A Spatial Circular Fitting Angle Precision Compensation Method for a Six-Axis Robot Arm

Abstract

The present invention provides a method for compensating the angular accuracy of spatial circular fitting for a six-axis robotic arm. All measurements are subjected to spatial plane fitting. A chord is obtained by connecting any two points within the spatial circle, and the corresponding mid-perpendicular plane equation is calculated through the straight-line equation of the chord. The fitting plane intersects with all spatial planes and there is only one intersection point, which is the center of the planar circle. Taking the fitted spatial plane as a constraint condition and the mid-perpendicular plane as an observation equation, the calculation equation for the center of the planar circle is derived using the indirect adjustment with constraint conditions as the basic function model. Then, the radius of the planar circle is calculated according to the algebraic distance equation from a point to the center of the circle. Based on the position of the circle radius and the center, the measured motion angle accuracy and the theoretical angle accuracy are used as compensation differences, and the compensation differences are used as the control basis for controlling the motion of the robotic arm for control correction. The present invention realizes high-precision and high-efficiency circular fitting, and improves the disadvantages of the traditional least-squares method for fitting spatial circles, which are complex and difficult to implement.

Description

Technical Field

[0001] The present invention belongs to the technical field of robotic arm control, and particularly relates to a method for compensating the angular accuracy of spatial circular fitting for a six-axis robotic arm. Background Art

[0002] For a six-axis robotic arm, only the repeat positioning accuracy of the robotic arm is included in its accuracy index, with the unit of mm, such as ±0.08 mm. However, in many actual application scenarios, the center point of the sixth-axis disc of the robotic arm will revolve around a virtual point in space as the center point to complete spatial circular motion in various postures. In actual robotic arm motion applications, the angular accuracy of the sixth axis (or single axis) of the line connecting the center point of this six-axis disc and the virtual center point in space often does not meet the usage requirements. The present invention is a compensation method for this angular accuracy to control the robotic arm through software compensation, so that the compensated angular accuracy reaches 0.01° (for the six-axis) or even 0.001° (for the single axis).

[0003] When the six axes of the robotic arm move jointly, the exact coordinate values (which can be accurate to 1 μm) of each point of the spatial circular motion can be measured by a laser tracker. How to use these coordinate values to accurately locate the center and radius of the actual circular motion and obtain the angular error of the circular motion is the key to realizing angular accuracy compensation. Currently, the following three methods are adopted for spatial circular fitting: 1) the average value method; 2) the weighted average method; 3) the least squares method. However, these three methods have the following problems respectively:

[0004] In practical problems, when the robotic arm drives an antenna to perform simulated circular angular motion around the antenna phase center (a virtual point in space), its motion accuracy exceeds the uniform distribution requirement of 0.01°. That is, the points measured by the laser tracker cannot be evenly distributed on the circle exactly, and most points cannot be distributed on a plane, but are three-dimensional discrete points in space. Although the average value method has a simple fitting process, it requires the known points to be evenly distributed. Currently, the circular motion of the robotic arm cannot achieve high precision, that is, the distribution is uneven, and the center and radius obtained by testing and fitting with the average value method cannot be used as the basic data for compensating the angular accuracy.

[0005] The weighted average method is slightly more complex, and the fitting effect is better than that of the average value method. However, it is only an approximation to a certain extent because when calculating the center coordinates and radius size, it should be the circumference of the circle, and the weights should be the sum of the arc lengths between the data acquisition points ( ) and the adjacent two points and the quotient. For high-precision compensation, this accuracy cannot meet the angular compensation requirements of 0.01° or a smaller angular accuracy.

[0006] The least squares method of spatial circle fitting is based on the mathematical model of the spatial sphere and the plane equation for data processing. The three-dimensional coordinate data is converted to the plane for plane circle fitting, and then the center and radius obtained by fitting are reversely converted to the circular coordinate system to obtain the center and radius of the spatial circle. The principle of this method is relatively complex, and it is difficult and cumbersome to implement. Many test data are not easy to obtain, and errors are added to reduce the accuracy of fitting and compensation. Summary of the invention

[0007] In view of the above technical problems, the present invention provides a spatial circular fitting angular accuracy compensation method for a six-axis robotic arm. When the six axes of the six-axis robotic arm move jointly, an antenna to be tested is installed at the end of the sixth axis. When azimuth motion, pitch motion, roll motion and other motion modes are performed with the phase center of the antenna to be tested (a virtual point in space) as the motion rotation center, the angular positioning accuracy of each motion mode after the six-axis joint motion cannot meet higher positioning angle accuracy requirements such as 0.01° or even 0.001°. It is necessary to compensate for the actual error value after spatial circular fitting to achieve an angular accuracy of 0.01°.

