Delta robot motion limiting calibration method

Abstract

The invention discloses a Delta robot motion limit calibration method, which is characterized in that a limit theoretical value of a tail end reachable working space range in a coordinate axis direction is obtained through robot structure parameters, then a cuboid space covering the limit theoretical value is generated, and three-dimensional grid division is carried out; and performing kinematics solution verification on the generated three-dimensional grid points, filtering out unreachable points to obtain a discrete point cloud set of a reachable space, then performing dimension reduction processing in a hierarchical slicing manner, extracting boundaries in a kernel density estimation manner, performing fitting to generate a boundary curve, and obtaining the three-dimensional motion limiting boundary of the tail end of the robot. According to the method, waste of effective space can be better reduced, dependence on human subjective experience is eliminated, and calibration repeatability and efficiency are improved.

Description

Technical Field

[0001] This invention relates to the field of Delta robot motion control technology, specifically to a Delta robot motion limit calibration method. Background Technology

[0002] Delta robots are high-speed, high-precision parallel robots with three degrees of freedom (or four degrees of freedom, including end-effector rotation). For example... Figure 1 As shown, its typical structure consists of a stationary platform 1 horizontally positioned above, a movable platform 2 (or end effector mounting platform) located below, and three identical motion chains evenly distributed circumferentially connecting the two. The motion chains include a driving arm 3 and a driven arm 4. One end of the driving arm is hinged to the outside of the stationary platform via a rotary joint and is directly driven by a servo motor, allowing it to swing in a vertical plane. The driven arm typically consists of a pair of parallel rods (forming a parallelogram mechanism), with both ends hinged to the free end of the driving arm and the movable platform, respectively. During operation, the actuator (i.e., the end effector) is mounted at the center of the movable platform.

[0003] The Delta robot's motion principle is as follows: three servo motors drive the three linked active arms to swing synchronously. Through the constraint of the parallelogram-shaped driven arms, the rotational motion of the active arms is converted into pure translational motion, maintaining a fixed posture on the moving platform in three-dimensional space. Due to the motion coupling of the three parallel motion chains, the spatial range (i.e., the reachable workspace) of the Delta robot's moving platform center position (end effector) is limited. Figure 2 As shown, in three-dimensional space, it presents an approximate spindle shape wrapped by multiple complex curved surfaces.

[0004] The operation and control of a Delta robot requires kinematic solutions. When controlling the Delta robot and solving its control kinematics, a coordinate system must first be established. The coordinate system's origin is the center of the Delta robot's stationary platform. The positive X-axis is the direction pointing from the plane of the stationary platform towards a single branch, and the positive Z-axis is the vertically upward direction of the stationary platform. The Y-axis is determined using the right-hand rule, thus establishing a reference coordinate system for the stationary platform. Within this coordinate system, the position coordinates of the center of the moving platform represent the motion position of the end effector.

[0005] In Delta robots, due to the strong coupling of the motion of the three branches, it is difficult to achieve the desired precise path (such as a straight line or arc) for the end effector by using only forward kinematics (i.e., calculating the end effector position given the rotation angles of each active arm) for trajectory control. Therefore, in industry, a trajectory control method based on inverse kinematics is usually adopted: first, the desired path of the end effector in three-dimensional space is planned, and then the required rotation angle commands of each active arm are calculated in real time using an inverse kinematics model. Due to this strong coupling, the actual reachable space range of the end effector is complex and not intuitive. Therefore, accurately defining its safe range of motion limits in advance is a fundamental prerequisite for achieving stable and reliable control of Delta robots.

[0006] Currently, the most commonly used motion limit calibration method in industry is the static calibration method based on the incorporation of a regular geometric body. The core of this method is to find a regularly shaped geometric body with the largest volume that can be incorporated within the irregularly accessible workspace of the Delta robot, and then set the boundary of this regular geometric body as the robot's motion limit boundary. The specific calibration process is typically as follows: First, based on specific task requirements, a desired regular shape for the workspace is preset (such as a cuboid of a specific size). Then, the spatial boundary parameters of this regular shape are substituted into the robot's kinematic model for containment verification. Next, it is determined whether the accessible space can completely accommodate the regular shape; if so, its boundary parameters are written into the controller as a limit. If not, the desired shape or robot layout needs to be manually adjusted, and step 2 is repeated until the requirements are met.

