Nonlinear robot system fixed-time sliding mode control method based on double-layer nested adaptive law

Abstract

The application discloses a kind of based on double-layer nested self-adaptive law nonlinear robot system fixed time sliding mode control method, comprising: the mathematical model of n degree of freedom nonlinear robot system is established;According to the mathematical model of n degree of freedom nonlinear robot system, the trajectory tracking error control equation of nonlinear robot system containing lumped unknown bounded disturbance is established;According to the trajectory tracking error equation of nonlinear robot system established, the sliding variable based on modified variable power function is designed;According to sliding variable, the fixed time approaching control law based on double-layer nested self-adaptive law is designed;According to sliding variable, the fixed time sliding mode controller of nonlinear robot system is designed based on double-layer nested self-adaptive law fixed time approaching control law.The application can realize the fast trajectory tracking control of nonlinear robot under different initial positions, ensure the safety and reliability of robot system under complex and changeable working environment.

Description

Technical Field

[0001] This invention relates to a fixed-time sliding mode control method for nonlinear robot systems based on a double-layer nested adaptive law, belonging to the field of stability and control of nonlinear robot systems. Background Technology

[0002] In the course of modern technological development, robots have demonstrated enormous potential and vitality in various fields such as industrial manufacturing, medicine, military, and entertainment. Whether in academic research or industrial applications, designing robust and effective control strategies is crucial to ensuring the safety and reliability of robot systems in complex and ever-changing working environments. To achieve this goal, researchers have developed many advanced control methods, such as active disturbance rejection control, adaptive control, intelligent control, model predictive control, and sliding mode control.

[0003] Sliding mode control is characterized by its simple structure and flexible design, providing researchers with a great deal of room for development. Therefore, the approach of combining sliding mode control with finite-time stability theory and fixed-time stability theory for controller design has received widespread attention in the field of control.

[0004] Significant progress has been made in the integration of finite-time stability theory and sliding mode control in the field of nonlinear system control. The literature “Shuanghe Yu, Xinghuo Yu, Bijan Shirinzadeh, et al. Continuous finite-time control for robotic manipulators with terminal sliding mode[J]. Automatica, 2005, Vol.41(11):1957-1964” proposes a finite-time terminal sliding mode control scheme, laying a theoretical foundation for improving the dynamic response speed and robustness of nonlinear systems. The literature “Chuanbin Sun, Shubo Wang, Haisheng Yu. Finite-time sliding mode control based on unknown system dynamics estimator for nonlinear robotic systems, IEEE Transactions on Circuits and Systems II: Express Briefs, Vol.70(7):2535-2539, 2023” designs a finite-time sliding mode controller based on an unknown system dynamics estimator, effectively enhancing the robustness of nonlinear systems to unknown bounded disturbances. Finite-time sliding mode control ensures that the system state converges to the equilibrium point within a certain time bound, but this convergence time bound changes with the initial state of the system, which reduces the application value of finite-time sliding mode control in engineering practice. To overcome this limitation, fixed-time stability theory, in which the convergence time bound does not depend on the initial state of the system, has been proposed. The combination of fixed-time stability theory and sliding mode control has been widely applied to nonlinear systems. The literature "Zuo, Zongyu. Non-singular fixed-time terminal sliding mode control of non-linear systems.[J].IET Control Theory&Applications,2015,Vol.9(4):545-552." proposes a non-singular fixed-time terminal sliding mode control scheme for nonlinear systems, which significantly improves the robustness of nonlinear systems. However, this scheme has the problem of conservative convergence time estimation.The paper "Moulay, Emmanuel, Lechappe, et al. Robust fixed-time stability: application to sliding-mode control. [J]. IEEE Transactions on Automatic Control, 2022, Vol. 67(2): 1061-1066." innovatively uses a power function to construct the sliding variable. Although this solves the singularity problem of the controller, its convergence characteristics still suffer from the deficiency of local dependence on the initial state of the system. The paper "Yunsong Hu, Huaicheng Yan, Hao Zhang, et al. Robust adaptive fixed-time sliding-mode control for uncertain robotic systems with inputsaturation [J]. IEEE Transactions on Cybernetics, 2023, Vol. 53(4): 2636-2646." proposes an adaptive fixed-time sliding mode control strategy, which handles the singularity problem of the controller through a switching function mechanism. However, this method may cause the controller to switch back and forth at the singular point near the switching point, affecting the stability of the system.

