Approximation damping reciprocal function feature solving method and system for singular point avoidance

Abstract

The invention relates to the technical field of mechanical arm control, and discloses an approximation damping reciprocal function feature solving method and system for singular point avoidance. The solving method is applied to mechanical arm control equipment, and specifically comprises the following steps that S101, the geometrical configuration and joint parameters of a mechanical arm are obtained, a complete kinematics model frame is established, a forward kinematics equation is established through a Denavit-Hartenberg parameter method, the mapping relation between the pose of an end effector and joint variables is determined, and the pose of the end effector is determined; deriving a speed level Jacobian matrix of the mapping relation by adopting a differential transformation method; s102, performing algebraic analysis and decomposition on the obtained Jacobian matrix, identifying key algebraic factors causing matrix singularity, and establishing corresponding singularity factor libraries for different mechanical arm configurations; according to the adaptive damping reciprocal function constructed based on the composite exponential function, the numerical stability in a singular point neighborhood is ensured by optimizing parameter combination, and the problem of joint velocity kick caused by damping mutation is thoroughly eliminated.

Description

Technical Field

[0001] This invention relates to the technical field of robotic arm control, and more specifically, to a method and system for solving the characteristics of the approximation damped inverse function for singularity avoidance. Background Technology

[0002] Singularity avoidance in robotic arms is one of the core problems in the field of robot motion control. Currently, the mainstream technical approaches in this field mainly include damped least squares (DLS) and damped inverse function methods based on fixed thresholds.

[0003] Damped least squares avoids numerical divergence when the matrix approaches singularity by introducing a damping factor in the calculation of the pseudo-inverse of the Jacobian matrix. Its basic idea is to regularize small singular values ​​based on singular value decomposition to obtain a stable inverse matrix solution. This method improves the control performance near singular points to some extent, but its calculation process heavily relies on singular value decomposition, resulting in a heavy computational burden in real-time control. This is especially true for high-degree-of-freedom robotic arms or scenarios requiring high control frequencies, making it difficult to meet real-time requirements.

[0004] Another existing technique is the fixed-threshold damped reciprocal function method. This method does not directly perform singular value decomposition, but instead uses a smoothing function to replace the reciprocal terms of the singular factors in the traditional inverse matrix, thus avoiding numerical divergence. However, the damping parameters in this method are usually selected based on fixed thresholds or empirical values, lacking adaptability to changes in the robot arm's structural parameters and task states. Furthermore, it cannot analyze the specific algebraic source of the localization singularity, limiting its performance optimization in complex application scenarios. Overall, while existing techniques provide feasibility for singularity avoidance at the fundamental level, they still have significant limitations in computational efficiency, singularity localization accuracy, parameter adaptability, and motion smoothness.

[0005] Regarding the above-mentioned technologies, the following technical problems still exist: Computational complexity: DLS relies on singular value decomposition, which leads to low efficiency in real-time control; Singularity identification is lacking: the Jacobian matrix structure cannot be decomposed analytically, the specific algebraic source of singularities cannot be located, and it is difficult to optimize specific mechanisms such as planar two-bar linkages. Poor damping adaptability: The parameters of the existing damping inverse function have no adaptive capability and cannot match the changes in the robot arm's structural parameters and task state; Avoiding insufficient smoothness: Sudden changes in damping factor can easily cause velocity jumps, trajectory discontinuities, or terminal acceleration oscillations, affecting mission safety.

[0006] Based on the above, there is an urgent need for an innovative method that can balance computational efficiency, positioning accuracy, adaptability, and motion smoothness. Summary of the Invention

[0007] The purpose of this invention is to provide a method and system for solving the characteristics of the approximate damped reciprocal function for singularity avoidance. By using an analytical singularity factor identification method instead of traditional singular value decomposition, the computational complexity is significantly reduced, and millisecond-level real-time response is achieved. Based on the adaptive damped reciprocal function constructed from the composite exponential function, the numerical stability in the neighborhood of the singularity is ensured by optimizing the parameter combination, aiming to solve the problems in the prior art.

[0008] This invention is implemented as follows: a method for solving the approximate damping inverse function for singularity avoidance, applied to robotic arm control equipment, specifically includes the following steps: S101: Obtain the geometric configuration and joint parameters of the robotic arm, establish a complete kinematic model framework, construct the forward kinematic equations through the Denavit-Hartenberg parametric method, establish the mapping relationship between the end effector pose and joint variables, and derive the velocity-level Jacobian matrix of the mapping relationship using the differential transformation method. S102: Perform algebraic analytical decomposition on the obtained Jacobian matrix, identify the key algebraic factors that cause the matrix to be singular, establish a corresponding singular factor library for different robotic arm configurations, and set an adaptively adjustable singular neighborhood threshold for the singular factor library based on the dynamic performance requirements of the robotic arm to achieve accurate singularity state detection and early warning. S103: Design a continuously differentiable damped reciprocal function based on a composite exponential function. By optimizing the determined parameter combination, ensure that the differentiable damped reciprocal function is within the threshold of the singular point neighborhood. When the differentiable damped reciprocal function is far from the singular region, it approximates the singular terms in the traditional inverse matrix. When it is close to the singular point, it automatically and smoothly transitions to a finite value. S104: The constructed damping reciprocal function is systematically embedded into the Jacobian matrix inversion operation process. By replacing the singular factors in the traditional inverse matrix, a numerically stable modified inverse matrix is ​​formed, ensuring that feasible inverse kinematic solutions can be obtained throughout the entire workspace. S105: In the real-time control loop, the change of singular factor value is continuously monitored. Based on the singular point neighborhood threshold, the damping parameter adjustment is automatically triggered to complete the smooth switching and realize the transition from standard inverse kinematics solution to singular avoidance mode, so as to achieve efficient and smooth singular point avoidance control.

