Bezier curve-based robot g3 continuous corner smoothing method

Abstract

The application is a robot G3 continuous corner smoothing method based on Bezier curve. Input the robot small line segment machining tool tip position data, adopt 6 times Bezier curve for switching and smoothing in the workpiece coordinate system, according to the maximum allowable value of tool tip position smoothing error and the condition constraint of G3 continuity at the curve connection, the control points of the Bezier curve are calculated; secondly, input the robot small line segment machining tool axis direction data, adopt 6 times Bezier curve for switching and smoothing in the RPY angle space of the workpiece coordinate system, according to the maximum allowable value of tool axis direction smoothing error and the condition constraint of G3 continuity at the curve connection, the control points of the Bezier curve are calculated; finally, the parameter of the tool axis direction and the tool tip position path after switching and smoothing is synchronized, and the remaining linear segment is replaced by B-spline straight line. The application can realize G3 continuous smoothing switching of the tool path analytically, without any iterative calculation, the smoothing error is controllable, and the processing real-time and stability are improved.

Description

Technical Field

[0001] The invention belongs to the technical field of robot arm trajectory smoothing, and in particular to a robot G3 continuous corner smoothing method based on Bezier curves. Background Art

[0002] In recent years, robots have been widely used in the processing of parts with complex curves and surfaces due to their advantages such as low cost and high flexibility. Due to the complexity of curves and surfaces, robots need to use software such as CAD and CAM for tool path planning when processing such parts. Usually, the robot processing path generated by CAM (computer-aided manufacturing software) is composed of a series of discrete small line segments. The tangential discontinuity at the junction of small line segments will cause sudden changes in the speed, acceleration, and jump of the robot's movement, resulting in mechanical vibration, which seriously affects the quality of the processed surface. In order to improve processing efficiency and processing quality, the industry mainly smoothes the small line segment path through the path smoothing algorithm. In addition, in order to avoid the problem of discontinuity in the angular velocity, angular acceleration, and angular jump of the tool axis rotation at the curve cross-segment during the movement of the tool axis direction following the tool tip position path, it is also necessary to synchronize the parameters of the tool axis direction and tool tip position path after the smoothing.

[0003] Currently, most robot path smoothing algorithms can only achieve curvature continuity at path transition points, known as G2 continuity. However, the rate of change of curvature is discontinuous, which results in poor smoothness of the robot's end-point motion. Secondly, to meet pose smoothing error constraints and pose synchronization requirements, most path smoothing algorithms use an iterative approach to calculate transition curve control points. This increases the robot's computational burden and seriously affects the real-time performance of machining. Summary of the Invention

[0004] To solve the above technical problems, the present invention provides a robot G3 continuous corner smoothing method based on Bezier curves, which can analytically realize the continuous and smooth transition of tool trajectory G3 (curvature differential continuity), thereby improving the real-time and smoothness of processing.

[0005] The technical solution of the present invention is described as follows in conjunction with the accompanying drawings:

[0006] A robot G3 continuous corner smoothing method based on Bezier curves includes the following steps:

[0007] Step 1: Input the tool tip position data for machining small line segments using the robot. Use a 6th-degree Bezier curve for smoothing in the workpiece coordinate system. Based on the maximum allowable tool tip position smoothing error and the G3 continuity constraints at the curve junction, the control points of the Bezier curve are analytically calculated. The smoothed path consists of the inserted Bezier curve and the remaining linear segments.

[0008] Step two, input the robot small line segment machining tool axis direction data, adopt 6 Bezier curve in RPY angle space under the workpiece coordinate system to connect and smooth, according to the maximum allowable value of tool axis direction smoothing error and the condition of G3 continuity at the curve connection, the control points of Bezier curve are calculated, and the smoothed path is composed of the inserted Bezier curve and the remaining linear segment;

[0009] Step three, parameter synchronization is carried out on the tool axis direction and the tool tip position path after connection and smoothing, the remaining linear segment is replaced by B spline straight line to ensure that the third order derivative of the tool axis direction relative to the tool tip position path arc length is continuous at the cross section, and then the pose parameter synchronization is realized, and finally the G3 continuous and parameter synchronized robot path is generated through the control points solved above.

