A method and device for modeling and identifying the stiffness of a mechanical arm used in nuclear power operation and maintenance

Abstract

This application belongs to the field of nuclear power plant operation and maintenance technology, aiming to solve the accuracy compensation problem of robotic arms after elastic deformation under heavy loads. It discloses a method and device for modeling and identifying the stiffness of robotic arms used in nuclear power plant operation and maintenance. The method uses a one-dimensional joint group fitted with arm deflection as a virtual joint group for a single arm. Coordinate systems are defined for all joints in the virtual joint group, and the arm deflection between any two joints is fitted. Kinematic modeling is performed on the robot, including the virtual joint group and the real joints, to obtain the Jacobian matrix characterizing the relationship between the robot's end effector velocity and the robot's joint velocities. A stiffness model of the robot is established, and the stiffness parameters of the stiffness model are optimized. The stiffness parameters are identified using the weighted least squares method. The device includes a robotic arm, a laser tracker, a force sensor, and a target mounting base. This application addresses the issue of deflection errors easily occurring in low-stiffness robotic arms under heavy loads by providing a virtual stiffness modeling method for robotic arms.

Description

Technical Field

[0001] This application belongs to the field of nuclear power operation and maintenance technology, and in particular relates to a method and device for modeling and identifying the stiffness of a robotic arm used in nuclear power operation and maintenance. Background Technology

[0002] The robotic arms used in nuclear power plant operation and maintenance scenarios differ from standardized industrial robots. To adapt to the non-standard characteristics of these scenarios, some robotic arms need to be compact, lightweight, and have a large load capacity. This leads to deformation of the robotic arm due to the heavy load and low rigidity, resulting in significant errors in its operational trajectory. Industrial-grade or collaborative robots, due to their higher structural rigidity or lower load capacity, experience far less error from external loads and structural flexibility compared to system parameter errors caused by machining and assembly errors. Therefore, the main calibration process for industrial robots involves calibrating system parameter errors such as link length, base coordinate system, and zero position.

[0003] Stiffness identification for robotic arms mainly includes the Finite Element Method (FEA), Matrix Analysis (MSA), and Virtual Joint Method (VJM). Among these, the most accurate is the finite element method (FEA), which considers the true shape and size of robot components. However, due to its high computational cost, this method is typically used only in the final design stage. Matrix Analysis combines the main ideas of finite element analysis with simplified elements—three-dimensional flexible beams—which obviously reduces computational load but cannot eliminate the drawbacks of finite element analysis. The Virtual Joint Method describes the elastic deformation of links, joints, and actuators by adding virtual joints (localized springs) to the traditional rigid model, offering a reasonable trade-off between model accuracy and computational complexity.

[0004] After completing the stiffness model of the robotic arm, it is necessary to identify and calculate its stiffness parameters. Existing methods often result in non-positive stiffness parameters due to the influence of measurement noise, which does not conform to physical properties. Furthermore, the large differences in the order of the stiffness parameters can affect the accuracy of identification. A complete elastic-static model of a robotic arm contains a large number of stiffness parameters, and the identification process is extremely difficult due to the influence of measurement noise. Summary of the Invention

[0005] The purpose of this application is to provide a method and device for stiffness modeling and identification of robotic arms for nuclear power plant operation and maintenance, so as to solve the problem of accuracy compensation after the robotic arm undergoes elastic deformation under heavy load.

[0006] To achieve the above objectives, this application provides the following technical solution:

[0007] Firstly, this application provides a method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance, including:

[0008] Step 1: Use the one-dimensional joint group for fitting the arm deflection as the virtual joint group of a single arm. Define the coordinate system for all joints in the virtual joint group based on the real joints to fit the arm deflection between any two joints.

[0009] Step 2: Based on the coordinate system definition, perform kinematic modeling on the robot including virtual joint groups and real joints to obtain the Jacobian matrix characterizing the relationship between the robot's end-effector velocity and the robot's joint velocities, which is used to establish the robot's stiffness model.

[0010] Step 3: Optimize the stiffness parameters of the stiffness model established in Step 2 based on a combination of algebraic and statistical methods, and identify the stiffness parameters by weighted least squares method.

[0011] As an feasible approach, the arm is equivalent to a cantilever beam. The spring fitted by the arm deflection is used as a 6 one-dimensional joint fitted by 6 links with a length of 0, including 3 translational joints and 3 rotational joints. The actual joints of the robot are simplified to one-dimensional springs. The stiffness matrix of the entire arm and joint combination is represented as a 7×7 stiffness matrix.

