A spray gun trajectory planning method for spraying robots

Abstract

The present invention discloses a spray gun trajectory planning method for a spray robot, comprising: obtaining coordinate data of each triangular facet of a surface to be sprayed by modeling; setting spray process parameters according to the film-forming characteristics of the spray gun spray paint film and the target paint film thickness of the surface to be sprayed; the spray process parameters include spray trajectory spacing, spray speed, spray height and spray time; screening out triangular facets that meet the spray trajectory conditions according to the spray trajectory spacing; solving the intersection points of the triangular facets that meet the spray trajectory conditions to obtain the intersection points of each key node of the spray gun; saving the normal vectors of the intersection points and the corresponding triangular facets into a normal vector matrix; solving the spray posture of the spray gun according to the normal vector matrix; and connecting each key node by linear interpolation and circular interpolation to obtain the spray gun spray trajectory. The above method generates a spray gun spray trajectory for a complex surface to be sprayed, which effectively improves the spray quality.

Description

Technical Field

[0001] The invention relates to the technical field of spraying robots, and in particular to a spray gun trajectory planning method for a spraying robot. Background Art

[0002] A spraying robot, also known as a spray painting robot, is an industrial robot that can automatically spray paint or other coatings.

[0003] At present, there are two main ways to generate spray trajectories for complex parts by spray robots: manual teaching and offline programming. Manual teaching has the disadvantages of low programming efficiency, poor trajectory accuracy, and low paint utilization, so it is only suitable for some simple spraying occasions. Although the existing offline programming software has greatly improved the efficiency, accuracy of spray trajectory, and adaptability compared with the traditional manual teaching method. However, for complex parts to be sprayed, the existing offline programming software has relatively fixed spray process packages, and the generated spray gun spray trajectory is relatively single, fixed, and poorly adaptable. Therefore, when using the existing offline programming software to directly generate the spray gun spray trajectory for complex parts to be sprayed, there may be some spraying areas where the spray quality is not high, thus failing to meet the requirements of paint film thickness. It is also possible that due to the large change in the curvature of the surface to be sprayed, there will be too many redundant points in the spray trajectory when using the offline programming software for offline programming, resulting in excessive calculation of the spray robot when performing motion solution. This in turn affects the continuity of the spray robot's movement, causing a sudden change in the spray robot's acceleration, thereby aggravating the shortening of the spray robot's service life and further deteriorating the spraying effect. Summary of the invention

[0004] In view of this, an embodiment of the present invention provides a spray gun trajectory planning method for a spraying robot to solve the problems in the prior art.

[0005] An embodiment of the present invention provides a spray gun trajectory planning method for a spraying robot, comprising:

[0006] Obtain the coordinate data of each triangular facet of the surface to be sprayed by modeling;

[0007] Set the spraying process parameters according to the film-forming characteristics of the spray gun and the target film thickness of the surface to be sprayed; the spraying process parameters include spraying track spacing, spraying speed, spraying height and spraying time;

[0008] According to the spraying track spacing, triangular facets meeting the spraying track conditions are selected;

[0009] Solve the intersection points of the triangular patches that meet the spray trajectory conditions to obtain the intersection points of each key node of the spray gun;

[0010] Save the intersection point and the normal vector of the corresponding triangle patch into the normal vector matrix;

[0011] Solve the spraying posture of the spray gun according to the normal vector matrix;

[0012] Each key node is connected by linear interpolation and circular interpolation to obtain the spraying trajectory of the spray gun.

[0013] Optionally, the coordinate data of each triangular facet of the surface to be sprayed is obtained by modeling, including:

[0014] Store the modeling file as an STL file in ASCII format;

[0015] Read the normal vector component information and vertex coordinate information of each triangle face from the STL file.

[0016] Optionally, the spraying process parameters are set according to the film-forming characteristics of the spray gun and the target film thickness of the surface to be sprayed, including:

[0017] Obtaining a track spacing between a center point of a spray gun corresponding to a first spray track and a center point of a spray gun corresponding to a second spray track;

[0018] Obtaining an overlap distance between an overlapped portion of a first spraying track and a second spraying track according to a track spacing and a long semi-axis of a spray gun;

[0019] According to the overlap distance, the long semi-axis of the spray torch, the short semi-axis of the spray torch, and the target coating thickness, the spray trajectory spacing, spray speed, spray height, and spray time are calculated.