[0010] Further, the specific method of step one is as follows:

[0011] 11) The robot small line segment machining tool tip position data refers to the position data P = [x, y, z] relative to the workpiece coordinate system, and the three tool tip points connected in sequence in the original tool tip position path are P i-1 , P i , P i+1 , and 6 Bezier curve is used as the transition curve at the corner formed by the three points; the curve expression is as follows:

[0012]

[0013] In the formula, P i (u) is the tool tip position transition curve inserted at the corner point P i ; B j,i represents the control points of the inserted 6 Bezier curve P i (u), wherein i is the number of the transition curve, j is the number of the control point, the number of control points is 7; u ∈ [0, 1] is the curve parameter; is the combination number;

[0014] 12) Set the conditions:

[0015]

[0016] In the formula, is the unit tangent vector from P i to P i-1 when u = 0; is the unit tangent vector from P i to P i+1 when u = 1; s is the arc length of the tool tip position path;

[0017] is to meet Insert the first three control points B of the 6th degree Bezier curve 0,i 、B 1,i 、B 2,i Should be located at the tool tip position path P i-1 P i On the segment, the last three control points B 4,i 、B 5,i 、B 6,i Should be located at the tool tip position path P i P i+1 on; to make Established, the following conditions are met:

[0018] Right now:

[0019]

[0020] Where B j,i is the Bezier curve P of the tool tip position i (u), where i is the transfer curve number and j is the control point number. According to the above formula, the length relationship between the control points is as follows:

[0021] To maintain symmetry, let ||B 2,i B 3,i ||=||B 3,i B 4,i ||=l p,i ;

[0022] 13) Set the smoothing parameters of the transfer p,i Requirements:

[0023]

[0024] Where, ε pos,max It is the maximum allowable value of the tool tip position smoothing error. It is generally believed in the industry that the deviation between the midpoint of the inserted curve segment and the original path is the largest.

[0025] 14) Determine the control point. The formula is as follows:

[0026]

[0027] Furthermore, the specific method of step 2 is as follows:

[0028] 21) The robot small segment processing tool axis direction data refers to the RPY angle data of the tool axis relative to the workpiece coordinate system φ = [α, β, γ]. Let the three sequentially connected direction points in the original tool axis direction path be φ i-1 、φ i 、φ i+1At the corner formed by these three points, a 6th-order Bezier curve is used as the transition curve; the curve expression is as follows:

[0029]

[0030] Where, φ i (u) is the angle at the corner point φ i The tool axis direction transition curve inserted at Q j,i is the inserted sixth degree Bezier curve φ i (u) control point, where i is the number of the transition curve, j is the number of the control point, and the number of control points is 7; u∈[0,1] is the curve parameter; is the number of combinations;

[0031] 22) Set conditions:

[0032]

[0033] Where, When u=0, φ i Pointing to φ i-1 The unit tangent vector of ; When u=1, φ i Pointing to φ i+1 The unit tangent vector of ; s is the arc length of the path in the direction of the tool axis;

[0034] To satisfy Insert the first three control points Q of the 6th degree Bezier curve 0,i , Q 1,i , Q 2,i Should be located in the tool axis direction path φ i-1 φ i On the segment, the last three control points Q 0,i , Q 1,i , Q 2,i Should be located in the tool axis direction path φ i φ i+1 In order to make To be established, the following conditions must be met:

[0035] Right now:

[0036]

[0037] Where Q j,i is the Bezier curve φ in the tool axis direction i (u), where i is the transfer curve number and j is the control point number. According to the above formula, the length relationship between the control points is as follows:

[0038] To maintain symmetry, let ||Q 2,i Q 3,i ||=||Q 3,i Q 4,i ||=l o,i ;

[0039] 23) Set the smoothing parameters of the transfer o,i The following conditions are met:

[0040]

[0041] Where, ε ori,max is the maximum allowable value of the tool axis direction smoothing error, and is used to limit the tool axis direction vector smoothing error ΔO within the allowable value. J o is the Jacobian matrix in the tool axis direction, specifically:

[0042]

[0043] Where S α 、S β 、S γ They are the abbreviations of sin(α), sin(β), and sin(γ), respectively. α 、C β 、C γ are the abbreviations of cos(α), cos(β), and cos(γ), respectively; α, β, and γ are the RPY angles of the tool axis relative to the workpiece coordinate system;

[0044] 24) Determine the control point. The formula is as follows:

[0045]

[0046] Furthermore, the specific method of step three is as follows:

[0047] 31) Replace the remaining linear segments in the smoothed tool tip position path and tool axis direction path with B-spline fitting, where the degree of B-spline is selected as 5, the number of control points is selected as 9, and the node vector is: U = [0, 0, 0, 0, 0, 0.5, 0.5, 0.5, 1, 1, 1, 1, 1];

[0048] 32) In order to achieve third-order parameter synchronization, the tool axis direction at the cross section should be continuous with respect to the third-order derivative of the tool tip position path arc length, and finally a G3 continuous and parameter synchronized robot path is generated by solving the control points.

[0049] Furthermore, the method for constructing the straight line of the B-spline fitting is as follows:

[0050] 311) The constructed spline line should meet the following conditions:

[0051] a. The straight line constructed using B-spline should maintain the same continuity as the original residual line, the control points should be on the original residual line, and it must also meet the following requirements:

[0052]

[0053] b. To ensure the synchronization of parameter analysis, it is necessary to ensure that the first-order, second-order, and third-order differentials of the tool axis direction relative to the arc length of the tool tip position path are continuous at the span, that is:

[0054]

[0055] 312) Solve the control points and perform B-spline fitting based on the obtained control points to generate a G3 continuous and parameter-synchronized robot path.

[0056] The beneficial effects of the present invention are:

[0057] 1) The present invention uses a 6-degree Bezier curve to perform G3 continuous local corner smoothing, which can improve the low-cost continuity at the connection of linear paths and achieve differential continuity of the motion path curvature, which is beneficial to avoid sudden changes in the robot's speed, acceleration, and acceleration direction;

[0058] 2) The present invention achieves parameter synchronization by replacing the remaining straight line segments with B-splines, which can avoid iterative calculations, improve calculation efficiency, and help improve the real-time and stability of robot processing. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other relevant drawings can be obtained based on these drawings without paying any creative work.

[0060] Figure 1 It is a flowchart of the present invention;

[0061] Figure 2 This is the tool tip position path smoothing diagram of the present invention;

[0062] Figure 3 This is the smoothing diagram of the tool axis direction path of the present invention;

[0063] Figure 4 It is the posture synchronization diagram of the present invention;

[0064] Figure 5 A curvature comparison diagram of the method of the present invention and the existing method;

[0065] Figure 6A comparison chart of the running time of the method of the present invention and the existing method;

[0066] Figure 7 2 is a comparison diagram of joint acceleration between the method of the present invention and the existing method. DETAILED DESCRIPTION

[0067] The present invention will be further described in detail below with reference to the accompanying drawings and examples. It will be understood that the specific embodiments described herein are intended only to illustrate the present invention and are not intended to limit the present invention. It should also be noted that, for ease of description, the accompanying drawings only illustrate portions relevant to the present invention, not all structures.