[0012] As an feasible approach, step 1 employs coordinate system definition rules, including:

[0013] Based on the coordinate system definition of the real joint, determine the initial coordinate system of the virtual joint of the preceding link connected to it;

[0014] The coordinate systems for the latter two rotational degrees of freedom are obtained by rotating 360 degrees around X and 360 degrees around Y, respectively.

[0015] The first translational degree of freedom and the last rotational degree of freedom use the same coordinate system;

[0016] The coordinate systems for the latter two translational degrees of freedom are obtained by rotating the coordinates by -360 degrees around the Y axis and -360 degrees around the X axis, respectively.

[0017] As an feasible approach, the orientation of the real joint coordinate system in the arm and joint assembly is consistent with the orientation of the last joint coordinate system in the arm's virtual joint group.

[0018] As an feasible approach, the entire 3R3T1R open-chain joint group is modeled to obtain the Jacobian matrix J(θ) that characterizes the relationship between the robot's end-effector velocity and the robot's joint velocities.

[0019] The stiffness matrix of all joints of the robot is expressed as K θ Based on the kinematics of the robotic arm and the relationship between force couples, the relationship between the external torque and the resulting end-effector displacement deviation is obtained:

[0020]

[0021] In the formula, Representing the flexibility of all virtual joint groups and real joints, and denoted by stiffness K. θ The reciprocal of J ξ Let F be the Jacobian matrix of the end effector in Cartesian space relative to the joint space, τ be the generalized torque of all joints of the robot, and F be the torque of the robot. f ξ represents the external load vector at the robot's end effector. X This indicates the deviation in the robotic arm's pose before and after it is loaded at the end effector;

[0022] Let the observation matrix be set. b i =ξ X The flexibility matrix is ​​solved using the least squares method, thus obtaining the stiffness K. θ :

[0023]

[0024] In the formula, N represents the N configurations of the robotic arm's motion. A is obtained by solving for the joint angles, end-effector forces, and end-effector pose deviations of the N configurations. i and b i .

[0025] As one feasible approach, step 3 includes:

[0026] Step 3.1: Perform singular value decomposition on the observation matrix and group the parameters according to the decomposition results;

[0027] Step 3.2: Perform a second decomposition of the submatrix with given parameters, extract key principal component columns through QR decomposition and add them to the identifiable group;

[0028] Step 3.3: Calculate the residuals of the current parameters, construct a weight matrix based on the residual degeneracy, and repeatedly update the parameters using weighted least squares until the residuals converge;

[0029] Step 3.4: Calculate and normalize the standard deviation of the parameters, remove parameters whose magnitude is less than the threshold, and repeat step 3.3 for the remaining parameters to obtain the final stiffness parameters.

[0030] As an feasible approach, step 3.1 involves optimizing the stiffness parameters using algebraic methods:

[0031] The observation matrix is ​​decomposed using SVD to obtain an orthogonal matrix:

[0032] V = [V1 V2 ... V] r V r+1 ... V m ]

[0033] In the formula, r corresponds to the number of non-zero elements on the diagonal of the S matrix, and m is the original number of stiffness parameters;

[0034] Based on the grouping of the orthogonal matrix V, the parameters in the stiffness vector are divided into three groups:

[0035] Group C1: V submatrix [V] r+1 ... V m The parameter index corresponding to the column number of the row vector that is equal to zero;

[0036] Group C2: V submatrix [V1 ... V] r The parameter index corresponding to the column number of the row vector that is equal to zero;

[0037] Group C3: Remaining parameters.

[0038] As a feasible approach, for the remaining stiffness parameter indices in group C3, select the columns of the observation matrix corresponding to the indices to form a matrix, perform SVD decomposition on this matrix again, and use the obtained orthogonal matrix V to calculate the following matrix L:

[0039]

[0040] In the formula, m′ represents the number of parameters in group C3, [V1 V2 ... V m ′] * To ensure full rank when extracting relevant rows from matrix V;

[0041] Perform QR decomposition on matrix L:

[0042] AP = QR(L)

[0043] The first n non-repeating row indices of the permutation matrix P represent linearly independent pivot columns in the matrix space of matrix L, and the corresponding parameters are found based on these indices.

