Robotic arm system fixed time tracking control method and system under random disturbance

Abstract

The application belongs to the technical field of mechanical arm system tracking control, and discloses a mechanical arm system fixed time tracking control method and system under random disturbance, which firstly establishes a mathematical model of the mechanical arm system under random disturbance affected by actuators and output constraints, and extends the mathematical model to a dynamic model of a multi-input multi-output random uncertain nonlinear system; then, a time-dependent barrier function is designed to process the output constraint, and a nonlinear transformation is designed for the nonlinear control input of the actuator; then, an equivalent system form is established, and a non-singular adaptive fixed time tracking controller is designed by using the backstepping method, so that the fixed time tracking control of the mechanical arm system is finally realized. The mechanical arm system fixed time tracking control method provided by the application can realize fast and accurate tracking of the preset reference signal under the conditions of random disturbance, actuator nonlinearity and output constraint, and can significantly improve the system robustness and control performance.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of mechanical arm system tracking control, and particularly relates to a mechanical arm system fixed time tracking control method and system under random disturbance. BACKGROUND

[0002] As a typical multi-degree-of-freedom electromechanical system, the dynamics structure of the double-link mechanical arm contains complex factors such as coupled inertia terms, Coriolis force / centrifugal force terms and external disturbance. In the actual running environment, random factors such as joint friction variation, load disturbance and sensor noise will inevitably affect the system dynamics, so that the model presents random nonlinear characteristics. Therefore, the double-link mechanical arm dynamics equation can be regarded as a typical special case of a multiple-input multiple-output (MIMO) random nonlinear system.

[0003] With the complication and intelligentization of industrial systems, MIMO random nonlinear systems are widely used in the fields of unmanned aerial vehicle formation control, robot operation, industrial process control and intelligent vehicle system. However, such systems are generally affected by multiple factors such as structural uncertainty, random disturbance, output constraint, and actuator input nonlinearity such as saturation, dead zone and friction characteristics, which makes stable control and performance guarantee face significant challenges.

[0004] Especially in complex engineering environments, random noise caused by external disturbance or measurement error generally exists. To characterize this randomness, a stochastic differential equation driven by a Wiener process, i.e. Brown motion, is commonly used for modeling. The Wiener process has been widely used to describe fluid disturbance, temperature fluctuation, communication signal jitter and financial price change, and can effectively characterize random disturbances such as sensor noise, wind speed variation and friction unevenness in control systems.

[0005] It should be noted that traditional adaptive control methods can only achieve asymptotic stability or finite time stability, and their convergence performance depends on the initial state of the system, which makes it difficult to guarantee error convergence within a predetermined time. In contrast, fixed-time control can achieve rapid steady-state convergence within a fixed time independent of the initial condition, and has significant advantages in disturbance suppression robustness.

[0006] Therefore, in the MIMO nonlinear system affected by random disturbance, actuator nonlinearity and output constraint, constructing an adaptive fixed-time control strategy that can ensure the boundedness of closed-loop signals and achieve precise tracking within a fixed time not only has important theoretical research value, but also has important engineering application significance for improving the reliability and accuracy of the system in complex environments. SUMMARY

[0007] The mechanical arm system fixed time tracking control method under random interference aims at a mechanical arm system affected by random disturbance, actuator nonlinearity and output constraint, and can ensure that the closed-loop signal is bounded while realizing adaptive fixed time control.

[0008] In order to achieve the above-mentioned purpose, the technical scheme is adopted as follows:

[0009] The mechanical arm system fixed time tracking control method under random interference comprises the following steps:

[0010] Step 1. A mathematical model of a mechanical arm system affected by actuator and output constraints under random interference is established, a dynamics model of the mechanical arm system in the Cartesian space is obtained by coordinate transformation, and the dynamics model is extended to a dynamics model of a multi-input multi-output random uncertain nonlinear system;

[0011] Step 2. A time-dependent barrier function is designed for processing the output constraint; a control input affected by a dead zone and saturation is established, and a nonlinear transformation is designed for the control input of the actuator nonlinearity;

[0012] Step 3. Based on the dynamics model of the multi-input multi-output random uncertain nonlinear system obtained in step 1, an equivalent system form is established, and an equivalent multi-input multi-output random uncertain nonlinear dynamics model is obtained;

[0013] Step 4. Based on the barrier function and the nonlinear transformation designed in step 2, and the equivalent multi-input multi-output random uncertain nonlinear dynamics model obtained in step 3, a non-singular adaptive fixed time tracking controller is designed by using the backstepping method;

[0014] Step 5. The non-singular adaptive fixed time tracking controller designed in step 4 is used to realize fixed time tracking control of the mechanical arm system.

[0015] In addition, on the basis of the mechanical arm system fixed time tracking control method under random interference, the present application further proposes a mechanical arm system fixed time tracking control system under random interference which is suitable for the mechanical arm system fixed time tracking control method under random interference, and the technical scheme is as follows:

[0016] The mechanical arm system fixed time tracking control system under random interference comprises:

[0017] The dynamics model construction module is used to establish a mathematical model of a mechanical arm system affected by actuator and output constraints under random interference, obtain a dynamics model of the mechanical arm system in the Cartesian space by coordinate transformation, and extend the dynamics model to a dynamics model of a multi-input multi-output random uncertain nonlinear system;

[0018] The output constraint processing and actuator nonlinear modeling module is used for designing a time-dependent barrier function for processing the output constraint; the control input affected by the dead zone and saturation is established, and a nonlinear transformation is designed for the control input of the actuator nonlinearity;

[0019] The equivalent transformation module is used for establishing an equivalent system form based on the dynamic model of the multi-input multi-output stochastic uncertain nonlinear system, and obtaining an equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model;

[0020] The controller design module is used for designing a non-singular adaptive fixed-time tracking controller by using the backstepping method based on the barrier function and the nonlinear transformation, and the equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model;

[0021] And the tracking control module is used for realizing the fixed-time tracking control of the manipulator system by using the non-singular adaptive fixed-time tracking controller.