[0020] Optionally, before solving the intersection points of the triangular facets meeting the spraying trajectory conditions to obtain the intersection points of each key node of the spray gun, the method includes:

[0021] Compare the height of the tangent plane corresponding to each triangular face with the minimum and maximum values ​​of the three vertices of the corresponding triangular face in the z-axis direction;

[0022] If the height of the tangent plane corresponding to the triangle patch is between the minimum and maximum values, the data of the corresponding triangle patch is retained; otherwise, the data of the corresponding triangle patch is discarded.

[0023] Optionally, the spray trajectory conditions include:

[0024] If the vertex coordinates of the adjacent first triangular face patch and the second triangular face patch do not have the same value in the z-axis direction, the intersection of the first triangular face patch and the second triangular face patch is solved, and the intersection and the corresponding normal vector are saved in the normal vector matrix;

[0025] If the tangent plane of a triangle patch is parallel to an edge of the corresponding triangle patch, and the current intersection point is the same as the previous intersection point, the intersection point is solved using the coordinate information of another vertex of the triangle patch on the z-axis, and the intersection point and the corresponding normal vector are saved in the normal vector matrix.

[0026] Optionally, the spraying trajectory conditions also include:

[0027] If the tangent plane of a triangular patch is tangent to a vertex of the corresponding triangular patch, the tangent vertex is saved as the intersection point, and the intersection point and the corresponding normal vector are saved in the normal vector matrix.

[0028] Optionally, the spraying trajectory conditions also include:

[0029] If the tangent plane of a triangle patch is tangent to an edge of the corresponding triangle patch, the corresponding triangle patch information is removed.

[0030] Optionally, solving the spraying posture of the spray gun according to the normal vector matrix includes:

[0031] Make the direction of the spray gun parallel to the normal vector at the corresponding position of the surface to be sprayed during spraying;

[0032] Make the long axis of the spray gun spray perpendicular to the tangential direction of the surface to be sprayed.

[0033] Optionally, solving the spraying posture of the spray gun according to the normal vector matrix also includes:

[0034] Get the rotation matrix and position matrix according to the normal vector matrix;

[0035] Get the homogeneous transformation matrix based on the rotation matrix and the position matrix;

[0036] The rotation angles of the coordinate system of the end of the spraying robot relative to each axis of the base coordinate system are calculated according to the homogeneous transformation matrix and the rotation angle of the spraying robot.

[0037] Beneficial effects of the embodiments of the present invention:

[0038] An embodiment of the present invention provides a method for planning a spray gun trajectory for a spray robot. For the generation of a spray gun spray trajectory for a complex curved surface to be sprayed, on the basis of satisfying the thickness of the spray paint film, the three-dimensional model of the complex curved surface is approximated to be composed of triangular facets, thereby avoiding the existence of redundant interpolation points caused by the complexity of the curved surface to be sprayed, thereby improving the solution speed of the spray robot while satisfying the requirement of the thickness of the spray paint film, and effectively improving the spraying quality. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] The features and advantages of the present invention will be more clearly understood by referring to the accompanying drawings, which are schematic and should not be construed as limiting the present invention in any way. In the accompanying drawings:

[0040] Figure 1 A flow chart of a spray gun trajectory planning method for a spraying robot in an embodiment of the present invention is shown;

[0041] Figure 2 A schematic diagram showing the superposition of two adjacent spray trajectory elliptical spray torches;

[0042] Figure 3 A schematic diagram showing the intersection of a slice plane and a triangular facet is shown;

[0043] Figure 4 A schematic diagram showing a slicing plane parallel to an edge of a triangular facet;

[0044] Figure 5 A schematic diagram showing a slicing plane being tangent to a vertex of a triangular face;

[0045] Figure 6 A schematic diagram showing that a slicing plane is tangent to an edge of a triangular face;

[0046] Figure 7 A flow chart of a method for generating a spray gun spray trajectory according to an embodiment of the present invention is shown;

[0047] Figure 8 A schematic diagram of a three-dimensional model of a complex curved surface to be sprayed is shown;

[0048] Fig. 9 A schematic diagram of a curved surface obtained by drawing the coordinate data of each triangular facet is shown;

[0049] Fig.10 A schematic diagram showing the intersection of a slice face and a triangular face patch;

[0050] Fig.11 A schematic diagram of the spraying trajectory of the spraying robot spray gun is shown;

[0051] Fig.12 A schematic diagram of the spraying trajectory planned by the offline programming software is shown. DETAILED DESCRIPTION

[0052] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present invention.

