Singular-avoiding method and system for robotic arm based on spectral risk fusion and selective damping

Abstract

The application discloses a mechanical arm singularity avoidance method and system based on spectral risk fusion and selective damping, relates to the technical field of mechanical hands, and solves the problem that the motion singularity of a mechanical arm in the prior art adopts a single evaluation index, a unified scalar damping factor is usually adopted in a damping least square method, and differentiated adjustment cannot be performed according to the degeneration degrees of directions, has the beneficial effect of guaranteeing accurate work of the mechanical arm, and the specific scheme is as follows: the mechanical arm singularity avoidance method based on spectral risk fusion and selective damping comprises the following steps: obtaining a Jacobian matrix of a mechanical arm, calculating singular value spectra of the Jacobian matrix, and obtaining three types of kinematic singularity detection indexes; obtaining a normalized minimum singular value index, a normalized reciprocal of a condition number index and a normalized overall motion ability index; dynamically weighting and fusing the three types of singularity risks; adopting the spectral risk fusion index to detect singularity of the mechanical arm, and determining whether to trigger damping; and constructing a direction-selective damping pseudo-inverse matrix.

Description

Technical Field

[0001] This invention relates to the field of robotic arm technology, and in particular to a method and system for avoiding singularities in robotic arms based on spectral risk fusion and selective damping. 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] To detect the kinematic singularity of robotic arms, various indicators are widely used in existing technologies to detect the kinematic singularity of robotic arms. However, these all use a single evaluation standard, which can easily lead to problems such as inaccurate or untimely test results and blind spots in the detection.

[0004] Damped least squares is a widely used kinematic singularity avoidance method in current technology. By introducing a damping factor into the Jacobian pseudoinverse, it can effectively suppress problems such as joint velocity amplification near singular configurations, thereby avoiding kinematic singularities. However, damped least squares typically uses a uniform scalar damping factor, applying the same damping to all motion directions, failing to differentiate according to the degree of degradation in each direction. When only a few directions degrade, uniformly increasing the damping can excessively suppress the motion capability in non-singular directions, resulting in decreased end-effector accuracy and dexterity. Furthermore, the damping factor is often determined using a hard threshold triggering mechanism. When the singularity index fluctuates around the threshold, the damping factor is prone to abrupt changes from zero to non-zero or frequent switching, leading to problems such as discontinuous joint velocities, robotic arm jitter, and decreased control stability. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the purpose of this invention is to provide a singularity avoidance method for robotic arms based on spectral risk fusion and selective damping, which effectively improves end-effector tracking accuracy and avoids problems such as sudden changes in joint velocity and motion jitter near singular configurations of the robotic arm.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solution: Singularity avoidance methods for robotic arms based on spectral risk fusion and selective damping include: Obtain the Jacobian matrix of the robotic arm, calculate the singular value spectrum of the Jacobian matrix, and obtain three types of kinematic singularity detection indices; Obtain the normalized minimum singular value index, the reciprocal of the normalized condition number index, and the normalized overall motion capability index; Dynamically weighted fusion of three types of singularity risks is used to construct a singular value spectrum risk fusion index. The singularity of the robotic arm is detected by spectral risk fusion index to determine whether damping is triggered: a global damping activation coefficient is constructed based on the damping factor intervention threshold, the high-risk damping factor saturation threshold and a positive number. A smoothing function is designed based on the global damping activation coefficient, and the value of the smoothing function is used to determine whether damping is triggered. An adaptive dual-threshold smoothing damping factor is designed, and a direction-selective damping pseudo-inverse matrix is ​​constructed: based on the pre-defined maximum damping value in the singular region, the smoothing function, and the first... The smoothing function of the nth motion direction is used to determine the th The damping factor in the direction of motion, according to the first... The singular values ​​of the i-th direction of motion, the left singular vector matrix, the right singular vector matrix, and the i-th... The damping factors in each motion direction construct a direction-selective damping pseudo-inverse matrix; Solve the inverse kinematics solution for the robotic arm to avoid singularities, and control the robotic arm to achieve singularity avoidance motion.

[0007] In the singularity avoidance method for robotic arms based on spectral risk fusion and selective damping described above, the global damping activation coefficient is obtained according to the following formula:

[0008] in, This represents a truncation function, whose function is to limit the output to a certain value. Inside, This represents the singular value spectrum risk fusion index. This indicates that the damping factor has begun to intervene at the threshold. This indicates the saturation threshold of the high-risk damping factor. Represent a positive number; Determine the smoothing function : ; when hour, ,thereby , indicating that the damping factor is not triggered; when hour, This indicates that the damping factor is smoothly intervened; when hour, ,thereby This indicates that the maximum damping factor trigger state has been entered.

[0009] The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping, as described above, determines the adaptive dual-threshold smoothing damping factor according to the following formula: in, Indicates the first Damping factor in each direction of motion, This represents the maximum damping value in the pre-defined singular region. Indicates the first A smoothing function for each direction of motion. Represents the truncation function The result value of the action, Indicates the first Singular values ​​in each direction of motion, and These represent the danger threshold and the safe threshold for singular values, respectively. Represent a positive number; For the first... One direction of movement: when hour, , This indicates that stronger damping is needed in that direction; when hour, , This indicates that no additional damping is needed in that direction; when hour, This indicates a smooth transition of damping in that direction; Construct the directional selective damping pseudo-inverse matrix according to the following formula. : ; in, , Let represent the left singular vector matrix and the right singular vector matrix, respectively.

[0010] The above-described method for singularity avoidance by a robotic arm based on spectral risk fusion and selective damping, which involves solving the inverse kinematics solution for singularity avoidance and controlling the robotic arm to achieve singularity avoidance motion, includes the following: The singular inverse solution of the robot arm's velocity-level inverse kinematics equations obtained by using the damped least squares method is as follows: in, The velocity vector represents the joint variable. This indicates the robotic arm end effector relative to the base coordinate system. The Cartesian velocity vector is represented. Representing the Jacobian matrix The fixed damping pseudo-inverse matrix, Indicates the damping factor. express The minimum singular value, This represents a pre-set threshold used to determine whether the robotic arm has reached a singular configuration. This represents the maximum damping value in the singular region.

[0011] The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping, as described above, obtains the normalized minimum singularity index according to the following: The danger threshold of the normalized minimum singularity index is determined using the percentile method in statistics. Safe threshold for normalized minimum singular value index ; Threshold normalization method is used for the minimum singular value index Perform dimensionless normalization: in, The normalized minimum singular value index; This means truncating the result to the interval. Operators in It represents a positive number.

