Aismc compensation control method for flexible joint robot arm under singular perturbation decomposition

Abstract

The application discloses an AISMC compensation control method for a flexible joint mechanical arm under singular perturbation decomposition and belongs to the field of flexible joint mechanical arm control, and comprises the following steps: S1, a dynamic model of the flexible joint mechanical arm containing uncertain composite disturbance is established, and is decoupled into a slow subsystem and a fast subsystem; S2, a lumped matching disturbance in the slow subsystem is estimated by using a compensation function observer; S3, for the slow subsystem, an integral sliding mode surface and an adaptive approaching law with a dead zone correction are constructed; for the fast subsystem, a linear quadratic regulator is designed, a performance index is defined, and Riccati equation is solved to obtain an optimal control law; S4, the control law of the slow subsystem and the control law of the fast subsystem are superposed to obtain an overall control input; and S5, simulation verification is performed. The AISMC compensation control method for the flexible joint mechanical arm under singular perturbation decomposition has wide application prospects by taking into account the trajectory tracking accuracy, robustness and stability.

Description

Technical Field

[0001] This invention relates to the field of flexible joint robotic arm control technology, and in particular to an AISMC compensation control method for flexible joint robotic arms under singular perturbation decomposition. Background Technology

[0002] With the rapid development of industrial automation, medical robotics, smart agriculture, and other fields, flexible joint robotic arms, with their advantages of lightweight, low energy consumption, and high compliance, have become core execution units in complex scenarios such as precision assembly, surgical assistance, and dynamic sorting. Their core requirements are to achieve high-precision trajectory tracking, effectively suppress elastic vibration, and withstand the influence of complex working conditions such as load changes and external disturbances.

[0003] However, the dynamic characteristics of flexible joint robotic arms determine the difficulty of control: the elastic deformation of the joints leads to strong nonlinearity and strong coupling characteristics of the system, and trajectory tracking and vibration suppression are mutually restrictive; in actual working conditions, there are both non-matching disturbances on the link side and matching disturbances on the motor side. The two types of disturbances have different channels of action and are difficult to handle in a unified manner; moreover, industrial production has put forward clear requirements for the real-time performance of control algorithms, the ease of parameter tuning, and the engineering feasibility, which further exacerbates the control challenges.

[0004] The mainstream control technologies for flexible joint robotic arms currently include the following categories:

[0005] 1. Traditional linear control technology: Represented by PID control, it achieves basic position control through proportional-integral-derivative adjustment and is widely used in scenarios with low precision and low dynamic requirements. It has the advantages of simple structure and easy implementation, but it has poor adaptability to the strong nonlinearity of flexible joints and time-varying parameters, insufficient robustness to complex disturbances, and cannot simultaneously meet the requirements of trajectory accuracy and vibration suppression, resulting in large control errors.

[0006] 2. Sliding Mode Control Techniques: Traditional sliding mode control (SMC) and integral sliding mode control (ISMC) rely on switching characteristics to improve robustness to uncertainties and are commonly used solutions for anti-interference control. However, traditional SMC suffers from high-frequency chattering, which can easily induce joint elastic vibration and damage actuators. Although ISMC eliminates the arrival phase, it lacks a parameter drift suppression mechanism, measurement noise can easily cause abnormal gain fluctuations, and it does not have system decoupling characteristics, making it difficult to handle strong coupling problems.

[0007] 3. Intelligent control technologies: such as neural network control and fuzzy control, which approximate complex dynamic models through nonlinear mapping to handle strongly nonlinear problems. Neural network control requires a large amount of sample data for training and is prone to overfitting; the rule formulation of fuzzy control relies on expert experience and lacks universality and systematicity; both technologies suffer from high computational load and insufficient real-time performance, making them difficult to implement in engineering. Summary of the Invention

[0008] The purpose of this invention is to provide an AISMC compensation control method for flexible joint robotic arms under singular perturbation decomposition, thereby solving the aforementioned technical problems.

[0009] To achieve the above objectives, this invention provides an AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition, comprising the following steps:

[0010] S1. Establish a dynamic model of a flexible joint manipulator with uncertain composite perturbations, and decouple the coupled dynamic system of the flexible joint manipulator with uncertain composite perturbations into a slow-varying subsystem and a fast-varying subsystem based on singular perturbation theory;

[0011] S2. Use the compensation function observer to estimate the lumped matching disturbance in the slow-varying subsystem, determine the observer gain parameter through pole placement and verify its exponential stability;

[0012] S3. Design an adaptive integral sliding mode controller for the slow-varying subsystem. Based on the lumped matching disturbance, construct the integral sliding surface and the adaptive reaching law with dead zone correction. Derive the control law combined with disturbance feedforward compensation and verify the asymptotic stability of the slow-varying subsystem.