[0022] In addition, on the basis of the above-mentioned manipulator system fixed-time tracking control method under random disturbance, the present application also proposes a computer device, which comprises a memory and one or more processors;

[0023] The memory stores executable code, and the processor executes the executable code to implement the steps of the above-mentioned manipulator system fixed-time tracking control method under random disturbance.

[0024] In addition, on the basis of the above-mentioned manipulator system fixed-time tracking control method under random disturbance, the present application also proposes a computer readable storage medium, which stores a program; the program is executed by the processor to implement the steps of the above-mentioned manipulator system fixed-time tracking control method under random disturbance.

[0025] The present application has the following advantages:

[0026] As described above, the present application discloses a manipulator system fixed-time tracking control method under random disturbance, which first constructs a general barrier function, breaks through the limitation that the traditional barrier Lyapunov function fails when passing through zero on the constraint boundary, not only can keep the system stable under the condition of meeting the output constraint, but also is still applicable in the case of no constraint, significantly improving the universality and adaptability of the control strategy.

[0027] Secondly, the method of the present application proposes a new input transformation mechanism for the dead zone and saturation nonlinearity of the actuator, and designs a non-singular fixed-time control method combined with a switching function, so as to ensure the convergence of tracking error in fixed time and effectively avoid the occurrence of singular problem.

[0028] Furthermore, to address unknown control gains and complex nonlinear terms, the method of this invention introduces a neural network of Nussbaum function and radial basis function for adaptive compensation and approximation, enabling the system to maintain closed-loop stability and boundedness of each signal even under random disturbances and parameter uncertainties.

[0029] In summary, the fixed-time tracking control method for robotic arm systems under random disturbances proposed in this invention not only achieves accurate and stable tracking of stochastic nonlinear systems within a fixed time, but also possesses strong robustness and broad engineering application value, demonstrating significant theoretical research value and promising prospects for practical application. Attached Figure Description

[0030] Figure 1 This is a flowchart of a fixed-time tracking control method for a robotic arm system under random disturbances, as described in an embodiment of the present invention.

[0031] Figure 2 This is a schematic diagram of the structure of the double-link robotic arm in an embodiment of the present invention.

[0032] Figure 3 The control input in this embodiment of the invention is affected by dead zone and saturation nonlinearity. A schematic diagram.

[0033] Figure 4 This is a state-time diagram of a stochastic nonlinear system based on output constraints and input nonlinearity under adaptive fixed-time tracking control in an embodiment of the present invention.

[0034] in, Figure 4 (a) in the figure represents the system state. That is, system output Reference signal and constraint boundaries and The time domain diagram, Figure 4 (b) in the figure represents the system state. The time-domain diagram.

[0035] Figure 5 This is the adaptive law for a stochastic nonlinear system based on output constraints and input nonlinearity under adaptive fixed-time tracking control in an embodiment of the present invention. and The time-domain diagram.

[0036] Figure 6 The control input for a stochastic nonlinear system based on output constraints and input nonlinearity under adaptive fixed-time tracking control in this embodiment of the invention is... and The time-domain diagram.

[0037] Figure 7The time-domain graph of (a) in the system state

[0038] wherein, Figure 7 The time-domain graph of (a) in the system state The time-domain graph of (b) in the system output The time-domain graph of (c) in the reference signal The time-domain graph of (d) in the constraint boundary And The time-domain graph of (e) in the system state Figure 7 The time-domain graph of (a) in the system state The time-domain graph of (b) in the system output

[0039] Figure 8 The adaptive law of the random nonlinear system based on output constraint and input nonlinearity under adaptive fixed-time tracking control in the embodiment of the application And The time-domain graph of (a) in the system state

[0040] Figure 9 The control input of the random nonlinear system based on output constraint and input nonlinearity under adaptive fixed-time tracking control in the embodiment of the application And The time-domain graph of (a) in the system state DETAILED DESCRIPTION

[0041] The application will be further described in detail below in combination with the drawings and specific embodiments:

[0042] Embodiment 1

[0043] In view of the problems of actuator nonlinearity and output constraint in the prior art, the embodiment proposes a fixed-time tracking control method for a double-link robot arm affected by actuator and output constraints under random disturbance, which focuses on the fixed-time stability when the actuator nonlinearity, i.e., the control input is affected by dead zone and saturation, and the output needs to meet the constraint, and is designed based on the adaptive tracking control method.

[0044] The application discloses a fixed-time tracking control method for a double-link mechanical arm under random interference and with actuator and output constraints.

[0045] The mechanical arm system fixed-time tracking control method under random interference is specifically introduced as follows.

[0046] As shown in Figure 1 The mechanical arm system fixed-time tracking control method under random interference specifically comprises the following steps.

[0047] Step 1. A mathematical model of a double-link mechanical arm system under random interference and with actuator and output constraints is established as shown in formula (1), a dynamics model of the double-link mechanical arm system in a Cartesian space is obtained by using coordinate transformation as shown in formula (2), and the research object is further extended to a general form of MIMO random nonlinear system, and a dynamics model of a multi-input multi-output random uncertain nonlinear system is obtained as shown in formula (7).

[0048] In this embodiment, step 1 is specifically as follows.

[0049] The dynamics equation of the mechanical arm system is an equation representing the relationship between force and motion, and the mathematical model of the mechanical arm system is constructed by using the Lagrange method, which is a method of constructing a model by analyzing the kinetic energy and potential energy of a system.

