Mechanical arm RBF (Radial Basis Function) network dynamic self-adaptive control method under constraint of time-varying mechanism

Abstract

The invention belongs to the technical field of robot control, and particularly relates to a mechanical arm RBF network dynamic self-adaptive control method under the constraint of a time-varying mechanism, which comprises the following steps of: constructing a mechanical arm dynamic model, determining a system specified time convergence standard, combining a joint motion reference trajectory of a mechanical arm, defining a trajectory tracking error, and determining a mechanical arm dynamic model based on a dynamic nominal model. Constructing a stable robust control law of the nominal dynamical model under the specified time; and selecting a radial basis function, and designing an RBF network adaptive control law in a specified time to dynamically fit a comprehensive nonlinear disturbance term in the system. The problem that the convergence time of a traditional method is uncontrollable is solved, the RBF neural network is adopted to dynamically approach comprehensive disturbance, the network weight is updated online through the adaptive law of the time-varying gain, the anti-jamming capability is remarkably improved, a model driving method and a data driving method are combined, and the convergence time of the model driving method and the convergence time of the data driving method are greatly improved through a dynamic model decoupling and feedforward compensation strategy. And more accurate and low-delay trajectory tracking control is realized.

Description

Technical Field

[0001] The present invention belongs to the field of robot technology control, and particularly relates to a dynamic adaptive control method for a robotic arm based on an RBF network under the constraint of a time-varying mechanism. Background Art

[0002] As one of the core research directions in the field of intelligent manufacturing, the robust control of intelligent agents. The robotic arm, as the core execution unit, is widely used in high-precision operation scenarios such as precision assembly, logistics handling, and welding. Currently, the commonly used control methods mainly include: PD feedback control based on an accurate dynamic model. Although this method can achieve accurate tracking performance of the robotic arm, its control performance significantly deteriorates when facing dynamic unmodeled disturbances; the sliding mode control method, which makes the system state converge within an asymptotic time by designing a sliding surface. Although it shows good robustness to external disturbances and parameter uncertainties, it has an inherent defect that the convergence time cannot be accurately guaranteed. For current traditional control methods, the robotic arm faces the following key challenges during actual operation: problems such as unmodeled dynamics, parameter uncertainties, external disturbance suppression, convergence speed, and time constraints.

[0003] Compared with the prior art, although the RBF neural network can approximate unknown non-linearities, the traditional weight update law has a slow convergence speed and lacks a time constraint mechanism; the existing time-varying gain methods (such as terminal sliding mode) can achieve finite-time convergence, but they require the upper bound of the known disturbance and have a singularity problem; the method combining the nominal model and the disturbance observer (such as ADRC) requires complex parameter tuning, and its engineering practicability is limited. Summary of the Invention

[0004] In order to effectively solve the problem of precise control of the robotic arm in a complex dynamic environment and provide a reliable solution for scenarios such as intelligent manufacturing and industrial automation that require millimeter-level response and synchronous control, the present invention proposes a dynamic adaptive control method for a robotic arm based on an RBF network under the constraint of a time-varying mechanism, including constructing a robotic arm dynamic model, determining the system's specified-time convergence criterion, combining the joint motion reference trajectory of the robotic arm, defining the trajectory tracking error, and constructing a stable and robust control law for the nominal dynamic model at the specified time based on the dynamic nominal model; selecting a radial basis function and designing an RBF network adaptive control law at the specified time to dynamically fit the comprehensive non-linear disturbance term in the system.

[0005] Furthermore, the RBF network dynamic adaptive control law at the specified time is expressed as:

[0006]

[0007] where, τ 总Represents the RBF network dynamic adaptive control law under the specified time; M0(q) is the nominal inertia matrix of the robotic arm system, and q represents the actual joint angle position vector of the robotic arm system; Represents the desired acceleration reference trajectory of the robotic arm system; k v Represents the second design control gain matrix; Represents the joint velocity tracking error scaled by the time-varying gain function; k p Represents the first design control gain matrix; η represents the joint angle tracking error signal scaled by the time-varying gain function; Is the nominal Coriolis force matrix of the robotic arm system; Represents the actual joint velocity vector of the robotic arm system; G0(q) is the nominal gravity vector of the robotic arm system; Is the real-time estimated output value of the RBF for the comprehensive disturbance nonlinear term.