[0044] As an implementable approach, in step 3.3, C is obtained through least squares. θ The residual E is obtained as follows:

[0045]

[0046] Calculate the covariance matrix of the residual E and obtain the weight matrix η:

[0047]

[0048] The weight matrix is ​​modified, and new compliance parameters are obtained through weighted least squares:

[0049]

[0050] Repeat the above iterative process, using the new compliance parameters to solve the residuals again, and obtain new weights, until the residuals are less than a certain value.

[0051] As an feasible approach, in step 3.4, the covariance matrix of the parameters is obtained by the following equation:

[0052]

[0053] Extracting diagonal parameters to estimate their standard deviation σ j For parameter vector C θ Each element C in i Divide by its standard deviation to obtain the normalization parameter v i :

[0054]

[0055] Using the normalized parameter v i Then, parameter optimization is performed, parameters whose normalized parameters are less than the threshold are removed, and step 3.3 is executed again on the remaining parameters to calculate the flexibility parameters of the robotic arm and obtain the final stiffness parameters.

[0056] Secondly, this application provides a robotic arm stiffness modeling and identification device for nuclear power plant operation and maintenance. Implementing the above-mentioned method, the device includes a robotic arm, a laser tracker, and a force sensor and a target mounting base installed at the end of the robotic arm. An end-effector pose target is installed on the target mounting base. The laser tracker measures the end-effector pose of the robotic arm when no load is applied. An end-effector load is installed at the end of the robotic arm, and the laser tracker measures the end-effector pose after the load is applied.

[0057] Compared with existing technologies, the method and device for modeling and identifying the stiffness of robotic arms for nuclear power plant operation and maintenance provided in this application have the following advantages:

[0058] This application addresses the issue of deflection errors in high-load, low-stiffness robotic arms by developing a virtual stiffness modeling method for robotic arms. For the established robotic arm stiffness model, a method for simplifying and identifying stiffness parameters is proposed. This application ensures the invertibility of the observation matrix during the identification process, eliminates redundant parameters and parameters whose influence is negligible, and simultaneously forms a flexible and implementable stiffness identification experimental device.

[0059] Furthermore, this application proposes a modeling method for virtual joints of robotic arms, extending the original modeling method that only considers the joints of robotic arms to a modeling method that considers the stiffness of the links. This method is universal for robotic arms with different configurations and has high computational efficiency.

[0060] Furthermore, considering the difficulty in identifying parameters in the stiffness model, this application instantiates a method combining algebra and probability statistics to eliminate and optimize the parameters, thus avoiding low identification accuracy due to measurement noise.

[0061] In addition, an experimental device was designed based on stiffness modeling and identification methods. This experimental device does not require additional loading equipment, can flexibly adjust the measurement configuration of the robotic arm, and quickly obtain end pose and force measurement data before and after load loading. Attached Figure Description

[0062] To more clearly illustrate the technical solution of this application, the accompanying drawings used in the technical description will be briefly introduced below.

[0063] Figure 1 A flowchart illustrating the method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance, as provided in this application;

[0064] Figure 2 A schematic diagram illustrating the method for defining virtual joints of a robotic arm provided in this application;

[0065] Figure 3 A structural schematic diagram of the robotic arm stiffness modeling and identification device for nuclear power plant operation and maintenance provided in this application;

[0066] Figure 4 This is a schematic diagram of the experimental procedure for measuring the stiffness of the robotic arm provided in this application.

[0067] Explanation of reference numerals in the attached figures:

[0068] 1. Robotic arm; 2. Force sensor; 3. End effector; 4. Target mount; 5. Laser tracker. Detailed Implementation

[0069] The following detailed description provides further details on specific implementation methods.

[0070] like Figure 1 As shown, this application provides a method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance, including:

[0071] Step 1: Use the one-dimensional joint group for fitting the arm deflection as the virtual joint group of a single arm. Define the coordinate system for all joints in the virtual joint group based on the real joints to fit the arm deflection between any two joints.

[0072] Step 2: Based on the coordinate system definition, perform kinematic modeling on the robot including virtual joint groups and real joints to obtain the Jacobian matrix characterizing the relationship between the robot's end-effector velocity and the robot's joint velocities, which is used to establish the robot's stiffness model.

[0073] Step 3: Optimize the stiffness parameters of the stiffness model based on a combination of algebraic and statistical methods, and identify the stiffness parameters by weighted least squares method.

