Adaptive control method for time-delay jumping flexible joint robot arm based on disturbance observation

Abstract

This invention discloses an adaptive control method for a time-delay jump-type flexible joint manipulator based on disturbance observation, belonging to the field of automatic control technology for manipulators. The method includes: each flexible joint of the flexible joint manipulator consists of a rigid link and a flexible actuator, the flexible actuator being represented by a linear spring-damped model; a dynamic model of the flexible joint manipulator is established based on the Lagrangian function, and the mapping relationship of the end-effector constraint forces is loaded into the joint space; control input time delay and jump signals are loaded into the dynamic model of the flexible joint manipulator, and disturbance observers for the rigid link and flexible joint are constructed respectively; the observer gain matrix, time delay compensation gain matrix, and time delay estimation error are configured; and the disturbance estimate is output as the observation error. This invention effectively suppresses system oscillations caused by communication delays or jumps by combining feedforward compensation and time delay estimation, thus improving anti-time delay capability.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology for robotic arms, and in particular to an adaptive control method for time-delay jump flexible joint robotic arms based on disturbance observation. Background Technology

[0002] Flexible joint robotic arms, with their combination of compliance, lightweight actuation, and safe interaction, are gradually being extended to high-precision task scenarios such as industrial automation, aerospace, and medical operations. Control research, focusing on the elastic vibration and strongly nonlinear coupled dynamics induced by flexible joints, has evolved from single-position tracking to unified modeling, stability analysis, and robust performance constraints for time delays and working mode switching (jumps). Furthermore, it integrates disturbance estimation and adaptive adjustment concepts to adapt to dynamic uncertainties under complex working conditions.

[0003] Existing methods have shortcomings. Time delays can alter system characteristics and potentially induce instability. Furthermore, when switching operating modes, the system matrix undergoes abrupt changes, and the dynamic parameters change with the transition signal, making it difficult for the control law to maintain consistent stability margins and performance boundaries across different operating modes. In addition, unknown disturbances such as external interference and model uncertainties coexist at rigid links and flexible joints. Relying solely on fixed robust gains makes it difficult to achieve quantitative compensation for disturbances and suppress error propagation. Under the premise of unified disturbance modeling and explicit introduction of dynamic equations, an observation mechanism is needed to estimate unknown disturbances online and feed back the disturbance estimates for compensation in order to maintain controllable error evolution characteristics under the superposition of time delays and transitions. Summary of the Invention

[0004] In view of the aforementioned existing problems, the present invention is proposed.

[0005] Therefore, this invention provides an adaptive control method for a time-delay jump flexible joint manipulator based on disturbance observation to solve the problems of difficulty in ensuring stability and difficulty in estimating and compensating for bilateral disturbances caused by time-delay jumps.

[0006] To solve the above-mentioned technical problems, the present invention provides the following technical solution: This invention provides an adaptive control method for a time-delay jump-type flexible joint manipulator based on disturbance observation. The method includes: each flexible joint of the flexible joint manipulator consists of a rigid link and a flexible actuator, the flexible actuator being represented by a linear spring-damping model; establishing a dynamic model of the flexible joint manipulator based on a Lagrangian function, and loading the mapping relationship of the end-effector constraint forces into the joint space; loading the control input time delay and jump signal into the dynamic model of the flexible joint manipulator, and constructing a rigid link disturbance observer and a flexible joint disturbance observer respectively, configuring the observer gain matrix, time delay compensation gain matrix, and time delay estimation error, outputting the disturbance estimate and forming the observation error; selecting state variables, deriving the system state equation based on the state variables and the observation error, generating a virtual control output based on the disturbance estimate and calculating the actual control input, constructing a state observer and a graded Lyapunov function, and deriving the motor control law.

[0007] As a preferred embodiment of the adaptive control method for a time-delay jump flexible joint manipulator based on disturbance observation described in this invention, wherein: the Lagrangian function and energy term of the flexible joint manipulator are constructed based on the Lagrangian method; The Lagrange function is the difference between the total kinetic energy and the total potential energy of the system, and its expression is: ; in, The Lagrangian function representing the flexible joint robotic arm in joint space; This represents the total kinetic energy of the system; This represents the total potential energy of the system; Substituting the Lagrange function into the Lagrange equation yields the result in flexible joints. Generalized active force The expression is, ; in, In a flexible joint robotic arm, the first... Joint angular displacement of a rigid link; Indicates the first Angular velocity of each joint; Represents the first in the Lagrange function Generalized coordinates The partial derivatives; Represents a continuous-time variable; Calculate the generalized active force The expression converted to vector form is: ; in, Represents the generalized angular displacement vector of a rigid link; This represents the angular velocity vector of the rigid link; Generalized driving force The vector form can be rearranged into a differential equation, expressed as: ; in, This represents the angular acceleration vector of the rigid link; Let represent the system inertia matrix, a symmetric positive definite matrix about the rigid link angle q; The Coriolis force and centrifugal force matrices are used to describe velocity-related nonlinear terms; Gravity term vector; This represents the control torque acting on the joints and actuators of the robotic arm; Due to the constraint forces at the end effector of the robotic arm, the expression for the dynamic model of the flexible joint robotic arm in joint space is as follows: ; in, This represents the equivalent total disturbance term acting on the joint dynamics of the robotic arm; Will Obtained by transforming to joint space. ; in, Representing the generalized coordinates of the end effector joints of a robotic arm Jacobian matrix The transpose of the matrix; Represents the Jacobian matrix; This represents the Lagrange multiplier vector corresponding to the end constraint force; This represents the matrix transpose symbol.

[0008] As a preferred embodiment of the time-delay jump flexible joint manipulator adaptive control method based on disturbance observation described in this invention, wherein: the total kinetic energy of the system is composed of the kinetic energy of the rigid link. and the kinetic energy of the flexible actuator Composition, the expression is, ; in, This represents the total kinetic energy of the system; This represents the kinetic energy of a rigid link; This represents the kinetic energy of the flexible actuator; The kinetic energy of the rigid link The expression is, ; in, The inertia matrix represents the rigid link side; Rigid link joint angular displacement; Rigid link joint angular velocity; The kinetic energy of the flexible actuator The expression is, ; in, This represents the angular velocity vector of the flexible actuator.