[0013] A linear quadratic regulator is designed for the fast variable subsystem. The performance index is defined and the Riccati equation is solved to obtain the optimal control law, thereby achieving optimal damping suppression of elastic vibration and verifying the asymptotic stability of the fast variable subsystem.

[0014] S4. The control law of the slow variable subsystem is superimposed with the control law of the fast variable subsystem to obtain the overall control input, and the asymptotic stability of the overall closed-loop system is verified based on Tikhonov's theorem.

[0015] S5. Simulation verification.

[0016] Therefore, the AISMC compensation control method for flexible joint robotic arms under singular perturbation decomposition described above has the following beneficial effects:

[0017] This invention utilizes singular perturbation theory and, through quasi-steady-state assumptions and variable transformations, successfully decouples the original complex system into two subsystems: a slowly variable one and a quickly variable one. This not only significantly reduces the complexity of controller design but also transforms unmatched disturbances into equivalent matched disturbances in the slowly variable subsystem. In the slowly variable subsystem, a CFO (Compensation Function Observer) is designed to accurately estimate and compensate for matched disturbances in real time. Simultaneously, an AISMC controller with dead-zone correction is designed, which not only avoids gain parameter drift but also improves the system's robustness to disturbances. In the quickly variable subsystem, an LQR (Linear Quadratic Regulator) is designed to suppress elastic vibrations caused by joint flexibility. Furthermore, the asymptotic stability of the closed-loop system is proved using Lyapunov stability theory.

[0018] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0019] Figure 1 This is a flowchart of the AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition as described in this invention.

[0020] Figure 2 To simulate and verify the effect of the disturbance estimation, the diagram is shown, where (a) represents the disturbance. The estimated curve (b) is the disturbance. Estimated curve;

[0021] Figure 3 To simulate and verify the effect of the disturbance estimation error, the following diagram is shown, where (a) represents the disturbance. The estimation error curve is shown in (b), which represents the disturbance. Estimated error curve;

[0022] Figure 4 To simulate and verify the adaptive switching gain dynamic response curve, where (a) represents the gain parameter. The curves showing the change of gain parameters are shown in (b). The change curve;

[0023] Figure 5 To simulate and verify the convergence effect of the sliding surface, (a) is the sliding surface. Convergence curve diagram, (b) is the sliding surface Convergence curve;

[0024] Figure 6 To simulate and verify the joint position tracking curves, (a) is the tracking curve of joint 1 and (b) is the tracking curve of joint 2.

[0025] Figure 7 To simulate and verify the joint position tracking error curves, (a) is the tracking error curve of joint 1 and (b) is the tracking error curve of joint 2. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the embodiments of the present invention and are not intended to limit the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of this application. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout.

[0027] It should be noted that the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion, such as a process, method, system, product, or server that includes a series of steps or units, not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such process, method, product, or device.

[0028] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0029] like Figure 1 As shown, the AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition includes the following steps:

[0030] S1. Establish a dynamic model of a flexible joint manipulator with uncertain composite perturbations, and decouple the coupled dynamic system of the flexible joint manipulator with uncertain composite perturbations into a slow-varying subsystem and a fast-varying subsystem based on singular perturbation theory;

[0031] S2. Use the compensation function observer to estimate the lumped matching disturbance in the slow-varying subsystem, determine the observer gain parameter through pole placement and verify its exponential stability;

[0032] S3. Design an adaptive integral sliding mode controller for the slow-varying subsystem. Based on the lumped matching disturbance, construct the integral sliding surface and the adaptive reaching law with dead zone correction. Derive the control law combined with disturbance feedforward compensation and verify the asymptotic stability of the slow-varying subsystem.

[0033] A linear quadratic regulator is designed for the fast variable subsystem. The performance index is defined and the Riccati equation is solved to obtain the optimal control law, thereby achieving optimal damping suppression of elastic vibration and verifying the asymptotic stability of the fast variable subsystem.

[0034] S4. The control law of the slow variable subsystem is superimposed with the control law of the fast variable subsystem to obtain the overall control input, and the asymptotic stability of the overall closed-loop system is verified based on Tikhonov's theorem.

[0035] S5. Simulation verification.

