Soft measurement method of end force based on Jacobian matrix of flexible hydraulic manipulator

Abstract

The present invention discloses a soft measurement method for the end force of a flexible hydraulic manipulator based on the Jacobian matrix, the method comprising: firstly establishing a dynamic model of the flexible hydraulic manipulator, establishing a nonlinear joint friction torque model, establishing a linear model of the minimum inertia parameter set of the flexible arm and its corresponding regression matrix, and generating an excitation trajectory; then running the excitation trajectory once, and calculating the dynamic parameters of the manipulator arm according to the obtained hydraulic driving torque τ and the regression matrix Y; then measuring data through an inclination sensor, solving the analytical expression of the deflection deformation of the arm end, and obtaining the Jacobian matrix of the flexible arm by taking the partial derivative of the end position of the flexible hydraulic manipulator based on the joint angle; finally, calculating the end force of the flexible hydraulic manipulator arm by combining the obtained dynamic parameters of the flexible arm with the Jacobian matrix of the flexible arm; the soft measurement method for the end force of the flexible hydraulic manipulator proposed by the present invention can solve the problem of inaccurate end force estimation of the flexible hydraulic manipulator arm due to arm deformation.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible hydraulic manipulator control, and in particular to a flexible hydraulic manipulator end force soft measurement method based on the Jacobian matrix. Background Art

[0002] At present, flexible hydraulic robotic arms have been widely used in contact operations such as large aircraft cleaning and tunnel drilling. Faced with the changing unstructured environment and the large inertia of the flexible hydraulic robotic arms themselves, frequent contact can easily cause damage to the end force sensor, affecting the measurement of the end force.

[0003] Currently, there are soft force measurement methods for hydraulic manipulator end points based on parameter identification. For example, patent publication number CN113977578A, titled "Soft Force Measurement Method for Hydraulic Manipulator End Points," was applied for and granted by our team on October 26, 2021. However, this soft force measurement method is only applicable to rigid hydraulic manipulators. Flexible hydraulic manipulators, due to the deformation characteristics of their boom structure during operation, have difficult-to-describe dynamic characteristics, resulting in inaccurate dynamic parameters. Therefore, it is not applicable to long, flexible, multi-joint hydraulic manipulators with arm deformation, and the impact of arm deformation on soft force measurement at the manipulator end point is not considered. When a force is applied to the end of a flexible hydraulic manipulator, it can be equated to the force applied by a hydraulic cylinder. The hydraulic cylinder's driving torque overcomes the end load force, so the end force can be decomposed into each joint using the Jacobian matrix. However, because the deflection deformation caused by arm flexibility affects the converted Jacobian matrix, directly using a rigid Jacobian matrix that does not account for deflection deformation will result in inaccurate end force calculations. Therefore, considering the deformation characteristics of the flexible arm, a method is proposed to solve the flexible arm Jacobian matrix based on the deflection deformation of each arm. Based on the flexible arm Jacobian matrix, a soft measurement method for the end force of the flexible hydraulic manipulator is proposed, which can solve the problem of inaccurate end force estimation caused by arm deformation. Summary of the Invention

[0004] To this end, the present invention provides a method for soft measurement of the end force of a flexible hydraulic manipulator based on the Jacobian matrix. By establishing a dynamic model of the flexible arm and adopting a nonlinear joint friction model for joint friction, a minimum inertia parameter set model for the flexible arm is established. Finite Fourier series are used to solve the excitation trajectory. Under the excitation trajectory, the minimum inertia parameter set model is used to solve the flexible arm's dynamic parameters. The analytical expression for the flexible arm's deflection deformation is calculated using an inclination sensor. Based on the flexible arm's kinematic characteristics, the flexible arm's Jacobian matrix, including the arm's end deflection, is solved to calculate the external force.

[0005] The technical solution adopted by the present invention is: a soft measurement method of the end force based on the Jacobian matrix of a flexible hydraulic manipulator, comprising the following steps:

[0006] Step 1: Establish a dynamic model of the flexible hydraulic manipulator. To address the problems of wide motion range and imprecise joint friction of the flexible hydraulic manipulator, a nonlinear joint friction torque model is established. Based on the flexible arm dynamic model, the relationship between the hydraulic drive torque and the generalized force of the three-joint flexible arm is determined. The dynamic model is linearized to establish a linear model of the minimum inertia parameter set of the flexible arm and its corresponding regression matrix.