[0050] In this embodiment, a double-link mechanical arm system is taken as the research object, and a structure diagram thereof is shown in Fig. 2. represents the number of links, The first link and the second link in the two-link manipulator system are defined as a first link and a second link respectively, and the first joint and the second joint are defined as a first joint and a second joint respectively.

[0051] In the two-link manipulator system, the lengths of the first link and the second link are and respectively, the masses of the first link and the second link are and respectively, the distance between the mass center of the first link and the first joint is , the distance between the mass center of the second link and the second joint is , and the gravitational acceleration is represented by .

[0052] The Lagrangian quantity is defined as the kinetic energy of the two-link manipulator system minus the potential energy : .

[0053] The Lagrangian equation of the manipulator system is constructed as:

[0054] (1)

[0055] wherein represents time; ; represents the angular displacement of the th joint in the manipulator system, represents the angular velocity of the th joint in the manipulator system; and represents the driving torque of the manipulator system.

[0056] The Lagrangian equation of the manipulator system is simplified, i.e. the dynamics model of the manipulator system in the Cartesian space is obtained by using coordinate transformation as shown in equation (2):

[0057] (2)

[0058] wherein , and represent the angular displacements of the first joint and the second joint respectively; , and represent the angular velocities of the first joint and the second joint respectively; , and represent the angular accelerations of the first joint and the second joint respectively; is a symmetric positive definite rigid body inertia matrix, . represents the Coriolis and centrifugal force matrix, ; represents the gravity vector; represents the external disturbance; is the control matrix, for representing the effective torque of the actuator, i.e. the driving motor, acting on the robot arm; represents the control torque, , and represent the control input torque of the first and second joint, respectively.

[0059] , , , , .

[0060] wherein, represents the equivalent moment of inertia of the first joint, represents the equivalent moment of inertia of the second joint, and represent the inertia coupling terms between the first and second joint; represents the Coriolis and centrifugal force terms of the first joint's own velocity and the coupling of the second joint's velocity to the first joint, represents the Coriolis and centrifugal force terms of the second joint's velocity to the first joint, represents the Coriolis and centrifugal force terms of the first joint's velocity to the second joint, represents the Coriolis and centrifugal force terms of the coupling of the first and second joint's velocity to the second joint; and represent the gravity torque acting in the direction of the first and second joint, respectively; and represent the driving gain acting on the first and second joint, respectively; and represent the external disturbance, i.e. the unknown disturbance, acting on the first and second joint, respectively.

[0061] Equation (2) is rearranged to:

[0062] (3)

[0063] define the variables , i.e. , ;

[0064] wherein, represents the angular displacement of the first link, This indicates the angular displacement of the second link. This represents the angular velocity of the first link. This indicates the angular velocity of the second link.

[0065] The dual-link robotic arm system is described by the nonlinear state equation shown in equation (4):

[0066] (4)

[0067] in, This refers to the natural dynamic behavior of a robotic arm system without control input, including coupling, nonlinearity, and gravitational effects. This represents the matrix representing the influence of control inputs on system dynamics. External interference experienced by the robotic arm system.

[0068] , , .

[0069] in, That is , That is , That is .

[0070] Considering , , , , .

[0071] Rearranging formula (4) yields:

[0072] (5)

[0073] in, and Indicates system output, and The angular displacements corresponding to the first joint respectively and angular velocity .

[0074] Considering the random disturbances that the dual-link robotic arm may experience during actual operation, such as friction fluctuations or changes in external load, a random disturbance term is introduced into the dynamic equation shown in formula (5), resulting in the dynamic model of the dual-link robotic arm system shown in formula (6):

[0075] (6)

[0076] in, Used to represent the effect of noise on the angle of the first joint. Used to represent the effect of noise on the second joint angle. The table is used to show the effect of noise on the angular velocity of the first joint. Used to represent the effect of noise on the angular velocity of the second joint. express Wiener standard process.

[0077] As a typical multi-degree-of-freedom electromechanical system, the dual-link manipulator is not only affected by random factors such as frictional changes, load disturbances, and sensor noise, but also generally faces physical nonlinearities of actuators, such as joint drive torque saturation, dead zone and energy limitations, as well as the motion range limitations of joint or end-effector outputs. These actuator constraints and output constraints cause the actual system to exhibit obvious nonlinear and limited capability characteristics in dynamic response, exacerbating the difficulty of control under random disturbances. Combining the aforementioned random factors, it can be seen that the dynamic model of the dual-link manipulator described by formula (5) can be regarded as a typical representative of a MIMO stochastic nonlinear system that is simultaneously affected by random disturbances, actuator nonlinearity and output constraints. Based on this understanding, this embodiment further extends the research object from this specific manipulator system to a more general form of MIMO stochastic uncertain nonlinear dynamic model, and converts it into a standardized model that is more convenient for controller design. Through this equivalent modeling, the proposed adaptive fixed-time tracking control method can not only cover the typical application scenario of the dual-link manipulator, but also be applicable to a wider range of engineering systems with random disturbances, actuator nonlinearity and output constraints.

[0078] To uniformly describe the effects of random disturbances, actuator nonlinearity, and output constraints on system dynamics, and to make the controller design more generalizable and universal, the dynamic model of the dual-link robotic arm system shown in Equation (6) is transformed into the dynamic model of the multi-input multi-output MIMO stochastic uncertain nonlinear system shown in Equation (7):

[0079] (7)

[0080] in, ; , Indicates the number of joints; express The first in the time-sensitive robotic arm system Angular displacement of each joint express The first in the time-sensitive robotic arm system Angular velocity of each joint; Represents the state of a nonlinear system. , to θ1, θ2, · · ·, θn denote the angular displacement of the 1st to the nth joint in the robotic arm system, respectively, denote the angular velocity of the 1st to the nth joint in the robotic arm system, respectively; denote the angular displacement of the 1st to the nth joint in the robotic arm system, respectively, denote the angular velocity of the 1st to the nth joint in the robotic arm system, respectively; denote the angular displacement of the 1st to the nth joint in the robotic arm system, respectively, denote the angular velocity of the 1st to the nth joint in the robotic arm system, respectively; denote the angular displacement of the 1st to the nth joint in the robotic arm system, respectively, denote the angular velocity of the 1st to the nth joint in the robotic arm system, respectively; denote the control input of the nonlinear system, i.e., the driving torque of the robotic arm system, .

