Positioning error compensation of a flexible track hybrid robot for aircraft assembly based on response surface methodology and experimental study

With the development of aviation industry, more stringent demands are put forward for the performance and manufacturing level of aircraft. Moreover, the automation and precision of aircraft assembly determine the efficiency and quality of aircraft production. In order to improve the positioning precision of the flexible track hybrid robots which are applied to the flexible automatic assembly of aircraft, a precision compensation method based on response surface methodology was proposed in this paper. Firstly, the global positioning error model, optimized by characteristics of error data, was constructed to predict the positioning errors of the flexible track hybrid robot. Secondly, the predicted errors are utilized to realize the compensation of the target points at drilling workspace on nose and front fuselage assembly areas. Finally, a series of experiments of the flexible track hybrid robot with no-load and drilling scenarios are implemented to validate the proposed precision compensation method. The experiment of a hybrid robot for aircraft assembly shows that the mean value of the absolute positioning precision of the end-effector was promoted from 0.081 mm to 0.025 mm, maximum error reduced from 0.143 mm to 0.039 mm, respectively, which means that the position accuracy of the robot is increased by 69.1% and 72.7% for two experimental conditions.


Introduction
In the process of aircraft manufacturing, the application of various robots has improved the production efficiency, automation level, and quality reliability of aircraft automatic assembly. In the process of automatic drilling and riveting with robot as carrier and end-effector, the requirements of positioning precision are top priority, otherwise the assembly quality of the aircraft will be seriously restricted [1]. Therefore, improving the positioning precision of robot and end-effector is the key measure to achieve high precision drilling process quality.
In order to improve the positioning precision of the robot, there are two main methods [2] for robot precision compensation: one is to add terminal feedback detection to achieve full closed-loop control; the other method is to improve the absolute positioning precision of the robot by off-line calibration.
The method of compensating positioning precision by adding terminal feedback detection is the most general, convenient, and stable way, and the feasibility is good. In this method, the robot, sensor, and control system are integrated by adding on-line measurement sensor, and the closed-loop real-time feedback compensation is formed [3]. However, for some complex non-standard and hybrid robots, it is difficult to arrange the online detection device and realize the online detection and real-time compensation of positioning error due to the narrow layout space [4], the constraints of design quality, and the closing of its own structure.
The error compensation method based on high-precision robot calibration first requires the establishment of a complex robot kinematics model [5] and uses advanced and precise measuring instruments and mature parameter identification methods to identify geometric parameters, so that the identified parameters can correct the kinematics model of the robot control system. This method requires very high precision of kinematics parameter identification, and it is difficult to quantify the influence of non-geometric parameters of the robot [6]. The neural network [7] needs to obtain the kinematics algorithm of the robot and calculate the precise joint coordinates of the robot according to the expected position and attitude. Flexible track hybrid robot system is a series-parallel hybrid structure with redundant degrees of freedom, which is easy to cause multiple solutions or no solutions. Especially for the large amount of computation, the neural network algorithm has slow response speed, which will restrict the efficiency of drilling.
Neural network algorithms need a lot of data to train, and training neural network algorithms will face two difficulties [8]. On the one hand, the data collection task of positioning error is heavy, and the acquisition of huge error data is difficult. On the other hand, if the mass positioning error is not accurate, it will affect the effect of algorithm training, and it is more difficult to judge the effect of algorithm training.