[0007] Step 2: Design the excitation trajectory using the finite Fourier series. With the goal of minimizing the condition number of the regression matrix in step 1, solve the finite Fourier series coefficients to generate the excitation trajectory.

[0008] Step 3: The flexible arm runs an excitation trajectory under no-load conditions, collects the pressure sensor values ​​of the two chambers of the hydraulic cylinder, and calculates the hydraulic driving torque τ based on the pressure of the two chambers of the hydraulic cylinder; the generalized modal angle (including joint angle and modal parameters), generalized angular velocity, and generalized angular acceleration at each moment are substituted into the regression matrix Y and merged into a total matrix;

[0009] Step 4, calculating the dynamic parameters of the flexible arm according to the hydraulic driving torque τ and the regression matrix Y obtained in step 3;

[0010] Step 5: Measure data using four inclination sensors installed at three locations on the arm segment and at the head end of the arm (near the base end), and solve the analytical expression for the deflection of the arm end.

[0011] Step 6: By analyzing the kinematic characteristics of the three-degree-of-freedom flexible arm, the end position of the flexible arm is solved. The Jacobian matrix of the flexible arm including the deflection deformation can be solved by taking the partial derivative of the end position with respect to the joint angle and combining it with the analytical expression of the deflection deformation of the arm end solved in step 5.

[0012] Step 7: Substitute the dynamic parameters obtained in step 4 into the linear model of step 1, and calculate the end force of the flexible hydraulic manipulator in combination with the flexible arm Jacobian matrix in step 6;

[0013] Furthermore, the dynamic model of the flexible hydraulic manipulator established in step 1 is as follows:

[0014]

[0015] Where p is the generalized angle of the flexible arm, p = [θ q] T , θ is the joint angle, q is the modal coordinate, is the generalized angular velocity, is the generalized angular acceleration, τ is the hydraulic driving torque, M(p) is the inertia matrix of the flexible arm, is the Coriolis force and centripetal force matrix of the flexible arm, K is the stiffness matrix of the flexible arm, G is the gravity matrix of the flexible arm, τ f is the nonlinear joint friction torque.

[0016] Furthermore, the nonlinear joint friction torque model established in step 1 is:

[0017]

[0018] Among them, k1, k2, k3, k4 are the coefficients of the Fourier series, k5, k6, k7, k8 are the coefficients of the third-order polynomial, and are the parameters to be identified.

[0019] Furthermore, the relationship between the hydraulic driving torque of the flexible arm described in step 1 and the generalized force acting on the flexible arm is:

[0020] Q I =τ i -τ i+1

[0021] Among them, Q I is the generalized force on the flexible arm, subscript I is the generalized force number, τ i is the joint torque acting on the flexible arm, and the subscript i is the joint number.

[0022] Furthermore, the linear model of the minimum inertia parameter set of the flexible arm described in step 1 is specifically as follows:

[0023]

[0024] in, represents the regression matrix, and L represents the minimum inertia parameter set.

[0025] Furthermore, the finite Fourier series expression described in step 2 is as follows:

[0026]

[0027] For the i-th joint, the number of sine terms and cosine terms is N, t represents the running time of the excitation trajectory, ω f =2πf f is the fundamental frequency, f f =1 / t,a l，i 、b l，i ,θ i0 To find the Fourier coefficients, we use the optimal method of multivariate function with constraints to solve. To ensure that the robot operates stably and within the safe range, the constraints are as follows:

[0028]

[0029] Among them, condmin (Y) represents the optimization objective of minimizing the condition number of the regression matrix Y, θ(t) represents the joint angle of the flexible arm at time t; θ0 is the initial joint angle of the flexible arm. When a cycle ends, t=t f , t f Indicates the termination moment, the flexible arm returns to the initial joint angle θ0 of the flexible arm, so that the next cycle can be continuously executed, with initial t = 0 and end t = t f The velocity and acceleration of θ are set to 0 to effectively avoid impact; min 、 θ max 、 They are the minimum and maximum values ​​of the angle, angular velocity and angular acceleration when the robotic arm moves; by limiting the upper and lower limits of the angle, angular velocity and angular acceleration, the flexible arm is guaranteed to move within a safe range.