[0053] The embodiment of the present invention provides a spray gun trajectory planning method for a spraying robot, such as Figure 1 As shown, including:

[0054] Step S10, obtaining coordinate data of each triangular facet of the surface to be sprayed by modeling.

[0055] In this embodiment, the complex curved surface is deconstructed into a model formed by fitting a number of triangular patches.

[0056] Step S20, setting spraying process parameters according to the film-forming characteristics of the spray gun and the target film thickness of the curved surface to be sprayed. The spraying process parameters include spraying track spacing, spraying speed, spraying height and spraying time.

[0057] In this embodiment, when the spraying trajectory is a bow shape, there is a problem of repeated spraying between adjacent trajectories. Taking into account the characteristics of the spray gun, the paint film at the center of the spray gun is thicker and the paint film at the edge is thinner. The repeated part at the edge position is set with corresponding spraying process parameters according to actual needs.

[0058] Step S30, selecting triangular facets that meet the spraying trajectory conditions according to the spraying trajectory spacing.

[0059] In this embodiment, taking into account the complexity of the surface to be sprayed, in order to reduce the amount of calculation and prevent each spraying trajectory from having to traverse all the triangular facets to solve whether there are intersections, the triangular facets that meet each spraying trajectory are quickly screened out according to the spray gun spraying trajectory spacing calculated in step S20.

[0060] Step S40, solving the intersection points of the triangular facets that meet the spraying trajectory conditions to obtain the intersection points of each key node of the spray gun.

[0061] In this embodiment, the intersection of the key nodes is a part of the spraying trajectory.

[0062] Step S50, saving the intersection point and the normal vector of the corresponding triangle face into a normal vector matrix.

[0063] In this embodiment, the normal vector matrix is ​​used to calculate the spraying posture of the painting robot.

[0064] Step S60, solving the spraying posture of the spray gun according to the normal vector matrix.

[0065] In this embodiment, the rotation matrix and position matrix of the spray gun are calculated based on the normal vector matrix, and the homogeneous transformation matrix of the spray gun at the intersection of key nodes is obtained by sorting out the rotation matrix and the position matrix. According to the homogeneous transformation matrix and the rotation angle corresponding to the spray robot, the rotation angle of the coordinate system of the end of the spray robot relative to each axis of the base coordinate system is calculated.

[0066] Step S70, connecting each key node by linear interpolation and circular interpolation to obtain the spraying trajectory of the spray gun.

[0067] The embodiment of the present invention aims at generating a spray gun spray trajectory for a complex curved surface to be sprayed. On the basis of satisfying the thickness of the sprayed paint film, the three-dimensional model of the complex curved surface is approximated to be composed of triangular facets, thereby avoiding the existence of redundant interpolation points caused by the complexity of the surface to be sprayed, thereby improving the solution speed of the spray robot while satisfying the requirement of the thickness of the sprayed paint film.

[0068] As an optional implementation, step S10 includes:

[0069] Step S11, storing the modeling file as an STL file in ASCII format.

[0070] Step S12, reading the normal vector component information and vertex coordinate information of each triangular face from the STL file.

[0071] In this embodiment, a three-dimensional model corresponding to the complex curved surface to be sprayed is built by existing three-dimensional modeling software, and then after setting a reasonable chord height and step length, the model is saved as an STL file in ASCII format. Since each triangular face of the STL file in ASCII format conforms to the right-hand rule to define the three components of the normal vector and the coordinates of the three vertices in sequence. By using MATLAB, it is read in this format in sequence, and each triangular face information is saved in a matrix A.