[0007] This existing motion limit calibration method has the following drawbacks: 1. Low space utilization: Because regular geometric shapes (such as cuboids and cylinders) cannot conform to the irregular shape boundaries of the Delta robot's actual reachable space, a large amount of effective theoretical motion space is conservatively excluded from the usable range, and the robot's performance is not fully utilized. 2. Poor adaptability and weak portability: The calibration results are strongly bound to the initially set "desired workspace". Once the task requirements change, or the calibration scheme needs to be transferred to Delta robots of different sizes and configurations, the entire "requirements-modeling-verification" process must be repeated based on the engineer's experience, lacking universality and parametric adaptive capabilities. 3. Subjective process and low efficiency: The entire calibration process relies heavily on the engineer's personal experience for "trial and error" and "adjustment", making the existing method lack objective and scientific judgment criteria and automated processes, resulting in low calibration efficiency, poor consistency, and difficulty in achieving standardized operations. Summary of the Invention

[0008] In view of the shortcomings of the prior art, the technical problem to be solved by the present invention is: how to provide a Delta robot motion limit calibration method that can better reduce the waste of effective space, eliminate the dependence on human subjective experience, and improve calibration repeatability and efficiency.

[0009] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0010] A Delta robot motion limit calibration method is characterized by obtaining the theoretical limit value of the reachable workspace range of the end effector in the coordinate axis direction through robot structural parameters, then generating a cuboid space covering the theoretical limit value and performing three-dimensional meshing, verifying the generated three-dimensional mesh points through kinematic solution, filtering out unreachable points to obtain a discrete point cloud set of reachable space, then performing dimensionality reduction processing by layer slicing, and using kernel density estimation to extract the boundary and then fitting to generate the boundary curve to obtain the three-dimensional motion limit boundary of the robot end effector.

[0011] This method can solve for motion limit boundaries that closely match the robot's actual reachable space shape, reducing wasted space and maximizing workspace utilization. Simultaneously, it provides an objective calibration process based on mathematical models and explicit algorithms, eliminating reliance on subjective human experience and improving the repeatability, consistency, and efficiency of calibration, achieving standardized and automated calibration. When applied, this method only requires inputting the robot's basic mechanism parameters to automatically adapt to the universal calibration of Delta robots of different sizes and series, possessing universality across various models.

[0012] Specifically, this method includes the following steps:

[0013] Step 1: Obtain the basic structural parameters of the robot and determine the motion positioning accuracy of the end effector (i.e., the center execution end of the moving platform);

[0014] The parameters include: static platform radius , radius of moving platform Length of active arm From boom length The accuracy of robot end-effector positioning under actual working conditions Positioning accuracy refers to the minimum distance that the end effector can resolve, such as the robot's ability to reach the target location. Then the end positioning accuracy The value is 0.1mm; the specific value can be obtained according to the design parameters or specific measurements during implementation.

[0015] Step 2: Locate the reachable workspace endpoints;

[0016] Based on the structural parameters obtained in step 1, using the forward kinematics model of the Delta robot, the angular range that the three active arms can rotate is divided into equal parts by a fixed value (the fixed value is usually selected in the range of 0.5-2°; too low a value increases the computational load, and too high a value reduces the accuracy; the optimal value is 1°). Then, by traversing each equal-divided angle of the three active arms, the initial shape of the robot's reachable workspace is obtained. Finally, based on the projection of the initially obtained reachable workspace onto the three orthogonal planes in the Cartesian coordinate system, the theoretical limit positions that the robot can reach in the three coordinate system axes are obtained, denoted as follows: , , , , and ; represent the minimum and maximum values ​​in the X-axis direction, the Y-axis direction, and the Z-axis direction, respectively; the Cartesian coordinate system described therein is consistent with the description method in the background technology; that is, the origin of the coordinate system is the center of the horizontally located static platform circle, the plane where the static platform is located is taken along the single branch direction as the positive X-axis, the vertical upward direction of the static platform is taken as the positive Z-axis, and the Y-axis is determined by the right-hand rule, thereby establishing the static platform reference coordinate system.

