A high-performance control method for cantilever Stewart parallel mechanism of sandblasting and rust removal robot

Abstract

The present invention discloses a high-performance control method for a cantilever Stewart parallel mechanism of a sandblasting and rust removal robot. First, an analytical method is used to perform kinematic analysis on the mechanism, obtain its kinematic inverse solution, and calculate the Jacobian transformation matrix from the end of the mechanism to the six branches, and establish a dynamic model of the cantilever Stewart parallel mechanism based on the Lagrangian method; secondly, in view of the random disturbances such as parameter perturbations and sandblasting jet reaction forces in the mechanism system, a filter-based disturbance estimator is designed to accurately estimate the disturbance when the upper limit of the disturbance amplitude is unknown and the sampling frequency is low, and feedforward compensation is performed to reduce the subsequent designed sliding mode control gain and reduce the sliding mode control chattering; finally, by sending the designed control quantity to the motor drivers of each branch of the Stewart parallel mechanism, the moving platform at the end of the mechanism moves according to the desired trajectory, thereby realizing high-performance control of the sandblasting and rust removal operation. The present invention realizes high-performance trajectory tracking control of the cantilever Stewart parallel mechanism.

Description

Technical Field

[0001] The invention discloses a sandblasting and rust removal technology for a bridge steel box girder, and in particular relates to a sandblasting and rust removal parallel robot control method based on a cantilever Stewart parallel mechanism. Background Art

[0002] In order to improve the efficiency of sandblasting and rust removal of steel box girders and ensure that their surfaces have a certain degree of cleanliness and roughness, a mobile automatic sandblasting and rust removal parallel robot was independently developed based on the cantilever Stewart parallel mechanism. It consists of a parallel mechanism, a lifting mechanism and a mobile platform, etc. It has the advantages of high rigidity, strong bearing capacity and good mobility, and can achieve arbitrary movement, free lifting, and six-degree-of-freedom posture precise motion operations. Since the cantilever Stewart parallel mechanism has a spatial closed-chain structure and has kinematic constraints, it is difficult to establish a model that can fully reflect its dynamic characteristics using the Lagrangian modeling method, so there are modeling errors, and the mechanism system has random disturbances such as parameter perturbations and sandblasting jet reaction forces, which affect the accuracy and stability of the mechanism's terminal trajectory tracking. Therefore, it is necessary to study the trajectory tracking control technology of the cantilever Stewart parallel mechanism.

[0003] Reference "Time-delay estimation adaptive sliding mode control of hybrid automobile electrophoretic coating conveying mechanism" (Gao Guoqin, Zhou Huihui, Fang Zhiming. Automotive Engineering, 2018, 40(12): 1405-1412.) In order to obtain a new mathematical model of a hybrid conveying mechanism with real-time control, the time-delay estimation technology is introduced in this paper to obtain the system model containing the unknown dynamics, friction and external disturbances of the hybrid mechanism system in real time. However, when the time-delay estimation technology is used, the system control signal has oscillation and delay problems. When the sampling frequency of the control system is low, the system control accuracy and stability performance will be weakened.

[0004] The paper "Finite-time Stable Robust Sliding Mode Dynamic Control for Parallel Robots" (Ye M, Gao G, Zhong J. International Journal of Control, Automation and Systems, 2021, 19 (9): 3026-3036.) studies a finite-time stable robust sliding mode control method for the tracking control problem of parallel robots in the presence of lumped disturbances such as modeling errors, friction and external interference. However, the convergence time of the finite-time convergent control system is related to the initial value of the system. Compared with the predetermined time control, its convergence performance is poor, and its convergence time cannot be given in advance. Summary of the invention

[0005] The purpose of the present invention is to overcome the shortcomings of the above-mentioned prior art, and in view of the structural characteristics of the sandblasting and rust removal parallel robot and the sandblasting and rust removal process requirements, a predetermined time sliding mode control method for a sandblasting and rust removal cantilever Stewart parallel mechanism based on disturbance estimation is proposed. The method can achieve rapid stabilization of the sandblasting and rust removal parallel robot cantilever Stewart parallel mechanism under the influence of random disturbances such as modeling errors and parameter perturbations, sandblasting jet reaction forces, etc., and can effectively improve the trajectory tracking accuracy of the parallel mechanism even when the sampling frequency of the control system is low.