[0081] Definition , .

[0082] is an unknown smooth nonlinear function, and satisfies .

[0083] wherein, denotes the corresponding function value in .

[0084] denote the control gain, and has a first-order continuous derivative, i.e., is an unknown function.

[0085] denote a bounded but unknown external disturbance, and there exists a constant such that , where .

[0086] is an unknown smooth nonlinear function, and satisfies , denotes the value at .

[0087] Assume that the output of the nonlinear system satisfies , where and denote smooth functions about and , and is a preset reference signal.

[0088] ​Step 2. The output constraint of the processing system and the nonlinearity of the actuator are considered, and a time-dependent barrier function is designed to deal with the output constraint; the control input affected by dead zone and saturation is established, and a nonlinear transformation is designed for the control input of the actuator nonlinearity.

[0089] In this embodiment, step 2 is specifically:

[0090] Step 2.1. Design a time-dependent barrier function to deal with the output constraint.

[0091] A general time-dependent barrier function is designed to ensure that the system output does not violate the given constraint boundary, and the control scheme designed based on the time-dependent barrier function is still effective even if the constraint boundary crosses the zero point or the constraint boundary does not exist.

[0092] Design a time-dependent barrier function to ensure that the system output does not violate the preset constraint boundary:

[0093] (8)

[0094] wherein, y represents the system output that needs to satisfy the lower boundary, y represents the system output that needs to satisfy the upper boundary; in the case of not causing singularity, use to represent , to represent .

[0095] Derive the time-dependent barrier function , and obtain:

[0096] .

[0097] wherein,

[0098] .

[0099] .

[0100] .

[0101] .

[0102] The control method based on the time-dependent barrier function shown in Equation (8) not only ensures that the system trajectory strictly satisfies the constraint boundary under time-varying asymmetric output constraints, but also guarantees the stability of the closed-loop system even if the constraint interval contains zeros or even when the constraints are temporarily absent. Therefore, the method of this invention demonstrates significant robustness and stronger applicability when dealing with complex and dynamically changing constraints.

[0103] Step 2.2. Modeling and processing of actuator nonlinearity.

[0104] Step 2.2.1. Establish a mathematical model of the control input affected by dead zone and saturation.

[0105] Establish control inputs affected by dead zone and saturation The mathematical model is shown in formula (9):

[0106] (9)

[0107] in, This represents the control input signal to be designed. and Breakpoints indicating dead zones and The slope of the dead zone interval is represented by the slope of the dead zone interval. and This indicates the saturation value. , , , , , It is a bounded constant. The control input is affected by dead zone and saturation nonlinearity. The illustration is as follows Figure 3 As shown.

[0108] Step 2.2.2. Design a nonlinear transformation for the nonlinear control input.

[0109] To overcome the effects of actuator nonlinearity, the following nonlinear transformation is used to effectively process it:

[0110] ;

[0111] in, The slope of the dead zone; and Represented as:

[0112] .

[0113] .

[0114] In view of practical application considerations, assume that the control input signal is a bounded signal, i.e., there exists a positive constant such that is always true; where , and denote the maximum and minimum values of , respectively.

[0115] Therefore, the function is a bounded function, i.e., there exists a constant such that is always true.

[0116] Thus, it can be verified that the following inequality is satisfied:

[0117] .

[0118] where is defined as , is defined as .

[0119] is always true, where is a piecewise bounded function and satisfies .

[0120] Based on the above, it is obtained that: .

[0121] Step 3. Based on the dynamic model of the multi-input multi-output stochastic uncertain nonlinear system obtained in step 1, an equivalent system form is established, and an equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model is obtained as shown in formula (10).

[0122] Based on the foregoing preparation work, the MIMO stochastic nonlinear dynamic model affected by the nonlinear of the actuator and the output constraint is equivalent converted into an equivalent uncertain nonlinear system dynamic model which can be directly studied for fixed-time tracking control method.

[0123] In this embodiment, step 3 is specifically:

[0124] The dynamic model of the MIMO stochastic uncertain nonlinear system shown in formula (7) is converted into an equivalent MIMO stochastic uncertain nonlinear dynamic model which can be directly studied for adaptive fixed-time tracking control method:

[0125] (10)

[0126] ​Step 4. Based on the barrier function and nonlinear transformation designed in Step 2, and the equivalent MIMO stochastic uncertain nonlinear dynamics model obtained in Step 3, a nonsingular adaptive fixed-time tracking controller is designed by using backstepping method.

[0127] Based on Step 1, Step 2 and Step 3, a nonsingular adaptive fixed-time tracking controller is designed by using backstepping method combined with Nussbaum function.

[0128] In this embodiment, Step 4 is specifically:

[0129] The coordinate transformation is introduced as:

[0130] ;

[0131] wherein, represents an error variable; represents a virtual control input, .

[0132] The switching function is introduced as:

[0133] (11)

[0134] wherein, is a constant.

[0135] For the error variable , a Lyapunov function as shown in formula (12) is constructed as:

[0136] (12)

[0137] wherein, is a constant, and .

[0138] .

[0139] wherein, is an estimate of , is an estimation error of , , represents an optimal weight vector.