[0005] Robot systems are typical nonlinear systems. Their motion is influenced by factors such as multi-joint coupling, time-varying inertia matrices, Coriolis forces, centrifugal forces, gravity, and nonlinear friction, resulting in a complex nonlinear relationship between input (torque) and output (motion). On the one hand, improving the rapid trajectory tracking control of nonlinear robot systems at different initial positions can enhance operational efficiency, reduce energy consumption, and meet the requirements of high efficiency and high precision. On the other hand, the singularity problem of robot controllers can lead to control torque divergence, state stagnation, increased high-frequency chattering, and stability degradation, severely affecting the response performance and safety of the robot system. Therefore, achieving rapid trajectory tracking control of nonlinear robot systems at different initial positions while ensuring the non-singularity of the robot controller and mitigating chattering of the robot system's control torque is crucial for the safety and reliability of robot systems in various working environments and remains a worthy research topic. Summary of the Invention

[0006] To eliminate the interference of nonlinear uncertainties and unknown bounded disturbances on the stable operation of robot systems, this invention provides a fixed-time sliding mode control method for nonlinear robot systems based on a double-layer nested adaptive law. This method enables rapid trajectory tracking control of the nonlinear robot at different initial positions, ensuring the safety and reliability of the robot system in complex and ever-changing working environments.

[0007] The technical solution of this invention is:

[0008] According to a first aspect of the present invention, a fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law is provided, comprising:

[0009] Step 1: Establish a mathematical model of an n-degree-of-freedom nonlinear robot system; based on the mathematical model of the n-degree-of-freedom nonlinear robot system, establish the trajectory tracking error control equation for the nonlinear robot system containing lumped unknown bounded disturbances;

[0010] Step 2: Based on the established trajectory tracking error equation of the nonlinear robot system, design a sliding variable based on a modified power-law function; based on the sliding variable, design a fixed-time approaching control law based on a double-nested adaptive law; based on the sliding variable and the fixed-time approaching control law based on the double-nested adaptive law, design a non-singular fixed-time sliding mode controller for the nonlinear robot system.

[0011] Furthermore, based on the established trajectory tracking error equation of the nonlinear robot system, a sliding variable s based on a modified power-law function is designed, specifically as follows:

[0012]

[0013] in, and Let represent the control gain in the sliding variable, and satisfy . and Let represent a positive constant and satisfy . λ1 represents a constant greater than 1; sign(e1) represents the sign function, e1 is the robot joint tracking error, and the first derivative of e1 is...

[0014] Furthermore, based on the sliding variable, a fixed-time approach control law based on a double-nested adaptive law is designed, specifically as follows:

[0015]

[0016] Among them, u r This indicates the approach control law. Indicates the first time-varying parameter. Represents a small positive integer. and This represents the control gain in the approach control law, and satisfies... Let represent a positive constant and satisfy . λ′1 represents a constant greater than 1; represents a double-nested adaptive law term; sign(s) represents the sign function, s is the sliding variable, and e represents the natural constant.