[0009] Furthermore, in S101, the geometric configuration and joint parameters of the robotic arm are obtained, a complete kinematic model framework is established, and the forward kinematic equations are constructed using the Denavit-Hartenberg parametric method, including: Based on the link length, joint offset, and torsion angle parameters of the robotic arm, a homogeneous transformation matrix sequence is constructed using the Denavit-Hartenberg parametric method to form the forward kinematic equations. , where q is the joint variable vector, including the rotational joint angle or the translational joint displacement, and x is the end effector pose vector, including the position coordinates and the pose matrix; The velocity-order Jacobian matrix is ​​derived using the differential transformation method. ,in Partial derivatives ensure that matrix elements are explicit functions of joint variables; For redundant robotic arms, the Jacobian matrix is ​​not a square matrix, so the effective sub-matrix needs to be determined by task space decomposition. After the model is established, the accuracy is verified by experimental calibration. The actual pose is measured using a laser tracker or vision system, and the geometric parameters are optimized by comparing it with the theoretical value.

[0010] Furthermore, in S102, the obtained Jacobian matrix is ​​subjected to algebraic analytical decomposition to identify the key algebraic factors that lead to the singularity of the matrix, including: The algebraic analytical decomposition is achieved by any one of the following methods: calculating the determinant of a matrix, analyzing the properties of submatrices, or extracting the roots of characteristic polynomials. The Jacobian matrix J(q) is analytically decomposed, and key algebraic factors are extracted by calculating the determinant det(J) or the determinant of the submatrix. ,in It is a scalar function that depends on the joint variable q; For a planar two-link robotic arm, the Jacobian matrix determinant is decomposed into det(J) = -L1L2sinq2, where L1 represents the length of the first link (from the robot's base to the rotation center of the second joint), and L2 represents the length of the second link (from the rotation center of the second joint to the end effector), thus identifying singular factors. =sinq2, when q2 = 0° or 180° →0 indicates a singular state; An adaptive singular neighborhood threshold ε is set and dynamically adjusted according to the robotic arm's motion state. When | When |≤ε, the system is determined to have entered a singular region. The singularity factor library is pre-built based on the robotic arm configuration, supports fast querying and real-time updates, and realizes accurate detection and early warning of singularity states.

[0011] Furthermore, an adaptive singular neighborhood threshold ε is set and dynamically adjusted according to the robotic arm's motion state. When | When |≤ε, the system is determined to have entered a singular region. A singularity factor library is pre-built based on the robotic arm configuration, supporting rapid querying and real-time updates, enabling accurate detection and early warning of singularity states, including: The dynamic adjustment of the adaptive singular neighborhood threshold ε is constructed based on the real-time motion state of the robotic arm end effector. By monitoring the end-effector velocity norm, acceleration rate of change, and task priority parameters, a functional mapping relationship between the threshold ε and the motion state is established. When the robotic arm is in high-speed motion or heavy load conditions, the ε value is automatically increased to provide early warning and enhance system stability. In precision operation or low-speed motion phases, the ε value is decreased accordingly to improve control accuracy. The singular factor library, pre-established based on the robotic arm configuration, adopts a hierarchical classification storage structure. It includes the unique singular factor analytical expressions and their corresponding singular configuration features for different types of robotic arms based on their topological configurations and kinematic characteristics. The singularity factor library supports fast querying and matching based on real-time joint angles, and continuously updates the parameterized representation of singularities through an online learning mechanism, achieving accurate singularity warnings from static configuration to dynamic evolution.

[0012] Furthermore, in S103, a continuously differentiable damped reciprocal function is designed based on a composite exponential function. The determined parameter combination is optimized to ensure that the differentiable damped reciprocal function remains within the singularity neighborhood threshold, including: Design a continuously differentiable damped reciprocal function based on a composite exponential function. First, construct the auxiliary function. ,in , The objective is determined through multi-objective optimization, which includes function smoothness, numerical stability, and computational efficiency. Define the reciprocal function of damping The inverse function of damping in When it is large, it is approximately equal to ,exist It converges to a finite value at →0°, where It is continuously differentiable, and both its first and second derivatives exist. a and b are calibrated online according to the structure and task requirements of the robotic arm, and different exotic scenarios are tested through simulation or experimentation to minimize joint speed error and end-effector tracking error.

