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

Abstract

The invention discloses a non-linear robot system fixed time sliding mode control method based on a double-layer nested adaptive law. The method comprises the following steps: establishing a mathematical model of an n-degree-of-freedom non-linear robot system; according to the mathematical model of the n-degree-of-freedom nonlinear robot system, establishing a trajectory tracking error control equation of the nonlinear robot system containing lumped unknown bounded disturbance; according to the established trajectory tracking error equation of the nonlinear robot system, designing a sliding variable based on a modified variable power exponent function; designing a fixed time approaching control law based on a double-layer nested adaptive law according to the sliding variable; and according to the sliding variable, a non-singular fixed time sliding mode controller of the nonlinear robot system is designed based on a fixed time approaching control law of a double-layer nested adaptive law. According to the method, rapid trajectory tracking control of the nonlinear robot at different initial positions can be realized, and the safety and reliability of a robot system in a complex and changeable working environment are ensured.

Description

Technical Field

[0001] The invention relates to a fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law, and belongs to the field of stabilization and control of nonlinear robot systems. Background Art

[0002] In the course of contemporary technological development, robots are demonstrating tremendous potential and vitality in various fields, including industrial manufacturing, healthcare, 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 robotic systems in complex and changing operating environments. To achieve this goal, researchers have developed numerous advanced control methods, such as active disturbance rejection control, adaptive control, intelligent control, model predictive control, and sliding mode control.

[0003] The sliding mode control method has the design characteristics of simple structure and flexible and changeable, which provides researchers of sliding mode control method with huge room for development. Therefore, the method of combining sliding mode control with finite time stability theory and fixed time stability theory to design controllers has received widespread attention in the control field.

[0004] In the field of nonlinear system control, significant progress has been made in the integration of finite-time stability theory and sliding mode control methods. The paper "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" proposed a finite-time terminal sliding mode control scheme, which laid a theoretical foundation for improving the dynamic response speed and robustness of nonlinear systems. The paper "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" designed a finite-time sliding mode controller based on an unknown system dynamics estimator, which effectively enhanced the robustness of nonlinear systems to unknown bounded disturbances. The finite-time sliding mode control method ensures that the system state converges to the equilibrium point within a time limit, but this convergence time limit varies 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, a fixed-time stability theory whose convergence time limit does not depend on the initial state of the system was proposed. The fixed-time stability theory was combined with the sliding mode control method and 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 exponential function to construct sliding variables. Although this solves the singularity problem of the controller, its convergence characteristics still have the disadvantage of being locally dependent 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 input saturation [J]. IEEE Transactions on Cybernetics, 2023, Vol. 53 (4): 2636-2646." proposes an adaptive fixed-time sliding mode control strategy, which handles the controller singularity problem through a switching function mechanism. However, this method may cause the controller to switch back and forth at the singularity 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 and centrifugal forces, gravity, and nonlinear friction, resulting in a complex nonlinear relationship between input (torque) and output (motion). Improving the rapid trajectory tracking control of nonlinear robot systems at different initial positions can improve the system's operating efficiency, reduce energy losses, and meet high-efficiency and high-precision requirements. On the other hand, the singularity of the robot controller can lead to control torque divergence, state stagnation, increased high-frequency chattering, and loss of stability, seriously affecting the robot system's responsiveness and safety. Therefore, while achieving rapid trajectory tracking control of nonlinear robot systems at different initial positions, ensuring the nonsingularity of the robot controller and reducing the chattering of the robot system's control torque are crucial to the safety and reliability of the robot system in various operating environments and remain a topic worthy of research. Summary of the Invention

[0006] In order to eliminate the interference of nonlinear uncertain unknown bounded disturbances on the stable operation of the robot system, the present invention provides a fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law, which realizes rapid trajectory tracking control of the nonlinear robot at different initial positions, ensuring the safety and reliability of the robot system in complex and changeable working environments.

[0007] The technical solution of the present 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 the n-DOF nonlinear robot system; Based on the mathematical model of the n-DOF nonlinear robot system, establish the trajectory tracking error control equation of the nonlinear robot system with lumped unknown bounded disturbance;

[0010] Step 2: Based on the established trajectory tracking error equation of the nonlinear robot system, design a sliding variable based on the modified variable power exponential function; based on the sliding variable, design a fixed-time approaching control law based on the double-layer nested adaptive law; based on the sliding variable and the fixed-time approaching control law based on the double-layer 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 variable power exponent function is designed, specifically:

[0012]

[0013] in, and Represents the control gain in the sliding variable and satisfies and represents a positive constant that satisfies λ1 represents a constant greater than 1; sign(e1) represents the sign function, e1 is the robot joint tracking error, and the first-order derivative of e1

[0014] Furthermore, according to the sliding variable, a fixed-time approaching control law based on a double-layer nested adaptive law is designed, specifically:

[0015]

[0016] Among them, u r represents the reaching control law, represents the first time-varying parameter, represents a small positive constant, and Represents the control gain in the approaching control law and satisfies represents a positive constant that satisfies λ′1 represents a constant greater than 1; represents a double-layer nested adaptive law term; sign(s) represents a sign function, s is a sliding variable, and e represents a natural constant.