[0017] Furthermore, the design of the double-nested adaptive law term is as follows:

[0018] Using low-pass filtering operation to approach the control law u r By approximation, we obtain u r approximation Its expression is:

[0019]

[0020] Where k represents the filtering time constant; Indicate u r Approximate value; express The first derivative;

[0021] u is obtained through low-pass filtering. r approximation Thus, the first time-varying parameter is constructed. The safety boundary ensures the first time-varying parameter. satisfy:

[0022]

[0023] in, Let represent a positive constant and satisfy . ε>0 represents a positive constant;

[0024] Based on the constructed first time-varying parameters The safety boundary is defined, and the error variable of the adaptive law is defined as follows: Its expression is as follows:

[0025]

[0026] By defining error variables Obtain the first time-varying parameters first derivative The expression is:

[0027]

[0028] in, Let represent a positive constant and satisfy . Φ(t) represents the safety boundary of the rate of change of the adaptive law, and its expression is:

[0029]

[0030] Second time-varying parameter first derivative It can be represented as:

[0031]

[0032] in, Let represent a positive constant and satisfy . sign(θ(t)) represents the sign function; θ(t) represents the rate of change of the adaptive law error, and its expression is:

[0033]

[0034] Where η represents a positive constant and satisfies η > 0; It represents the upper bound of the first derivative of the lumped unknown bounded perturbation.

[0035] Furthermore, the non-singular fixed-time sliding mode controller τ = τ of the robot system eq +τ sw ; where τ eq The feedback term represents compensation for nonlinear dynamics; τ sw It is a convergence controller used to handle lumped unknown bounded disturbances in nonlinear robot systems.

[0036] According to a second aspect of the present invention, a fixed-time sliding mode control system for a nonlinear robot system based on a double-layer nested adaptive law is provided, comprising:

[0037] The first module is used to establish the mathematical model of an n-degree-of-freedom nonlinear robot system; based on the mathematical model of the n-degree-of-freedom nonlinear robot system, the trajectory tracking error control equation of the nonlinear robot system containing lumped unknown bounded disturbances is established.

[0038] The second module is used to design a sliding variable based on a modified power-law function, based on the established trajectory tracking error equation of the nonlinear robot system; to design a fixed-time approaching control law based on a double-nested adaptive law based on the sliding variable; and to design a non-singular fixed-time sliding mode controller for the nonlinear robot system based on the sliding variable and the fixed-time approaching control law based on the double-nested adaptive law.

[0039] According to a third aspect of the present invention, a processor is provided for running a program, wherein the program, when running, executes a fixed-time sliding mode control method for a nonlinear robot system based on a double-nested adaptive law as described in any one of the preceding claims.

[0040] The beneficial effects of this invention are as follows: This invention designs a fixed-time sliding mode control method based on a double-nested adaptive law, which features fast convergence speed, non-singularity, and strong robustness. It enables rapid trajectory tracking control of a nonlinear robot system at different initial positions, ensuring its safety and reliability in complex and variable working environments. Specifically, this invention establishes the trajectory tracking error control equation for a nonlinear robot system containing lumped unknown bounded disturbances based on the mathematical model of an n-degree-of-freedom nonlinear robot system. Furthermore, a modified variable exponential function is used instead of a constant exponential function in the sliding variable, avoiding the singularity problem present in traditional fixed-time terminal sliding mode controllers. In the approach control law design, a time-varying double-nested adaptive law term is used instead of a fixed control gain term. This not only avoids overestimation of unknown disturbances and enables online adjustment of the control gain as the unknown disturbance changes, but also weakens system chattering. The fixed-time stability of the closed-loop system is proven using Lyapunov's theorem. Attached Figure Description

[0041] Figure 1 This is a flowchart of the invention;

[0042] Figure 2 This is a six-degree-of-freedom robot experimental platform provided according to embodiments of the present invention;

[0043] Figure 3 This is the position tracking response of joint 5 provided according to the embodiments of the present invention;

[0044] Figure 4 This refers to the position tracking error of joint 5 provided in the embodiments of the present invention;

[0045] Figure 5 It is the control input of joint 5 provided according to the embodiments of the present invention. Detailed Implementation

[0046] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of this application can be arbitrarily combined with each other.