[0072] A=[x1 y1 z1,x2 y2 z2,x3 y3 z3,n x n y n z ] n×12 (1)

[0073] As an optional implementation, the spraying process parameters are set according to the film-forming characteristics of the spray gun and the target film thickness of the curved surface to be sprayed, including:

[0074] Obtaining a track spacing between a center point of a spray gun corresponding to a first spray track and a center point of a spray gun corresponding to a second spray track;

[0075] Obtaining an overlap distance between an overlapped portion of a first spraying track and a second spraying track according to a track spacing and a long semi-axis of a spray gun;

[0076] According to the overlap distance, the long semi-axis of the spray torch, the short semi-axis of the spray torch, and the target coating thickness, the spray trajectory spacing, spray speed, spray height, and spray time are calculated.

[0077] In this embodiment, the characteristics of the paint film formed by the spray gun are as follows: Figure 1As shown, the calculation method of spraying process parameters such as spraying track spacing, spraying speed, spraying height, etc. of the spray gun is as follows;

[0078]

[0079]

[0080] In the formula,

[0081]

[0082]

[0083]

[0084]

[0085]

[0086] Among them, W s1 (x), W s2 (x) represents the coating thickness swept by the spraying at point W on the adjacent trajectory; t1 and t2 represent the spraying time at point W on the adjacent spraying path, x1 and x2 represent the distance from the spraying point W to the center point of the spray gun of the two adjacent spraying trajectories, t is the time taken for the projection from point O to point W to the spraying trajectory, W L is the ideal coating thickness of the paint film, V represents the spraying speed of the spray gun, a is the long semi-axis of the spray gun torch, b is the short semi-axis of the spray gun torch, D is the track spacing between the center point of the spray gun corresponding to the first spraying track and the center point of the spray gun corresponding to the second spraying track, and d is the overlap range of the edges of adjacent spraying tracks.

[0087] In this embodiment, the spraying trajectory of the spray robot's spray gun not only takes into account the position of the spray gun, but also takes into account that the spray gun can satisfy the normal vector of the spray gun parallel to the surface to be sprayed during the spraying process; the long axis of the paint spray width sprayed by the spray gun can be perpendicular to the tangent vector of the surface to be sprayed, thereby greatly improving the quality of the spray paint film.

[0088] As an optional implementation, before step S40, the following steps are included:

[0089] Compare the height of the tangent plane corresponding to each triangular face with the minimum and maximum values ​​of the three vertices of the corresponding triangular face in the Z-axis direction;

[0090] If the height of the tangent plane corresponding to the triangle patch is between the minimum and maximum values, the data of the corresponding triangle patch is retained; otherwise, the data of the corresponding triangle patch is discarded.

[0091] In this embodiment, the specific implementation is as follows:

[0092] 1) Input the height h of each slice plane i ;

[0093] 2) Find the minimum value Z in the 3rd, 6th, and 9th columns of each row from matrix A min With the maximum value Z max ;

[0094] 3) If Z min ≤h i ≤Z max , then save all the data in this row into another matrix B j ; Otherwise, the data of this row will not be taken;

[0095] 4) Repeat steps 1) to 3) in sequence until all slice planes are calculated.

[0096] The intersection of each selected triangle face is calculated to find each key node intersection of the spray gun. Each intersection is calculated using formula (3).

[0097]

[0098] As an optional implementation, the spraying trajectory conditions include:

[0099] If the vertex coordinates of the adjacent first triangular face patch and the second triangular face patch do not have the same value in the Z-axis direction, the intersection of the first triangular face patch and the second triangular face patch is solved, and the intersection point and the corresponding normal vector are saved in the normal vector matrix.

[0100] like Figure 3 As shown, the Z of adjacent triangle patches min With Z max There are no two identical values. In this case, directly use formula (2) to solve the intersection point, and save the solved intersection point and the normal vector of the triangle patch into matrix C. Then save the two vertices of the solved intersection point and delete the triangle patch.

[0101] If the tangent plane of a triangle patch is parallel to an edge of the corresponding triangle patch, and the current intersection point is the same as the previous intersection point, the intersection point is solved using the coordinate information of another vertex of the triangle patch on the Z axis, and the intersection point and the corresponding normal vector are saved in the normal vector matrix.

[0102] like Figure 4 As shown, the slice plane is parallel to one edge of the triangle patch. In this case, the intersection point solved may be the same as the previous intersection point. If you need to select another Z min or Z max, and then solve the intersection point. And save the solved intersection point and the normal vector of the triangle patch into the matrix C. Then save the two vertices of the solved intersection point and delete the triangle patch.