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

[0018] Using low-pass filtering operation to approach the control law u r Approximate and get u r Approximate value of Its expression is:

[0019]

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

[0021] Get u by low-pass filtering r Approximate value of Thus constructing the first time-varying parameter The safety margin of the first time-varying parameter satisfy:

[0022]

[0023] in, represents a positive constant that satisfies ε>0 indicates a positive constant;

[0024] According to the first time-varying parameter constructed The safety boundary of the adaptive law is defined as Its expression is as follows:

[0025]

[0026] By defining the error variable Get the first time-varying parameter The first derivative of The expression is:

[0027]

[0028] in, represents a positive constant that satisfies Φ(t) represents the safety margin of the adaptive law’s rate of change, and its expression is:

[0029]

[0030] The second time-varying parameter The first derivative of It can be expressed as:

[0031]

[0032] in, represents a positive constant that satisfies sign(θ(t)) represents the sign function; θ(t) represents the error change rate of the adaptive law, which is expressed as:

[0033]

[0034] Wherein, η represents a positive constant and satisfies η>0; represents an upper bound on the first-order derivative of the lumped unknown bounded perturbation.

[0035] Furthermore, the non-singular fixed-time sliding mode controller of the robot system τ=τ eq +τ sw ; where τ eq represents the feedback term that compensates for nonlinear dynamics; τ sw It is a reaching controller for handling lumped unknown bounded disturbances in nonlinear robotic systems.

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

[0037] The first module is used to 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, 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 variable power exponential function according to the established trajectory tracking error equation of the nonlinear robot system; based on the sliding variable, design a fixed-time approaching control law based on a double-layer nested adaptive law; based on the sliding variable and the fixed-time approaching control law based on the double-layer nested adaptive law, design a non-singular fixed-time sliding mode controller for the nonlinear robot system.

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

[0040] The beneficial effects of the present invention are as follows: the present invention designs a fixed-time sliding mode control method based on a double-layer nested adaptive law, which has the characteristics of fast convergence speed, non-singularity, and strong robustness, and realizes fast trajectory tracking control of a nonlinear robot system under different initial positions, ensuring its safety and reliability in a complex and changeable working environment. Specifically: based on the mathematical model of an n-degree-of-freedom nonlinear robot system, the present invention establishes a trajectory tracking error control equation for a nonlinear robot system containing lumped unknown bounded disturbances; further, a modified variable power exponential function is used in place of a constant exponential function in the sliding variable, thereby avoiding the singularity problem existing in the traditional fixed-time terminal sliding mode controller; in the design of the approaching control law, a time-varying double-layer nested adaptive law term is used in place of a fixed control gain term, which not only avoids overestimation of the unknown disturbance and realizes online adjustment of the control gain as the unknown disturbance changes, but also weakens the system's chattering, and uses Lyapunov's theorem to prove the fixed-time stability of the closed-loop system. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a flow chart of the present invention;

[0042] Figure 2 A six-degree-of-freedom robot experimental platform provided according to an embodiment of the present invention;

[0043] Figure 3 is the position tracking response of the joint 5 provided in accordance with an embodiment of the present invention;

[0044] Figure 4 is the position tracking error of the joint 5 provided according to the embodiment of the present invention;

[0045] Figure 5 It is the control input of the joint 5 provided according to the embodiment of the present invention. DETAILED DESCRIPTION

[0046] To make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in 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 ordinary technicians in this field without making creative work are within the scope of protection of the present invention. It should be noted that, in the absence of conflict, the embodiments in this application and the features in the embodiments can be combined with each other in any way.

[0047] Example 1: Figure 1-Figure 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 disturbances, and based on the Lagrangian modeling method, a mathematical model of the n-degree-of-freedom nonlinear robot system is established. The expression is as follows:

[0049]

[0050] in, Represent the robot joint position, joint angular velocity and joint angular acceleration respectively; M(q)∈R n×n represents the inertia matrix of the robot system, Denotes the centrifugal torque of the robot system, G(q)∈R n represents the gravitational torque of the robot system; τ∈R n Represents the control torque of the robot system; d∈R n represents the unknown bounded disturbance to the robot system, R n×n represents an n×n real matrix, R n Represents an n-dimensional real vector.