[0009] As a preferred embodiment of the time-delay jump flexible joint manipulator adaptive control method based on perturbation observation described in this invention, the total potential energy of the system is composed of the elastic potential energy of the flexible actuator. and gravitational potential energy composition; Elastic potential energy of flexible actuator The expression is, ; in, This represents the elastic potential energy of the flexible actuator; Represents the equivalent elastic stiffness matrix of a flexible joint; Indicates the angular displacement of the flexible actuator; gravitational potential energy The expression is, ; in, Represents gravitational potential energy; Represents gravity; Represents the angular displacement vector of a rigid link joint. Configuration functions; The expansion of the Lagrange function is, ; in, Transpose of the rigid link joint angular velocity vector matrix; This represents the transpose of the angular velocity vector of the flexible actuator.

[0010] As a preferred embodiment of the adaptive control method for a time-delay jump flexible joint manipulator based on disturbance observation described in this invention, the following steps are taken: The partial derivatives of the Lagrange function are calculated and substituted into the Lagrange equation to obtain the dynamic equation of the flexible joint manipulator, expressed as follows: ; ; in, Indicates the angular acceleration of a flexible joint; This represents the control torque acting on the rigid link; Represents the gravity term; Represents the matrix of Coriolis force and centrifugal force on the rigid connecting rod side; The inertia matrix represents the rigid link side; This represents the joint angular acceleration of a rigid link; This represents the control torque acting on the flexible joint; This represents the equivalent moment of inertia matrix on the motor side; Control input and There is a time delay The actual control input acting on the system is and ; The transition signal belongs to a finite state set. At this point, the dynamic equations of the flexible joint robotic arm change, and the expression on the load side becomes: ; in, This indicates the joint angular displacement on the rigid link side; Indicates the angular velocity of the joint on the rigid link side; This indicates the joint angular acceleration on the side of the rigid link; Jump signal; Indicates the time delay of the control input; The equivalent elastic stiffness matrix of the flexible joint, as a function of the jump signal. Switch; This represents the joint angular displacement on the motor side after considering the time delay effect; The inertia matrix represents the rigid link side and varies with the jump signal. Switch; The matrix representing the Coriolis force and centrifugal force on the rigid connecting rod side, as a function of the jump signal. Switch; The gravity term on the rigid link side changes with the jump signal. Switch; The expression on the motor side is, ; in, This represents the equivalent moment of inertia matrix on the motor side, which changes with the jump signal. Switch; This represents the equivalent damping matrix on the motor side, and it varies with the jump signal. Switch; Indicates the control input torque on the motor side; Indicates the angular velocity of the motor-side joint; Indicates the joint angular acceleration on the motor side; All disturbances in the actual system are uniformly represented as Substituting this into the dynamic equations of the flexible joint robotic arm, the expression is: ; ; in, This represents a disturbance acting on a rigid link; This indicates a disturbance acting on a flexible joint; Indicates association The rigid link side inertia matrix, with the jump signal Switch; Indicates association The rigid connecting rod Coriolis and centrifugal force matrix; Indicates association The gravitational term on the rigid connecting rod changes with the jump signal. Switch; This indicates the joint angular displacement of the flexible actuator side considering time lag; This represents the control input torque considering the time lag on the rigid link side; This indicates the control input torque considering the time lag on the flexible actuator side.

[0011] As a preferred embodiment of the time-delay jump flexible joint manipulator adaptive control method based on disturbance observation described in this invention, wherein: the disturbance observer includes a rigid link disturbance observer and a flexible joint disturbance observer; The expression for the rigid linkage disturbance observer is as follows: ; The expression for the flexible joint perturbation observer is as follows: ; ; in, This represents the estimated disturbance acting on the rigid link; This represents the estimated perturbation acting on the flexible joint; and These represent the observation errors for rigid links and flexible joints, respectively. and These represent the first derivatives of the observation errors with respect to time on the rigid link side and the flexible joint side, respectively. and This represents the gain matrix of the rigid link and flexible joint of the observer, as a function of the jump signal. Switch; and This represents the time-delay compensation gain matrix for rigid links and flexible joints, which varies with the jump signal. Switch; This represents the actual control input delay time in the system; This represents the online estimate of the control input time delay; This indicates the time delay estimation error; The dynamic equation for the observation error is: ; ; in, This represents the first derivative of the observation error of the side disturbance of the rigid link with respect to time; This represents the first derivative of the observation error of the flexible joint side perturbation with respect to time.

[0012] As a preferred embodiment of the adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation described in this invention, wherein: the expression for the state variable is, ; , , , ; in, Indicates the angle of a rigid link; Indicates the angular velocity of a rigid link; Indicates the angular velocity of the flexible joint; This indicates the observation error of the rigid link disturbance; This indicates the observation error of flexible joint perturbation; The dynamic equations of the flexible joint robotic arm can be rewritten as a system of first-order differential equations, expressed as follows: , ; ; ; in, Indicates joint angular displacement The rigid link side inertia matrix, with the jump signal Switch; Indicates joint angular displacement angular acceleration Coriolis force and centrifugal force on the rigid connecting rod side; Indicates joint angular displacement The rigid connecting rod side gravity term; Solve and The expression is, ; ; in, The inverse matrix representing the lateral inertia of a rigid link; The inverse matrix representing the equivalent rotational inertia on the motor side; This represents the elastic restoring torque caused by the relative angular displacement between the flexible actuator side and the rigid connecting rod side; Because system parameters change with the jump signal When a switch occurs, and the control input exhibits time-delay uncertainty and unknown disturbances, the dynamic equation for the observation error can be expressed as follows: ; ; ; ; in, This represents the time-delay compensation gain matrix on the rigid link side, which varies with the jump signal. A switchover has occurred; This represents the time-delay compensation gain matrix on the flexible joint side, which varies with the jump signal. A switchover has occurred; This represents the adaptive gain coefficient for time-delay estimation, which varies with the jumping signal. A switchover has occurred; The time-delay estimated damping coefficient varies with the jump signal. A switch has occurred.

[0013] As a preferred embodiment of the adaptive control method for a time-delay jump flexible joint manipulator based on disturbance observation described in this invention, wherein: the first-order differential equation system and the dynamic equation of the modified observation error are substituted into the derivative expression of the state variables to obtain the system state equation, which is expressed as follows: ; When the system switches operating modes, the parameters of the system state equations change abruptly. ; in, Indicates the jump pattern index; This indicates the total number of available operating modes in the system; Indicates the first The start time of each working mode; Indicates that the system starts from the first The switching point from one working mode to the next working mode; The state is continuous at the moment of switching working modes, but the system matrix changes abruptly. ; in, Represents the state vector As time approaches the switching moment The left limit; Represents the state vector As time approaches the switching moment The right limit.