[0036] Step S1 specifically includes the following steps:

[0037] S11. Based on the simplified model of the Spong flexible joint manipulator, incorporate link-side mismatch disturbances. Matching disturbances with the motor side ,Establish Dynamic model of a linkage flexible joint robotic arm:

[0038] ;

[0039] In the formula, Let represent the positive definite inertia matrix, and ; , , Let these represent the displacement, velocity, and acceleration vectors of the joint angle, respectively. ; Represents the Coriolis force and centrifugal force matrices, and ; This represents the gravitational vector force, and ; Let represent the diagonal positive definite matrix of joint stiffness coefficients, and ; , These represent the rotor angular displacement and angular acceleration of the motor after the action of the reducer, respectively. ; Let represent the moment of inertia matrix of the joint motor, and ; This represents the torque control input vector of the joint motor, and ;

[0040] S12, Introducing the requirement to satisfy Singular perturbation parameters Define the scaled joint stiffness matrix and joint torque And set the joint angular displacement For slow sub-variables, joint torques Substituting the tachyon variable into the dynamic model established by S11, we obtain the coupling equations for the slow and tachyon variables:

[0041] ;

[0042] S13, Order and slow subsystem control input Replace the original joint motor torque control input vector , obtain tachyon variables Quasi-steady-state value :

[0043] ;

[0044] S14, quasi-steady-state value Substituting into the coupling equations, we obtain the slowly varying subsystem dominated by trajectory tracking after decoupling:

[0045] ;

[0046] in,

[0047] ;

[0048] In the formula, This represents the lumped-matching perturbation, and ;

[0049] S15. Define the quasi-steady-state deviation of the fast sub-variable. The quasi-steady-state deviation of the fast sub-variable and quasi-steady-state value Substituting into the coupling equation, we obtain about Equations for the fast variable subsystem:

[0050] ;

[0051] In the formula, Represents the quasi-steady-state deviation of the tachyon variable The second derivative with respect to time; Indicates the quasi-steady-state value The second derivative with respect to time;

[0052] S16, Control input of the fast converter subsystem At the same time, a fast time scale is introduced. satisfy This yields a fast-changing subsystem dominated by elastic vibration suppression:

[0053] .

[0054] Through the singular perturbation decomposition described above, the flexible joint robotic arm is decomposed into two subsystems at two time scales: one is a slowly varying subsystem described by a quasi-steady-state model, which can be equivalent to a rigid body model, and its reduced-order characteristics simplify the design of the trajectory tracking controller; the other is a rapidly varying subsystem described by a boundary layer model, which reflects the flexible vibration characteristics of the system and is suitable for vibration suppression control. Furthermore, through model decomposition and transformation, the unmatched and matched disturbances in the original system are integrated into a lumped matched disturbance in the slowly varying subsystem, providing structural convenience for disturbance estimation and compensation control.

[0055] Step S2 specifically includes the following steps:

[0056] S21, Let the state variables of the slowly varying subsystem be... The slow-varying subsystem described in step S14 is rewritten in the form of the following state-space equations:

[0057] ;

[0058] In the formula, Represents the state variables of a slowly varying subsystem The first derivative with respect to time; Represents the state variables of a slowly varying subsystem The first derivative with respect to time; Let represent the equivalent inertia matrix of the slowly varying subsystem, and ; Represents the Coriolis force and centrifugal force matrices;

[0059] S22, Denote the disturbance term And considering it as an extended state of the system, the following extended state equations are constructed:

[0060] ;

[0061] In the formula, Represents extended state variables The first derivative with respect to time, and ; Indicates the disturbance term The first derivative with respect to time;

[0062] S23. Construct a compensation function observer based on the extended state equation, the expression of which is as follows:

[0063] ;

[0064] In the formula, Represents the system state variables of the observer, and ; System state variables representing the observer The first derivative with respect to time; Let represent the state error feedback gain matrix of the observer, and , This represents the diagonal matrix constructor. and These represent the gain matrices respectively. The One diagonal element; This represents the estimation error of the system state, and , State variables State corresponding to the observer Deviation; State variables State corresponding to the observer deviation, Indicates transpose; Let represent the perturbation estimation gain matrix of the observer, and , This represents the diagonal matrix constructor. Represents the perturbation estimation gain matrix of the observer. The One diagonal element; Indicates the disturbance term The estimated value, and ;

[0065] S24. Define estimation error Subtracting the extended state equation from the observer equation from the observer equation from the observer equation described in step S22 yields the following set of equations for the observer error:

[0066] ;

[0067] In the formula, , and These represent the observer error components. , and The first derivative with respect to time;

[0068] S25. Describe the observer error vector. The equations for the observer error described in step S24 are summarized as follows:

[0069] ;

[0070] in,

[0071] ;

[0072] ;

[0073] In the formula, and Representing block diagonal matrices respectively and The Middle Sub-block matrices, and ; Represents the observer error vector The first derivative with respect to time, and , ;

[0074] get The characteristic equation is as follows:

[0075]

[0076] In the formula, The complex frequency variable represents the characteristic equation; and These represent the state error feedback gain matrix of the observer in the compensation function. The first in One element;

[0077] S26, will The pole placement in the characteristic equation is as follows The gain parameter is obtained through the following pole placement relationship:

[0078] ;

[0079] In the formula, The scaling factor representing pole placement; Indicates the observer bandwidth, and ;

[0080] S27. Verify the stability of the observer: when the condition is met... At that time, the Routh-Hurwitz criterion proves that the observer of the compensation function is exponentially stable and the error of the disturbance steady-state estimation is zero.

