Method and system for visually analyzing kinematics singularity in working space of mechanical arm

Abstract

The invention discloses a method and system for visually analyzing kinematics singularity in a working space of a mechanical arm. The problem that in the prior art, kinematics singularity distribution information in the reachable working space of the mechanical arm cannot be accurately and reliably obtained is solved. Comprising the steps that voxelization information of a mechanical arm operation space is obtained, and a reachable work space of the mechanical arm is established according to the voxelization information of the mechanical arm operation space, and the voxelization information comprises reachable voxels and unreachable voxels; a kinematics singularity index containing an operability index and a condition number reciprocal index is obtained, and the kinematics singularity of each reachable voxel envelope area in the mechanical arm voxelization reachable working space is detected in a traversal mode through the kinematics singularity index; and color coding is conducted on each reachable voxel in the mechanical arm voxelization reachable working space according to the kinematics singularity information of the envelope area of the reachable voxel, a result is visually displayed, and a kinematics singularity map of the mechanical arm is constructed and obtained.

Description

Technical Field

[0001] This invention relates to the field of kinematic singularity analysis technology for robotic arms, and in particular to a method and system for visually analyzing kinematic singularities in the workspace of a robotic arm. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Kinematic singularities are an inherent property of robotic arm kinematics, manifesting when the Jacobian matrix is ​​not full rank. At singular configurations, the robotic arm loses one or more degrees of freedom, significantly reducing the motion performance of the end effector. More critically, as the robotic arm approaches a singular configuration, its joint angular velocity, angular acceleration, and joint torque undergo abrupt changes, tending towards infinity. These problems not only introduce significant errors into the robotic arm's task space but may also lead to loss of control, posing safety risks and ultimately causing task failure. Therefore, effectively avoiding the potential problems caused by kinematic singularities is crucial for ensuring the safe and reliable execution of various tasks by robotic arms. To achieve this, a thorough analysis of the kinematic singularities of robotic arms is essential.

[0004] In the existing technical field, the analysis methods and techniques for kinematic singularities of robotic arms mainly focus on how to efficiently and reliably derive the conditions that prevent the Jacobian matrix of the robotic arm from being incompletely rank, i.e., the kinematic singularity conditions. These conditions can be used to identify specific singular configurations within the reachable workspace of the robotic arm. However, a significant shortcoming of current technology is the lack of analytical methods for the overall distribution of kinematic singularities within the reachable workspace of the robotic arm. Specifically, existing technologies cannot accurately and reliably obtain information on the distribution of singularities within the reachable workspace of the robotic arm, thus failing to identify singular regions at the workspace boundaries, singular regions within the workspace, and non-singular regions, thereby failing to ensure the safety and reliability of the robotic arm during task execution. Furthermore, commonly used kinematic singularity detection metrics in existing technologies include operability and condition number. Both metrics are based on the geometric properties of the operable ellipsoid to detect singular states of the robotic arm configuration. However, it is worth noting that the operability index is not sensitive to the detection of singular states in manipulator configurations corresponding to operability ellipsoids with extremely significant differences between their major and minor axes, while the condition number index cannot effectively detect singular states in manipulator configurations corresponding to operability ellipsoids with insignificant differences between their major and minor axes. Therefore, a single singularity index can only detect singularities in some configurations within the reachable workspace of the manipulator, and cannot accurately and comprehensively describe the singular states of all manipulator configurations.

[0005] In conclusion, there is an urgent need to supplement and improve the methods for in-depth analysis of the inherent kinematic singularities of robotic arms. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention aims to provide a method and system for visually analyzing the kinematic singularities in the workspace of a robotic arm, obtaining a voxelized reachable workspace of the robotic arm to ensure the accuracy of the analysis; based on this, a kinematic singularity index is set, and the kinematic singularities of each region within the voxelized reachable workspace are analyzed using this index; finally, based on the analysis results, the voxelized reachable workspace is color-coded and visualized, thereby constructing a kinematic singularity map of the robotic arm.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: A method for visually analyzing kinematic singularities in the workspace of a robotic arm includes the following: Obtain voxelized information of the robotic arm's operating space. Based on the voxelized information of the robotic arm's operating space, which includes reachable and inaccessible voxels, establish the reachable workspace of the robotic arm. Obtain the kinematic singularity index, which includes the operability index and the inverse of the condition number index, and use the kinematic singularity index to traverse and detect the kinematic singularity of each reachable voxel envelope region within the voxelized reachable workspace of the robotic arm. Each reachable voxel within the voxelized reachable workspace of the robotic arm is color-coded based on the kinematic singularity information of its envelope region, and the results are visualized to obtain the kinematic singularity map of the robotic arm.

[0008] The method described above for visually analyzing the kinematic singularities in the workspace of a robotic arm employs an improved Monte Carlo method and a robotic arm self-collision detection method. It combines the position-level forward kinematic equations of the robotic arm with the voxelized information of the robotic arm's operating space. The reachable workspace of the robotic arm is then established by fitting the inscribed sphere of all voxels contained within the reachable workspace and visualizing the results.