[0047] Example 1: As Figures 1-5 As shown, a fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law includes:

[0048] Step 1: Considering that the actual robot system will be affected by unknown bounded perturbations, and based on the Lagrange modeling method, establish the mathematical model of the n-degree-of-freedom nonlinear robot system, as shown in the following expression:

[0049]

[0050] in, Let M(q) represent the robot's joint position, joint angular velocity, and joint angular acceleration, respectively; M(q)∈R n×n The inertia matrix of the robot system is represented by... G(q)∈R represents the centrifugal torque of the robot system. n The gravitational torque of the robot system is represented by τ∈R. n The control torque of the robot system is represented by d∈R. n R represents the unknown bounded perturbation experienced by the robot system. n×n Let R represent an n×n real matrix. n It represents an n-dimensional real vector.

[0051] By rewriting the mathematical model of an n-degree-of-freedom nonlinear robot system, the acceleration equations of the robot joints can be obtained as follows:

[0052]

[0053] To achieve trajectory tracking error control in a nonlinear robot system, the tracking error of the robot joint is defined as e1 = [e 11 , ..., e 1m ] T ∈R n The first derivative of the robot joint tracking error e1 Then the tracking error e of robot joint i 1i and its first derivative e 2i It can be represented as:

[0054]

[0055] Among them, e 11 e represents the tracking error of joint 1 of the robot. 1n e represents the tracking error of the robot's nth joint. 21 The first derivative of the robot's joint tracking error, e 2n q represents the first derivative of the tracking error at joint n of the robot. i Indicates the position of joint i of the robot, q di This represents the desired position of the robot's i-joint. This represents the angular velocity of the i-th joint of the robot. This represents the desired angular velocity of the robot's i-th joint.

[0056] Based on the acceleration equation of the robot joint in equation (2), the second derivative of the robot joint tracking error e1 can be derived, and its expression is:

[0057]

[0058] in, This represents the angular acceleration of the robot's joints. This represents the desired angular acceleration of the robot's joints.

[0059] The tracking error e of robot joint i in equation (3) 1i and its first derivative e 2i The second derivative expression of the robot joint tracking error e1 in equation (4) can be rearranged to obtain the trajectory tracking error equation of the nonlinear robot system containing lumped unknown bounded disturbances, as follows:

[0060]

[0061] in, Let represent the lumped unknown bounded perturbation of an n-DOF nonlinear robot system, and satisfy ... It represents the upper bound of the ensemble of unknown bounded perturbations. It represents the upper bound of the first derivative of the lumped unknown bounded perturbation.

[0062] The trajectory tracking error equation for a nonlinear robot system containing lumped unknown bounded disturbances, established based on the formula in step 1, is used to eliminate lumped unknown bounded disturbances. To address the impact on the stable and safe operation of the robot system, a non-singular fixed-time sliding mode controller for the nonlinear robot system will be designed to ensure the safety and reliability of the robot system under various working environments.

[0063] Step 2: Based on the established trajectory tracking error equation of the nonlinear robot system, design a sliding variable based on a modified power-law function; based on the sliding variable, design a fixed-time approaching control law based on a double-nested adaptive law; based on the sliding variable and the fixed-time approaching control law based on the double-nested adaptive law, design a non-singular fixed-time sliding mode controller for the nonlinear robot system. Specifically, this includes:

[0064] Step 2.1: Based on the trajectory tracking error equation of the nonlinear robot system established by the formula in Step 1, the robot joint tracking error e1 is obtained, and the following sliding variable s based on the modified power-law function is designed:

[0065]

[0066] in, and Represents the control gain, and satisfies and Let represent a positive constant and satisfy . λ1 represents a constant greater than 1; sign(e1) represents the sign function, e1 is the robot joint tracking error. When e1 > 0, sign(e1) = 1; when e1 = 0, sign(e1) = 0; when e1 < 0, sign(e1) = -1. It should be noted that the other sign functions sign(·) in this invention are similar.

