of industrial robots with abrasive belts. Zou et al. [1] used abrasive belt floating compensation technology to polish aluminum alloy blades to improve the polishing accuracy and surface roughness R a ≤ 0.4 m . Li et al. [2] used abrasive belts to polish nickel-based superalloy blades, adopted parameter control methods and established a multi-parameter experimental platform (polishing force, vibration, and temperature), obtained optimized surface parameters, and improved the polished surface quality. 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.
The above literature use robot abrasive belts to polish the blades. Although the abrasive belts have the certain flexibility, the conformal characteristics and tool integration of its flexible contact are not as good as the abrasive cloth wheel (ACW). The ACW can reach any narrow working space with small size and simple structure. It is flexible and can realize the conformal contact between the blade and the surface of the ACW. The contact deformation generated in the local area can adapt to the geometric contour of the blade surface, effectively avoiding the phenomenon of "under-polishing" and "over-polishing" at the leading and trailing edges. Huai et al. [5] used the ACW to polish the leading and trailing edges of the integral blisk to predict and optimize the surface roughness and concluded that the influencing factors were the granularity of the ACW, speed, contact compression, and robot feed speed, and surface roughness R a ≤ 0.4 m after polishing. Lin et al. [6] optimized the process parameters for polishing the integral blisk of the ACW and obtained the optimized stable polishing interval by orthogonal experiment, the compression amount is 0.6-0.8 mm, the spindle speed is 5000-6000 r/min, and the feed speed is 200-400 mm/min. The above integral blisk with the ACW polishing can better adapt to the geometric profile of leading and trailing edges and obtain better polishing surface quality, but its application is the five-axis CNC machine tool, and the flexibility of the machine tool movement space is small; the size and variety of processing blades are small. Therefore, this paper tries to adopt a new method of robot-assisted abrasive cloth wheel (ACW) polishing blade, which well combines the large working space and low cost of the robot and the flexibility and with-shape contact of the abrasive cloth wheel (ACW). A reasonable trajectory planning can realize the grinding of different blades with good surface accuracy and stability of the polishing process.
Many scholars have studied the trajectory planning of blade free-form polishing. Ji et al. [7][8][9] investigated surface interpolation of blade surfaces, using adaptive NURBS curves, cubic B-sample, seventh B-sample, and other trajectory interpolation algorithms to ensure that the velocity, acceleration, and jerk of robot trajectories are continuous and bounded, and that good tracking performance can be obtained. He et al. [10] used the circular path interpolation method to ensure the G1 continuity of the connection points. Ma et al. [11,12] selected key contact points on the 3D model and refined to produce sensitive target points, so that the polishing track points and curves evenly cover the complex free-form surface, and the experiment verified that the precision polishing effect was achieved. The trajectory planning of the robot polishing free-form surface mentioned above is analyzed from the perspective of graphics and geometry, and it lacks consideration the impact of dynamic contact force and material removal in the polishing process. Ji and Wang [13][14][15][16][17] analyzed the contact particle movement to determine the contact pressure distribution, considering the elastic modulus and Poisson's ratio of the contact wheel, and its root mean square error (RMSE) and mean absolute percentage error (MAPE) values significantly reduced. Wu and Tong [18,19] used the flexible inclined polishing disk to study the pressure and velocity distribution about the material removal area under three conditions of flat, convex, and concave surfaces. Simulations and experiments verified the accuracy of the method.
Although the above literature analyzed the pressure and velocity distribution in the contact area, they lack the guidance about CAE analysis results to guide the generation of robot trajectory planning. Therefore, this paper considers the polishing contact deformation and material removal amount and adopts the method of robot-assisted abrasive cloth wheel polishing blades. Firstly, the contact process between the blade and the abrasive cloth wheel is modeled with ABAQUS software, and the contact profile and pressure distribution law of the contact area are analyzed. Then calculated the polishing step length by considering the flexible deformation of the abrasive cloth wheel, calculated the Fig. 1 Flow chart of robot-assisted abrasive cloth wheel polishing blade trajectory planning line spacing by considering the contact area profile and the amount of material removal. The trajectory points of the blade polishing area are generated by offline programming software, imported the generated program into the robot control cabinet through post-processing, the external PLC performs linkage control to complete the polishing of the blade entire area. Finally, compared the surface roughness, micro surface profile, and polishing efficiency at blade different polishing areas when use the rigid trajectory and the flexible adaptive trajectory. Figure 1 shows the flow chart of the robot-assisted abrasive cloth wheel polishing blade.