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

[0052]

[0053] In order to realize the trajectory tracking error control of nonlinear robot system, the tracking error of robot joint is defined as e1=[e 11 ,…,e 1m ] T ∈R n , the first-order derivative of the robot joint tracking error e1 Then the tracking error e of robot joint i is 1i and its first-order derivative e 2i It can be expressed as:

[0054]

[0055] Among them, e 11 represents the tracking error of robot 1 joint, e 1n represents the tracking error of the robot's n joints. e 21 represents the first-order derivative of the tracking error of robot 1 joint, e 2n Represents the first-order derivative of the tracking error of the robot n joint. i represents the position of the robot's i joint, q di represents the desired position of joint i of the robot. represents the angular velocity of the robot's joint i, represents the desired angular velocity of joint i of the robot.

[0056] According to the acceleration equation of the robot joint in formula (2), the second-order derivative of the robot joint tracking error e1 can be derived, and its expression is:

[0057]

[0058] in, represents the angular acceleration of the robot joint, represents the desired angular acceleration of the robot joints.

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

[0060]

[0061] in, represents the lumped unknown bounded disturbance of an n-DOF nonlinear robot system and satisfies represents the upper bound of the lumped unknown bounded disturbance, represents an upper bound on the first-order derivative of the lumped unknown bounded perturbation.

[0062] According to the formula in step 1, the trajectory tracking error equation of the nonlinear robot system with lumped unknown bounded disturbance is established to eliminate the lumped unknown bounded disturbance. In order to understand the impact of the nonlinear robot system on the stable and safe operation, a non-singular fixed-time sliding mode controller will be designed to ensure the safety and reliability of the robot system in 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 variable power exponential function; based on the sliding variable, design a fixed-time approaching control law based on a double-layer nested adaptive law; based on the sliding variable and the fixed-time approaching control law based on the double-layer nested adaptive law, design a non-singular fixed-time sliding mode controller for the nonlinear robot system. Specifically, it includes:

[0064] Step 2.1: Based on the trajectory tracking error equation of the nonlinear robot system established in step 1, the robot joint tracking error e1 is obtained, and the sliding variable s based on the modified variable power exponential function is designed as follows:

[0065]

[0066] in, and represents the control gain and satisfies and represents a positive constant that satisfies λ1 represents a constant greater than 1; sign(e1) represents a 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 other sign functions sign(·) in the present invention are similar.

[0067] Step 2.2: To avoid the lumped unknown bounded disturbance The overestimation of is used to reduce the high-frequency chattering of the robot system. Based on the sliding variable s obtained in the formula in step 2.1, the following fixed-time approaching control law based on the double-layer nested adaptive law is designed:

[0068]

[0069] Among them, u r represents the reaching control law, represents the first time-varying parameter, represents a small positive constant, and represents the control gain and satisfies represents a positive constant that satisfies λ′1 represents a constant greater than 1; Represents a double-layer nested adaptive law term. sign(s) represents a sign function, s is a sliding variable, when s>0, sign(s)=1; when s=0, sign(s)=0; when s<0, sign(s)=-1.

[0070] Step 2.3: The reaching control law u obtained by formula (7) r in The term contains the sign function sign(s), which leads to the inequality It is not always true. Using low-pass filtering to approach the control law u r Approximate and get u r Approximate value of Its expression is:

[0071]

[0072] Where κ represents the filtering time constant. Indicates u r The approximate value of express The first derivative of .

[0073] From formula (8), we can get u by low-pass filtering. r Approximate value of Thus constructing the first time-varying parameter The safety margin of the first time-varying parameter satisfy:

[0074]

[0075] in, represents a positive constant that satisfies ε>0 represents a positive constant.

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

[0077]

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

[0079]

[0080] in, represents a positive constant that satisfies when hour, when hour, when hour, Φ(t) represents the safety margin of the adaptive law’s rate of change, and its expression is:

[0081]

[0082] The second time-varying parameter The first derivative of It can be expressed as:

[0083]

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

[0085]

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

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

[0088] τ=τ eq +τ sw (15)

[0089] Among them, τ eq represents the feedback term that compensates for nonlinear dynamics; τ sw It is a reaching controller for handling lumped unknown bounded disturbances in nonlinear robotic systems.