[0013] Furthermore, in S104, the constructed damping reciprocal function is systematically embedded into the Jacobian matrix inversion process. By replacing the singular factors in the traditional inverse matrix, a numerically stable modified inverse matrix is ​​formed, including: The inverse function of damping The process of embedding the Jacobian matrix inverse calculation replaces the traditional inverse matrix calculation. The corrected inverse matrix is ​​calculated as follows: ,in, In matrix algebra operations, it is used as... Replace the corresponding The results were obtained using a numerically stable algorithm. The modified inverse matrix ensures numerical stability near singular points, while degenerating into a standard inverse matrix far from singular points, thus guaranteeing control accuracy and ensuring the feasibility and continuity of the inverse kinematics solution.

[0014] Furthermore, in S105, damping parameter adjustment is automatically triggered based on the singularity neighborhood threshold to complete a smooth switch, including: In the real-time control loop, singular factors are continuously monitored. The numerical change is automatically triggered by adjusting the damping parameters based on the preset singular neighborhood threshold ε; When | When |>ε, the damping decreases, and the algorithm degenerates into a standard inverse kinematics solution. The joint velocities are then calculated as follows: ,in For the desired terminal velocity; when | When |≤ε, the damping automatically increases, and the joint velocity is calculated using the corrected Jacobian inverse matrix: ; The control process integrates sensor feedback to correct joint status in real time, avoids sudden jumps in joint speed or discontinuous trajectory through smooth switching, and uses a low-pass filter to smooth the output. It supports multimodal operations, including trajectory scaling or joint redirection, to ensure end-point tracking accuracy and mission safety.

[0015] Furthermore, the control process integrates sensor feedback to correct joint states in real time, avoids sudden jumps in joint speed or discontinuous trajectory through smooth switching, and uses a low-pass filter to smooth the output, including: The sensor feedback integration mechanism achieves real-time correction of joint status through multi-source data fusion. It continuously collects encoder data, torque sensor readings, and end-effector visual positioning information installed on the joint motor. The extended Kalman filter algorithm is used to fuse the data and generate high-precision estimates of joint angle, speed, and load torque. This process operates in a kilohertz-level control loop, ensuring millisecond-level response to the dynamic state of the robotic arm, providing accurate and reliable real-time feedback for subsequent evasive control, thereby effectively compensating for model errors and external disturbances; Based on the continuous variation of singular factors, a non-disruptive transition logic is designed to achieve smooth switching. When the system is detected to be approaching the singular region, the controller gradually adjusts the weight ratio of the damping coefficient and the Jacobian inverse matrix according to the preset smoothing function to achieve a natural transition between the standard algorithm and the avoidance algorithm. At the same time, an adjustable low-pass filter is introduced at the joint speed command output to filter out high-frequency components caused by algorithm switching or sensor noise, and the phase response characteristics of the system are maintained through a feedforward compensation mechanism.

[0016] Compared with existing technologies, the approximation damping inverse function characteristic solution method and system for singularity avoidance provided by this invention have the following advantages: 1. By adopting an analytical singular factor identification method instead of traditional singular value decomposition, the computational complexity is significantly reduced, achieving millisecond-level real-time response; the adaptive damping reciprocal function constructed based on the composite exponential function ensures numerical stability in the neighborhood of singular points through optimized parameter combinations, completely eliminating the joint velocity jump problem caused by damping mutation; the singular factor library management mechanism and adaptive threshold adjustment strategy enable the system to accurately identify various singular configurations and achieve smooth transition control; 2. The performance boundaries of singularity avoidance for robotic arms have been expanded. The modular system architecture organically integrates kinematic modeling, singularity recognition, function construction, matrix calculation, and control execution. Multi-sensor fusion and real-time filtering technologies enhance the system's anti-interference capability. The parameterized damping function supports dynamic parameter tuning according to different robotic arm configurations, enhancing the method's versatility and adaptability. The hardware acceleration integration scheme ensures that the algorithm can meet the real-time requirements of highly dynamic scenarios while maintaining control accuracy, providing a complete solution for precise control of robotic arms under complex working conditions.

[0017] A system for solving the approximate damping inverse function characteristic for singularity avoidance, used to execute the above-described approximate damping inverse function characteristic solution method, the system comprising: The kinematic modeling module is used to construct the kinematic framework of the robotic arm. It establishes kinematic equations through parameter analysis and coordinate transformation, and generates velocity-level Jacobian matrices. The singular factor identification module is used to perform algebraic decomposition of the Jacobian matrix, extract singular factors and manage the singular factor library, and realize dynamic monitoring of singular states. The damping function construction module is used to generate continuously differentiable damping functions and ensures the smooth transition characteristics of the function in singular regions through parameter optimization. The inverse matrix calculation module is used to construct the corrected inverse matrix, and the stability of the matrix is ​​ensured through damping function replacement and numerical optimization. The control execution module is used to perform real-time avoidance control and smoothly switch motion modes through status monitoring and parameter adjustment.