[0006] The technical solution of the present invention is: a predetermined time sliding mode control method of a cantilever Stewart parallel mechanism for sandblasting and rust removal, comprising the following steps:

[0007] 1) The kinematics of the cantilever Stewart parallel mechanism for sandblasting and rust removal were analyzed by analytical method, and the inverse kinematic solution was obtained, and the Jacobian matrix from the end moving platform to the six-branch chain was further calculated;

[0008] 2) According to the requirements of the sandblasting and rust removal process for steel box girders, determine the expected motion trajectory of the terminal moving platform of the sandblasting and rust removal parallel robot;

[0009] 3) According to the cantilever structure characteristics, the lateral translation kinetic energy of the moving platform and the potential energy of the electric cylinder legs are calculated, and the dynamic model of the cantilever Stewart parallel mechanism for sandblasting and rust removal is established;

[0010] 4) Based on the dynamic model established in step 3), a disturbance estimator is designed. The estimator can estimate the lumped disturbance of the sandblasting and rust removal parallel robot system by designing a filter and using convolution operation when the upper limit of the disturbance amplitude is unknown and the sampling frequency is low. By using the disturbance estimation value to perform system feedforward compensation, the sliding mode control gain designed subsequently can be reduced, and the sliding mode control chattering can be reduced;

[0011] 5) Based on step 4), a predetermined time sliding mode controller based on disturbance estimation is designed; the controller is a predetermined time convergent sliding mode trajectory tracking controller based on the predetermined time stability theory and combined with the non-singular terminal sliding mode control design, which can utilize the robustness of the sliding mode control and the predetermined time parameter T c , so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the vicinity of the origin within a predetermined time according to actual operation requirements;

[0012] 6) A distributed structure, i.e., a "host computer + slave computer" structure, is used to construct a predetermined time sliding mode control system for a cantilever Stewart parallel mechanism for sandblasting and rust removal based on disturbance estimation, wherein the model of the host computer is IPC-610L and the model of the slave computer is CK3M;

[0013] 7) The calculated control quantities of each active joint of the cantilever Stewart parallel mechanism for sandblasting and rust removal are sent to each motor driver to control its terminal moving platform to move according to the desired trajectory.

[0014] Further, in step 3), establishing a dynamic model of a cantilever Stewart parallel mechanism for sandblasting and rust removal includes:

[0015] The moving platform at the end of the cantilever structure has the characteristic of lateral translation, and the potential energy direction of the six electric cylinder legs changes;

[0016] The lateral translation kinetic energy of the operating end of the moving platform is:

[0017]

[0018] in, are the velocities of the three components of the moving platform in the coordinate system, m p is the mass of the moving platform;

[0019] The potential energy of the electric cylinder legs of the Stewart parallel mechanism is transformed by rotation to obtain the sum of the potential energy of the six electric cylinder legs:

[0020]

[0021] In the formula, m t ,m r are the masses of the cylinder barrel and the push rod of the electric cylinder respectively, and its structure diagram is as follows Figure 4 As shown, g is the acceleration due to gravity, r gi Indicates the center of gravity of the electric cylinder leg. r tc is the distance from the center of gravity of the electric cylinder to the rotation center of the lower hinge point. Similarly, r rc It is the distance from the center of gravity of the electric cylinder push rod to the center of rotation of the upper hinge point. represents the coordinate components of the rotation center of the hinge point on the moving platform of the Stewart parallel mechanism in the motion coordinate system {p}, α, β, γ are the rotation angles of the moving platform around the X, Y, and Z axes, respectively, which are called the roll angle, pitch angle, and yaw angle;

[0022] Considering that the dynamic model of the cantilever Stewart parallel mechanism has modeling errors and is affected by random disturbances such as parameter perturbations and the reaction force of the sandblasting jet in the actual working environment, the dynamic model of the cantilever Stewart parallel mechanism of the sandblasting and rust removal parallel robot is established as follows:

[0023]

[0024] Where M0(q) is the mass matrix, is the Coriolis / centripetal force coefficient matrix, G(q) is the gravity matrix, ΔM(q), and ΔG(q) represents its modeling error, d ext represents random disturbance, Q is the generalized force in the workspace corresponding to the generalized coordinate q;

[0025] Treat modeling errors and random disturbances as lumped disturbances

[0026] Further, in step 4), a filter-based disturbance estimator is designed to estimate the lumped disturbance existing in the cantilever Stewart parallel mechanism system for sandblasting and rust removal, characterized in that:

[0027] The estimated value of this disturbance estimator is:

[0028]

[0029] In the formula, * is the convolution operator, G f is the filter g f The frequency domain expression of L -1 is the inverse Laplace transform.