[0017] Step 3: Generate a set of discrete points in three-dimensional space;

[0018] Calculated in step 2 , , To define the range, a rectangular regular space is generated, which covers the theoretical limit values ​​in each direction (usually exceeding 10mm-20mm in each positive and negative direction along the X, Y, and Z axes). The rectangular regular space is then cut using a three-dimensional orthogonal mesh to obtain a uniformly arranged three-dimensional mesh point set.

[0019] Furthermore, the size of the three-dimensional orthogonal mesh and the end-effector positioning accuracy... Consistent. Specific implementation details can be as follows: If the value is too large, the final calculation accuracy will decrease; if it is too small, the calculation amount will be too large.

[0020] Step 4: Reverse filtering to construct a reachable space point cloud;

[0021] All the 3D mesh points generated in step 3 are substituted into the inverse kinematics model of the Delta robot for verification; all points that meet the conditions for the existence of the inverse kinematics model and whose joint angles are within the physical constraints are retained, and unreachable points are filtered out; finally, a set of 3D discrete point clouds that accurately describes the reachable space of the robot's end effector is output.

[0022] Step 5: Slice the 3D point cloud into layers;

[0023] like Figure 11 As shown, along the Z-axis, the three-dimensional point cloud obtained in step 4 is sliced ​​into layers at fixed height intervals to obtain a series of two-dimensional point cloud slices.

[0024] Furthermore, the fixed height interval is 0.1mm. If it is too large, the final calculation accuracy will be reduced; if it is too small, the calculation workload will be too large.

[0025] Step 6: Reduce the dimensionality of the 2D point cloud to a 1D radial distribution;

[0026] Utilizing the rotational symmetry of the Delta robot's workspace about the Z-axis, points in each 2D slice are transformed from their 2D coordinates... Transform to polar coordinates , where radial distance Extract the radial distance of all points. This forms a discrete dataset of the next-dimensional radial distance of the slice height. ;

[0027] Step 7: Kernel density estimation to obtain the radial distribution function;

[0028] For the radial distance dataset Di at each slice height, the probability density function is estimated using the kernel density estimation (KDE) method; the kernel density estimation formula is as follows:

[0029]

[0030] in:

[0031] —At point The probability density value obtained at that location;

[0032] —The independent variable represents any point in the range of values;

[0033] — The number of discrete data points at the slice height of this layer;

[0034] —The radiation range of a bandwidth data point in its neighborhood;

[0035] ——No. The radial distance value of each;

[0036] This density function can be reflected in the radius. The "possibility" or "density" of the existence of reachable points is determined; ultimately, the radial distribution function curve is obtained. Due to structural constraints, the distribution curve usually exhibits a shape that first rises and then falls.

[0037] Step 8: Identify the effective boundary radius of each slice layer;

[0038] The kernel density estimation curves for each height layer are analyzed; the radial location of the highest density is determined by finding its extreme points.

[0039] The process is as follows: calculate the first and second derivatives of the probability density function, and find the maximum point where the first derivative is 0 and the second derivative is less than 0. This point is considered to be the radial boundary with the densest reachable points on this height layer, and is used as the motion limit for this layer.

[0040] More specifically, the maximum point appears at the point where the first derivative of the function is 0 and the second derivative is less than 0. The kernel density function in the previous step is twice continuously differentiable. First, find the point where the first derivative of the probability density function is 0. Then, substitute the independent variable at this point into the second derivative of the probability density function to determine if it is the required maximum point. If this point is the required maximum point, then combine its radial distance with the floor height to form a data pair, and use this radial distance as the motion limit for that floor.

[0041] Step 9: Fit and generate a continuous axial boundary curve;

[0042] The radii obtained from each layer are paired into data pairs, and the discrete mathematical expression of the limit boundary curve is obtained by fitting. The obtained radii and z-height are paired into data pairs and displayed in the form of a scatter plot in a coordinate system with z-height as the x-axis and radius r as the y-axis. The points on adjacent layer heights are connected by fitting a straight line, and finally the limit boundary curve is obtained.

[0043] Step 10: Construct the three-dimensional motion limiting boundary surface;

[0044] Rotate the two-dimensional boundary curve obtained in step 9 around the Z-axis by 360 degrees to form a surface of revolution with the Z-axis as the central axis; this surface is the three-dimensional motion limit boundary marked by the present invention (which is highly consistent with the actual reachable space shape of the robot).