[0140] The infinitesimal differential operator of the Lyapunov function is calculated, and is arranged by using Young inequality, and represents an unknown nonlinear function appearing in the process, which is approximated by using a radial basis function neural network RBF-NN to obtain:

[0141] .

[0142] in, This represents the input vector of the neural network. , This represents the desired angular displacement of the first joint. This represents the desired angular velocity of the first joint; For the basis function vector, To approximate the error, and satisfy... , It is a constant.

[0143] Design a singular virtual controller as shown in formula (13). And adaptive law :

[0144] (13)

[0145] in, This represents the Nussbaum function; , , , , , It is a constant and satisfies , , , , , .

[0146] For error variables ,in Construct Lyapunov functions for:

[0147] (14)

[0148] in, It is a constant, and .

[0149] .

[0150] in, for The estimate, for The estimation error, , This represents the weight vector.

[0151] For Lyapunov functions The infinitesimal differential operator is sought, arranged by using Young inequality, and the RBF-NN is used to approximate the nonlinear function appearing in the process:

[0152] .

[0153] where is the input vector of the neural network, , are the estimation errors of ; is the basis function vector, is the approximation error, and satisfies , is a constant.

[0154] The nonsingular virtual controller and the adaptive law are designed as shown in formula (15):

[0155] (15)

[0156] where is the Nussbaum function; , , , , is a constant, and , , , , .

[0157] The Lyapunov function is constructed for the error variable as follows:

[0158] (16)

[0159] where is a positive number;

[0160] .

[0161] where is the estimation of , is the estimation error of , , is the weight vector.

[0162] ​​​Lyapunov function The infinitesimal differential operator is obtained, and the Young inequality is used to arrange and use The nonlinear function appearing in the expression process is represented, and the RBF-NN is used to approximate to obtain:

[0163] .

[0164] wherein, is an input vector of the neural network, , to respectively represent the estimation errors of to ; is a basis function vector, is an approximation error, and satisfies , is a constant.

[0165] The non-singular control input as shown in formula (17) is designed and the adaptive law are as follows:

[0166] (17)

[0167] wherein, represents a Nussbaum function; , , , , is a positive number.

[0168] Finally, the non-singular adaptive fixed-time tracking controller is obtained.

[0169] In step 4, after the design of the non-singular adaptive fixed-time tracking controller, the stability and convergence of the manipulator system controlled by the non-singular adaptive fixed-time tracking controller are analyzed.

[0170] In this embodiment, the stability and convergence of the MIMO random uncertain nonlinear system are analyzed, and the stability criterion of the MIMO random uncertain system is proposed. On this basis, the closed-loop stability of the designed adaptive fixed-time tracking control scheme is analyzed, and the specific process is as follows:

[0171] A stability theorem of the MIMO random nonlinear system affected by the actuator nonlinearity and output constraint under the designed adaptive fixed-time tracking control method is proposed, that is, a stability criterion of the uncertain nonlinear system based on the influence of the actuator and output constraint is proposed. On this basis, the closed-loop stability of the designed adaptive fixed-time tracking control scheme is analyzed.

[0172] A stability theorem for MIMO stochastic nonlinear systems with output constraints and input nonlinearities is proposed under the adaptive fixed-time tracking control method.

[0173] Theorem 1. For the MIMO stochastic nonlinear system with asymmetric input dead-zone and saturation as shown in formula (10), the controller and adaptive law as shown in formula (13), formula (15), formula (17) proposed can guarantee the following performance indicators in fixed time:

[0174] (i) The system output always does not exceed the given constraint boundary.

[0175] (ii) The tracking error can converge to a small neighborhood containing the origin in fixed time.

[0176] (iii) All signals in the closed-loop system are bounded in the sense of probability.

[0177] The proof shows that the designed non-singular adaptive fixed-time tracking controller can guarantee that the system output can stably track the preset reference signal in probability and will not violate the given constraint boundary in fixed time, and all signals in the closed-loop system are bounded, thereby verifying the effectiveness and reliability of the control method proposed in the application.

[0178] That is, it is proved that in the manipulator system controlled by the designed non-singular adaptive fixed-time tracking controller, the system output not only does not violate the preset constraint boundary, but also the system output can track the reference signal in fixed time, and all signals of the closed-loop system are bounded.

[0179] In this embodiment, the process of convergence analysis of the manipulator system controlled by the non-singular adaptive fixed-time tracking controller is specifically:

[0180] Let the Lyapunov function be:

[0181] .

[0182] Take the infinitesimal differential operator of the Lyapunov function , and get:

[0183] (18)

[0184] Wherein, , , , where is a constant, a mathematical model of the manipulator system under the random disturbance shown in formula (1) and the influence of the actuator and output constraints is obtained, and the semi-global fixed-time stability in the probability sense is achieved.

[0185] According to the definition of Lyapunov function , it is obtained that and are bounded in the fixed time, and and represent error variables and estimation errors, respectively.

[0186] Let the constant satisfy , and it is obtained that , that is , and the reference signal is bounded.

[0187] By reasonably selecting the control parameters, the system output can track the reference signal in the fixed time, and the preset constraint boundaries and are not violated.

[0188] Since and are bounded, it can be further deduced that is bounded, and the virtual controller remains bounded.

[0189] By repeating the above derivation process, it can be obtained that all signals in the manipulator system controlled by the non-singular adaptive fixed-time tracking controller remain bounded in the probability sense.

[0190] Step 5. Using the non-singular adaptive fixed-time tracking controller designed in step 4, the fixed-time tracking control of the manipulator system is realized.

[0191] In addition, in order to verify the effectiveness of the adaptive fixed-time tracking control method proposed in the present application, a general MIMO nonlinear numerical model with random disturbance, actuator nonlinearity and output constraint is selected for simulation in the present embodiment. The model has commonality with the double-link manipulator in terms of dynamic coupling, control difficulty and response characteristics, thereby proving the universality and wide applicability of the method in the double-link manipulator, flexible manipulator, robot joint system, vehicle suspension system and other similar MIMO systems.