[0009] The method for visually analyzing kinematic singularities in the workspace of a robotic arm, as described above, includes obtaining voxelized information about the robotic arm's operational space, comprising the following: The robotic arm's operating space is divided into multiple cubes, which are then processed column-by-column, row-by-row, and layer-by-layer. Individual elements are numbered, and the resulting numbers are stored in a created one-dimensional matrix. ,right Voxels are classified into reachable voxels and unreachable voxels. Voxels that can be reached by the end effector of the robotic arm are defined as reachable voxels, while the remaining voxels are defined as unreachable voxels.

[0010] The method described above for visually analyzing the kinematic singularities in the workspace of a robotic arm establishes the reachable workspace of the robotic arm, including the following: According to the storage in the one-dimensional matrix The numbers in the sequence are traversed and calculated. The coordinates of the center of the inscribed sphere of each voxel are calculated, and the results are stored in a created one-dimensional matrix according to the voxel number order. middle; Traversal and inspection The number of reachable workspace points that a voxel falls into is stored in a created one-dimensional matrix, along with the numbers of voxels with a non-zero count. middle; In the matrix Extracting the matrix The coordinates of the center of the inscribed sphere of the voxel corresponding to the number in the table are given, and the radius of each inscribed sphere is given. All extracted inscribed spheres are visualized and drawn to obtain the voxelized reachable workspace of the robotic arm. Create database Based on the numbering order of reachable voxels in the established voxelized workspace of the robotic arm, the voxels containing each voxel are... The position vector information of each workspace point, as well as the corresponding robot arm joint angle position vector information when these workspace points were generated, are stored in this database.

[0011] The method described above for visually analyzing the kinematic singularities in the workspace of a robotic arm, wherein the kinematic singularity index is... According to the weighting coefficients Operability indicators Conditional reciprocal index To obtain the weighting coefficients According to the Jacobian matrix Minimum singular value and maximum singular value To obtain.

[0012] The above describes a method for visually analyzing kinematic singularities in the workspace of a robotic arm, wherein the operability index... According to the Jacobian matrix of the robotic arm Obtain the square root of the determinant of its product with its transpose; The inverse index of the condition number According to the minimum singular value of the Jacobian matrix and maximum singular value The ratio is used to obtain the value.

[0013] The method described above for visually analyzing the kinematic singularities in the workspace of a robotic arm uses the vector product method and combines the position-level forward kinematic equations of the robotic arm to derive the analytical expression of the Jacobian matrix.

[0014] The method described above for visually analyzing kinematic singularities in the workspace of a robotic arm, wherein the kinematic singularity is detected by traversing each reachable voxel envelope region within the voxelized reachable workspace using kinematic singularity indices, includes the following: Based on the forward kinematics model of the robotic arm, the Jacobian matrix of the robotic arm is derived. Combined with database The information stored in the database is used to iterate and calculate the data contained in each reachable voxel, according to the numbering order of the reachable voxels in the established voxelized reachable workspace of the robotic arm. The value of the kinematic singularity index of the robotic arm configuration.

[0015] The method described above for visually analyzing the kinematic singularities in the workspace of a robotic arm involves color-coding each reachable voxel within the voxelized reachable workspace of the robotic arm based on the kinematic singularity information of its envelope region, and visualizing the results to obtain a kinematic singularity map of the robotic arm, including the following: The average kinematic singularity index of all reachable voxels in the voxelized reachable workspace of the robotic arm is calculated, and the maximum value of the average value in the entire voxelized reachable workspace of the robotic arm is obtained. Set the numerical range The data is then divided into multiple numerical intervals, and the numbers corresponding to the reachable voxels falling into different numerical intervals are stored in a multi-row, multi-column matrix. In this matrix, the number of rows is equal to the number of divisions of the numerical intervals mentioned above, and the number of columns is equal to the number of reachable voxels falling into each numerical interval. In the matrix Extracting matrices sequentially Each row in the database stores the coordinates of the center of the inscribed sphere of the reachable voxel corresponding to its voxel number, combined with the radius of the inscribed sphere. The extracted inscribed spheres are visualized and drawn, and different colors are assigned to the inscribed spheres in different rows to distinguish them, thus obtaining the kinematic singularity map of the robotic arm.

[0016] Secondly, the present invention also provides a construction system for visually analyzing the kinematic singularities in the workspace of a robotic arm, characterized in that it includes a computing device, which is configured to: Obtain voxelized information of the robotic arm's operating space. Based on the voxelized information of the robotic arm's operating space, which includes reachable and inaccessible voxels, establish the reachable workspace of the robotic arm. Obtain the kinematic singularity index, which includes the operability index and the inverse of the condition number index, and use the kinematic singularity index to traverse and detect the kinematic singularity of each reachable voxel envelope region within the voxelized reachable workspace of the robotic arm. Each reachable voxel within the voxelized reachable workspace of the robotic arm is color-coded based on the kinematic singularity information of its envelope region, and the results are visualized to construct the kinematic singularity map of the robotic arm.

[0017] The beneficial effects of the present invention are as follows: 1) The method provided by this invention can intuitively present the precise reachable workspace of a robotic arm considering self-collision, and simultaneously store precise information related to the kinematic singularity distribution within this space to support detailed visualization and analysis of the kinematic singularity distribution within the reachable workspace. Compared with existing methods that identify specific singular configurations within the reachable workspace by deriving the kinematic singularity conditions of the robotic arm, the proposed method not only comprehensively presents the overall distribution of singularities within the reachable workspace, but can also be used to identify internal boundary singularity regions, external boundary singularity regions, and optimal operation regions for tasks within the robotic arm's workspace. Furthermore, it can determine the singularity situation of the robotic arm reaching a given position based on the target task's placement information. This is crucial for ensuring that the robotic arm completes various operational tasks safely, reliably, and with high precision.