[0045] Furthermore, the present invention also includes step 11: writing and applying limit boundary parameters;

[0046] The mathematical form of the discrete point interpolation table of the three-dimensional motion limit boundary surface obtained in step 10 is written into the motion controller of the Delta robot as a software limit constraint for the motion of its end effector.

[0047] The above process reveals three core innovations of this method: 1. A discrete point cloud generation and filtering method based on inverse kinematics; this method generates a 3D point set with end-effector positioning accuracy as the grid resolution, filters actually reachable points through inverse kinematics, and constructs an accurate reachable space point cloud. 2. A boundary intelligent recognition method based on kernel density estimation; this method slices the 3D point cloud into layers, analyzes the radial distance data of each layer through kernel density estimation, and can automatically identify density peaks as effective motion boundaries. 3. A parameterized adaptive calibration process; only basic robot mechanism parameters need to be input to automatically generate the optimal limit surface, without manual intervention or preset shape rules.

[0048] The present invention can have the following beneficial effects.

[0049] 1. This invention extracts boundaries through discrete point clouds, layered slicing, and kernel density estimation, enabling accurate fitting of the irregular real-world shape of a robot's reachable space. This maximizes the utilization of the theoretical reachable space, avoids the space waste caused by traditional inscribed regular geometry methods, and significantly improves workspace utilization.

[0050] This method fully parameterizes and algorithms the calibration process. Users only need to input the robot's basic mechanism parameters. and accuracy indicators Subsequent boundary calculations, extractions, and fitting are all completed automatically by computer programs. This completely eliminates the subjective process of relying on engineers' personal experience for "trial and error" and "adjustments," making the calibration process highly repeatable, consistent, and reliable. Therefore, this method achieves a standardized and automated calibration process. This allows different personnel to obtain completely consistent and optimal limit results when calibrating the same robot, or when applying this method to the same model of robot on different production lines, greatly improving production efficiency and quality stability.

[0051] 3. The calibration logic of this invention is based on the robot's universal kinematic model and reachable space characteristics, and does not rely on any preset regular shape, thus possessing excellent universality and adaptability. Therefore, by simply changing the input basic mechanism parameters, this method can be automatically applied to Delta robots of any size, series, and configuration without modifying the core algorithm. Furthermore, when the robot's task requirements or workspace layout change, only one rerun of this calibration process is needed to quickly obtain the new optimal limits, demonstrating strong parametric adaptability and task migration capabilities, significantly reducing the cost and time required for equipment reuse and production line modification.

[0052] In summary, this invention can better reduce the waste of effective space, eliminate reliance on human subjective experience, and improve calibration repeatability and efficiency. Attached Figure Description

[0053] Figure 1 This is a schematic diagram of the Delta robot involved in the present invention.

[0054] Figure 2 for Figure 1 A schematic diagram of the Delta robot's end effector reaching the workspace.

[0055] Figure 3 This is a schematic diagram of the method flow steps of the present invention.

[0056] Figure 4 This is a schematic diagram of the Delta robot and its corresponding Cartesian coordinate system in this invention.

[0057] Figure 5 This is a schematic diagram of the Delta robot's end effector's reachable workspace projection and its extreme positions on the three axes in a Cartesian coordinate system.

[0058] Figure 6 This is a schematic diagram of the regular space of the cuboid generated in step 3.

[0059] Figure 7 A schematic diagram illustrating a set of three-dimensional discrete point clouds that can be accessed by the robot's end effector.

[0060] Figure 8 for Figure 7 The three-dimensional point cloud is sliced ​​layer by layer, resulting in a series of schematic diagrams of two-dimensional point cloud slices.

[0061] Figure 9 This is a schematic diagram of a single two-dimensional slice.

[0062] Figure 10 This is a schematic diagram of obtaining the radial distribution function curve from a single two-dimensional slice point cloud.

[0063] Figure 11 This is a schematic diagram showing the extreme points of the curves and their corresponding radii for each layer slice.

[0064] Figure 12 This is the limit boundary curve obtained in step 9.

[0065] Figure 13 This is a schematic diagram of the three-dimensional motion limiting boundary surface obtained in step 10. Detailed Implementation

[0066] The present invention will now be described in further detail with reference to specific embodiments.