The factors that affect positioning precision of endeffector of drilling system include but are not limited to the straightness of the guide rail [9], screw pitch error [10], ambient temperature, and lubrication conditions [11], etc. Therefore, the precision compensation method is more complicated. At present, the precision compensation methods for end-effector are generally divided into two kinds [12]: one is the internal parameter error modeling method of the drive shaft, and the other is the empirical interpolation method [13]. The internal parameter error modeling method of drive shaft needs to calculate the error models of straightness of guide rail and screw pitch [14], which will involve the identification of internal parameters such as the ball stiffness of linear guide rail, the torsional stiffness of screw, and screw pitch. Therefore, the precision of identification parameters determines the precision of the error model [15], which puts forward strict requirements for parameter identification, which is difficult to calculate and difficult to evaluate the identification precision. The principle of empirical interpolation method is to detect the actual position reached by system during the execution of target instruction through highprecision measuring tools, calculate the positioning error of the terminal in the whole trip, and write the positioning error data into numerical control program [16]. During the execution of target instruction, the interpolation compensation can be carried out automatically by looking up the table. Although this method is simple in real time, quick in response, and does not require any operation [17]; it does not establish error model for the whole trip and cannot predict the positioning error of any point [18]. If this method wants to improve the interpolation compensation precision, it needs to increase the number of samples as much as possible, and the workload is heavy [19]. It is difficult to find an accurate and universal adapted error model and compensation method because of the structure complexity and the difference of different control systems, the influence factors of complex and alternating positioning error of CNC equipment, and the different application scenarios [20].
To solve the above problems, this paper proposed a position precision compensation method based on response surface for end-effector of flexible track hybrid drilling robot. This method sampled and analyzed the positioning error data of actuator working area and fitted out global positioning error response surface algorithm, predicted the end of the actuator working travel all targets within expectations point positioning error, so as to realize the improvement of positioning precision by compensating the target errors with instructions.

Requirements of aircraft assembly
Aircraft assembly process mainly has two obvious characteristics. First, the connection technology involved in aircraft assembly is a large of variety and number. Most of the connecting methods of aircraft are riveting, screw joint, and glue, etc. These connecting technologies themselves have the characteristics of large deformation and difficult to control the precision of the connection. The shape and size of aircraft structure are determined by assembly process, so the accuracy of assembly technology directly determines the assembly quality of aircraft. Second, the aircraft assembly process uses a large number of complex and non-general tooling, fixture. Also, the requirements of aircraft assembly technology directly affect the process of aircraft manufacturing. Aircraft assembly has two main stringent requirements. The first is the strength of aircraft connection area. In the whole process of aircraft assembly, the first requirement of aircraft assembly quality is to meet the strength of any joint surface. No matter what kind of connection is adopted, a slight deviation in technological quality will cause serious defects of the whole aircraft and even lead to serious accidents. The second is the accuracy of aircraft connection. Aircraft skin, frame, and purlin must be connected in such a way as to ensure accuracy. On the outer surface of aircraft, if rivets protrude too high or submerge too low of the countersunk socket, which not only affects aerodynamic layout of the aircraft but also causes uneven stress in riveted area, and even causes the skin tearing in serious cases.
The assembly and technological process of aircraft nose and front fuselage are shown in Fig. 1. The aircraft nose and the front fuselage are accurately connected by auxiliary tooling. The ribband is mounted across the joint area, connecting the nose and the front fuselage. The auxiliary fasteners are used to tighten aircraft skin and ribband. According to technical requirements of assembly process, precise drilling, countersink, and deburring are carried out along both sides of the joint ring seam. Then, aircraft skin and ribband are riveted together to finally realize the assembly of aircraft nose and the front fuselage.
According to the aircraft assembly process requirements and assembly process, it is the first step to ensure the drilling quality of the fuselage skin to realize the fuselage assembly. It is also a crucial step. The quality of drilling in the joint area of aircraft fuselage determines the quality of riveting, thus the quality of fuselage assembly, and then the manufacturing level, quality, and service life of aircraft. The contradiction between the high precision requirement of aircraft assembly task and the low precision of automatic drilling and riveting equipment gives rise to the precision compensation method. Aircraft assembly task is the application scene of precision compensation, and automatic drilling and riveting equipment is the object of precision compensation. The research of precision compensation method is aimed at solving the quality problem of aircraft assembly process.

Flex track hybrid robot for drilling
In order to improve production efficiency and assembly quality, aircraft assembly technology is developing toward automation, flexibility, and low cost. Digital assembly technology is adopted in large component assembly to reduce the use of assembly tools such as mold frame and realize automatic flexible assembly.