[0067] Step 2.2: To avoid bounded perturbations to the lumped set of unknowns Overestimation weakens the high-frequency chattering of the robot system. Based on the sliding variable s obtained from the formula in step 2.1, a fixed-time approaching control law based on a double-nested adaptive law is designed as follows:

[0068]

[0069] Among them, u r This indicates the approach control law. Indicates the first time-varying parameter. Represents a small positive integer. and Represents the control gain, and satisfies Let represent a positive constant and satisfy . λ′1 represents a constant greater than 1; This represents a double-nested adaptive law term. `sign(s)` represents the sign function, where `s` is the sliding variable. When `s>0`, `sign(s)` = 1; when `s=0`, `sign(s)` = 0; when `s<0`, `sign(s)` = -1.

[0070] Step 2.3: Due to the approach control law u obtained from formula (7) r In The term contains the sign function sign(s), leading to the inequality This is not always true. Using a low-pass filter operation on the approaching control law u... r By approximation, we obtain u r approximation Its expression is:

[0071]

[0072] Where κ represents the filtering time constant. Indicate u r Approximate value, express The first derivative.

[0073] u is obtained from formula (8) through a low-pass filter operation. r approximation Thus, the first time-varying parameter is constructed. The safety boundary ensures the first time-varying parameter. satisfy:

[0074]

[0075] in, Let represent a positive constant and satisfy . ε > 0 represents a positive constant.

[0076] Construct the first time-varying parameter according to formula (9) The safety boundary is defined, and the error variable of the adaptive law is defined as follows: Its expression is as follows:

[0077]

[0078] Error variable defined by formula (10) The first time-varying parameter can be obtained. first derivative The expression is:

[0079]

[0080] in, Let represent a positive constant and satisfy . when hour, when hour, when hour, Φ(t) represents the safety boundary of the rate of change of the adaptive law, and its expression is:

[0081]

[0082] Second time-varying parameter first derivative It can be represented as:

[0083]

[0084] in, Let represent a positive constant and satisfy . θ(t) represents the rate of change of the error of the adaptive law, and its expression is:

[0085]

[0086] Here, η represents a positive constant, and η > 0.

[0087] Step 2.4: The sliding variable s in formula (6) and the approach control law u in formula (7) r A non-singular fixed-time sliding mode controller τ for a robot system can be designed, and its expression is as follows:

[0088] τ=τ eq +τ sw (15)

[0089] Where, τ eq The feedback term represents compensation for nonlinear dynamics; τ sw It is a convergence controller used to handle lumped unknown bounded disturbances in nonlinear robot systems.

[0090] In formula (15), the feedback term τ eq The expression is:

[0091]

[0092] In formula (15), the approach controller τ sw The expression is:

[0093]

[0094] Based on the designed non-singular fixed-time sliding mode controller τ of the robot system, the control torque required for robot joint movement can be calculated. The calculated control torque command is sent to the robot system's integrated controller via the CAN bus. The integrated controller converts the received control signal into drive signals for the robot joint motors, controlling the motor speed and direction to drive the robot joint movement. The position, velocity, and acceleration of the robot joints are monitored in real-time by the host computer software. If the robot system is subjected to unknown disturbances, causing a deviation between the robot joint position and the desired reference position, the host computer can adjust the control torque command in real-time according to the designed non-singular fixed-time sliding mode controller τ to reduce robot joint tracking errors and ensure the stability and accuracy of robot joint trajectory tracking control.

[0095] Step 3: Based on the formula in Step 2, design a non-singular fixed-time sliding mode controller τ for the robot system. Analyze and explain the stability of the non-singular fixed-time sliding mode controller τ. Specific implementation steps include:

[0096] Step 3.1, Convergence analysis of the sliding variable s:

[0097] To analyze the dynamic characteristics of the sliding variable s in formula (6), we can obtain the first derivative of the sliding variable s:

[0098]

[0099] Substituting the trajectory tracking error equation of the nonlinear robot system into formula (18), the first derivative of the sliding variable is... From this, we can obtain:

[0100]