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

[0091]

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

[0093]

[0094] The control torque required for robot joint motion is calculated based on the designed nonsingular fixed-time sliding mode controller τ for the robot system. 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 a drive signal for the robot joint motors, controlling the speed and direction of the motors, thereby driving the robot joint motion. The host computer software monitors the robot joint's state variables, such as position, velocity, and acceleration, in real time. If the robot system is subject to unknown disturbances, resulting in a deviation between the robot joint position and the desired reference position, the host computer can adjust the control torque command in real time based on the designed nonsingular fixed-time sliding mode controller τ to reduce the robot joint tracking error and ensure the stability and accuracy of the robot joint trajectory tracking control.

[0095] Step 3: Based on the non-singular fixed-time sliding mode controller τ of the robot system designed according to the formula in step 2, the stability of the non-singular fixed-time sliding mode controller τ is analyzed and explained. The specific implementation steps include:

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

[0097] In order to analyze the dynamic characteristics of the sliding variable s in formula (6), the first-order derivative of the sliding variable s is obtained:

[0098]

[0099] Substitute the trajectory tracking error equation of the nonlinear robot system into the first-order derivative of the sliding variable in formula (18): In the above equation, we can get:

[0100]

[0101] Substitute the non-singular fixed-time sliding mode controller τ designed in step 2 into the first-order derivative of the sliding variable in formula (19): In the above equation, we can get:

[0102]

[0103] Select Lyapunov function Its first-order derivative can be expressed as:

[0104]

[0105] When the parameters in formula (21) are selected to satisfy Lyapunov function V s The first derivative of Can be changed to:

[0106]

[0107] From the Lyapunov function V in formula (22) s The first derivative of It can be seen that the robustness of the sliding mode controller is mainly determined by When the double-layer nested adaptive terms Greater than the perturbation upper bound When, we must ensure Therefore, the controller proposed in this paper not only has strong robustness but also uses a double-layer nested adaptive law term to replace the fixed control gain term, achieving online adjustment of the control gain as the unknown disturbance changes. This avoids overestimation of the aggregate unknown bounded disturbance and weakens the high-frequency chattering of the robot system.

[0108] At this time, the sliding variable s of formula (6) and the error variable of the adaptive law of formula (10) are The error change rate θ(t) of the adaptive law of formula (14) will be s Converges to the origin, convergence time T 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, that is, s = 0, then:

[0112]

[0113] Select Lyapunov function Its first-order derivative can be expressed as:

[0114]

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

[0116]

[0117] At this time, the robot joint tracking error e1 will be From |e1|≥1 to |e1|<1, the convergence time It can be expressed as:

[0118]

[0119] 2) When the robot joint tracking error |e1| is less than 1, there is Then we can get the following inequality:

[0120]

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

[0122]

[0123] At this time, the robot joint tracking error e1 will be From |e1|<1, it converges to the origin, and the convergence time It can be expressed as:

[0124]

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

[0126]

[0127] By the upper bound of the total convergence time T max From the expression, we can see that the time upper bound T max It does not depend on the initial conditions of the robot system, that is, no matter what the initial value is, the robot joint tracking error e1 is always max Converges 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 the present invention to control the robot joint tracking error e1, a comparative experiment was carried out on the robot platform. The robot experimental platform is as follows: Figure 2 The real-time implementation of the control algorithm uses an integrated controller based on the TMS320F28335 digital signal processor (DSP). Each joint module is equipped with a 20,000-line optical encoder that acquires position and velocity information from the frameless torque motor at a sampling frequency of 1 kHz.

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

[0130] The present invention takes the 5 joints 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 setting of the fixed-time sliding mode controller based on the double-layer nested adaptive law term is:

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

[0132] The experimental results of the control performance comparison of the 5 joints of the nonlinear robot are as follows: Figure 3-Figure 5 shown.

[0133] from Figure 3 Position tracking response and Figure 4It can be seen from the tracking error results in that the fixed-time sliding mode control based on the double-layer nested adaptive law proposed in the present invention achieves satisfactory tracking performance and convergence speed. In comparison, 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.” 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 the robot joint position and a large tracking error of the robot joint.

[0134] from Figure 5 It can be seen that the fixed-time sliding mode control based on the double-layer nested adaptive law term in the present invention not only ensures the rapid tracking response of the robot joint position, but also reduces the robot joint tracking error and control input amplitude, thereby weakening the robot system vibration 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-layer nested adaptive law, comprising: a first module for establishing a mathematical model of an n-degree-of-freedom nonlinear robot system; establishing a trajectory tracking error control equation for the nonlinear robot system containing a lumped unknown bounded disturbance based on the mathematical model of the n-degree-of-freedom nonlinear robot system; a second module for designing a sliding variable based on a modified variable power exponential function based on the established trajectory tracking error equation for the nonlinear robot system; designing a fixed-time approaching control law based on the double-layer nested adaptive law based on the sliding variable; and designing a nonsingular 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-layer nested adaptive law. Although the system described in the above embodiment is preferably implemented in software, implementation in hardware, or a combination of software and hardware, is also possible and contemplated. For portions not described in detail in each module, please refer to the relevant descriptions of other embodiments.