[0018] 2) The method provided by this invention can construct a kinematic singularity map of a robotic arm. This map can not only identify singular regions at the internal and external boundaries of the robotic arm's workspace, as well as the optimal operation area for the task, but also accurately determine the singularity status of the robotic arm at the target task's placement position based on that position information. This characteristic is crucial for ensuring that the robotic arm can safely, reliably, and with high precision complete various tasks. Furthermore, the method for constructing the kinematic singularity map is simple, reliable, and universal. In summary, this map is expected to become a universal tool for in-depth analysis of the kinematic singularities of various types of cascaded robotic arms.

[0019] 3) The kinematic singularity detection index designed in this invention is a dynamic weighted combination of the operability index and the reciprocal of the condition number index. While effectively combining the advantages of the currently widely used operability index and the condition number index, it overcomes the problem that when the above two indices are used alone, they can only detect the singularity of some configurations in the reachable workspace of the robot arm, and cannot accurately and comprehensively describe the singular states of all configurations of the robot arm. This makes it possible to obtain more accurate and reliable results when using the designed index to analyze the kinematic singularity in the reachable workspace of the robot arm. This is of great importance for the development of kinematic singularity analysis of robot arms and singularity avoidance motion planning. Attached Figure Description

[0020] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0021] Figure 1 A schematic diagram of a seven-degree-of-freedom redundant robotic arm and its forward kinematics model according to one or more embodiments of the present invention; Figure 2 To Figure 1 A schematic diagram of the voxelization process used in the robotic arm's operating space; Figure 3 To Figure 2 A schematic diagram showing the ordered numbering of voxels obtained after voxelization. Figure 4 Established for the method of the present invention Figure 1 A schematic diagram of the kinematic singularity map of a robotic arm; Figure 5 for Figure 4 Half-section view of the kinematic singularity map; Figure 6 The method of the present invention identifies Figure 1 A half-section view of the kinematic singularity region in the reachable workspace of the robotic arm; Figure 7 The method of the present invention identifies Figure 1 A half-section view of the singularity region at the outer boundary of the workspace accessible by the robotic arm; Figure 8 The method of the present invention identifies Figure 1 A half-section view of the singularity region of the internal boundary of the workspace accessible by the robotic arm; Figure 9 Identified by the inventive method Figure 1 A half-section view of the optimal working area within the workspace that the robotic arm can reach. Detailed Implementation

[0022] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0023] A method for visually analyzing kinematic singularities in the workspace of a robotic arm includes the following steps: Step 1: Establish the forward kinematics model of the robotic arm and derive the position-level forward kinematics equations. The forward kinematics model of the robotic arm should be established using the method described in John J. Craig's "Introduction to Robotics," in which the coordinate system... and coordinate system These represent the base coordinate system and the end effector's coordinate system, respectively. Indicates the first The fixed coordinate system of each joint Let the number of joints be the origin. The derived position-level forward kinematic equations are: (1) in, The position vector representing the joint variables of the robotic arm. Indicates the first There are several joint variables, with the superscript indicating the transpose of the vector. Let be the pose matrix representing the end effector coordinate system relative to the base coordinate system. Representing the coordinate system Relative to coordinate system The pose matrix is ​​represented.

[0024] Step 2: Voxelize the robotic arm's operating space, and then number, classify, and store the resulting voxels. Specifically, Step 21: Using a side length The large cube encloses the operating space of the robotic arm in step one, where... This represents the reach of the robotic arm. The body center of the large cube and the base coordinate system of the robotic arm. The origin of the large cube coincides with the position of the base coordinate system, and the length, width, and height of the large cube are aligned with the direction of the base coordinate system. , , The axes are parallel. The coordinates of the origin of the robot arm's base coordinate system are represented as follows: Then the coordinates of the eight vertices of the large cube relative to the base coordinate system are as follows: (2) Step 22: Divide the large cube obtained in Step 21 into equal parts along its length, width, and height. The large cube is divided into 10 equal-spaced parts. A small cube, the side length of the small cube This process is called voxelization, and each small cube is called a voxel. Steps Two and Three: Starting from the first voxel in the lower left corner of the large cube in Step Two and Two, process the obtained voxels column by column, row by row, and layer by layer. Individual elements are numbered, and the resulting numbers are stored in a created one-dimensional matrix. In the middle, then to Voxels are classified into "reachable voxels" and "unreachable voxels" based on whether they can be reached by the end effector of the robotic arm.