[0192] Consider the following form of MIMO stochastic uncertain nonlinear system:

[0193] (19)

[0194] (20)

[0195] The reference signal is selected as:

[0196] , .

[0197] The initial value is:

[0198] , , , .

[0199] , , , .

[0200] The system output and The constraint boundary that needs to be met is:

[0201] .

[0202] .

[0203] The adaptive fixed-time tracking controller designed in combination with the application is based on the system state , , , and the constraint boundary , , , , the reference signal , , the adaptive law , , , , the trajectory diagram as shown in Figure 4 , Figure 5 , Figure 7 , Figure 8 . The required control input of the system and , and the control input and affected by the dead zone and saturation, the time domain diagram as shown in Figure 6 , Figure 9It can be seen that the tracking controller based on the backstepping method and the Nussbaum function can successfully realize the fixed-time tracking control of the manipulator system, and it is proved that the adaptive fixed-time tracking control scheme designed in the embodiment is feasible and effective.

[0204] The application provides a fixed-time tracking control method for a manipulator system under random disturbance.

[0205] The application provides an adaptive fixed-time tracking control method for a double-link manipulator under random disturbance and affected by actuator and output constraints.

[0206] The method can effectively deal with complex problems such as output constraints, actuator nonlinearities and unknown control gains, has good robustness and wide engineering applicability. Compared with the prior art, the present application has the following advantages: first, for the problem of system output constraint, a general barrier function is constructed, which breaks through the limitation that the traditional barrier Lyapunov function is invalid when the constraint boundary passes through zero, not only can keep the system stable under the condition of meeting the output constraint, but also is applicable in the case of no constraint, which significantly improves the universality and adaptability of the control strategy. Secondly, for the dead zone and saturation nonlinear characteristics of the actuator, a new input transformation mechanism is proposed to eliminate the adverse effects of input nonlinearity on system control performance. On this basis, a fixed-time control method without singularity is designed by combining the switching function, so that the system output not only does not violate the given constraint boundary, but also can realize the convergence of tracking error to an arbitrary small neighborhood containing the origin within a fixed time, while effectively avoiding the singularity problem caused by the power term in the fixed-time control, thereby ensuring the convergence of tracking error within a fixed time and effectively avoiding the generation of singularity problem. In addition, in order to solve the problem of unknown control gain and complex nonlinear term in the system, Nussbaum function is introduced in the controller design to cope with the sign uncertainty of control gain, and radial basis function neural network RBF-NN is used to approximate and adaptively compensate the unknown nonlinear dynamics, thereby significantly enhancing the robustness of the system. Through random Lyapunov stability analysis, it is proved that the method can realize that the output does not violate the constraint boundary within a fixed time, the tracking error converges to a small neighborhood containing the origin, and the signals of the closed-loop system are bounded in the sense of probability. Therefore, the control strategy proposed in the present application can effectively deal with multiple coupled problems such as random disturbance, output constraint and input nonlinearity, has good robustness and wide engineering applicability.

[0207] Embodiment 2

[0208] The embodiment 2 describes a mechanical arm system fixed time tracking control system under random disturbance, which is based on the same inventive concept as the mechanical arm system fixed time tracking control method under random disturbance in embodiment 1.

[0209] Specifically, the mechanical arm system fixed time tracking control system under random disturbance includes the following modules:

[0210] The dynamics model construction module is used to establish the mathematical model of the mechanical arm system under random disturbance and affected by the actuator and output constraint, and the dynamics model of the mechanical arm system in the Cartesian space is obtained by coordinate transformation, and it is extended to the dynamics model of the multi-input multi-output random uncertain nonlinear system.

[0211] In this embodiment, the dynamic model construction module is used to establish a system model based on the typical dynamic structure of the double-link robot arm. Considering that the robot arm is often affected by random disturbances, actuator nonlinearities (such as saturation and dead zone), and joint output range limitations in actual operation, the module further introduces random terms, input constraints, and output constraints into the original dynamic equation, making it expand into a more engineering-applicable MIMO stochastic uncertain nonlinear system model, providing a unified modeling framework for the subsequent design of the adaptive fixed-time tracking controller.

[0212] The output constraint processing and actuator nonlinearity construction module is used to design a time-dependent barrier function for processing output constraints, establish control inputs affected by dead zones and saturation, and design a nonlinear transformation for actuator nonlinear control inputs.

[0213] In this embodiment, the output constraint processing and actuator nonlinearity construction module includes:

[0214] The general barrier function construction module is used to ensure that the system output always remains within the given constraint range.

[0215] The actuator nonlinearity processing module, since the system input is nonlinear, is not only subject to saturation constraints, but also cannot be driven when the control input signal is small, so a nonlinear transformation is performed on the control input to facilitate subsequent controller design work.

[0216] The equivalent conversion module is used to establish an equivalent system form based on the dynamic model of the MIMO stochastic uncertain nonlinear system, obtaining an equivalent MIMO stochastic uncertain nonlinear dynamic model.

[0217] The controller design module is used to design a non-singular adaptive fixed-time tracking controller using backstepping based on the barrier function and nonlinear transformation, as well as the equivalent MIMO stochastic uncertain nonlinear dynamic model.

[0218] In this embodiment, the controller design module is based on coordinate transformation and backstepping, and uses a switching function and Nussbaum function to design a non-singular adaptive fixed-time tracking controller.

[0219] The tracking control module is used to implement fixed-time tracking control of the robot arm system using the non-singular adaptive fixed-time tracking controller.