[0014] As a preferred embodiment of the time-delay jump flexible joint manipulator adaptive control method based on disturbance observation described in this invention, wherein: the state variables of the system are reconstructed based on the input and output information of the flexible joint manipulator, and the state observer of the flexible joint manipulator is obtained as follows: ; in, Represents the estimated value of the actual state variable; The first derivative of the estimated value of the actual state variable; This represents the state observer gain matrix; Represents the system state matrix; Represents the system state matrix; Represents the system output matrix; Indicates the actual output of the system; This represents the estimated output value of the system; Indicates system control input; Furthermore, the state observer of the flexible joint robotic arm satisfies the formula: ; , ; ; in, Represents the true state vector of the system; This indicates the state estimation error; This represents the estimation error of the first state component; This represents the estimation error of the second state component; This represents the expected joint angle displacement trajectory; This represents the desired joint angular velocity trajectory; Indicates the first-level error feedback gain; Based on the state observer of the flexible joint robotic arm satisfying the formula, a first-level Lyapunov function is constructed, and selected... And by taking the derivative, we get ; in, This represents a first-level Lyapunov function; Represents Lyapunov functions The first derivative with respect to time; Denotes the Euclidean norm; Represents the error vector The square norm; Represents the error vector and The inner product term; This represents a positive control design gain coefficient; Indicates the desired joint angular velocity; Representing vectors transpose; Constructing the second-level virtual control law includes, Define a new Lyapunov function with the expression: ; , ; in, This represents a second-order Lyapunov function; Representing vectors transpose; Represents a symmetric positive definite weighted matrix; This represents the system's equivalent total disturbance; Indicates the estimated disturbance value; This indicates the error in perturbation estimation; This represents the transpose of the disturbance estimation error; Represents the adaptive gain matrix; The inverse matrix of the adaptive gain matrix; right Taking the derivative, we get ; in, This represents the derivative of a second-order Lyapunov function; This represents the derivative of a first-order Lyapunov function; This represents the actual disturbance vector; Represents the second-level state error vector The first derivative with respect to time; Indicates the disturbance estimate The first derivative with respect to time; Transpose of the perturbation estimation error matrix; The dynamic equation Substitute have to: ; Multiply both sides by have to: ; Substitution get, ; in, This represents a virtual control input, replacing the original expression. ; Indicates a unified equivalent perturbation; This represents the desired joint angular velocity trajectory; Denotes the inverse of a symmetric positive definite weighted matrix; make Tracking target , Used in the rigid linkage side control design stage, as a reference target for subsequent actual control input on the motor side, it is derived from Lyapunov stability analysis and serves as a virtual control input. satisfy, ; , ; in, This represents the expected value of the virtual control input; Represents a diagonal matrix Positive real number elements in; Indicates the diagonalization operator; By selecting offset and coupling terms, and introduce and To ensure error convergence; Update the disturbance estimate. ,but, ; If we assume If it is bounded, then it can be solved by appropriate selection. Guarantee convergence; Construct a third-level motor control law, based on definition, ; ; ; in, This represents the position error vector on the third-stage motor side; This represents the desired trajectory of the joint angular displacement on the motor side; definition The dynamic formula for the motor side is given: ; in, Represents the inertia matrix on the motor side; where, This represents the equivalent disturbance on the motor side; ; in, This represents the position error vector on the fourth-stage motor side; This represents the expected joint angular velocity on the motor side; Construct a fourth-level Lyapunov function. ; in, This represents the transpose of the position error vector on the third-stage motor side; This shows the transpose of the fourth-stage motor-side position error vector; Multiply by both sides And substitute it into motor dynamics, ; ; use have to: ; The final actual control input is, ; in, This represents the expected joint angle acceleration on the motor side; Indicates error The first derivative with respect to time; This represents the gain coefficient of the third-level error feedback; Indicates error The first derivative with respect to time; Substituting into the formula for the final actual control input, we get: ; ; in, This represents the transpose of the fourth-stage motor-side position error vector; This represents the gain coefficient for the fourth-level error feedback. Combination Conclusion: ; in, express , Bounded residuals caused by the rate of change of constant disturbances; This represents the derivative of a fourth-order Lyapunov function; Represents the error vector The square norm; Intersection terms can be defined by the standard inequality, and residual terms are bounded.

[0015] As a preferred embodiment of the time-delay jump flexible joint manipulator adaptive control method based on disturbance observation described in this invention, the complete control system structure includes an extended state observer, a virtual control output, and an actual control input. The extended state observer refers to the estimation of unknown disturbances as extended states based on the state observer. Based on the stability analysis of the complete control system structure using Lyapunov, a global Lyapunov function is constructed, with the expression: ; Differentiating along the system trajectory, we get: ; in, Represents the error vector The square norm; Represents the error vector The square norm; Indicates the gain of the third-level error feedback; This indicates the gain of the fourth-level error feedback; This indicates the gain of the fifth-level error feedback. when When it is bounded, The system is uniformly bounded eventually, and all errors tend to the zero neighborhood. Both position tracking error and force tracking error converge exponentially to a small neighborhood near the origin.

[0016] The beneficial effects of this invention are as follows: By combining feedforward compensation and time delay estimation, it effectively suppresses system oscillations caused by communication delays or jumps, improving anti-time delay capability; the extended state observer estimates the total disturbance (including modeling error, external interference, and time delay nonlinearity) in real time, enhancing the system's adaptability to external changes and strengthening robustness; only some prior information (such as inertial range) is required, and the remaining parameters are automatically adjusted through an adaptive mechanism, eliminating the need for precise modeling; it is suitable for compliant operation tasks in constrained environments, achieving unified control of trajectory tracking and contact force adjustment, taking into account both motion and force control; the algorithm structure is clear, easy to deploy on DSP / FPGA platforms, suitable for embedded control systems of industrial robots, and has strong engineering practicality. Attached Figure Description

[0017] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a flowchart of an adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation.

[0019] Figure 2 The graph shows the actual and expected changes in the angular velocity of joint 1 of the robotic arm.

[0020] Figure 3 The graph shows the actual and expected changes in the angle of joint 1 of the robotic arm.

[0021] Figure 4 This is a graph showing the position tracking error of joint 1 of the robotic arm.

[0022] Figure 5 This is a graph showing the actual and expected values ​​of the joint angles of the robotic arm's joint 2.

[0023] Figure 6 This is a graph showing the position tracking error of joint 2 of the robotic arm. Detailed Implementation

[0024] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0025] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0026] Secondly, the term "one embodiment" or "embodiment" as used herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of the present invention. The phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it a single or selective embodiment that is mutually exclusive with other embodiments.

[0027] Reference Figures 1-6 This is one embodiment of the present invention, which provides an adaptive control method for a time-delay jump flexible joint manipulator based on disturbance observation, including the following steps: S1. Each flexible joint of the flexible joint robotic arm consists of a rigid link and a flexible actuator, which is represented by a linear spring-damping model.