[0074] In this application, since the virtual joint is fitted as a six-dimensional spring, and the six-dimensional spring can essentially be regarded as a one-dimensional joint in six degrees of freedom, the six-dimensional spring is split into six one-dimensional springs.

[0075] The linkage model of a robot can be approximated as a cantilever beam. According to EB beam theory, the stiffness matrix of a cantilever beam is a 6×6 matrix, where the six degrees of freedom correspond to three translational and three rotational stiffnesses. The deformation angles of these stiffnesses are mainly generated about the following six axes:

[0076] Translational degrees of freedom:

[0077] Translational stiffness along the X-axis corresponds to the translational deformation of the beam in the transverse X direction;

[0078] Translational stiffness along the Y-axis corresponds to the translational deformation of the beam in the direction perpendicular to the Y-axis;

[0079] Translational stiffness along the Z-axis corresponds to the translational deformation of the beam in the longitudinal Z-direction (along the beam axis).

[0080] Rotational degrees of freedom:

[0081] The rotational stiffness about the X-axis corresponds to the torsional deformation of the beam about the transverse axis, and usually involves the torsional stiffness of the beam.

[0082] The rotational stiffness about the Y-axis corresponds to the bending deformation of the beam about the vertical axis, and mainly affects the lateral bending of the beam.

[0083] The rotational stiffness about the Z-axis corresponds to the bending deformation of the beam about the longitudinal axis, which affects the bending of the beam in the vertical direction.

[0084] For the virtual spring at the motor, its six-dimensional virtual spring has a much smaller stiffness in the direction of motor rotation than in other directions, so the virtual spring at the motor can be regarded as a one-dimensional spring. The spring for fitting the arm deflection is regarded as a set of six one-dimensional joints fitted by six links with a length of 0, including three translational joints and three rotational joints.

[0085] The one-dimensional joint set for fitting arm deflection is defined as the virtual joint set of a single arm. For the deflection of a single robot arm, a coordinate system transformation definition for the virtual joint set is proposed to fit the arm deflection between any two joints. The details are as follows:

[0086] Determine the initial coordinate system of the virtual joint of the preceding link connected to it based on the coordinate system definition of the real joint;

[0087] The coordinate systems for the latter two rotational degrees of freedom are obtained by rotating 360 degrees around X and 360 degrees around Y, respectively.

[0088] The first translational degree of freedom and the last rotational degree of freedom use the same coordinate system;

[0089] The coordinate systems for the latter two translational degrees of freedom are obtained by rotating the coordinates by -360 degrees around the Y axis and -360 degrees around the X axis, respectively.

[0090] The order of this transformation is variable, but the pose of the real joint coordinate system must remain consistent with the pose of the last joint coordinate system in the virtual joint group. This coordinate system definition rule can be expressed as follows: Figure 2 As shown in the figure, the green arrows represent the Y-axis direction of each joint coordinate system, the red arrows represent the X-axis direction of each joint coordinate system, and the blue arrows represent the Z-axis direction of each joint coordinate system. The figure shows the coordinate system definition of a group of arm combinations. The real joints are the reference for the coordinate system definition of the virtual joint group of the arm. The definition of virtual joint 1 is consistent with that of the real joint. The coordinate system definitions of virtual joints 2 to 6 are obtained through coordinate system transformation. The definition of virtual joint 6 is consistent with that of the real joint. In particular, the base is the first arm that starts to virtualize the joint.

[0091] Model the entire 3R3T1R open-chain joint group to obtain the Jacobian matrix J(θ) that characterizes the relationship between the robot's end-effector velocity and the robot's joint velocities.

[0092] The stiffness matrix of all joints of the robot is expressed as K θ Based on the kinematics of the robotic arm and the relationship between force couples, the relationship between the external torque and the resulting end-effector displacement deviation is obtained:

[0093]

[0094] In the formula, Representing the flexibility of all virtual joint groups and real joints, and denoted by stiffness K. θ The reciprocal of J ξ Let F be the Jacobian matrix of the end effector in Cartesian space relative to the joint space, τ be the generalized torque of all joints of the robot, and F be the torque of the robot. f ξ represents the external load vector at the robot's end effector. X This indicates the deviation in the robotic arm's pose before and after it is loaded at the end effector;

[0095] Let the observation matrix be set. b i =ξ X The flexibility matrix is ​​solved using the least squares method, thus obtaining the stiffness:

[0096]

[0097] In the formula, N represents the N configurations of the robotic arm's motion. A is obtained by solving for the joint angles, end-effector forces, and end-effector pose deviations of the N configurations. i and b i .