[0025] Step 3: Using an improved Monte Carlo method and a self-collision detection method for the robotic arm, combined with the position-level forward kinematics equations of the robotic arm and the voxelized information of its operating space from Step 1, a precise reachable workspace for the robotic arm is established. The improved Monte Carlo method should adopt the method proposed in the journal article "Xu Zhenbang, Zhao Zhiyuan, He Shuai, et al. Improvement of Monte Carlo method and volume calculation for robot workspace solution [J]. Optics and Precision Engineering, 2018, 26(11):2703-2713." The self-collision detection method for the robotic arm should adopt the method proposed in the invention patent "Jiang Zainan, Liang Mengde. A fast collision detection method for a space robotic arm: 202010325643.9 [P]. 2022-05-17." Using the above improved Monte Carlo method and the self-collision detection method for the robotic arm, and combining the position-level positive kinematic equation of the robotic arm obtained in step one and the voxelization information in step two, a precise reachable workspace for the robotic arm is established. In order to ensure that each region in the established reachable workspace can be accurately described, a precision threshold is set. This represents the number of end-effector pose points that fall into each "reachable voxel" of the robotic arm as defined in steps two and three. Specifically, during the establishment of the reachable workspace, when generating an end-effector pose point in each "reachable voxel" using the improved Monte Carlo method, a self-collision detection method is used to determine whether the corresponding robotic arm configuration is a self-collision configuration. If it is a self-collision configuration, the pose point is discarded and regenerated, and the above process is repeated until a pose point is generated in each "reachable voxel". Each pose point and its corresponding robotic arm configuration is a non-collision robotic arm configuration.

[0026] Step 4: Fit the reachable workspace using the inscribed spheres of all voxels contained within the reachable workspace and visualize the results to obtain the voxelized reachable workspace. Create a database to store information related to this reachable workspace.

[0027] Specifically, Step 41: Store the data in the one-dimensional matrix as described in steps 2 and 3. The numbers in the sequence are used to traverse and calculate the results obtained in step two. The coordinates of the center of the inscribed sphere of each voxel are calculated, and the results are stored in a created one-dimensional matrix according to the voxel number order. In the middle, the first Liede Line number The voxel number corresponding to the layer is The center of the tangent sphere of this voxel Relative to the robot arm's base coordinate system The coordinates are: (3) Step 42: Traverse and check The number of reachable workspace points that a voxel falls into is stored in a created one-dimensional matrix. Voxels with a non-zero count, i.e., the "reachable voxels" defined in steps two and three, are represented by their corresponding numbers. middle; Step 43: In the matrix Extracting the matrix The coordinates of the center of the inscribed sphere of the voxel corresponding to the number in the table are given, and the radius of each inscribed sphere is given. The extracted inscribed spheres were visualized using MATLAB software (mathematical modeling), and the resulting voxelized reachable workspace of the robotic arm was obtained. Step 44: Create a database Based on the numbering order of the "reachable voxels" in the reachable workspace of the robotic arm voxelization established in step four-three, the voxels containing... The position vector information of each workspace point, as well as the corresponding robot arm joint angle position vector information when these workspace points were generated, are stored in this database.

[0028] Step 5: Design a kinematic singularity index that is dynamically weighted by an operability index and a condition number reciprocal index. Use this index to traverse and detect the kinematic singularity of each reachable voxel envelope region within the voxelized reachable workspace, and store the detection results.

[0029] Specifically, Step 51: To address the problem that existing technologies using a single kinematic singularity index cannot accurately and comprehensively detect all configurational singularities within the reachable workspace of a robotic arm, this invention designs a kinematic singularity index that is dynamically weighted by a combination of an operability index and a condition number reciprocal index. Its specific expression is as follows: (4) (5) (6) (7) in, Indicates the weighting coefficient. and These represent the operability index and the reciprocal of the condition number index, respectively. The Jacobian matrix represents the robotic arm. Represents the determinant of a matrix. and Let them represent the Jacobian matrix respectively. The minimum and maximum singular values; As can be seen from formulas (4) and (5), the designed kinematic singularity index Through weighting coefficients Dynamic adjustment and The weights between them solve the problem that it is difficult to accurately assess the singular state of a robotic arm when using the above two indicators alone. They combine the advantages of the two indicators, thus enabling a comprehensive and accurate detection of kinematic singularities in the reachable workspace of the robotic arm. Step 52: To calculate the designed kinematic singularity index The value of requires the use of the robotic arm's Jacobian matrix. The vector product method should be used, combined with the forward kinematics model of the robotic arm from step one, to derive the result. The analytical expression is derived as follows: (8) (9) in, express The List, Representing the coordinate system of Axis in base coordinate system The unit vector represented in the figure. for The zero vector of dimension, Represents the coordinate system of the end effector The origin relative to the coordinate system The position vector is transformed to the base coordinate system. The representation in; Step 53: Obtain the Jacobian matrix After parsing the expression, combine it with the database in step four. The information stored in the system is used to iterate and calculate the information contained in each "reachable voxel" according to the numbering order of the "reachable voxels" in the reachable workspace of the robotic arm voxelization established in step four-three. Kinematic singularity index of a robotic arm configuration The value is calculated by first substituting the joint angle position vectors of each group into formulas (8) and (9) to obtain the Jacobian matrix. The value, and then through the value of, and then through the Perform singular value decomposition to find the maximum singular value. and minimum singular value Next , and Substituting into formulas (5) to (7), we can calculate the results. , and The values ​​are then substituted into formula (4) to calculate the result. The value, A larger value indicates that the robotic arm is further away from the singular configuration. A value of 0 indicates that the robotic arm has reached a singular configuration; Step 54: Based on the calculation results in Step 53, further iterate and calculate the value within each "reachable voxel". Kinematic singularity index of a robotic arm configuration average Then, all calculation results are stored in the database created in step four in voxel number order. middle. The value represents the degree of kinematic singularity of the region enclosed by each "reachable voxel". The larger the value, the farther the enclosed region is from the singular region. The closer the value is to zero, the closer the enclosed region is to the singular region. The specific calculation formula is as follows: (10) in, Indicates the first "reachable voxel" in each voxel The values ​​of the kinematic singularity index corresponding to each robotic arm configuration.