[0101] Substituting the nonsingular fixed-time sliding mode controller τ designed in step 2 into the first derivative of the sliding variable in formula (19) From this, we can obtain:

[0102]

[0103] Choosing Lyapunov functions Its first derivative can be expressed as:

[0104]

[0105] When the parameter selection in formula (21) satisfies Lyapunov function V s first derivative It can be changed to:

[0106]

[0107] From formula (22), the Lyapunov function V s first derivative It can be seen that the robustness of the sliding mode controller is mainly due to This is reflected in the double-nested adaptive law terms. Greater than the upper bound of the disturbance At that time, it will definitely be able to ensure This holds true. Therefore, the controller proposed in this invention, while possessing strong robustness, employs a double-layer nested adaptive law term instead of a fixed control gain term, enabling online adjustment of the control gain as unknown disturbances change. This avoids overestimation of lumped unknown bounded disturbances and reduces high-frequency chattering in the robot system.

[0108] At this point, the sliding variable s in formula (6) and the error variable of the adaptive law in formula (10) are... The rate of change of error θ(t) of the adaptive law of formula (14) will be at a fixed time T. s The convergence time is T, which converges to the origin. s The expression is:

[0109]

[0110] Step 3.2, Convergence Analysis of Robot Joint Tracking Error:

[0111] When the sliding variable reaches the sliding surface, i.e., s = 0, then:

[0112]

[0113] Choosing Lyapunov functions Its first derivative can be expressed as:

[0114]

[0115] 1) When the robot joint tracking error |e1| ≥ 1, The Lyapunov function of formula (25) holds true. first derivative It can be rewritten as:

[0116]

[0117] At this time, the robot joint tracking error e1 will be within a fixed time. Within the range, the convergence time is from |e1|≥1 to |e1|<1. It can be represented as:

[0118]

[0119] 2) When the robot joint tracking error |e1| < 1, we have Then we can obtain the following inequality:

[0120]

[0121] Lyapunov function of formula (25) first derivative It can be represented as:

[0122]

[0123] At this time, the robot joint tracking error e1 will be within a fixed time. Within the range, the convergence from |e1|<1 to the origin takes time. It can be represented as:

[0124]

[0125] Based on the above convergence analysis, according to the sliding mode variable s in formula (6) and the convergence control law u in formula (7) r The designed non-singular fixed-time sliding mode controller τ enables the robot joint tracking error e1 to achieve global fixed-time stability. Simultaneously, the upper bound T of the total time for the robot joint tracking error e1 to converge to the origin is also determined. max It can be represented as:

[0126]

[0127] The upper bound of the total convergence time T max From the expression, we can see that the upper bound of time T max The robot joint tracking error e1 is independent of the initial conditions of the robot system; that is, regardless of the initial value, the error remains constant in time T. max It converges inward to the origin.

[0128] Step 4: To verify the effectiveness and feasibility of the fixed-time sliding mode controller based on the double-layer nested adaptive law designed in this invention for controlling the robot joint tracking error e1, a comparative experiment was conducted on a robot platform. The robot experimental platform is as follows: Figure 2 As shown. The real-time implementation of the control algorithm employs an integrated controller based on the TMS320F28335 digital signal processor (DSP). Each joint module is equipped with a 20,000-line optical encoder, acquiring position and speed information from the frameless torque motor at a sampling frequency of 1 kHz.

[0129] The implementation process of robot joint tracking error e1 control includes designing trajectory reference signals for the robot joints in the host computer software, generating control commands, and sending them to the integrated controller via JTAG or UART interface. The controller receives the commands and distributes them to each joint of the robot via the CAN bus. Each joint of the robot executes the corresponding movement to achieve the predetermined trajectory. Simultaneously, the host computer monitors the motion status in real time and adjusts the control commands as needed to ensure the accuracy and stability of the motion. The entire process involves closed-loop control from trajectory reference signal design to integrated controller execution and host computer monitoring to ensure the stability and accuracy of robot joint trajectory tracking control.