[0012] In the singularity avoidance method for robotic arms based on spectral risk fusion and selective damping described above, the reciprocal of the normalized condition number is obtained according to the following: The danger threshold of the reciprocal index of the normalized condition number is determined using the percentile method in statistics. The safety threshold of the reciprocal of the normalized condition number. ; The reciprocal index of the condition number is obtained by using the threshold normalization method. Perform the following processing:

[0013] in, The reciprocal of the normalization condition number is the index. It represents a positive number.

[0014] The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping, as described above, obtains the normalized overall motion capability index according to the following: The danger threshold of the normalized overall athletic ability index was determined using the percentile method in statistics. Safety threshold of normalized overall athletic ability index ; Threshold normalization method is used to analyze the overall athletic performance index Perform the following processing:

[0015] in, To normalize the overall athletic performance index, It represents a positive number.

[0016] The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping, as described above, involves dynamically weighting and fusing three types of singularity risks to construct a singularity value spectral risk fusion index, which includes the following: Constructing a singular value spectrum risk fusion index:

[0017] (19) (20) (twenty one) in, This represents a singular value spectrum risk fusion index; , and These respectively represent the risk of deterioration in mobility in the most dangerous direction, the risk of distortion of the operability ellipsoid shape, and the risk of overall mobility decline. , and These represent the weighting coefficients for the three types of risks mentioned above. This indicates the penalty index for weaknesses.

[0018] The above-described method for avoiding singularities in a robotic arm based on spectral risk fusion and selective damping involves obtaining the Jacobian matrix of the robotic arm, calculating the singular value spectrum of the Jacobian matrix, and obtaining three types of kinematic singularity detection indices, including the following: A forward kinematics model of the robotic arm is established, and the position-level forward kinematics equations are obtained; The Jacobian matrix of the robotic arm is derived, and the forward and inverse kinematic equations of the velocity level are obtained. The singular value spectrum of the Jacobian matrix is ​​calculated to obtain three types of kinematic singularity detection indices: The singular value spectrum is a Jacobian matrix A numerical distribution formed by arranging all singular values ​​in order of magnitude, for Singular value decomposition yields: in, Represents a singular value matrix. for singular values, for The zero vector of dimension, This represents the dimension of the operation space. This is the singular value spectrum of the Jacobian matrix under the current configuration.

[0019] Secondly, this invention discloses a singularity avoidance system for a robotic arm based on spectral risk fusion and selective damping, comprising a computing device that performs the following: Obtain the Jacobian matrix of the robotic arm, calculate the singular value spectrum of the Jacobian matrix, and obtain three types of kinematic singularity detection indices; Obtain the normalized minimum singular value index, the reciprocal of the normalized condition number index, and the normalized overall motion capability index; Dynamically weighted fusion of three types of singularity risks is used to construct a singular value spectrum risk fusion index. The singularity of the robotic arm is detected by spectral risk fusion index to determine whether damping is triggered: a global damping activation coefficient is constructed based on the damping factor intervention threshold, the high-risk damping factor saturation threshold and a positive number. A smoothing function is designed based on the global damping activation coefficient, and the value of the smoothing function is used to determine whether damping is triggered. An adaptive dual-threshold smoothing damping factor is designed, and a direction-selective damping pseudo-inverse matrix is ​​constructed: based on the pre-defined maximum damping value in the singular region, the smoothing function, and the first... The smoothing function of the nth motion direction is used to determine the th The damping factor in the direction of motion, according to the first... The singular values ​​of the i-th direction of motion, the left singular vector matrix, the right singular vector matrix, and the i-th... The damping factors in each motion direction construct a direction-selective damping pseudo-inverse matrix; Solve the inverse kinematics solution for the robotic arm to avoid singularities, and control the robotic arm to achieve singularity avoidance motion.

[0020] The beneficial effects of the present invention are as follows: This invention applies the singular value spectrum risk fusion index to the problem of kinematic singularity avoidance in robotic arms. Since this index can comprehensively evaluate the overall kinematic singularity risk of the current configuration, it is more suitable as the triggering basis for the damping factor in singularity avoidance methods compared with single indices such as operability, minimum singular value or the reciprocal of the condition number. Meanwhile, by designing an adaptive dual-threshold smoothing damping factor, the problem of joint velocity mutation and shaking that is easily caused by traditional hard threshold triggering damping factors near singular configurations can be effectively avoided, which is conducive to improving joint velocity continuity and control stability. The strategy of selectively introducing damping in singular directions involves constructing a direction-selective damping pseudo-inverse matrix. This means that strong damping is applied only to singular directions of motion, while weaker damping or no damping is applied to non-singular directions. Compared with the existing method of applying the same damping to all directions of motion, this approach can improve the flexibility and stability of the robotic arm in singular regions, while ensuring that its end effector tracks the task trajectory with higher precision. This is of great significance for ensuring that the robotic arm can perform tasks accurately, stably, and safely.

[0021] Based on the singular value spectrum of the Jacobian matrix, which reflects the complete motion capability information of a robotic arm, threshold normalization is performed on spectral features with different dimensions and numerical ranges to give all features a unified dimensionless evaluation scale. Then, through a fusion mechanism, a singular value spectrum risk fusion index that can comprehensively detect the kinematic singularities of a robotic arm is constructed. This index can simultaneously respond to the risks of motion capability degradation in the most dangerous direction, ellipsoidal distortion of operability, and overall motion capability decay. This overcomes the blind spot problem in existing technologies that use only minimum singular value, operability, or the reciprocal of condition number as indicators to detect the kinematic singularities of a robotic arm. Finally, the more comprehensive, accurate, and reliable singularity state detection results obtained can provide strong support for applications such as singular motion avoidance control of robotic arms. Attached Figure Description

[0022] 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.

[0023] 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 for Figure 1 A schematic diagram of a robotic arm performing a straight-line path tracking task in Cartesian space; Figure 3 To obtain using the pseudo-inverse method Figure 1 A schematic diagram of the joint speeds of a robotic arm; Figure 4 To obtain using the damped least squares method Figure 1 A schematic diagram of the joint speeds of a robotic arm; Figure 5 To achieve the singularity avoidance method of this invention, Figure 1 A schematic diagram of the joint speeds of a robotic arm; Figure 6 The singularity avoidance method and damped least squares method of this invention are obtained Figure 1 A schematic diagram comparing the position tracking errors at the end effector of a robotic arm; Figure 7The singularity avoidance method and damped least squares method of this invention are obtained Figure 1 A schematic diagram comparing the attitude tracking errors at the end effector of a robotic arm. Detailed Implementation

[0024] It should be noted that the following detailed description is illustrative and intended to provide further explanation of the invention. Unless otherwise specified, all technical and scientific terms used in this invention have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0025] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, unless otherwise expressly indicated by the invention, the singular form is also intended to include the plural form. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof. As introduced in the background section, existing technologies use a single evaluation index for the motion singularity of robotic arms, and the damping least squares method usually uses a uniform scalar damping factor, which cannot be adjusted differently according to the degree of degradation in each direction. In order to solve the above technical problems, this invention proposes a robotic arm singularity avoidance method based on spectral risk fusion and selective damping.