[0030] Further, step 5) designs a predetermined time sliding mode controller based on disturbance estimation, so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the neighborhood of the origin within a predetermined time according to actual operation requirements, thereby reducing the convergence time, specifically:

[0031] For the cantilever Stewart parallel mechanism for sandblasting and rust removal, the position error of each active joint motion is first defined:

[0032] e1=qq d (5)

[0033] In the formula, the vector q d and q are the expected position and actual position of each active joint of the Stewart parallel mechanism, respectively;

[0034] The velocity error of each active joint motion:

[0035]

[0036] Design a non-singular terminal sliding surface:

[0037]

[0038] Where, parameter n=[n1,n2,…,n n ] T , Parameter c1 = 1 + n, parameter c2 = -n, parameter T cis the predetermined convergence time parameter, l is the convergence accuracy, and ε represents the disturbance estimation error;

[0039] Design a scheduled time sliding mode controller based on disturbance estimation as:

[0040]

[0041] In the formula,

[0042]

[0043] G1-G5 are intermediate variables; λ is a diagonal matrix, exp(k) represents the exponential function of the natural number e, ε is the estimation error, |ε|≤ε max ,ε max >0, M0 is the mass matrix, e1 is the position error vector of each active joint motion, e2 is the velocity error vector of each active joint motion, T d is the scheduled time parameter, m=[m1,m2,…,m n ] T ,

[0044]

[0045] The present invention proposes for the first time a sliding mode control method for a cantilever Stewart parallel mechanism for sandblasting and rust removal based on disturbance estimation, which realizes high-performance control of a cantilever Stewart parallel mechanism for a sandblasting and rust removal robot. The characteristics and beneficial effects are as follows:

[0046] 1) According to the cantilever structure characteristics of the mechanism, the Lagrangian function method is used to establish the dynamic model of the cantilever Stewart parallel mechanism. The established dynamic model better describes the dynamic characteristics of the cantilever structure of the Stewart parallel mechanism.

[0047] 2) Design a disturbance estimator for random disturbances such as parameter perturbations and sandblasting jet reaction forces in the mechanism system. This estimator can estimate the lumped disturbance of the sandblasting and rust removal parallel robot system by designing a filter and using convolution operations when the upper limit of the disturbance amplitude is unknown and the sampling frequency is low. By using the disturbance estimate value for system feedforward compensation, the sliding mode control gain designed later can be reduced, and the sliding mode control chattering can be reduced.

[0048] 3) Based on the predetermined time stability theory and combined with non-singular terminal sliding mode control, a predetermined time convergence sliding mode trajectory tracking controller is designed. By utilizing the robustness of sliding mode control and the predeterminable time parameters, the tracking errors of the active joints of the cantilever Stewart parallel mechanism can converge quickly and stably to the neighborhood of the origin within the predetermined time according to actual operation requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0050] Figure 1 This is a real picture of the sandblasting and rust removal parallel robot.

[0051] Figure 2 This is a common structural diagram of a large steel box girder.

[0052] Figure 3 This is a physical picture of the cantilever Stewart parallel mechanism of the sandblasting and rust removal robot.

[0053] Figure 4 It is a schematic diagram of the electric cylinder leg structure.

[0054] Figure 5 It is the kinematic geometric model of the cantilever Stewart parallel mechanism.

[0055] Figure 6 It is the expected trajectory diagram of each active joint of the cantilever Stewart parallel mechanism.

[0056] Figure 7 It is a schematic diagram of a high-performance control method for a cantilever Stewart parallel mechanism of a sandblasting and rust removal robot.

[0057] Figure 8 It is a scheduled time sliding mode control system of a sandblasting and rust removal parallel robot based on disturbance estimation.

[0058] Figure 1 In: 1-sandblasting gun, 2-Stewart parallel mechanism for sandblasting and rust removal-mobile platform drive motor, 3-sandblasting hose, 4-mobile platform, 5-lifting mechanism, 6-electrical control cabinet, 7-host computer. DETAILED DESCRIPTION