[0008] Furthermore, the real-time estimated output value of the RBF for the comprehensive disturbance nonlinear term Is expressed as:

[0009]

[0010] Where, Is the on-line estimated value of the RBF neural network weight; h(x) = [h1(x),..., h m (x)] T Is the output vector of the radial basis function, Represents the RBF neural network input state vector, q represents the current joint angle position vector of the robotic arm system, Represents the current joint motion velocity vector of the robotic arm system, Represents the current joint motion acceleration vector of the robotic arm system, and m represents the number of nodes in the hidden layer of the RBF neural network.

[0011] Furthermore, when calculating the RBF network dynamic adaptive control law under the specified time, the approximation error of the RBF neural network for the comprehensive nonlinear disturbance term is less than or equal to the supremum of the approximation error of the RBF neural network for the comprehensive nonlinear disturbance term. The approximation error of the RBF neural network for the comprehensive nonlinear disturbance term is:

[0012]

[0013] Where, σ(x) is the approximation error of the RBF neural network for the comprehensive nonlinear disturbance term; Is the comprehensive nonlinear disturbance term, expressed as ΔM represents the inertia parameter uncertainty of the robotic arm system, ΔC represents the modeling error of the Coriolis force matrix of the robotic arm system, ΔG represents the gravity compensation residual of the robotic arm system, d is the vector of unknown external environmental disturbances; w * represents the ideal weight of the RBF neural network.

[0014] Furthermore, the online estimated value of the RBF neural network weights is adaptively updated by the following law:

[0015]

[0016] where γ is the adaptive learning rate; μ(t) is the time-varying gain function; h(x) = [h1(x),..., h m (x)] T is the output vector of the radial basis function; is the error state vector, η represents the joint angle tracking error scaled by the time-varying gain function, represents the joint velocity tracking error scaled by the time-varying gain function; P is the solution matrix of the Lyapunov function; B represents the inertia matrix.

[0017] Furthermore, the calculation of the error state vector includes:

[0018] η(t) = μ(t)e(t)

[0019]

[0020] where η(t) is the joint angle tracking error scaled by the time-varying gain function at time t, e(t) is the error between the actual joint angle trajectory at time t and the desired joint angle trajectory at time t; is the joint velocity tracking error scaled by the time-varying gain function at time t; is the difference between the actual joint velocity trajectory at time t and the desired joint velocity trajectory at time t, is the derivative of the time-varying gain function with respect to time at time t.

[0021] Furthermore, the solution matrix P of the Lyapunov function satisfies the following conditions:

[0022] A T P + PA = -Q

[0023] where is expressed as the designed control gain parameter matrix, I n represents an n×n identity matrix, n is the degree of freedom of the robotic arm system, and Q represents a positive definite symmetric matrix.

[0024] Furthermore, the first designed control gain matrix is expressed as The second design control gain matrix is expressed as α is a design positive constant.

[0025] Furthermore, when the preset convergence time is T, the Lyapunov function of the system is bounded on the interval [0, T), and the time convergence criterion is defined as:

[0026]

[0027] where represents the derivative of the system Lyapunov function with respect to time; k and θ are positive constants, V(t) represents the system Lyapunov function; d(t) represents the unknown disturbance of the system.

[0028] Furthermore, the dynamic model of a robotic arm with n degrees of freedom is expressed as:

[0029]

[0030] where is the positive definite inertia matrix; is the Coriolis force matrix; is the gravity vector; d is the vector of unknown external environmental disturbances; is the input torque of the joint, q represents the current joint angle position vector of the robotic arm system, represents the current joint motion velocity vector of the robotic arm system, represents the current joint motion acceleration vector of the robotic arm system.

[0031] In view of the unmodeled dynamics and external disturbances faced by the robotic arm in actual working conditions, the present invention proposes a prescribed-time RBF neural network architecture. By introducing a time-varying gain mechanism, the following technical breakthroughs are achieved: precise time convergence control, the designed control algorithm ensures that the approximation error of the system nonlinear model globally converges within the user-prescribed time, solving the problem of uncontrollable convergence time in traditional methods; intelligent disturbance compensation, using an RBF neural network to dynamically approximate the combined disturbance, and online updating the network weights through the adaptive law of time-varying gain, significantly improving the anti-interference ability; innovatively combining model-driven and data-driven methods, and through dynamic model decoupling and feedforward compensation strategies, achieving more accurate and low-latency trajectory tracking control. Compared with the prior art, the proposed prescribed-time RBF network dynamic adaptive control achieves precise convergence within the user-prescribed time and significantly enhances the anti-interference ability. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 is a flowchart of a method for dynamic adaptive control of a robotic arm RBF network under the constraint of a time-varying mechanism according to the present invention;