Precision flexible automatic drilling equipment and technology have been widely used and developed in aircraft assembly. It mainly includes but is not limited to industrial robot drilling system, flexible drilling system, parallel machine flexible drilling equipment, portable automatic drilling system, etc., which can cover any production and assembly site to achieve precision drilling task, which lays a foundation for digital assembly of aircraft.
Comparing with industrial robot drilling system, machine tool drilling system, and other automatic systems, the flexible track hybrid automatic drilling system independently developed by Nanjing University of Aeronautics and Astronautics mainly has the advantages of flexible, compact, maintainable, and suitable for different shapes and sizes of the fuselage skin. It makes up for the defects of other automatic drilling systems such as large volume, poor versatility, and tedious maintenance and also meets the requirements of high-precision automatic drilling for the assembly of large parts of aircraft fuselage. The basic structure of flexible track hybrid robot for drilling in abutting joint between the nose and the front fuselage of large aircraft components is shown in Fig. 1.
Flexible track hybrid drilling robot is composed of flexible track, parallel base, cross slide table, and end-effector, etc. Powerful vacuum suckers are uniformly deployed at the bottom of the flexible track, which are adsorbed on the skin surface of the aircraft. Both sides of flexible track are clamped by a kind of V-wheel at the bottom of parallel base and connected as a whole. The parallel base is composed of four column legs and a rectangular inner frame, which connects the four legs to form a closed and stable parallel mechanism. Each column leg is equipped with a linear guide rail and ball screw, and a servo motor is installed on the top to realize independent drive of each leg. The end-effector comprises a mechanical spindle, a pressure angle shaft, a laser detector, and a grating ruler, etc. Both of drilling feeding and pose adjustment are realized by four-axis synchronous feeding and relative motion of parallel base. Four-axis parallel servo motor drives the whole parallel base downward along the direction of parallel legs, so as to drilling. Based on normal deviation detected by pressure angle of end-effector, the motor driving instructions of each leg of four shafts are calculated through inverse kinematics algorithm, and different vectors are executed by servo motors of each shaft, so as to adjust the normal deviation between drilling tool and aircraft skin and reduce it to target range (Fig. 2).

Analysis of position error cause
The process layout and equipment working station of flexible track hybrid robot for fuselage assembly are shown in Fig. 3. The technics design of aircraft assembly is to complete drilling task of ribband and aircraft skin on joint area between the nose and the front of aircraft around the fuselage. Therefore, flexible track is laid around fuselage joint area, paralleling to fuselage joint ring seam area. X-axis direction of cross slide table is parallel to the direction of flexible track laying along the joint area. The design size of X-axis stroke is consistent with the distance between two purlins in joint area, so that the drilling range in each station is between the two purlins. When performing the drilling task, automatic drilling robot moves to the designated station along the flexible track that has been laid. According to prefabricated benchmark hole on the ribband between two purlins, the drilling task is carried out.
Apparently, the technical requirements of assembly docking area between the front fuselage and the nose of aircraft and the application scenarios of flexible track hybrid robot for automatic drilling put forward strict requirements for positioning precision of X-axis. According to the process requirements, there are only four holes in the Y direction, while there are eight rows of holes in the X direction. In established process station, the high-precision positioning of hole location is realized only by relying on X and Y axes of the cross slide table to drive the spindle and cutter to specified position. Obviously, if the positioning precision of X-axis is not enough, then the precision of automatic hole drilling task which executed after detection of benchmark is more difficult to guarantee. The longer the length in hole drilling area, the greater the cumulative error. Not only the positioning precision of a single hole, but also the straightness deviation of row hole location, which will seriously restrict the quality of riveting in the area. Meanwhile, the structure characteristics and the way of drive shaft layout of flexible track hybrid robot for automatic drilling restrict positioning accuracy of X-axis, due to the comprehensive factors such as heavy load, bias single drive, and lever action. At the beginning of design, in order to reduce overall size and mass of equipment, compact structure, reduce the center of gravity, balance the counterweight for easy lifting, and other factors, the X direction drive shaft cannot be arranged in the middle of the cross slide table. Moreover, due to the constraints of spatial layout and overall quality, the dual-axis layout and synchronous driving method cannot be adopted. Eventually, the X-axis servo motor with the lead screw is mounted on the outside of the bottom of the inner frame. However, the X-axis is required to drive the entire end-effector and cross slide table, and the lead screw is subjected to much heavier drag than the Y-axis. At the same time, because of the driving force operating point is not at the central position, the parallel linear guide rail on both sides of the cross slide table bears a certain torsional moment, resulting in the positioning precision of X-axis showing nonlinear characteristics.