[0130] This invention takes the 5th joint of a 6-DOF robot as an example, and sets the trajectory reference signal of the 5th joint of the robot as q. d = (π / 6)sin(0.2πt). The parameter settings for the fixed-time sliding mode controller based on the double-nested adaptive law term are:

[0131] λ1=1.2, λ′1=1.2, κ = 0.02, ε = 0.001,

[0132] Comparative experimental results of the control performance of a 5-joint nonlinear robot are as follows: Figures 3-5 As shown.

[0133] from Figure 3 Location tracking response and Figure 4The tracking error results show that the fixed-time sliding mode control based on the double-layer nested adaptive law proposed in this invention achieves satisfactory tracking performance and convergence speed. In contrast, the finite-time terminal sliding mode control proposed in the literature "Shuanghe Yu, Xinghuo Yu, Bijan Shirinzadeh, et al. Continuous finite-time control for robotic manipulators with terminal sliding mode[J].Automatica,2005,Vol.41(11):1957-1964." and the non-singular fixed-time terminal sliding mode control proposed in "Zuo,Zongyu. Non-singular fixed-time terminal sliding mode control of non-linear systems.[J].IET Control Theory&Applications,2015,Vol.9(4):545-552." are different. The fixed-time sliding mode control method proposed in the literature "Moulay, Emmanuel, Lechappe, et al. Robust fixed-time stability: application to sliding-mode control.[J].IEEE Transactions on Automatic Control,2022,Vol.67(2):1061-1066." has a slow convergence speed of robot joint positions and a large robot joint tracking error.

[0134] from Figure 5 As can be seen, the fixed-time sliding mode control based on the double-layer nested adaptive law term in this invention not only ensures the rapid tracking response of the robot joint position, but also reduces the robot joint tracking error and the amplitude of the control input, thereby weakening the chattering of the robot system and verifying the effectiveness of the proposed double-layer nested adaptive law.

[0135] Example 2: A fixed-time sliding mode control system for a nonlinear robot system based on a double-nested adaptive law, comprising: a first module for establishing a mathematical model of an n-DOF nonlinear robot system; and establishing a trajectory tracking error control equation for the nonlinear robot system containing lumped unknown bounded disturbances based on the mathematical model of the n-DOF nonlinear robot system; a second module for designing a sliding variable based on a modified variable power function based on the established trajectory tracking error equation of the nonlinear robot system; designing a fixed-time approaching control law based on a double-nested adaptive law based on the sliding variable; and designing a non-singular fixed-time sliding mode controller for the nonlinear robot system based on the sliding variable and the fixed-time approaching control law based on the double-nested adaptive law. Although the system described in the above embodiment is preferably implemented in software, hardware implementation, or a combination of software and hardware, is also possible and contemplated. For parts of each module not described in detail, please refer to the relevant descriptions in other embodiments.

[0136] Example 3: A processor for running a program, wherein the program executes a fixed-time sliding mode control method for a nonlinear robot system based on a double-nested adaptive law as described in any one of Examples 1. The processor executes the program by performing the following steps: Step 1: Establishing a mathematical model of an n-DOF nonlinear robot system; based on the mathematical model of the n-DOF nonlinear robot system, establishing a trajectory tracking error control equation for the nonlinear robot system containing lumped unknown bounded disturbances; Step 2: Based on the established trajectory tracking error equation of the nonlinear robot system, designing a sliding variable based on a modified variable power function; based on the sliding variable, designing a fixed-time approaching control law based on a double-nested adaptive law; based on the sliding variable and the fixed-time approaching control law based on a double-nested adaptive law, designing a non-singular fixed-time sliding mode controller for the nonlinear robot system.