Blade structure and related data
The blade adopts certain gas turbine blade, the length is the total length of the blade body and the tenon, and the width is the maximum width of the cross-section of the blade body. Its structural parameters are shown in Fig. 2a. The configuration design of the blade is completed by lofting and fitting according to the given contour cross-section (such as contour 1-contour 4). The design parameters are shown in Fig. 2b. The blade length is 205 mm, width is 77 mm, leading edge thickness is 3 mm, and trailing edge is 1.3 mm. The stack point in the figure is the closed curve's geometric center of the blade section, and this point is defined as the origin. Chord length is the length of the blade's cross-section, which is perpendicular to the basic chord line through the leading edge pole and the trailing edge pole. Maximum thickness is defined as the maximum thickness of the blade section, which is usually obtained by calculating the diameter of the largest inscribed circle of the section. As shown in Fig. 2b, the diameter of the circle is the maximum thickness of the blade. The leading and trailing edges are variable arc curves. In the manufacturing process, the leading edge is regarded as a circular arc curve, as shown by R1 in Fig. 2b. The surface roughness value R a of the blade surface measured by PermeterM2 is between 1.0 and 1.5 μm, and the surface roughness S a measured by Leica super optical profiler 5X is about 3.353 μm.

Structure and characteristics of the ACW
The ACW is an abrasive tool formed by sticking sand cloth closely arranged and at a certain angle with the central axis on the mandrel. There are various sizes and can be customized. The polishing material removal is completed by the polishing action of the abrasive grains on the sheet, as shown in Fig. 3. The industrial robot-assisted the ACW polishing process has the following advantages:  Fig. 3, the emery cloth sheet of the ACW is in a bent state when it is stationary. When rotating at high speed, the emery cloth sheet stretches and the radius increases. Under the action of the polishing force, the sheet is bent and deformed. The base material of the sheet is flexible. High-strength cloth base material, the adhesive has elasticity, and the polishing flexibility is greater. 2. Good spatial accessibility. The Kawasaki industrial robot RS020N has a maximum arm span of 1725 mm, six degrees of rotation, and can reach any direction and position of the machining coordinate system. It adopts trajectory planning to effectively avoid interference and is suitable for polishing of blades with complex surfaces. 3. Cold polishing. When the ACW rotates at a high speed, as shown in Fig. 3, the gap between the emery cloth sheets is formed, which has many cooling grooves, can function as a fan, and it is not easy to cause the surface of the blade to burn. 4. High efficiency. The flexibility of the ACW increases the contact area between them, and the contour of the abrasive grains on the emery cloth sheet makes the grinding ability stronger than that of the grinding wheel. Although the linear velocity of the ACW is lower than the abrasive belt during polishing, its flexibility is greater than that of the abrasive belt, which can increase the contact area with the blade, and its polishing efficiency is about 2/3 of the abrasive belt.

Contact model establishment and simulation
The key process parameters about this simulation between the abrasive cloth wheel and blades are shown in Table 1.
The CAE simulation software ABAQUS 6.14-4 is used to analyze the polishing contact process between the blade and ACW. Firstly, established the three-dimensional model of the ACW and the blade, and set the related material properties of the blade and the ACW, as shown in Table 1. Through operations such as translation and rotation to complete the contact assembly of the blade and the ACW. Then analyzed by the statics general analysis step, take the blade convex and the ACW outer surface as the mutual contact surface, created coupling constraint between the central axis of the ACW and the emery cloth sheet, applied load to the ACW central axis, fixed the blade concave, and divided the mesh. Finally, the simulation is completed, and the contact profile of the contact area and the polishing pressure are analyzed. The simulation model and results are shown in Fig. 4.
Analyzing Fig. 4, it can be seen that the contact area is elliptical, with its semi-major axis is a, and semi-minor axis is b. The simulation analysis shows that the long semi-axis of the ellipse is related to the polishing contact pressure, the elastic modulus of the ACW and blade, the curvature radius of the blade, and the width of the ACW. The pressure at the center of the ellipse is the largest, that is, the unit material removal is the largest, and the pressure decreases toward the boundary of the ellipse, and the edge pressure is zero, that is, the unit material removal is zero.