[0026] Example 1 In a typical embodiment of the present invention, reference is made to Figure 1 As shown, the singularity avoidance method for robotic arms based on spectral risk fusion and selective damping includes the following: Step 1: Establish the forward kinematics model of the robotic arm and obtain the position-level forward kinematics equations. The forward kinematics model of the robotic arm was established using the method described in Professor John J. Craig's textbook "Introduction to Robotics," and a coordinate system was selected in the established model. and coordinate system The coordinate systems are respectively used as the base coordinate system and the coordinate system of the end effector, and the coordinate systems are selected simultaneously. As the first A fixed coordinate system for each joint, wherein... This represents the number of joints. Based on the above coordinate system setting and the meaning of the homogeneous transformation matrix, the position-level forward kinematics equations of the robotic arm are derived as follows: (1) in, It is the function name representing forward kinematics. This represents the position vector of the robotic arm's joint variables; the superscript indicates vector transpose. Indicates the first The position variables of each joint Let be the pose matrix representing the end effector coordinate system relative to the base coordinate system. coordinate system Relative to coordinate system The pose matrix is ​​represented.

[0027] Step 2: Derive the Jacobian matrix of the robotic arm to obtain the forward and inverse kinematic equations of the velocity level. Specifically, Step 21: The Jacobian matrix of the robotic arm establishes a linear mapping relationship between joint velocities and the Cartesian velocity of the end effector. Therefore, to derive the forward and inverse kinematic equations of the robotic arm's velocity levels, it is first necessary to derive the Jacobian matrix. In this embodiment, the vector product method is used to derive the robotic arm's velocity relative to the base coordinate system. The Jacobian matrix represented The analytical expression is derived as follows: (2) (3) 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 the text, Represents the vector cross product; Step 22: The Jacobian matrix is ​​derived in Step 21. Then, the forward kinematic equations of the robot arm's velocity level can be expressed in the following form: (4) in, This indicates the robotic arm end effector relative to the base coordinate system. The Cartesian velocity vector is represented. The velocity vector represents the joint variable; Steps two and three: When the Jacobian matrix When the matrix is ​​a full-rank row matrix, the pseudo-inverse method is used to solve for the velocity vectors of the joint variables in the forward kinematic equations of the velocity level obtained in step 22. The inverse kinematic equations of the robot arm's velocity level are obtained as follows: (5) in, express The pseudo-inverse matrix.

[0028] Step 3: Calculate the singular value spectrum of the Jacobian matrix to obtain three types of kinematic singularity detection indices. Specifically, Step 31: The singular value spectrum is a Jacobian matrix The set of numerical distributions formed by arranging all singular values ​​in order of magnitude is used to calculate the singular value spectrum. Singular value decomposition yields: (6) (7) in, , and Let these represent the left singular vector matrix, the right singular vector matrix, and the singular value matrix, respectively. for singular values, for The zero vector of dimension, This represents the dimension of the operation space. This is the singular value spectrum of the Jacobian matrix under the current configuration; Step 32: The singular value spectrum reflects complete information about the robotic arm's motion capabilities in each principal motion direction under its current configuration. In existing technologies, it is used to separately... Minimum singular value in Minimum singular value With the maximum singular value The ratio of the singular values ​​and the product of all singular values ​​are defined as the minimum singular value index. The reciprocal index of the condition number and operability indicators All three are used to detect the kinematic singularities of robotic arms.

[0029] Step 4: Derive the normalized minimum singular value index to reflect the risk of motion degradation in the most dangerous direction. Specifically, Step 41: Minimum Singularity Index When used to detect kinematic singularities of robotic arms, it primarily reflects the risk of degradation in the weakest direction of motion at the end effector. The closer the value is to zero, the higher the risk of degradation in the weakest direction of motion, and the closer the robotic arm is to a kinematic singularity. However, The value of is easily affected by the dimensions of the robotic arm structure, the scale of the Jacobian matrix, and the selection of the dimensions of the robotic arm's end effector position and attitude. To address this issue, a threshold normalization method is used to optimize the minimum singular value index. Perform dimensionless normalization: (8) (9) in, The normalized minimum singularity index reflects the risk of degradation of movement capability in the most dangerous direction. The closer its value is to 0, the closer the robotic arm is to a singular state. This means truncating the result to the interval. Operators in This represents the danger threshold of the normalized minimum singular value index. The safe threshold for the normalized minimum singularity index; For a very small positive number, for example, take Its function is to prevent the denominator from equaling 0; Step 42: Apply the normalized minimum singular value index derived in Step 41 in practice. At this time, the danger threshold of the normalized minimum singular value index needs to be known. Safety threshold for normalized minimum singular value index To reasonably determine the values ​​of these two parameters, an offline calibration method using workspace sampling was adopted. First, random sampling was performed within the joint space of the robotic arm. Position vector of joint variables , N Let be natural numbers, and each set of position vectors corresponds to a set of robotic arm configurations. Then, use the expression derived in step 21 to calculate the Jacobian matrix of each configuration, and use the singular value decomposition method in step 31 to find the minimum singular value of each configuration. Finally, for indivual The sample was composed of [variables], and the percentile method in statistics was used to determine [the percentiles]. and Value: (10) (11) in, Indicates the 5th percentile. This represents the 80th percentile.

[0030] Step 5: Derive the reciprocal index of the normalized condition number, which reflects the risk of ellipsoidal distortion in operability. Specifically, Step 51: Reciprocal index of the condition number When used to detect kinematic singularities of robotic arms, it primarily reflects the risk of distortion in the ellipsoidal shape of the robotic arm's end effector maneuverability. The closer the value is to zero, the higher the risk of distortion in the operability ellipsoid shape, and the closer the robotic arm is to a singular state. However, It only reflects the ratio between the minimum and maximum singular values, resulting in a detection blind zone when used to detect kinematic singularities of robotic arms. To facilitate... Integrating with other risk indicators determined by singular value spectra to improve detection performance, the reciprocal of the condition number index... The following processing is performed using a threshold normalization method: (12) in, The reciprocal of the normalized condition number reflects the risk of distortion of the operability ellipsoid shape. The closer its value is to 0, the closer the robotic arm is to a singular state. The danger threshold is represented by the reciprocal of the normalization condition number. The safety threshold representing the reciprocal of the normalization condition number; Step 52: Use the reciprocal index of the normalized condition number derived in Step 51. At that time, it is necessary to know and The appropriate values ​​for both were determined using a workspace sampling offline calibration method.