[0067] Detailed implementation: A Delta robot motion limit calibration method is characterized by obtaining the theoretical limit value of the reachable workspace range of the end effector in the coordinate axis direction through robot structural parameters, then generating a cuboid space covering the theoretical limit value and performing three-dimensional mesh division, verifying the generated three-dimensional mesh points through kinematic solution, filtering out unreachable points to obtain a discrete point cloud set of reachable space, then performing dimensionality reduction processing by layer slicing, and using kernel density estimation to extract the boundary and then fitting to generate the boundary curve to obtain the three-dimensional motion limit boundary of the robot end effector.

[0068] This method can solve for motion limit boundaries that closely match the robot's actual reachable space shape, reducing wasted space and maximizing workspace utilization. Simultaneously, it provides an objective calibration process based on mathematical models and explicit algorithms, eliminating reliance on subjective human experience and improving the repeatability, consistency, and efficiency of calibration, achieving standardized and automated calibration. When applied, this method only requires inputting the robot's basic mechanism parameters to automatically adapt to the universal calibration of Delta robots of different sizes and series, possessing universality across various models.

[0069] Specifically, this method includes the following steps: See Figure 3 ;

[0070] Step 1: Obtain the robot's basic parameters and determine the motion positioning accuracy of the end effector (i.e., the center execution end of the moving platform):

[0071] Measure and acquire the key structural dimensional parameters of the current Delta robot, including: static platform radius R, moving platform radius r, active arm length La, and driven arm length Lb. Simultaneously, determine or measure the end-effector positioning accuracy of the robot under actual working conditions. Positioning accuracy refers to the minimum distance that the end effector can distinguish, such as the robot's ability to reach the target location. Then the end positioning accuracy The value is 0.1mm; the specific value can be obtained according to the design parameters or specific measurements during implementation.

[0072] Step 2: Locate the reachable workspace endpoints;

[0073] Based on the structural parameters obtained in step 1, and using the forward kinematics model of the Delta robot, the angular range that the three active arms can rotate is divided into equal parts by a fixed value of 1°. Then, by traversing each of the three active arms at each of these equal parts, the initial shape of the robot's reachable workspace is obtained. Finally, based on the projection of this initially obtained reachable workspace onto three orthogonal planes in the Cartesian coordinate system, the theoretical limit positions that the robot can reach in the three coordinate system axes are obtained, denoted as follows: , , , , and (like Figure 5 (As shown); representing the minimum and maximum values ​​in the X-axis, Y-axis, and Z-axis directions respectively; the Cartesian coordinate system described therein is consistent with the description method in the background technology; that is, with the center of the horizontally located static platform as the origin of the coordinate system, the plane where the static platform is located along a single branch direction as the positive X-axis, the vertical upward direction of the static platform as the positive Z-axis, and the Y-axis determined by the right-hand rule, thus establishing the static platform reference coordinate system (see...). Figure 4 ).

[0074] Step 3: Generate a set of discrete points in three-dimensional space;

[0075] Calculated in step 2 , , To define the range, generate a regular space of a cuboid (e.g., ...). Figure 6 As shown), this regular space covers all theoretical limit values ​​in each direction (usually more than 10mm-20mm in each positive and negative direction of the X, Y, and Z axes). The regular space of the cuboid covers all theoretical limit values ​​in each direction. Then, the regular space is cut using a three-dimensional orthogonal mesh to obtain a uniformly arranged three-dimensional mesh point set.

[0076] During implementation, the size of the three-dimensional orthogonal mesh and the end-effector positioning accuracy are... Consistent, specifically If the value is too large, the final calculation accuracy will decrease; if it is too small, the calculation amount will be too large.

[0077] Step 4: Reverse filtering to construct a reachable space point cloud;

[0078] All 3D mesh points generated in step 3 are substituted point by point into the inverse kinematics model of the Delta robot for verification. Points satisfying the inverse kinematics existence condition (i.e., joint angles within physical constraints) are retained, while unreachable points are filtered out. The final output is as follows: Figure 7 The figure shows a set of three-dimensional discrete point clouds that accurately describe the space reachable by the robot's end effector.

[0079] Step 5: Slice the 3D point cloud into layers;

[0080] like Figure 11 As shown, along the Z-axis, the 3D point cloud obtained in step 4 is divided into points at fixed height intervals. By performing layered slicing, a series of two-dimensional point cloud slices are obtained, see [link to relevant documentation]. Figure 8 ;

[0081] During implementation, the fixed height interval and The value should be consistent, specifically 0.1 mm. If it is too large, the final calculation accuracy will be reduced; if it is too small, the calculation workload will be too large.