Contact area analysis
According to the simulation analysis in Fig. 4 and combined with the Preston equation, the material removal rate is proportional to the polishing pressure and the relative speed, and its expression is formula (1).
where k p is the Preston coefficient, p is the pressure distribution at the contact area between ACW and blade, v r is the relative speed between ACW and blade and is given by v r = v s ± v f , and v s represents the linear velocity of ACW, unit is m/s; v f represents the robot feed speed, unit is mm/s, " ± " represents the same/reverse direction between ACW steering and the robot feed speed. Assuming within dt time, the actual arc length of the industrial robot end polished at the feed speed v f is dl, then dl = v f ⋅ dt , formula (1) becomes formula (2). Analysis formula (2) shows that the material removal rate per unit length is proportional to the polishing pressure, the relative speed between ACW and blade, and inversely proportional to the feed speed of the robot. According to Fig. 4, it can be seen that the polishing pressure distribution in the contact area is approximately elliptical, and the contact process satisfies the conditions of Hertz's law [20]. The formula for the boundary conditions is: In formulas (3)-(9), a and b are the length of long and short semi-axes of the ellipse, k is the ratio of long and short semi-axes, (k) and (k) are the first and second type of elliptic integral, and F n is the normal contact force between ACW and blade, R is the relative radius of curvature, k 1 and k 2 are the relative principal curvature of the ACW and blade, unit (mm), E c is the relative contact elastic modulus, unit (MPa); where E 1 and v 1 , E 2 and v 2 represent the elastic modulus and Poisson's ratio of the ACW and blade, respectively, R 1 and R 1 ′ are the main radius of curvature of the ACW at the contact point, R 2 and R 2 ′ is the main curvature radius of the blade at the contact point, γ is the angle between the direction where the main curvature of the ACW and the blade is in contact. x i and y i are the coordinate values of any point in the contact area. According to Hertz contact theory, the pressure distribution of any point in the contact area is expressed as formula (10).

Material removal profile analysis
The polishing removal depth of the blade surface directly affects the material removal rate of the blade surface, and the shape of the contact area is approximately elliptical. The micro element in the elliptical contact area is selected for analysis. As shown in Fig. 5, the material removal amount at the polishing contact point A is: Integrate formula (11) and substitute formulas (4) and (10) into formula (11) to get the polishing depth of the contact point.

Fig. 5 Distribution of material removal elements in the contact area
The depth of material removal at the center contact point of the polishing area is the largest, and the maximum can be expressed as: Analysis formula (13) shows that the maximum removal depth of the contact point O is related to the polishing contact normal force, relative speed, robot feed speed, relative elastic modulus, and relative principal curvature, which are basically consistent with the simulation results in Fig. 4. For the selected polishing ACW and blade, the relative elastic modulus E c and the main curvature of the ACW are fixed. For a small range, the curvature radius of the blade is considered to be approximately constant. Comprehensive analysis shows that the factors affecting the maximum material removal depth are normal contact force, relative speed, and robot feed speed.

Trajectory planning algorithm
When planning the trajectory of the blade, it is necessary to consider the material removal rate in the contact area of the blade. A reasonable step length and line spacing can obtain a uniform material removal on the blade surface.