[0068] Example 1

[0069] See Figure 1 This embodiment provides a robot G3 continuous corner smoothing method based on Bezier curves, comprising the following steps:

[0070] Step 1: Input the tool tip position data for machining a small line segment using the robot. Use a 6th-order Bezier curve for smoothing in the workpiece coordinate system. Under the constraints of the maximum allowable tool tip position smoothing error and G3 continuity (curvature differential continuity) at the curve junction, calculate the control points of the Bezier curve analytically. The smoothed path consists of the inserted Bezier curve and the remaining linear segments, as follows:

[0071] 11) The tool tip position data of the robot for small line segment machining refers to the position data P = [x, y, z] relative to the workpiece coordinate system (WCS), such as Figure 2 As shown, let the three consecutive tool tip points in the original tool tip position path be P i-1 、P i 、P i+1 At the corner formed by these three points, a 6th-order Bezier curve is used as the transition curve; the curve expression is as follows:

[0072]

[0073] Where, P i (u) is the angle at the corner point P i The tool tip position transition curve inserted at B j,i Indicates the inserted sixth-degree Bezier curve P i (u) control point, where i is the number of the transition curve, j is the number of the control point, and the number of control points is 7; u∈[0,1] is the curve parameter; is the number of combinations;

[0074] 12) To make the transition curve G3 continuous (curvature differential continuity) at the tool tip position path junction, the following conditions must be met:

[0075]

[0076] like Figure 2 As shown, where When u=0, P i Point to P i-1 The unit tangent vector of ; When u=1, P i Point to P i+1 The unit tangent vector of the tool tip; s is the arc length of the tool tip position path;

[0077] To satisfy Insert the first three control points B of the 6th degree Bezier curve 0,i 、B 1,i 、B 2,i Should be located at the tool tip position path P i-1 P i On the segment, the last three control points B 4,i 、B 5,i 、B 6,i Should be located at the tool tip position path P i P i+1 on; to make Established, the following conditions are met:

[0078] Right now:

[0079]

[0080] Where B j,i is the Bezier curve P of the tool tip position i (u), where i is the transfer curve number and j is the control point number. According to the above formula, the length relationship between the control points is as follows:

[0081] To maintain symmetry, let ||B 2,i B 3,i ||=||B 3,i B 4,i ||=l p,i ;

[0082] 13) In order to avoid overlapping and intersecting of the constructed transfer curves and ensure that the smoothing error is within the maximum allowable value of the tool tip position smoothing error, while retaining a certain residual segment to adjust the tool axis direction and tool tip position synchronization, the transfer smoothing parameter l p,i The following conditions must be met:

[0083]

[0084] Where, ε pos,max is the maximum allowable value of the tool tip position smoothing error, and the industry generally believes that the deviation between the midpoint of the inserted curve segment and the original path is the largest, such as Figure 2 As shown;

[0085] 14) According to the transfer smoothing parameter l p,i And the above geometric relationship determines the control points, the formula is as follows:

[0086]

[0087] Step 2: Input the tool axis direction data for machining small line segments of the robot. Use the 6th degree Bezier curve for smoothing in the RPY angular space of the workpiece coordinate system. Based on the maximum allowable value of the tool axis direction smoothing error and the G3 continuity constraints at the curve connection, the control points of the Bezier curve are calculated analytically. The smoothed path consists of the inserted Bezier curve and the remaining linear segments, as shown below:

[0088] 21) The tool axis direction data of the robot small line segment processing refers to the RPY angle data φ = [α, β, γ] of the tool axis relative to the workpiece coordinate system (WCS), such as Figure 3 As shown, let the three sequentially connected direction points in the original tool axis direction path be φ i-1 、φ i 、φ i+1 At the corner formed by these three points, a 6th-order Bezier curve is used as the transition curve; the curve expression is as follows:

[0089]

[0090] Where, φ i (u) is the angle at the corner point φ i The tool axis direction transition curve inserted at Q j,i is the inserted sixth degree Bezier curve φ i (u) control point, where i is the number of the transition curve, j is the number of the control point, and the number of control points is 7; u∈[0,1] is the curve parameter; is the number of combinations;