[0031] The calculation in step four two is obtained Minimum singular value of the configuration of the robotic arm Based on this, continue to calculate Maximum singularity of configuration Then calculate the reciprocal index value of the condition number corresponding to each configuration. Finally, for indivual The sample was composed of [variables], and the percentile method in statistics was used to determine [the percentiles]. and Value: (13) (14) Step Six: Derive the normalized overall athletic ability index to reflect the risk of overall athletic ability decline. Specifically, Step 61: Operability Indicators When used to detect kinematic singularities, it primarily reflects the risk of degradation in the overall motion capability of the robotic arm in all directions of the task space. The closer the value is to 0, the higher the risk of overall motion capacity degradation, and the closer the robotic arm is to a kinematic singularity. However, in practical use... At that time, the following problems will inevitably be encountered: On the one hand, Defining it as a direct product of all singular values ​​can easily lead to problems with values ​​that are too large or too small; on the other hand... Related to all singular values, its numerical range varies significantly with the number of degrees of freedom of the robotic arm and the scale of the Jacobian matrix.

[0032] To address the aforementioned issues, the geometric mean of the product of all singular values ​​is first used to equivalently characterize the overall motion capability of the robotic arm, and this is defined as the overall motion capability index. Then, the threshold normalization method was used to analyze the overall athletic performance index. Perform dimensionless normalization: (15) in, The value of the normalized overall motion capability index is as follows: the lower the value, the higher the risk of the overall motion capability of the robotic arm. A value of 0 indicates that the robotic arm is in a singular state. and These represent the danger threshold and the safety threshold of the normalized overall athletic ability index, respectively. Step 62: Use the normalized overall athletic ability index derived in Step 61. At this time, the danger threshold of the normalized overall athletic ability index needs to be known. Safety threshold of normalized overall athletic ability index The appropriate values ​​for both can also be determined using the workspace sampling offline calibration method. This is calculated in step four-two. After obtaining the Jacobian matrix of the robot arm configuration, the singular value spectrum of each configuration is obtained using the singular value decomposition method in step 3.1. Next, calculate the overall motion capability index value corresponding to each configuration. Finally, for indivual The sample was composed of [variables], and the percentile method in statistics was used to determine [the percentiles]. and Value: (16) (17) Step 7: Dynamically weight and fuse the three types of singularity risks to construct a singular value spectrum risk fusion index. Specifically, Step 71: As can be seen from the descriptions in Steps 4 to 6, the minimum singular value index... The reciprocal index of the condition number With overall athletic ability indicators Different spectral features from the singular value spectrum are used to detect the kinematic singularities of the robotic arm. However, since none of these indicators fully utilize all spectral features in the singular value spectrum, using any single indicator alone to detect kinematic singularities can easily lead to inaccurate, untimely, and blind-zone detection problems. To address these issues, a threshold normalization method is used to... , and Based on processing to map them all to the same dimensionless interval, a fusion mechanism with bottleneck penalty characteristics and risk weight adjustment capabilities is adopted to normalize the minimum singular value index. The reciprocal index of the normalized condition number and normalized overall athletic performance index The three types of kinematic singularity risks reflected are fused to construct a singularity value spectrum risk fusion index that can be used to comprehensively detect the kinematic singularities of robotic arms, as shown below: (18) (19) (20) (twenty one) in, This represents the singular value spectrum risk fusion index. The closer its value is to 1, the closer the current configuration of the robotic arm is to a kinematic singular state. The closer its value is to 0, the safer the robotic arm configuration is. , and These respectively represent the risk of deterioration in mobility in the most dangerous direction, the risk of distortion of the operability ellipsoid shape, and the risk of overall mobility decline. , and , , and These represent the weighting coefficients of the three types of risks mentioned above, and they are related. ; This indicates the penalty index for shortcomings, which meets the following criteria. ; By employing a fusion mechanism with both short-board penalty characteristics and risk weight adjustment capabilities, a singularity spectrum risk fusion index is constructed that can comprehensively detect the kinematic singularities of a robotic arm. This index can simultaneously respond to the risks of motion capability degradation in the most dangerous direction, ellipsoidal distortion of operability, and overall motion capability decay. This overcomes the blind spot problem in existing technologies that use only minimum singularity, operability, or the reciprocal of the condition number as indicators to detect the kinematic singularities of a robotic arm, ultimately obtaining more comprehensive, accurate, and reliable singularity state detection results.

[0033] Step 72: Apply the singular value spectrum risk fusion index obtained in Step 71 in practice. When testing the kinematic singularities of a robotic arm, the short-board penalty index and weighting coefficients , and The value can be determined through offline calibration and performance optimization, or the default parameter value can be used directly. , , and This default parameter set ensures sensitivity to the risk of motion degradation in the most dangerous direction while also considering the risks of ellipsoidal shape distortion and overall motion capability decline. The penalty effect of addressing weaknesses improves the ability to respond to local high-risk factors.

[0034] Step 8: Use spectral risk fusion indicators to detect the singularity of the robotic arm and determine whether to trigger damping. As can be seen from the description of step 71, compared to operability, minimum singular value, or the reciprocal of the condition number, the singular value spectrum risk fusion index is superior. It can comprehensively detect the kinematic singularity risk of the current configuration of the robotic arm. Therefore, this index is used as the triggering basis for the damping factor in the damped least squares method and its improvement method, so as to avoid the kinematic singularity problem of the robotic arm with higher performance.

[0035] To achieve the above objectives, two singularity risk thresholds are first defined. , ,in This indicates that the damping factor begins to intervene at the threshold. This indicates the saturation threshold of the high-risk damping factor. and Satisfying Relationship: Then, based on the singular value spectrum risk fusion index Construct the global damping activation coefficients as shown below. : (twenty two) in, This represents a truncation function, whose function is to limit the output to a certain value. Inside; Furthermore, to ensure that the damping factor is derived from the singular value spectrum risk fusion index In actual triggering, design the smoothing function as shown below. : (twenty three) Combining formulas (22) and (23), we can see that: when hour, ,thereby , indicating that the damping factor is not triggered; when hour, This indicates that the damping factor is smoothly intervened; when hour, ,thereby This indicates that the maximum damping factor trigger state has been entered.