[0008] The specific technical solution is:

[0009] Fix the laser target ball collected by the laser tracker and the six-axis of the robotic arm through the tooling;

[0010] Then fix the counterweight fixture to the six axes of the robot arm. The center of mass position and mass of the counterweight fixture must be consistent with the device to be tested. The mass and center of mass position of the robot arm with the load will have a large error with the mechanical empty load, so compensation calibration must be performed after the load test.

[0011] Control the robot arm to move along the designed azimuth, pitch and roll arc trajectories, and record the three-dimensional coordinate value of each sampling point in the spatial position every 1° (or other sampling interval) during the movement interval;

[0012] Utilizing these spatial three-dimensional coordinate values, the compensation method of the present invention is used to perform spatial arc motion angle accuracy compensation;

[0013] After software compensation, the robot arm is controlled to perform six-axis linkage (or single-axis circular) spatial arc motion, and the position of the target ball is recorded by the laser tracker to verify whether the accuracy of the arc motion meets the angular accuracy requirement of 0.01° (or higher). If the accuracy requirement is met, the compensation is successful, otherwise iterative compensation is required until it converges to the required angular accuracy.

[0014] According to the collected spatial coordinate data, a spatial circular fitting angle accuracy compensation method for a six-axis robot arm of the present invention is used, comprising the following steps:

[0015] (1) All the discrete 3D spatial point coordinates collected by measurement are mapped into the spatial plane and the spatial plane is fitted;

[0016] (2) Connect any two points within the spatial circle to obtain a chord, calculate the corresponding mid-perpendicular plane equation through the straight-line equation of this chord, measure point N, and obtain N - 1 linearly independent chords, thereby obtaining N - 1 linearly independent mid-perpendicular plane equations;

[0017] (3) According to the characteristics of the spatial circle, the fitting plane intersects with all spatial planes and there is only one intersection point, and this intersection point is the center of the planar circle;

[0018] (4) Taking the fitted spatial plane as the constraint condition and the mid-perpendicular plane as the observation equation, use the indirect adjustment with constraints as the basic function model to derive the calculation equation of the center of the planar circle, and then calculate the radius of the planar circle according to the algebraic distance equation from the point to the center of the circle;

[0019] (5) Project the input spatial point data onto the fitted plane equation;

[0020] (6) Specify the initial point, calculate the included angle degree between other projected points and the initial point through the cosine formula, calculate the difference between the measured value and the theoretical value, and this difference is the compensation calibration angle value of the azimuth and pitch angles;

[0021] (7) Repeat the angular accuracy measurement, record the spatial coordinate points (laser tracker coordinate system) and the robotic arm axis coordinates, calculate the angular accuracy of the robotic arm azimuth and pitch movements. If it meets the expected error range (for example, 0.01°), the compensation ends, and the compensation table is stored in the software for future use when actually operating the robotic arm movement. If the compensation result does not meet the expectation, iterative compensation is performed until the error meets the angular measurement accuracy requirements.

[0022] In step (4), use the spatial vector thinking to derive the mid-perpendicular plane equation between any two points on the spherical surface;

[0023] Vector is ( ) Let the center of the circle be , passing through and The direction vector of the line connecting the midpoint and the center of the circle is . Since the two spatial vectors are perpendicular, we get:

[0024] (1)

[0025] Equation (1) is simplified to the following equation:

[0026] (2)

[0027] In the formula, and Let (x1, y1, z1) and (x2, y3, z3) be any two points in space, and these are the spatial coordinate values of these two points. Due to the correlation of the equation of the mid-perpendicular plane of the spatial sphere, n - 1 linearly independent mid-perpendicular equations can be listed from the coordinates of n observation points, resulting in the error equation:

[0028] (3)

[0029] The above equation is simplified to:

[0030] (4)

[0031] At this time, the weight matrix P is the identity matrix; it is determined that the center of the circle must be on the fitted spatial plane. Based on this as a constraint condition, the calculation is carried out according to the conditional indirect adjustment. The constraint condition is Equation (5), and the least-squares solution of the center of the circle can be obtained by deriving the equation.