Calculate the step length and line spacing
The surface of the blade is a free-form surface; the expression of the surface parameter equation is expressed p = p(u, v) . The blade free-form surface can be represented as a rectangular area the parameter plane represented by the change interval of the uv two parameters. In this way, a curved surface with four boundaries is obtained. The points in the parameter domain and the points on the surface form one-to-one mapping relationship, that is, the convex surface of the blade is considered to be a given known parameter surface. One direction u of the parametric surface is defined as the transverse direction of the blade in this experiment and the other v is defined as the longitudinal direction of the blade. Take the speed tangent direction of the abrasive cloth wheel line and the contact point of the blade as the u direction, and the same direction as the abrasive cloth wheel axis direction as the v direction.
According to the contact model established during the simulation in Fig. 4, the step length and the line spacing are calculated by transverse polishing. Use equal chord height error to calculate the step length in the u direction, and use equal residual height to calculate the line spacing in the v direction. During the contact process, the polishing contact point will produce visible elastic deformation. The magnitude of the elastic deformation is related to the normal contact force, which determines the material removal profile, therefore, it is necessary to calculate the polishing step length and the line spacing based on the contact deformation of the ACW. Figure 6 is a schematic diagram of the step length calculation.
It can be seen from Fig. 6 that the material removal at the polishing contact point A is the largest, ′ is the maximum removal depth of the blade, its size is h max , can expressed as: In formula (14), δ is the material removal depth when the blade is in initial contact, ε is the material removal depth when the ACW is in contact with the blade at the (14) � = + Fig. 6 Step-length calculation considering flexible deformation of the wheel and ε′ is the material removal depth when the ACW deformation is considered. can be expressed as [21]: Substitute formulas (13) and (15) into formula (14) to obtain formula (16).
The calculation of the polishing step length when the ACW and the blade are in rigid contact is shown in Fig. 7, and the formula is (17).
In RtΔABO , according to the Pythagorean theorem, s can be expressed as: Simplify formula (17) to get Substitute formula (16) into formula (18) to obtain the calculation formula (19) of polishing line spacing considering the elastic deformation of the ACW.
According to formula (19), the following can be obtained: assuming that the blade and ACW material, the spindle speed, the robot feed speed, the relative principal curvature of the contact point, and other process parameters are known, the step length is related to the normal contact force of the polishing. The greater the normal force, the greater the deformation of the ACW and the higher the efficiency, regardless of the polishing accuracy. The most notable feature of industrial robot-assisted ACW polishing is flexible polishing, and the influence of elastic deformation needs to be considered when calculating the polishing line spacing. Figure 8 is the calculation of the polishing line spacing, w is the width of the ACW, L is the polishing line spacing, R i−1 /R i /R i+1 is the curvature radius of the blade adjacent points, 2θ is the angle of the blade corresponding to the width of ACW, and ε is the compression of the ACW perpendicular to the direction of the polishing vector.
In RtΔDBO, The relationship between the width of the ACW and the polishing line spacing is formula (21).
Substituting formulas (15) and (20) into formula (21) can obtain formula (22). Fig. 7 Step-length calculation of rigid polishing wheel Analyzing formula (22), given the relative elastic modulus of the polishing contact and the curvature of the polishing contact point, the polishing line spacing is related to the normal contact force F n , the major semi-axis of the ellipse in the contact area, and the central angle of the adjacent polishing contact points. In the v direction, the blade curvature change is small and almost zero, so cos ≈ 1 . Therefore, the polishing line spacing is related to the normal contact force. After theoretical calculation and simulation analysis, as shown in formula (10) and Fig. 4, it can be seen that the normal polishing force at the center of the ellipse circle is the largest, and the material removal is the largest, with zero edge. In order to improve the blade surface quality, it is necessary to improve the consistency of the polishing contact area. Since the average polishing force F nm in the contact area is π⁄4 ≈ 0.785 times of the maximum polishing force, that is, F nm = 0.785F nmmax , and the formula (23) as the polishing line spacing in simulation and experiment.
Formulas (19) and (22) are the calculation formulas for the polishing step length and line spacing considering the elastic contact deformation of the ACW. Under the premise that the polishing contact force and elliptic long and short semi-axes are known, the step length and line spacing can be obtained. Table 2 shows the process parameters during polishing. The main curvature of the blade concave is selected as k 1 = 1 230 mm , k 2 = 1 320 mm ; the main curvature of the ACW is k 1 = 1 110 mm , k 2 = 0 as the research object. Table 3 is the calculated comparison value of the step length and line spacing before and after the improvement of this point.
Analyzing Table 3 shows that, under the premise of not affecting the polishing efficiency, compared with the rigid tool to calculate the step length and line spacing, the flexible adaptive trajectory planning algorithm is adopted to select the known polishing point on the blade, The polishing step length is increased from 9.25 to 13.09 mm, and the line spacing is reduced from 20 to 15 mm.

Constant feed rate interpolation calculation
Let P(u) to be the spline curve, and the time function u is the curve parameter, let u t i = u i , u t i+1 = u i+1 .
Perform a second-order Taylor expansion of the parameter u i to time t to get the next interpolation point u i+1 .
In the formula, u i -the curve parameter i corresponding to the first interpolation point; u i+1 -the curve parameter i + 1 corresponding to the first interpolation point; t i -the interpolation time i corresponding to the first interpolation point; t i+1 -the interpolation time i + 1 corresponding to the first interpolation point; and H.O.T-high-order infinitesimals.
The interpolation speed of the servo motors of each joint of the robot is: According to formula (26),

Derivation of formula (27) can be obtained formula (28).
NURBS curve interpolation period t s is the difference between adjacent interpolation points, t s = t i+1 − t i , substituting formulas (27) and (28) into formula (25), ignoring high-order infinitesimals, and obtaining NURBS curve  Table 3 Step-length/line-spacing at that point before/after improvement Parameters Before improvement After improvement Step-length 9.25 mm 13.09 mm Line-spacing 20 mm 15 mm second-order approximate expression of interpolation (29), the second-order accuracy can satisfies the interpolation accuracy of industrial robots.