[0137] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law, characterized in that, include: Step 1: Establish a mathematical model for an n-degree-of-freedom nonlinear robot system; Based on the mathematical model of an n-degree-of-freedom nonlinear robot system, a trajectory tracking error control equation for a nonlinear robot system containing lumped unknown bounded disturbances is established. Step 2: Based on the established trajectory tracking error equation of the nonlinear robot system, design a sliding variable based on a modified power-law function; Based on the sliding variable, a fixed-time approach control law based on a double-nested adaptive law is designed; Based on the sliding variable and the fixed-time approaching control law based on the double-nested adaptive law, a non-singular fixed-time sliding mode controller for a nonlinear robot system is designed. Based on the established trajectory tracking error equation of the nonlinear robot system, a sliding variable based on a modified power-law function is designed. Specifically: in, , and Let represent the control gain in the sliding variable, and satisfy . , , ; and Let represent a positive constant and satisfy . ; Represents a constant greater than 1; Represents a symbolic function. It is the robot joint tracking error. first derivative .

2. The fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law according to claim 1, characterized in that, Based on the sliding variable, a fixed-time approach control law based on a double-nested adaptive law is designed, specifically as follows: in, This indicates the approach control law. Indicates the first time-varying parameter. Represents a small positive integer. and This represents the control gain in the approach control law, and satisfies... , ; , , Let represent a positive constant and satisfy . , Represents a constant greater than 1; This indicates a double-nested adaptive law term; Represents a symbolic function. It is a sliding variable. Represents the natural constant.

3. The fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law according to claim 2, characterized in that, The design of the double-nested adaptive law term is as follows: Using low-pass filtering operation for the approach control law By approximating, we obtain approximation Its expression is: in, Indicates the filter time constant; express Approximate value; express The first derivative; Obtained through low-pass filtering. approximation Thus, the first time-varying parameter is constructed. The safety boundary ensures the first time-varying parameter. satisfy: in, Let represent a positive constant and satisfy . ; Represents a positive constant; Based on the constructed first time-varying parameters The safety boundary is defined, and the error variable of the adaptive law is defined as follows: Its expression is as follows: By defining error variables The first time-varying parameter is obtained. first derivative The expression is: in, Let represent a positive constant and satisfy . ; The safety boundary representing the rate of change of the adaptive law is expressed as: Second time-varying parameter first derivative It can be represented as: in, Let represent a positive constant and satisfy . ; Represents a symbolic function; The rate of change of error of the adaptive law is expressed as: in, Let represent a positive constant and satisfy . ; It represents the upper bound of the first derivative of the lumped unknown bounded perturbation.

4. The fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law according to claim 1, characterized in that, The non-singular fixed-time sliding mode controller of the robot system ;in, The feedback term represents compensation for nonlinear dynamics; It is a convergence controller used to handle lumped unknown bounded disturbances in nonlinear robot systems.

5. A fixed-time sliding mode control system for a nonlinear robot system based on a double-layer nested adaptive law, characterized in that, include: The first module is used to establish the mathematical model of an n-degree-of-freedom nonlinear robot system; Based on the mathematical model of an n-degree-of-freedom nonlinear robot system, a trajectory tracking error control equation for a nonlinear robot system containing lumped unknown bounded disturbances is established. The second module is used to design a sliding variable based on a modified power-law function, according to the established trajectory tracking error equation of the nonlinear robot system. Based on the sliding variable, a fixed-time approach control law based on a double-nested adaptive law is designed; Based on the sliding variable and the fixed-time approaching control law based on the double-nested adaptive law, a non-singular fixed-time sliding mode controller for a nonlinear robot system is designed. Based on the established trajectory tracking error equation of the nonlinear robot system, a sliding variable based on a modified power-law function is designed. Specifically: in, , and Let represent the control gain in the sliding variable, and satisfy . , , ; and Let represent a positive constant and satisfy . ; Represents a constant greater than 1; Represents a symbolic function. It is the robot joint tracking error. first derivative .

6. A processor, characterized in that: The processor is used to run a program, wherein the program executes the fixed-time sliding mode control method for nonlinear robot systems based on double-layer nested adaptive laws as described in any one of claims 1-4.

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