[0028] It should be noted that the invention was verified on a two-degree-of-freedom planar robotic arm experimental platform, wherein the link length, ,quality Joint stiffness Motor inertia Time delay setting, and in Apply a ±0.01s jump at the specified point; control parameters, .

[0029] The flexible joint robotic arm has several flexible joints, each consisting of a rigid link and a flexible actuator. The flexible joint robotic arm satisfies the following assumptions during its movement: 1. The link is rigid, and the elastic deformation of the link is ignored; 2. The elastic force of the flexible actuator is proportional to the elastic deformation of the joint, and the damping force is proportional to the relative angular velocity of the joint; 3. The movement of the flexible joint robotic arm takes place in an inertial coordinate system, and the effects of external disturbances and friction are ignored.

[0030] S2. Establish a dynamic model of the flexible joint manipulator based on the Lagrangian function, and load the mapping relationship of the end-effector constraint forces into the joint space.

[0031] It should be noted that the Lagrangian function and energy term of the flexible joint robotic arm are constructed based on the Lagrangian method.

[0032] The Lagrange function is the difference between the total kinetic energy and the total potential energy of the system, and its expression is: ; in, The Lagrangian function representing the flexible joint robotic arm in joint space; This represents the total kinetic energy of the system; It represents the total potential energy of the system.

[0033] Substituting the Lagrange function into the Lagrange equation yields the generalized active force Qj on the flexible joint j, expressed as follows: ; in, In a flexible joint robotic arm, the first... Joint angular displacement of a rigid link; Indicates the first Angular velocity of each joint; Represents the first in the Lagrange function Generalized coordinates The partial derivatives; This represents a continuous-time variable.

[0034] Calculate the generalized active force The expression converted to vector form is: ; in, Represents the generalized angular displacement vector of a rigid link; This represents the angular velocity vector of the rigid link.

[0035] Generalized driving force The vector form can be rearranged into a differential equation, expressed as: ; in, This represents the angular acceleration vector of the rigid link; Let represent the system inertia matrix, a symmetric positive definite matrix about the rigid link angle q; The Coriolis force and centrifugal force matrices are used to describe velocity-related nonlinear terms; Gravity term vector; This represents the control torque acting on the joints and actuators of the robotic arm.

[0036] Due to the constraint forces at the end effector of the robotic arm, the expression for the dynamic model of the flexible joint robotic arm in joint space is as follows: ; in, This represents the equivalent total disturbance term acting on the joint dynamics of the robotic arm.

[0037] Will Obtained by transforming to joint space. ; in, Representing the generalized coordinates of the end effector joints of a robotic arm Jacobian matrix The transpose of the matrix; Represents the Jacobian matrix; This represents the Lagrange multiplier vector corresponding to the end constraint force; This represents the matrix transpose symbol.

[0038] The total kinetic energy of the system consists of the kinetic energy of the rigid connecting rod. and the kinetic energy of the flexible actuator Composition, the expression is, ; in, This represents the total kinetic energy of the system; This represents the kinetic energy of a rigid link; This represents the kinetic energy of the flexible actuator.

[0039] Kinetic energy of a rigid link The expression is, ; in, The inertia matrix represents the rigid link side; Rigid link joint angular displacement; Angular velocity of rigid connecting rod joint.

[0040] Kinetic energy of flexible actuator The expression is, ; in, This represents the angular velocity vector of the flexible actuator.

[0041] The total potential energy of the system consists of the elastic potential energy of the flexible actuator. and gravitational potential energy composition.

[0042] Elastic potential energy of flexible actuator The expression is, ; in, This represents the elastic potential energy of the flexible actuator; Represents the equivalent elastic stiffness matrix of a flexible joint; This indicates the angular displacement of the flexible actuator.

[0043] gravitational potential energy The expression is, ; in, Represents gravitational potential energy; Represents gravity; Represents the angular displacement vector of a rigid link joint. Configuration functions.

[0044] The expansion of the Lagrange function is, ; in, Transpose of the rigid link joint angular velocity vector matrix; This represents the transpose of the angular velocity vector of the flexible actuator.

[0045] S3. Load the control input time delay and jump signal into the dynamic model of the flexible joint manipulator, and construct the rigid link disturbance observer and the flexible joint disturbance observer respectively. Configure the observer gain matrix, time delay compensation gain matrix and time delay estimation error, output the disturbance estimate and form the observation error.

[0046] It should be noted that by taking the partial derivative of the Lagrange function and substituting it into the Lagrange equation, the dynamic equation of the flexible joint robotic arm is obtained, expressed as follows: ; ; in, Indicates the angular acceleration of a flexible joint; This represents the control torque acting on the rigid link; Represents the gravity term; Represents the matrix of Coriolis force and centrifugal force on the rigid connecting rod side; The inertia matrix represents the rigid link side; This represents the joint angular acceleration of a rigid link; This represents the control torque acting on the flexible joint; This represents the equivalent moment of inertia matrix on the motor side.

[0047] In practical systems, signal transmission and processing involve a certain time delay, which affects system stability and control performance. Assuming the control input... and There is a time delay The actual control input acting on the system is and ; The transition signal belongs to a finite set of states. This represents a switch in the system's operating mode. At this time, the dynamic equations of the flexible joint robotic arm change, and the expression on the load side becomes... ; in, This indicates the joint angular displacement on the rigid link side; Indicates the angular velocity of the joint on the rigid link side; This indicates the joint angular acceleration on the side of the rigid link; Jump signal; Indicates the time delay of the control input; The equivalent elastic stiffness matrix of the flexible joint, as a function of the jump signal. Switch; This represents the joint angular displacement on the motor side after considering the time delay effect; The inertia matrix represents the rigid link side and varies with the jump signal. Switch; The matrix representing the Coriolis force and centrifugal force on the rigid connecting rod side, as a function of the jump signal. Switch; The gravity term on the rigid link side changes with the jump signal. Switch.

[0048] The expression on the motor side is, ; in, This represents the equivalent moment of inertia matrix on the motor side, which changes with the jump signal. Switch; This represents the equivalent damping matrix on the motor side, and it varies with the jump signal. Switch; Indicates the control input torque on the motor side; Indicates the angular velocity of the motor-side joint; This indicates the joint angular acceleration on the motor side.