[0018] Specifically, the control execution module includes: The data processing unit is used to integrate feedback information from multiple sensor sources and correct the joint state in real time through an extended Kalman filter algorithm; Transition unit, designed with non-perturbative transition logic based on continuous variation of singular factors; The function adjustment unit uses a smooth function to adjust the damping coefficient and the weight ratio of the Jacobian inverse matrix, and introduces an adjustable low-pass filter at the joint speed command output to ensure smooth joint speed and continuous trajectory. Attached Figure Description

[0019] Figure 1 This is a flowchart illustrating the method for solving the approximate damping inverse function characteristics for singularity avoidance proposed in this invention. Figure 2 The flowchart illustrates the process of setting an adaptive singular neighborhood threshold ε in the approximation damped reciprocal function feature solution method for singularity avoidance proposed in this invention, and the singularity factor library is pre-established based on the robotic arm configuration, supporting fast query and real-time update, to achieve accurate detection and early warning of singularity states. Figure 3 This is a schematic diagram comparing the approximate damping reciprocal curve with the traditional reciprocal curve in the approximate damping reciprocal function characteristic solution method for singular point avoidance proposed in this invention. Figure 4 This is a schematic diagram of the system for solving the characteristic of the approximation damped inverse function for singularity avoidance proposed in this invention. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0021] The implementation of the present invention will be described in detail below with reference to specific embodiments.

[0022] In the accompanying drawings of this embodiment, the same or similar reference numerals correspond to the same or similar components. In the description of this invention, it should be understood that if terms such as "upper," "lower," "left," and "right" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting this invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0023] Reference Figure 1-3 As shown, the approximation damping inverse function characteristic solution method for singularity avoidance, applied to robotic arm control equipment, specifically includes the following steps: S101: Obtain the geometric configuration and joint parameters of the robotic arm, establish a complete kinematic model framework, construct the forward kinematic equations through the Denavit-Hartenberg parametric method, establish the mapping relationship between the end effector pose and joint variables, and derive the velocity-level Jacobian matrix of the mapping relationship using the differential transformation method. Establishing a kinematic model of a robotic arm involves creating its forward kinematic equations based on its geometric parameters and joint configuration (whether rotational, translational, or hybrid). ,in, Let x be the joint variable and x be the pose vector of the end effector in the task space. Calculate its velocity Jacobian matrix: In the formula, The Jacobian matrix describes the mapping relationship between joint velocities and end-effector velocities; S102: Perform algebraic analytical decomposition on the obtained Jacobian matrix, identify the key algebraic factors that cause the matrix to be singular, establish a corresponding singular factor library for different robotic arm configurations, and set an adaptively adjustable singular neighborhood threshold for the singular factor library based on the dynamic performance requirements of the robotic arm to achieve accurate singularity state detection and early warning. Singularity detection and singular factor identification involve analytically decomposing the determinant, submatrix determinant, or equivalent algebraic form of the Jacobian matrix to identify the algebraic factors that lead to singularity, denoted as . When one or more When the system enters a singular state, for redundant robotic arms, an effective submatrix corresponding to the task dimension can be selected for discrimination; for arbitrary mechanisms, the general expression for the singularity factor can be used: It represents the neighborhood range of a singular region; S103: Design a continuous differentiable damped reciprocal function based on a composite exponential function. By optimizing the determined parameter combination, the differentiable damped reciprocal function is ensured to be within the threshold of the singular point neighborhood. When the differentiable damped reciprocal function is far from the singular region, it approaches the singular terms in the traditional inverse matrix and automatically and smoothly transitions to a finite value when it is close to the singular point. The construction of the damping reciprocal function is to avoid the divergence of the inverse solution in singular regions due to the singularity factor approaching zero. This invention proposes a continuously differentiable damping reciprocal function to replace the reciprocal term in the traditional inverse matrix.

[0024] For those appearing in the inverse matrix The damping reciprocal function is used in the following general form: :

[0025]

[0026] in , .

[0027] S104: The constructed damping reciprocal function is systematically embedded into the Jacobian matrix inversion operation process. By replacing the singular factors in the traditional inverse matrix, a numerically stable modified inverse matrix is ​​formed, ensuring that feasible inverse kinematic solutions can be obtained throughout the entire workspace. The calculation of the modified Jacobian inverse matrix involves embedding the aforementioned damping reciprocal function into the calculation process of the Jacobian matrix inverse, thus forming the modified inverse matrix:

[0028] in, In matrix algebra operations, it is used as... Replace the corresponding The results obtained.

[0029] S105: In the real-time control loop, continuously monitor the changes in the singular factor value, automatically trigger the adjustment of damping parameters based on the singular point neighborhood threshold, complete the smooth switching, realize the transition from standard inverse kinematics solution to singular avoidance mode, and achieve efficient and smooth singular point avoidance control. The singularity avoidance solution utilizes the modified Jacobian inverse to calculate joint velocities.

[0030] in, To achieve the desired end velocity, the system continuously detects singular factors during trajectory interpolation or real-time control loops. And adjust the damping factor in real time according to its size. This enables adaptive singularity avoidance.