[0036] Step 9: Design an adaptive dual-threshold smoothing damping factor and construct a direction-selective damping pseudo-inverse matrix. Specifically, Step 91: Following the method in Step 8, utilize the singular value spectrum risk fusion index. After determining the trigger damping factor, the robotic arm enters a kinematic singularity avoidance state. Unlike existing technologies that uniformly set damping factors for all motion directions, this design sets corresponding damping factors based on the degree of degradation in each motion direction. Therefore, the following adaptive dual-threshold smoothing damping factor is designed: (twenty four) (25) (26) in, Indicates the first Damping factor in each direction of motion, This represents the maximum damping value in the pre-defined singular region. Indicates the first A smoothing function for each direction of motion. Represents the truncation function The result value of the action, Indicates the first Singular values ​​in each direction of motion, and These represent the danger threshold and the safe threshold for singular values, respectively. Combining formulas (24) to (26), it can be seen that for the first... One direction of movement: when hour, , This indicates that stronger damping is needed in that direction; when hour, , This indicates that no additional damping is needed in that direction; when hour, This indicates a smooth transition of damping in that direction; Step 92: In order to calculate the result of step 91... The values ​​of the damping factors in each direction of motion need to be known. and To achieve reasonable values ​​for both, the same workspace sampling offline calibration method as in step four, two, was used. The smallest singular value After forming the sample, the percentile method in statistics was used to determine the percentiles. and Value: (27) (28) in, Indicates the 20th percentile; Step 93: After calculating the damping factors corresponding to each direction of motion of the robotic arm according to Step 91 and Step 92, construct the pseudo-inverse matrix of the Jacobian matrix so as to apply these damping factors to each direction of motion.

[0037] To this end, a directional selective damping pseudo-inverse matrix is ​​constructed. Its expression is as follows: (29) Step 10: Solve the inverse kinematics solution for the robotic arm to avoid singularities, and control the robotic arm to achieve singularity avoidance motion. Specifically, Step 11: To obtain the singularity avoidance inverse kinematics solution for the robotic arm, the damped least squares method is used to avoid kinematic singularities. Using this method, the singularity avoidance inverse solution of the robotic arm velocity-level inverse kinematics equations derived in Step 22 is obtained as follows: (30) (31) (32) in, Representing the Jacobian matrix The fixed damping pseudo-inverse matrix, Indicates the damping factor. express The minimum singular value, and These represent the pre-set threshold for determining whether the robotic arm has reached a singular configuration and the maximum damping value in the singular region, respectively. From formulas (31) and (32), it can be seen that the damping least squares method... Triggering a unified damping factor This will apply the same damping to all motion directions of the robotic arm's end effector, potentially excessively suppressing non-singular direction motion, thereby reducing end effector motion capability and pose tracking accuracy. Furthermore, as shown in formula (32), The value is determined using a hard threshold triggering mechanism, i.e., when... hour , and when hour ,lead to When the robotic arm approaches a singular configuration, a step abrupt change from zero to a non-zero value will occur, which can easily lead to problems such as sudden changes in joint velocity and motion jitter. In contrast, the adaptive dual-threshold smoothing damping factor designed in step 91 can effectively avoid the problems caused by the hard threshold triggering mechanism mentioned above, and is conducive to improving the continuity of joint velocity and control smoothness. Step 12: To overcome the problems of the damped least squares method in solving the singular inverse kinematics of the robotic arm, the direction-selective damping pseudo-inverse matrix constructed in Step 93 is used. Replace the formula (30) Thus, the singular inverse solution of the inverse kinematics equations for the robot arm's velocity level is obtained as follows: (33) As can be seen from formula (33), the singularity avoidance inverse solution method based on the direction-selective damping pseudo-inverse matrix can apply large damping only to the singular direction of motion, while maintaining small damping or not introducing damping for the non-singular direction. Therefore, compared with the damped least squares method, it can improve the motion flexibility and stability of the robotic arm in the singularity region while ensuring that the end effector tracks the task trajectory with higher accuracy. Step 13: Solve for the velocity vectors of the joint variables of the robotic arm obtained in Step 12. Then, by integrating it with respect to time, the joint position vector can be calculated. Then, servo control is used to make each joint of the robotic arm track precisely. By determining the angles corresponding to each joint, the robotic arm can be controlled to achieve singularity avoidance. Since all kinematic singularity avoidance methods based on Jacobi damping achieve singularity avoidance by sacrificing end-effector pose tracking accuracy, the performance of singularity avoidance methods can be verified by analyzing the pose tracking accuracy of the robotic arm's end effector on the desired trajectory in Cartesian space during singularity avoidance.

[0038] To achieve the above analytical objectives, a function for calculating the end-effector position tracking error is derived. and attitude tracking error function The expression. Assume the joint position vector and end-effector pose matrix corresponding to the current configuration of the robotic arm are respectively expressed using... and It means that among them By Substituting these equations into the positive kinematics equations of the robotic arm's position level derived in step one, we can calculate the result as follows: The end-effector pose matrix corresponding to the desired configuration of the robotic arm is given as a known quantity and can be used... Indicate, then and The following formulas were used to calculate the results: (34) (35) in, , and The end coordinate systems representing the current configuration of the robotic arm are respectively... of , and Axis in base coordinate system The unit vector represented in the figure. This represents the position vector of the origin of the coordinate system at the end of this configuration relative to the base coordinate system; , and The end coordinate systems representing the desired configuration of the robotic arm are respectively... of , and Axis in base coordinate system The unit vector represented in the figure. This represents the position vector of the origin of the coordinate system at the end of this configuration relative to the base coordinate system.

[0039] In this embodiment, the singular value spectrum risk fusion index is applied to the kinematic singularity avoidance problem of the robotic arm as the triggering basis for the damping factor in the singularity avoidance method. At the same time, an adaptive double threshold smoothing damping factor is designed according to the degree of degradation of each motion direction of the robotic arm, and targeted damping suppression is only implemented for the singular motion direction. Then, the direction-selective damping pseudo-inverse matrix of the Jacobian matrix is ​​constructed, and finally it is used to solve the singularity avoidance inverse kinematic solution of the inverse kinematic equation of the robotic arm velocity level, thereby controlling the robotic arm to achieve high-performance singularity avoidance motion.

[0040] Specifically, with 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 John J. Craig method, as follows: Figure 1 As shown, the coordinate system and coordinate system These are the base coordinate system and the end effector's coordinate system, respectively. For the first The fixed coordinate system of each joint. Based on the above settings, the position-level forward kinematics equations of the robotic arm are derived as follows:

[0041] in, The position vector representing the joint variables of the robotic arm. Indicates the first The position variables of each joint This is the pose matrix representing the end effector coordinate system relative to the base coordinate system; Combination Figure 1 As shown in Table 1, the DH parameters corresponding to the forward kinematics model of the robotic arm are obtained by combining the parameter values ​​in Table 1 with the position vectors of the joint variables corresponding to the current configuration of the robotic arm. Substituting these values ​​into the derived position-level forward kinematics equations, the pose matrix is ​​calculated. The specific value.