To sum up, positioning error compensation should be carried out for X-axis. The positioning precision of X-axis directly determines the positioning precision of global hole drilling, the distribution of row holes, and the precision of hole arrangement. Compensating the X-axis positioning error is to compensate the comprehensive positioning error and improve the comprehensive positioning precision. Just like Cannikin Law, the so-called compensation is to complement the shortest board, that is, compensation of the worst precision of the drive shaft.

Method of positioning error compensation
The positioning precision compensation principle based on response surface methodology is to use statistical method to collect part of the positioning error data, draw the positioning error curve, and analyze the characteristics of the positioning error. Then, the fitting method corresponding to the characteristics of positioning error is selected to establish the prediction model of global positioning error, and the compensation of position coordinates is realized through the prediction of positioning error at last. The process of compensation method is shown in Fig. 4.
Response surface methodology [21] is an optimization method that integrates experimental design and mathematical modeling. It conducts experiments at representative local points, regresses the functional relationship between factors and results in the global scope, and finally obtains the optimal solution of each factor. By collecting the set of sample points in the design space, the global approximation of output variable (system response) is fitted, which approximates the real response surface. In the engineering optimization design, response surface method is widely used. It can not only establish the relationship between the target response and the design variables but also obtain the optimization method [22], so that the objective function can reach the optimal.
At present, there are polynomial, exponential function, logarithmic function fitting, and neural network approximate methods to construct response surface methods. Among them, polynomial regression approximates complex function relations with relatively simple polynomials, especially in nonlinear regression [23], which shows the characteristics of simplicity and small computation. Usually, a low-order polynomial approximation is used within a certain range of design variables. In general, a second-order polynomial approximation model is [24] where 0 , i , ij are the unknown coefficients.
In order to distinguish, let x 1 denote the X-axis position coordinate variable, x 2 denote the Y-axis position coordinate variable, x 3 denote the z-axis position coordinate variable, then the design variable is write it in matrix form: The key of polynomial model fitting is to solve the unknown coefficient vector. The response surface and unknown coefficient vector can be solved by analyzing the unknown coefficient vector by least square method [25].
Generally, the fitting effect of the response surface was evaluated by coefficient of determination ( r 2 ) and root  [26]. The root mean square deviation (RMSD) represents the sample standard deviation of the difference between the predicted value and the observed value. The determination coefficient is used to evaluate the goodness of fit of the regression model coefficients. The calculation formula [27] for both is as follows: For the selection of appropriate sample size to construct the response surface, the working plane of end-effector is evenly partitioned according to the established grid size, as shown in Fig. 5. Where, the grid side length is l ; each theoretical coordinate position (x i , y i , z i ) is equally divided according to the grid step size. The actual target position is detected by the laser tracker, and the difference between the actual coordinate position and the theoretical coordinate position is the corresponding positioning error Δx.