[0220] In addition, the embodiment also includes a stability theorem and a controller effectiveness verification module, which is used to propose a fixed-time stability theorem for MIMO stochastic nonlinear systems under uncertainty, and to verify the effectiveness of the designed adaptive fixed-time tracking controller.

[0221] It should be noted that the functions and effects of each functional module in the fixed-time tracking control system of the robotic arm system under random disturbance are specifically described in the implementation process of the corresponding steps in the method of Embodiment 1, and will not be repeated here.

[0222] Embodiment 3

[0223] This embodiment 3 describes a computer device, which includes a memory and one or more processors.

[0224] The executable code is stored in the memory, and when the processor executes the executable code, the steps of the fixed-time tracking control method of the robotic arm system under random disturbance in Embodiment 1 above are implemented.

[0225] The computer device in this embodiment is any device or apparatus with data processing capability, which will not be repeated here.

[0226] Embodiment 4

[0227] This embodiment 4 describes a computer readable storage medium, which stores a program, and when the program is executed by a processor, the steps of the fixed-time tracking control method of the robotic arm system under random disturbance are implemented.

[0228] The computer readable storage medium can be an internal storage unit of any device or apparatus with data processing capability, such as a hard disk or a memory, or an external storage device of any device with data processing capability, such as a plug-in hard disk, a smart media card (SMC), an SD card, a flash card, etc.

[0229] Of course, the above description is only for the preferred embodiments of the present application, and the present application is not limited to the above-mentioned embodiments. It should be noted that any person skilled in the art, under the guidance of this specification, all equivalent replacements, obvious modifications made, fall within the scope of the present application, and should be protected by the present application.

Claims

1. A fixed-time tracking control method for a robotic arm system under random disturbances, characterized in that, Includes the following steps: Step 1. Establish a mathematical model of the robotic arm system under random disturbances and the influence of actuator and output constraints. Use coordinate transformation to obtain the dynamic model of the robotic arm system in Cartesian space, and extend it to the dynamic model of a multi-input multi-output random uncertain nonlinear system. Step 2. Design a time-dependent barrier function to handle output constraints; establish control inputs affected by dead zones and saturation, and design nonlinear transformations for control inputs with nonlinear actuators; Step 3. Based on the dynamic model of the multi-input multi-output stochastic uncertain nonlinear system obtained in Step 1, establish an equivalent system form to obtain an equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model; Step 4. Based on the obstacle function and nonlinear transformation designed in Step 2, and the equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model obtained in Step 3, a singular-free adaptive fixed-time tracking controller is designed using the backstepping method. Step 5. Using the singularity-free adaptive fixed-time tracking controller designed in Step 4, realize fixed-time tracking control of the robotic arm system; Step 2 specifically involves: Design a time-dependent barrier function This ensures that the system output does not violate the preset constraint boundaries. (8) in, , This indicates the number of links. Taking a two-link robotic arm system as the research object, this refers to the number of links. ; As a preset reference signal, Indicates system output The lower boundary that needs to be satisfied, Indicates system output The upper bound that needs to be satisfied; using without causing singularities. express , express , and Indicates about and A smooth function, Indicates time; Time-dependent barrier function Taking the derivative, we get: ； in: ； ； ； ； Establish control inputs affected by dead zone and saturation The mathematical model is shown in formula (9): (9) in, Indicates the control input signal. and Breakpoints indicating dead zones and The slope of the dead zone interval is represented by the slope of the dead zone interval. and Indicates the saturation value. and It is a bounded constant; To overcome the effects of actuator nonlinearity, a nonlinear transformation is designed: ； in, The slope of the dead zone; and Represented as: ； ； Assuming control input signal It is a bounded signal, meaning that there exists a positive constant. , making Heng is established; in, , and They represent The maximum and minimum values; get It is a bounded function, that is, there exists a constant. , making Established; The following inequalities must be satisfied: ； in, Defined as , Defined as ; Established, among which It is a piecewise bounded function, and satisfies ; And thus obtain ; In step 4, after completing the design of the non-singular adaptive fixed-time tracking controller, a convergence analysis is performed on the robotic arm system controlled by the non-singular adaptive fixed-time tracking controller. In step 4, the process of performing convergence analysis on the robotic arm system controlled by the non-singular adaptive fixed-time tracking controller is as follows: Let the Lyapunov function for: ； in, Indicates the number of joints. Indicates the error variable Constructed Lyapunov function: (16) in, It is a positive number; ； in, for The estimate, for The estimation error, , Represents the weight vector; For Lyapunov functions Finding infinitesimal differential operators ,get: (18) in, , , , It is a constant and satisfies Thus, the mathematical model of the robotic arm system, i.e. the robotic arm system affected by the actuator and output constraints under random disturbances as shown in formula (1), is obtained, which has semi-global fixed-time stability in a probabilistic sense. According to Lyapunov functions The definition is obtained. and It is bounded within a fixed time period. and These represent the error variable and the estimation error, respectively. Let constant And satisfy ,get ,Right now Reference signal It is bounded, in which This represents the angular displacement of the first joint in the robotic arm system. This represents the angular velocity of the first joint in the robotic arm system; By selecting control parameters, the system output can be guaranteed. Tracking reference signal within a fixed time period And it will not violate the preset constraint boundaries. and ; because and Bounded, capable of further derivation Bounded, thus obtaining the virtual controller Maintaining boundedness; where, , for The estimate, for The estimation error, , Represents the weight vector; In a robotic arm system controlled by a singular adaptive fixed-time tracking controller, all signals remain bounded in a probabilistic sense.