[0031] In step S101 of this embodiment, the geometric configuration and joint parameters of the robotic arm are obtained, a complete kinematic model framework is established, and the forward kinematic equations are constructed using the Denavit-Hartenberg parametric method, including: Based on the link length, joint offset, and torsion angle parameters of the robotic arm, a homogeneous transformation matrix sequence is constructed using the Denavit-Hartenberg parametric method to form the forward kinematic equations. , where q is the joint variable vector, including the rotational joint angle or the translational joint displacement, and x is the end effector pose vector, including the position coordinates and the pose matrix; The velocity-order Jacobian matrix is ​​derived using the differential transformation method. ,in Partial derivatives ensure that matrix elements are explicit functions of joint variables; For redundant robotic arms, the Jacobian matrix is ​​not a square matrix, so the effective sub-matrix needs to be determined by task space decomposition. After the model is established, the accuracy is verified by experimental calibration. The actual pose is measured using a laser tracker or vision system, and the geometric parameters are optimized by comparing it with the theoretical value.

[0032] In S102 of this embodiment, the obtained Jacobian matrix is ​​subjected to algebraic analytical decomposition to identify key algebraic factors that lead to the singularity of the matrix, including: Algebraic analytic decomposition can be achieved by any of the following methods: calculating the determinant of a matrix, analyzing the properties of submatrices, or extracting the roots of characteristic polynomials. The Jacobian matrix J(q) is analytically decomposed, and key algebraic factors are extracted by calculating the determinant det(J) or the determinant of the submatrix. ,in It is a scalar function that depends on the joint variable q; For a planar two-link robotic arm, the Jacobian matrix determinant is decomposed into det(J) = -L1L2sinq2, where L1 represents the length of the first link (from the robot's base to the rotation center of the second joint), and L2 represents the length of the second link (from the rotation center of the second joint to the end effector), thus identifying singular factors. =sinq2, when q2 = 0° or 180° →0 indicates a singular state; An adaptive singular neighborhood threshold ε is set and dynamically adjusted according to the robotic arm's motion state. When | When |≤ε, the system is determined to have entered a singular region. A singularity factor library is pre-built based on the robotic arm configuration, supporting rapid querying and real-time updates, enabling accurate detection and early warning of singularity states, including: The dynamic adjustment of the adaptive singular neighborhood threshold ε is constructed based on the real-time motion state of the robotic arm end effector. By monitoring the end-effector velocity norm, acceleration rate of change, and task priority parameters, a functional mapping relationship between the threshold ε and the motion state is established. When the robotic arm is in high-speed motion or heavy load conditions, the ε value is automatically increased to provide early warning and enhance system stability. In precision operation or low-speed motion phases, the ε value is decreased accordingly to improve control accuracy. The singular factor library, pre-established based on the robotic arm configuration, adopts a hierarchical classification storage structure. It includes the unique singular factor analytical expressions and their corresponding singular configuration features for different types of robotic arms based on their topological configurations and kinematic characteristics. The singularity factor library supports fast querying and matching based on real-time joint angles, and continuously updates the parameterized representation of singularities through an online learning mechanism, achieving accurate singularity warnings from static configuration to dynamic evolution.

[0033] In S103 of this embodiment, a continuously differentiable damped reciprocal function is designed based on a composite exponential function. The optimized parameter combination ensures that the differentiable damped reciprocal function remains within the singularity neighborhood threshold, including: Design a continuously differentiable damped reciprocal function based on a composite exponential function. First, construct the auxiliary function. ,in , The objective is determined through multi-objective optimization, which includes function smoothness, numerical stability, and computational efficiency. Define the reciprocal function of damping The inverse function of damping in When it is large, it is approximately equal to ,exist It converges to a finite value at →0°, where It is continuously differentiable, and both its first and second derivatives exist. a and b are calibrated online according to the structure and task requirements of the robotic arm, and different exotic scenarios are tested through simulation or experimentation to minimize joint speed error and end-effector tracking error.

[0034] In S104 of this embodiment, the constructed damping reciprocal function is systematically embedded into the Jacobian matrix inversion operation process. By replacing the singular factors in the traditional inverse matrix, a numerically stable modified inverse matrix is ​​formed, including: The inverse function of damping The process of embedding the Jacobian matrix inverse calculation replaces the traditional inverse matrix calculation. The corrected inverse matrix is ​​calculated as follows: , among which, refer to Figure 3 The curve shown, In matrix algebra operations, it is used as... Replace the corresponding The results were obtained using a numerically stable algorithm. The modified inverse matrix ensures numerical stability near singular points, while degenerating into a standard inverse matrix far from singular points, thus guaranteeing control accuracy and ensuring the feasibility and continuity of the inverse kinematics solution.

[0035] In S105 of this embodiment, the damping parameter adjustment is automatically triggered based on the singularity neighborhood threshold to complete the smooth switching, including: In the real-time control loop, singular factors are continuously monitored. The numerical change is automatically triggered by adjusting the damping parameters based on the preset singular neighborhood threshold ε; When | When |>ε, the damping decreases, and the algorithm degenerates into a standard inverse kinematics solution. The joint velocities are then calculated as follows: ,in For the desired terminal velocity; when | When |≤ε, the damping automatically increases, and the joint velocity is calculated using the corrected Jacobian inverse matrix: ; The control process integrates sensor feedback to correct joint status in real time, avoids sudden jumps in joint speed or discontinuous trajectory through smooth switching, and uses a low-pass filter to smooth the output. It supports multimodal operations, including trajectory scaling or joint redirection, to ensure end-point tracking accuracy and mission safety.