[0059] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings. Figure 1 As shown, 1-sandblasting gun, 2-sandblasting and rust removal Stewart parallel mechanism-mobile platform drive motor, 3-sandblasting hose, 4-mobile platform, 5-lifting mechanism, 6-electrical control cabinet, 7-host computer. The sandblasting and rust removal parallel robot is mainly composed of the following parts: mobile platform, lifting platform, Stewart parallel mechanism, steel sand delivery hose, straight rigid support cylinder rod, sandblasting gun. Its main function is to perform sandblasting and rust removal on steel box girders. Steel box girders are the structural form of large-span bridges. Because they look like boxes, they are called steel box girders. The structural diagram is as follows Figure 2As shown, it is large in size, trapezoidal in shape and composed of many U-shaped ribs. In order to meet the requirements of sandblasting and rust removal, firstly, the analytical method is used to perform kinematic inverse analysis on the Stewart parallel mechanism for sandblasting and rust removal, and then the Jacobian matrix J of the Stewart parallel mechanism is obtained. lq ; Secondly, according to the requirements of the sandblasting and rust removal process of steel box girders, the expected motion trajectory of the sandblasting and rust removal parallel robot spray gun is determined; then, the Lagrangian method is used to establish the dynamic model of the cantilever Stewart parallel mechanism and a filter-based disturbance estimator is designed to estimate the disturbance. Furthermore, a non-singular terminal sliding mode controller with a predetermined time convergence is designed to utilize the robustness of the sliding mode control and the predetermined time parameter T c , so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the origin neighborhood within a predetermined time according to the actual operation requirements. Finally, a sandblasting and rust removal parallel robot control system is constructed using a distributed structure, and the control quantity is sent to the motor driver to control the spray gun to move along the desired trajectory.

[0060] The specific method is as follows:

[0061] 1. The analytical method is used to perform kinematic analysis on the cantilever Stewart parallel mechanism for sandblasting and rust removal, and the inverse kinematic solution is obtained, and the Jacobian matrix from the end moving platform to the six-branch chain is further calculated.

[0062] Select the posture parameter q=[x, y, z, α, β, γ] of the midpoint of the Stewart parallel mechanism for sandblasting and rust removal T As the generalized coordinates of the system, x, y, z are the displacements of the moving platform in the x-axis, y-axis, and z-axis directions (unit: m); α, β, and γ are the rotation angles of the moving platform around the X-axis, Y-axis, and Z-axis (unit: rad). The inverse kinematics equation of the mechanism is analyzed by analytical method, and the inverse solution coefficient matrix is ​​the Jacobian matrix, which is expressed as:

[0063]

[0064] In the formula, is the velocity vector of the midpoint of the moving platform, The unit is m / s, w=[w x w y w z ] T , the generalized velocity vector in formula (1) The relationship between the angular velocity vector w can be expressed as:

[0065]

[0066] In the formula, is the matrix composed of the reciprocating linear motion speeds of the six electric cylinders, J lq is the Jacobian matrix.

[0067] 2. Determine the expected motion trajectory of the sandblasting and rust removal parallel robot spray gun based on the sandblasting and rust removal process requirements of the steel box girder.

[0068] According to the sandblasting and rust removal process requirements of the spray gun sweeping left and right and moving up and down, determine the expected motion trajectory q of the Stewart parallel mechanism moving platform for sandblasting and rust removal d .

[0069] 3. The dynamic equation of the Stewart parallel mechanism of the sandblasting and rust removal parallel robot is:

[0070]

[0071] In the formula, is the inertia matrix, are the Coriolis force and centrifugal force terms; is the gravity term; q is the pose vector of the end effector, is the velocity vector of the end effector, is the acceleration vector of the end effector, J is the Jacobian matrix of the parallel mechanism; τ is the driving force acting on each active joint (in N·m).

[0072] The end moving platform of the cantilever structure has the characteristic of lateral translation, and the potential energy direction of the six electric cylinder legs changes. The kinetic energy of the end moving platform is divided into lateral translation kinetic energy and rotational kinetic energy. The lateral translation kinetic energy is related to the speed of the moving platform, and the rotational kinetic energy is related to the angular velocity of the moving platform. According to the kinetic energy theorem, the kinetic energy expression of the moving platform operation end is as follows:

[0073]

[0074] Where ω is the angular velocity vector of the moving platform, I P is the unit matrix, m p It is the mass of the moving platform.

[0075]

[0076] The potential energy of the operating end of the moving platform is:

[0077] P p =m p gz=[0 0 m p g 0 0 0] T q (6)

[0078] Kinetic energy and potential energy of electric cylinder outriggers:

[0079]

[0080]

[0081] Where M l Indicates the mass of the electric cylinder outrigger, L ni is the axial unit vector of the i-th electric cylinder, z is the matrix composed of the unit vectors of the rotation centers of the six upper hinges on the Z axis of the motion coordinate system, where z ai Indicates A i The value on the Z axis. The rotation change is:

[0082]

[0083] From equations (8) and (9), the potential energy of the six electric cylinder legs of the cantilever Stewart parallel mechanism can be obtained as:

[0084]

[0085] The gravity term of the electric cylinder outrigger is:

[0086]

[0087] In the formula, m t ,m r are the masses of the cylinder barrel and the push rod of the electric cylinder respectively, and its structure diagram is as follows Figure 4 As shown, g is the acceleration due to gravity, r gi Indicates the center of gravity of the electric cylinder leg. It represents the coordinate components of the rotation center of the hinge on the moving platform of the Stewart parallel mechanism in the motion coordinate system. α, β, and γ are the rotation angles of the moving platform around the X, Y, and Z axes, respectively, which are called the roll angle, pitch angle, and yaw angle.