[0030] Furthermore, the matrix of the regression matrix Y merged in step 3 is first calculated at each moment of the regression matrix Y k , k is the number of time interval points, and then all regression matrices Y k Merge into a total matrix Y:

[0031]

[0032] Furthermore, the actual hydraulic driving torque τ described in step 3 i for:

[0033] τ i =(P ai A ai -P bi A bi )·r i

[0034] Where i is the joint number, P ai , P bi A is the pressure of the rodless chamber and the rod chamber of the hydraulic cylinder, measured by the pressure sensor. ai , A bi is the area of ​​the rodless cavity and the rod cavity of the hydraulic cylinder, r i It is the effective force arm of the hydraulic cylinder.

[0035] Furthermore, the dynamic parameters described in step 4 are solved as follows, and the minimum inertia parameter set L is obtained after the solution:

[0036] L=(Y T Y) -1 Y T Q

[0037] Furthermore, the expression for the deflection deformation of the arm end of the flexible hydraulic manipulator arm measured by the inclination sensor in step 5 is:

[0038]

[0039] in,

[0040]

[0041]

[0042]

[0043] Among them, n is the serial number of each arm, a, b, c are the installation positions of the inclination sensor, γ a , γ b , γ c It is the angle measured by the inclination sensor at each installation point.

[0044] Furthermore, in step 6, the end position expressions of the three-degree-of-freedom flexible hydraulic manipulator in the X and Y directions are:

[0045] X arm =l1cosθ1+w1 sinθ1+l2 cos(θ1+θ2)+w2 sin(θ1+θ2)+l3 cos(θ1+θ2+θ3)+w3 sin(θ1+θ2+θ3)

[0046] Y arm =l1sinθ1-w1cosθ1+l2sin(θ1+θ2)-w2 cos(θ1+θ2)+l3 sin(θ1+θ2+θ3)-w3 cos(θ1+θ2+θ3)

[0047] Furthermore, in step 6, the partial derivative of the end position of the flexible hydraulic manipulator with respect to the joint angle is obtained as follows:

[0048]

[0049] in,

[0050] J 11 =-l1sinθ1+w1cosθ1-l2sin(θ1+θ2)+w2cos(θ1+θ2)-l3sin(θ1+θ2+θ3)+w3cos(θ1+θ2+θ3)

[0051] J 12 =-l2sin(θ1+θ2)+w2cos(θ1+θ2)-l3sin(θ1+θ2+θ3)+w3cos(θ1+θ2+θ3)

[0052] J 13=-l3sin(θ1+θ2+θ3)+w3cos(θ1+θ2+θ3)

[0053] J 21 =l1cosθ1+w1sinθ1+l2cos(θ1+θ2)+w2sin(θ1+θ2)+l3cos(θ1+θ2+θ3)+w3sin(θ1+θ2+θ3)

[0054] J 22 =l2cos(θ1+θ2)+w2sin(θ1+θ2)+l3cos(θ1+θ2+θ3)+w3sin(θ1+θ2+θ3)

[0055] J 23 =l3cos(θ1+θ2+θ3)+w3sin(θ1+θ2+θ3)

[0056] Furthermore, the end force of the flexible hydraulic manipulator described in step 7 is solved as follows:

[0057]

[0058] Where F is the end force of the manipulator; J(p) is the Jacobian matrix of the flexible hydraulic manipulator, represents the regression matrix, L represents the minimum inertia parameter set, τ f is the nonlinear joint friction torque.

[0059] Beneficial effects of the present invention:

[0060] (1) The soft measurement method of the end force of the flexible hydraulic manipulator proposed in the present invention can replace the end force sensor, solving the problem that the force sensor is easily damaged under complex working conditions and thus cannot accurately perceive the end force.