[0033] Figure 2 This is the Simulink framework diagram for the RBF network dynamic adaptive control of the present invention;

[0034] Figure 3 This is a schematic diagram showing the actual angular trajectory of robotic arm joint 1 and robotic arm joint 2 tracking the desired angular trajectory at a specified time of 1 s in the present invention;

[0035] Figure 4 This is a schematic diagram showing the actual velocity trajectory of robotic arm joint 1 and robotic arm joint 2 tracking the desired velocity trajectory at a specified time of 1 s in the present invention;

[0036] Figure 5 This is a schematic diagram of the control input of robotic arm joint 1 and robotic arm joint 2 at a specified time of 1 s in the present invention;

[0037] Figure 6 This is a schematic diagram of the RBF approximation of the nonlinear disturbance of robotic arm joint 1 and robotic arm joint 2 at a specified time of 1 s in the present invention;

[0038] Figure 7 This is a schematic diagram showing the actual angular trajectory of robotic arm joint 1 and robotic arm joint 2 tracking the desired angular trajectory at a specified time of 0.5 s in the present invention. Detailed implementation manners

[0039] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0040] The present invention proposes a robotic arm RBF network dynamic adaptive control method under the constraint of a time-varying mechanism, including constructing a robotic arm dynamics model, determining the system's specified-time convergence criterion, combining the joint motion reference trajectory of the robotic arm, defining the trajectory tracking error, and constructing a stable and robust control law for the nominal dynamics model at the specified time based on the dynamic nominal model; selecting a radial basis function and designing an RBF network adaptive control law at the specified time to dynamically fit the comprehensive nonlinear disturbance term in the system.

[0041] This embodiment proposes a robotic arm RBF network dynamic adaptive control method under the constraint of a time-varying mechanism, as Figure 1 shown, the method includes the following contents:

[0042] S1: Construct a robotic arm dynamics model and determine the system's specified-time convergence criterion.

[0043] This embodiment considers a robotic arm with n degrees of freedom, and its dynamic model can be expressed as:

[0044]

[0045] where, is a positive definite inertia matrix; is the Coriolis force matrix (including centrifugal force and Coriolis force); is the gravity vector; d is the vector of unknown external environmental disturbances; is the input torque of the joint, that is, the control input vector, q represents the current joint angle position vector of the robotic arm system, represents the current joint motion velocity vector of the robotic arm system, represents the current joint motion acceleration vector of the robotic arm system.

[0046] When the preset convergence time is T, that is, the actual control of the robotic arm and the desired control reach agreement within the convergence time at the latest, the system Lyapunov function is bounded on the interval [0, T), and the specified time convergence judgment criterion is:

[0047]

[0048] where, represents the derivative of the system Lyapunov function with respect to time; k, θ are positive constants greater than 0, μ(t) represents the time-varying gain function, V(t) represents the system Lyapunov function; d(t) represents the unknown disturbance of the system.

[0049] Furthermore, considering the time-varying gain function μ(t), if there exists a smooth function V: [0, T) → [0, +∞) that satisfies the inequality:

[0050] For d(t) being an unknown disturbance and k, θ > 0 being positive constants, then V(t) is bounded on the time interval [0, T), that is

[0051] where, when the time t in the system control process approaches the preset convergence time T (i.e., t → T - ), the time-varying gain function μ(t) → ∞. To prevent unbounded gain, the expression of the time-varying gain function μ(t) is further expressed as:

[0052]

[0053] where, t represents the time in the control process of the robotic arm system, T represents the convergence time specified by the user, δ represents the coefficient of the safe transition interval, μ max represents the upper limit to prevent the unbounded growth of the time-varying gain function, μ mindenotes the lower bound of the stable time-varying gain, p represents the order of the robotic arm system, and β represents a positive design parameter.