Algorithm simulation
Use offline programming software to verify the correctness of the algorithm. First, imported the Kawasaki industrial robot RS020N, the four-station polishing platform, the blade, and the ACW into the simulation environment. Calibration of the ACW and blades to make the simulation environment consistent with the real machining environment. Then extract the blade surface NURBS curve, set process parameters and machining environment. The simulation verifies the over-limit, unreachable, and collision and sets the attitude of the polishing trajectory of the ACW and blade to the normal direction of the contact point. Figures 9a, b represent the simulation diagram of rigid and flexible adaptive trajectory planning for NURBS curves, respectively. Contrast literature u=u i [10] using spatial linear interpolation/space circular interpolation, discretized the NURBS curve into short straight segments or arc curves, destroyed the continuity of the first derivative of the blade profile curve, affect the smoothness and the surface quality of the blade surface. In addition, a large number of tiny line segments are used to approximate the contour curve of the blade, and there are automatic acceleration and deceleration functions at the start and end points of the linear interpolation, which makes it difficult for the processing feed speed of the blade to reach the feed speed set by the robot. Figure 10 is the comparison effect diagram of robot NURBS curve interpolation and space circular arc interpolation. It can be seen that when space circular arc interpolation is used, the surface track point error is larger and the accuracy is low. NURBS curve adopts parametric interpolation technology, which surface accuracy is high.
Analyzing the simulation results of the trajectory planning in Fig. 11, after the algorithm is improved, the polishing time of the blade convex is shortened, and the polishing trajectory points are reduced.
The polishing time of the blade convex before the algorithm improvement is 339 s and 215 s after the improvement. The polishing efficiency is increased by 36.58%. The number of polishing nodes is 614 before the improvement and 364 after the improvement, and the number of nodes is reduced by 40.72%. The polishing time of the blade concave is 263 s before the algorithm is improved and 160 s after the improvement. The polishing efficiency is increased by 39.13%. The number of polishing nodes is 478 before the improvement and 368 after the improvement, and the number of nodes is reduced by 23.01%. The polishing time of the leading edge is 291 s before the algorithm improvement and is 347 s after the improvement. The polishing time is increased. This is because the curvature of the adjacent line spacing of the leading and trailing edges changes greatly, that is, the central angle of adjacent curvatures is larger and the line spacing is small. Therefore, the number of polishing nodes is 528 before the improvement and 665 after the improvement, the trailing edge polishing time before the algorithm is improved is 371 s and after the improvement is 423 s, the number of polishing nodes is 678 before the improvement and 768 after the improvement, the total polishing time before the improvement is 1264 s and after the improvement is 1145 s, the total number of polishing nodes before the improvement is 2298, the total number of polishing nodes after the improvement is 2165, the number of nodes is reduced by 5.79%, and total polishing time reduced by 9.4% after algorithm improvement.

Selection of the ACW
The diameter, width, and abrasive of the ACW are selected according to the blade curvature and the polishing process. The curvature radius of the blade convex is between 14 and 1988 mm, the concave is between 15 and 1316 mm, the leading edge is about 2.1-3.5 mm, and the trailing edge is about 1.3 mm. The curvature radius of the blade convex and concave varies greatly. The red dividing line divides the curvature radius of the blade convex and concave into two parts, as shown in Fig. 12. The curvature radius of near the red dividing line of the blade convex is about 80 mm. Considering the influence of polishing surface quality and polishing efficiency, the diameter of the ACW is ∅ 80 mm; the width is 20 mm. The coarse polishing abrasive number is 320 # and the finely polishing is 600 # . The ACW with the same process parameters at the leading and trailing edge of the blade.
In consideration the principle of avoiding interference in blade concave polishing, the diameter of the ACW selected according to the minimum curvature is ∅ 30 mm, the width is 20 mm, the coarse polishing abrasive is 320 # , and the fine   Figure 14a is an experimental device for the robot to polish complex free-form blades, including the Kawasaki RS020N six-degree-of-freedom industrial robots, with terminal load of 20 kg, working space range is 1725 mm, and repeat positioning accuracy is 0.05 mm. The six-dimensional force/torque sensor (ATI delta SI330-30) is installed at the end of the sixth joint flange of the robot to measure the contact force during the polishing process. The data processing module uses TCP communication to transmit the polishing force to the robot control cabinet. The machine uses the force control algorithm to adjust the robot's end pose in real time to maintain constant polishing normal contact force. The blade clamp is installed at the end of the force sensor, the blade is installed in the clamp, and the ACW is used as the polishing tool. The four-station wheel changing platform adopts PLC control to replace the polishing wheel in real time according to the different parts of the polishing blade. The leading and trailing edge polishing force is about 10 N; the blade concave and concave polishing    Fig. 14b, the robot-assisted abrasive cloth wheel polishing system, five Cartesian space coordinate systems are defined, including the robot base coordinate, which is fixed. The tool coordinate system tool is fixed on the sixth axis, and the default is the flange center of the sixth axis. Wobj is the workpiece coordinate system, which is connected with the tool coordinate system tool and defined on the fixture of the workpiece and Tool0 is the coordinate system defined on the turning edge of the abrasive cloth wheel. The coordinate system work is the work coordinate system, which is defined relative to the workpiece coordinate system on the workpiece, which is equivalent to the coordinate system defined by the polishing point on the workpiece.