[0030] Step Six: Color-code each reachable voxel within the voxelized reachable workspace based on the kinematic singularity information of its envelope region, and visualize the results to construct a kinematic singularity map of the robotic arm. This map can be used for visual analysis of kinematic singularities in the robotic arm's workspace. Specifically, Step 61: After calculating the average kinematic singularity index of all "reachable voxels" in the voxelized reachable workspace of the robotic arm in Step 54... To determine the entire reachable workspace maximum value Then set the interval The data is then divided into multiple numerical intervals, and the corresponding numbers of the "reachable voxels" falling into different numerical intervals are stored in a newly created multi-row, multi-column matrix. In this matrix, the number of rows is equal to the number of divisions of the numerical intervals mentioned above, and the number of columns is equal to the number of "reachable voxels" falling into each numerical interval. Step 62: Based on the results in Step 62, in the matrix of Step 41 Extracting matrices sequentially Each row stores the coordinates of the center of the inscribed sphere of the "reachable voxel" corresponding to the voxel number, combined with the radius of the inscribed sphere. MATLAB software was used to visualize all the extracted inscribed spheres, and different colors were assigned to distinguish the inscribed spheres in different rows. The result is a kinematic singularity map that can be used to visualize and analyze the kinematic singularities in the workspace of the robotic arm. It should be noted that during visualization, a half-section view is recommended to clearly see the internal structure of the singularity map. Since the entire reachable workspace of the robotic arm is symmetrically distributed, the section view will not affect the singularity analysis. In the constructed kinematic singularity map, The area covered by the "reachable voxel" corresponding to the minimum value interval is the kinematic singularity region in the reachable workspace of the robotic arm, while other regions are non-singular regions. The reachable workspace region corresponding to the maximum value interval is the optimal working area for the robotic arm to perform its tasks.

[0031] Step 7: Using a kinematic singularity map, identify the internal boundary singularity regions, external boundary singularity regions, and the optimal operation area for the task within the robotic arm's workspace. The kinematic singularity map of the robotic arm constructed in step six-two allows for a visual analysis of the distribution of kinematic singularities within the reachable workspace. In this map, red voxels correspond to regions where the minimum singularity measurement value approaches zero, representing the area reached by the robotic arm's singular configuration; while blue voxels represent regions with the maximum singularity measurement value, indicating that the robotic arm is far from the singular configuration when reaching that region. Furthermore, throughout the reachable workspace, the singularity measurement value increases as the robotic arm extends from its base position, reaching a maximum value, and then rapidly decreases to near zero. The minimum value is concentrated in the region near the robotic arm base and in the area accessible when the robotic arm is fully extended, while the maximum value occurs in the region approximately halfway through the robotic arm's reach.

[0032] Based on the information presented in the singularity map described above, the internal boundary singularity region, the external boundary singularity region, and the optimal operation region for the task can be identified within the robotic arm's workspace. Specifically, the internal boundary singularity region is the area enclosed by the red voxels at the inner boundary of the accessible workspace, the external boundary singularity region is the area enclosed by the red voxels at the outer boundary of the accessible workspace, and the optimal operation region for the task is the area enclosed by the blue voxels within the accessible workspace. In practical applications, the desired motion trajectory of the robotic arm should be planned within the blue voxel region as much as possible to ensure the safety and reliability of task execution.

[0033] Step 8: Using a kinematic singularity map, determine the singularity of the robotic arm reaching the target position based on the placement information. Specifically, Step 81: Based on the placement vector information of the target location In step four, the corresponding "reachable voxel" number is located in the voxelized reachable workspace established in step four. , specifically, It can be calculated using the formula shown below: (11) (12) (13) (14) in, , and Representing rows, columns, and layers respectively. Represents the absolute value symbol. Indicates the rounding up symbol; Step 82: Obtain the "reachable voxel" number in Step 81. Then, in the database Find the average value of the kinematic singularity index corresponding to the voxel. The specific value is such that if the absolute value of the value is close to zero, it indicates that the robotic arm may encounter a kinematic singularity problem when it reaches the vicinity of the target position; conversely, if the absolute value of the value is much greater than zero, it indicates that the robotic arm can safely reach the target position.