[0032] Constraint condition:

[0033] (5)

[0034] In the formula ; the equation is:

[0035] (6)

[0036] In the formula is the correlation coefficient vector of the constraint condition; the least-squares solution is obtained:

[0037] (7)

[0038] Then, according to the solved center coordinates of the circle, the distances from each observation point to the center of the spatial circle are calculated:

[0039] , (i = 1, 2,..., n) (8)

[0040] The fitting radius r of the circle is the average value of these distances, is the roundness of these points.

[0041] The present invention realizes high-precision and high-efficiency circle fitting through a brand-new spatial circle fitting algorithm, and improves the disadvantages of the traditional least-squares method for fitting a spatial circle, which is complex and not easy to implement. Brief Description of the Drawings

[0042] Figure 1 Connection block diagram during the actual spatial circle fitting compensation angle accuracy test;

[0043] Figure 2 Connection relationship between the actual laser tracker target ball and the robotic arm through the tooling;

[0044] Figure 3 Schematic diagram of the principle of the new method of space fitting circle;

[0045] Figure 4 Schematic diagram of the movement of each axis of the robotic arm;

[0046] Figure 5 Schematic diagram of the six-axis linkage of the robotic arm to realize azimuth and pitch motion with the virtual point in space as the center of the circle. DETAILED DESCRIPTION

[0047] The specific technical solution of the present invention is explained in conjunction with embodiments.

[0048] The present invention uses a new method of spatial circular fitting to achieve angular accuracy compensation, such as Figure 1 and Figure 2 As shown in the figure, this method is combined with a laser tracker to obtain high-precision position coordinates, and the robot arm is controlled to first achieve the expected angular motion of the azimuth circle, pitch circle, and roll circle, and the accurate angular error is obtained through the new method of spatial circular fitting, and then compensation is performed. After compensation, the robot arm moves again and achieves 0.01° high-precision angular motion.

[0049] Based on the least square method, the present invention utilizes the characteristics of the space circle set and mathematical theorem, that is, the perpendicular median plane corresponding to multiple chord lengths in the plane circle intersects with the plane of the space circle and has only one intersection point, and this point is the center point of the plane circle, and a mathematical calculation model is established based on the perpendicular median plane and the space vector concept, and the calculation equation of the center of the circle is deduced according to the spacing adjustment with restrictive conditions, and then the radius, flatness, roundness and other characteristic quantities of the space circle are inversely calculated. Figure 3 As shown, the specific embodiments are as follows:

[0050] (1) If Figure 4 As shown, the robot arm is controlled to carry the azimuth or pitch movement of the test piece. The laser tracker captures a set of spatial coordinate point positions through the laser target ball installed on the robot arm, records the spatial coordinate data (laser tracker coordinate system) and the tool axis coordinates of the robot arm, and uses this set of spatial point data as input for the fitting compensation calibration algorithm input;

[0051] (2) Based on the spatial point data, the least squares method ( ) Calculate the normal vector of the fitted plane, and the plane equation ax+by+cz-l=0;

[0052] (3) Fitting space circle, vector for( ), center coordinates for( ), then the vector and (The midplane, coordinates are ( , , ) is vertical ( is the midpoint of), the two vectors are perpendicular, it can be known that the dot product (scalar product) of vector and vector is 0, ( ). ( , , ) = 0; From this, it can be deduced that ( ) * + ( ) * + ( ) * - ( = 0; With this formula, the coordinates of the center of the circle can be calculated by the least squares method;

[0053] (4) Project the input space point data onto the fitted plane equation;

[0054] (5) Specify the initial point, calculate the included angle degree between other projection points and the initial point through the cosine formula, calculate the difference between the measured value and the theoretical value, and this difference is the compensation calibration angle value of the azimuth and pitch angles;

[0055] (6) Repeat the operation in step (1), record the space coordinate points (laser tracker coordinate system) and the manipulator axis coordinates, calculate the angular accuracy of the manipulator azimuth and pitch movements, as Figure 5 shown, if it is within the expected error range (for example, 0.01°), the compensation ends, and the compensation table is stored in the software for future use when actually operating the manipulator movement. If the compensation result does not meet the expectation, continue the compensation until the error meets the accuracy requirement.