[0081] The proof is as follows:

[0082] when conditions When satisfied, The characteristic equations all have negative real parts, therefore Satisfies Routh-Hurwitz stability, and The calculation table for Rolls is as follows:

[0083]

[0084] For the observer estimation error equation described in step S25, let... They represent exist The value at time, where, , This indicates the sampling interval length. Assume... for Infinitesimals of order ( ),but exist The Taylor expansion of time is:

[0085] ;

[0086] In the formula, , , , They represent the disturbance terms respectively. exist The value of time, the first derivative with respect to time, the second derivative, and The first derivative, and when hour , , , All are constants; This indicates the order of the corresponding derivative term in the Taylor expansion; It represents an infinitesimal quantity.

[0087] right conduct Taking the derivative, we get... Since the state-space equation described in step S21 belongs to a second-order system, it can be known that... It belongs to the third-order infinitesimal, that is... The observer estimation error equation described in step S25 is obtained by taking the derivative twice:

[0088]

[0089] In the formula, and Represents the observer error vector The second and third derivatives with respect to time; Represents lumped matching perturbation The third derivative with respect to time;

[0090] because Satisfying Routh-Hurwitz stability, for observation errors... ,Right now ;in, express The second derivative with respect to time; therefore, when the matrix any sub-block The characteristic equation satisfies the condition At that time, the compensation function observer is exponentially stable, and for The steady-state estimation error is 0.

[0091] Step S3 specifically includes the following steps:

[0092] S31. Design an adaptive integral sliding mode controller to realize trajectory tracking control of a slow-varying subsystem;

[0093] S311. Define the tracking error vector of the slowly varying subsystem. , :

[0094] ;

[0095] In the formula, Indicates the tracking error of the slowly varying subsystem The first derivative with respect to time; Desired trajectory representing joint angle The derivative; and These represent the tracking error vectors respectively. and The One component;

[0096] S312. Substitute the expression from step S311 into the state-space equation from step S21 to obtain the system tracking error dynamic equation:

[0097] ;

[0098] In the formula, Indicates the tracking error of the slowly varying subsystem The second derivative with respect to time; Represents the desired trajectory The second derivative with respect to time;

[0099] S313. Constructing the sliding surface function vector , Represents the sliding surface function vector The Each component, and , and Both represent the positive sliding surface gain, which satisfies , The characteristic polynomial of the sliding surface is represented. The complex frequency variable in the characteristic polynomial, This represents the bandwidth of the adaptive integral sliding mode controller; Represents a time variable;

[0100] S314. Design an adaptive reaching law with dead-zone correction:

[0101] ;

[0102] ;

[0103] In the formula, and They represent and The first derivative with respect to time; and Both represent the dead zone threshold, and ; and They represent and The benchmark adjustment factor;

[0104] S315, regarding the steps described in S313 Perform a derivative and substitute it into the compensation function observer expression described in step S23, the system tracking error dynamic equation described in step S312, and... The control law of the slow subsystem is obtained. :

[0105] ;

[0106] in, This is an estimate of the lumped disturbance; ;

[0107] In the formula, Represents the sliding surface function The first derivative with respect to time; Denotes a saturation function, and ; and Both represent the gain matrix of sliding mode control, and , , and They represent and The One adaptive gain; Represents the sliding surface function The saturation limit;

[0108] S316. Verify the stability of the slowly varying subsystem: Construct the Lyapunov function of the slowly varying subsystem. :

[0109] ,

[0110] in,

[0111] ;

[0112] ;

[0113] In the formula, Indicates adaptive gain Compared with actual gain Deviation terms; Indicates adaptive gain Compared with actual gain Deviation terms; and Both represent the positive weighting coefficients of the adaptive error term in the Lyapunov function, and ;

[0114] By proof First derivative with respect to time Verify the finite-time convergence of the sliding surface and the asymptotic stability of the system;

[0115] The proof is as follows: First derivative with respect to time The expression is as follows:

[0116] ;

[0117] In the formula, and They represent and The first derivative with respect to time;

[0118] Will Substituting into the above equation, we get:

[0119] ;

[0120] Combining the adaptive reaching law with dead-zone correction described in step S314, we obtain:

[0121] ;

[0122] It can be seen that for the above formula, there always exists and ,Right now Make The Lyapunov stability condition is satisfied. This indicates that the sliding surface... It can converge to zero in a finite time, and the tracking error of the slow-varying subsystem asymptotically converges to zero when reaching the sliding surface.