[0088] Considering that the dynamic model of the cantilever Stewart parallel mechanism has modeling errors and is affected by random disturbances such as parameter perturbations and the reaction force of the sandblasting jet in the actual working environment, the dynamic model of the cantilever Stewart parallel mechanism of the sandblasting and rust removal parallel robot is established as follows:

[0089]

[0090] In the formula, ΔM(q), and ΔG(q) represents the modeling error of the system, d ext represents random disturbance, and Q is the generalized force in the workspace corresponding to the generalized coordinate q.

[0091] Treat modeling errors and random disturbances as lumped disturbances

[0092] 4. According to step 4), a filter-based disturbance estimator is designed to estimate the lumped disturbance existing in the system.

[0093] Its estimated value is:

[0094]

[0095] In the formula, * is the convolution operator, G f is the filter g f The frequency domain expression of L -1 is the inverse Laplace transform.

[0096] 5. According to step 5), a predetermined time sliding mode controller based on disturbance estimation is designed so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the neighborhood of the origin within the predetermined time according to actual operation requirements.

[0097] First, define the position error of each active joint motion:

[0098] e1=qq d (14)

[0099] In the formula, the vector q d and q are the expected position and actual position of each active joint of the Stewart parallel mechanism, respectively.

[0100] The velocity error of each active joint motion:

[0101]

[0102] Design a non-singular terminal sliding surface:

[0103]

[0104] Where η=[η1,η2,…,η n ] T , c1=1+η, T c is the predetermined convergence time parameter, l is the convergence accuracy, and ε represents the disturbance estimation error.

[0105] Design a scheduled-time non-singular terminal sliding mode controller based on disturbance estimation as:

[0106]

[0107] In the formula,

[0108]

[0109] In the formula λ is a diagonal matrix, exp(k) represents the exponential function of the natural number e, ε is the estimation error, |ε|≤ε max ,ε max >0, T d is the scheduled time parameter, m=[m1,m2,…,m n ] T ,

[0110] 6. A distributed structure is used to construct a predetermined time sliding mode control system for a cantilever Stewart parallel mechanism for sandblasting and rust removal based on disturbance estimation.

[0111] The control system adopts a distributed structure of "upper computer industrial computer + lower computer Omron CK3M multi-axis motion controller" to construct a cantilever Stewart parallel mechanism scheduled time sliding mode control system based on disturbance estimation for the sandblasting and rust removal parallel robot.

[0112] 7. Send the calculated control quantities of each active joint of the cantilever Stewart parallel mechanism for sandblasting and rust removal to each motor driver to control its end moving platform to move according to the desired trajectory.

[0113] According to formula (17) in step 5, the drive control quantity is calculated and programmed by the host computer. Figure 6 The control system shown is sent to the motor drivers of each active joint of the cantilever Stewart parallel mechanism for sandblasting and rust removal to drive the robot to move along the desired trajectory.

[0114] An embodiment of the present invention is provided below:

[0115] Example 1

[0116] The control method of the present invention is mainly based on a disturbance estimation, and designs a predetermined time sliding mode control algorithm for the cantilever Stewart parallel mechanism of the sandblasting and rust removal, so as to realize high-performance control of the cantilever Stewart parallel mechanism of the sandblasting and rust removal robot. The specific implementation method of the method is as follows:

[0117] 1. The analytical method is used to perform kinematic analysis on the cantilever Stewart parallel mechanism for sandblasting and rust removal, and the inverse kinematic solution is obtained, and the Jacobian matrix from the end moving platform to the six-branch chain is further calculated.

[0118] exist Figure 4 In the moving coordinate system, any vector Transformed into vector A in the base coordinate system by coordinate transformation method i , thus obtaining:

[0119]

[0120] Where t = [xyz] T O is the origin of the moving coordinate system p The vector representation in the base coordinate system is: 3×3 is the rotation transformation matrix from the moving coordinate system to the base coordinate system. Different attitude angles, positive directions, and rotation orders result in different rotation transformation matrices. The following definitions are given for the above parameters:

[0121] (1) α, β, and γ represent the rotation angles around the X, Y, and Z axes of the coordinate system, namely the roll angle, pitch angle, and yaw angle, respectively;

[0122] (2) The rotation order is XYZ. The rotation matrix R can be expressed as:

[0123]

[0124] Therefore, the vector equation of the electric cylinder leg can be expressed as:

[0125]

[0126] In the formula, B i Coordinate description of the rotation center of the hinge point under the fixed platform in the reference coordinate system.