[0103] As an optional implementation, the spraying trajectory condition also includes:

[0104] If the tangent plane of a triangular patch is tangent to a vertex of the corresponding triangular patch, the tangent vertex is saved as the intersection point, and the intersection point and the corresponding normal vector are saved in the normal vector matrix.

[0105] like Figure 5 As shown in the figure, the slice plane is tangent to a vertex of the triangle patch. The intersection position information of the point and the normal vector information of the triangle patch are retained.

[0106] As an optional implementation, the spraying trajectory condition also includes:

[0107] If the tangent plane of a triangle patch is tangent to an edge of the corresponding triangle patch, the corresponding triangle patch information is removed.

[0108] like Figure 6 As shown in Figure 1, the slice plane is tangent to one edge of the triangle patch. In this case, since there are countless intersection points, this situation should be excluded.

[0109] As an optional implementation, solving the spraying posture of the spray gun according to the normal vector matrix includes: making the direction of the spray gun parallel to the normal vector at the corresponding position of the surface to be sprayed during spraying; making the long axis of the spray gun spray width perpendicular to the tangential direction of the surface to be sprayed.

[0110] As an optional implementation, solving the spraying posture of the spray gun according to the normal vector matrix also includes:

[0111] Get the rotation matrix and position matrix based on the normal vector matrix.

[0112] Get the homogeneous transformation matrix based on the rotation matrix and the position matrix.

[0113] The rotation angles of the coordinate system of the end of the spraying robot relative to each axis of the base coordinate system are calculated according to the homogeneous transformation matrix and the rotation angle of the spraying robot.

[0114] In this embodiment, it is assumed that the current path intersection is w1, the next key path intersection is w2, and:

[0115] w1=[w x1 w y1 w z1 n x1 n y1 n z1 ]

[0116] w2=[w x2 w y2 w z2 n x2 n y2 n z2 ]

[0117] Normal vector negation [-n x1 -n y1 -n z1 ] as the Z-axis direction vector of the spray gun. Take the two adjacent intersection points w1w2 as the unit vector in the X-axis direction; the unit vector in the Y-axis direction is: Y=X×Z.

[0118]

[0119] Arrange the calculated rotation matrix and position matrix of the spray gun to get the homogeneous transformation matrix of the spray gun at this point: Among them, the rotation matrix:

[0120]

[0121] Position Matrix:

[0122]

[0123] Homogeneous transformation matrix:

[0124]

[0125] In a specific embodiment, Figure 7 to Figure 12 As shown, the spray gun trajectory planning method for a spraying robot provided in an embodiment of the present invention is applied to perform trajectory planning on a specific example:

[0126] Step 1: Use 3D modeling software to create a complex surface to be sprayed, such as Figure 8 As shown. Set reasonable chord height and step length for the 3D model and save it as an STL file;

[0127] Step 2: Read the STL file in MATLAB according to certain rules, and store the data read from each triangular patch into matrix A. Then use the plot command to draw the complex surface to be sprayed with the coordinate data of the triangular patch, such as Fig. 9 shown.

[0128] Step 3: According to the film thickness requirement of the complex curved surface to be sprayed, the spray track spacing of the spray gun is calculated according to formula (2), and then the tangent plane, that is, the intersection of part of the spray track and the triangular surface patch is obtained through steps S20 to S40, such as Fig.10 As shown;

[0129] Step 4: Finally, by determining the spraying height of the spray gun, the generation of the spraying trajectory of the spraying robot spray gun is completed according to steps S60 to S70, such as Fig.11 shown.

[0130] like Fig.12 The figure shows the spraying trajectory of the spray gun planned by the offline programming software. Fig.11 and Fig.12 By comparing the results, it can be clearly seen that there are too many redundant points in the offline programming software when planning the spraying trajectory of complex surfaces, which will cause the speed of the spraying robot to drop significantly when performing motion calculation, thereby indirectly affecting the spraying quality.

[0131] Although the embodiments of the present invention have been described in conjunction with the accompanying drawings, those skilled in the art may make various modifications and variations without departing from the spirit and scope of the present invention, and such modifications and variations are all within the scope defined by the appended claims.