[0123] S32. Design a linear quadratic regulator to suppress elastic vibration of a fast-changing subsystem;

[0124] S321. Perform state modeling on the fast-changing subsystem obtained in step S16, and define the state vector of the fast-changing subsystem. Construct the state equations of the rapidly changing subsystem:

[0125] ;

[0126] in,

[0127] ;

[0128] ;

[0129] In the formula, Let represent the state matrix of the rapidly changing subsystem, and ; Let represent the control input matrix of the fast-changing subsystem, and ; Denotes the identity matrix, and ;

[0130] S322. Define the quadratic performance index of the linear quadratic regulator. :

[0131] ;

[0132] In the formula, Let represent the state weight matrix, and ; This represents the control input weight matrix, and ;

[0133] S323. Solving for the optimal control law: Solving the Riccati matrix equation A positive definite solution is obtained. This leads to the optimal control law for the fast variable subsystem. :

[0134] ;

[0135] In the formula, K f For optimal feedback gain, and ;

[0136] In this embodiment, the state weight matrix mentioned in step S322 It can also be set to ,in, Represents the state weight matrix The sub-block matrix, and ; Then by solving ,available At fast timescales, due to the quasi-steady-state value The changes are slow, so it is considered the one that comes frequently, that is... At this point, the control law of the tachy subsystem can also be expressed as: .

[0137] S324. Verify the stability of the fast-changing subsystem: Construct the Lyapunov function of the fast-changing subsystem:

[0138] ;

[0139] By proof The asymptotic stability of the fast variable subsystem was verified.

[0140] The specific proof process is as follows:

[0141] right On a fast timescale Differentiating further, we get:

[0142] ;

[0143] Will Substituting into the Riccati matrix equation, we get:

[0144] ;

[0145] Substitute the above formula into From the derivative formula, we get:

[0146] ;

[0147] because and Therefore Therefore, At this point, according to Lyapunov's stability proof theory, the fast variable subsystem is asymptotically stable under the optimal feedback control law.

[0148] S51. Synthesize the overall control input: Input the control law of the slowly varying subsystem obtained in step S315. The optimal control law for the fast-changing subsystem obtained in step S323 By superimposing the data, the overall control input for the flexible joint robotic arm is obtained. ;

[0149] S52. Verify the overall system stability: Based on Tikhonov's theorem and combined with the stability conclusions of the slow-changing subsystem and the fast-changing subsystem verified in step S4, prove that the overall closed-loop system is asymptotically stable.

[0150] Simulation verification

[0151] To verify the effectiveness and superiority of the Compensation Function Observer (CFO) and the control algorithm proposed in this invention, this simulation experiment uses a two-bar flexible joint robotic arm as the controlled object. We designed comparative simulation experiments for uncertain composite disturbance observation and joint trajectory tracking, respectively.

[0152] Table 1 Parameters of Flexible Joint Robotic Arm

[0153]

[0154] 1. Observational comparison simulation experiment of uncertain composite disturbance;

[0155] To verify the performance advantages of the CFO proposed in this invention in the face of complex disturbances with trajectory uncertainty, this simulation experiment uses ESO as the benchmark for comparative testing, and selects the following disturbances as the observation objects:

[0156] ;

[0157] In the design process of the aforementioned CFO, the object of observation is the disturbance term. In the disturbance observation comparative simulation, the comparison object is the lumped disturbance. ,Right now ,in This is an estimate of the disturbance term.

[0158] Bandwidth of CFO and ESO observers The expected extreme point is set as and ,Right now This leads to the gain parameter of CFO. The characteristic equation of ESO is The calculated gain parameter of ESO is: As can be seen from the comparison, under the same bandwidth, the gain of ESO is higher than that of CFO.

[0159] At the same time, such as Figure 2 As shown, under the same bandwidth, the observation response of ESO (blue dashed line) is significantly lagging and deviates greatly from the true value (black solid line); while CFO (red dashed line) responds quickly and can closely track the actual disturbance, showing better observation performance.

[0160] like Figure 3 As shown, the observation error of ESO fluctuates drastically during the steady-state phase. The peak value is 120, while CFO is 40. The maximum steady-state error of ESO is about 3 times that of CFO.

[0161] In summary, the CFO proposed in this invention is superior to the traditional ESO in terms of observation performance and accuracy.