[0098] The stiffness model established above contains a large number of stiffness parameters. Taking a six-axis robotic arm as an example, there are at least 42 stiffness parameters. Moreover, even with simplification, the calculated stiffness parameters may be non-positive. There are also parameters with relatively small absolute values. Measurement noise leads to low recognition accuracy, and the resulting deformation has a very small impact on the entire robotic arm, which can be ignored. These parameters should be eliminated through parameter optimization.

[0099] The following is an example process for parameter optimization and identification:

[0100] Step 3.1: Optimize stiffness parameters using algebraic methods.

[0101] The observation matrix is ​​decomposed using SVD to obtain an orthogonal matrix:

[0102] V = [V1 V2 ... V] r V r+1 ... V m ]

[0103] In the formula, r corresponds to the number of non-zero elements on the diagonal of the S matrix, and m is the original number of stiffness parameters.

[0104] Based on the grouping of the orthogonal matrix V, the parameters in the stiffness vector can be divided into three groups:

[0105] The first group, C1, V submatrix [V r+1 ... V m The parameter index corresponding to the column number of the row vector that is equal to zero is the identifiable parameter;

[0106] The second group, C2, contains V submatrices [V1 ... V] r The parameter index corresponding to the column number of the row vector that is equal to zero is an unrecognizable parameter and is eliminated.

[0107] The third group, C3, requires further evaluation of the remaining parameters.

[0108] Step 3.2: Parameter optimization based on QR decomposition.

[0109] For the remaining stiffness parameter indices in group C3, select the observation matrix columns corresponding to the indices to form a matrix, perform SVD decomposition on this matrix again, and use the obtained orthogonal matrix V to calculate the following matrix L:

[0110]

[0111] In the formula, m′ represents the number of parameters in group C3, [V1 V2 ... V m ′] * To ensure full rank when extracting relevant rows from matrix V.

[0112] Perform QR decomposition on matrix L:

[0113] AP = QR(L)

[0114] The first n non-repeating row indices of the permutation matrix P represent linearly independent principal columns in the matrix space of matrix L. The corresponding parameters are found based on these indices. After adding them to group C1, the rank of the observation matrix is ​​increased accordingly, which avoids the phenomenon that noise disturbances are amplified when the matrix is ​​not in full rank.

[0115] Step 3.3: Parameter optimization based on dimensional balance.

[0116] C obtained through least squares θ The residual E is obtained as follows:

[0117]

[0118] Find the covariance matrix of the residual E, diagonalize and square root the covariance matrix, and take the inverse of the covariance matrix. This inverse matrix is ​​the weight matrix η.

[0119]

[0120] The weight matrix is ​​modified, and new compliance parameters are obtained through weighted least squares:

[0121]

[0122] Repeat the above iterative process, using the new compliance parameters to solve the residuals again, and obtain new weights, until the residuals are less than a certain value.

[0123] Step 3.4: Parameter identification strategy based on probabilistic statistical methods.

[0124] The compliance parameters obtained at this stage may vary significantly in magnitude. The covariance matrix of the parameters is obtained using the following formula, and the diagonal parameters are extracted to estimate their standard deviation σ. j For parameter vector C θ Each element C in i Divide by its standard deviation to obtain the normalized parameter.

[0125]

[0126] Using this normalization parameter v iThis process can identify which parameters are unidentifiable and which are identifiable. Larger parameters indicate higher identification accuracy and represent significant flexibility of the robotic arm at that joint. After removing the smaller-order-of-magnitude normalized parameters, the remaining identifiable parameters are processed again using step 3.3 to calculate the robotic arm's flexibility parameters, thereby obtaining the stiffness parameters.