In the range of (X, Y) plane travel, the X-axis positioning error at any point can be predicted by the response surface fitted by each detection error. The specific steps of constructing response surface to predict positioning errors are as follows: (1) Detect the positioning errors at all grid intersections within the entire travel range; according to the established grid intersection point coordinates as the instruction position (x i , y i , z i ) , the actual position coordinates are detected and compared with the instruction , and the error distribution curve of the ball screw on the full travel is calculated; (2) according to the characteristics of the error distribution curve, the corresponding response surface fitting method was selected, and the location coordinates of X, Y, and Z axes were taken as the design variables, and the positioning error was taken as the response target to calculate the regression equation of the positioning error surface; (3) Calculate the error prediction value of the instruction position coordinates by the fitting positioning error regression equation input from any instruction position coordinates: Ŷ (Δa, Δb, Δc).
Not only to make more accurate fitting by the response surface, but also to meet the precision requirement of conditions as far as possible reduce the number of sampling points, putting forward a method of optimal gridding, to assess the effect of grid length and fitting precision, so that to make the optimal analysis of grid length detection area. The grid size is not used to evaluate goodness of fit, but as an optimal solution for sampling density. Before establishing the regression equation between positioning error and coordinate position, a lot of positioning error data need to be taken. How to adopt accurate and convenient sample number needs boundary conditions to evaluate. Therefore, a method to evaluate sampling density based on grid size is proposed. In this method, the regression equation between sample density and fitting precision is established, which is used as boundary condition to evaluate and optimize sample density. The specific method is as follows: in the whole travel area, the positioning error data are collected by groups according to the length of different grids; then, the least square regression method was used to fit the error data of different grids, and the residual mean value of each group of regression equation was calculated based on the regression equation of positioning error data fitting. Finally, by comparing the fitting residuals of different grid partitioning, the optimal grid size length with relatively small fitting residuals and relatively small grid partitioning number was selected.
Relationship of different grid steps for the positioning error of the regression model fitting the data is shown in Fig. 6. As we can see, the grid size is smaller, and the better the fitting effect, the smaller the offset of the measured data with the regression model, that is, to the least squares fitting error of the positioning error of forecasting results and actual values between the residual error is smaller, the more fully and accurately predict the change trend of actual positioning error.
Residuals of the regression model fitted with different grid sizes are shown in Fig. 7. The residual value of regression fitting increases with the increase of grid size. In all sample points with positioning errors, the residual values within the grid size of 20 mm are all less than 0.02 mm, which has little influence on the goodness of fit of the response surface. The finer the grid dividing step is, the smaller the residual value is, and the better the fitting effect is. Meanwhile, considering the test detection error, the influence on the fitting precision when the residual value is less than 0.02 mm can be ignored. Therefore, in order to meet the fitting precision and relatively reduce the number of samples, the grid size of 20 mm was selected as the relative optimal solution.

Design of experiment
In the test, the end-effector of the hybrid robot assembly for the fuselage was used as the carrier, and the position of the end-effector was measured by API-T3 laser tracker. The test site is shown in Fig. 8. The laser tracking target spherically mounted reflector (SMR) is mounted on the spindle of the end-effector of the robot. During the measurement process, the position and attitude of the orbit robot remain fixed and the end-effector is only driven by the motor instruction.
Both the sampling point and the instruction coordinate point are relative to the robot coordinate system, and the point coordinates measured by the laser tracker are also relative to the robot coordinate system. The steps to establish the robot coordinate system are as follows: (1) The end-effector returns to zero and collects the zero position as the origin of the base coordinate system O.
(2) Drive the X-axis to move a certain distance, use the laser tracker to measure and obtain the point set, and then fit the X-axis. (3) Similarly, Y-axis and Z-axis are driven, respectively, according to the same method. Laser tracker is used to measure and obtain point sets, and Y-axis and Z-axis are fitted. (4) Three features (coordinate origin, X-, Y-, and Z-axis) are obtained through the above steps to establish the robot coordinate system.
After the robot coordinate system is established, the working travel is divided according to the optimal grid, and the positioning error within the full travel is detected and calculated. The experimental detection steps are shown in Fig. 9.