[0127] (3) Jacobian matrix of cantilever Stewart parallel mechanism:

[0128] The generalized velocity of the cantilever Stewart parallel mechanism moving platform can be described as:

[0129]

[0130] Where ω is the angular velocity vector of the motion platform in the reference coordinate system, ω=[ω x ,ω y ,ω z ] T , is the velocity representation of t.

[0131] In the reference coordinate system, according to the angular velocity synthesis theorem of space motion, the conversion relationship between the angular velocity vector of the moving platform and the differential of its attitude angle with respect to time can be expressed as:

[0132]

[0133] By differentiating equation (19), we can obtain the matrix form of the generalized velocity of each upper hinge point in the reference coordinate system:

[0134]

[0135] Where v a is the matrix composed of the velocity vectors of each upper hinge point in the reference coordinate system, is the antisymmetric spiral matrix of each upper hinge point, I is the third-order unit matrix, J a,q is the Jacobian matrix of velocity transformation from the end moving platform to each upper hinge point.

[0136] Projecting the hinge point velocity to the direction of the electric cylinder, the expression of the telescopic velocity matrix of the six electric cylinders is:

[0137]

[0138] In the formula is the matrix composed of the extension and contraction velocity vectors of the six electric cylinders along the axis direction, L n is the matrix composed of the six electric cylinder axial unit vectors, J lq It is the Jacobian matrix from the generalized velocity of the moving platform to the extension and retraction velocity of the electric cylinder.

[0139] 2. Determine the expected motion trajectory of the sandblasting and rust removal parallel robot spray gun based on the sandblasting and rust removal process requirements of the steel box girder.

[0140] According to the sandblasting and rust removal process requirements of the spray gun sweeping left and right and moving up and down, determine the expected motion trajectory q of the Stewart parallel mechanism moving platform for sandblasting and rust removal d .

[0141] 3. The dynamic equation of the cantilever Stewart parallel mechanism of the sandblasting and rust removal parallel robot is:

[0142]

[0143] In the formula, is the inertia matrix, are the Coriolis force and centrifugal force terms; is the gravity term; q is the pose vector of the end effector, is the velocity vector of the end effector, is the acceleration vector of the end effector, J is the Jacobian matrix of the parallel mechanism; τ is the driving force acting on each active joint (in N·m).

[0144] The end moving platform of the cantilever structure has the characteristic of lateral translation, and the potential energy direction of the six electric cylinder legs changes. The kinetic energy of the end moving platform is divided into lateral translation kinetic energy and rotational kinetic energy. The translational kinetic energy is related to the speed of the moving platform, and the rotational kinetic energy is related to the angular velocity of the moving platform. According to the kinetic energy theorem, the kinetic energy expression of the moving platform operation end is as follows:

[0145]

[0146] The kinetic energy of the operating end of the dynamic platform is:

[0147]

[0148] The potential energy of the operating end of the moving platform is:

[0149] P p =m p gz=[0 0 m p g 0 0 0] T q (29)

[0150] Kinetic energy and potential energy of electric cylinder outriggers:

[0151]

[0152]

[0153] Where, L ni is the axial unit vector of the i-th electric cylinder, z is the matrix composed of the unit vectors of the rotation centers of the six upper hinges on the Z axis of the motion coordinate system, where z ai Indicates A i The value on the Z axis is obtained by rotating:

[0154]

[0155] From equation (31) and equation (32), we can get:

[0156]

[0157] The gravity term of the electric cylinder outrigger is:

[0158]

[0159] Considering that the dynamic model of the cantilever Stewart parallel mechanism has modeling errors and is affected by random disturbances such as the reaction force of the sandblasting jet and parameter perturbations in the actual working environment, the dynamic model of the cantilever Stewart parallel mechanism of the sandblasting and rust removal parallel robot is established as follows:

[0160]

[0161] In the formula, ΔM(q), and ΔG(q) represents the modeling error of the system, d ext represents random disturbance, and Q is the generalized force in the workspace corresponding to the generalized coordinate q.

[0162] Treat modeling errors and random disturbances as lumped disturbances

[0163] 4. According to step 4), a filter-based disturbance estimator is designed to estimate the lumped disturbance existing in the system.

[0164] Its estimated value is:

[0165]

[0166] In the formula, * is the convolution operator, G f is the filter g f The frequency domain expression of L -1 is the inverse Laplace transform.