[0162] 2. Simulation experiment comparing joint trajectory tracking;

[0163] To verify the effectiveness and superiority of the algorithm proposed in this invention, this simulation experiment, based on CFO perturbation compensation, employs three controllers for the slowly varying subsystem: AISMC (as described in this invention), ISMC, and SMC (the sliding surface selection for ISMC is as follows). SMC Sliding Surface Selection ;and , (The sliding mode reaching law and its values ​​are the same as in this invention), and a comparative experiment on trajectory tracking was conducted. During the experiment, a lumped matching perturbation was added. And keep the LQR controller of the fast variable subsystem unchanged.

[0164] Set initial position Reference trajectory , and The specific values ​​are as follows: ; .like Figure 4 As shown, it can be seen that and It responds quickly, reaching a stable value in about 0.46s, thus avoiding gain parameter drift and gain overestimation issues.

[0165] Table 2 Controller Parameters

[0166]

[0167] The sliding surface convergence performance of the three controllers is as follows: Figure 5 As shown, compared to ISMC and SMC, the AIISMC proposed in this invention has a faster convergence speed, with its sliding surface converging to zero in approximately 0.46 s, while ISMC and SMC require approximately 0.6 s and 0.8 s respectively. Furthermore, SMC fails to achieve complete convergence and exhibits a significant steady-state deviation. Combined with... Figure 4 It can be seen that when the sliding surface of AISMC reaches zero, the adaptive gain... and The fact that it stabilizes at 0.46s indicates that the adaptive law can suppress the overestimation of the gain and avoid severe chattering caused by gain overshoot.

[0168] like Figure 6 As shown, under the same bandwidth, SMC and ISMC deviate significantly from the desired trajectory, while AISMC closely tracks the desired trajectory with the smallest deviation, exhibiting superior tracking performance. Furthermore, in the initial stage, SMC and ISMC respond slowly, with SMC having the longest settling time, while AISMC responds quickly with the shortest settling time, approximately twice as fast as SMC and ISMC. Figure 7 As shown, SMC's tracking error fluctuates more, while AISMC's error curve is the flattest and its robust performance is the best. Overall, AISMC is significantly better than SMC and ISMC in terms of response speed, tracking performance, and tracking accuracy.

[0169] To quantitatively analyze the control performance of the three controllers, a settling time is introduced. Maximum steady-state error Mean absolute error and root mean square steady-state error As an evaluation indicator.

[0170] in,

[0171] ;

[0172] ;

[0173] In the formula, This indicates the total number of error data points included in the statistics; Indicates the first The systematic error components corresponding to each data point; Indicates the integration time interval; The period of the steady-state error curve is represented. This represents the point in time after the system has reached steady state;

[0174] In this embodiment, the steady-state point of the SMC controller is used. and steady-state period As a benchmark That is, SMC enters steady state at 4s, and the period of the tracking error curve is 3s. These are the quantitative performance indicators of the three controllers.

[0175] Table 3 Performance Indicators of Three Controllers

[0176]

[0177] As shown in Table 3, the AISMC proposed in this invention achieved the fastest response speed during the transient phase. Taking joint 1 as an example (joint 2 is similar), the settling time of the AISMC is... The steady-state error is as low as 1.2s, while ISMC and SMC are 2.4s and 2.8s, respectively; in the steady-state phase, the maximum steady-state error of the AISMC proposed in this invention is... The mean absolute error (MAE) is the smallest of the three at 0.0542 rad, significantly lower than ISMC's 0.1130 rad and SMC's 0.4261 rad. Furthermore, comparing the performance metrics of the entire trajectory tracking process, namely the mean absolute error... and root mean square steady-state error Taking joint 2 as an example (joint 1 is similar), the proposed AISMC index values ​​are as low as 0.0694 rad and 0.0177 rad, respectively, which are 29% and 63% higher than ISMC's 0.0978 rad and 0.0520 rad, and 24% and 72% higher than SMC's 0.0916 rad and 0.0630 rad, respectively. In summary, when facing complex disturbances, the AISMC method proposed in this invention exhibits excellent performance in terms of response speed, steady-state error, and overall tracking accuracy.

[0178] In summary, the simulation results show that, compared with traditional ISMC and SMC, the method proposed in this invention has significant improvements in settling time, steady-state error, and trajectory tracking accuracy, verifying the superiority and practicality of the control strategy.