Analysis of experimental data
Before constructing the response surface approximation model, the relationship between the design variables and the analysis target should be preliminarily analyzed qualitatively, so as to choose the appropriate functional form to quantitatively describe the relationship between the design variables and the response target. According to the detection scheme, a total of 180 grid intersecting point coordinates positioning errors were obtained within the travel range, which were plotted into a broken line diagram as shown in Fig. 10. In the plane of the end-effector (X, Y), motors and ball screws of the same specifications are used for both Positive Negative X Y Fig. 3 Process requirements of the joint area between nose and front fuselage X and Y axes. However, the absolute positioning error of X-axis is much larger than that of Y-axis, and the absolute positioning error of X-axis is nearly ten times larger than that of Y-axis and Z-axis. Obviously, the absolute positioning precision of X-axis determines the comprehensive positioning precision. Since the X-axis positioning error plays  In order to show the changing trend of positioning error more directly, three-dimensional views of positioning error in the X-axis and Y-axis plane are drawn with the positioning error value as the ordinate, as shown in Fig. 11. Taking the abscissa as the X-axis coordinate position, the ordinate as the positioning error value, and different Y-axis coordinate positions as groups, a line chart of positioning error along the X-axis direction is drawn, as shown in Fig. 12. Within the working stroke of (X, Y) axis, the whole positioning error data presents a certain fluctuation along the X-axis. The variation trend of positioning error data of different groups along the X-axis is basically the same. The minimum value of positioning error is in the Y = 0 mm position coordinate, and the maximum value is in the Y = 220 mm position coordinate. Residual (mm) Step of gridding length (mm) Fig. 7 Relationship between grid size and fitted residuals Positioning error data are laid out in X-and Y-axis planes, representing by different color aberrations, as shown in Fig. 13. It can be seen that, obviously, the area with high positioning accuracy is mainly near the drive shaft. According to the 180 positioning error data detected above, the response surface of global positioning error was constructed, with combining the nonlinear characteristics of sampling positioning error and incremental feature with coordinate position and selecting the method of polynomial fitting and error offset term increasing where is the deviation term of fitting error. The response surface determination coefficient of fitting is 99.6%, and the root mean square deviation is 0.006. The fitting precision of the response surface model is good, which is sufficient to meet the requirements of accurate prediction of positioning errors.

Effect of precision compensation
According to the regression equation fitted by the positioning error data, the X-axis positioning precision compensation algorithm was programmed into numerical control system. Based on same experiment conditions, the compensated positioning error was measured with a laser tracker. During the experiment, the positions of laser  Table 1.
After compensation, the precision of 180 grid intersections is greatly improved, and the absolute positioning precision is all within 0.04 mm. The maximum absolute positioning error is 0.039 mm, which is 72.7% lower than that of 0.143 mm before compensation. The mean value was 0.025 mm, which decreased by 69.1% compared with 0.081 mm before compensation. Compared with before compensation, the standard deviation is reduced by 25% and the error offset is reduced.
Through no-load experiment, the precision compensation method is verified to improve the positioning accuracy of the equipment. However, the no-load experiment is to verify the accuracy of the equipment drive shaft driving according to the command, not the accuracy of the riveting hole position of the aircraft. The automatic drilling experiment is to investigate the final effect of precision compensation. The fundamental purpose of precision compensation research is to improve the accuracy of aircraft assembly and to solve the quality problem of aircraft assembly process. Therefore, further automatic drilling experiments are needed to verify the final improvement effect of assembly accuracy.
In order to fully verify the effect of the precision compensation algorithm on final drilling task, two sets of drilling experiments were designed to compare the hole position accuracy before and after precision compensation. There is a set of datum holes on both sides of the drilling end. Firstly,

Start
Return to zero Instruction: X axis moves by the step Measure & record X-axis return to zero If X-axis reaches a limit End Y-axis moves the step If Y-axis reaches a limit N Y N Y Fig. 9 Absolute positioning error detection program ▸ the datum holes on both sides are detected by an industrial camera to obtain the location of drilling area. Then along the X-axis, the first group of evenly spaced hole drilling tasks are executed according to the instructions after precision compensation. After completing the first group of drilling tasks, the precision compensation algorithm is shielded and the second drilling task is executed according to original instructions. The material of experiments is aviation aluminum 2024-T3. The drilling process is lubricated with trace oil mist and the ambient temperature is 21 °C. With the same set of datum holes as benchmark, in two different groups of drilling methods, one group adopts the compensation algorithm, and the other group does not use the compensation algorithm. The results of drilling are shown in Fig. 17.