[0036] This technical solution significantly reduces computational complexity and achieves millisecond-level real-time response by replacing traditional singular value decomposition with an analytical singular factor identification method. The adaptive damping reciprocal function constructed based on the composite exponential function ensures numerical stability in the neighborhood of singular points through optimized parameter combinations, completely eliminating the joint velocity jump problem caused by damping mutation. The singular factor library management mechanism and adaptive threshold adjustment strategy enable the system to accurately identify various singular configurations and achieve smooth transition control.

[0037] In this embodiment, the control process integrates sensor feedback to correct the joint state in real time, avoids sudden jumps in joint speed or discontinuous trajectory through smooth switching, and uses a low-pass filter to smooth the output, including: The sensor feedback integration mechanism achieves real-time correction of joint status through multi-source data fusion. It continuously collects encoder data, torque sensor readings, and end-effector visual positioning information installed on the joint motor. The extended Kalman filter algorithm is used to fuse the data and generate high-precision estimates of joint angle, speed, and load torque. This process operates in a kilohertz-level control loop, ensuring millisecond-level response to the dynamic state of the robotic arm, providing accurate and reliable real-time feedback for subsequent evasive control, thereby effectively compensating for model errors and external disturbances; Based on the continuous variation of singular factors, a non-disruptive transition logic is designed to achieve smooth switching. When the system is detected to be approaching the singular region, the controller gradually adjusts the weight ratio of the damping coefficient and the Jacobian inverse matrix according to the preset smoothing function to achieve a natural transition between the standard algorithm and the avoidance algorithm. At the same time, an adjustable low-pass filter is introduced at the joint speed command output to filter out high-frequency components caused by algorithm switching or sensor noise, and the phase response characteristics of the system are maintained through a feedforward compensation mechanism.

[0038] Reference Figure 3 A system for solving the approximate damping inverse function characteristics for singularity avoidance, used to execute the above-mentioned approximate damping inverse function characteristic solving method, the system includes: The kinematic modeling module constructs the kinematic framework of the robotic arm, establishing kinematic equations through parameter analysis and coordinate transformation, and generating a velocity-level Jacobian matrix. The singularity factor identification module performs algebraic decomposition of the Jacobian matrix, extracts singular factors, and manages the singularity factor library, enabling dynamic monitoring of singular states. The damping function construction module generates continuously differentiable damping functions, ensuring smooth transitions in singular regions through parameter optimization. The inverse matrix calculation module constructs a modified inverse matrix, ensuring matrix stability through damping function replacement and numerical optimization. The control execution module performs real-time avoidance control. By achieving smooth switching of motion modes through state monitoring and parameter adjustment, the performance boundaries of the robotic arm's singularity avoidance are expanded. The modular system architecture organically integrates kinematic modeling, singularity recognition, function construction, matrix calculation, and control execution. Multi-sensor fusion and real-time filtering technologies enhance the system's anti-interference capability. The parameterized damping function supports dynamic parameter adjustment according to different robotic arm configurations, enhancing the method's versatility and adaptability. The hardware acceleration integration scheme ensures that the algorithm can meet the real-time requirements of highly dynamic scenarios while maintaining control accuracy, providing a complete solution for precise control of robotic arms under complex working conditions.

[0039] In this embodiment, the control execution module includes: a data processing unit for integrating feedback information from multiple sensors and correcting joint states in real time using an extended Kalman filter algorithm; a transition unit for designing disturbance-free transition logic based on continuous changes in singular factors; a function adjustment unit for adjusting the damping coefficient and the weight ratio of the Jacobian inverse matrix using a smooth function, and introducing an adjustable low-pass filter at the joint speed command output to ensure smooth joint speed and continuous trajectory; an adaptive damping reciprocal function constructed based on a composite exponential function, which ensures numerical stability in the neighborhood of singular points through optimized parameter combinations, completely eliminating the problem of sudden joint speed jumps caused by damping mutations; and a singular factor library management mechanism and an adaptive threshold adjustment strategy, enabling the system to accurately identify various singular configurations and achieve smooth transition control.

[0040] In this embodiment, the entire operation process can be automated by computer control. In each operation stage, sensors can be set up to provide signal feedback and ensure that the steps are performed sequentially. These are all conventional knowledge of current automation control, and will not be elaborated on in this embodiment.