[0082] Step 6: Reduce the dimensionality of the 2D point cloud to a 1D radial distribution;

[0083] Utilizing the rotational symmetry of the Delta robot's workspace about the Z-axis, points in each 2D slice (see...) Figure 9 ), from its two-dimensional coordinates Transform to polar coordinates , where radial distance Extract the radial distance of all points. This forms a discrete dataset of the next-dimensional radial distance of the slice height. ;

[0084] Step 7: Kernel density estimation to obtain the radial distribution function;

[0085] For the radial distance dataset Di at each slice height, the probability density function is estimated using kernel density estimation (KDE). The kernel density estimation formula is as follows:

[0086]

[0087] in:

[0088] —At point The probability density value obtained at that location;

[0089] —The independent variable represents any point in the range of values;

[0090] — The number of discrete data points at the slice height of this layer;

[0091] —The radiation range of a bandwidth data point in its neighborhood;

[0092] ——No. The radial distance value of each;

[0093] This density function can be reflected in the radius. The "probability" or "density" of the existence of reachable points is determined. This ultimately yields the radial distribution function curve, as shown below. Figure 10 As shown, due to structural limitations, the distribution curve usually exhibits a shape of first rising and then falling;

[0094] Step 8: Identify the effective boundary radius of each slice layer;

[0095] like Figure 11 As shown, the kernel density estimation curves for each height layer are analyzed. By finding the extreme points, the radial location of the highest density is determined.

[0096] The process is as follows: calculate the first and second derivatives of the probability density function, and find the maximum point where the first derivative is 0 and the second derivative is less than 0. This point is considered to be the radial boundary with the densest reachable points on this height layer, and is used as the motion limit for this layer.

[0097] More specifically, the maximum point appears at the point where the first derivative of the function is 0 and the second derivative is less than 0. The kernel density function in the previous step is twice continuously differentiable. First, find the point where the first derivative of the probability density function is 0. Then, substitute the independent variable at this point into the second derivative of the probability density function to determine if it is the required maximum point. If this point is the required maximum point, then combine its radial distance with the floor height to form a data pair, and use this radial distance as the motion limit for that floor.

[0098] Step 9: Fit and generate a continuous axial boundary curve;

[0099] The radii obtained from each layer are paired, and a discrete mathematical expression for the limit boundary curve is derived through fitting. The obtained radii and z-height are paired and displayed as a scatter plot in a coordinate system with z-height as the x-axis and radius r as the y-axis. Points on adjacent layer heights are connected by straight-line fitting, as shown below. Figure 12 As shown, the final limit boundary curve diagram is obtained;

[0100] Step 10: Construct the three-dimensional motion limiting boundary surface;

[0101] like Figure 13 As shown, the two-dimensional boundary curve obtained in step 9 is rotated 360 degrees around the Z-axis to form a surface of revolution with the Z-axis as the central axis. This surface is the three-dimensional motion limiting boundary defined in this invention (which closely matches the actual reachable space shape of the robot).

[0102] Step 11: Writing and applying limit boundary parameters;

[0103] The mathematical form of the discrete point interpolation table of the three-dimensional motion limit boundary surface obtained in step 10 is written into the motion controller of the Delta robot as a software limit constraint for the motion of its end effector.

[0104] This invention extracts boundaries through discrete point clouds, layered slicing, and kernel density estimation, enabling accurate fitting of the irregular real-world shape of a robot's reachable space. This maximizes the utilization of the theoretically reachable space, avoiding the space waste caused by traditional methods involving inscribed regular geometry, and significantly improves workspace utilization.

[0105] To further verify the effectiveness of the present invention, the applicant used an existing robot as an example and performed motion limit calibration according to both the existing calibration method described in the background art and the method of the present invention. The dimensional parameters of the robot are shown in Table 1 below:

[0106]

[0107] Detailed data obtained using existing technical solutions and the method of this invention are shown in Table 2 below:

[0108]

[0109] The existing technical solution achieves a space utilization rate of only 29.6%, while the space utilization rate of the method of the present invention is 76.7%. The usable space volume of the method of the present invention is increased by 56.272L compared with the original method, and the improvement rate is 159.22%.