Taking the fitted center of datum hole as origin of coordinate system, a plane coordinate system was established to measure the coordinates of different holes. The geometric tolerance of each hole was detected by coordinate measuring machine, and the two groups of data were fitted into one group to compare the positioning errors before and after compensation, as shown in Figs. 18 and 19. The equipment used for measurement and test conditions are shown in Table 2.
According to the origin of datum hole, the coordinates of each hole are measured and fitted. The difference value between each two adjacent holes and the target instruction value is the positioning error. The maximum positioning error was only 0.039 mm, which was 81.6% lower than the maximum value of 0.212 mm before compensation. Besides, the average error was 0.026 mm, which was 80.3% lower than the average error which was 0.132 mm before compensation. Obviously, the error tends to be flat, small, and stable after compensation. The straightness of continuous drilling is expressed by deviation of the hole position Y-axis coordinate. Relatively speaking, the straighter the error curve is, the more accurate trajectory of continuous automatic drilling can be. Before and after compensation, the deviation of the hole location in the direction of Y-axis has no significant change, but the overall trend is more gentle. The mean positioning error of Y-axis before and after compensation is reduced from 0.036 mm to 0.024 mm, which is 33.3% lower than that. The positioning error variation before and after compensation has obvious position correlation: the positioning error deviation of the first half of the drilling position is small, but in the second half of the drilling position, the positioning error difference before and after compensation gradually increases. This verifies the correlation between the end-effector positioning error and its load position.
In no-load experiment, the positioning error was reduced by 69.1%. Meanwhile, the drilling experiment proved that the positioning error was reduced by 80.3%. In both Location on Y axis(*20mm) X Y experimental schemes, the positioning accuracy of endeffector and drilling process is improved by compensation algorithm, and the correlation between positioning error and end-effector position change is verified. The results showed that the positioning precision of the hybrid robot compensated by response surface method was significantly improved, and the precision was more than doubled compared with that before the compensation. Moreover, the fluctuation range of the compensated positioning error was smaller and the error tends to be more stable.

Conclusion
(1) A precision compensation method based on response surface is proposed. Based on the analysis of the distribution characteristics of global positioning errors, a binary second-order response surface model was established to predict the X-and Y-axis plane positioning errors.
(2) A gridding length optimization method is proposed. The least square method was used to fit the positioning errors of different grid sizes in the same area, and the relationship between different grid sizes and fitting residuals was revealed. The goodness of fit of response surface was improved by the grid size optimization method.
(3) After compensation, the absolute positioning precision of the hybrid robot is improved to 0.05 mm, which is about 70% higher than before; the maximum positioning error is reduced to 0.039 mm, which is reduced by 70% and the mean square error is reduced by 25%. The error fluctuation is more stable, which is more suitable for the application scenario of the hybrid robot in the fuselage docking and assembling the configuration hole task. Absolute positioning error(mm) The number of sampling After Before (4) The results of experiments show that the method based on response surface precision compensation and optimization method of least square grid size for preci-sion effect is remarkable. This method is accurate, simple, and has a small amount of computation, which effectively improves the application range of the hybrid robot for positioning precision. (5) When constructing the response surface, the positioning error data of a single drive shaft, which has the greatest influence on the comprehensive positioning error, are adopted, while the pose error of the end-effector is ignored. The pose error compensation method of the endeffector still needs further study. Absolute positioning error(mm) The number of sampling After Before

Data availability
The data sets supporting the results of this article are included within the article.
Code availability Not applicable.