[0041] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for solving the characteristic of the approximate damped inverse function for singularity avoidance, characterized in that, Applied to robotic arm control equipment, it specifically includes the following steps: S101: Obtain the geometric configuration and joint parameters of the robotic arm, establish a complete kinematic model framework, construct the forward kinematic equations through the Denavit-Hartenberg parametric method, establish the mapping relationship between the end effector pose and joint variables, and derive the velocity-level Jacobian matrix of the mapping relationship using the differential transformation method. S102: Perform algebraic analytical decomposition on the obtained Jacobian matrix, identify the key algebraic factors that cause the matrix to be singular, establish a corresponding singular factor library for different robotic arm configurations, and set an adaptively adjustable singular neighborhood threshold for the singular factor library based on the dynamic performance requirements of the robotic arm to achieve accurate singularity state detection and early warning. S103: Design a continuously differentiable damped reciprocal function based on a composite exponential function. By optimizing the determined parameter combination, ensure that the differentiable damped reciprocal function is within the threshold of the singular point neighborhood. When the differentiable damped reciprocal function is far from the singular region, it approximates the singular terms in the traditional inverse matrix. When it is close to the singular point, it automatically and smoothly transitions to a finite value. S104: The constructed damping reciprocal function is systematically embedded into the Jacobian matrix inversion operation process. By replacing the singular factors in the traditional inverse matrix, a numerically stable modified inverse matrix is ​​formed, ensuring that feasible inverse kinematic solutions can be obtained throughout the entire workspace. S105: In the real-time control loop, the change of singular factor value is continuously monitored. Based on the singular point neighborhood threshold, the damping parameter adjustment is automatically triggered to complete the smooth switching and realize the transition from standard inverse kinematics solution to singular avoidance mode, so as to achieve efficient and smooth singular point avoidance control.

2. The method for solving the approximate damped inverse function characteristic for singularity avoidance as described in claim 1, characterized in that, In S101, the geometric configuration and joint parameters of the robotic arm are obtained, a complete kinematic model framework is established, and the forward kinematic equations are constructed using the Denavit-Hartenberg parametric method, including: Based on the link length, joint offset, and torsion angle parameters of the robotic arm, a homogeneous transformation matrix sequence is constructed using the Denavit-Hartenberg parametric method to form the forward kinematic equations. , where q is the joint variable vector, including the rotational joint angle or the translational joint displacement, and x is the end effector pose vector, including the position coordinates and the pose matrix; The velocity-order Jacobian matrix is ​​derived using the differential transformation method. ,in Partial derivatives ensure that matrix elements are explicit functions of joint variables; For redundant robotic arms, the Jacobian matrix is ​​not a square matrix, so the effective sub-matrix needs to be determined by task space decomposition. After the model is established, the accuracy is verified by experimental calibration. The actual pose is measured using a laser tracker or vision system, and the geometric parameters are optimized by comparing it with the theoretical value.

3. The method for solving the approximate damped inverse function characteristic for singularity avoidance as described in claim 2, characterized in that, In S102, the obtained Jacobian matrix is ​​subjected to algebraic analytical decomposition to identify the key algebraic factors that lead to the singularity of the matrix, including: The algebraic analytical decomposition is achieved by any one of the following methods: calculating the determinant of a matrix, analyzing the properties of submatrices, or extracting the roots of characteristic polynomials. The Jacobian matrix J(q) is analytically decomposed, and key algebraic factors are extracted by calculating the determinant det(J) or the determinant of the submatrix. ,in It is a scalar function that depends on the joint variable q; For a planar two-link robotic arm, the Jacobian matrix determinant is decomposed into det(J) = -L1L2sinq2, where L1 represents the length of the first link (from the robot's base to the rotation center of the second joint), and L2 represents the length of the second link (from the rotation center of the second joint to the end effector), thus identifying singular factors. =sinq2, when q2 = 0° or 180° →0 indicates a singular state; An adaptive singular neighborhood threshold ε is set and dynamically adjusted according to the robotic arm's motion state. When | When |≤ε, the system is determined to have entered a singular region. The singularity factor library is pre-built based on the robotic arm configuration, supporting fast query and real-time updates, and realizing accurate detection and early warning of singularity states.

4. The method for solving the approximate damped inverse function characteristic for singularity avoidance as described in claim 3, characterized in that, An adaptive singular neighborhood threshold ε is set and dynamically adjusted according to the robotic arm's motion state. When | When |≤ε, the system is determined to have entered a singular region. A singularity factor library is pre-built based on the robotic arm configuration, supporting rapid querying and real-time updates, enabling accurate detection and early warning of singularity states, including: The dynamic adjustment of the adaptive singular neighborhood threshold ε is constructed based on the real-time motion state of the robotic arm end effector. By monitoring the end-effector velocity norm, acceleration rate of change, and task priority parameters, a functional mapping relationship between the threshold ε and the motion state is established. When the robotic arm is in high-speed motion or heavy load conditions, the ε value is automatically increased to provide early warning and enhance system stability. In precision operation or low-speed motion phases, the ε value is decreased accordingly to improve control accuracy. The singular factor library, pre-established based on the robotic arm configuration, adopts a hierarchical classification storage structure. It includes the unique singular factor analytical expressions and their corresponding singular configuration features for different types of robotic arms based on their topological configurations and kinematic characteristics. The singularity factor library supports fast querying and matching based on real-time joint angles, and continuously updates the parameterized representation of singularities through an online learning mechanism, achieving accurate singularity warnings from static configuration to dynamic evolution.