[0034] A method for visually analyzing kinematic singularities in the workspace of a robotic arm, to Figure 1 Taking the seven-DOF redundant robotic arm shown as an example, the specific implementation steps are as follows: Step 1: Combining Figure 1 The seven-DOF redundant robotic arm shown is modeled using the method described in John J. Craig's "Introduction to Robotics". Figure 1 As shown, in the established model, the coordinate system and coordinate system These represent the base coordinate system and the end effector's coordinate system, respectively. Indicates the first The position-level forward kinematic equations are derived from the fixed coordinate system of each joint as follows:

[0035] in, The position vector representing the joint variables of the robotic arm. Indicates the first One joint variable, , Indicates along Shaft from Axis moves to Distance between axes Indicates circling Shaft from The axis rotates to Angle of axis Indicates along Shaft from Axis moves to Distance between axes; Combination Figure 1 As shown in Table 1, the DH parameters corresponding to the forward kinematics model of the robotic arm can be calculated by substituting the values ​​of each parameter in the table into the position-level forward kinematics equations. The specific value.

[0036] Table 1. Robotic Arm DH Parameters

[0037] Step Two: Combining Figure 2 As shown, a side length is used. Large cubic envelope Figure 1The operating space of the robotic arm, the body center of the cube and Figure 1 Base coordinate system of the robotic arm The origin of the coordinate system coincides with the origin of the coordinate system, and the length, width, and height of the coordinate system are aligned with the direction of the base coordinate system. , , The axes are parallel, and the coordinates of the eight vertices of the large cube relative to the base coordinate system are as follows:

[0038] Divide the large cube equally along its length, width, and height into three parts. The large cube is divided into 512,000 voxels by equidistant sections, with each voxel having a side length of... This process is called voxelization, and the result is as follows: Figure 3 As shown; Starting with the first voxel at the bottom left corner of the large cube, the 512,000 voxels are numbered sequentially by column, row, and layer, corresponding to the robot arm's base coordinate system. middle The axis alignment direction is the column direction, and... The axis alignment direction is the row direction, and... If the axis alignment direction is the layer direction, then the first... Liede Line number The voxel number corresponding to the layer is A diagram illustrating the numbering process is shown below. Figure 3 As shown, the obtained numbers are stored in the created one-dimensional matrix. In the middle, then to Voxels are classified into "reachable voxels" and "unreachable voxels" based on whether they can be reached by the end effector of the robotic arm.

[0039] Step 3: Combining Figure 1 and Figure 3 As shown, based on the definition of the reachable workspace of the robotic arm, the improved Monte Carlo method proposed in "Xu Zhenbang, Zhao Zhiyuan, He Shuai, et al. Improved Monte Carlo method and volume calculation for solving robot workspace [J]. Optics and Precision Engineering, 2018, 26(11):2703-2713" and the self-collision detection method of the robotic arm proposed in "Jiang Zainan, Liang Mengde. A fast collision detection method for space robotic arms: 202010325643.9 [P]. 2022-05-17" are adopted. Combined with the position-level forward kinematics equations of the seven-degree-of-freedom redundant robotic arm obtained in step one and the voxelization information in step two, the solution is obtained. Figure 1 The precise reachable workspace of the robotic arm, and the accuracy threshold set during the solution process. .

[0040] Step 4: Combining Figure 3 As shown, according to a one-dimensional matrix The coordinates of the centers of the inscribed spheres of the 512,000 voxels are calculated sequentially by iterating through the voxel numbers stored in the matrix, and the results are stored in the created one-dimensional matrix. In the middle, the first Liede Line number The voxel number corresponding to the layer is The center of the tangent sphere of this voxel Relative to the robot arm's base coordinate system The coordinates are:

[0041] Then, according to the one-dimensional matrix The stored numbers are sequentially traversed to check the number of reachable workspace points within the 512000 voxels, and the numbers corresponding to voxels with non-zero counts are stored in the created one-dimensional matrix. middle; Next, in the matrix Extracting the matrix The coordinates of the center of the inscribed sphere of the voxel corresponding to the number in the figure, and the radius of the inscribed sphere are... The extracted inscribed spheres were visualized using MATLAB software, and the results were obtained. Figure 1 The voxelized workspace of the robotic arm shown is accessible. Finally, create a database. , the results Figure 1 The contents of each reachable voxel within the reachable workspace of the robotic arm voxelization The position vector information of each workspace point, along with the corresponding robot arm joint angle position vector information when these points were generated, is stored in the database. middle.

[0042] Step 5: Combining Figure 1 The DH parameters in Table 1 are derived using formulas (8) and (9). Figure 1 The robotic arm shown is relative to the base coordinate system Jacobian matrix The parsing expression; Next, according to the database The information stored in it is traversed and calculated. Figure 1 Each "reachable voxel" in the voxelized reachable workspace of the robotic arm contains Kinematic singularity index of a robotic arm configuration The value is calculated by first substituting the joint angle position vectors of each group into formulas (8) and (9) to obtain the Jacobian matrix. The value, and then through the value of, and then through the Perform singular value decomposition to find the maximum singular value. and minimum singular value Next , and Substituting into formulas (5) to (7), we can calculate the results. , and The values ​​are then substituted into formula (4) to calculate the result. The value; Based on the above calculation results, further calculations are performed. Figure 1 Each "reachable voxel" in the voxelized reachable workspace of the robotic arm contains Kinematic singularity index of a robotic arm configuration average Then, all calculation results are stored in the database in voxel number order. middle. The value represents the kinematic singularity information of the region enclosed by each "reachable voxel". The larger the value, the farther the enclosed region is from the singularity region. The closer the value is to zero, the closer the enclosed region is to the singularity region. The specific calculation formula is as follows:

[0043] Step Six: Combining Figure 4 and Figure 5 As shown, according to the database Stored in Figure 1 The average kinematic singularity index of all "reachable voxels" in the voxelized reachable workspace of the robotic arm. The range of values ​​for is determined to obtain the value range within the entire reachable workspace. The maximum value is Then set the interval The values ​​were divided into 11 intervals, and the results are shown in Table 3. Each interval in the table represents the degree of proximity to the kinematic singularity. Next, the numbers corresponding to the "reachable voxels" falling into different numerical ranges in Table 3 are stored in a multi-row, multi-column matrix. In this system, there are 11 rows, and the number of columns is equal to the number of "reachable voxels" falling into each numerical range. Finally, in the matrix Extracting matrices sequentially Each row stores the coordinates of the center of the inscribed sphere of the "reachable voxel" corresponding to the voxel number, combined with the radius of the inscribed sphere. The extracted inscribed spheres were visualized using MATLAB software, and different colors were assigned to distinguish the inscribed spheres in different rows. The RGB values ​​of the 11 colors used are shown in column 3 of Table 3. The result is a kinematic singularity map that can visualize and analyze the kinematic singularities in the workspace of the robotic arm, as shown below. Figure 4 As shown. To clearly see the internal structure of the obtained kinematic singularity diagram, a half-sectional view can be used. Figure 5 That is Figure 4 A half-section view.

[0044] Table 3. Mean values ​​of kinematic singularity indices Interval partitioning results and RGB color values

[0045] Step Seven: Combining Figures 6-9 As shown, the material obtained in step six is ​​used. Figure 1 The kinematic singularity map of the robotic arm identifies the singular regions of the internal and external boundaries of the workspace, as well as the optimal operating region for the task. All results are represented using a half-section view. First, according to Figure 1 The region enveloped by red voxels in the reachable workspace of the robotic arm was voxelized, and singular regions existing in the reachable workspace of the robotic arm were identified. The results are as follows: Figure 6 As shown, tasks should be avoided in this area during practical applications; Then, according to Figure 1 Voxelization of the robotic arm reaches the region enclosed by the red voxels at the boundary of the workspace, identifying singular regions at the boundary of the workspace, such as... Figure 7 As shown; Next, according to Figure 1 The voxelization of the robotic arm reaches the region enclosed by the red voxels at the outer boundary of the workspace, identifying the singular region at the outer boundary of the workspace, such as... Figure 8 As shown; Finally, according to Figure 1 The robotic arm's voxelization can reach the region enclosed by the blue voxels in the workspace, identifying the optimal operating area for the task. The results are as follows: Figure 9 As shown, in actual tasks, by planning the desired motion trajectory of the robotic arm within the optimal operating area of ​​the task, the safety and reliability of task execution can be guaranteed.

[0046] Step 8: Use the material obtained in Step 6 Figure 1The kinematic singularity map of the robotic arm can also determine the singularity of the robotic arm reaching the target position based on the target's placement information. Assume the vector information of the target position is... First, use formula (11) to calculate the number of the "reachable voxel" in the voxelized reachable workspace of the robotic arm where the target position falls. The calculation results are as follows:

[0047]

[0048]

[0049]

[0050] Next, based on the calculated numbering information, in the database... Find the average value of the kinematic singularity index corresponding to the voxel. The specific value is 0.0870, which indicates that the robotic arm will not encounter kinematic singularity problems when it reaches the area where the target position is located.

[0051] Example 2 This embodiment provides a system for constructing a visualization analysis system for kinematic singularities in the workspace of a robotic arm, including a computing device, which is a computer, and is configured as follows: The robotic arm can accurately reach the workspace by taking into account self-collision information. The reachable workspace is obtained by fitting the inscribed sphere of all voxels contained within the reachable workspace and visualizing it, and the relevant information of the space is stored. Obtain the kinematic singularity index, which is dynamically weighted by the operability index and the inverse of the condition number index. Combined with the relevant information of the stored voxelized reachable workspace, calculate the kinematic singularity information of the envelope region of each reachable voxel in the reachable workspace of the robotic arm, and store the calculation results. Based on the distribution of kinematic singularity information in different numerical ranges in the stored reachable workspace, all reachable voxels in the voxelized reachable workspace are color-coded and the results are visualized to obtain a singularity map that can visualize and analyze the kinematic singularities in the robot arm's workspace. Using the obtained kinematic singularity map, the internal boundary singularity region, the external boundary singularity region, and the optimal operation region for the task are identified in the workspace of the robotic arm. Using the obtained kinematic singularity map, the singularity of the robotic arm reaching the target position is determined based on the placement information of the target position.

[0052] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for visually analyzing kinematic singularities in the workspace of a robotic arm, characterized in that, Includes the following: Obtain voxelized information of the robotic arm's operating space. Based on the voxelized information of the robotic arm's operating space, which includes reachable and inaccessible voxels, establish the reachable workspace of the robotic arm. Obtain the kinematic singularity index, which includes the operability index and the inverse of the condition number index, and use the kinematic singularity index to traverse and detect the kinematic singularity of each reachable voxel envelope region within the voxelized reachable workspace of the robotic arm. Each reachable voxel within the voxelized reachable workspace of the robotic arm is color-coded based on the kinematic singularity information of its envelope region, and the results are visualized to obtain the kinematic singularity map of the robotic arm.

2. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 1, characterized in that, An improved Monte Carlo method and a self-collision detection method for robotic arms are adopted. Combining the position-level forward kinematics equations of the robotic arm and the voxelized information of the robotic arm's operating space, the reachable workspace of the robotic arm is established by fitting the inscribed sphere of all voxels contained in the reachable workspace and visualizing the results.

3. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 2, characterized in that, The process of obtaining voxelized information about the robotic arm's operating space includes the following: The robotic arm's operating space is divided into multiple cubes, which are then processed column-by-column, row-by-row, and layer-by-layer. Individual elements are numbered, and the resulting numbers are stored in a created one-dimensional matrix. ,right Voxels are classified into reachable voxels and unreachable voxels. Voxels that can be reached by the end effector of the robotic arm are defined as reachable voxels, while the remaining voxels are defined as unreachable voxels.

4. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 3, characterized in that, Establishing the accessible workspace of the robotic arm includes the following: According to the storage in the one-dimensional matrix The numbers in the sequence are traversed and calculated. The coordinates of the center of the inscribed sphere of each voxel are calculated, and the results are stored in a created one-dimensional matrix according to the voxel number order. middle; Traversal and inspection The number of reachable workspace points that a voxel falls into is stored in a created one-dimensional matrix, along with the numbers of voxels with a non-zero count. middle; In the matrix Extracting the matrix The coordinates of the center of the inscribed sphere of the voxel corresponding to the number in the table are given, and the radius of each inscribed sphere is given. All extracted inscribed spheres are visualized and drawn to obtain the voxelized reachable workspace of the robotic arm. Create database Based on the numbering order of reachable voxels in the established voxelized workspace of the robotic arm, the voxels containing each voxel are... The position vector information of each workspace point, as well as the corresponding robot arm joint angle position vector information when these workspace points were generated, are stored in this database.

5. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 1, characterized in that, The kinematic singularity index Based on weighting coefficients Operability indicators Conditional reciprocal index To obtain the weighting coefficients According to the Jacobian matrix Minimum singular value and maximum singular value To obtain. The inverse index of the condition number According to the minimum singular value of the Jacobian matrix and maximum singular value The ratio is used to obtain the value.

6. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 5, characterized in that, The operability index According to the Jacobian matrix of the robotic arm Obtain the square root of the determinant of its product with its transpose; The inverse index of the condition number According to the minimum singular value of the Jacobian matrix and maximum singular value The ratio is used to obtain the value.

7. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 6, characterized in that, By employing the vector product method and combining it with the position-level forward kinematics equations of the robotic arm, the analytical expression for the Jacobian matrix is ​​derived.

8. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 4, characterized in that, The method of using kinematic singularity indices to traverse and detect the kinematic singularity of each reachable voxel envelope region within the voxelized reachable workspace includes the following: Based on the forward kinematics model of the robotic arm, the Jacobian matrix of the robotic arm is derived. Combined with database The information stored in the database is used to iterate and calculate the data contained in each reachable voxel, according to the numbering order of the reachable voxels in the established voxelized reachable workspace of the robotic arm. The value of the kinematic singularity index of the robotic arm configuration.

9. The method for visually analyzing kinematic singularities in the workspace of a robotic arm according to claim 4, characterized in that, The process involves color-coding each reachable voxel within the voxelized reachable workspace of the robotic arm based on the kinematic singularity information of its envelope region, and visualizing the results to obtain the kinematic singularity map of the robotic arm, which includes the following: The average kinematic singularity index of all reachable voxels in the voxelized reachable workspace of the robotic arm is calculated, and the maximum value of the average value in the entire voxelized reachable workspace of the robotic arm is obtained. Set the numerical range The data is then divided into multiple numerical intervals, and the numbers corresponding to the reachable voxels falling into different numerical intervals are stored in a multi-row, multi-column matrix. In this matrix, the number of rows is equal to the number of divisions of the numerical intervals mentioned above, and the number of columns is equal to the number of reachable voxels falling into each numerical interval. In the matrix Extracting matrices sequentially Each row in the database stores the coordinates of the center of the inscribed sphere of the reachable voxel corresponding to its voxel number, combined with the radius of the inscribed sphere. The extracted inscribed spheres are visualized and drawn, and different colors are assigned to the inscribed spheres in different rows to distinguish them, thus obtaining the kinematic singularity map of the robotic arm.

10. A system for constructing a visualization analysis system for kinematic singularities in the workspace of a robotic arm, characterized in that, Includes a computing device, which is configured as follows: Obtain voxelized information of the robotic arm's operating space. Based on the voxelized information of the robotic arm's operating space, which includes reachable and inaccessible voxels, establish the reachable workspace of the robotic arm. Obtain the kinematic singularity index, which includes the operability index and the inverse of the condition number index, and use the kinematic singularity index to traverse and detect the kinematic singularity of each reachable voxel envelope region within the voxelized reachable workspace of the robotic arm. Each reachable voxel within the voxelized reachable workspace of the robotic arm is color-coded based on the kinematic singularity information of its envelope region, and the results are visualized to construct the kinematic singularity map of the robotic arm.

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