[0054] The dynamic model of the robotic arm system includes a nominal part and an unmodeled (unknown part), which are specifically described as:

[0055] ΔM = M0 - M

[0056] ΔC = C0 - C

[0057] ΔG = G0 - G

[0058] Among them, respectively represent the nominal inertia matrix, nominal Coriolis force matrix (including Coriolis force and centrifugal force), and nominal gravity vector of the robotic arm system. ΔM, ΔC, and ΔG respectively represent the inertia parameter uncertainty, Coriolis force matrix uncertainty, and gravity compensation residual of the robotic arm system. M(q) is the positive definite inertia matrix of the robotic arm system; is the Coriolis force matrix of the robotic arm system; G(q) is the gravity vector of the robotic arm system.

[0059] Based on the clear dynamic characteristics, the following assumption conditions are introduced:

[0060] Assumption 1: For any time The system satisfies that the reference trajectory and its derivative are continuous and bounded, that is:

[0061]

[0062] Among them, q d (t), respectively represent the desired joint angle, velocity, and acceleration trajectories, are known positive constants, and ||·|| represents the two-norm.

[0063] The unknown disturbance d(t) is bounded, that is:

[0064]

[0065] Among them, represents an unknown but bounded positive constant.

[0066] S2: Define the trajectory tracking error.

[0067] Based on the reference trajectory of the robotic arm joint motion, define the robotic arm joint angle tracking error e(t) and the joint velocity tracking error respectively as:

[0068]

[0069] Among them, Denoted as the actual joint angle position vector at time \(t\), Denoted as the desired joint angle trajectory at time \(t\), Denoted as the actual joint velocity vector at time \(t\), Denoted as the desired joint velocity trajectory at time \(t\).

[0070] Based on the prescribed-time control theory, the tracking error is dynamically scaled according to the time-varying gain function \(\mu(t)\). Therefore, the scaled joint angle tracking error signal \(\eta(t)\) and the velocity tracking error signal Can be respectively expressed as:

[0071] \(\eta(t)=\mu(t)e(t)\)

[0072]

[0073] where \(\mu(t)\) represents the time-varying scaling function and can be expressed as The derivative of the time-varying gain function with respect to time Can be expressed as:

[0074]

[0075] where \(p\) represents the order of the robotic arm system, and \(\beta\) represents a positive design parameter.

[0076] S3: Based on the dynamic nominal model, construct a stable robust control law for the nominal dynamic model under the prescribed time.

[0077] Substituting the nominal model described in Step 1 into the original dynamic model can be expressed as:

[0078]

[0079] where, Represents the comprehensive nonlinear perturbation term, including the unmodeled dynamics and the external perturbation \(d\).

[0080] Based on the nominal dynamic model, the prescribed-time control law in the case where the comprehensive nonlinear perturbation term is known is:

[0081]

[0082] where, Represents the joint control input vector based on the nominal dynamic model, Respectively represent the nominal inertia matrix of the robotic arm system, the nominal Coriolis force matrix (including Coriolis force and centrifugal force), and the nominal gravity vector, Represents the desired acceleration reference trajectory, Represents the actual joint velocity vector, \(k\) p Represents the first design control gain matrix and can be expressed as α > 0 represents a positive constant, k v represents the second design control gain matrix, which can be expressed as is the comprehensive nonlinear perturbation term, which can be expressed as ΔM, ΔC, and ΔG respectively represent the inertia parameter uncertainty of the robotic arm system, the modeling error of the Coriolis force matrix, and the gravity compensation residual. d represents an unknown but bounded perturbation, and η and respectively represent the joint angle tracking error signal and the joint velocity tracking error scaled by the time-varying gain function.

[0083] S4: Select the radial basis function and design the RBF network adaptive control law at the specified time to dynamically fit the comprehensive nonlinear perturbation in the system.

[0084] Due to the comprehensive nonlinear perturbation term is often unknown in practical engineering. Using the radial basis RBF neural network estimator for approximation and compensation can be expressed as:

[0085]

[0086] where represents the input state vector, represents the real-time estimated output value vector of the RBF for the comprehensive perturbation nonlinear term, is the online estimated value of the RBF neural network weights, is the output vector of the Gaussian radial basis function. Each Gaussian basis function can be expressed as:

[0087]

[0088] In the formula, and b i > 0 respectively represent the center vector and width parameter of the i-th radial basis function, and m represents the number of nodes in the hidden layer.