[0167] 5. Based on step 5), a predetermined time sliding mode controller based on disturbance estimation is designed so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the neighborhood of the origin within the predetermined time according to actual operation requirements.

[0168] First, define the position error of each active joint motion:

[0169] e1=qq d (37)

[0170] In the formula, the vector q d and q are the expected position and actual position of each active joint of the Stewart parallel mechanism, respectively.

[0171] The velocity error of each active joint motion:

[0172]

[0173] Design a non-singular terminal sliding surface:

[0174]

[0175] Where η=[η1,η2,…,η n ] T , c1=1+η,c2=-η, T c is the predetermined convergence time parameter, l is the convergence accuracy, and ε represents the disturbance estimation error.

[0176] The scheduled-time non-singular terminal sliding mode controller based on disturbance estimation is designed as:

[0177]

[0178] In the formula,

[0179]

[0180] In the formula λ is a diagonal matrix, n}, exp(k) represents the exponential function of the natural number e, ε is the estimation error, |ε|≤ε max ,ε max >0, T d is the scheduled time parameter, m=[m1,m2,…,m n ] T ,

[0181] 6. A distributed structure is used to construct a predetermined time sliding mode control system for a cantilever Stewart parallel mechanism for sandblasting and rust removal based on disturbance estimation.

[0182] The control system adopts a distributed structure of "upper computer industrial computer + lower computer Omron CK3M multi-axis motion controller" to construct a cantilever Stewart parallel mechanism scheduled time sliding mode control system based on disturbance estimation for the sandblasting and rust removal parallel robot.

[0183] 7. Send the calculated control quantities of each active joint of the cantilever Stewart parallel mechanism for sandblasting and rust removal to each motor driver to control its end moving platform to move according to the desired trajectory.

[0184] In summary, the present invention discloses a predetermined time sliding mode trajectory tracking control method based on disturbance estimation for a cantilever Stewart parallel mechanism of a bridge steel box girder sandblasting and rust removal robot. Firstly, an analytical method is used to perform kinematic analysis on the mechanism to obtain its kinematic inverse solution, and the Jacobian transformation matrix from the end of the mechanism to the six branches is calculated, and a dynamic model of the cantilever Stewart parallel mechanism is established based on the Lagrangian method; secondly, in view of the random disturbances such as parameter perturbations and sandblasting jet reaction forces in the mechanism system, a filter-based disturbance estimator is designed to accurately estimate the disturbance when the upper bound of the disturbance amplitude is unknown and the sampling frequency is low, and feedforward compensation is performed to reduce the sliding mode control gain designed subsequently and reduce the sliding mode control jitter; further, a predetermined time convergent non-singular terminal sliding mode controller is designed to utilize the robustness of the sliding mode control and the predetermined time parameter T c, so that the tracking error of the cantilever Stewart parallel mechanism can be stably converged to the neighborhood of the origin within a predetermined time according to the actual operation requirements; finally, by sending the designed control quantity to the motor drivers of each branch of the Stewart parallel mechanism, the moving platform at the end of the mechanism moves according to the desired trajectory, thereby realizing high-performance control of the sandblasting and rust removal operation. The present invention is aimed at the cantilever Stewart parallel mechanism for sandblasting and rust removal, and the proposed sliding mode trajectory tracking control method based on disturbance estimation and predetermined time can estimate and compensate for random disturbances such as parameter perturbations and sandblasting jet reaction forces in the mechanism system, and at the same time can achieve the predetermined time stable convergence performance of the sandblasting and rust removal control system, thereby effectively improving the control accuracy of the sandblasting operation, reducing the convergence time, and realizing high-performance trajectory tracking control of the cantilever Stewart parallel mechanism.