[0042] Table 1. Robotic Arm DH Parameters

[0043] Step Two: Combining Figure 1 The DH parameters of the robotic arm in Table 1 can be derived from formulas (2) and (3). Figure 1 robotic arm relative to the base coordinate system Jacobian matrix The analytical expression for its velocity level is derived, and its forward kinematic equations are obtained as follows:

[0044] when When the matrix is ​​a full-rank matrix, the pseudo-inverse method is used to obtain... Figure 1 Solving for the velocity vector of joint variables in the forward kinematics equations of the robotic arm's velocity level The inverse kinematic equations for the velocity level of the robotic arm can be obtained as follows:

[0045] Step 3: Combining Figure 1 Compared with the DH parameters in Table 1, Figure 1Substituting the position vector of the joint variable corresponding to the current configuration of the robotic arm into the derivation obtained in step two... In the analytical expression, it can be calculated that The specific value is then used to calculate the value using formula (4). Perform singular value decomposition to calculate the left singular vector matrix. Right singular vector matrix and singular value matrix Finally All the singular values ​​are arranged in order of magnitude to form a numerical distribution. This is the Jacobian matrix of the robotic arm. The singular value spectrum; This reflects complete information about the robotic arm's motion capabilities in each principal motion direction under its current configuration, including the minimum singular value. , With the maximum singular value The ratio of the singular values ​​and the product of all singular values , respectively, correspond to the minimum singular value index The reciprocal index of the condition number and operability indicators All three are used to detect the kinematic singularities of robotic arms.

[0046] Step 4: Combining Figure 1 Based on the robot arm DH parameters in Table 1, derive... Figure 1 Normalized minimum singular value index for robotic arms.

[0047] First, random sampling is performed within the joint space of the robotic arm. Position vector of joint variables Each set of position vectors corresponds to a set of robotic arm configurations; then, the Jacobian matrix of each set of robotic arm configurations is calculated using formulas (2) and (3). And use formula (6) to perform singular value decomposition to obtain the minimum singular value. Then on indivual The constituent samples were determined using the percentile method shown in formulas (10) and (11). and Finally, substituting the two thresholds into formulas (8) and (9) yields the results. Figure 1 The normalized minimum singular value index of the robotic arm is:

[0048] Among them, the normalized minimum singular value index It reflects the risk of degradation in the most dangerous direction of movement. The closer its value is to 0, the closer the robotic arm is to a singular state. It is a very small positive number, which in this embodiment is 1. .

[0049] Step 5: Combining Figure 1 Based on the robot arm DH parameters in Table 1, derive... Figure 1 The reciprocal of the normalized condition number of the robotic arm.

[0050] First, the calculation in step four is obtained Minimum singular value of the configuration of the robotic arm Based on this, the maximum singular value of all configurations is calculated. Then calculate the reciprocal index value of the condition number corresponding to each configuration. Then on indivual The constituent samples were determined using the percentile method shown in formulas (13) and (14). and Finally, substituting the two thresholds into formula (12) yields the result. Figure 1 The reciprocal of the normalized condition number of the robotic arm is:

[0051] Among them, the reciprocal index of the normalized condition number It reflects the risk of ellipsoidal distortion in operability; the closer its value is to 0, the closer the robotic arm is to a singular state.

[0052] Step Six: Combining Figure 1 Based on the robot arm DH parameters in Table 1, derive... Figure 1 Normalized overall motion capability index of robotic arm.

[0053] Calculated in step four Jacobian matrix of the robot arm configuration Then, the singular value spectrum of each configuration is calculated using formula (6). Then calculate the overall motion capability index value corresponding to each configuration. Then on indivual The threshold is determined using the percentile method shown in formulas (16) and (17) for the sample. and Finally, substituting the two thresholds into formula (15) yields the result. Figure 1 The normalized overall motion capability index of the robotic arm is:

[0054] in, The value is a normalized overall motion capability index. The lower the value, the higher the risk of the overall motion capability of the robotic arm. A value of 0 indicates that the robotic arm is in a strange state.

[0055] Step Seven: Combining Steps Four, Five, and Six, we can construct... Figure 1 Singularity spectrum risk fusion index for robotic arms.

[0056] After obtaining the normalized index , and Subsequently, a fusion mechanism with weakest link penalty characteristics and risk weight adjustment capabilities is adopted to fuse the three types of kinematic singularity risks reflected by the three indicators, constructing a singularity value spectrum risk fusion index that can comprehensively detect the kinematic singularities of the robotic arm:

[0057] in, The value is a singular value spectrum risk fusion index. The closer its value is to 1, the closer the current configuration of the robotic arm is to a kinematic singular state. The closer its value is to 0, the safer the configuration of the robotic arm. , and These represent the risk of degradation of motion capability in the weakest direction, the risk of distortion of the operability ellipsoid shape, and the risk of overall motion capability decay, respectively. , and These represent the weighting coefficients for the three types of risks mentioned above; This indicates the penalty index for weaknesses.

[0058] Step 8: Combining Figure 1 Together with step seven, the resulting singular value spectrum risk fusion index Applied to Figure 1 The kinematic singularity avoidance problem of robotic arms serves as the triggering basis for the damping factor in singularity avoidance methods.

[0059] Specifically, the global damping activation coefficient is first obtained using formula (22). The expression:

[0060] in, This represents the truncation function; This indicates the threshold at which the damping factor begins to intervene; in this embodiment, the value is [value to be filled in]. ; This represents the saturation threshold of the high-risk damping factor, with a value of [value missing]. ; Then, to ensure that the damping factor is fused from the singular value spectrum risk index In actual triggering, the smoothing function is obtained using formula (23). The expression:

[0061] Now, combined and From the expression, we can see that for Figure 1 Arbitrary configuration of the robotic arm: when hour, , indicating that the damping factor is not triggered; when hour, This indicates that the damping factor is smoothly intervened; when hour, This indicates that the maximum damping factor trigger state has been entered.

[0062] Step Nine: Combining Figure 1 In accordance with the DH parameters in Table 1 and step eight, the indicators are used. After determining the trigger damping factor, the robotic arm enters a kinematic singularity avoidance state.