[0089] Based on the universal approximation theory of the RBF neural network, for any given approximation accuracy ε > 0 and the unknown comprehensive nonlinear perturbation term there exists an ideal weight matrix such that:

[0090]

[0091] where is the compact set of the system state space, and the ideal weight w * is determined by the minimum-maximum approximation error criterion

[0092] Define the approximation error of the RBF neural network for the comprehensive nonlinear perturbation term as:

[0093]

[0094] Among them, is the approximation error vector of the comprehensive disturbance non - linear term, w *T h(x) is expressed as the optimal neural network output, and there exists a positive constant σ0 such that:

[0095]

[0096] where the upper bound σ0 is defined as

[0097] In summary, the adaptive control law of the RBF dynamic network under the specified time can be expressed as:

[0098]

[0099] Among them, is the real - time estimated output value of the RBF for the comprehensive disturbance non - linear term, which can be expressed as is the on - line estimated value of the RBF neural network weights.

[0100] Substituting the adaptive control law into the original dynamic model of the manipulator, the following closed - loop system characteristics are obtained:

[0101]

[0102] Among them, the left - hand side of the equation is the ideal second - order error dynamics, and the right - hand side is the neural network approximation residual.

[0103] The above equation is transformed into the error state - space matrix form (i.e., the updated speed of the scaled joint angle and velocity error):

[0104]

[0105] Among them, is the error state vector, is expressed as the designed control gain parameter matrix, I n represents an n×n identity matrix; represents the inertia matrix, M o (q) is expressed as the nominal inertia matrix of the manipulator system, is expressed as the weight estimation error matrix, is the bounded approximation error of the RBF network.

[0106] Based on the Lyapunov stability theory and the time - varying gain function μ(t), the adaptive update law of the neural network weights under the specified time is designed as:

[0107]

[0108] Among them, represents the RBF network weight adaptive update law, and γ > 0 represents the adaptive learning rate; P = P T > 0 represents the solution matrix of the Lyapunov function, satisfying Q is a positive definite symmetric matrix, used to specify the system convergence performance, indicating that Q is a positive definite matrix.

[0109] S5: Integrate the above parts to achieve high-precision position and speed tracking of the robotic arm within the user-specified time.

[0110] Consider a fully actuated system of an n-degree-of-freedom robotic arm with unmodeled and dynamic external disturbances. Given the joint angle q d and velocity of the reference trajectory, stable tracking control within the user-defined time T can be achieved through the following controller, and its control law can be expressed as:

[0111]

[0112] Based on the above control law, the tracking control of the joint angle and joint velocity of the robotic arm can ultimately be achieved within the user-specified time.

[0113] As Figure 2 shown, a dynamic adaptive control method for a robotic arm RBF network under time-varying mechanism constraints includes a reference trajectory generator Input module, a controller ctrl module including error transformation, RBF neural network disturbance approximation within a specified time, and a neural network weight adaptive estimation module, and an actuator plant module.

[0114] The reference trajectory generator Input module mainly generates the desired angle and desired velocity trajectories of the control system. The controller Ctrl module mainly transforms the desired angle and desired velocity trajectories through error transformation, RBF neural network disturbance approximation within a specified time, and neural network weight adaptive estimation to generate the control input torque of the entire system. The actuator plant module mainly inputs the control torque into the entire robotic arm system to achieve stable tracking of the desired trajectory of the robotic arm within the specified time.

[0115] To verify the effectiveness of the method of the present invention, simulation experiments are carried out through matlab2018b and its simulink module, specifically as follows:

[0116] Through selecting appropriate model parameters to construct a numerical simulation experiment of a two-degree-of-freedom joint robotic arm system to verify the effectiveness of a dynamic adaptive control method for a robotic arm RBF network under time-varying mechanism constraints proposed by the present invention. The dynamic parameters are shown in Table 1, and the control parameters are shown in Table 2.

[0117] Table 1 Kinetic related parameters

[0118]

[0119] Among them, M 11 represents the first symmetric positive definite inertia matrix of the two-degree-of-freedom robotic arm, M 12 represents the second symmetric positive definite inertia matrix of the two-degree-of-freedom robotic arm, M 21 represents the third positive definite inertia symmetric matrix of the two-degree-of-freedom robotic arm, M 22 represents the fourth positive definite inertia symmetry of the two-degree-of-freedom robotic arm; C 11 represents the first Coriolis force of the two-degree-of-freedom robotic arm, C 12 represents the first centrifugal force of the two-degree-of-freedom robotic arm, C 21 represents the second Coriolis force of the two-degree-of-freedom robotic arm, C 22 represents the second centrifugal force of the two-degree-of-freedom robotic arm; v represents the equivalent inertia moment of the first joint of the robotic arm, q 01 represents the inertia moment of the second joint, q 02 represents the coupling inertia coefficient, and g represents the standard value of the gravitational acceleration.