[0091] 22) To make the transition curve G3 continuous (curvature differential continuity) at the path junction in the tool axis direction, the following conditions must be met:

[0092]

[0093] like Figure 3 As shown, where When u=0, φ i Pointing to φi-1 The unit tangent vector of ; When u=1, φ i Pointing to φ i+1 The unit tangent vector of ; s is the arc length of the path in the direction of the tool axis;

[0094] To satisfy like Figure 3 As shown, insert the first three control points Q of the 6-degree Bezier curve 0,i , Q 1,i , Q 2,i Should be located in the tool axis direction path φ i-1 φ i On the segment, the last three control points Q 0,i , Q 1,i , Q 2,i Should be located in the tool axis direction path φ i φ i+1 paragraph above; to make To be established, the following conditions must be met:

[0095] Right now:

[0096]

[0097] Where Q j,i is the Bezier curve φ in the tool axis direction i (u), where i is the transfer curve number and j is the control point number. According to the above formula, the length relationship between the control points is as follows:

[0098] To maintain symmetry, let ||Q 2,i Q 3,i ||=||Q 3,i Q 4,i ||=l o,i ;

[0099] 23) In order to avoid overlapping and intersecting of the constructed transfer curves and to ensure that the smoothing error is within the maximum allowable value of the smoothing error in the tool axis direction, while retaining a certain residual segment to adjust the tool axis direction and tool tip position synchronization, the transfer smoothing parameter l o,i The following conditions must be met:

[0100]

[0101] Where, ε ori,max is the maximum allowable value of the tool axis direction smoothing error, and is used to limit the tool axis direction vector smoothing error ΔO within the allowable value. J o is the Jacobian matrix in the tool axis direction, specifically:

[0102]

[0103] Where S α 、S β 、S γ They are the abbreviations of sin(α), sin(β), and sin(γ), respectively. α 、C β 、C γ are the abbreviations of cos(α), cos(β), and cos(γ), respectively; α, β, and γ are the RPY angles of the tool axis relative to the workpiece coordinate system;

[0104] 24) According to the transfer smoothing parameter l o,i And the above geometric relationship determines the control points, the formula is as follows:

[0105]

[0106] Step 3: Synchronize the parameters of the tool axis direction and tool tip position path after the smoothing process. Replace the remaining linear segments with B-spline lines to ensure that the third-order derivative of the tool axis direction relative to the tool tip position path arc length at the cross-segment is continuous, and then analytically synchronize the pose parameters. Finally, a G3 continuous and parameter-synchronized robot path is generated through the control points solved above. The purpose of replacing the remaining linear segments is to analytically synchronize the parameters. The control points obtained through the above three steps are fitted with B-spline to obtain the path, as follows:

[0107] 31) Replace the remaining linear segments in the smoothed tool tip position path and tool axis direction path with B-spline fitting, where the degree of B-spline is selected as 5, the number of control points is selected as 9, and the node vector is: U = [0, 0, 0, 0, 0, 0.5, 0.5, 0.5, 1, 1, 1, 1, 1];

[0108] The method for constructing the straight line of the B-spline fitting is as follows:

[0109] 311) The constructed spline line should meet the following conditions:

[0110] a. The straight line constructed using B-spline should maintain the same continuity as the original residual line, the control points should be on the original residual line, and it must also meet the following requirements:

[0111]

[0112] b. Ensure that the first, second, and third order differentials of the tool axis direction relative to the tool tip position arc length are continuous at the span, that is:

[0113]

[0114] 312) Solve the control points and then obtain the B-spline fitting straight line to achieve analytical synchronization of posture parameters.

[0115] 32) Perform B-spline fitting based on the required control points to generate a G3 continuous and parameter-synchronized robot path.