Result analysis
The comparison diagrams of the blades before and after polishing are shown in Figs. 15, 16 and 17. Figure 15 is the comparison diagram of the convex and concave of the gas turbine blades before and after polishing. Figure 16 is the comparison diagram of the leading and trailing edges of gas turbine blades before and after polishing. Figure 17 is the comparison diagram of aero-engine blade convex and concave before and after polishing. The mirror polishing effect is achieved, and the surface roughness of the aero-engine blade after polishing is measured. The blade convex and concave of surface roughness R a ≤ 0.3μm , and the blade leading and trailing edges of surface roughness R a ≤ 0.2μm . The polishing experiment of aero-engine turbine blades further verified the rigor and repeatability of the trajectory planning method. Figure 18 shows the measurement of gas turbine blades surface topography by using the Leica ultra-depth of field profiler. Figure 18a is the surface topography comparison about before and after polishing of the leading and trailing edges of the blade, the convex and concave of the blade, and   Fig. 18b is the three-dimensional topography comparison about before and after polishing of the leading and trailing edges of the blade, the convex and concave of the blade. Before polishing, there are obvious wave crests and trough strips on the leading and trailing edges of the blade. After polishing, the uniformity of the blade surface is improved, and the fringe bands of wave crests and troughs disappeared, the maximum surface roughness S a ≤ 60μm.
Before polishing, the surface of the blade convex and the concave is uneven, and after polishing the maximum surface roughness S a ≤ 70μm. Figure 19 compares the surface roughness values R a of various parts of the blade before and after polishing, the measured surface roughness values R a ≥ 1.0μm of the blade convex/concave before polishing, the surface roughness value before the algorithm is improved is about R a ≤ 0.4μm , and the surface roughness value after the algorithm is improved approximately R a ≤ 0.3μm , the polishing efficiency is increased by approximately 22%. The surface roughness value before the leading edge polishing is about R a ≥ 0.5μm , the surface roughness before the algorithm is improved is about R a ≤ 0.3μm , and the surface roughness after the algorithm is improved is about R a ≤ 0.2μm , and the polishing efficiency is reduced by 20%. The surface roughness of the trailing edge before polishing is about R a ≥ 0.6μm , the surface roughness before the algorithm is improved is about R a ≤ 0.3μm , and the surface roughness after the algorithm is improved is about R a ≤ 0.2μm , and the polishing efficiency is reduced by 12%. The test results fully proved that when considering the material removal in the flexible contact of contact area about the ACW, the surface roughness of the blade is reduced when the total polishing efficiency is increased by 9.4%.

Conclusion
In the robot-assisted abrasive cloth wheel flexible polishing blade, the CAD trajectory planning is guided by using CAE simulation analysis, and the influence factors of the deformation about the abrasive cloth wheel and the material removal in the contact area are obtained by ABAQUS simulation analysis. Good surface roughness and high polishing efficiency can be obtained by considering the above factors in trajectory planning.
1. For gas turbine blades, the polishing time of blade concave and convex is reduced from 602 to 375 s, and the leading edge and trailing edge is increased from 662 to 770 s by using flexible adaptive trajectory planning algorithm. The total polishing efficiency increased by 9.4%. Availability of data and material All data generated or analyzed during this study are included in this published paper.

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