[0120] Table 2 Control parameters

[0121]

[0122] As Figure 2 shown, α represents the first control augmentation matrix and the second designed control gain matrix of the positive constant, γ represents the adaptive learning rate, k1 represents the design parameter, Q represents the positive definite inertia matrix, c i represents the center vector of the i-th radial basis function of the RBF neural network, b i represents the width of the i-th radial basis function, μ max represents the upper bound of the adjustment of the time-varying gain function to avoid unbounded growth of the time-varying gain, μ min represents the steady-state lower limit of the time-varying gain, and δ represents the safety process interval coefficient of the time-varying gain.

[0123] The dynamic model of the two-degree-of-freedom robotic arm can be expressed as:

[0124]

[0125] Among them:

[0126]

[0127] Let the desired tracking commands of the joint angle and angular velocity be:

[0128]

[0129] The external disturbance of the system is where d1 = 2, d2 = 3, d3 = 6, and the initial state of the system is [q1 q2 q3 q4] T = [0.6 0.3 0.5 0.5] T , where q1, q2, q3, and q4 represent the initial angle of the first joint of the robotic arm, the initial angle of the second joint, the initial angular velocity of the first joint, and the initial angular velocity of the second joint, respectively. It is assumed that the unmodeled parts of the dynamic model are ΔM = 0.2M, ΔC = 0.2C, and ΔG = 0.2G, which represent 20% modeling error in the inertia matrix, 20% error in the Coriolis force matrix, and 20% deviation in the gravity compensation, respectively.

[0130] Let the total simulation time of the system be set to 20 s, i.e., t total = 20 s, the sampling period is Δt = 0.01 s, the initial weight of the RBF neural network is w = 0.1, the time-varying gain functions are p = 1 and β = 2, and other design parameters refer to Table 1 and Table 2. Using the RBF network dynamic adaptive control law (described by S5) under the specified time proposed by the present invention, the simulation results are as follows Figures 3 to 7 .

[0131] Figure 3 and Figure 4 respectively show the position angle and speed tracking errors of Joint 1 and Joint 2 of the robotic arm. It can be clearly seen from the figure that the time-varying gain mechanism effectively bounds the fluctuation range of the error, and the actual trajectory of the robotic arm joints coincides with the desired trajectory within the user-defined time T = 1 s, verifying the high precision and fast response ability of the system. Figure 5 Shows the control input signal of the robotic arm. Under the action of the RBF network dynamic adaptation rate, it can ensure that the control input is stable within a certain range and no high-frequency chattering phenomenon occurs, verifying the engineering feasibility of the algorithm. Figure 6 Shows the approximation degree of the RBF neural network to the system's non-linear disturbance. Although the fluctuation amplitude is very large at the beginning, with the combined action of the time-varying gain and RBF, accurate approximation within the user-specified time is achieved, improving the ability to cope with dynamic disturbances in actual working conditions. From Figure 3 and Figure 7 comparison, it can be seen that as the preset time is shortened to 0.5 s, the joint angles of the robotic arm can be well tracked within the specified time, verifying the adaptability of the proposed control method to strict time constraints.

[0132] Although embodiments of the present invention have been shown and described, it will be understood by those of ordinary skill in the art that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A dynamic adaptive control method for a robotic arm based on a radial basis function (RBF) network under time-varying mechanism constraints, including constructing a robotic arm dynamics model, determining the system's prescribed-time convergence criterion, combining the joint motion reference trajectory of the robotic arm, defining the trajectory tracking error, and constructing a stable and robust control law for the nominal dynamics model at the prescribed time based on the nominal dynamics model. It is characterized in that: Select the radial basis function and design the RBF network adaptive control law at the specified time to dynamically fit the comprehensive nonlinear disturbance term in the system.

2. The method according to claim 1, characterized in that, The RBF network dynamic adaptive control law at the specified time is expressed as: Among them, τ 总 represents the RBF network dynamic adaptive control law under the specified time; M0(q) is the nominal inertia matrix of the manipulator system, and q represents the actual joint angle vector of the manipulator system; represents the desired acceleration reference trajectory of the manipulator system; k v represents the second designed control gain matrix; represents the joint velocity tracking error scaled by the time-varying gain function; k p represents the first designed control gain matrix; η represents the joint angle tracking error signal scaled by the time-varying gain function; is the nominal Coriolis force matrix of the manipulator system; represents the actual joint velocity vector of the manipulator system; G0(q) is the nominal gravity vector of the manipulator system; is the real-time estimated output value of the RBF for the comprehensive disturbance nonlinear term.