[0056] Table 1 shows the accuracy of each axis of the manipulator with load.

[0057] Table 1 Measured angular accuracy of each axis of the manipulator with load before compensation

[0058]

[0059] The compensated set angular accuracy needs to meet ±0.01°, and Table 2 shows the actual angular accuracy data after compensation.

[0060] Table 2 Measured angular accuracy of each axis of the manipulator with load after compensation

[0061]

[0062] Practice has proved that the angular accuracy after compensation has increased by 5-10 times. At that time, the algorithm was set so that the angular accuracy after compensation meets 0.01°. The data before compensation is shown in Table 1, and the data after compensation is shown in Table 2, indicating that the compensation effect is obvious.

Claims

1. A method for compensating the angular accuracy of spatial circular fitting for a six-axis robotic arm, characterized in that, It includes the following steps: (1) Map the discrete three-dimensional space point coordinates collected by all measurements into a space plane and perform space plane fitting; (2) Connect any two points within the space circle to obtain a chord, calculate the corresponding mid-perpendicular plane equation through the straight line equation of this chord, measure point N, and obtain N - 1 linearly independent chords, thereby obtaining N - 1 linearly independent mid-perpendicular plane equations; (3) According to the characteristics of the space circle, the fitting plane intersects with all space planes and there is only one intersection point, and this intersection point is the center of the plane circle; (4) Take the fitted space plane as a constraint condition and the mid-perpendicular plane as an observation equation, and use the indirect adjustment with constraints as the basic function model to derive the calculation equation for the center of the plane circle, and then calculate the radius of the plane circle according to the algebraic distance equation from a point to the center of the circle; (5) Project the input space point data onto the fitted plane equation; (6) Specify an initial point, calculate the included angle degree between other projected points and the initial point through the cosine formula, calculate the difference between the measured value and the theoretical value, and this difference is the compensation calibration angle value of the azimuth and pitch angles; (7) Repeat the angular accuracy measurement, record the space coordinate points and the coordinates of the robotic arm axis, calculate the angular accuracy of the azimuth and pitch movements of the robotic arm. If it is within the expected error range, the compensation ends, and the compensation table is stored in the software for future use when actually operating the robotic arm movement; if the compensation result does not meet the expectation, iterative compensation is performed until the error meets the angular measurement accuracy requirement.

2. A method for compensating the angular accuracy of spatial circular fitting for a six-axis robotic arm according to claim 1, characterized in that, In step (4), use the spatial vector thinking to derive the mid-perpendicular plane equation between any two points on the spherical surface; vector is ( ), let the center of the circle be , passing through and the midpoint of and the direction vector of the line connecting the center of the circle is , since the two spatial vectors are perpendicular, it is obtained that: （1） Equation (1) is simplified to the following equation: （2） In the formula, , due to the correlation of the perpendicular bisecting plane equations of the spatial sphere, n - 1 linearly independent perpendicular bisecting equations can be listed from the coordinates of n observation points, and the error equation is obtained: （3） The above formula is simplified to: (4) At this time, the weight matrix P is the identity matrix; it is determined that the center of the circle must be on the fitted space plane, and this is used as a constraint condition, and the calculation is performed according to the indirect adjustment with conditions. The constraint condition is equation (5), and the least squares solution of the center of the circle can be derived from the derived equation; Restriction conditions: (5) wherein ; The equation is: （6） wherein is the correlation coefficient vector of the constraint condition; and the least squares solution is obtained as follows: （7） Then, according to the solved center coordinates of the circle, calculate the distances from each observation point to the center of the space circle: , (i = 1, 2, …, n) (8) The fitting radius r of the circle is the average of these distances, which is the roundness of these points.

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