Claims

1. A spray gun trajectory planning method for a spray robot, characterized in that: include: Obtain the coordinate data of each triangular facet of the surface to be sprayed by modeling; Setting spraying process parameters according to the film-forming characteristics of the spray gun and the target film thickness of the curved surface to be sprayed; the spraying process parameters include spraying track spacing, spraying speed, spraying height and spraying time; According to the spraying track spacing, triangular facets meeting the spraying track conditions are selected; Solving the intersection points of the triangular facets that meet the spraying trajectory conditions to obtain the intersection points of each key node of the spray gun; The intersection point and the corresponding normal vector of the triangular face are saved in a normal vector matrix; Solving the spraying posture of the spray gun according to the normal vector matrix; The connection of each of the key nodes is achieved by linear interpolation and circular interpolation to obtain a spraying trajectory of the spray gun; Wherein, the spraying process parameters are set according to the film-forming characteristics of the spray gun and the target film thickness of the curved surface to be sprayed, including: Obtaining a track spacing between a center point of a spray gun corresponding to a first spray track and a center point of a spray gun corresponding to a second spray track; Obtaining an overlap distance of an overlapped portion between the first spraying trajectory and the second spraying trajectory according to the trajectory spacing and the long semi-axis of the spray gun; According to the overlap distance, the long semi-axis of the torch, the short semi-axis of the torch, and the target paint film coating thickness, the spray track spacing, the spray speed, the spray height, and the spray time are calculated; Before solving the intersection points of the triangular facets meeting the spraying trajectory conditions to obtain the intersection points of each key node of the spray gun, the method includes: Compare the height of the tangent plane corresponding to each triangular face patch with the minimum and maximum values ​​of the three vertices of the triangular face patch in the z-axis direction; If the height of the tangent plane corresponding to the triangular face patch is between the minimum value and the maximum value, the data corresponding to the triangular face patch is retained; otherwise, the data corresponding to the triangular face patch is discarded; The spraying trajectory conditions include: Define any two adjacent triangular face patches as a first triangular face patch and a second triangular face patch, and if the vertex coordinates of the first triangular face patch and the second triangular face patch do not have the same value in the z-axis direction, solve the intersection of the first triangular face patch and the second triangular face patch, and save the intersection and the corresponding normal vector into the normal vector matrix; If the tangent plane of the triangular patch is parallel to an edge of the corresponding triangular patch, and the currently solved intersection is the same as the previous intersection, the intersection is solved using the coordinate information of another vertex of the triangular patch on the z-axis, and the intersection and the corresponding normal vector are saved in the normal vector matrix.

2. The spray gun trajectory planning method for a spraying robot according to claim 1, characterized in that: The coordinate data of each triangular patch of the surface to be sprayed is obtained through modeling, including: Store the modeling file as an STL file in ASCII format; The normal vector component information and vertex coordinate information of each triangular facet are read from the STL file.

3. The spray gun trajectory planning method for a spraying robot according to claim 1, characterized in that: The spraying trajectory conditions also include: If the tangent plane of the triangular face is tangent to a vertex of the corresponding triangular face, the tangent vertex is saved as an intersection point, and the intersection point and the corresponding normal vector are saved in the normal vector matrix.

4. The spray gun trajectory planning method for a spraying robot according to claim 3, characterized in that: The spraying trajectory conditions also include: If the tangent plane of the triangular patch is tangent to an edge of the corresponding triangular patch, the corresponding triangular patch information is discarded.

5. The spray gun trajectory planning method for a spraying robot according to claim 1, characterized in that: Solving the spraying posture of the spray gun according to the normal vector matrix includes: Make the direction of the spray gun parallel to the normal vector at the corresponding position of the curved surface to be sprayed during spraying; The long axis of the spray width of the spray gun is perpendicular to the tangential direction of the curved surface to be sprayed.

6. The spray gun trajectory planning method for a spraying robot according to claim 1, characterized in that: Solving the spraying posture of the spray gun according to the normal vector matrix also includes: Obtaining a rotation matrix and a position matrix according to the normal vector matrix; Obtaining a homogeneous transformation matrix according to the rotation matrix and the position matrix; The rotation angles of the coordinate system of the end of the spraying robot relative to the axes of the base coordinate system are calculated according to the homogeneous transformation matrix and the rotation angle of the spraying robot.

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