[0061] (2) Considering that application scenarios such as large aircraft cleaning, tunnel spraying, and glass wall wiping require a robotic arm with a large working range, the arm has the characteristic of flexible deformation because of its length. Especially when force is applied at the end, the deformation will be more obvious. Therefore, it is necessary to take the flexible deformation of the arm into account when measuring the end force, based on this actual reference scenario. The present invention aims at the deformation characteristics of the arm of a flexible hydraulic robotic arm and proposes a Jacobian matrix containing the deformation characteristics of the flexible arm, which is suitable for soft measurement of the end force of the flexible hydraulic robotic arm. Compared with the rigid hydraulic robotic arm disclosed previously, the difference is that the flexible hydraulic robotic arm has arm deformation, and when calculating the end force, it is necessary to use the Jacobian matrix to convert the end force to each joint. The Jacobian matrix of the rigid arm does not include the deformation part, and the Jacobian matrix of the flexible arm includes the deformation part. Since the arm will have a slight deformation when force is applied to the end, the present invention solves the expression of the end position of the flexible arm by analyzing the kinematic characteristics of the flexible arm after deformation. The deformation in the expression can be measured by an inclination sensor, eliminating the need to solve the Jacobian matrix for calculating the modal parameters, and the arm deformation measured by the inclination sensor is more accurate than the theoretically calculated arm deformation, and has more practical application value. The deformation is considered in the Jacobian matrix, and the accuracy of the end force perception is improved.

[0062] (3) The present invention proposes a universal end force soft measurement method for flexible hydraulic manipulators, and the dynamic parameters obtained through the excitation trajectory can meet the needs of different application scenarios.

[0063] (4) Compared with the rigid hydraulic manipulator arm disclosed previously, the technical solution disclosed in the present invention is not a conventional replacement for those skilled in the art, because the flexible hydraulic manipulator arm is a typical complex system with multiple inputs, multiple outputs, high nonlinearity, and rigid-flexible coupling. The multi-link flexible hydraulic manipulator arm is much more complex and more difficult to model and analyze dynamically than the rigid hydraulic manipulator arm. The flexible deformation of the arm has a greater impact on motion and control. It is particularly important for the flexible arm to describe its arm flexible deformation during the dynamic modeling process, while the rigid arm only needs conventional dynamic modeling and does not need to consider the influence of arm deformation. This article introduces modal parameters in the dynamic modeling process to characterize the deformation characteristics of the flexible arm. Therefore, the calculation process of dynamic modeling and identification is more complex than that of the rigid arm, which requires a lot of time and creative labor. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 This is the flow chart of soft force measurement at the end of the flexible hydraulic manipulator.

[0065] Figure 2 Schematic diagram of the installation position of the tilt sensor of the flexible hydraulic robotic arm.

[0066] Figure 3Schematic diagram of the kinematic deformation of the flexible hydraulic manipulator. DETAILED DESCRIPTION

[0067] In order to better understand the present invention, the content of the present invention is further explained below with reference to the accompanying drawings and embodiments, but the content of the present invention is not limited to the following embodiments.

[0068] Example 1

[0069] See Figure 1 This embodiment proposes a soft measurement method of the end force based on the Jacobian matrix of a flexible hydraulic manipulator. The specific implementation steps are as follows:

[0070] Step 1: Establish a dynamic model of the flexible hydraulic manipulator. In order to solve the problems of wide motion range and inaccurate joint friction of the flexible hydraulic manipulator, a nonlinear joint friction torque model is established. According to the dynamic model of the flexible arm, the relationship between the hydraulic drive torque of the flexible arm and the generalized force is determined. The dynamic model is linearized to establish a linear model of the minimum inertia parameter set of the flexible arm and its corresponding regression matrix.

[0071] The mechanical arm dynamics model:

[0072]

[0073] Where p is the generalized angle of the flexible arm, p = [θ q] T , θ is the joint angle, q is the modal coordinate, is the generalized angular velocity, is the generalized angular acceleration, τ is the hydraulic driving torque, M(p) is the inertia matrix of the flexible arm, K is the stiffness matrix of the flexible arm, is the Coriolis force and centripetal force matrix of the flexible arm, G is the gravity matrix of the flexible arm, τ f is the nonlinear joint friction torque.

[0074] The expression of the established nonlinear joint friction torque model is as follows:

[0075]

[0076] Among them, k1, k2, k3, k4 are the coefficients of the Fourier series, k5, k6, k7, k8 are the coefficients of the third-order polynomial, and are the parameters to be identified.