[0063] To ensure high-performance singularity avoidance control, the adaptive dual-threshold smoothing damping factor is first obtained using equations (24) to (26). The expression is:

[0064]

[0065] in, This represents the maximum damping value in the pre-defined singular region, which in this embodiment is taken as [value]. ; and These represent the danger threshold and the safety threshold for singular values, respectively, and are calculated in step four. indivual After assembling the sample, formulas (27) and (28) can be used to determine the... and ; Then, in order to obtain the damping factor mentioned above Apply to Figure 1 The direction-selective damping pseudo-inverse matrix of the Jacobian matrix is ​​obtained by constructing the Jacobian matrix for each direction of motion of the robotic arm using formula (29). :

[0066] Step 10: Combining Figure 1 and Figure 2 As shown, to verify the effectiveness and advancement of the singularity avoidance method based on spectral risk fusion and selective damping proposed in this embodiment, a control... Figure 1The task of a robotic arm to track a straight path in its Cartesian space, such as Figure 2 As shown. In this task, the robotic arm maintains the orientation of the end effector constant, moving from a position relative to the base coordinate system. The coordinates are represented as The starting point, along Figure 2 The straight path moves to the coordinates of The endpoint position. To ensure the robotic arm can perform tasks safely, reliably, and with high precision, any potential singular configurations encountered along the path must be avoided; Combination Figures 3 to 5 As shown, the method based on the pseudo-inverse of the Jacobian matrix, as shown in formula (5), is first used to control the robotic arm to track a straight path. Figure 3 This demonstrates the results obtained using this method. Figure 1 The joint velocity trajectory of the robotic arm shows that the robotic arm encountered a kinematic singularity at approximately 32 seconds, at which point the joint velocity increased sharply, causing the task to fail. Then, the same path-tracking task is performed using the damped least squares method shown in equations (30) to (32), with the relevant parameters in the method set to... and , Figure 4 This demonstrates the results obtained using this method. Figure 1 The joint velocity trajectories of the robotic arm show continuity and smoothness near singular configurations, indicating that the damped least squares method can be applied by introducing a fixed damping term. Suppressing the joint velocity amplification phenomenon that occurs when the Jacobian matrix approaches singularity, enabling the robotic arm to pass through singularity regions more smoothly; Finally, the same straight path tracing task is performed using the singularity avoidance method proposed in this embodiment, as shown in formula (33). Figure 5 This demonstrates the results obtained using this method. Figure 1 The results show that the method of this embodiment can ensure that the velocity trajectories of each joint are continuous and smooth even when the robotic arm is close to a kinematic singular configuration. This indicates that the method of this embodiment can effectively suppress the influence of kinematic singularity on joint movement and ensure that the robotic arm completes the preset path tracking task.

[0067] Further comparison Figure 4 and Figure 5It can be seen that, under the same path tracking task, the joint velocity amplitude obtained by the method in this embodiment is generally smaller than that of the damped least squares method, especially near the kinematic singular configuration, where the peak value of the joint velocity is lower and the change is more gradual. This result indicates that, compared to the damped least squares method which uniformly introduces damping in all motion directions, the method in this embodiment, by applying strong damping only to singular motion directions and applying less damping or no damping to non-singular motion directions, can effectively reduce the risk of joint velocities reaching or exceeding physical limits while ensuring end-effector tracking capability and joint velocity continuity, thereby improving the smoothness and reliability of the robotic arm motion control process.

[0068] Combination Figure 6 and Figure 7 As shown, by comparing the pose tracking errors of the robotic arm end effector when performing the same task using different singularity avoidance methods, the performance of the singularity avoidance method can be verified. Figure 6 The damping least squares method calculated using formula (34) and the method of this invention are demonstrated in tracking... Figure 3 The results obtained during the task of the straight path shown Figure 1 Comparison results of end-effector position tracking errors. Figure 7 The comparison results of the end attitude tracking errors of the two methods calculated using formula (35) are shown. Figure 6 and Figure 7 Solid lines represent results obtained by the method of this invention, while dashed lines represent results obtained by the damped least squares method. Figure 6 and Figure 7 It is evident that, because the damped least squares method employs a uniform scalar damping factor, applying the same damping to all directions of motion, it excessively suppresses the motion capability in non-singular directions, leading to... Figure 1 The pose tracking accuracy of the robotic arm's end effector decreased significantly. In contrast, the method in this embodiment employs a strategy of selectively introducing damping in singular directions, thereby achieving higher end effector pose tracking accuracy.

[0069] Compared with the singularity avoidance methods widely used in the prior art, the method proposed in this embodiment can effectively suppress the amplification of joint velocity near singular configurations while reducing the impact on non-singular motion directions, thereby taking into account the stability, flexibility and end-effector trajectory tracking accuracy of the robotic arm, which is conducive to ensuring that the robotic arm performs its tasks accurately, stably and safely.

[0070] Example 2 This embodiment discloses a singularity avoidance system for a robotic arm based on spectral risk fusion and selective damping, comprising a computing device that performs the following: Obtain the Jacobian matrix of the robotic arm, calculate the singular value spectrum of the Jacobian matrix, and obtain three types of kinematic singularity detection indices; Obtain the normalized minimum singular value index, the reciprocal of the normalized condition number index, and the normalized overall motion capability index; Dynamically weighted fusion of three types of singularity risks is used to construct a singular value spectrum risk fusion index. The singularity of the robotic arm is detected by spectral risk fusion index to determine whether damping is triggered: a global damping activation coefficient is constructed based on the damping factor intervention threshold, the high-risk damping factor saturation threshold and a positive number. A smoothing function is designed based on the global damping activation coefficient, and the value of the smoothing function is used to determine whether damping is triggered. An adaptive dual-threshold smoothing damping factor is designed, and a direction-selective damping pseudo-inverse matrix is ​​constructed: based on the pre-defined maximum damping value in the singular region, the smoothing function, and the first... The smoothing function of the nth motion direction is used to determine the th The damping factor in the direction of motion, according to the first... The singular values ​​of the i-th direction of motion, the left singular vector matrix, the right singular vector matrix, and the i-th... The damping factors in each motion direction construct a direction-selective damping pseudo-inverse matrix; Solve the inverse kinematics solution for the robotic arm to avoid singularities, and control the robotic arm to achieve singularity avoidance motion.

[0071] Specifically, see Embodiment 1 for relevant details, and the computing device is specifically a computer.

[0072] 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 singularity avoidance method for robotic arms based on spectral risk fusion and selective damping, characterized in that, Includes the following: Obtain the Jacobian matrix of the robotic arm, calculate the singular value spectrum of the Jacobian matrix, and obtain three types of kinematic singularity detection indices; Obtain the normalized minimum singular value index, the reciprocal of the normalized condition number index, and the normalized overall motion capability index; Dynamically weighted fusion of three types of singularity risks is used to construct a singular value spectrum risk fusion index. The singularity of the robotic arm is detected by spectral risk fusion index to determine whether damping is triggered: a global damping activation coefficient is constructed based on the damping factor intervention threshold, the high-risk damping factor saturation threshold and a positive number. A smoothing function is designed based on the global damping activation coefficient, and the value of the smoothing function is used to determine whether damping is triggered. An adaptive dual-threshold smoothing damping factor is designed, and a direction-selective damping pseudo-inverse matrix is ​​constructed: based on the pre-defined maximum damping value in the singular region, the smoothing function, and the first... The smoothing function of the nth motion direction is used to determine the th The damping factor in the direction of motion, according to the first... The singular values ​​of the i-th direction of motion, the left singular vector matrix, the right singular vector matrix, and the i-th... The damping factors in each motion direction construct a direction-selective damping pseudo-inverse matrix; Solve the inverse kinematics solution for the robotic arm to avoid singularities, and control the robotic arm to achieve singularity avoidance motion.

2. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 1, characterized in that, The global damping activation coefficient is obtained according to the following formula: in, This represents a truncation function, whose function is to limit the output to a certain value. Inside, This represents the singular value spectrum risk fusion index. This indicates that the damping factor begins to intervene at the threshold. This indicates the saturation threshold of the high-risk damping factor. Represent a positive number; Determine the smoothing function : ； when hour, ,thereby , indicating that the damping factor is not triggered; when hour, This indicates that the damping factor is smoothly intervened; when hour, ,thereby This indicates that the maximum damping factor trigger state has been entered.

3. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 2, characterized in that, The adaptive dual-threshold smoothing damping factor is determined according to the following formula: in, Indicates the first Damping factor in each direction of motion, This represents the maximum damping value in the pre-defined singular region. Indicates the first A smoothing function for each direction of motion. Represents the truncation function The result value of the action, Indicates the first Singular values ​​in each direction of motion, and These represent the danger threshold and the safe threshold for singular values, respectively. For the first... One direction of movement: when hour, , This indicates that stronger damping is needed in that direction; when hour, , This indicates that no additional damping is needed in that direction; when hour, This indicates a smooth transition of damping in that direction; Construct the directional selective damping pseudo-inverse matrix according to the following formula. : ； in, , Let represent the left singular vector matrix and the right singular vector matrix, respectively.

4. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 1, characterized in that, The process of solving the singularity avoidance inverse kinematics solution for the robotic arm and controlling the robotic arm to achieve singularity avoidance motion includes the following: The singular inverse solution of the robot arm's velocity-level inverse kinematics equations obtained by using the damped least squares method is as follows: in, The velocity vector represents the joint variable. This indicates the robotic arm end effector relative to the base coordinate system. The Cartesian velocity vector is represented. Representing the Jacobian matrix The fixed damping pseudo-inverse matrix, Indicates the damping factor. express The minimum singular value, This represents a pre-set threshold used to determine whether the robotic arm has reached a singular configuration. This represents the maximum damping value in the singular region.

5. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 1, characterized in that, The normalized minimum singular value index is obtained based on the following: The danger threshold of the normalized minimum singularity index is determined using the percentile method in statistics. Safe threshold for normalized minimum singular value index ; Threshold normalization method is used for the minimum singular value index Perform dimensionless normalization: in, The normalized minimum singular value index; This indicates that the result will be truncated to an interval. Operators in It represents a positive number.

6. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 1, characterized in that, The reciprocal of the normalized condition number is obtained based on the following: The danger threshold of the reciprocal index of the normalized condition number is determined using the percentile method in statistics. The safety threshold of the reciprocal of the normalized condition number. ; The reciprocal index of the condition number is obtained by using the threshold normalization method. Perform the following processing: in, The reciprocal of the normalization condition number is the index. It represents a positive number.

7. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 1, characterized in that, The normalized overall athletic ability index is obtained based on the following: The danger threshold of the normalized overall athletic ability index was determined using the percentile method in statistics. Safety threshold of normalized overall athletic ability index ; Threshold normalization method is used to analyze the overall athletic performance index Perform the following processing: in, To normalize the overall athletic performance index, It represents a positive number.

8. The method for avoiding singularities in a robotic arm based on spectral risk fusion and selective damping according to claim 1, characterized in that, The dynamic weighted fusion of the three types of singularity risks constructs a singular value spectrum risk fusion index, including the following: Constructing a singular value spectrum risk fusion index: （19） （20） （21） in, This represents a singular value spectrum risk fusion index; , and These respectively represent the risk of deterioration in mobility in the most dangerous direction, the risk of distortion of the operability ellipsoid shape, and the risk of overall mobility decline. , and These represent the weighting coefficients for the three types of risks mentioned above. This indicates the penalty index for weaknesses.

9. The singularity avoidance method for robotic arms based on spectral risk fusion and selective damping according to claim 1, characterized in that, The process involves obtaining the Jacobian matrix of the robotic arm, calculating the singular value spectrum of the Jacobian matrix, and obtaining three types of kinematic singularity detection indices, including the following: A forward kinematics model of the robotic arm is established, and the position-level forward kinematics equations are obtained; The Jacobian matrix of the robotic arm is derived, and the forward and inverse kinematic equations of the velocity level are obtained. The singular value spectrum of the Jacobian matrix is ​​calculated to obtain three types of kinematic singularity detection indices: The singular value spectrum is a Jacobian matrix A numerical distribution formed by arranging all singular values ​​in order of magnitude, for Singular value decomposition yields: in, Represents a singular value matrix. for singular values, for The zero vector of dimension, This represents the dimension of the operation space. This is the singular value spectrum of the Jacobian matrix under the current configuration.

10. A singularity avoidance system for a robotic arm based on spectral risk fusion and selective damping, characterized in that, Includes a computing device, which performs the following: Obtain the Jacobian matrix of the robotic arm, calculate the singular value spectrum of the Jacobian matrix, and obtain three types of kinematic singularity detection indices; Obtain the normalized minimum singular value index, the reciprocal of the normalized condition number index, and the normalized overall motion capability index; Dynamically weighted fusion of three types of singularity risks is used to construct a singular value spectrum risk fusion index. The singularity of the robotic arm is detected by spectral risk fusion index to determine whether damping is triggered: a global damping activation coefficient is constructed based on the damping factor intervention threshold, the high-risk damping factor saturation threshold and a positive number. A smoothing function is designed based on the global damping activation coefficient, and the value of the smoothing function is used to determine whether damping is triggered. An adaptive dual-threshold smoothing damping factor is designed, and a direction-selective damping pseudo-inverse matrix is ​​constructed: based on the pre-defined maximum damping value in the singular region, the smoothing function, and the first... The smoothing function of the nth motion direction is used to determine the th The damping factor in the direction of motion, according to the first... The singular values ​​of the i-th direction of motion, the left singular vector matrix, the right singular vector matrix, and the i-th... The damping factors in each motion direction construct a direction-selective damping pseudo-inverse matrix; Solve the inverse kinematics solution for the robotic arm to avoid singularities, and control the robotic arm to achieve singularity avoidance motion.

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