[0116] In summary, through the above three steps, a high-order continuous motion path can be generated, and parameter synchronization does not require iterative calculation; the present invention uses a technical solution of 6-order Bezier curve for G3 continuous local corner smoothing, which can improve the low-cost continuity at the connection of linear paths and realize the differential continuity of the motion path curvature, which is beneficial to avoid sudden changes in the robot speed, acceleration and acceleration direction. In addition, by using B-spline to replace the remaining straight line segments to achieve parameter synchronization, iterative calculation can be avoided, the calculation efficiency is improved, and it is beneficial to improve the real-time and stability of robot processing.

[0117] Example 2

[0118] In order to verify the effectiveness and superiority of the method proposed in the invention, an existing method was selected to conduct a comparative verification with the present invention;

[0119] Depend on Figure 5 It can be seen that in terms of curvature, the robot G3 continuous corner smoothing method based on Bezier curve proposed in the present invention has a smaller curvature extreme value, which means that the motion path constructed by the present invention is more continuous and smoother, which is beneficial to improving the robot processing stability.

[0120] Depend on Figure 6 It can be seen that in terms of running time, the robot G3 continuous corner smoothing method based on Bezier curve proposed in the present invention has a relatively short running time and high running efficiency, which can improve the real-time performance and efficiency of robot processing.

[0121] Depend on Figure 7 It can be seen that in terms of joint acceleration, the robot G3 continuous corner smoothing method based on Bezier curve proposed in the present invention has relatively smaller fluctuations and mutations in joint jerk, which further verifies that the method of the present invention can improve the robot's motion performance.

[0122] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

Claims

1. A robot G3 continuous corner smoothing method based on Bezier curve, characterized in that: The following steps are included Step 1: Input the tool tip position data for machining small line segments using the robot. Use a 6th-degree Bezier curve for smoothing in the workpiece coordinate system. Based on the maximum allowable tool tip position smoothing error and the G3 continuity constraints at the curve junction, the control points of the Bezier curve are analytically calculated. The smoothed path consists of the inserted Bezier curve and the remaining linear segments. Step 2: Input the tool axis direction data for machining small line segments of the robot. Use a 6th-order Bezier curve for smoothing in the RPY angular space of the workpiece coordinate system. Based on the maximum allowable value of the tool axis direction smoothing error and the G3 continuity constraints at the curve connection, the control points of the Bezier curve are analytically calculated. The smoothed path consists of the inserted Bezier curve and the remaining linear segments. Step 3: Synchronize the parameters of the tool axis direction and tool tip position path after the smoothing process. Replace the remaining linear segments with B-spline lines to ensure that the third-order derivative of the tool axis direction relative to the tool tip position path at the cross-segment is continuous. Then, analytically synchronize the pose parameters. Finally, generate a G3 continuous and parameter-synchronized robot path using the control points solved above. The specific method of step one is as follows: 11) The tool tip position data of the robot for processing small line segments refers to the position data P = [x, y, z] relative to the workpiece coordinate system. Let the three consecutive tool tip points in the original tool tip point position path be P i-1 、P i 、P i+1 At the corner formed by these three points, a 6th-order Bezier curve is used as the transition curve; the curve expression is as follows: Where, P i (u) is the angle at the corner point P i The tool tip position transfer curve inserted at B j,i Indicates the inserted sixth-degree Bezier curve P i (u) control point, where i is the number of the transition curve, j is the number of the control point, and the number of control points is 7; u∈[0,1] is the curve parameter; is the number of combinations; 12) Set conditions: Where, When u=0, P i Point to P i-1 The unit tangent vector of ; When u=1, P i Point to P i+1 The unit tangent vector of the tool tip; s is the arc length of the tool tip position path; To satisfy Insert the first three control points B of the 6th degree Bezier curve 0,i 、B 1,i 、B 2,i Should be located at the tool tip position path P i-1 P i On the segment, the last three control points B 4,i 、B 5,i 、B 6,i Should be located at the tool tip position path P i P i+1 on; to make Established, the following conditions are met: Right now: Where B j,i is the Bezier curve P of the tool tip position i (u), where i is the transfer curve number and j is the control point number. According to the above formula, the length relationship between the control points is as follows: To maintain symmetry, let ||B 2,i B 3,i ||=||B 3,i B 4,i ||=l p,i ; 13) Set the smoothing parameters of the transfer p,i Requirements: Where, ε pos,max is the maximum allowable value of tool tip position smoothing error; 14) Determine the control point. The formula is as follows:

2. A robot G3 continuous corner smoothing method based on Bezier curve according to claim 1, characterized in that: The specific method of step 2 is as follows: 21) The robot small segment processing tool axis direction data refers to the RPY angle data φ = [α, β, γ] of the tool axis relative to the workpiece coordinate system. Let the three sequentially connected direction points in the original tool axis direction path be 、 、 At the corner formed by these three points, a 6th-order Bezier curve is used as the transition curve; the curve expression is as follows: Where, For the corner The tool axis direction transition curve inserted at Q j,i The inserted 6th degree Bezier curve The control points of , where i is the number of the transition curve, j is the number of the control point, and the number of control points is 7; u∈[0,1] is the curve parameter; is the number of combinations; 22) Set conditions: Where, When u=0, point to The unit tangent vector of ; When u=1, point to The unit tangent vector of ; s is the arc length of the path in the direction of the tool axis; To satisfy Insert the first three control points Q of the 6th degree Bezier curve 0,i , Q 1,i , Q 2,i Should be located in the tool axis direction path On the segment, the last three control points Q 0,i , Q 1,i , Q 2,i Should be located in the tool axis direction path In order to make To be established, the following conditions must be met: Right now: Where Q j,i Bezier curve in the tool axis direction where i is the transfer curve number and j is the control point number. According to the above formula, the length relationship between the control points is as follows: To maintain symmetry, let ||Q 2,i Q 3,i ||=||Q 3,i Q 4,i ||=l o,i ; 23) Set the smoothing parameters of the transfer o,i The following conditions are met: Where, ε ori,max is the maximum allowable value of the tool axis direction smoothing error, and is used to limit the tool axis direction vector smoothing error △O within the allowable value. J o is the Jacobian matrix in the tool axis direction, specifically: Where S α 、S β 、S γ They are the abbreviations of sin(α), sin(β), and sin(γ), respectively. α 、C β 、C γ are the abbreviations of cos(α), cos(β), and cos(γ), respectively; α, β, and γ are the RPY angles of the tool axis relative to the workpiece coordinate system; 24) Determine the control point. The formula is as follows:

3. The robot G3 continuous corner smoothing method based on Bezier curve according to claim 2, characterized in that: The specific method of step three is as follows: 31) Replace the remaining linear segments in the smoothed tool tip position path and tool axis direction path with B-spline fitting, where the degree of B-spline is selected as 5, the number of control points is selected as 9, and the node vector is: U = [0, 0, 0, 0, 0, 0.5, 0.5, 0.5, 1, 1, 1, 1, 1]; 32) In order to achieve third-order parameter synchronization, the tool axis direction at the cross section should be continuous with respect to the third-order derivative of the tool tip position path arc length, and finally a G3 continuous and parameter synchronized robot path is generated by solving the control points.

4. The robot G3 continuous corner smoothing method based on Bezier curve according to claim 3, characterized in that: The method for constructing the straight line of the B-spline fitting is as follows: 311) The constructed spline line should meet the following conditions: a. The straight line constructed with B-spline should maintain the same continuity as the original residual line, the control points should be on the original residual line, and it must also meet the following requirements: b. Ensure that the first, second, and third order differentials of the tool axis direction relative to the tool tip position path arc length are continuous at the span, that is: 312) Solve for the control points and obtain the straight line of B-spline fitting.

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