2. The fixed-time tracking control method for a robotic arm system under random disturbances according to claim 1, characterized in that, In step 1, the Lagrange quantity Defined as the kinetic energy of a two-bar linkage robotic arm system Subtract potential energy : ; The Lagrange equation of the robotic arm system is constructed, that is, the mathematical model of the robotic arm system under random disturbance and affected by actuator and output constraints is established as shown in formula (1): (1) in, Indicates the first in the robotic arm system Angular displacement of each joint Indicates the first in the robotic arm system Angular velocity of each joint; This represents the driving torque of the robotic arm system; The dynamic model of the robotic arm system in Cartesian space is obtained by simplifying the Lagrange equations of the robotic arm system, i.e., by using coordinate transformation, as shown in equation (2): (2) in, , and These represent the angular displacements of the first and second joints, respectively. , and These represent the angular velocities of the first and second joints, respectively. , and These represent the angular accelerations of the first and second joints, respectively. The inertia matrix of a symmetric positive definite rigid body is... ; Represents the Coriolis force and centrifugal force matrices. ; Represents the gravity vector; Indicates external disturbance; For control matrix; Indicates control torque. , and These represent the control input torques of the first and second joints, respectively. Extending to the dynamic model of a multi-input multi-output (MIMO) stochastic uncertain nonlinear system as shown in equation (7): (7) in, ; express The first in the time-sensitive robotic arm system Angular displacement of each joint express The first in the time-sensitive robotic arm system Angular velocity of each joint; Represents the state of a nonlinear system. , to These represent the first joint to the second joint in the robotic arm system. Angular displacement of each joint to These represent the first joint to the second joint in the robotic arm system. Angular velocity of each joint; Indicates the first in the robotic arm system Angular displacement of each joint Indicates the first in the robotic arm system Angular velocity of each joint; This represents the control input of a nonlinear system, specifically the driving torque of the robotic arm system. ; definition , ; It is a smooth nonlinear function and satisfies ; Indicates control gain, and It has a continuous first derivative; This represents a bounded external disturbance with a constant. Make ,in ; It is a smooth nonlinear function and satisfies , Representation function exist The value at time; Assume the output of the nonlinear system satisfy: .

3. The fixed-time tracking control method for a robotic arm system under random disturbances according to claim 2, characterized in that, Step 3 specifically involves: The dynamic model of the MIMO stochastic uncertain nonlinear system shown in Equation (7) is transformed into an equivalent MIMO stochastic uncertain nonlinear dynamic model that can be directly used for research on adaptive fixed-time tracking control methods: (10)。 4. The fixed-time tracking control method for a robotic arm system under random disturbances according to claim 3, characterized in that, Step 4 specifically involves: Introducing coordinate transformation as follows: ； in, Indicates the error variable; Indicates virtual control input. ; Introducing switching functions for: (11) in, It is a constant; For error variables Construct the Lyapunov function as shown in formula (12). for: (12) in, It is a constant, and ; For Lyapunov functions Find the infinitesimal differential operator, simplify using Young's inequality, and then use... The nonlinear functions that appear during the representation process are approximated using a radial basis function neural network (RBF-NN). ； in, This represents the input vector of the neural network. , This represents the desired angular displacement of the first joint. This represents the desired angular velocity of the first joint; For the basis function vector, To approximate the error, and satisfy... , It is a constant; Design a singular virtual controller as shown in formula (13). And adaptive law : (13) in, This represents the Nussbaum function; , , , , It is a constant and satisfies , , , , ; For error variables ,in Construct Lyapunov functions for: (14) in, It is a constant, and ; ； in, for The estimate, for The estimation error, , Represents the weight vector; For Lyapunov functions Find the infinitesimal differential operator, simplify using Young's inequality, and then use... The nonlinear functions that appear during the representation process are approximated using RBF-NN: ； in, This represents the input vector of the neural network. , to They represent to The estimation error; For the basis function vector, To approximate the error, and satisfy... , It is a constant; Design a singular virtual controller as shown in formula (15). And adaptive law : (15) in, This represents the Nussbaum function; , , , , It is a constant, and , , , , ; For Lyapunov functions Find the infinitesimal differential operator, simplify using Young's inequality, and then use... The nonlinear functions that appear during the representation process are approximated using RBF-NN: ； in, This is the input vector of the neural network. , to They represent to The estimation error; For the basis function vector, To approximate the error, and satisfy... , It is a constant; Design a singular control input as shown in formula (17). And adaptive law for: (17) in, This represents the Nussbaum function; , , , , It is a positive number.

5. A fixed-time tracking control system for a robotic arm system under random disturbances for implementing the fixed-time tracking control method for a robotic arm system under random disturbances as described in claim 1, characterized in that, The fixed-time tracking control system for the robotic arm system under random disturbances includes: The dynamic model construction module is used to establish a mathematical model of a robotic arm system affected by actuator and output constraints under random disturbances. It uses coordinate transformation to obtain the dynamic model of the robotic arm system in Cartesian space and extends it to the dynamic model of multi-input multi-output random uncertain nonlinear systems. The output constraint processing and actuator nonlinearity building module is used to design time-dependent barrier functions to handle output constraints; establish control inputs affected by dead zones and saturation, and design nonlinear transformations for control inputs with actuator nonlinearity; The equivalent transformation module is used to establish an equivalent system form based on the dynamic model of a multi-input multi-output stochastic uncertain nonlinear system, and obtain an equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model. The controller design module is used to design a singular-free adaptive fixed-time tracking controller based on the barrier function, nonlinear transformation, and equivalent multi-input multi-output stochastic uncertain nonlinear dynamic model using the backstepping method. And a tracking control module, used to achieve fixed-time tracking control of the robotic arm system using a singular adaptive fixed-time tracking controller.

6. A computer device comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that, When the processor executes the executable code, it implements the steps of the fixed-time tracking control method for a robotic arm system under random disturbance as described in any one of claims 1 to 4.

7. A computer-readable storage medium having a program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the fixed-time tracking control method for a robotic arm system under random disturbances as described in any one of claims 1 to 4.

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