[0110] Furthermore, in implementation, based on the above solution, some modifications and substitutions can be made to certain local steps to generate alternative solutions. However, each of these alternatives has its limitations, highlighting the superiority of the present invention from different perspectives. The following are some possible alternative solutions:

[0111] Alternative Solution 1: Boundary Extraction Based on Geometric Envelope Analytical Calculation

[0112] Solution Description: Instead of using discrete point clouds and statistical methods, this approach directly solves the branch intersections of the Delta robot using analytical geometry. By solving the boundary conditions of its kinematic constraint equations, implicit or parametric surface equations describing the workspace boundary are derived.

[0113] Compared to the present invention, this method theoretically offers better continuity and accuracy. However, the workspace boundary equations of the Delta robot are extremely complex, often making it difficult to obtain closed and concise analytical solutions. The computational process is complex and prone to numerical instability. The discrete point cloud reverse verification and statistical method employed in this invention avoids complex analytical derivations, resulting in simpler implementation, stronger robustness, and ease of programming and engineering.

[0114] Alternative Solution 2: Boundary Extraction Based on Image Processing Techniques (such as OpenCV)

[0115] Solution Description: Each 2D point cloud slice at each height level is treated as a binary image (reachable points are foreground, inaccessible points are background), and image processing is performed using computer vision libraries (such as OpenCV). First, the outer contour of the point cloud is obtained through edge detection or contour extraction algorithms. Then, the maximum inscribed circle is calculated through morphological processing to obtain the inscribed radius of each data volume slice.

[0116] In contrast to this invention, this method utilizes mature image processing tools, making its implementation relatively intuitive. However, it essentially transforms the boundary recognition problem of point clouds into an image processing problem, typically evolving into solving for the maximum inscribed circle of a sliced ​​point cloud. This transformation process itself adds extra, complex steps (such as point cloud to image conversion and image resolution setting) and introduces parameter tuning problems specific to the field of image processing (such as edge detection thresholds, morphological operation kernel sizes, etc.). The stability and optimality of the results are highly dependent on the selection of these parameters. This invention directly performs statistical analysis on one-dimensional radial distance data through kernel density estimation, extracting the probability density peak. This method is inherent in the mathematical statistical framework and does not require problem transformation using tools from other fields. It utilizes the probabilistic characteristics of the data's own distribution to more directly and accurately capture the densest "effective" boundary of the point cloud, rather than simply searching for a geometrically maximum inscribed shape, thus achieving a more refined and self-consistent spatial utilization in principle.

Claims

1. A method for calibrating the motion limit of a Delta robot, characterized in that, The theoretical limit of the reachable workspace of the end effector in the coordinate axis direction is obtained by using the robot's structural parameters. Then, a cuboid space covering the theoretical limit is generated and a three-dimensional mesh is generated. The generated three-dimensional mesh points are verified by kinematic solution. Unreachable points are filtered out to obtain a discrete point cloud set of reachable space. Then, dimensionality reduction is performed by layer slicing. The boundary is extracted by kernel density estimation and then fitted to generate boundary curves to obtain the three-dimensional motion limit boundary of the robot end effector.