[0077] The relationship between the hydraulic driving torque of the flexible arm and the generalized force acting on the flexible arm is:

[0078] Q I =τ i -τ i+1

[0079] For a three-degree-of-freedom hydraulic flexible arm, the relationship between the hydraulic driving torque and the generalized force acting on the flexible arm is:

[0080] Q1=τ1-τ2, Q2=τ2-τ3, Q3=τ3,

[0081] Among them, Q I (I=1, ..., 9) is the generalized force on the flexible arm, τ i (i=1,…,3) is the joint torque on the flexible arm, l i (i=1, ..., 3) is the arm length of the flexible arm.

[0082] The minimum inertia parameter set linear model of the flexible arm is as follows:

[0083]

[0084] in, Represents the regression matrix, and L represents the minimum inertia parameter set. The minimum inertia parameter dynamic model is obtained by linearizing the dynamic parameters of the manipulator and then merging and reorganizing the linear correlation terms. For a three-degree-of-freedom flexible hydraulic manipulator, the minimum inertia parameter set dynamic model is:

[0085]

[0086] in,

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094] y 520 =8π 4 q 12 , y 721 =8π 4 q 22 , y 922 =8π 4 q 32 .

[0095] Step 2: Design the excitation trajectory using the finite Fourier series. With the goal of minimizing the condition number of the regression matrix in step 1, solve the finite Fourier series coefficients to generate the excitation trajectory.

[0096] The finite Fourier series expression is as follows:

[0097]

[0098] For the i-th joint, the number of sine terms and cosine terms is N, t represents the running time of the excitation trajectory, ω f =2πf f is the fundamental frequency, f f =1 / t,a l，i 、 bl，i ,θ i0 To find the Fourier coefficients, we use the optimal method of multivariate function with constraints to solve. To ensure that the robot operates stably and within the safe range, the constraints are as follows:

[0099]

[0100] Among them, cond min (Y) represents the optimization objective of minimizing the condition number of the regression matrix Y, θ(t) represents the joint angle of the flexible arm at time t; θ0 is the initial joint angle of the flexible arm. When a cycle ends, t=t f , t f Indicates the termination moment, the flexible arm returns to the initial joint angle θ0 of the flexible arm, so that the next cycle can be continuously executed, with initial t = 0 and end t = t f The velocity and acceleration of θ are set to 0 to effectively avoid impact; min 、 θ max 、 They are the minimum and maximum values ​​of the angle, angular velocity and angular acceleration when the robotic arm moves; by limiting the upper and lower limits of the angle, angular velocity and angular acceleration, the flexible arm is guaranteed to move within a safe range.

[0101] Step 3: The flexible arm runs an excitation trajectory under no-load conditions, collects the pressure sensor values ​​of the two chambers of the hydraulic cylinder, and calculates the hydraulic driving torque τ through the pressure of the two chambers of the hydraulic cylinder; the generalized modal angle (including joint angle and modal parameters), generalized angular velocity, and generalized angular acceleration at each moment are brought into the regression matrix Y and merged into a total matrix.

[0102] The matrix of the regression matrix Y is calculated by first calculating the regression matrix Y at each moment. k , k is the number of time interval points, and then all regression matrices Y k Merge into a total matrix Y:

[0103]

[0104] The actual hydraulic drive torque τ i for:

[0105] τ i =(P ai A ai -P bi A bi )·r i

[0106] Where i is the joint number, P ai , P bi A is the pressure of the rodless chamber and the rod chamber of the hydraulic cylinder, measured by the pressure sensor. ai , A bi is the area of ​​the rodless cavity and the rod cavity of the hydraulic cylinder, r i It is the effective force arm of the hydraulic cylinder.

[0107] Step 4: Calculate the dynamic parameters of the flexible arm based on the hydraulic driving torque τ and regression matrix Y obtained in step 3

[0108] The dynamic parameters are solved as follows, and the minimum inertia parameter set L is obtained after the solution:

[0109] L=(Y T Y) -1 Y T Q

[0110] Step 5, reference Figure 2Taking the second link of the flexible hydraulic robotic arm as an example, data was measured by four inclination sensors installed at three locations of the arm segment and the inclination sensor at the head end of the arm (near the base end), and the analytical expression of the deflection deformation of the arm end was solved.

[0111] The expression of the deflection deformation of the arm end of the flexible hydraulic manipulator based on the measurement of the inclination sensor is:

[0112]

[0113] in,

[0114]

[0115]

[0116]

[0117] Among them, n is the serial number of each arm, a, b, c are the installation positions of the inclination sensor, γ a , γ b , γ c It is the angle measured by the inclination sensor at each installation point.