[0049] In real-world systems, time delays and various unknown disturbances exist, such as external interference and model uncertainties. All disturbances in a real-world system can be uniformly represented as… This is then incorporated into the dynamic equations of the flexible joint robotic arm, with the expression being: ; ; in, This represents a disturbance acting on a rigid link; This indicates a disturbance acting on a flexible joint; Indicates association The rigid link side inertia matrix, with the jump signal Switch; Indicates association The rigid connecting rod Coriolis and centrifugal force matrix; Indicates association The gravitational term on the rigid connecting rod changes with the jump signal. Switch; This indicates the joint angular displacement of the flexible actuator side considering time lag; This represents the control input torque considering the time lag on the rigid link side; This indicates the control input torque considering the time lag on the flexible actuator side.

[0050] The disturbance observers include rigid linkage disturbance observers and flexible joint disturbance observers; The expression for the rigid link disturbance observer is: ; The expression for the flexible joint perturbation observer is as follows: ; ; in, This represents the estimated disturbance acting on the rigid link; This represents the estimated perturbation acting on the flexible joint; and These represent the observation errors for rigid links and flexible joints, respectively. and These represent the first derivatives of the observation errors with respect to time on the rigid link side and the flexible joint side, respectively. and This represents the gain matrix of the rigid link and flexible joint of the observer, as a function of the jump signal. Switch; and This represents the time-delay compensation gain matrix for rigid links and flexible joints, which varies with the jump signal. Switch; This represents the actual control input delay time in the system; This represents the online estimate of the control input time delay; This indicates the time delay estimation error.

[0051] The dynamic equation for the observation error is: ; ; in, This represents the first derivative of the observation error of the side disturbance of the rigid link with respect to time; This represents the first derivative of the observation error of the flexible joint side perturbation with respect to time.

[0052] S4. Select state variables, derive the system state equation based on the state variables and observation errors, generate virtual control output based on disturbance estimates and calculate actual control input, construct state observer and graded Lyapunov function, and derive motor control law.

[0053] It should be noted that the expression for the state variable is, ; , , , ; in, Indicates the angle of a rigid link; Indicates the angular velocity of a rigid link; Indicates the angular velocity of the flexible joint; This indicates the observation error of the rigid link disturbance; This indicates the observation error of the flexible joint perturbation.

[0054] The dynamic equations of the flexible joint robotic arm can be rewritten as a system of first-order differential equations, expressed as follows: , ; ; ; in, Indicates joint angular displacement The rigid link side inertia matrix switches with the jump signal σ(t); Indicates joint angular displacement angular acceleration Coriolis force and centrifugal force on the rigid connecting rod side; Indicates joint angular displacement The rigid connecting rod has a side gravity term.

[0055] Solve and The expression is, ; ; in, The inverse matrix representing the lateral inertia of a rigid link; The inverse matrix representing the equivalent rotational inertia on the motor side; This represents the elastic restoring torque caused by the relative angular displacement between the flexible actuator side and the rigid connecting rod side.

[0056] Because the jump signal contains disturbances and time delays, the dynamic equation for the observation error becomes: ; ; ; ; in, This represents the time-delay compensation gain matrix on the rigid link side, which varies with the jump signal. A switchover has occurred; This represents the time-delay compensation gain matrix on the flexible joint side, which varies with the jump signal. A switchover has occurred; This represents the adaptive gain coefficient for time-delay estimation, which varies with the jumping signal. A switchover has occurred; The time-delay estimated damping coefficient varies with the jump signal. A switch has occurred.

[0057] Substituting the first-order differential equations and the dynamic equations of the modified observation error into the derivative expressions of the state variables, we obtain the system state equations, which are expressed as follows: ; The system state equations introduce displacement and velocity state variables from the rigid link side and the flexible actuator side, as well as disturbance observation error state and input time delay estimation state, to uniformly represent the dynamic behavior of the flexible joint manipulator under the conditions of system parameter jumps, input time delays and unknown disturbances as a first-order extended state space form.

[0058] The first four components of the state vector correspond to the displacement and velocity states of the rigid link side and the flexible actuator side of the robotic arm. Their dynamics are jointly determined by the system inertia, Coriolis and centrifugal force terms, elastic coupling terms, and gravity terms, and are driven by control inputs containing time delays and equivalent disturbances. The middle two components are the disturbance observation error states of the rigid link side and the flexible joint side, which are used to describe the evolution process of the disturbance estimation error under the influence of system parameter jumps and time delay uncertainties. The last component is the adaptive estimation state of the input time delay, which is used to correct the time delay estimate online, thereby compensating for the impact of time delay uncertainty on system stability and control performance.

[0059] By using system state equation modeling, mechanical dynamics, disturbance observation, and time delay estimation are all incorporated into the same state space framework, providing a unified mathematical foundation for subsequent stability analysis and adaptive control law design based on the Lyapunov method.

[0060] When the system switches operating modes, the parameters of the system state equations change abruptly. , ; in, Indicates the jump pattern index; This indicates the total number of available operating modes in the system; Indicates the first The start time of each working mode; Indicates that the system starts from the first The time point at which a work mode switches to the next work mode.

[0061] The state is continuous at the moment of switching working modes, but the system matrix changes abruptly. ; in, Represents the state vector As time approaches the switching moment The left limit; Represents the state vector As time approaches the switching moment The right limit.

[0062] The system state equations contain nonlinear terms, such as the inertia matrix. Coriolis force and centrifugal force matrix These factors complicate the analysis and control of the system.

[0063] The existence of time delay can cause changes in the characteristic roots of the system, which may make the system unstable. Therefore, the impact of time delay needs to be considered when designing the controller, and appropriate methods should be adopted to compensate for the time delay.

[0064] The system's abrupt changes increase its complexity. Under different operating modes, the system's dynamic parameters will change, which requires the controller to have adaptive capabilities and be able to adjust the control strategy in real time according to the system's operating mode.

[0065] The introduction of a disturbance observer can effectively estimate unknown disturbances in the system, thereby improving the system's anti-interference capability. By feeding the disturbance estimate back to the controller, the disturbance can be compensated for, thus improving the system's control performance.

[0066] By using the input and output information of the flexible joint manipulator to reconstruct the system's state variables, the state observer of the flexible joint manipulator is obtained. ; in, Represents the estimated value of the actual state variable; The first derivative of the estimated value of the actual state variable; This represents the state observer gain matrix; Represents the system state matrix; Represents the system state matrix; Represents the system output matrix; Indicates the actual output of the system; This represents the estimated output value of the system; This indicates the system control input.

[0067] Furthermore, the state observer of the flexible joint robotic arm satisfies the formula: ; , , ; ; in, Represents the true state vector of the system; This indicates the state estimation error; This represents the estimation error of the first state component; This represents the estimation error of the second state component; This represents the expected joint angle displacement trajectory; This represents the desired joint angular velocity trajectory; This represents the first-level error feedback gain.