3. The method according to claim 2, wherein The real-time estimation output value of the RBF for the comprehensive disturbance non-linear term is expressed as: Among them, is the online estimated value of the weights of the RBF neural network; h(x) = [h1(x),..., h m (x)] T is the output vector of the radial basis function, represents the input state vector of the RBF network, q represents the current joint angular position vector of the robotic arm system, represents the current joint motion velocity vector of the robotic arm system, represents the current joint motion acceleration vector of the robotic arm system, and m represents the number of nodes in the hidden layer of the RBF neural network.

4. The method according to claim 3, characterized in that, When calculating the RBF network dynamic adaptive control law at the specified time, the approximation error of the RBF neural network for the comprehensive nonlinear disturbance term is less than or equal to the supremum of the approximation error of the RBF neural network for the comprehensive nonlinear disturbance term. The approximation error of the RBF neural network for the comprehensive nonlinear disturbance term is: Among them, σ(x) is the approximation error of the RBF neural network for the comprehensive nonlinear disturbance term; is the comprehensive nonlinear disturbance term, expressed as ΔM represents the inertia parameter uncertainty of the robotic arm system, ΔC represents the modeling error of the Coriolis force matrix of the robotic arm system, ΔG represents the gravity compensation residual of the robotic arm system, and d is the unknown external environmental disturbance vector; w * represents the ideal weight of the RBF neural network.

5. The method according to any one of claims 1 to 4, characterized in that, Online estimated value of the weights of the RBF neural network The adaptive update law is as follows: where γ is the adaptive learning rate; μ(t) is the time-varying gain function; h(x)=[h1(x),...,h m (x)] T is the output vector of the radial basis function, represents the input state vector of the RBF neural network, q represents the current joint angle position vector of the robotic arm system, represents the current joint motion velocity vector of the robotic arm system, represents the current joint motion acceleration vector of the robotic arm system, m represents the number of nodes in the hidden layer of the RBF neural network; is the error state vector, η represents the joint angle tracking error scaled by the time-varying gain function, represents the joint velocity tracking error scaled by the time-varying gain function; P is the solution matrix of the Lyapunov function; B represents the inertia matrix.

6. The method according to claim 5, characterized in that, The calculation of the error state vector includes: η(t) = μ(t)e(t) Among them, η(t) is the joint angle tracking error scaled by the time-varying gain function at time t, and e(t) is the error between the actual joint angle trajectory and the desired joint angle trajectory at time t; is the joint velocity tracking error scaled by the time-varying gain function at time t; is the difference between the actual joint velocity trajectory and the desired joint velocity trajectory at time t, is the derivative of the time-varying gain function with respect to time.

7. The method according to claim 5, characterized in that, The solution matrix P of the Lyapunov function satisfies the following conditions: A T P + PA = -Q Among them, is expressed as the design control gain parameter matrix, k p represents the first design control gain matrix, k v represents the second design control gain matrix, I n represents an n×n identity matrix, and Q represents a positive definite symmetric matrix.

8. The method according to claim 2 or 7, characterized in that, The first design control gain matrix is expressed as The second design control gain matrix is expressed as α is a design positive constant.

9. The method according to claim 1, wherein When the preset convergence time is T, the system Lyapunov function is bounded on the interval [0, T). The specified-time convergence criterion is: wherein, represents the derivative of the system Lyapunov function with respect to time; k and θ are positive constants, μ(t) represents a time-varying gain function, V(t) represents the system Lyapunov function; d(t) represents an unknown disturbance of the system.

10. The method according to claim 1, wherein The dynamic model of the n-degree-of-freedom manipulator is expressed as: wherein, is a positive definite inertia matrix; is a Coriolis force matrix; is a gravity vector; d is an unknown external environmental disturbance vector; is the input torque of the joint; $\boldsymbol{q}$ represents the current joint angular position vector of the robotic arm system, $\dot{\boldsymbol{q}}$ represents the current joint motion velocity vector of the robotic arm system, $\ddot{\boldsymbol{q}}$ represents the current joint motion acceleration vector of the robotic arm system.

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