[0127] In response to the aforementioned methods for modeling and identifying the stiffness of robotic arms used in nuclear power plant operation and maintenance, this application provides a device (experimental apparatus) for modeling and identifying the stiffness of robotic arms used in nuclear power plant operation and maintenance, such as... Figure 3 As shown, the device includes a robotic arm 1, a force sensor 2, an end effector load 3, a target mounting base 4, an end effector pose target, a laser tracker 5, and a workstation. The end effector pose target is used in conjunction with the tracker to acquire the end effector pose, and the end effector load 3 is an arbitrarily applied load component. The end effector pose target is mounted on the target mounting base 4. The workstation is communicatively connected to the force sensor 2 and the laser tracker 5; the signal lines of the force sensor 2 and the laser tracker 5 are connected to the workstation for acquiring data from the force sensor 2 and the laser tracker 5.

[0128] Force sensor 2 and end load 3 are installed at the end of robotic arm 1, and target mount 4 is installed at the end of robotic arm 1. The installation sequence is: first install force sensor 2, then install target mount 4, and then install end load 3. The advantage of this is that whether or not end load 3 is installed will not affect the installation relationship between target mount 4 and robotic arm, that is, target mount 4 will not be removed because of the need to install or unload end load 3.

[0129] The laser tracker 5 is the main posture data measuring instrument, used to measure the end-effector posture changes of the robotic arm 1 before and after the end-effector load 3 is applied. The end-effector posture target is installed at the end of the robotic arm and has a fixed relative relationship with the end of the robotic arm. It works with the laser tracker 5 to reflect the end-effector posture changes. The force sensor 2 is installed at the end of the robotic arm to measure the force on the end when the load is applied. The end-effector load 3 is also installed at the end of the robotic arm. The load can be easily increased step by step through the connection method such as screws. The workstation is used to receive the measurement data from the force sensor 2 and the laser tracker 5 and to identify the stiffness parameters using the method described above. This device does not require additional loading equipment, can flexibly adjust the measurement configuration of the robotic arm 1, and quickly obtain the end-effector posture and force measurement data before and after the load is applied.

[0130] like Figure 4 As shown, the main process of using this device is as follows:

[0131] (1) Install and deploy all measuring instruments, and remove the end-point load;

[0132] (2) By using the hand-eye calibration method of the robotic arm, the transformation relationship between the end pose target and the end flange of the robotic arm is obtained, as well as the transformation relationship between the base coordinate system of the robotic arm 1 and the measurement coordinate system of the laser tracker 5.

[0133] (3) Calibrate the force sensor 2 at the end of the robotic arm;

[0134] (4) Plan the measurement configuration of the robotic arm 1, and measure and record the end pose of the robotic arm 1 when no load is applied by the laser tracker 5, and at the same time check whether the value of the force sensor 2 is zero.

[0135] (5) Install end-effector load 3 on the robotic arm;

[0136] (6) The robotic arm 1 moves to all the measurement positions, and the end pose of the robotic arm 1 after the load is applied is measured and recorded by the laser tracker 5;

[0137] (7) Obtain the stiffness parameters of the robotic arm through stiffness modeling and identification methods.

[0138] The above description is only a specific embodiment of this application, but the protection scope of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the protection scope of this application.

Claims

1. A method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance, characterized in that, include: Step 1: Use the one-dimensional joint group for fitting the arm deflection as the virtual joint group of a single arm. Define the coordinate system for all joints in the virtual joint group based on the real joints to fit the arm deflection between any two joints. step The coordinate system definition rules adopted in section 1 include: Based on the coordinate system definition of the real joint, determine the initial coordinate system of the virtual joint of the preceding link connected to it; The coordinate systems for the latter two rotational degrees of freedom are obtained by rotating 360 degrees around X and 360 degrees around Y, respectively. The first translational degree of freedom and the last rotational degree of freedom use the same coordinate system; The coordinate systems for the latter two translational degrees of freedom are obtained by rotating the coordinates by -360 degrees around the Y axis and by -360 degrees around the X axis, respectively. Step 2: Based on the coordinate system definition, perform kinematic modeling on the robot including virtual joint groups and real joints to obtain the Jacobian matrix representing the relationship between the robot's end-effector velocity and the robot's joint velocities, which is used to establish the robot's stiffness model; model the entire 3R3T1R open chain joint group to obtain the Jacobian matrix J(θ) representing the relationship between the robot's end-effector velocity and the robot's joint velocities. The stiffness matrix of all joints of the robot is expressed as follows Based on the kinematics of the robotic arm and the relationship between force couples, the relationship between the external torque and the resulting end-effector displacement deviation is obtained: In the formula, Represents the flexibility of all virtual joint groups and real joints, and is the stiffness. The reciprocal, Let be the Jacobian matrix of the robotic arm's end effector in Cartesian space relative to the joint space. For the generalized torque of all joints of the robot, This represents the external load force vector at the robot's end effector. This indicates the deviation in the robotic arm's pose before and after it is loaded at the end effector; Let the observation matrix be set. , The flexibility matrix is ​​solved using the least squares method, thus obtaining the stiffness: In the formula, N represents the N configurations of the robotic arm's motion. A is obtained by solving for the joint angles, end-effector forces, and end-effector pose deviations of the N configurations. i and b i ; Step 3: Optimize the stiffness parameters of the stiffness model using a combination of algebraic and statistical methods, and identify the stiffness parameters using weighted least squares. Step 3 includes: Step 3.1: Perform singular value decomposition on the observation matrix and group the parameters according to the decomposition results; Step 3.2: Perform a second decomposition of the submatrix with given parameters, extract key principal component columns through QR decomposition and add them to the identifiable group; Step 3.3: Calculate the residuals of the current parameters, construct a weight matrix based on the residual degeneracy, and repeatedly update the parameters using weighted least squares until the residuals converge; Step 3.4: Calculate and normalize the standard deviation of the parameters, remove parameters whose magnitude is less than the threshold, and repeat step 3.3 for the remaining parameters to obtain the final stiffness parameters.