[0179] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. An AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition, characterized in that: Includes the following steps: S1. Establish a dynamic model of a flexible joint manipulator with uncertain composite perturbations, and decouple the coupled dynamic system of the flexible joint manipulator with uncertain composite perturbations into a slow-varying subsystem and a fast-varying subsystem based on singular perturbation theory; S2. Use the compensation function observer to estimate the lumped matching disturbance in the slow-varying subsystem, determine the observer gain parameter through pole placement and verify its exponential stability; S3. Design an adaptive integral sliding mode controller for the slow-varying subsystem. Based on the lumped matching disturbance, construct the integral sliding surface and the adaptive reaching law with dead zone correction. Derive the control law combined with disturbance feedforward compensation and verify the asymptotic stability of the slow-varying subsystem. A linear quadratic regulator is designed for the fast variable subsystem. The performance index is defined and the Riccati equation is solved to obtain the optimal control law, thereby achieving optimal damping suppression of elastic vibration and verifying the asymptotic stability of the fast variable subsystem. S4. The control law of the slow variable subsystem is superimposed with the control law of the fast variable subsystem to obtain the overall control input, and the asymptotic stability of the overall closed-loop system is verified based on Tikhonov's theorem. S5. Simulation verification; Step S1 specifically includes the following steps: S11. Based on the simplified model of the Spong flexible joint manipulator, incorporate link-side mismatch disturbances. Matching disturbances with the motor side ,Establish Dynamic model of a linkage flexible joint robotic arm: ； In the formula, Let represent the positive definite inertia matrix, and ; , , Let these represent the displacement, velocity, and acceleration vectors of the joint angle, respectively. ; Represents the Coriolis force and centrifugal force matrices, and ; This represents the gravitational vector force, and ; Let represent the diagonal positive definite matrix of joint stiffness coefficients, and ; , These represent the rotor angular displacement and angular acceleration of the motor after the action of the reducer, respectively. ; Represents the moment of inertia matrix of the joint motor, and ; This represents the torque control input vector of the joint motor, and ; S12, Introducing the requirement to satisfy Singular perturbation parameters Define the scaled joint stiffness matrix and joint torque And set the joint angular displacement For slow sub-variables, joint torques Substituting the tachyon variable into the dynamic model established by S11, we obtain the coupling equations for the slow and tachyon variables: ； S13, Order And use the slow subsystem to control the input. Replace the original joint motor torque control input vector , obtain tachyon variables Quasi-steady-state value : ； S14, quasi-steady-state value Substituting into the coupling equations, we obtain the slowly varying subsystem dominated by trajectory tracking after decoupling: ； in, ； In the formula, This represents the lumped-matching perturbation, and ; S15. Define the quasi-steady-state deviation of the fast sub-variable. The quasi-steady-state deviation of the fast sub-variable and quasi-steady-state value Substituting into the coupling equation, we obtain about Equations for the fast variable subsystem: ； In the formula, Represents the quasi-steady-state deviation of the tachyon variable The second derivative with respect to time; Indicates the quasi-steady-state value The second derivative with respect to time; S16, Control input of fast converter subsystem At the same time, a fast time scale is introduced. satisfy This yields a fast-changing subsystem dominated by elastic vibration suppression: 。 2. The AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition according to claim 1, characterized in that: Step S2 specifically includes the following steps: S21, Let the state variables of the slowly varying subsystem be... The slow-varying subsystem described in step S14 is rewritten in the form of the following state-space equations: ； In the formula, Represents the state variables of a slowly varying subsystem The first derivative with respect to time; Represents the state variables of a slowly varying subsystem The first derivative with respect to time; Let represent the equivalent inertia matrix of the slowly varying subsystem, and ; Represents the Coriolis force and centrifugal force matrices; S22, Denote the disturbance term And considering it as an extended state of the system, the following extended state equations are constructed: ； In the formula, Represents extended state variables The first derivative with respect to time, and ; Indicates the disturbance term The first derivative with respect to time; S23. Construct a compensation function observer based on the extended state equation, the expression of which is as follows: ； In the formula, Represents the system state variables of the observer, and ; System state variables representing the observer The first derivative with respect to time; Let represent the state error feedback gain matrix of the observer, and , This represents the diagonal matrix constructor. and These represent the gain matrices respectively. The One diagonal element; This represents the estimation error of the system state, and , State variables State corresponding to the observer Deviation; State variables State corresponding to the observer deviation, Indicates transpose; Let represent the perturbation estimation gain matrix of the observer, and , The perturbation estimation gain matrix of the observer represents the perturbation estimation matrix. The One diagonal element; Indicates the disturbance term The estimated value, and ; S24. Define estimation error Subtracting the extended state equation from the observer equation from the observer equation from the observer equation described in step S22 yields the following set of equations for the observer error: ； In the formula, , and These represent the observer error components. , and The first derivative with respect to time; S25. Describe the observer error vector. The equations for the observer error described in step S24 are summarized as follows: ； in, ； ； In the formula, and Representing block diagonal matrices respectively and The Middle Sub-block matrices, and ; Represents the observer error vector The first derivative with respect to time, and , ; get The characteristic equation is as follows: ； In the formula, The complex frequency variable represents the characteristic equation; and These represent the state error feedback gain matrix of the observer in the compensation function. The first in One diagonal element; S26, will The pole placement in the characteristic equation is as follows The gain parameter is obtained through the following pole placement relationship: ； In the formula, The scaling factor representing pole placement; Indicates the observer bandwidth, and ; S27. Verify the stability of the observer: when the condition is met... At that time, the Routh-Hurwitz criterion proves that the observer of the compensation function is exponentially stable and the error of the disturbance steady-state estimation is zero.