5. The method for solving the approximate damped inverse function characteristic for singularity avoidance as described in claim 4, characterized in that, In S103, a continuously differentiable damped reciprocal function is designed based on a composite exponential function. The optimized parameter combination ensures that the differentiable damped reciprocal function remains within the singularity neighborhood threshold, including: Design a continuously differentiable damped reciprocal function based on a composite exponential function. First, construct the auxiliary function. ,in , The objective is determined through multi-objective optimization, which includes function smoothness, numerical stability, and computational efficiency. Define the reciprocal function of damping The inverse function of damping in When it is large, it is approximately equal to ,exist It converges to a finite value at →0°, where It is continuously differentiable, and both its first and second derivatives exist. a and b are calibrated online according to the structure and task requirements of the robotic arm, and different exotic scenarios are tested through simulation or experimentation to minimize joint speed error and end-effector tracking error.

6. The method for solving the approximate damped inverse function characteristic for singularity avoidance as described in claim 5, characterized in that, In S104, the constructed damping reciprocal function is systematically embedded into the Jacobian matrix inversion process. By replacing the singular factors in the traditional inverse matrix, a numerically stable modified inverse matrix is ​​formed, including: The inverse function of damping The process of embedding the Jacobian matrix inverse calculation replaces the traditional inverse matrix calculation. The corrected inverse matrix is ​​calculated as follows: ,in, In matrix algebra operations, it is used as... Replace the corresponding The results were obtained using a numerically stable algorithm. The modified inverse matrix ensures numerical stability near singular points, while degenerating into a standard inverse matrix far from singular points, thus guaranteeing control accuracy and ensuring the feasibility and continuity of the inverse kinematics solution.

7. The method for solving the approximate damped inverse function characteristic of singularity avoidance as described in claim 6, characterized in that, In S105, damping parameter adjustment is automatically triggered based on the singularity neighborhood threshold to achieve smooth switching, including: In the real-time control loop, singular factors are continuously monitored. The numerical change is automatically triggered by adjusting the damping parameters based on the preset singular neighborhood threshold ε; When | When |>ε, the damping decreases, and the algorithm degenerates into a standard inverse kinematics solution. The joint velocities are then calculated as follows: ,in For the desired terminal velocity; when | When |≤ε, the damping automatically increases, and the joint velocity is calculated using the corrected Jacobian inverse matrix: ; The control process integrates sensor feedback to correct joint status in real time, avoids sudden jumps in joint speed or discontinuous trajectory through smooth switching, and uses a low-pass filter to smooth the output. It supports multimodal operations, including trajectory scaling or joint redirection, to ensure end-point tracking accuracy and mission safety.

8. The method for solving the approximate damping inverse function characteristic for singularity avoidance as described in claim 7, characterized in that, The control process integrates sensor feedback to correct joint status in real time, avoids sudden jumps in joint speed or discontinuous trajectory through smooth switching, and uses a low-pass filter to smooth the output, including: The sensor feedback integration mechanism achieves real-time correction of joint status through multi-source data fusion. It continuously collects encoder data, torque sensor readings, and end-effector visual positioning information installed on the joint motor. It uses the extended Kalman filter algorithm to fuse the data and generate high-precision estimates of joint angle, speed, and load torque. This process operates in a kilohertz-level control loop, ensuring millisecond-level response to the dynamic state of the robotic arm, providing accurate and reliable real-time feedback for subsequent evasive control, thereby effectively compensating for model errors and external disturbances; Based on the continuous variation of singular factors, a non-disruptive transition logic is designed to achieve smooth switching. When the system is detected to be approaching the singular region, the controller gradually adjusts the weight ratio of the damping coefficient and the Jacobian inverse matrix according to the preset smoothing function to achieve a natural transition between the standard algorithm and the avoidance algorithm. At the same time, an adjustable low-pass filter is introduced at the joint speed command output to filter out high-frequency components caused by algorithm switching or sensor noise, and the phase response characteristics of the system are maintained through a feedforward compensation mechanism.

9. A system for solving the approximate damped inverse function characteristics for singularity avoidance, characterized in that, The system is used to perform the method for solving the characteristic of the approximation damped inverse function according to any one of claims 1-8, the system comprising: The kinematic modeling module is used to construct the kinematic framework of the robotic arm. It establishes kinematic equations through parameter analysis and coordinate transformation, and generates velocity-level Jacobian matrices. The singular factor identification module is used to perform algebraic decomposition of the Jacobian matrix, extract singular factors and manage the singular factor library, and realize dynamic monitoring of singular states. The damping function construction module is used to generate continuously differentiable damping functions and ensures the smooth transition characteristics of the function in singular regions through parameter optimization. The inverse matrix calculation module is used to construct the corrected inverse matrix, and the stability of the matrix is ​​ensured through damping function replacement and numerical optimization. The control execution module is used to perform real-time avoidance control and smoothly switch motion modes through status monitoring and parameter adjustment.

10. The approximation damped inverse function characteristic solution system for singularity avoidance as described in claim 9, characterized in that, The control execution module includes: The data processing unit is used to integrate feedback information from multiple sensor sources and correct the joint state in real time through an extended Kalman filter algorithm; Transition unit, designed with non-perturbative transition logic based on continuous variation of singular factors; The function adjustment unit uses a smooth function to adjust the damping coefficient and the weight ratio of the Jacobian inverse matrix, and introduces an adjustable low-pass filter at the joint speed command output to ensure smooth joint speed and continuous trajectory.

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