2. The method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance according to claim 1, characterized in that, The arm is equivalent to a cantilever beam. The spring fitted by the arm deflection is used as a 6 one-dimensional joint fitted by 6 links with a length of 0, including 3 translational joints and 3 rotational joints. The actual joints of the robot are simplified to one-dimensional springs. The stiffness matrix of the entire arm and joint combination is represented as a 7×7 stiffness matrix.

3. The method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance according to claim 1, characterized in that, In step 3.1, stiffness parameters are optimized using algebraic methods: The observation matrix is ​​decomposed using SVD to obtain an orthogonal matrix: In the formula, The number of non-zero elements on the diagonal of matrix S. This represents the original number of stiffness parameters; Based on the grouping of the orthogonal matrix V, the parameters in the stiffness vector are divided into three groups: Group C1: V submatrix The parameter index corresponding to the column number of the row vector that is equal to zero; Group C2: V submatrix The parameter index corresponding to the column number of the row vector that is equal to zero; Group C3: Remaining parameters.

4. The method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance according to claim 3, characterized in that, For the remaining stiffness parameter indices in group C3, select the observation matrix columns corresponding to the indices to form a matrix, perform SVD decomposition on this matrix again, and use the obtained orthogonal matrix V to calculate the following matrix L: In the formula, This indicates the number of parameters in group C3. To ensure full rank when extracting relevant rows from matrix V; Perform QR decomposition on matrix L: The first n non-repeating row indices of the permutation matrix P represent linearly independent pivot columns in the matrix space of matrix L, and the corresponding parameters are found based on these indices.

5. The method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance according to claim 1, characterized in that, In step 3.3, the result obtained through least squares The residual E is obtained as follows: Calculate the covariance matrix of the residual E and then obtain the weight matrix. : The weight matrix is ​​modified, and new compliance parameters are obtained through weighted least squares: Repeat the above iterative process, using the new compliance parameters to solve the residuals again, and obtain new weights, until the residuals are less than a certain value.

6. The method for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance according to claim 1, characterized in that, In step 3.4, the covariance matrix of the parameters is obtained using the following formula: Extracting diagonal parameters to estimate their standard deviation For parameter vectors Each element in Divide by its standard deviation to obtain the normalized parameter. : Using normalized parameters Then, parameter optimization is performed, parameters whose normalized parameters are less than the threshold are removed, and step 3.3 is executed again on the remaining parameters to calculate the flexibility parameters of the robotic arm and obtain the final stiffness parameters.

7. A device for modeling and identifying the stiffness of a robotic arm used in nuclear power plant operation and maintenance, characterized in that, Based on the method of any one of claims 1 to 6, the device includes a robotic arm (1), a laser tracker (5), and a force sensor (2) and a target mounting base (4) installed at the end of the robotic arm (1), wherein an end pose target is installed on the target mounting base (4); the end pose of the robotic arm (1) when no load is applied is measured by the laser tracker (5); an end load (3) is installed at the end of the robotic arm (1); and the end pose after the load is applied is measured by the laser tracker (5).

Images