[0068] Based on the state observer of the flexible joint robotic arm satisfying the formula, a first-level Lyapunov function is constructed, and selected... And by taking the derivative, we get ; in, This represents a first-level Lyapunov function; Represents Lyapunov functions The first derivative with respect to time; Denotes the Euclidean norm; Represents the error vector The square norm; Represents the error vector and The inner product term; This represents a positive control design gain coefficient; Indicates the desired joint angular velocity; Representing vectors The transpose of .

[0069] Constructing the second-level virtual control law includes, Define a new Lyapunov function with the expression: ; , ; in, This represents a second-order Lyapunov function; Representing vectors transpose; Represents a symmetric positive definite weighted matrix; This represents the system's equivalent total disturbance; Indicates the estimated disturbance value; This indicates the error in perturbation estimation; This represents the transpose of the disturbance estimation error; Represents the adaptive gain matrix; This represents the inverse of the adaptive gain matrix.

[0070] right Taking the derivative, we get ; in, This represents the derivative of a second-order Lyapunov function; This represents the derivative of a first-order Lyapunov function; This represents the actual disturbance vector; Represents the second-level state error vector The first derivative with respect to time; Indicates the disturbance estimate The first derivative with respect to time; Transpose of the perturbation estimation error matrix.

[0071] The dynamic equation Substitute have to: ; Multiply both sides by have to: ; Substitution get, ; in, This represents a virtual control input, replacing the original expression. ; Indicates a unified equivalent perturbation; This represents the desired joint angular velocity trajectory; It represents the inverse of a symmetric positive definite weighted matrix.

[0072] make Tracking target , Used in the rigid linkage side control design stage, as a reference target for subsequent actual control input on the motor side, it is derived from Lyapunov stability analysis and serves as a virtual control input. satisfy, ; , ; in, This represents the expected value of the virtual control input; Represents the positive real elements in a diagonal matrix L; This represents the diagonalization operator.

[0073] By selecting offset and coupling terms, and introduce and To ensure error convergence; Update the disturbance estimate. ,but, ; If we assume If it is bounded, then it can be solved by appropriate selection. Ensure convergence.

[0074] Construct a third-level motor control law, based on definition, ; ; ; in, This represents the position error vector on the third-stage motor side; This represents the desired trajectory of the joint angular displacement on the motor side.

[0075] definition The dynamic formula for the motor side is given: ; in, Represents the inertia matrix on the motor side; This represents the equivalent disturbance on the motor side.

[0076] ; in, This represents the position error vector on the fourth-stage motor side; This represents the expected joint angular velocity on the motor side.

[0077] Construct a fourth-level Lyapunov function. ; in, This represents the transpose of the position error vector on the third-stage motor side; This shows the transpose of the fourth-stage motor-side position error vector.

[0078] Multiply by both sides And substitute it into motor dynamics, ; ; use have to: ; The final actual control input is, ; in, This represents the expected joint angle acceleration on the motor side; Indicates error The first derivative with respect to time; This represents the gain coefficient of the third-level error feedback; Indicates error The first derivative with respect to time.

[0079] Substituting into the formula for the final actual control input, we get: ; ; in, This represents the transpose of the fourth-stage motor-side position error vector; This represents the gain coefficient for the fourth-level error feedback.

[0080] Combination Conclusion: ; in, express and Bounded residuals caused by the rate of change of constant disturbances; This represents the derivative of a fourth-order Lyapunov function; Represents the error vector The square norm.

[0081] Intersection terms can be defined by the standard inequality, and residual terms are bounded.

[0082] The complete control system architecture includes an extended state observer, a virtual control output, and actual control inputs.

[0083] An extended state observer is an example of an observer that estimates unknown disturbances as extended states, based on the existing state observer.

[0084] Based on the stability analysis of the complete control system structure using Lyapunov, a global Lyapunov function is constructed, with the expression: ; Differentiating along the system trajectory, we get: ; in, Represents the error vector The square norm; Represents the error vector The square norm; Indicates the gain of the third-level error feedback; This indicates the gain of the fourth-level error feedback; This indicates the gain of the fifth-level error feedback.

[0085] when When it is bounded, The system is uniformly bounded eventually, and all errors tend to the zero neighborhood. Both position tracking error and force tracking error converge exponentially to a small neighborhood near the origin.

[0086] Simulation results show that the method of the present invention can still maintain stability under time-delay jump conditions, with a position tracking error of less than 0.005 rad and a force control deviation of less than 2 N, which is superior to conventional sliding mode control and PID method.

[0087] In summary, this invention effectively suppresses system oscillations caused by communication delays or jumps by combining feedforward compensation with time delay estimation, thereby improving anti-time delay capability; it extends the state observer to estimate the total disturbance (including modeling errors, external disturbances, and time delay nonlinearity) in real time, enhancing the system's adaptability to external changes and strengthening robustness; it requires only some prior information (such as inertial range), with other parameters automatically adjusted through an adaptive mechanism, eliminating the need for precise modeling; it is suitable for compliant operation tasks in constrained environments, achieving unified control of trajectory tracking and contact force adjustment, taking into account both motion and force control; the algorithm structure is clear, easy to deploy on DSP / FPGA platforms, suitable for embedded control systems of industrial robots, and has strong engineering practicality.

[0088] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. An adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation, characterized in that: include, Each flexible joint of the flexible joint robotic arm consists of a rigid link and a flexible actuator, which is represented by a linear spring-damping model; A dynamic model of a flexible joint robotic arm is established based on the Lagrangian function, and the mapping relationship of the end-effector constraint forces is loaded into the joint space. In the dynamic model of the flexible joint manipulator, the control input time delay and jump signal are loaded, and the rigid link disturbance observer and the flexible joint disturbance observer are constructed respectively. The observer gain matrix, time delay compensation gain matrix and time delay estimation error are configured, and the disturbance estimate is output and the observation error is formed. Select state variables, derive system state equations based on state variables and observation errors, generate virtual control outputs based on disturbance estimates and calculate actual control inputs, construct state observers and graded Lyapunov functions, and derive motor control laws.

2. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 1, characterized in that: Constructing the Lagrangian function and energy term of a flexible joint robotic arm based on the Lagrangian method; The Lagrange function is the difference between the total kinetic energy and the total potential energy of the system, and its expression is: ； in, The Lagrangian function representing the flexible joint robotic arm in joint space; This represents the total kinetic energy of the system; This represents the total potential energy of the system; Substituting the Lagrange function into the Lagrange equation yields the result in flexible joints. Generalized active force The expression is, ； in, In a flexible joint robotic arm, the first... Joint angular displacement of a rigid link; Indicates the first Angular velocity of each joint; Represents the first in the Lagrange function Generalized coordinates The partial derivatives; Represents a continuous-time variable; Calculate the generalized active force The expression converted to vector form is: ； in, Represents the generalized angular displacement vector of a rigid link; This represents the angular velocity vector of the rigid link; Generalized driving force The vector form can be rearranged into a differential equation, expressed as: ； in, This represents the angular acceleration vector of the rigid link; Let represent the system inertia matrix, a symmetric positive definite matrix about the rigid link angle q; The Coriolis force and centrifugal force matrices are used to describe velocity-related nonlinear terms; Gravity term vector; This represents the control torque acting on the joints and actuators of the robotic arm; Due to the constraint forces at the end effector of the robotic arm, the expression for the dynamic model of the flexible joint robotic arm in joint space is as follows: ； in, This represents the equivalent total disturbance term acting on the joint dynamics of the robotic arm; Will Obtained by transforming to joint space. ； in, Representing the generalized coordinates of the end effector joints of a robotic arm Jacobian matrix The transpose of the matrix; Represents the Jacobian matrix; This represents the Lagrange multiplier vector corresponding to the end constraint force; This represents the matrix transpose symbol.

3. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 2, characterized in that: The total kinetic energy of the system consists of the kinetic energy of the rigid connecting rod. and the kinetic energy of the flexible actuator Composition, the expression is, ； in, This represents the total kinetic energy of the system; This represents the kinetic energy of a rigid link; This represents the kinetic energy of the flexible actuator; The kinetic energy of the rigid link The expression is, ； in, The inertia matrix represents the rigid link side; Rigid link joint angular displacement; Rigid link joint angular velocity; The kinetic energy of the flexible actuator The expression is, ； in, This represents the angular velocity vector of the flexible actuator.

4. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 2, characterized in that: The total potential energy of the system consists of the elastic potential energy of the flexible actuator. and gravitational potential energy composition; Elastic potential energy of flexible actuator The expression is, ； in, This represents the elastic potential energy of the flexible actuator; Represents the equivalent elastic stiffness matrix of a flexible joint; Indicates the angular displacement of the flexible actuator; gravitational potential energy The expression is, ； in, Represents gravitational potential energy; Represents gravity; Represents the angular displacement vector of a rigid link joint. Configuration functions; The expansion of the Lagrange function is, ； in, Transpose of the rigid link joint angular velocity vector matrix; This represents the transpose of the angular velocity vector of the flexible actuator.

5. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 4, characterized in that: Taking the partial derivatives of the Lagrange function and substituting them into the Lagrange equation, we obtain the dynamic equation of the flexible joint robotic arm, expressed as follows: ； ； in, Indicates the angular acceleration of a flexible joint; This represents the control torque acting on the rigid link; Represents the gravity term; Represents the matrix of Coriolis force and centrifugal force on the rigid connecting rod side; The inertia matrix represents the rigid link side; This represents the joint angular acceleration of a rigid link; This represents the control torque acting on the flexible joint; This represents the equivalent moment of inertia matrix on the motor side; Control input and There is a time delay The actual control input acting on the system is and ; The transition signal belongs to a finite state set. At this point, the dynamic equations of the flexible joint robotic arm change, and the expression on the load side becomes: ； in, This indicates the angular displacement of the joint on the rigid link side; Indicates the angular velocity of the joint on the rigid link side; This indicates the joint angular acceleration on the side of the rigid link; Jump signal; Indicates the time delay of the control input; The equivalent elastic stiffness matrix of the flexible joint, as a function of the jump signal. Switch; This represents the joint angular displacement on the motor side after considering the time delay effect; The inertia matrix represents the rigid link side and varies with the jump signal. Switch; The matrix representing the Coriolis force and centrifugal force on the rigid connecting rod side, as a function of the jump signal. Switch; The gravity term on the rigid link side changes with the jump signal. Switch; The expression on the motor side is, ； in, This represents the equivalent moment of inertia matrix on the motor side, which changes with the jump signal. Switch; This represents the equivalent damping matrix on the motor side, and it varies with the jump signal. Switch; Indicates the control input torque on the motor side; Indicates the angular velocity of the motor-side joint; Indicates the joint angular acceleration on the motor side; All disturbances in the actual system are uniformly represented as Substituting this into the dynamic equations of the flexible joint robotic arm, the expression is: ； ； in, This represents a disturbance acting on a rigid link; This indicates a disturbance acting on a flexible joint; Indicates association The rigid link side inertia matrix, with the jump signal Switch; Indicates association The rigid connecting rod Coriolis and centrifugal force matrix; Indicates association The gravitational term on the rigid connecting rod changes with the jump signal. Switch; This indicates the joint angular displacement of the flexible actuator side considering time lag; This represents the control input torque considering the time lag on the rigid link side; This indicates the control input torque considering the time lag on the flexible actuator side.

6. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 1, characterized in that: The disturbance observer includes a rigid linkage disturbance observer and a flexible joint disturbance observer; The expression for the rigid linkage disturbance observer is as follows: ； The expression for the flexible joint perturbation observer is as follows: ； ； in, This represents the estimated disturbance acting on the rigid link; This represents the estimated perturbation acting on the flexible joint; and These represent the observation errors for rigid links and flexible joints, respectively. and These represent the first derivatives of the observation errors with respect to time on the rigid link side and the flexible joint side, respectively. and This represents the gain matrix of the rigid link and flexible joint of the observer, as a function of the jump signal. Switch; and This represents the time-delay compensation gain matrix for rigid links and flexible joints, which varies with the jump signal. Switch; This represents the actual control input delay time in the system; This represents the online estimate of the control input time delay; This indicates the time delay estimation error; The dynamic equation for the observation error is: ； ； in, This represents the first derivative of the observation error of the side disturbance of the rigid link with respect to time; This represents the first derivative of the observation error of the flexible joint side perturbation with respect to time.

7. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 1, characterized in that: The expression for the state variable is, ； , ， ， ； in, Indicates the angle of a rigid link; Indicates the angular velocity of a rigid link; Indicates the angular velocity of the flexible joint; This indicates the observation error of the rigid link disturbance; This indicates the observation error of flexible joint perturbation; The dynamic equations of the flexible joint robotic arm can be rewritten as a system of first-order differential equations, expressed as follows: ， ； ； ； in, Indicates joint angular displacement The rigid link side inertia matrix, with the jump signal Switch; Indicates joint angular displacement angular acceleration Coriolis force and centrifugal force on the rigid connecting rod side; Indicates joint angular displacement The rigid connecting rod has a side gravity term; Solve and The expression is, ； ； in, The inverse matrix representing the lateral inertia of a rigid link; The inverse matrix representing the equivalent rotational inertia on the motor side; This represents the elastic restoring torque caused by the relative angular displacement between the flexible actuator side and the rigid connecting rod side; Because system parameters change with the jump signal When a switch occurs, and the control input exhibits time-delay uncertainty and unknown disturbances, the dynamic equation for the observation error can be expressed as follows: ； ； ； ； in, This represents the time-delay compensation gain matrix on the rigid link side, which varies with the jump signal. A switchover has occurred; This represents the time-delay compensation gain matrix on the flexible joint side, which varies with the jump signal. A switchover has occurred; This represents the adaptive gain coefficient for time-delay estimation, which varies with the jumping signal. A switchover has occurred; The time-delay estimated damping coefficient varies with the jump signal. A switch has occurred.

8. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 7, characterized in that: Substituting the first-order differential equations and the dynamic equations of the modified observation error into the derivative expressions of the state variables, we obtain the system state equations, which are expressed as follows: ； When the system switches operating modes, the parameters of the system state equations change abruptly. ； in, Indicates the jump pattern index; Indicates the total number of available operating modes in the system; Indicates the first The start time of each working mode; Indicates that the system starts from the first The switching point from one working mode to the next working mode; The state is continuous at the moment of switching working modes, but the system matrix changes abruptly. ； in, State vector As time approaches the switching moment The left limit; Represents the state vector As time approaches the moment of transition The right limit.

9. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 8, characterized in that: Based on the input and output information of the flexible joint manipulator, the system's state variables are reconstructed, resulting in the state observer of the flexible joint manipulator. ； in, Represents the estimated value of the actual state variable; The first derivative of the estimated value of the actual state variable; This represents the state observer gain matrix; Represents the system state matrix; Represents the system state matrix; Represents the system output matrix; Indicates the actual output of the system; This represents the estimated output value of the system. Indicates system control input; Furthermore, the state observer of the flexible joint robotic arm satisfies the formula: ； ， ； ； in, Represents the true state vector of the system; This indicates the state estimation error; This represents the estimation error of the first state component; This represents the estimation error of the second state component; This represents the expected joint angle displacement trajectory; This represents the desired joint angular velocity trajectory; Indicates the first-level error feedback gain; Based on the state observer of the flexible joint robot, a first-level Lyapunov function is constructed according to the formula. And by taking the derivative, we get ； in, This represents a first-level Lyapunov function; Represents Lyapunov functions The first derivative with respect to time; Denotes the Euclidean norm; Represents the error vector The square norm; Represents the error vector and The inner product term; Indicates a positive control design gain coefficient; Indicates the desired joint angular velocity; Representing vectors transpose; Constructing the second-level virtual control law includes, Define a new Lyapunov function with the expression: ； ， ； in, This represents a second-order Lyapunov function; Representing vectors transpose; Represents a symmetric positive definite weighted matrix; This represents the system's equivalent total disturbance; Indicates the estimated disturbance value; This indicates the error in perturbation estimation; This represents the transpose of the disturbance estimation error; Represents the adaptive gain matrix; The matrix representing the inverse of the adaptive gain matrix; right Taking the derivative, we get ； in, This represents the derivative of a second-order Lyapunov function; This represents the derivative of a first-order Lyapunov function; This represents the actual disturbance vector; Represents the second-level state error vector The first derivative with respect to time; Indicates the disturbance estimate The first derivative with respect to time; Transpose of the perturbation estimation error matrix; The dynamic equation Substitute have to: ； Multiply both sides by have to: ； Substitution get, ； in, This represents a virtual control input, replacing the original expression. ; Indicates a unified equivalent perturbation; This represents the desired joint angular velocity trajectory; Denotes the inverse of a symmetric positive definite weighted matrix; make Tracking target , Used in the rigid linkage side control design stage, as a reference target for subsequent actual control input on the motor side, it is derived from Lyapunov stability analysis and serves as a virtual control input. satisfy, ； ， ； in, This represents the expected value of the virtual control input; Represents a diagonal matrix Positive real number elements in; Indicates the diagonalization operator; By selecting offset and coupling terms, and introduce and To ensure error convergence; Update the disturbance estimate. ,but, ； If we assume If it is bounded, then it can be solved by appropriate selection. Guarantee convergence; Construct a third-level motor control law, based on definition, ； ； ； in, This represents the position error vector on the third-stage motor side; This represents the desired trajectory of the joint angular displacement on the motor side; definition The dynamic formula for the motor side is given: ； in, Represents the inertia matrix on the motor side; where, This represents the equivalent disturbance on the motor side; ； in, This represents the position error vector on the fourth-stage motor side; This represents the expected joint angular velocity on the motor side; Construct a fourth-level Lyapunov function. ； in, This represents the transpose of the position error vector on the third-stage motor side; This shows the transpose of the fourth-stage motor-side position error vector; Multiply by both sides And substitute it into motor dynamics, ； ； use have to: ； The final actual control input is, ； in, This represents the expected joint angle acceleration on the motor side; Indicates error The first derivative with respect to time; This represents the gain coefficient of the third-level error feedback; Indicates error The first derivative with respect to time; Substituting into the formula for the final actual control input, we get: ； ； in, This represents the transpose of the fourth-stage motor-side position error vector; This represents the gain coefficient for the fourth-level error feedback. Combination Conclusion: ； in, express , Bounded residuals caused by the rate of change of constant disturbances; This represents the derivative of a fourth-order Lyapunov function; Represents the error vector The square norm; Intersection terms can be defined by the standard inequality, and residual terms are bounded.

10. The adaptive control method for a time-delay jump flexible joint robotic arm based on disturbance observation as described in claim 9, characterized in that: A complete control system architecture includes an extended state observer, a virtual control output, and actual control inputs; The extended state observer refers to the estimation of unknown disturbances as extended states based on the state observer. Based on the stability analysis of the complete control system structure using Lyapunov, a global Lyapunov function is constructed, with the expression: ； Differentiating along the system trajectory, we get: ； in, Represents the error vector The square norm; Represents the error vector The square norm; Indicates the gain of the third-level error feedback; This indicates the gain of the fourth-level error feedback; This indicates the gain of the fifth-level error feedback. when When it is bounded, The system is uniformly bounded eventually, and all errors tend to the zero neighborhood. Both position tracking error and force tracking error converge exponentially to a small neighborhood near the origin.

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