3. The AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition according to claim 2, characterized in that: Step S3 specifically includes the following steps: S31. Design an adaptive integral sliding mode controller to realize trajectory tracking control of a slow-varying subsystem; S311. Define the tracking error vector of the slowly varying subsystem. , : ； In the formula, Indicates the tracking error of the slowly varying subsystem The first derivative with respect to time; Desired trajectory representing joint angle The derivative; and These represent the tracking error vectors respectively. and The One component; S312. Substitute the expression from step S311 into the state-space equation from step S21 to obtain the system tracking error dynamic equation: ； In the formula, Indicates the tracking error of the slowly varying subsystem The second derivative with respect to time; Represents the desired trajectory The second derivative with respect to time; S313. Constructing the sliding surface function vector , Represents the sliding surface function vector The Each component, and , and Both represent the positive sliding surface gain, which satisfies , The characteristic polynomial of the sliding surface is represented. The complex frequency variable in the characteristic polynomial, This represents the bandwidth of the adaptive integral sliding mode controller; Represents a time variable; S314. Design an adaptive reaching law with dead-zone correction: ； ； In the formula, and They represent and The first derivative with respect to time; and Both represent the dead zone threshold, and ; and They represent and The benchmark adjustment factor; S315, regarding the steps described in S313 Perform a derivative and substitute it into the compensation function observer expression described in step S23, the system tracking error dynamic equation described in step S312, and... The control law of the slow subsystem is obtained. : ； in, This is an estimate of the lumped disturbance; ; In the formula, Represents the sliding surface function The first derivative with respect to time; Denotes a saturation function, and ; and Both represent the gain matrix of sliding mode control, and , , and They represent and The One adaptive gain; Represents the sliding surface function The saturation limit; S316. Verify the stability of the slowly varying subsystem: Construct the Lyapunov function of the slowly varying subsystem. : ， in, ； ； In the formula, Indicates adaptive gain Compared with actual gain Deviation terms; Indicates adaptive gain Compared with actual gain Deviation terms; and Both represent the positive weighting coefficients of the adaptive error term in the Lyapunov function, and ; By proof First derivative with respect to time Verify the finite-time convergence of the sliding surface and the asymptotic stability of the system; S32. Design a linear quadratic regulator to suppress elastic vibration of a fast-changing subsystem; S321. Perform state modeling on the fast-changing subsystem obtained in step S16, and define the state vector of the fast-changing subsystem. Construct the state equations of the rapidly changing subsystem: ;in, ； ； In the formula, Let represent the state matrix of the rapidly changing subsystem, and ; Let represent the control input matrix of the fast-changing subsystem, and ; Denotes the identity matrix, and ; Indicates a fast time scale; S322. Define the quadratic performance index of the linear quadratic regulator. : ； In the formula, Let represent the state weight matrix, and ; This represents the control input weight matrix, and ; S323. Solving for the optimal control law: Solving the Riccati matrix equation A positive definite solution is obtained. This leads to the optimal control law for the fast variable subsystem. : ； In the formula, K f For optimal feedback gain, and ; S324. Verify the stability of the fast-changing subsystem: Construct the Lyapunov function of the fast-changing subsystem: ； By proof The asymptotic stability of the fast variable subsystem was verified.

4. The AISMC compensation control method for a flexible joint robotic arm under singular perturbation decomposition according to claim 3, characterized in that: Step S4 Specifically, the following steps are included: S41. Synthesize the overall control input: Use the slow-varying subsystem control law obtained in step S315... The optimal control law for the fast-changing subsystem obtained in step S323 By superimposing the data, the overall control input for the flexible joint robotic arm is obtained. ; S42. Verify the overall system stability: Based on Tikhonov's theorem, and combined with the stability conclusions of the slow-changing subsystem and the fast-changing subsystem verified in step S4, prove that the overall closed-loop system is asymptotically stable.

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