Claims

1. A method for controlling the predetermined time sliding mode of a cantilever Stewart parallel mechanism for sandblasting and rust removal, characterized in that: The following steps are involved: 1) The kinematics of the cantilever Stewart parallel mechanism for sandblasting and rust removal were analyzed by analytical method, and the inverse kinematic solution was obtained, and the Jacobian matrix from the end moving platform to the six-branch chain was further calculated; 2) According to the requirements of the sandblasting and rust removal process for steel box girders, determine the expected motion trajectory of the terminal moving platform of the sandblasting and rust removal parallel robot; 3) According to the cantilever structure characteristics, the lateral translation kinetic energy of the moving platform and the potential energy of the electric cylinder legs are calculated, and the dynamic model of the cantilever Stewart parallel mechanism for sandblasting and rust removal is established; 4) Based on the dynamic model established in step 3), a disturbance estimator is designed. The estimator can estimate the lumped disturbance of the sandblasting and rust removal parallel robot system by designing a filter and using convolution operation when the upper limit of the disturbance amplitude is unknown and the sampling frequency is low. By using the disturbance estimation value to perform system feedforward compensation, the sliding mode control gain designed subsequently can be reduced, and the sliding mode control chattering can be reduced; 5) designing a scheduled time sliding mode controller based on disturbance estimation based on step 4); The controller is a time-scheduled convergent sliding mode trajectory tracking controller based on the time-scheduled stability theory and combined with the non-singular terminal sliding mode control design. It can utilize the robustness of sliding mode control and the predictable time parameter T c , so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the vicinity of the origin within a predetermined time according to actual operation requirements; 6) A distributed structure, i.e., "host computer + slave computer" structure, is used to construct a cantilever Stewart parallel mechanism scheduled time sliding mode control system for sandblasting and rust removal based on disturbance estimation, wherein the host computer model is IPC-610L and the slave computer model is CK3M; 7) Send the calculated control quantities of each active joint of the cantilever Stewart parallel mechanism for sandblasting and rust removal to each motor driver to control its end moving platform to move according to the desired trajectory; In step 4), a filter-based disturbance estimator is designed to estimate the lumped disturbance existing in the cantilever Stewart parallel mechanism system for sandblasting and rust removal, characterized in that: The estimated value of this disturbance estimator is: In the formula, * is the convolution operator, G f is the filter g f The frequency domain expression of L -1 is the inverse Laplace transform; d is the lumped disturbance; Step 5) Design a preset time sliding mode controller based on disturbance estimation so that the tracking error of the cantilever Stewart parallel mechanism can converge quickly and stably to the neighborhood of the origin at a preset time according to actual operation requirements, thereby reducing the convergence time. Specifically, For the cantilever Stewart parallel mechanism for sandblasting and rust removal, the position error of each active joint motion is first defined: e1=qq d (5) In the formula, the vector q d and q are the expected position and actual position of each active joint of the Stewart parallel mechanism, respectively; The velocity error of each active joint motion: Design a non-singular terminal sliding surface: Where, parameter n=[n1,n2,…,n n ] T , Parameter c1 = 1 + n, parameter c2 = -n, parameter T c is the predetermined convergence time parameter, l is the convergence accuracy, and ε represents the disturbance estimation error; Design a scheduled time sliding mode controller based on disturbance estimation as: In the formula, G1-G5 are intermediate variables; λ is a diagonal matrix, exp(k) represents the exponential function of the natural number e, ε is the estimation error, |ε|≤ε max ,ε max >0, M0 is the mass matrix, e1 is the position error vector of each active joint motion, e2 is the velocity error vector of each active joint motion, T d is the scheduled time parameter, m=[m1,m2,…,m n ] T , 2. The method according to claim 1, characterized in that: In step 3), the dynamic model of the cantilever Stewart parallel mechanism for sandblasting and rust removal is established, including: The moving platform at the end of the cantilever structure has the characteristic of lateral translation, and the potential energy direction of the six electric cylinder legs changes; The lateral translation kinetic energy of the operating end of the moving platform is: in, are the velocities of the three components of the moving platform in the coordinate system, m p is the mass of the moving platform; The potential energy of the electric cylinder legs of the Stewart parallel mechanism is transformed by rotation to obtain the sum of the potential energy of the six electric cylinder legs: In the formula, m t ,m r are the masses of the cylinder barrel and the push rod of the electric cylinder, g is the acceleration due to gravity, r gi Indicates the center of gravity of the electric cylinder leg. r tc is the distance from the center of gravity of the electric cylinder to the rotation center of the lower hinge point. Similarly, r rc It is the distance from the center of gravity of the electric cylinder push rod to the center of rotation of the upper hinge point. represents the coordinate components of the rotation center of the hinge point on the moving platform of the Stewart parallel mechanism in the motion coordinate system {p}, α, β, γ are the rotation angles of the moving platform around the X, Y, and Z axes, respectively, which are called the roll angle, pitch angle, and yaw angle; Considering that the dynamic model of the cantilever Stewart parallel mechanism has modeling errors and is affected by random disturbances such as parameter perturbations and the reaction force of the sandblasting jet in the actual working environment, the dynamic model of the cantilever Stewart parallel mechanism of the sandblasting and rust removal parallel robot is established as follows: Where M0(q) is the mass matrix, is the Coriolis / centripetal force coefficient matrix, G(q) is the gravity matrix, ΔM(q), and ΔG(q) represents its modeling error, d ext represents random disturbance, Q is the generalized force in the workspace corresponding to the generalized coordinate q; Treat modeling errors and random disturbances as lumped disturbances

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