[0136] Embodiment 3: A processor, configured to run a program, wherein the program, when run, executes the fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law as described in any one of Embodiments 1. When the processor executes the program, the following steps are implemented: Step 1: Establishing 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, establishing a trajectory tracking error control equation for the nonlinear robot system containing a lumped unknown bounded disturbance; Step 2: Based on the established trajectory tracking error equation for the nonlinear robot system, designing a sliding variable based on a modified variable power exponential function; Based on the sliding variable, designing a fixed-time approaching control law based on the double-layer nested adaptive law; Based on the sliding variable and the fixed-time approaching control law based on the double-layer nested adaptive law, designing a nonsingular fixed-time sliding mode controller for the nonlinear robot system.

[0137] The specific embodiments of the present invention are described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made within the knowledge of ordinary technicians in this field without departing from the scope 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 of the n-DOF 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 with lumped unknown bounded disturbance is established; Step 2: Based on the established trajectory tracking error equation of the nonlinear robot system, design the sliding variable based on the modified variable power exponent function; According to the sliding variable, a fixed-time approaching control law based on a double-layer nested adaptive law is designed; A non-singular fixed-time sliding mode controller for a nonlinear robot system is designed based on a fixed-time approaching control law with sliding variables and a double-layer nested adaptive law.

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 is characterized in that: According to the established trajectory tracking error equation of the nonlinear robot system, the sliding variable s based on the modified variable power exponential function is designed, specifically: in, and Represents the control gain in the sliding variable and satisfies and represents a positive constant that satisfies λ1 represents a constant greater than 1; sign(e1) represents the sign function, e1 is the robot joint tracking error, and the first-order derivative of e1 3. 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: According to the sliding variable, a fixed-time approaching control law based on a double-layer nested adaptive law is designed, specifically: Among them, u r represents the reaching control law, represents the first time-varying parameter, represents a small positive constant, and Represents the control gain in the approaching control law and satisfies represents a positive constant that satisfies λ′1 represents a constant greater than 1; represents a double-layer nested adaptive law term; sign(s) represents a sign function, s is a sliding variable, and e represents a natural constant.

4. The fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law according to claim 3 is characterized in that: The design of the double-layer nested adaptive law term is specifically as follows: Using low-pass filtering operation to approach the control law u r Approximate and get u r Approximate value of Its expression is: Where k represents the filtering time constant; Indicates u r Approximate value of express The first derivative of ; Get u by low-pass filtering r Approximate value of Thus constructing the first time-varying parameter The safety margin of the first time-varying parameter satisfy: in, represents a positive constant that satisfies ε>0 indicates a positive constant; According to the first time-varying parameter constructed The safety margin of the adaptive law is defined as the error variable Υ(t), which is expressed as follows: By defining the error variable Υ(t), we can get the first time-varying parameter The first derivative of The expression is: in, represents a positive constant that satisfies Φ(t) represents the safety margin of the adaptive law’s rate of change, and its expression is: The second time-varying parameter The first derivative of It can be expressed as: in, represents a positive constant that satisfies sign(θ(t)) represents the sign function; θ(t) represents the error change rate of the adaptive law, which is expressed as: Wherein, η represents a positive constant and satisfies η>0; represents an upper bound on the first-order derivative of the lumped unknown bounded perturbation.

5. 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 τ=τ eq +τ sw ; where τ eq represents the feedback term that compensates for nonlinear dynamics; τ sw It is a reaching controller for handling lumped unknown bounded disturbances in nonlinear robotic systems.

6. A fixed-time sliding mode control system for a nonlinear robotic system based on a double-layer nested adaptive law, characterized in that: include: The first module is used to 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, the trajectory tracking error control equation of the nonlinear robot system with lumped unknown bounded disturbance is established; The second module is used to design a sliding variable based on a modified variable power exponent function according to the established trajectory tracking error equation of the nonlinear robot system; According to the sliding variable, a fixed-time approaching control law based on a double-layer nested adaptive law is designed; A non-singular fixed-time sliding mode controller for a nonlinear robot system is designed based on a fixed-time approaching control law with sliding variables and a double-layer nested adaptive law.

7. A processor, characterized in that: The processor is used to run a program, wherein the program, when running, executes the fixed-time sliding mode control method for a nonlinear robot system based on a double-layer nested adaptive law according to any one of claims 1 to 5.

Images