[0118] Step 6, reference Figure 3 By analyzing the kinematic characteristics of the three-degree-of-freedom flexible hydraulic manipulator, the end position of the flexible arm is solved. The Jacobian matrix of the flexible hydraulic manipulator containing the deflection deformation can be solved by taking the partial derivative of the joint angle with the end position and combining it with the analytical expression of the deflection deformation of the arm end solved in step 5.

[0119] The position expression of the end position of the three-degree-of-freedom flexible hydraulic manipulator is:

[0120] X arm =l1cosθ1+w1sinθ1+l2cos(θ1+θ2)+w2sin(θ1+θ2)+l3 cos(θ1+θ2+θ3)+w3 sin(θ1+θ2+θ3)

[0121] Y arm =l1sinθ1-w1cosθ1+l2sin(θ1+θ2)-w2 cos(θ1+θ2)+l3sin(θ1+θ2+θ3)-w3 cos(θ1+θ2+θ3)

[0122] The partial derivative of the end position of the flexible hydraulic manipulator with respect to the joint angle is as follows:

[0123]

[0124] in,

[0125] J11 =-l1sinθ1+w1cosθ1-l2sin(θ1+θ2)+w2cos(θ1+θ2)-l3sin(θ1+θ2+θ3)+w3cos(θ1+θ2+θ3)

[0126] J 12 =-l2sin(θ1+θ2)+w2cos(θ1+θ2)-l3sin(θ1+θ2+θ3)+w3cos(θ1+θ2+θ3)

[0127] J 13 =-l3sin(θ1+θ2+θ3)+w3cos(θ1+θ2+θ3)

[0128] J 21 =l1cosθ1+w1sinθ1+l2cos(θ1+θ2)+w2sin(θ1+θ2)+l3cos(θ1+θ2+θ3)+w3sin(θ1+θ2+θ3)

[0129] J 22 =l2cos(θ1+θ2)+w2sin(θ1+θ2)+l3cos(θ1+θ2+θ3)+w3sin(θ1+θ2+θ3)

[0130] J 23 =l3cos(θ1+θ2+θ3)+w3sin(θ1+θ2+θ3)

[0131] In step 7, the dynamic parameters obtained in step 4 are brought into the linear model of step 1, and the force at the end of the flexible hydraulic manipulator is calculated in combination with the flexible arm Jacobian matrix in step 6.

[0132] The end force of the flexible hydraulic manipulator arm is solved as follows:

[0133]

[0134] Where F is the end force of the manipulator; J(p) is the Jacobian matrix of the flexible hydraulic manipulator.

Claims

1. A soft measurement method of the end force based on the Jacobian matrix of a flexible hydraulic manipulator, characterized in that: The following steps are involved: Step 1: Establish a dynamic model of the flexible hydraulic manipulator. To address the problems of wide motion range and inaccurate joint friction of the flexible hydraulic manipulator, a nonlinear joint friction torque model is established. Based on the flexible arm dynamic model, the relationship between the hydraulic drive torque and the generalized force of the flexible arm is determined. The dynamic model is linearized to establish a linear model of the minimum inertia parameter set of the flexible arm and its corresponding regression matrix Y. Step 2: Design the excitation trajectory using the finite Fourier series. Minimize the condition number of the regression matrix Y in step 1, solve the finite Fourier series coefficients, and generate the excitation trajectory. Step 3: The flexible arm runs an excitation trajectory under no-load conditions, collects the pressure sensor values ​​of the two chambers of the hydraulic cylinder, and calculates the hydraulic driving torque through the pressure of the two chambers of the hydraulic cylinder. τ ; Bring the generalized modal angle, generalized angular velocity, and generalized angular acceleration at each moment into the regression matrix Y and merged into a total matrix, the generalized modal angle includes the joint angle and modal parameters; Step 4: The hydraulic driving torque obtained in step 3 τ and the regression matrix Y Calculate the dynamic parameters of the flexible arm; Step 5: Using the data from four inclination sensors installed at three locations on the arm segment and the inclination sensor at the head of the arm near the end of the base, the analytical formula for the deflection of the arm end is solved. Step 6: By analyzing the kinematic characteristics of the three-degree-of-freedom flexible arm, the end position of the flexible arm is solved. By taking the partial derivative of the joint angle in the end position and combining it with the analytical expression of the arm end deflection deformation solved in step 5, the Jacobian matrix of the flexible arm including the deflection deformation can be solved; Step 7: Substitute the dynamic parameters obtained in step 4 into the linear model of step 1, and calculate the end force of the flexible hydraulic manipulator in combination with the flexible arm Jacobian matrix in step 6; in, is the end force of the robotic arm; is the Jacobian matrix of the flexible hydraulic manipulator, represents the regression matrix Y, represents the minimum inertia parameter set, is the nonlinear joint friction torque, is the joint torque on the flexible arm, subscript is the joint number, represents the generalized angle of the flexible arm, , is the joint angle, are the modal coordinates, is the generalized angular velocity, represents the generalized angular acceleration.

2. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The dynamic model of the flexible hydraulic manipulator described in step 1 is as follows: in, is the generalized modal angle of the flexible arm, , is the joint angle, are the modal coordinates, is the generalized angular velocity, is the generalized angular acceleration, is the hydraulic driving torque, is the inertia matrix of the flexible arm, is the Coriolis force and centripetal force matrix of the flexible arm, K is the flexible arm stiffness matrix, is the gravity matrix of the flexible arm.

3. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The nonlinear joint friction torque model established in step 1 is: in, 、 、 、 are the coefficients of the Fourier series, 、 、 、 are the coefficients of the third-order polynomial and are the parameters to be identified; is the joint angle; for The first derivative with respect to time represents the joint angular velocity.

4. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The relationship between the hydraulic driving torque of the flexible arm described in step 1 and the generalized force acting on the flexible arm is: in, is the generalized force on the flexible arm, subscript I is the generalized force number, is the joint torque on the flexible arm, subscript i is the joint number.

5. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The linear model of the minimum inertia parameter set of the flexible arm described in step 1 is as follows: in, represents the minimum inertia parameter set.

6. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The finite Fourier series expression described in step 2 is as follows: represents a finite Fourier series, Indicates the length of each arm. i joints, the number of sine and cosine terms is N , t represents the running time of the excitation trajectory, is the fundamental frequency, , 、 、 To find the Fourier coefficients, we use the optimal method of multivariate function with constraints to solve it. To ensure that the robot operates stably and within the safe range, the constraints are as follows: in, Represented by regression matrix The optimization goal is to minimize the condition number. is the initial joint angle of the flexible arm. When a cycle ends, , Indicates the termination moment, the flexible arm returns to the initial joint angle of the flexible arm , so that the next cycle can be executed continuously, the initial and end The speed and acceleration are set to 0 to effectively avoid impact; 、 、 、 、 、 They are the minimum and maximum values ​​of the angle, angular velocity and angular acceleration when the robotic arm moves; by limiting the upper and lower limits of the angle, angular velocity and angular acceleration, the flexible arm is guaranteed to move within a safe range.

7. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The regression matrix described in step 3 The merged matrix is ​​to first calculate the regression matrix at each moment , k is the number of time interval points, and then all regression matrices Merge into a total matrix : 。 8. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: Based on step 3, calculate the actual hydraulic driving torque for: in, i is the joint number, , is the pressure of the rodless chamber and the rod chamber of the hydraulic cylinder, measured by the pressure sensor, , is the area of ​​the rodless cavity and the rod cavity of the hydraulic cylinder, It is the effective force arm of the hydraulic cylinder.

9. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The dynamic parameters described in step 4 are solved as follows, and the minimum inertia parameter set is obtained after solving : Q is the linear model of the minimum inertia parameter set of the flexible arm.

10. The method for soft measurement of end force based on Jacobian matrix of flexible hydraulic manipulator according to claim 1, characterized in that: The expression of the deflection deformation of the flexible hydraulic manipulator arm end measured by the inclination sensor in step 5 is: in, in, n is the serial number of each arm, , b , c The installation position of the tilt sensor. , , The angle measured by the inclination sensor at each installation point; In step 6, the flexible hydraulic manipulator X Direction and Y The end position expression of the direction is: ； represents the angle of joint 1, represents the angle of joint 2, represents the angle of joint three; In step 6, the partial derivative of the end position of the flexible hydraulic manipulator with respect to the joint angle is calculated as follows: 。

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