2. The Delta robot motion limit calibration method as described in claim 1, characterized in that, This method includes the following steps. Step 1: Obtain the basic structural parameters of the robot and determine the positioning accuracy of the end effector. The parameters include: static platform radius , radius of moving platform Active arm length From boom length The accuracy of robot end-effector positioning under actual working conditions ; Step 2: Locate the reachable workspace endpoints; Based on the structural parameters obtained in step 1, and using the forward kinematics model of the Delta robot, the angular range that the three active arms can rotate is divided into equal parts by a fixed value. By traversing each equal-divided angle of the three active arms, the preliminary shape of the robot's reachable workspace is obtained. Then, based on the projection of the preliminary reachable workspace onto the three orthogonal planes in the Cartesian coordinate system, the theoretical limit positions that the robot can reach in the three coordinate system axes are obtained, denoted as follows: , , , , and ; represent the minimum and maximum values ​​in the X-axis direction, the Y-axis direction, and the Z-axis direction, respectively; Step 3: Generate a set of discrete points in three-dimensional space; Calculated in step 2 , , To define the range, a rectangular regular space is generated, which covers the theoretical limit values ​​in each direction. The regular space is then cut using a three-dimensional orthogonal mesh to obtain a uniformly arranged set of three-dimensional mesh points. Step 4: Reverse filtering to construct a reachable space point cloud; All the 3D mesh points generated in step 3 are substituted into the inverse kinematics model of the Delta robot for verification; all points that meet the conditions for the existence of the inverse kinematics model and whose joint angles are within the physical constraints are retained, and unreachable points are filtered out; finally, a set of 3D discrete point clouds that can accurately describe the reachable space of the robot's end effector is output. Step 5: Slice the 3D point cloud into layers; Along the Z-axis, the three-dimensional point cloud obtained in step 4 is sliced ​​at fixed height intervals to obtain a series of two-dimensional point cloud slices. Step 6: Reduce the dimensionality of the 2D point cloud to a 1D radial distribution; Utilizing Delta robot workspace regarding The rotational symmetry of the axis will change the position of each point in the two-dimensional slice from its two-dimensional coordinates. Transform to polar coordinates , where radial distance Extract the radial distance of all points. This forms a discrete dataset of the next-dimensional radial distance of the slice height. ; Step 7: Kernel density estimation to obtain the radial distribution function; Radial distance dataset at each slice height The probability density function is estimated using the kernel density estimation (KDE) method; the kernel density estimation formula is as follows: in: —At point The probability density value obtained at; —The independent variable represents any point in the range of values; — The number of discrete data points at the slice height of this layer; —The radiation range of a bandwidth data point in its neighborhood; ——No. The radial distance value of each; This density function can be reflected in the radius. The "possibility" or "density" of the existence of reachable points is determined; ultimately, the radial distribution function curve is obtained. Due to structural constraints, the distribution curve usually exhibits a shape that first rises and then falls. Step 8: Identify the effective boundary radius of each slice layer; The kernel density estimation curves for each height layer are analyzed; the radial location of the highest density is determined by finding its maximum point. The process is as follows: calculate the first and second derivatives of the probability density function, and find the maximum point where the first derivative is 0 and the second derivative is less than 0. This point is considered to be the radial boundary with the densest reachable points on this height layer, and is used as the motion limit for this layer. Step 9: Fit and generate a continuous axial boundary curve; The radii obtained from each layer are paired into data pairs, and a discrete mathematical expression for the limit boundary curve is derived by fitting. The obtained radii and Highly composed data pairs and in Height is Axis, radius for The coordinate system of the axis is displayed in the form of a scatter plot. Points on adjacent floor heights are connected by a straight line fitting method to finally obtain the limit boundary curve. Step 10: Construct the three-dimensional motion limiting boundary surface; The two-dimensional boundary curve obtained in step 9 is wrapped around The axis rotates 360 degrees to form a circle. The axis is a surface of revolution with the central axis as the axis; this surface is the three-dimensional motion limit boundary defined in this invention.

3. The Delta robot motion limit calibration method as described in claim 2, characterized in that, In step 1, each value is obtained based on specific measurements.

4. The Delta robot motion limit calibration method as described in claim 2, characterized in that, The coordinate system described in step 2 takes the center of the horizontally positioned static platform as its origin, the plane containing the static platform along a single branch as the positive X-axis, the vertically upward direction of the static platform as the positive Z-axis, and the Y-axis as determined by the right-hand rule, thus establishing the reference coordinate system for the static platform.

5. The Delta robot motion limit calibration method as described in claim 2, characterized in that, The fixed division value mentioned in step 2 is .

6. The Delta robot motion limit calibration method as described in claim 2, characterized in that, In step 3, the regular space of the cuboid shape exceeds the theoretical limit value by 10-20 mm in each direction.

7. The Delta robot motion limit calibration method as described in claim 2, characterized in that, In step 3, the size of the three-dimensional orthogonal mesh and the end-effector positioning accuracy are consistent.

8. The Delta robot motion limit calibration method as described in claim 2, characterized in that, In step 5, the fixed height interval is 0.1 mm.

9. The Delta robot motion limit calibration method as described in claim 2, characterized in that, It also includes step 11: writing and applying limit boundary parameters; The mathematical form of the discrete point interpolation table of the three-dimensional motion limit boundary surface obtained in step 10 is written into the motion controller of the Delta robot as a software limit constraint for the motion of its end effector.

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